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 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
4. 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:
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 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.
8. 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%
9. 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.
10. 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
11. 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
12. 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?
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 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
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 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.
17. IMPORTANT
Slide 1
Chapter 3
Time Value of
Time Value of
Money
Money
3-1
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18. Slide 2
The Time Value of Money
The Interest Rate
Simple Interest
Compound Interest
Amortizing a Loan
3-2
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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