BR
Uploaded byBhavanams Rao
359 views

Lecture 05

The document discusses the time value of money, which refers to the concept that money has greater value when received now rather than in the future due to opportunity costs, inflation, and uncertainty. It provides formulas for calculating future value, present value, and interest rates. It also discusses compound interest and how money can double over time depending on the interest rate and compounding periods. Examples are provided to demonstrate calculations for simple vs compound interest and different compounding periods.

Embed presentation

Downloaded 15 times
UNIT ONE CHAPTER THREE THE TIME VALUE OF MONEY
Lesson 5 Chapter 3 The time value of money Unit 1 Core concepts in financial management After reading this lesson you will be able to: - Understand what is meant by "the time value of money." Describe how the interest rate can be used to adjust the value of cash flows to a single point in time. Calculate the future value of an amount invested today. Calculate the present value of a single future cash flow. Understand the relationship between present and future values. Understand in what period of time money doubles Understand shorter compounding periods Calculate & understand the relationship between effective & nominal interest rate. Use the interest factor tables and understand how they provide a short cut to calculating present and future values. You all instinctively know that money loses its value with time. Why does this happen? What does a Financial Manager have to do to accommodate this loss in the value of money with time? In this section, we will take a look at this very interesting issue. Why should financial managers be familiar with the time value of money? The time value of money shows mathematically how the timing of cash flows, combined
with the opportunity costs of capital, affect financial asset values. A thorough understanding of these concepts gives a financial manager powerful tool to maximize wealth. What is the time value of money? The time value of money serves as the foundation for all other notions in finance. It impacts business finance, consumer finance and government finance. Time value of money results from the concept of interest. This overview covers an introduction to simple interest and compound interest, illustrates the use of time value of money tables, shows a approach to solving time value of money problems and introduces the concepts of intra year compounding, annuities due, and perpetuities. A simple introduction to working time value of money problems on a financial calculator is included as well as additional resources to help understand time value of money. Time value of money The universal preference for a rupee today over a rupee at some future time is because of the following reasons: - Alternative uses/ Opportunity cost Inflation Uncertainty The manner in which these three determinants combine to determine the rate of interest can be represented symbolically as Nominal or market rate of interest rate = Real rate of interest + Expected rate of Inflation + Risk of premiums to compensate uncertainty
Basics Evaluating financial transactions requires valuing uncertain future cash flows. Translating a value to the present is referred to as discounting. Translating a value to the future is referred to as compounding The principal is the amount borrowed. Interest is the compensation for the opportunity cost of funds and the uncertainty of repayment of the amount borrowed; that is, it represents both the price of time and the price of risk. The price of time is compensation for the opportunity cost of funds and the price of risk is compensation for bearing risk. Interest is compound interest if interest is paid on both the principal and any accumulated interest. Most financial transactions involve compound interest, though there are a few consumer transactions that use simple interest (that is, interest paid only on the principal or amount borrowed). Under the method of compounding, we find the future values (FV) of all the cash flows at the end of the time horizon at a particular rate of interest. Therefore, in this case we will be comparing the future value of the initial outflow of Rs. 1,000 as at the end of year 4 with the sum of the future values of the yearly cash inflows at the end of year 4. This process can be schematically represented as follows: PROCESS OF DISCOUNTING Under the method of discounting, we reckon the time value of money now, i.e. at time 0 on the time line. So, we will be comparing the initial outflow with the sum of the present values (PV) of the future inflows at a given rate of interest. Translating a value back in time -- referred to as discounting -- requires determining what a future amount or cash flow is worth today. Discounting is used in valuation because we often want to determine the value today of future value or cash flows. The equation for the present value is:
Present value = PV = FV / (1 + i) n Where: PV = present value (today's value), FV = future value (a value or cash flow sometime in the future), i = interest rate per period, and n = number of compounding periods And [(1 + i) n] is the compound factor. We can also represent the equation a number of different, yet equivalent ways: Where PVIFi,n is the present value interest factor, or discount factor. In other words future value is the sum of the present value and interest: Future value = Present value + interest From the formula for the present value you can see that as the number of discount periods, n, becomes larger, the discount factor becomes smaller and the present value becomes less, and as the interest rate per period, i, becomes larger, the discount factor becomes smaller and the present value becomes less. Therefore, the present value is influenced by both the interest rate (i.e., the discount rate) and the numbers of discount periods. Example Suppose you invest 1,000 in an account that pays 6% interest, compounded annually. How much will you have in the account at the end of 5 years if you make no withdrawals? After 10 years?
Solution FV5 = Rs 1,000 (1 + 0.06) 5 = Rs 1,000 (1.3382) = Rs 1,338.23 FV10 = Rs 1,000 (1 + 0.06) 10 = Rs 1,000 (1.7908) = Rs 1,790.85 What if interest was not compounded interest? Then we would have a lower balance in the account: FV5 = Rs 1,000 + [Rs 1,000(0.06) (5)] = Rs 1,300 FV10 = Rs 1,000 + [Rs 1,000 (0.06)(10)] = Rs 1,600 Simple interest is the product of the principal, the time in years, and the annual interest rate. In compound interest the principal is more than once during the time of the investment. Compound interest is another matter. It's good to receive compound interest, but not so good to pay compound interest. With compound interest, interest is calculated not only on the beginning interest, but also on any interest accumulated in the meantime. I hope you have understood the concept of simple interest and compound interest. It is explained with the help of a graph, which is self-explanatory.
Now let us solve a problem for Compound Interest vs. Simple Interest Example Suppose you are faced with a choice between two accounts, Account A and Account B. Account A provides 5% interest, compounded annually and Account B provides 5.25% simple interest. Consider a deposit of Rs 10,000 today. Which account provides the highest balance at the end of 4 years? Solution Account A: FV4 = Rs 10,000 (1 + 0.05) 4 = Rs 12,155.06 Account B: FV4 = Rs 10,000 + (Rs 10,000 (0.0525)(4)] = Rs 12,100.00 Account A provides the greater future value.
Present value is simply the reciprocal of compound interest. Another way to think of present value is to adopt a stance out on the time line in the future and look back toward time 0 to see what was the beginning amount. Present Value = P0 = Fn / (1+I) n Table A-3 shows present value factors: Note that they are all less than one. Therefore, when multiplying a future value by these factors, the future value is discounted down to present value. The table is used in much the same way as the other time value of money tables. To find the present value of a future amount, locate the appropriate number of years and the appropriate interest rate, take the resulting factor and multiply it times the future value. How much would you have to deposit now to have Rs 15,000 in 8 years if interest is 7%? = 15000 X .582 = 8730 Rs Consider a case in which you want to determine the value today of $ 1,000 to be received five years from now. If the interest rate (i.e., discount rate) is 4%, Problem Suppose that you wish to have Rs 20,000 saved by the end of five years. And suppose you deposit funds today in account that pays 4% interest, compounded annually. How much must you deposit today to meet your goal? Solution Given: FV = Rs 20,000; n = 5; i = 4%
PV = Rs 20,000/(1 + 0.04) 5 = Rs 20,000/1.21665 PV = Rs 16,438.54 Q. If you want to have Rs 10,000 in 3 years and you can earn 8%, how much would you have to deposit today? Rs 7938.00 Rs 25,771 Rs 12,597 Using Tables to Solve Future Value Problems A-1 for future value at the end of n yrs A-3 for present value at the beginning of the year Compound Interest tables have been calculated by figuring out the (1+I) n values for various time periods and interest rates. Look at Time Value of Money Future Value Factors. This table summarizes the factors for various interest rates for various years. To use the table, simply go down the left-hand column to locate the appropriate number of years. Then go out along the top row until the appropriate interest rate is located. For instance, to find the future value of Rs100 at 5% compound interest, look up five years on the table, and then go out to 5% interest. At the intersection of these two values, a factor of 1.2763 appears. Multiplying this factor times the beginning value of Rs100.00 results in Rs127.63, exactly what was calculated using the Compound Interest Formula. Note, however, that there may be slight differences between using the formula and tables due to rounding errors.
An example shows how simple it is to use the tables to calculate future amounts. You deposit Rs2000 today at 6% interest. How much will you have in 5 years? =2000*1.338=2676 The following exercise should aid in using tables to solve future value problems. Please answer the questions below by using tables 1. You invest Rs 5,000 today. You will earn 8% interest. How much will you have in 4 years? (Pick the closest answer) Rs 6,802.50 Rs 6,843.00 Rs 3,675 2.You have Rs 450,000 to invest. If you think you can earn 7%, how much could you accumulate in 10 years? ? (Pick the closest answer) Rs 25,415 Rs 722,610 Rs 722,610 3.If a commodity costs Rs500 now and inflation is expected to go up at the rate of 10% per year, how much will the commodity cost in 5 years? Rs 805.25 Rs 3,052.55 Cannot tell from this information
Now we will talk about the cases when the interest is given semi annually, quarterly, monthly…. The interest rate per compounding period is found by taking the annual rate and dividing it by the number of times per year the cash flows are compounded. The total number of compounding periods is found by multiplying the number of years by the number of times per year cash flows is compounded. The formula for this shorter compounding period is FVn = PV0 (1+i/m)n*m Consider the following example. You deposited Rs 1000 for 5 yrs in a bank that offers 10% interest p.a. compounded semiannually, what will be the future value. =1000 (1+. 10/2) 5*2 For instance, suppose someone were to invest Rs 5,000 at 8% interest, compounded semiannually, and hold it for five years. The interest rate per compounding period would be 4%, (8% / 2) The number of compounding periods would be 10 (5 x 2) To solve, find the future value of a single sum looking up 4% and 10 periods in the Future Value table. FV = PV (FVIF) FV = Rs 5,000(1.480) FV = Rs 7,400 Now let us solve a problem for Frequency of Compounding
Problem Suppose you invest Rs 20,000 in an account that pays 12% interest, compounded monthly. How much do you have in the account at the end of 5 years? Solution FV = Rs 20,000 (1 + 0.01) 60 = Rs 20,000 (1.8167) = Rs 36,333.93 In what period of time money will be doubled? Investor most of the times wants to know that in what period of time his money will be doubled. For this the “rule of 72” is used. Suppose the rate of interest is 12%, the doubling period will be 72/12=6 yrs. Apart from this rule we do use another rule, which gives better results, is the “rule of 69” = .35 + 69 int rate = .35 + 69 12 = .35 + 5.75 = 6.1 yrs Practice Problems What is the balance in an account at the end of 10 years if Rs 2,500 is deposited today and the account earns 4% interest, compounded annually? Quarterly? If you deposit Rs10 in an account that pays 5% interest, compounded annually, how much will you have at the end of 10 years? 50 years? 100 years? How much will be in an account at the end of five years the amount deposited today is Rs 10,000 and interest is 8% per year, compounded semi-annually?
Answers 1.Annual compounding: FV = Rs 2,500 (1 + 0.04) 10 = Rs 2,500 (1.4802) = Rs 3,700.61 Quarterly compounding: FV = Rs 2,500 (1 + 0.01) 40 = Rs 2,500 (1.4889) = Rs3,722.16 2. 10 years: FV = Rs10 (1+0.05) 10 = Rs10 (1.6289) = Rs16.29 50 years: FV = Rs10 (1 + 0.05) 50 = Rs10 (11.4674) = Rs114.67 100 years: FV = Rs10 (1 + 0.05) 100 = Rs10 (131.50) = Rs 1,315.01 3. FV = Rs 10,000 (1+0.04) 10 = Rs10,000 (1.4802) = Rs14,802.44 For example, assume you deposit Rs. 10,000 in a bank, which offers 10% interest per annum compounded semi-annually which means that interest is paid every six months. Now, amount in the beginning = Rs. 10,000 Rs. Interest @ 10% p.a. for first six = 500 0.1 Months 10000 x =10500 2 Interest for second 0.1 6 months = 10500 x = 525 2 Amount at the end of the year = 11,025
Instead, if the compounding is done annually, the amount at the end of the year will be 10,000 (1 + 0.1) = Rs, 11000. This difference of Rs. 25 is because under semi-annual compounding, the interest for first 6 moths earns interest in the second 6 months. The generalized formula for these shorter compounding periods is mxn  K FVn = PV 1 +   M Where FVn = future value after ‘n’ years PV = cash flow today K = Nominal Interest rate per annum M = Number of times compounding is done during a year N = Number of years for which compounding is done. Example Under the Vijaya Cash Certificate scheme of Vijaya Bank, deposits can be made for periods ranging from 6 months to 10 years. Every quarter, interest will be added on to the principal. The rate of interest applied is 9% p.a. for periods form 12 to 13 months and 10% p.a. for periods form 24 to 120 months. An amount of Rs. 1,000 invested for 2 years will grow to mn  K Fn = PV 1 +   M Where m = frequency of compounding during a year
8  0.10  = 1000 1 +   4  = 1000 (1.025)8 = 1000 x 1.2184 = Rs. 1218 Effective vs. Nominal Rate of interest We have seen above that the accumulation under the semi-annual compounding scheme exceeds the accumulation under the annual compounding scheme compounding scheme, the nominal rate of interest is 10% per annum, under the scheme where compounding is done semi annually, the principal amount grows at the rate of 10.25 percent per annum. This 1025 percent is called the effective rate of interest which is the rate of interest per annum under annual compounding that produces the same effect as that produced by an interest rate of 10 percent under semi – annual compounding. The general relationship between the effective an nominal rates of interest is as follows: m  k = 1 +  − 1  m where r = effective rate of interest k = nominal rate of interest m = frequency of compounding per year. Example Find out the effective rate of interest, if the nominal rate of interest is 12% and is quarterly compounded?
Effective rate of interest k m = (1 + ) –1 m 0.12 4 = (+ ) –1 4 = (1 + 0.03)4 -1 = 1.126 -1 = 0.126 = 12.6% p.a. compounded quarterly By now you should have clear understanding of Compounding Discounting Doubling period (Rule of 72) Doubling period (Rule of 69) Shorter compounding periods Effective vs. Nominal Rate of interest By now you should be an expert in using the following two tables: A-1 The Compound Sum of one rupee FVIF A-3 The Present Value of one rupee PVIF IMPORTANT The inverse of FVIF is PVIF i.e. inverse of FVIF is PVIF.
IMPORTANT Slide 1 Chapter 3 Time Value of Time Value of Money Money 3-1 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 2 The Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan 3-2 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 3 The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years? years Obviously, $10,000 today. today You already recognize that there is TIME VALUE TO MONEY!! MONEY 3-3 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 4 Why TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST. INTEREST 3-4 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 5 Types of Interest Simple Interest Interest paid (earned) on only the original amount, or principal borrowed (lent). Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). 3-5 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 6 Simple Interest Formula Formula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods 3-6 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 7 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? SI = P0(i)(n) = $1,000(.07)(2) = $140 3-7 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 8 Simple Interest (FV) What is the Future Value (FV of the FV) deposit? FV = P0 + SI = $1,000 + $140 = $1,140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. 3-8 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 9 Simple Interest (PV) What is the Present Value (PV of the PV) previous problem? The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest 3-9 rate. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 10 Why Compound Interest? Future Value of a Single $1,000 Deposit Future Value (U.S. Dollars) 20000 10% Simple 15000 Interest 7% Compound 10000 Interest 5000 10% Compound Interest 0 1st Year 10th 20th 30th Year Year Year 3-10 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 11 Future Value Single Deposit (Graphic) Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. years 0 1 2 7% $1,000 FV2 3-11 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 12 Future Value Single Deposit (Formula) FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. 3-12 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 13 Future Value Single Deposit (Formula) FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $1,000(1.07)(1.07) $1,000 = P0 (1+i) 2 = $1,000(1.07)2 $1,000 = $1,144.90 You earned an EXTRA $4.90 in Year 2 with 3-13 compound over simple interest. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 14 General Future Value Formula FV1 = P0(1+i)1 FV2 = P0(1+i)2 etc. General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) -- See Table I 3-14 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 15 Valuation Using Table I FVIFi,n is found on Table I at the end of the book or on the card insert. Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 3-15 5 1.338 1.403 1.469 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 16 Using Future Value Tables FV2 = $1,000 (FVIF7%,2) FVIF = $1,000 (1.145) = $1,145 [Due to Rounding] Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469 3-16 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 17 Story Problem Example Julie Miller wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. years 0 1 2 3 4 5 10% $10,000 FV5 3-17 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 18 Story Problem Solution Calculation based on general formula: FVn = P0 (1+i)n FV5 = $10,000 (1+ 0.10)5 = $16,105.10 Calculation based on Table I: FV5 = $10,000 (FVIF10%, 5) FVIF = $10,000 (1.611) = $16,110 [Due to Rounding] 3-18 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 19 Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? We will use the “Rule-of-72”. Rule- of- 72” 3-19 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 20 The “Rule-of-72” “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? Approx. Years to Double = 72 / i% 72 / 12% = 6 Years [Actual Time is 6.12 Years] 3-20 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 21 Present Value Single Deposit (Graphic) Assume that you need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 2 7% $1,000 PV0 PV1 3-21 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 22 Present Value Single Deposit (Formula) PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2 = FV2 / (1+i)2 = $873.44 0 1 2 7% $1,000 PV0 3-22 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 23 General Present Value Formula PV0 = FV1 / (1+i)1 PV0 = FV2 / (1+i)2 etc. General Present Value Formula: PV0 = FVn / (1+i)n or PV0 = FVn (PVIFi,n) -- See Table II 3-23 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 24 Valuation Using Table II PVIFi,n is found on Table II at the end of the book or on the card insert. Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681 3-24 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 25 Using Present Value Tables PV2 = $1,000 (PVIF7%,2) = $1,000 (.873) = $873 [Due to Rounding] Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 3-25 5 .747 .713 .681 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 26 Story Problem Example Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%. 0 1 2 3 4 5 10% $10,000 PV0 3-26 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 27 Story Problem Solution Calculation based on general formula: PV0 = FVn / (1+i)n PV0 = $10,000 / (1+ 0.10)5 = $6,209.21 Calculation based on Table I: PV0 = $10,000 (PVIF10%, 5) PVIF = $10,000 (.621) = $6,210.00 [Due to Rounding] 3-27 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 28 Types of Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. Ordinary Annuity: Payments or receipts Annuity occur at the end of each period. Annuity Due: Payments or receipts Due occur at the beginning of each period. 3-28 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 29 Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings 3-29 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 30 Parts of an Annuity (Ordinary Annuity) End of End of End of Period 1 Period 2 Period 3 0 1 2 3 $100 $100 $100 Today Equal Cash Flows Each 1 Period Apart 3-30 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 31 Parts of an Annuity (Annuity Due) Beginning of Beginning of Beginning of Period 1 Period 2 Period 3 0 1 2 3 $100 $100 $100 Today Equal Cash Flows Each 1 Period Apart 3-31 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 32 Overview of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 n n+1 i% . . . R R R R = Periodic Cash Flow FVAn FVAn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0 3-32 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 33 Example of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,070 $1,145 FVA3 =$1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 $3,215 = FVA3 = $1,145 + $1,070 + $1,000 = $3,215 3-33 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 34 Hint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period. 3-34 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 35 Valuation Using Table III FVAn = R (FVIFAi%,n) FVA3 = $1,000 (FVIFA7%,3) = $1,000 (3.215) = $3,215 Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867 3-35 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 36 Overview View of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 n -1 n i% . . . R R R R R FVADn = R(1+i)n + R(1+i)n-1 + FVADn ... + R(1+i)2 + R(1+i)1 = FVAn (1+i) 3-36 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 37 Example of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,070 $1,145 $1,225 FVAD3 = $1,000(1.07)3 + $3,440 = FVAD3 $1,000(1.07)2 + $1,000(1.07)1 = $1,225 + $1,145 + $1,070 = $3,440 3-37 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 38 Valuation Using Table III FVADn = R (FVIFAi%,n)(1+i) FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440 Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867 3-38 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 39 Overview of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 n n+1 i% . . . R R R R = Periodic Cash Flow PVAn PVAn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n 3-39 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 40 Example of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $ 934.58 $ 873.44 $ 816.30 $2,624.32 = PVA3 PVA3 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3 = $934.58 + $873.44 + $816.30 3-40 = $2,624.32 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 41 Hint on Annuity Valuation The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the end of the first cash flow period. 3-41 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 42 Valuation Using Table IV PVAn = R (PVIFAi%,n) PVA3 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624 Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993 3-42 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 43 Overview of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 n -1 n i% . . . R R R R R: Periodic PVADn Cash Flow PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAn (1+i) 3-43 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 44 Example of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7% $1,000.00 $1,000 $1,000 $ 934.58 $ 873.44 $2,808.02 = PVADn PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02 3-44 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 45 Valuation Using Table IV PVADn = R (PVIFAi%,n)(1+i) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808 Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993 3-45 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 46 Steps to Solve Time Value Steps to Solve Time Value of Money Problems of Money Problems 1. Read problem thoroughly 2. Determine if it is a PV or FV problem 3. Create a time line 4. Put cash flows and arrows on time line 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) 3-46 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 47 Mixed Flows Example Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%? 10% 0 1 2 3 4 5 10% $600 $600 $400 $400 $100 PV0 3-47 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 48 How to Solve? 1. Solve a “piece-at-a-time” by piece-at- time discounting each piece back to t=0. 2. Solve a “group-at-a-time” by first group-at- time breaking problem into groups of annuity streams and any single cash flow group. Then discount each group back to t=0. 3-48 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 49 “Piece-At-A-Time” “Piece-At-A-Time” 0 1 2 3 4 5 10% $600 $600 $400 $400 $100 $545.45 $495.87 $300.53 $273.21 $ 62.09 $1677.15 = PV0 of the Mixed Flow 3-49 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 50 “Group-At-A-Time” (#1) “Group-At-A-Time” 0 1 2 3 4 5 10% $600 $600 $400 $400 $100 $1,041.60 $ 573.57 $ 62.10 $1,677.27 = PV0 of Mixed Flow [Using Tables] $600(PVIFA10%,2) = $600(1.736) = $1,041.60 $400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57 $100 (PVIF10%,5) = $100 (0.621) = $62.10 3-50 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 51 “Group-At-A-Time” (#2) “Group-At-A-Time” 0 1 2 3 4 $400 $400 $400 $400 $1,268.00 0 1 2 PV0 equals Plus $200 $200 $1677.30. $347.20 0 1 2 3 4 5 Plus $100 $62.10 3-51 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 52 Frequency of Compounding General Formula: FVn = PV0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today 3-52 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 53 Impact of Frequency Julie Miller has $1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2) 1,000 = 1,254.40 Semi FV2 = 1,000(1+ [.12/2])(2)(2) 1,000 = 1,262.48 3-53 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 54 Impact of Frequency Qrtly FV2 = 1,000(1+ [.12/4])(4)(2) 1,000 = 1,266.77 Monthly FV2 = 1,000(1+ [.12/12])(12)(2) 1,000 = 1,269.73 Daily FV2 = 1,000(1+[.12/365])(365)(2) 1,000 = 1,271.20 3-54 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 55 Effective Annual Interest Rate Effective Annual Interest Rate The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. (1 + [ i / m ] )m - 1 3-55 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 56 BW’s Effective Annual Interest Rate Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)? EAR EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%! 3-56 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 57 Steps to Amortizing a Loan 1. Calculate the payment per period. 2. Determine the interest in Period t. (Loan balance at t-1) x (i% / m) 3. Compute principal payment in Period t. (Payment - interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal payment from Step 3) 5. Start again at Step 2 and repeat. 3-57 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 58 Amortizing a Loan Example Julie Miller is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PV0 = R (PVIFA i%,n) $10,000 = R (PVIFA 12%,5) $10,000 = R (3.605) R = $10,000 / 3.605 = $2,774 3-58 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
Slide 59 Amortizing a Loan Example End of Paym ent In te re s t P rin c ip a l E n d in g Year B a la n c e 0 --- --- --- $ 1 0 ,0 0 0 1 $ 2 ,7 7 4 $ 1 ,2 0 0 $ 1 ,5 7 4 8 ,4 2 6 2 2 ,7 7 4 1 ,0 1 1 1 ,7 6 3 6 ,6 6 3 3 2 ,7 7 4 800 1 ,9 7 4 4 ,6 8 9 4 2 ,7 7 4 563 2 ,2 1 1 2 ,4 7 8 5 2 ,7 7 5 297 2 ,4 7 8 0 $ 1 3 ,8 7 1 $ 3 ,8 7 1 $ 1 0 ,0 0 0 [Last Payment Slightly Higher Due to Rounding] 3-59 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________

More Related Content

PDF
Knowledge varsity dec 2011 study_session_2
byfinexcel
 
PDF
Time value of money
byAtif Hossain
 
PPTX
The Time Value of Money
byJyoti Yadav
 
PPT
Time value of money
byDelwin Arikatt
 
PPTX
Time value concepts
byAswin prakash i , Xantus Technologies
 
PPTX
Time value of money
bySimran Kaur
 
PPTX
القيمة الزمنية للنقود Tvm
byMahmoud Al Saftawi
 
PPTX
Time value of money
byM.i. Jamil
 
Knowledge varsity dec 2011 study_session_2
byfinexcel
 
Time value of money
byAtif Hossain
 
The Time Value of Money
byJyoti Yadav
 
Time value of money
byDelwin Arikatt
 
Time value concepts
byAswin prakash i , Xantus Technologies
 
Time value of money
bySimran Kaur
 
القيمة الزمنية للنقود Tvm
byMahmoud Al Saftawi
 
Time value of money
byM.i. Jamil
 

What's hot

PPTX
Time value of money ppt.
bypriya sinha
 
PPTX
Time value of money
byMahesh Kadam
 
PDF
Class I finance
byReuben Ray
 
PPT
Time value of money
bymishra21
 
PPT
Time value of money
bydomsr
 
PPT
Time Value of Money
byMaged Elsakka
 
PPTX
Time value of money
byBFSI academy
 
PPTX
Time Value Of Money F F M
byZoha Qureshi
 
PPT
Lecture 4 tvm
bysohag sikder
 
PPT
time value of money
byashfaque75
 
PPTX
Time Value of Money
byJoseph Winthrop Godoy
 
PPTX
Interest rates
byGiga B
 
PDF
Textual notes time value of money
bypriyameril
 
PPT
Transtutors time value of money
byTransweb Global Inc
 
PPT
Time value of money
byhanumandas88
 
RTF
Time value of money 1
bySandeep Surya
 
PPT
Time value of money
byJubayer Alam Shoikat
 
PPTX
Time value of money
byVijay R. Joshi
 
Time value of money ppt.
bypriya sinha
 
Time value of money
byMahesh Kadam
 
Class I finance
byReuben Ray
 
Time value of money
bymishra21
 
Time value of money
bydomsr
 
Time Value of Money
byMaged Elsakka
 
Time value of money
byBFSI academy
 
Time Value Of Money F F M
byZoha Qureshi
 
Lecture 4 tvm
bysohag sikder
 
time value of money
byashfaque75
 
Time Value of Money
byJoseph Winthrop Godoy
 
Interest rates
byGiga B
 
Textual notes time value of money
bypriyameril
 
Transtutors time value of money
byTransweb Global Inc
 
Time value of money
byhanumandas88
 
Time value of money 1
bySandeep Surya
 
Time value of money
byJubayer Alam Shoikat
 
Time value of money
byVijay R. Joshi
 

Similar to Lecture 05

PPT
TIME VALUE OF MONEY IS VERY IMPORTANT.ppt
byVijaylaxmiSingh21
 
PPTX
TIME VALUE OF MONEY BUSINESS ADMINISTRATION
byladyapril1
 
PPTX
finman-ppt.pptx TIME VALUE OF MONEY BUSINESS AD
byladyapril1
 
DOC
Time value of money
byJubayer Alam Shoikat
 
PPTX
Tov
bysneha dalvi
 
PPTX
Time Value of Money
bySultan Islam
 
PPT
Valuation 1 (1)
byEzgi Kurt
 
PPTX
Time value of money
bysriharshini makam
 
PPTX
Time value of money
byEsmatullah Mohammady
 
PPTX
FM-I Chap. 3& 4 for Finance and Ac.pptx
bybelaywube
 
PPTX
Time value of money means of a unit of money
byabhishekdhabalad
 
PDF
Corporate Financial Management (Time value of Money)
byAcharya Institute of Graduate Studies
 
DOCX
Time Value of Money and Bond Valuation  Please respond to the foll.docx
byamit657720
 
PPT
ch06 Accounting and Time Value of Money.ppt
byPida8
 
PPTX
Time Value of Money... Business Finance.pptx
byTAWSI1
 
PPTX
cf-iNTRO
byDr Ajay Kumar
 
PPT
Finan Mgmt I time value of money (1).ppt
bybelaywube
 
PPT
Iii A Time Value Of Money
byrajeevgupta
 
PPT
Iii A Time Value Of Money
byrajeevgupta
 
PDF
CHAPTER_3_TIME_VALUE_OF_MONEY Models.pdf
byyabsirayabuman
 
TIME VALUE OF MONEY IS VERY IMPORTANT.ppt
byVijaylaxmiSingh21
 
TIME VALUE OF MONEY BUSINESS ADMINISTRATION
byladyapril1
 
finman-ppt.pptx TIME VALUE OF MONEY BUSINESS AD
byladyapril1
 
Time value of money
byJubayer Alam Shoikat
 
Tov
bysneha dalvi
 
Time Value of Money
bySultan Islam
 
Valuation 1 (1)
byEzgi Kurt
 
Time value of money
bysriharshini makam
 
Time value of money
byEsmatullah Mohammady
 
FM-I Chap. 3& 4 for Finance and Ac.pptx
bybelaywube
 
Time value of money means of a unit of money
byabhishekdhabalad
 
Corporate Financial Management (Time value of Money)
byAcharya Institute of Graduate Studies
 
Time Value of Money and Bond Valuation  Please respond to the foll.docx
byamit657720
 
ch06 Accounting and Time Value of Money.ppt
byPida8
 
Time Value of Money... Business Finance.pptx
byTAWSI1
 
cf-iNTRO
byDr Ajay Kumar
 
Finan Mgmt I time value of money (1).ppt
bybelaywube
 
Iii A Time Value Of Money
byrajeevgupta
 
Iii A Time Value Of Money
byrajeevgupta
 
CHAPTER_3_TIME_VALUE_OF_MONEY Models.pdf
byyabsirayabuman
 

Lecture 05

  • 1.
    UNIT ONE CHAPTER THREE THETIME VALUE OF MONEY
  • 2.
    Lesson 5 Chapter 3 The time value of money Unit 1 Core concepts in financial management After reading this lesson you will be able to: - Understand what is meant by "the time value of money." Describe how the interest rate can be used to adjust the value of cash flows to a single point in time. Calculate the future value of an amount invested today. Calculate the present value of a single future cash flow. Understand the relationship between present and future values. Understand in what period of time money doubles Understand shorter compounding periods Calculate & understand the relationship between effective & nominal interest rate. Use the interest factor tables and understand how they provide a short cut to calculating present and future values. You all instinctively know that money loses its value with time. Why does this happen? What does a Financial Manager have to do to accommodate this loss in the value of money with time? In this section, we will take a look at this very interesting issue. Why should financial managers be familiar with the time value of money? The time value of money shows mathematically how the timing of cash flows, combined
  • 3.
    with the opportunitycosts of capital, affect financial asset values. A thorough understanding of these concepts gives a financial manager powerful tool to maximize wealth. What is the time value of money? The time value of money serves as the foundation for all other notions in finance. It impacts business finance, consumer finance and government finance. Time value of money results from the concept of interest. This overview covers an introduction to simple interest and compound interest, illustrates the use of time value of money tables, shows a approach to solving time value of money problems and introduces the concepts of intra year compounding, annuities due, and perpetuities. A simple introduction to working time value of money problems on a financial calculator is included as well as additional resources to help understand time value of money. Time value of money The universal preference for a rupee today over a rupee at some future time is because of the following reasons: - Alternative uses/ Opportunity cost Inflation Uncertainty The manner in which these three determinants combine to determine the rate of interest can be represented symbolically as Nominal or market rate of interest rate = Real rate of interest + Expected rate of Inflation + Risk of premiums to compensate uncertainty
  • 4.
    Basics Evaluating financial transactionsrequires valuing uncertain future cash flows. Translating a value to the present is referred to as discounting. Translating a value to the future is referred to as compounding The principal is the amount borrowed. Interest is the compensation for the opportunity cost of funds and the uncertainty of repayment of the amount borrowed; that is, it represents both the price of time and the price of risk. The price of time is compensation for the opportunity cost of funds and the price of risk is compensation for bearing risk. Interest is compound interest if interest is paid on both the principal and any accumulated interest. Most financial transactions involve compound interest, though there are a few consumer transactions that use simple interest (that is, interest paid only on the principal or amount borrowed). Under the method of compounding, we find the future values (FV) of all the cash flows at the end of the time horizon at a particular rate of interest. Therefore, in this case we will be comparing the future value of the initial outflow of Rs. 1,000 as at the end of year 4 with the sum of the future values of the yearly cash inflows at the end of year 4. This process can be schematically represented as follows: PROCESS OF DISCOUNTING Under the method of discounting, we reckon the time value of money now, i.e. at time 0 on the time line. So, we will be comparing the initial outflow with the sum of the present values (PV) of the future inflows at a given rate of interest. Translating a value back in time -- referred to as discounting -- requires determining what a future amount or cash flow is worth today. Discounting is used in valuation because we often want to determine the value today of future value or cash flows. The equation for the present value is:
  • 5.
    Present value =PV = FV / (1 + i) n Where: PV = present value (today's value), FV = future value (a value or cash flow sometime in the future), i = interest rate per period, and n = number of compounding periods And [(1 + i) n] is the compound factor. We can also represent the equation a number of different, yet equivalent ways: Where PVIFi,n is the present value interest factor, or discount factor. In other words future value is the sum of the present value and interest: Future value = Present value + interest From the formula for the present value you can see that as the number of discount periods, n, becomes larger, the discount factor becomes smaller and the present value becomes less, and as the interest rate per period, i, becomes larger, the discount factor becomes smaller and the present value becomes less. Therefore, the present value is influenced by both the interest rate (i.e., the discount rate) and the numbers of discount periods. Example Suppose you invest 1,000 in an account that pays 6% interest, compounded annually. How much will you have in the account at the end of 5 years if you make no withdrawals? After 10 years?
  • 6.
    Solution FV5 = Rs 1,000 (1 + 0.06) 5 = Rs 1,000 (1.3382) = Rs 1,338.23 FV10 = Rs 1,000 (1 + 0.06) 10 = Rs 1,000 (1.7908) = Rs 1,790.85 What if interest was not compounded interest? Then we would have a lower balance in the account: FV5 = Rs 1,000 + [Rs 1,000(0.06) (5)] = Rs 1,300 FV10 = Rs 1,000 + [Rs 1,000 (0.06)(10)] = Rs 1,600 Simple interest is the product of the principal, the time in years, and the annual interest rate. In compound interest the principal is more than once during the time of the investment. Compound interest is another matter. It's good to receive compound interest, but not so good to pay compound interest. With compound interest, interest is calculated not only on the beginning interest, but also on any interest accumulated in the meantime. I hope you have understood the concept of simple interest and compound interest. It is explained with the help of a graph, which is self-explanatory.
  • 7.
    Now let ussolve a problem for Compound Interest vs. Simple Interest Example Suppose you are faced with a choice between two accounts, Account A and Account B. Account A provides 5% interest, compounded annually and Account B provides 5.25% simple interest. Consider a deposit of Rs 10,000 today. Which account provides the highest balance at the end of 4 years? Solution Account A: FV4 = Rs 10,000 (1 + 0.05) 4 = Rs 12,155.06 Account B: FV4 = Rs 10,000 + (Rs 10,000 (0.0525)(4)] = Rs 12,100.00 Account A provides the greater future value.
  • 8.
    Present value issimply the reciprocal of compound interest. Another way to think of present value is to adopt a stance out on the time line in the future and look back toward time 0 to see what was the beginning amount. Present Value = P0 = Fn / (1+I) n Table A-3 shows present value factors: Note that they are all less than one. Therefore, when multiplying a future value by these factors, the future value is discounted down to present value. The table is used in much the same way as the other time value of money tables. To find the present value of a future amount, locate the appropriate number of years and the appropriate interest rate, take the resulting factor and multiply it times the future value. How much would you have to deposit now to have Rs 15,000 in 8 years if interest is 7%? = 15000 X .582 = 8730 Rs Consider a case in which you want to determine the value today of $ 1,000 to be received five years from now. If the interest rate (i.e., discount rate) is 4%, Problem Suppose that you wish to have Rs 20,000 saved by the end of five years. And suppose you deposit funds today in account that pays 4% interest, compounded annually. How much must you deposit today to meet your goal? Solution Given: FV = Rs 20,000; n = 5; i = 4%
  • 9.
    PV = Rs20,000/(1 + 0.04) 5 = Rs 20,000/1.21665 PV = Rs 16,438.54 Q. If you want to have Rs 10,000 in 3 years and you can earn 8%, how much would you have to deposit today? Rs 7938.00 Rs 25,771 Rs 12,597 Using Tables to Solve Future Value Problems A-1 for future value at the end of n yrs A-3 for present value at the beginning of the year Compound Interest tables have been calculated by figuring out the (1+I) n values for various time periods and interest rates. Look at Time Value of Money Future Value Factors. This table summarizes the factors for various interest rates for various years. To use the table, simply go down the left-hand column to locate the appropriate number of years. Then go out along the top row until the appropriate interest rate is located. For instance, to find the future value of Rs100 at 5% compound interest, look up five years on the table, and then go out to 5% interest. At the intersection of these two values, a factor of 1.2763 appears. Multiplying this factor times the beginning value of Rs100.00 results in Rs127.63, exactly what was calculated using the Compound Interest Formula. Note, however, that there may be slight differences between using the formula and tables due to rounding errors.
  • 10.
    An example showshow simple it is to use the tables to calculate future amounts. You deposit Rs2000 today at 6% interest. How much will you have in 5 years? =2000*1.338=2676 The following exercise should aid in using tables to solve future value problems. Please answer the questions below by using tables 1. You invest Rs 5,000 today. You will earn 8% interest. How much will you have in 4 years? (Pick the closest answer) Rs 6,802.50 Rs 6,843.00 Rs 3,675 2.You have Rs 450,000 to invest. If you think you can earn 7%, how much could you accumulate in 10 years? ? (Pick the closest answer) Rs 25,415 Rs 722,610 Rs 722,610 3.If a commodity costs Rs500 now and inflation is expected to go up at the rate of 10% per year, how much will the commodity cost in 5 years? Rs 805.25 Rs 3,052.55 Cannot tell from this information
  • 11.
    Now we willtalk about the cases when the interest is given semi annually, quarterly, monthly…. The interest rate per compounding period is found by taking the annual rate and dividing it by the number of times per year the cash flows are compounded. The total number of compounding periods is found by multiplying the number of years by the number of times per year cash flows is compounded. The formula for this shorter compounding period is FVn = PV0 (1+i/m)n*m Consider the following example. You deposited Rs 1000 for 5 yrs in a bank that offers 10% interest p.a. compounded semiannually, what will be the future value. =1000 (1+. 10/2) 5*2 For instance, suppose someone were to invest Rs 5,000 at 8% interest, compounded semiannually, and hold it for five years. The interest rate per compounding period would be 4%, (8% / 2) The number of compounding periods would be 10 (5 x 2) To solve, find the future value of a single sum looking up 4% and 10 periods in the Future Value table. FV = PV (FVIF) FV = Rs 5,000(1.480) FV = Rs 7,400 Now let us solve a problem for Frequency of Compounding
  • 12.
    Problem Suppose you investRs 20,000 in an account that pays 12% interest, compounded monthly. How much do you have in the account at the end of 5 years? Solution FV = Rs 20,000 (1 + 0.01) 60 = Rs 20,000 (1.8167) = Rs 36,333.93 In what period of time money will be doubled? Investor most of the times wants to know that in what period of time his money will be doubled. For this the “rule of 72” is used. Suppose the rate of interest is 12%, the doubling period will be 72/12=6 yrs. Apart from this rule we do use another rule, which gives better results, is the “rule of 69” = .35 + 69 int rate = .35 + 69 12 = .35 + 5.75 = 6.1 yrs Practice Problems What is the balance in an account at the end of 10 years if Rs 2,500 is deposited today and the account earns 4% interest, compounded annually? Quarterly? If you deposit Rs10 in an account that pays 5% interest, compounded annually, how much will you have at the end of 10 years? 50 years? 100 years? How much will be in an account at the end of five years the amount deposited today is Rs 10,000 and interest is 8% per year, compounded semi-annually?
  • 13.
    Answers 1.Annual compounding: FV= Rs 2,500 (1 + 0.04) 10 = Rs 2,500 (1.4802) = Rs 3,700.61 Quarterly compounding: FV = Rs 2,500 (1 + 0.01) 40 = Rs 2,500 (1.4889) = Rs3,722.16 2. 10 years: FV = Rs10 (1+0.05) 10 = Rs10 (1.6289) = Rs16.29 50 years: FV = Rs10 (1 + 0.05) 50 = Rs10 (11.4674) = Rs114.67 100 years: FV = Rs10 (1 + 0.05) 100 = Rs10 (131.50) = Rs 1,315.01 3. FV = Rs 10,000 (1+0.04) 10 = Rs10,000 (1.4802) = Rs14,802.44 For example, assume you deposit Rs. 10,000 in a bank, which offers 10% interest per annum compounded semi-annually which means that interest is paid every six months. Now, amount in the beginning = Rs. 10,000 Rs. Interest @ 10% p.a. for first six = 500 0.1 Months 10000 x =10500 2 Interest for second 0.1 6 months = 10500 x = 525 2 Amount at the end of the year = 11,025
  • 14.
    Instead, if thecompounding is done annually, the amount at the end of the year will be 10,000 (1 + 0.1) = Rs, 11000. This difference of Rs. 25 is because under semi-annual compounding, the interest for first 6 moths earns interest in the second 6 months. The generalized formula for these shorter compounding periods is mxn  K FVn = PV 1 +   M Where FVn = future value after ‘n’ years PV = cash flow today K = Nominal Interest rate per annum M = Number of times compounding is done during a year N = Number of years for which compounding is done. Example Under the Vijaya Cash Certificate scheme of Vijaya Bank, deposits can be made for periods ranging from 6 months to 10 years. Every quarter, interest will be added on to the principal. The rate of interest applied is 9% p.a. for periods form 12 to 13 months and 10% p.a. for periods form 24 to 120 months. An amount of Rs. 1,000 invested for 2 years will grow to mn  K Fn = PV 1 +   M Where m = frequency of compounding during a year
  • 15.
    8  0.10  = 1000 1 +   4  = 1000 (1.025)8 = 1000 x 1.2184 = Rs. 1218 Effective vs. Nominal Rate of interest We have seen above that the accumulation under the semi-annual compounding scheme exceeds the accumulation under the annual compounding scheme compounding scheme, the nominal rate of interest is 10% per annum, under the scheme where compounding is done semi annually, the principal amount grows at the rate of 10.25 percent per annum. This 1025 percent is called the effective rate of interest which is the rate of interest per annum under annual compounding that produces the same effect as that produced by an interest rate of 10 percent under semi – annual compounding. The general relationship between the effective an nominal rates of interest is as follows: m  k = 1 +  − 1  m where r = effective rate of interest k = nominal rate of interest m = frequency of compounding per year. Example Find out the effective rate of interest, if the nominal rate of interest is 12% and is quarterly compounded?
  • 16.
    Effective rate ofinterest k m = (1 + ) –1 m 0.12 4 = (+ ) –1 4 = (1 + 0.03)4 -1 = 1.126 -1 = 0.126 = 12.6% p.a. compounded quarterly By now you should have clear understanding of Compounding Discounting Doubling period (Rule of 72) Doubling period (Rule of 69) Shorter compounding periods Effective vs. Nominal Rate of interest By now you should be an expert in using the following two tables: A-1 The Compound Sum of one rupee FVIF A-3 The Present Value of one rupee PVIF IMPORTANT The inverse of FVIF is PVIF i.e. inverse of FVIF is PVIF.
  • 17.
    IMPORTANT Slide 1 Chapter 3 Time Value of Time Value of Money Money 3-1 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 18.
    Slide 2 The Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan 3-2 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 19.
    Slide 3 The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years? years Obviously, $10,000 today. today You already recognize that there is TIME VALUE TO MONEY!! MONEY 3-3 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 20.
    Slide 4 Why TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST. INTEREST 3-4 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 21.
    Slide 5 Types of Interest Simple Interest Interest paid (earned) on only the original amount, or principal borrowed (lent). Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). 3-5 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 22.
    Slide 6 Simple Interest Formula Formula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods 3-6 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 23.
    Slide 7 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? SI = P0(i)(n) = $1,000(.07)(2) = $140 3-7 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 24.
    Slide 8 Simple Interest (FV) What is the Future Value (FV of the FV) deposit? FV = P0 + SI = $1,000 + $140 = $1,140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. 3-8 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 25.
    Slide 9 Simple Interest (PV) What is the Present Value (PV of the PV) previous problem? The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest 3-9 rate. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 26.
    Slide 10 Why Compound Interest? Future Value of a Single $1,000 Deposit Future Value (U.S. Dollars) 20000 10% Simple 15000 Interest 7% Compound 10000 Interest 5000 10% Compound Interest 0 1st Year 10th 20th 30th Year Year Year 3-10 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 27.
    Slide 11 Future Value Single Deposit (Graphic) Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. years 0 1 2 7% $1,000 FV2 3-11 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 28.
    Slide 12 Future Value Single Deposit (Formula) FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. 3-12 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 29.
    Slide 13 Future Value Single Deposit (Formula) FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070 FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $1,000(1.07)(1.07) $1,000 = P0 (1+i) 2 = $1,000(1.07)2 $1,000 = $1,144.90 You earned an EXTRA $4.90 in Year 2 with 3-13 compound over simple interest. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 30.
    Slide 14 General Future Value Formula FV1 = P0(1+i)1 FV2 = P0(1+i)2 etc. General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) -- See Table I 3-14 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 31.
    Slide 15 Valuation Using Table I FVIFi,n is found on Table I at the end of the book or on the card insert. Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 3-15 5 1.338 1.403 1.469 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 32.
    Slide 16 Using Future Value Tables FV2 = $1,000 (FVIF7%,2) FVIF = $1,000 (1.145) = $1,145 [Due to Rounding] Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469 3-16 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 33.
    Slide 17 Story Problem Example Julie Miller wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. years 0 1 2 3 4 5 10% $10,000 FV5 3-17 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 34.
    Slide 18 Story Problem Solution Calculation based on general formula: FVn = P0 (1+i)n FV5 = $10,000 (1+ 0.10)5 = $16,105.10 Calculation based on Table I: FV5 = $10,000 (FVIF10%, 5) FVIF = $10,000 (1.611) = $16,110 [Due to Rounding] 3-18 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 35.
    Slide 19 Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? We will use the “Rule-of-72”. Rule- of- 72” 3-19 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 36.
    Slide 20 The “Rule-of-72” “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? Approx. Years to Double = 72 / i% 72 / 12% = 6 Years [Actual Time is 6.12 Years] 3-20 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 37.
    Slide 21 Present Value Single Deposit (Graphic) Assume that you need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 2 7% $1,000 PV0 PV1 3-21 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 38.
    Slide 22 Present Value Single Deposit (Formula) PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2 = FV2 / (1+i)2 = $873.44 0 1 2 7% $1,000 PV0 3-22 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 39.
    Slide 23 General Present Value Formula PV0 = FV1 / (1+i)1 PV0 = FV2 / (1+i)2 etc. General Present Value Formula: PV0 = FVn / (1+i)n or PV0 = FVn (PVIFi,n) -- See Table II 3-23 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 40.
    Slide 24 Valuation Using Table II PVIFi,n is found on Table II at the end of the book or on the card insert. Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681 3-24 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 41.
    Slide 25 Using Present Value Tables PV2 = $1,000 (PVIF7%,2) = $1,000 (.873) = $873 [Due to Rounding] Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 3-25 5 .747 .713 .681 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 42.
    Slide 26 Story Problem Example Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%. 0 1 2 3 4 5 10% $10,000 PV0 3-26 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 43.
    Slide 27 Story Problem Solution Calculation based on general formula: PV0 = FVn / (1+i)n PV0 = $10,000 / (1+ 0.10)5 = $6,209.21 Calculation based on Table I: PV0 = $10,000 (PVIF10%, 5) PVIF = $10,000 (.621) = $6,210.00 [Due to Rounding] 3-27 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 44.
    Slide 28 Types of Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. Ordinary Annuity: Payments or receipts Annuity occur at the end of each period. Annuity Due: Payments or receipts Due occur at the beginning of each period. 3-28 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 45.
    Slide 29 Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings 3-29 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 46.
    Slide 30 Parts of an Annuity (Ordinary Annuity) End of End of End of Period 1 Period 2 Period 3 0 1 2 3 $100 $100 $100 Today Equal Cash Flows Each 1 Period Apart 3-30 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 47.
    Slide 31 Parts of an Annuity (Annuity Due) Beginning of Beginning of Beginning of Period 1 Period 2 Period 3 0 1 2 3 $100 $100 $100 Today Equal Cash Flows Each 1 Period Apart 3-31 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 48.
    Slide 32 Overview of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 n n+1 i% . . . R R R R = Periodic Cash Flow FVAn FVAn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0 3-32 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 49.
    Slide 33 Example of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,070 $1,145 FVA3 =$1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 $3,215 = FVA3 = $1,145 + $1,070 + $1,000 = $3,215 3-33 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 50.
    Slide 34 Hint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period. 3-34 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 51.
    Slide 35 Valuation Using Table III FVAn = R (FVIFAi%,n) FVA3 = $1,000 (FVIFA7%,3) = $1,000 (3.215) = $3,215 Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867 3-35 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 52.
    Slide 36 Overview View of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 n -1 n i% . . . R R R R R FVADn = R(1+i)n + R(1+i)n-1 + FVADn ... + R(1+i)2 + R(1+i)1 = FVAn (1+i) 3-36 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 53.
    Slide 37 Example of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,070 $1,145 $1,225 FVAD3 = $1,000(1.07)3 + $3,440 = FVAD3 $1,000(1.07)2 + $1,000(1.07)1 = $1,225 + $1,145 + $1,070 = $3,440 3-37 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 54.
    Slide 38 Valuation Using Table III FVADn = R (FVIFAi%,n)(1+i) FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440 Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867 3-38 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 55.
    Slide 39 Overview of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 n n+1 i% . . . R R R R = Periodic Cash Flow PVAn PVAn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n 3-39 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 56.
    Slide 40 Example of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $ 934.58 $ 873.44 $ 816.30 $2,624.32 = PVA3 PVA3 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3 = $934.58 + $873.44 + $816.30 3-40 = $2,624.32 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 57.
    Slide 41 Hint on Annuity Valuation The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the end of the first cash flow period. 3-41 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 58.
    Slide 42 Valuation Using Table IV PVAn = R (PVIFAi%,n) PVA3 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624 Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993 3-42 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 59.
    Slide 43 Overview of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 n -1 n i% . . . R R R R R: Periodic PVADn Cash Flow PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAn (1+i) 3-43 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 60.
    Slide 44 Example of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7% $1,000.00 $1,000 $1,000 $ 934.58 $ 873.44 $2,808.02 = PVADn PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02 3-44 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 61.
    Slide 45 Valuation Using Table IV PVADn = R (PVIFAi%,n)(1+i) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808 Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993 3-45 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 62.
    Slide 46 Steps to Solve Time Value Steps to Solve Time Value of Money Problems of Money Problems 1. Read problem thoroughly 2. Determine if it is a PV or FV problem 3. Create a time line 4. Put cash flows and arrows on time line 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) 3-46 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 63.
    Slide 47 Mixed Flows Example Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%? 10% 0 1 2 3 4 5 10% $600 $600 $400 $400 $100 PV0 3-47 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 64.
    Slide 48 How to Solve? 1. Solve a “piece-at-a-time” by piece-at- time discounting each piece back to t=0. 2. Solve a “group-at-a-time” by first group-at- time breaking problem into groups of annuity streams and any single cash flow group. Then discount each group back to t=0. 3-48 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 65.
    Slide 49 “Piece-At-A-Time” “Piece-At-A-Time” 0 1 2 3 4 5 10% $600 $600 $400 $400 $100 $545.45 $495.87 $300.53 $273.21 $ 62.09 $1677.15 = PV0 of the Mixed Flow 3-49 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 66.
    Slide 50 “Group-At-A-Time” (#1) “Group-At-A-Time” 0 1 2 3 4 5 10% $600 $600 $400 $400 $100 $1,041.60 $ 573.57 $ 62.10 $1,677.27 = PV0 of Mixed Flow [Using Tables] $600(PVIFA10%,2) = $600(1.736) = $1,041.60 $400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57 $100 (PVIF10%,5) = $100 (0.621) = $62.10 3-50 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 67.
    Slide 51 “Group-At-A-Time” (#2) “Group-At-A-Time” 0 1 2 3 4 $400 $400 $400 $400 $1,268.00 0 1 2 PV0 equals Plus $200 $200 $1677.30. $347.20 0 1 2 3 4 5 Plus $100 $62.10 3-51 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 68.
    Slide 52 Frequency of Compounding General Formula: FVn = PV0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today 3-52 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 69.
    Slide 53 Impact of Frequency Julie Miller has $1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2) 1,000 = 1,254.40 Semi FV2 = 1,000(1+ [.12/2])(2)(2) 1,000 = 1,262.48 3-53 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 70.
    Slide 54 Impact of Frequency Qrtly FV2 = 1,000(1+ [.12/4])(4)(2) 1,000 = 1,266.77 Monthly FV2 = 1,000(1+ [.12/12])(12)(2) 1,000 = 1,269.73 Daily FV2 = 1,000(1+[.12/365])(365)(2) 1,000 = 1,271.20 3-54 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 71.
    Slide 55 Effective Annual Interest Rate Effective Annual Interest Rate The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. (1 + [ i / m ] )m - 1 3-55 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 72.
    Slide 56 BW’s Effective Annual Interest Rate Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)? EAR EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%! 3-56 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 73.
    Slide 57 Steps to Amortizing a Loan 1. Calculate the payment per period. 2. Determine the interest in Period t. (Loan balance at t-1) x (i% / m) 3. Compute principal payment in Period t. (Payment - interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal payment from Step 3) 5. Start again at Step 2 and repeat. 3-57 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 74.
    Slide 58 Amortizing a Loan Example Julie Miller is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PV0 = R (PVIFA i%,n) $10,000 = R (PVIFA 12%,5) $10,000 = R (3.605) R = $10,000 / 3.605 = $2,774 3-58 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
  • 75.
    Slide 59 Amortizing a Loan Example End of Paym ent In te re s t P rin c ip a l E n d in g Year B a la n c e 0 --- --- --- $ 1 0 ,0 0 0 1 $ 2 ,7 7 4 $ 1 ,2 0 0 $ 1 ,5 7 4 8 ,4 2 6 2 2 ,7 7 4 1 ,0 1 1 1 ,7 6 3 6 ,6 6 3 3 2 ,7 7 4 800 1 ,9 7 4 4 ,6 8 9 4 2 ,7 7 4 563 2 ,2 1 1 2 ,4 7 8 5 2 ,7 7 5 297 2 ,4 7 8 0 $ 1 3 ,8 7 1 $ 3 ,8 7 1 $ 1 0 ,0 0 0 [Last Payment Slightly Higher Due to Rounding] 3-59 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________