Chapter 6: Time Value of Money Concepts Time value of money concepts, specifically future value and present value, are essential in a variety of accounting situations. These concepts and the related computational procedures are the subjects of this chapter. Present values and future values of single amounts and present values and future values of annuities (series of equal periodic payments) are described separately but shown to be interrelated.
The time value of money means that money can be invested today to earn interest and grow to a larger dollar amount in the future. Time value of money concepts are useful in valuing several assets and liabilities. Interest is the amount of money paid or received in excess of the amount borrowed or lent. Simple interest is computed by multiplying an initial investment times both the applicable interest rate and the period of time for which the money is used. Assume you invest $1,000 at 6% simple interest for 3 years. You would earn $180 interest.
Compound interest includes interest not only on the initial investment but also on the accumulated interest in previous periods. Assume we deposit $1,000 in a bank that earns 6% interest compounded annually. What is the balance in our account at the end of three years?
Each year we earn interest on the initial investment amount plus any previously earned interest. As a result, at the end of the three years, we have a total of $1,191.02.
The future value of a single amount is the amount of money that a dollar will grow to at some point in the future. Recall our previous example where we assume we deposit $1,000 for three years that earns 6% interest compounded annually.
Writing in a more efficient way, we can say $1,000 times 1.06 to the 3 rd power. In fact, the future value of any invested amount can be determined using the concise formula shown on the screen: PV times 1 plus the interest rate raised to the power of the number of compounding periods.
Another way to find the future value is to use tables that contain the future value of $1 invested for various periods of time and at various interest rates. Table 1 in your textbook is the Future Value of $1 table. Using the Future Value of $1 Table, we find the factor for 6% and 3 periods is 1.19102. To solve our problem, we just multiply $1,000 times the table factor of 1.19102.
Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount. This is a present value question. A present value of a single amount is today’s equivalent to a particular amount in the future.
Using our previous equation for future value, we can solve for present value by dividing the future value by 1 plus the interest rate raised to the number of compounding periods. Another way to solve for the present value is to use the Present Value of $1 table in your textbook. In your textbook, Table 2 is the Present Value of $1 table. Does the format look familiar?
Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time. What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?
To solve this question, we multiply the future value of $20,000 by the present value factor for 8% for 5 periods (.68058) from the present value of $1 Table. If you deposit $13,611.60 now, at 8% annual interest, you will have $20,000 at the end of 5 years.
There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined by using a little algebra.
Part I Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? Part II First, we must determine the present value factor we are looking to find. We divide the present value by the future value and find a factor of .91575. Using the present value table, we look for this factor in row 2 since the loan period in this example is for 2 years. This leads us to the answer which is 4.5%.
Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. However, even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest. (No one will loan you money interest free except, perhaps, your parents!) For these noninterest-bearing notes, we impute an appropriate interest rate .
Statement of Financial Accounting Concepts Number 7, “ Using Cash Flow Information and Present Value in Accounting Measurements,” provides a framework for using future cash flows in accounting measurements. The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability. This new expected cash flow approach incorporates specific probabilities of cash flows into the analysis. The expected cash flow is multiplied by the company’s credit-adjusted risk-free rate of interest to arrive at the present value.
An annuity is a series of equal periodic payments. If you make a car payment or a house payment, you likely pay the same amount each month. Both of these are examples of a series of equal periodic payments or an annuity.
An annuity with payments at the end of the period is known as an ordinary annuity.
An annuity with payments at the beginning of the period is known as an annuity due.
To find the future value of an ordinary annuity, multiply the amount of the annuity by the future value of an ordinary annuity factor found in Table 3 in your textbook.
Part I We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded interest annually, on all invested funds. What will be the fund balance at the end of 10 years? Part II The future value of the ordinary annuity is $36,216.50.
To find the future value of an annuity due, multiply the amount the annuity by the future value of an annuity due factor found in Table 5 in your textbook.
Part I Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually. Part II The future value of the annuity due is $46,371.
First, let’s look at the present value of an ordinary annuity. You wish to withdraw $10,000 at the end of each of the next 4 years from a bank account that pays 10% interest compounded annually. How much do you need to invest today to meet this goal?
Here is a graphic that depicts the annuity payments. It may help you to draw a similar graphic when you are working time value of money problems. This graphic illustrates that we are determining the present value of each ordinary annuity.
Using the Present Value of $1 table, we can find the present value of each ordinary annuity as illustrated. We can then add up the present values to arrive at the present value of the entire annuity stream. So, to answer our question, if you invest $31,698.60 today you will be able to withdraw $10,000 at the end of each of the next four years.
An alternative way to solve this problem is to use the Present Value of Ordinary Annuity of $1 table. We can find the 3.16986 factor on this table at the intersection of the 4 th row and the 10% column.
Now, let’s look at the present value of an annuity due. Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. To solve this problem, we need to find the factor from the Present Value of Annuity Due of $1 Table at the intersection of the 4 th row and the 6% column. The present value of this annuity due is $36,730.
In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement.
On January 1, 2011, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on your investments, how much are you willing to pay for this investment? As the graphic illustrates, the annuity is deferred for two periods. One way to solve this problem is to determine the present value of the individual payments using the factors found at the intersection of the 3 rd and 4 th periods and 12%.
A more efficient computation is as follows: Calculate the PV of the annuity as of the beginning of the annuity period. Discount the single value amount calculated in (1) to its present value as of today. Let’s see how this works for this problem.
Part I First, let’s calculate the PV of the annuity as of the beginning of the annuity period. Our $12,500 annuity is for 2 periods at 12% so our factor from the Present Value of Ordinary Annuity of $1 table Is 1.69005. The present value of the annuity is $21,126. Part II Next, let’s discount the $21,126 calculated in part 1 to its present value as of today. The present value factor for 2 periods at 12% is .79719. The present value of this annuity stream 2 years deferred is $16,841.
In present value problems involving annuities, there are four variables: 1. Present value of an ordinary annuity or Present value of an annuity due 2. The amount of the annuity payment 3. The number of periods, and 4. The interest rate. If you know any three of these, the fourth can be determined.
Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made to repay the loan in four years?
In this problem we know the present value of $700, the interest rate of 8%, and the number of periods of 4. So, we are solving for the amount of the ordinary annuity. Before we can go much further, we need to know the factor from the 4 th period and 8% intersection on the Present Value of Ordinary Annuity of $1 table. This factor is 3.31213. To solve this problem we divide the present value by the factor to determine the amount of the annuity is $211.34.
Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. Some common examples include long-term bonds, long-term leases, and pension obligations.
Part I When determining the present value of bonds, we must consider two cash flow streams: 1. Calculate the Present Value of the Lump-sum Maturity Payment (Face Value). 2. Calculate the Present Value of the Annuity Payments (Interest). On June 30, 2011, Ebsen Electric issued 10% stated rate bonds with a face value of $1 million. The bonds mature in 5 years. The market rate of interest for similar issues was 12%. Interest is paid semiannually beginning on December 31, 2011. What is the price of the bond issue? Part II To determine the present value of the face value of bonds, we use the Present Value of $1 table. To determine the present value of the interest payments, we use the Present Value of Ordinary Annuity of $1 table. By adding together these two present value amounts, we arrive at the present value, or selling price, of the bonds of $926,405.
Certain long-term leases require the recording of an asset and corresponding liability at the present value of future lease payments.
On January 1, 2011, Todd Furniture Company signed a 20-year non-cancelable lease for a new retail showroom. The lease agreement calls for annual payments of $25,000 for 20 years beginning on January 1, 2011. The appropriate rate of interest for this long-term lease is 8%. Calculate the value of the asset acquired and the liability assumed by Todd (the present value of an annuity due at 8% for 20 years). The nature of this lease requires it to be treated in a manner similar to an installment purchase. Since the first lease payment occurs at the beginning of the first year, this is an annuity due. To solve this problem, we need to find the interest factor from the Present Value of Annuity Due of $1 Table at the intersection of the 25 th row and the 8% column. The present value of this annuity due is $265,090.
Some pension plans create obligations during employees’ service periods that must be paid during their retirement periods. The amounts contributed during the employment period are determined using present value computations of the estimate of the future amount to be paid during retirement.
On January 1, 2011, Todd Furniture Company hired a new sales manger for the new showroom. The sales manager is expected to work 30 years before retirement on December 31, 2040. Annual retirement benefits will be paid at the end of each year of retirement, a period that is expected to be 25 years. The sales manager will earn $2,500 in annual retirement benefits for the first year worked, 2011. How much must Todd contribute to the company pension fund in 2011 to provide for $2,500 in annual pension benefits for 25 years that are expected to begin in 30 years. Todd’s pension fund is expected to earn 5%.
This is a two part calculation. The first part requires the computation of the present value of a 25-year ordinary annuity of $2,500 as of December 31, 2040. Next we calculate the present value of the December 31, 2040 amount. This second present value is the amount Todd will contribute for 2011 to fund the retirement benefit earned by the sales manager in 2011. The present value of the ordinary annuity of $2,500 is $35,234.85 as of December 31, 2040. The present value of that amount in 2011 is $8,152.64, the amount Todd must contribute to the pension fund for the benefits earned in 2011.
2.
Simple Interest Interest amount = P × i × n Assume you invest $1,000 at 6% simple interest for 3 years. You would earn $180 interest. ($1,000 × .06 × 3 = $180) ( or $60 each year for 3 years )
3.
Compound Interest <ul><li>Assume we deposit $1,000 in a bank that earns 6% interest compounded annually. </li></ul>What is the balance in our account at the end of three years?
5.
Future Value of a Single Amount <ul><li>The future value of a single amount is the amount of money that a dollar will grow to at some point in the future. </li></ul><ul><li>Assume we deposit $1,000 for three years that earns 6% interest compounded annually. </li></ul>$1,000.00 × 1.06 = $1,060.00 and $1,060.00 × 1.06 = $1,123.60 and $1,123.60 × 1.06 = $1,191.02
6.
Future Value of a Single Amount <ul><li>Writing in a more efficient way, we can say . . . . </li></ul><ul><li>$1,191.02 = $1,000 × [1.06] 3 </li></ul>FV = PV (1 + i ) n Future Value Amount Invested at the Beginning of the Period Interest Rate Number of Compounding Periods
7.
Future Value of a Single Amount Using the Future Value of $1 Table, we find the factor for 6% and 3 periods is 1.19102. So, we can solve our problem like this. . . FV = $1,000 × 1.19102 FV = $1,191.02
8.
Present Value of a Single Amount <ul><li>Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount. </li></ul><ul><li>This is a present value question. </li></ul><ul><li>Present value of a single amount is today’s equivalent to a particular amount in the future. </li></ul>
9.
Present Value of a Single Amount <ul><li>Remember our equation? </li></ul><ul><li>FV = PV (1 + i) n </li></ul><ul><li>We can solve for PV and get . . . . </li></ul>FV (1 + i ) n PV =
10.
Present Value of a Single Amount <ul><li>Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time. </li></ul><ul><li>What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually? </li></ul>
11.
Present Value of a Single Amount <ul><li>i = .08, n = 5 </li></ul><ul><li>Present Value Factor = .68058 </li></ul><ul><li>$20,000 × .68058 = $13,611.60 </li></ul>If you deposit $13,611.60 now, at 8% annual interest, you will have $20,000 at the end of 5 years. Present Value of $1 Table
12.
Solving for Other Values FV = PV (1 + i ) n Future Value Present Value Interest Rate Number of Compounding Periods There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined.
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Determining the Unknown Interest Rate <ul><li>Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? </li></ul><ul><li>a. 3.5% </li></ul><ul><li>b. 4.0% </li></ul><ul><li>c. 4.5% </li></ul><ul><li>d. 5.0% </li></ul>Present Value of $1 Table $1,000 = $1,092 × ? $1,000 ÷ $1,092 = .91575 Search the PV of $1 table in row 2 (n=2) for this value.
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<ul><li>Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes . </li></ul>Accounting Applications of Present Value Techniques—Single Cash Amount Even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest . We impute an appropriate interest rate for noninterest-bearing notes.
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Expected Cash Flow Approach Statement of Financial Accounting Concepts No. 7 “ Using Cash Flow Information and Present Value in Accounting Measurements” The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability.
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Basic Annuities An annuity is a series of equal periodic payments. Period 1 Period 2 Period 3 Period 4 $10,000 $10,000 $10,000 $10,000
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<ul><li>An annuity with payments at the end of the period is known as an ordinary annuity . </li></ul>Ordinary Annuity End of year 1 $10,000 $10,000 $10,000 $10,000 End of year 2 End of year 3 End of year 4 1 2 3 4 Today
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Annuity Due <ul><li>An annuity with payments at the beginning of the period is known as an annuity due . </li></ul>Beginning of year 1 $10,000 $10,000 $10,000 $10,000 Beginning of year 2 Beginning of year 3 Beginning of year 4 1 2 3 4 Today
19.
Future Value of an Ordinary Annuity <ul><li>To find the future value of an ordinary annuity , multiply the amount of the annuity by the future value of an ordinary annuity factor. </li></ul>
20.
Future Value of an Ordinary Annuity <ul><li>We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded interest annually, on all invested funds. </li></ul><ul><li>What will be the fund balance at the end of 10 years? </li></ul>
21.
Future Value of an Annuity Due <ul><li>To find the future value of an annuity due , multiply the amount of the annuity by the future value of an annuity due factor. </li></ul>
22.
Future Value of an Annuity Due <ul><li>Compute the future value of $10,000 invested at the beginning of each of the next four years with interest at 6% compounded annually. </li></ul>
23.
Present Value of an Ordinary Annuity <ul><li>You wish to withdraw $10,000 at the end of each of the next 4 years from a bank account that pays 10% interest compounded annually. </li></ul><ul><li>How much do you need to invest today to meet this goal? </li></ul>
24.
Present Value of an Ordinary Annuity PV1 PV2 PV3 PV4 $10,000 $10,000 $10,000 $10,000 1 2 3 4 Today
25.
Present Value of an Ordinary Annuity <ul><li>If you invest $31,698.60 today you will be able to withdraw $10,000 at the end of each of the next four years. </li></ul>
26.
Present Value of an Ordinary Annuity <ul><li>Can you find this value in the Present Value of Ordinary Annuity of $1 table? </li></ul>More Efficient Computation $10,000 × 3.16986 = $31,698.60
27.
Present Value of an Annuity Due <ul><li>Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. </li></ul>
28.
Present Value of a Deferred Annuity <ul><li>In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement. </li></ul>
29.
Present Value of a Deferred Annuity <ul><li>On January 1, 2011, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on your investments, how much are you willing to pay for this investment? </li></ul>1/1/11 12/31/11 12/31/12 12/31/13 12/31/14 12/31/15 Present Value? $12,500 $12,500 1 2 3 4
30.
Present Value of a Deferred Annuity <ul><li>More Efficient Computation </li></ul><ul><li>Calculate the PV of the annuity as of the beginning of the annuity period. </li></ul><ul><li>Discount the single value amount calculated in (1) to its present value as of today. </li></ul>On January 1, 2011, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on your investments, how much are you willing to pay for this investment? 1/1/11 12/31/11 12/31/12 12/31/13 12/31/14 12/31/15 Present Value? $12,500 $12,500 1 2 3 4
31.
Present Value of a Deferred Annuity On January 1, 2011, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on your investments, how much are you willing to pay for this investment? 1/1/11 12/31/11 12/31/12 12/31/13 12/31/14 12/31/15 Present Value? $12,500 $12,500 1 2 3 4
32.
Solving for Unknown Values in Present Value Situations <ul><li>In present value problems involving annuities, there are four variables: </li></ul>Present value of an ordinary annuity or Present value of an annuity due The amount of the annuity payment The number of periods The interest rate If you know any three of these, the fourth can be determined.
33.
Solving for Unknown Values in Present Value Situations <ul><li>Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. </li></ul><ul><li>What is the required annual payment that must be made (the annuity amount) to repay the loan in four years? </li></ul>Today End of Year 1 Present Value $700 End of Year 2 End of Year 3 End of Year 4
34.
Solving for Unknown Values in Present Value Situations Assume that you borrow $700 from a friend and intend to repay the amount in four equal annual installments beginning one year from today. Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made (the annuity amount) to repay the loan in four years?
35.
Accounting Applications of Present Value Techniques—Annuities <ul><li>Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. </li></ul>Long-term Bonds Long-term Leases Pension Obligations
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Valuation of Long-term Bonds Calculate the Present Value of the Lump-sum Maturity Payment (Face Value) Calculate the Present Value of the Annuity Payments (Interest) On June 30, 2011, Ebsen Electric issued 10% stated rate bonds with a face value of $1 million. The bonds mature in 5 years. The market rate of interest for similar issues was 12%. Interest is paid semiannually beginning on December 31, 2011. What was the price of the bond issue?
37.
Valuation of Long-term Leases Certain long-term leases require the recording of an asset and corresponding liability at the present value of future lease payments.
38.
Valuation of Long-term Leases On January 1, 2011, Todd Furniture Company signed a 20-year non-cancelable lease for a new retail showroom. The lease agreement calls for annual payments of $25,000 for 20 years beginning on January 1, 2011. The appropriate rate of interest for this long-term lease is 8%. Calculate the value of the asset acquired and the liability assumed by Todd (the present value of an annuity due at 8% for 20 years).
39.
Valuation of Pension Obligations Some pension plans create obligations during employees’ service periods that must be paid during their retirement periods. The amounts contributed during the employment period are determined using present value computations of the estimate of the future amount to be paid during retirement.
40.
Valuation of Pension Obligations On January 1, 2011, Todd Furniture Company hired a new sales manger for the new showroom. The sales manager is expected to work 30 years before retirement on December 31, 2040. Annual retirement benefits will be paid at the end of each year of retirement, a period that is expected to be 25 years. The sales manager will earn $2,500 in annual retirement benefits for the first year worked, 2011. How much must Todd contribute to the company pension fund in 2011 to provide for $2,500 in annual pension benefits for 25 years that are expected to begin in 30 years. Todd’s pension fund is expected to earn 5%.
41.
Valuation of Pension Obligations This is a two part calculation. The first part requires the computation of the present value of a 25-year ordinary annuity of $2,500 as of December 31, 2040. Next we calculate the present value of the December 31, 2040 amount. This second present value is the amount Todd will contribute in 2011 to fund the retirement benefit earned by the sales manager in 2011.
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