Time Value of Money Concepts
Chapter
6
Time Value of Money Concepts
Learning Objectives
After studying this chapter, you should be able to:
LO6-1Explain the difference between simple and compound interest.
LO6-2Compute the future value of a single amount.
LO6-3Compute the present value of a single amount.
LO6-4Solve for either the interest rate or the number of compounding periods when present
value and future value of a single amount are known.
LO6-5Explain the difference between an ordinary annuity and an annuity due situation.
LO6-6Compute the future value of both an ordinary annuity and an annuity due.
LO6-7Compute the present value of an ordinary annuity, an annuity due, and a deferred
annuity.
LO6-8Solve for unknown values in annuity situations involving present value.
LO6-9Briefly describehow the concept of the time value of money is incorporated into the
valuation of bonds, long-term leases, and pension obligations.
Chapter Highlights
Part A: basic concepts
The time value of moneymeans that money can be invested today to earn interest and grow to a larger dollar amount in the future. Interest is the rent paid for the use of money for some period of time. For example, if you invested $10,000 today in a savings account, it would grow to a larger dollar amount in the future.
Future Value of a Single Amount
Simple interest is computed by multiplying the initial investment times both the applicable interest rate and the period of time for which the money is used. For example, if the $10,000 above were invested in a savings account at 10% for one year, the interest would be $10,000 x 10%, or $1,000. This interest would increase the investment to $11,000, referred to as the future value of $10,000 invested at 10% for 1 year.
Compound interestincludes interest not only on the initial investment but also on the accumulated interest in previous periods. As shown above, thefuture value of $10,000 invested at 10% for one year is $10,000 x (1.10), or $11,000. If invested for two years, the future value is $10,000 x (1.10) x (1.10), or $12,100, and if invested for three years, the future value is $10,000 x (1.10) x (1.10) x (1.10), or $13,310. The future value of any amount invested for n periods at interest rate I is computed as that amounttimes (1+i )n.
Table 1, located at the back of the text, contains the future value (FV) of $1 invested for various periods of time, n, and at various rates, i. The future value of any invested amount, I, can be easily determined by multiplying that amount by the table value. For example, in the 10% column (i = 10%) and 3 period row (n = 3), Table 1 shows the future value of $1 to be 1.331. Multiplying this future value by $10,000 yields $13,310:
FV= I x FV factor
FV= $10,000 x 1.331= $13,310
Future value of $1: n=3, i=10% (from Table 1)
Present Value of a Single Amount
Because the future value of a present amount is the present amount times (1 + i), logically, then, that computation can be reversed to find the present value of a future amount to be the future amount divided by (1 + i). In the future value formula, FV + I x (1+i )n, we substitute present value (PV) for I (invested amount) and solve for PV:
PV = FV
(1 + i)
In our example,
PV = $13,310 = $13,310 = $10,000
(1 + .10)1.331
As with future value, these computations are simplified by using present value tables. Of course, dividing by (1 + i)is the same as multiplying by its reciprocal, . Table 6 in the text provides the solutions of for various interest rates (i) and compounding periods (n). These amounts represent the present value (PV) of $1 to be received at the end of the different periods. The present value factor for i= 10% and n = 3 is .75131 and we can simply multiply this amount by future value to determine present value:
PV= FV x PV factor
PV= $13,310 x .75131= $10,000
Present value of $1: n=3, i=10% (from Table 2)
Graphically, the relation between the present value and the future value can be viewed this way:
End of End of End of
0 year 1 year 2 year 3
$1,000 $1,100 $1,210
$10,000 $13,310
PV _____ FV
To better understand the relationship between future value and present value, consider the following two illustrations:
Illustration
Future Value
Assume you deposit $10,000 in a savings account that pays 8% interest at the end of each year. What will be your account balance after 3 years?
Computation:
FV=$10,000x1.25971=$12,597 (rounded)
invested
amount
Future value of $1: n = 3, i= 8% (from Table 1)
The accuracy of this computation is shown in the following demonstration:
Year-end
Date InterestBalance
Initial deposit$10,000
Year 1 $10,000 x 8% = $80010,800
Year 2 10,800 x 8% = $86411,664
Year 3 11,664 x 8% = $933 (rounded)12,597
Illustration
Present Value
Assume you are offered an automobile for which the salesperson says you can pay $12,597, at the end of three years. This amount includes interest at 8% annually. What price would you be paying for the automobile if you accept the offer?
Computation:
PV=$12,597x.79383=$10,000 (rounded)
future
amount
Present value of $1: n = 3, i= 8% (from Table 2)
The $12,597 you would be paying for the $10,000 automobile includes $2,597 for three years' interest at 8%.
In some situations, the interest rate or the number of periods may be the unknown value. The solution requires that the future value be divided by the present value (or vice versa) to derive a table value. Locating this table value in Table 1 (or Table 2) relative to the known value, n ori, determines the corresponding unknown value, n ori.
Expected Cash Flow Approach
Present value measurement has long been integrated with accounting valuation and is specifically addressed in several accounting standards. SFAC No. 7 provides a framework for using future cash flows as the basis for accounting measurement and asserts that the objective in valuing an asset or liability using present value is to approximate the fair value of that asset or liability. Key to that objective is determining the present value of future cash flows associated with the asset or liability, taking into account any uncertainty concerning the amounts and timing of the cash flows. Although future cash flows in many instances are contractual and certain, the amounts and timing of cash flows are less certain in other situations.
Traditionally, the way uncertainty has been considered in present value calculations has been by discounting the “best estimate” of future cash flows applying a discount rate that has been adjusted to reflect the uncertainty or risk of those cash flows. With the approach described by SFAC No. 7 the adjustment for uncertainty or risk of cash flows is applied to the cash flows, not the discount rate. This expected cash flow approach incorporates specific probabilities of cash flows into the analysis.
Illustration
Expected Cash Flow Approach
Trident Exploration Corporation faces the likelihood of having to pay an uncertain amount in four years for the restoration of land in connection with a mining operation. The future cash flow estimate is in the range of $130 million to $240 million with the following estimated probabilities:
Loss amountProbability
$130 million30%
$200 million45%
$240 million25%
Computation:
The expected cash flow is $189 million:
$130 X 30% =$ 39 million
200 X 45% = 90 million
240 X 25% = 60 million
$189 million
Assuming a risk-free interest rate of 6%, Trident Exploration would report a liability for the expected restoration costs of $149.7 million, the present value of the expected cash outflow:
$189 million X .79209* = $149.7 million
* Present value of $1, n = 4, i = 6% (from Table 2)
Part B: basic annuities
An annuityis a series of equal-sized cash flows occurring over equal intervals of time. In an ordinary annuity cash flows occur at the end of each period. In an annuity due, cash flowsoccur at the beginning of each period. The following time diagrams illustrate the distinction between an ordinary annuity and an annuity due for a three-period, $10,000 annuity beginning on 1/1/13. This annuity is then used to illustrate future and present values of annuities.
1/1/1312/31/1312/31/14 12/31/15
Ordinary Annuity
______
$10,000 $10,000 $10,000
1st2nd 3rd
payment paymentpayment
1/1/1312/31/1312/31/14 12/31/15
Annuity Due
______
$10,000 $10,000 $10,000
1st2nd 3rd
payment payment payment
Future Value of an Ordinary Annuity
Table 3 in the text contains the future values of ordinary annuities (FVA) of $1 invested at the end of n periods at various interest rates, i. The future value of the $10,000 annuity depicted in the ordinary annuity graph above is calculated as follows, assuming an interest rate of 10%:
FVA=$10,000 x 3.31
annuity
amount
Future value of an ordinary annuity of $1: n=3, i=10% (from Table 3)
FVA=$33,100
If you deposited $10,000 at the end of each of three years in a savings account paying 10% annually, the account balance at the end of the three years would grow to $33,100. Of course, since the third deposit is made on the last day of year three, it earns no interest. This can be seen by determining the total future value by calculating the future value of each of the individual deposits as follows:
FV of $1 Future Value
Deposit i=10% (at the end of year 3) n
First deposit $10,000x 1.21 = $12,1002
Second deposit 10,000x 1.10 = 11,0001
Third deposit 10,000x 1.00 = 10,0000
Total 3.31 $33,100
Future Value of an Annuity Due
Table 5 in the text contains the future values of annuities due (FVAD) of $1 invested at the beginning of n periods at various interest rates, i. The future value of the $10,000 annuity depicted in the annuity due graph above is calculated as follows, assuming an interest rate of 10%:
FVAD=$10,000 x 3.641
annuity
amount
Future value of an annuity due of $1: n=3, i=10% (from Table 5)
FVAD=$36,410
The account balance at the end of three years, the future value, is higher than in the ordinary annuity case because the deposits are made at the beginning of the year thus earning higher interest. This same amount can be determined by calculating the future value of each of the individual deposits as follows:
FV of $1 Future Value
Deposit i=10% (at the end of year 3) n
First deposit $10,000x 1.331 = $13,3103
Second deposit 10,000x 1.210 = 12,1002
Third deposit 10,000x 1.100 = 11,0001
Total 3.641 $36,410
Present Value of an Ordinary Annuity
Table 4 in the text contains the present values of ordinary annuities (PVA) of $1 invested at the end of n periods at various interest rates, i. The present value of the $10,000 annuity depicted in the ordinary annuity graph above is calculated as follows, assuming an interest rate of 10%:
PVA=$10,000 x 2.48685
annuity
amount
Present value of an ordinary annuity of $1: n=3, i=10% (from Table 4)
PVA=$24,868.50
The present value can be interpreted as the amount you would need to deposit into a savings account today, earning 10% annually, in order to withdraw $10,000 from the account at the end of each of the next three years. This same total present value can be determined by calculating the present value of each of the individual withdrawals as follows:
PV of $1 Present Value
Withdrawali=10% (at the beginning of year 1) n
First withdrawal $10,000x .90909 = $ 9,090.901
Second withdrawal 10,000x .82645 = 8,264.502
Third withdrawal 10,000x .75131 = 7,513.103
Total 2.48685 $24,868.50
Present Value of an Annuity Due
Table 6 in the text contains the present values of annuities due (PVAD) of $1 invested at the beginning of n periods at various interest rates, i. The present value of the $10,000 annuity depicted in the annuity due graph above is calculated as follows, assuming an interest rate of 10%:
PVAD=$10,000 x 2.73554
annuity
amount
Present value of an annuity due of $1: n=3, i=10% (from Table 6)
PVAD=$27,355.40
The present value can be interpreted as the amount you would need to deposit into a savings account today, earning 10% compounded annually, in order to withdraw $10,000 from the account at the beginning of each of the next three years. This same present value can be determined by calculating the present value of each of the individual withdrawals as follows:
PV of $1 Present Value
Withdrawali=10% (at the beginning of year 1) n
First withdrawal $10,000x1.00000= $10,000.000
Second withdrawal 10,000x.90909= 9,090.901
Third withdrawal 10,000x .82645= 8,264.502
Total2.73554 $27,355.40
To better understand the relationship between future value and present value and between an ordinary annuity and an annuity due, consider the following illustrations.
Illustration
Future Value of an Ordinary Annuity
Assume that you deposit $10,000 at the end of each of the next five years to a savings account that pays annual interest at the rate of 8% annually. What will be your account balance at the end of the five-year period?
Computation:
FVA=$10,000x5.8666=$58,666
invested
amount
Future value of an ordinary annuity of $1: n = 5, i= 8% (from Table 3)
Illustration
Future Value of an Annuity Due
Assume your annual deposits in the previous illustration are made at the beginning of each year. What will be your account balance at the end of the five-year period?
Computation:
FVAD=$10,000x6.3359=$63,359
invested
amount
Future value of an annuity due of $1: n = 5, i= 8% (from Table 5)
Notice that the account balance in this illustration is greater than in the previous illustration. This is due to the fact that deposits are made sooner, and thus earn interest for a longer period of time.
Illustration
Present Value of an Ordinary Annuity
Assume that a friend offered to sell you a lakeside cabin in exchange for five, $10,000 installment payments to be made at the end of each of the next five years. These payments include interest at 8%. What price would you be paying for the cabin?
Computation:
PVA=$10,000x3.99271= $39,927 (rounded)
annuity
amount
Present value of an ordinary annuity of $1: n = 5, i= 8% (from Table 4)
Since you would pay $50,000 over the five years for the $39,927 cabin, the remaining $10,073 represents interest paid for financing the purchase over the five years.
Illustration
Present Valueof an Annuity Due
Assume your installment payments in the previous illustration are made at the beginning of each year. What would be the price of the cabin?
Computation:
PVAD=$10,000x4.31213=$43,121 (rounded)
annuity
amount
Present value of an annuity due of $1: n = 5, i= 8% (from Table 6)
Notice that the price of the cabin in this illustration is greater than in the previous illustration. This is due to the fact that payments are made sooner and thus you are giving up more.
Present Value of a Deferred Annuity
A deferred annuity exists when the first cash flow occurs more than one period after the date the agreement begins. The following time diagram depicts a situation where a three-period, $10,000 annuity agreement begins on 1/1/13 requiring annual cash flows to begin on 12/31/15:
Present
Value
?
1/1/1312/31/1312/31/1412/31/1512/31/1612/31/17
Deferred
Annuity
______
$10,000 $10,000$10,000
n = 2, n =3
The present value of this annuity can be calculated numerous ways. For example, we could sum the present values of the three individual cash flows each discounted to 1/1/13. Or, a more efficient way involves a two-step process that first calculates the present value of the annuity as of the beginning of the annuity period (12/31/14) and then discounts this single amount to its present value as of today (1/1/13). Assuming an interest rate of 10%, this two-step process determines the present value to be $20,553:
PVA= $10,000 x 2.48685=$24,868.50
annuity
amount
Present value of an ordinary annuity of $1: n=3, i=10% (from Table 4)
This is the present value as of December 31, 2014. This single amount is then reduced to present value as of January 1, 2013, by making the following calculation:
PV= $24,868.50 x .82645=$20,553 (rounded)
future
amount
Present value of $1: n=2, i=10% (from Table 2)
In some cases, the present value (or future value) is known and the annuity payments are to be determined. For example, in a later chapter you will determine the lease payments required in each of a specified number of periods to recover a specified investment, a present value. In these cases, the known present value (or future value) is divided by the table value (for the appropriate number of periods and the appropriate interest rate) to derive the required periodic payments. In other situations, the interest rate or the number of periods may be the unknown value. The solution requires that the present value (or future value) be divided by the payment amount to derive a table value. Locating this table value in Table 4 (or Table 3) relative to the known value, n ori, determines the corresponding unknown value, n ori.
Financial Calculators and Excel
Financial calculators can be used to solve future and present value problems. Also, many professionals choose to use spreadsheet software, such as Excel, to solve time value of money problems. These spreadsheets can be used in a variety of ways. A template can be created using the formulas or you can use the software’s built-in financial functions. For example, Excel has a function called PV that calculates the present value of an ordinary annuity. To use the function, you would select the pull-down menu for “Insert,” click on “Function” and choose the category called “Financial.” Scroll down to PV and double click. You will then be asked to input the necessary variables - interest rate, the number of periods, and the payment amount
Accounting Applications of Present Value Techniques - Annuities
The time value of money has many applications in accounting. Most of these applications involve the concept of present value. Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations. In later chapters you will use present value concepts to value long-term notes, bonds, leases, and postretirement obligations.
Summary
The following table summarizes the time value of money concepts discussed in Chapter 6:
Concept / Summary / Formula / TableFuture value (FV) of $1 / The amount of money that a dollar will grow to at some point in the future / FV = $1(1 + i) / 1
Present value (PV) of $1 / The amount of money today that is equivalent to a given amount to be received or paid in the future / PV = / 2
Future value of an ordinary annuity (FVA) of $1 / The future value of a series of equal-sized cash flows with the first payment taking place at the end of the first compounding period / FVA = / 3
Present value of an ordinary annuity (PVA) of $1 / The present value of a series of equal-sized cash flows with the first payment taking place at the end of the first compounding period / PVA = / 4
Future value of an annuity due (FVAD) of $1 / The future value of a series of equal-sized cash flows with the first payment taking place at the beginning of the annuity period / FVAD = [] x (1+i) / 5
Present value of an annuity due (PVAD) of $1 / The present value of a series of equal sized-cash flows with the first payment taking place at the beginning of the annuity period / PVAD = [] x (1+i) / 6
Self-Study Questions And Exercises
Concept Review
1.The of money means that money can be invested today to earn interest and grow to a larger dollar amount in the future.
2. is the amount of money paid or received in excess of the amount borrowed or lent.
3. 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.
4. interest includes interest not only on the initial investment but also on the accumulated interest in previous periods.
5.The is the rate at which money actually will grow during a full year.
6.The value of a single amount is the amount of money that a dollar will grow to at some point in the future.
7.The value of a single amount is today’s equivalent to a particular amount in the future.