study objectives
After studying this appendix, you should be able to: 1 Distinguish between simple and compound interest. 2 Solve for future value of a single amount.
3 Solve for future value of an annuity.
4 Identify the variables fundamental to solving present value problems.
appendix
Time Value of Money
C
5Solve for present value of a single amount. 6Solve for present value of an annuity.
7Compute the present value of notes and bonds. 8Use a financial calculator to solve time value of
money problems.
C-1
Would you rather receive $1,000 today or a year from now? You should prefer to receive the $1,000 today because you can invest the $1,000 and earn interest on it. As a result, you will have more than $1,000 a year from now. What this example illustrates is the concept of the time value of money. Everyone prefers to receive money today rather than in the future because of the interest factor.
Nature of Interest
Interestis payment for the use of another person’s money. It is the difference between the amount borrowed or invested (called the principal) and the amount repaid or collected. The amount of interest to be paid or collected is usually stated as a rate over a specific period of time. The rate of interest is generally stated as an annual rate.
The amount of interest involved in any financing transaction is based on three elements:
1. Principal (p): The original amount borrowed or invested. 2. Interest Rate (i): An annual percentage of the principal.
3. Time (n): The number of years that the principal is borrowed or invested.
SIMPLE INTEREST
Simple interestis computed on the principal amount only. It is the return on the principal for one period. Simple interest is usually expressed as shown in Illustration C-1.
Distinguish between simple and compound interest.
1
Interest ⴝ ⴛ ⴛ
For example, if you borrowed $5,000 for 2 years at a simple interest rate of 12% annually, you would pay $1,200 in total interest computed as follows:
Interest p i n $5,000 .12 2 $1,200 Time n Rate i Principal p Illustration C-1 Interest computation study objective
COMPOUND INTEREST
Compound interestis computed on principal and on any interest earned that has not been paid or withdrawn. It is the return on (or growth of) the principal for two or more time periods. Compounding computes interest not only on the principal but also on the interest earned to date on that principal, assuming the interest is left on deposit.
To illustrate the difference between simple and compound interest, assume that you deposit $1,000 in Bank Two, where it will earn simple interest of 9% per year, and you deposit another $1,000 in Citizens Bank, where it will earn compound interest of 9% per year compounded annually. Also assume that in both cases you will not withdraw any cash until three years from the date of de-posit. Illustration C-2 shows the computation of interest to be received and the accumulated year-end balances.
Illustration C-2 Simple versus compound interest
Simple Interest Calculation Year 1 Year 2 Year 3 $1,000.00 × 9% $1,000.00 × 9% $1,000.00 × 9% $ $ 90.00 90.00 90.00 270.00 $1,090.00 $1,180.00 $1,270.00 $25.03 Difference Simple Interest Accumulated Year-end Balance Bank Two Compound Interest Calculation Year 1 Year 2 Year 3 $1,000.00 × 9% $1,090.00 × 9% $1,188.10 × 9% $ $ 90.00 98.10 106.93 295.03 $1,090.00 $1,188.10 $1,295.03 Compound Interest Accumulated Year-end Balance Citizens Bank
Note in Illustration C-2 that simple interest uses the initial principal of $1,000 to compute the interest in all three years. Compound interest uses the accumulated balance (principal plus interest to date) at each year-end to compute interest in the succeeding year—which explains why your compound interest account is larger. Obviously, if you had a choice between investing your money at simple in-terest or at compound inin-terest, you would choose compound inin-terest, all other things—especially risk—being equal. In the example, compounding provides $25.03 of additional interest income. For practical purposes, compounding as-sumes that unpaid interest earned becomes a part of the principal, and the ac-cumulated balance at the end of each year becomes the new principal on which interest is earned during the next year.
Illustration C-2 indicates that you should invest your money at a bank that com-pounds interest. Most business situations use compound interest. Simple interest is generally applicable only to short-term situations of one year or less.
Future Value of a Single Amount
The future value of a single amount is the value at a future date of a given amount invested, assuming compound interest. For example, in Illustration C-2, $1,295.03 is the future value of the $1,000 investment earning 9% for three
section one
Future Value Concepts
Solve for future value of a single amount.
2
years. The $1,295.03 could be determined more easily by using the following formula:
Future Value of a Single Amount C-3
Illustration C-3 Formula for future value
FV ⴝ p ⴛ (1 ⴙ i)n
where:
FV future value of a single amount
p principal (or present value; the value today) i interest rate for one period
n number of periods
The $1,295.03 is computed as follows:
FV p (1 i)n
$1,000 (1 .09)3
$1,000 1.29503 $1,295.03
The 1.29503 is computed by multiplying (1.09 1.09 1.09). The amounts in this example can be depicted in the time diagram shown in Illustration C-4.
Present Value (p) 0 $1,000 1 2 3 $1,295.03 i = 9% Future Value n = 3 years Illustration C-4 Time diagram
Another method used to compute the future value of a single amount in-volves a compound interest table. This table shows the future value of 1 for n periods. Table 1 on the next page is such a table.
In Table 1, n is the number of compounding periods, the percentages are the periodic interest rates, and the 5-digit decimal numbers in the respective columns are the future value of 1 factors. In using Table 1, you would multiply the prin-cipal amount by the future value factor for the specified number of periods and interest rate. For example, the future value factor for two periods at 9% is 1.18810. Multiplying this factor by $1,000 equals $1,188.10—which is the accu-mulated balance at the end of year 2 in the Citizens Bank example in Illustra-tion C-2. The $1,295.03 accumulated balance at the end of the third year can be calculated from Table 1 by multiplying the future value factor for three periods (1.29503) by the $1,000.
The demonstration problem in Illustration C-5 (page C-4) shows how to use Table 1.
TABLE 1 Future Value of 1 (n) Periods 4% 5% 6% 8% 9% 10% 11% 12% 15% 0 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1 1.04000 1.05000 1.06000 1.08000 1.09000 1.10000 1.11000 1.12000 1.15000 2 1.08160 1.10250 1.12360 1.16640 1.18810 1.21000 1.23210 1.25440 1.32250 3 1.12486 1.15763 1.19102 1.25971 1.29503 1.33100 1.36763 1.40493 1.52088 4 1.16986 1.21551 1.26248 1.36049 1.41158 1.46410 1.51807 1.57352 1.74901 5 1.21665 1.27628 1.33823 1.46933 1.53862 1.61051 1.68506 1.76234 2.01136 6 1.26532 1.34010 1.41852 1.58687 1.67710 1.77156 1.87041 1.97382 2.31306 7 1.31593 1.40710 1.50363 1.71382 1.82804 1.94872 2.07616 2.21068 2.66002 8 1.36857 1.47746 1.59385 1.85093 1.99256 2.14359 2.30454 2.47596 3.05902 9 1.42331 1.55133 1.68948 1.99900 2.17189 2.35795 2.55803 2.77308 3.51788 10 1.48024 1.62889 1.79085 2.15892 2.36736 2.59374 2.83942 3.10585 4.04556 11 1.53945 1.71034 1.89830 2.33164 2.58043 2.85312 3.15176 3.47855 4.65239 12 1.60103 1.79586 2.01220 2.51817 2.81267 3.13843 3.49845 3.89598 5.35025 13 1.66507 1.88565 2.13293 2.71962 3.06581 3.45227 3.88328 4.36349 6.15279 14 1.73168 1.97993 2.26090 2.93719 3.34173 3.79750 4.31044 4.88711 7.07571 15 1.80094 2.07893 2.39656 3.17217 3.64248 4.17725 4.78459 5.47357 8.13706 16 1.87298 2.18287 2.54035 3.42594 3.97031 4.59497 5.31089 6.13039 9.35762 17 1.94790 2.29202 2.69277 3.70002 4.32763 5.05447 5.89509 6.86604 10.76126 18 2.02582 2.40662 2.85434 3.99602 4.71712 5.55992 6.54355 7.68997 12.37545 19 2.10685 2.52695 3.02560 4.31570 5.14166 6.11591 7.26334 8.61276 14.23177 20 2.19112 2.65330 3.20714 4.66096 5.60441 6.72750 8.06231 9.64629 16.36654 0 $20,000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 i = 6% Value = ?Future Present Value (p) n = 18 years
John and Mary Rich invested $20,000 in a savings account paying 6% interest at the time their son, Mike, was born. The money is to be used by Mike for his college education. On his 18th birthday, Mike withdraws the money from his savings account. How much did Mike withdraw from his account?
Answer: The future value factor from Table 1 is 2.85434 (18 periods at 6%). The future value of $20,000 earning 6% per year for 18 years is $57,086.80 ($20,000 × 2.85434).
Future Value of an Annuity
The preceding discussion involved the accumulation of only a single principal sum. Individuals and businesses frequently encounter situations in which a
seriesof equal dollar amounts are to be paid or received periodically, such as loans or lease (rental) contracts. Such payments or receipts of equal dollar amounts are referred to as annuities.
Illustration C-5 Demonstration problem— Using Table 1 for FV of 1
Solve for future value of an annuity.
3
The future value of an annuity is the sum of all the payments (receipts) plus the accumulated compound interest on them. In computing the future value of an annuity, it is necessary to know (1) the interest rate, (2) the number of compounding periods, and (3) the amount of the periodic payments or receipts. To illustrate the computation of the future value of an annuity, assume that you invest $2,000 at the end of each year for three years at 5% interest compounded annually. This situation is depicted in the time diagram in Illustration C-6.
Future Value of an Annuity C-5
Illustration C-6 Time diagram for a three-year annuity 0 1 2 3 $2,000 Present Value $2,000 $2,000 i = 5% Future Value = ? n = 3 years
The $2,000 invested at the end of year 1 will earn interest for two years (years 2 and 3), and the $2,000 invested at the end of year 2 will earn interest for one year (year 3). However, the last $2,000 investment (made at the end of year 3) will not earn any interest. The future value of these periodic payments could be com-puted using the future value factors from Table 1, as shown in Illustration C-7.
Invested at Number of
End of Compounding Amount Future Value of Future
Year Periods Invested ⴛ 1 Factor at 5% ⴝ Value
1 2 $2,000 1.10250 $2,205
2 1 $2,000 1.05000 2,100
3 0 $2,000 1.00000 2,000
3.15250 $6,305
Illustration C-7 Future value of periodic payment computation
The first $2,000 investment is multiplied by the future value factor for two periods (1.1025) because two years’ interest will accumulate on it (in years 2 and 3). The second $2,000 investment will earn only one year’s interest (in year 3) and therefore is multiplied by the future value factor for one year (1.0500). The fi-nal $2,000 investment is made at the end of the third year and will not earn any interest. Thus n 0 and the future value factor is 1.00000. Consequently, the future value of the last $2,000 invested is only $2,000 since it does not accumulate any interest.
Calculating the future value of each individual cash flow is required when the periodic payments or receipts are not equal in each period. However, when the periodic payments (receipts) are the same in each period, the future value can be computed by using a future value of an annuity of 1 table. Table 2 (page C-6) is such a table.
Table 2 shows the future value of 1 to be received periodically for a given number of periods. It assumes that each payment is made at the end of each period. We can see from Table 2 that the future value of an annuity of 1 factor for three periods at 5% is 3.15250. The future value factor is the total of the three individual future value factors was shown in Illustration C-7. Multiplying this amount by the annual investment of $2,000 produces a future value of $6,305. The demonstration problem in Illustration C-8 shows how to use Table 2.
TABLE 2 Future Value of an Annuity of 1 (n) Periods 4% 5% 6% 8% 9% 10% 11% 12% 15% 1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 2 2.04000 2.05000 2.06000 2.08000 2.09000 2.10000 2.11000 2.12000 2.15000 3 3.12160 3.15250 3.18360 3.24640 3.27810 3.31000 3.34210 3.37440 3.47250 4 4.24646 4.31013 4.37462 4.50611 4.57313 4.64100 4.70973 4.77933 4.99338 5 5.41632 5.52563 5.63709 5.86660 5.98471 6.10510 6.22780 6.35285 6.74238 6 6.63298 6.80191 6.97532 7.33592 7.52334 7.71561 7.91286 8.11519 8.75374 7 7.89829 8.14201 8.39384 8.92280 9.20044 9.48717 9.78327 10.08901 11.06680 8 9.21423 9.54911 9.89747 10.63663 11.02847 11.43589 11.85943 12.29969 13.72682 9 10.58280 11.02656 11.49132 12.48756 13.02104 13.57948 14.16397 14.77566 16.78584 10 12.00611 12.57789 13.18079 14.48656 15.19293 15.93743 16.72201 17.54874 20.30372 11 13.48635 14.20679 14.97164 16.64549 17.56029 18.53117 19.56143 20.65458 24.34928 12 15.02581 15.91713 16.86994 18.97713 20.14072 21.38428 22.71319 24.13313 29.00167 13 16.62684 17.71298 18.88214 21.49530 22.95339 24.52271 26.21164 28.02911 34.35192 14 18.29191 19.59863 21.01507 24.21492 26.01919 27.97498 30.09492 32.39260 40.50471 15 20.02359 21.57856 23.27597 27.15211 29.36092 31.77248 34.40536 37.27972 47.58041 16 21.82453 23.65749 25.67253 30.32428 33.00340 35.94973 39.18995 42.75328 55.71747 17 23.69751 25.84037 28.21288 33.75023 36.97351 40.54470 44.50084 48.88367 65.07509 18 25.64541 28.13238 30.90565 37.45024 41.30134 45.59917 50.39593 55.74972 75.83636 19 27.67123 30.53900 33.75999 41.44626 46.01846 51.15909 56.93949 63.43968 88.21181 20 29.77808 33.06595 36.78559 45.76196 51.16012 57.27500 64.20283 72.05244 102.44358 Illustration C-8 Demonstration problem— Using Table 2 for FV of an annuity of 1 0 1 2 3 4 $2,500 Present Value $2,500 $2,500 $2,500 i = 6% Future Value = ? n = 4 years
John and Char Lewis’ daughter, Debra, has just started high school. They decide to start a college fund for her and will invest $2,500 in a savings account at the end of each year she is in high school (4 payments total). The account will earn 6% interest compounded annually. How much will be in the college fund at the time Debra graduates from high school?
Answer: The future value factor from Table 2 is 4.37462 (4 periods at 6%). The future value of $2,500 invested each year for 4 years at 6% interest is $10,936.55 ($2,500 × 4.37462).
Present Value of a Single Amount C-7
Present Value Variables
The present valueis the value now of a given amount to be paid or received in the future, assuming compound interest. The present value, like the future value, is based on three variables: (1) the dollar amount to be received (future amount), (2) the length of time until the amount is received (number of periods), and (3) the interest rate (the discount rate). The process of determining the present value is referred to as discounting the future amount.
In this textbook, we use present value computations in measuring several items. For example, Chapter 10 computed the present value of the principal and interest payments to determine the market price of a bond. In addition, deter-mining the amount to be reported for notes payable and lease liabilities involves present value computations.
Present Value of a Single Amount
To illustrate present value, assume that you want to invest a sum of money today that will provide $1,000 at the end of one year. What amount would you need to invest today to have $1,000 one year from now? If you want a 10% rate of return, the investment or present value is $909.09 ($1,000 1.10). The for-mula for calculating present value is shown in Illustration C-9.
section two
Present Value Concepts
Illustration C-9 Formula for present value
Present Value ⴝ Future Value ⴜ (1 ⴙ i)n
The computation of $1,000 discounted at 10% for one year is as follows:
PV FV (1 i)n
$1,000 (1 .10)1
PV $1,000 1.10 PV $909.09
The future amount ($1,000), the discount rate (10%), and the number of pe-riods (1) are known. The variables in this situation can be depicted in the time diagram in Illustration C-10.
Identify the variables fundamental to solving present value problems.
4
study objective
Solve for present value of a single amount.
5
Illustration C-10 Finding present value if discounted for one period i = 10% n = 1 year Present Value (?) $909.09 Future Value $1,000
If the single amount of $1,000 is to be received in two years and discounted at 10% [PV $1,000 (1 .10)2], its present value is $826.45 [($1,000 1.21),
depicted as shown in Illustration C-11 on the next page.
The present value of 1 may also be determined through tables that show the present value of 1 for n periods. In Table 3, n is the number of discounting pe-riods involved. The percentages are the periodic interest rates or discount rates, and the 5-digit decimal numbers in the respective columns are the present value of 1 factors.
When using Table 3, the future value is multiplied by the present value fac-tor specified at the intersection of the number of periods and the discount rate.
Illustration C-11 Finding present value if discounted for two periods
i = 10% 1 Present Value (?) 0 Future Value 2 n = 2 years $826.45 $1,000
TABLE 3 Present Value of 1 (n) Periods 4% 5% 6% 8% 9% 10% 11% 12% 15% 1 .96154 .95238 .94340 .92593 .91743 .90909 .90090 .89286 .86957 2 .92456 .90703 .89000 .85734 .84168 .82645 .81162 .79719 .75614 3 .88900 .86384 .83962 .79383 .77218 .75132 .73119 .71178 .65752 4 .85480 .82270 .79209 .73503 .70843 .68301 .65873 .63552 .57175 5 .82193 .78353 .74726 .68058 .64993 .62092 .59345 .56743 .49718 6 .79031 .74622 .70496 .63017 .59627 .56447 .53464 .50663 .43233 7 .75992 .71068 .66506 .58349 .54703 .51316 .48166 .45235 .37594 8 .73069 .67684 .62741 .54027 .50187 .46651 .43393 .40388 .32690 9 .70259 .64461 .59190 .50025 .46043 .42410 .39092 .36061 .28426 10 .67556 .61391 .55839 .46319 .42241 .38554 .35218 .32197 .24719 11 .64958 .58468 .52679 .42888 .38753 .35049 .31728 .28748 .21494 12 .62460 .55684 .49697 .39711 .35554 .31863 .28584 .25668 .18691 13 .60057 .53032 .46884 .36770 .32618 .28966 .25751 .22917 .16253 14 .57748 .50507 .44230 .34046 .29925 .26333 .23199 .20462 .14133 15 .55526 .48102 .41727 .31524 .27454 .23939 .20900 .18270 .12289 16 .53391 .45811 .39365 .29189 .25187 .21763 .18829 .16312 .10687 17 .51337 .43630 .37136 .27027 .23107 .19785 .16963 .14564 .09293 18 .49363 .41552 .35034 .25025 .21199 .17986 .15282 .13004 .08081 19 .47464 .39573 .33051 .23171 .19449 .16351 .13768 .11611 .07027 20 .45639 .37689 .31180 .21455 .17843 .14864 .12403 .10367 .06110
For example, the present value factor for one period at a discount rate of 10% is .90909, which equals the $909.09 ($1,000 .90909) computed in Illustration C-10. For two periods at a discount rate of 10%, the present value factor is .82645, which equals the $826.45 ($1,000 .82645) computed previously.
Note that a higher discount rate produces a smaller present value. For ex-ample, using a 15% discount rate, the present value of $1,000 due one year from now is $869.57 versus $909.09 at 10%. Also note that the further removed from the present the future value is, the smaller the present value. For example, us-ing the same discount rate of 10%, the present value of $1,000 due in five years is $620.92. The present value of $1,000 due in one year is $909.09, a difference of $288.17.
The following two demonstration problems (Illustrations C-12, C-13) illus-trate how to use Table 3.
Present Value of an Annuity C-9
Illustration C-12 Demonstration problem— Using Table 3 for PV of 1
Illustration C-13 Demonstration problem— Using Table 3 for PV of 1
Solve for present value of an annuity.
6
i = 8% 2 PV = ? Today $10,000 3 years 1Suppose you have a winning lottery ticket and the state gives you the option of taking $10,000 three years from now or taking the present value of $10,000 today. The state uses an 8% rate in discounting. How much will you receive if you accept your winnings today?
Answer: The present value factor from Table 3 is .79383 (3 periods at 8%). The present value of $10,000 to be received in 3 years discounted at 8% is $7,938.30 ($10,000 × .79383). n = 3 i = 9% 3 PV = ? Today $5,000 4 years 1
Determine the amount you must deposit today in your SUPER savings account, paying 9% interest, in order to accumulate $5,000 for a down payment 4 years from now on a new car.
Answer: The present value factor from Table 3 is .70843 (4 periods at 9%). The present value of $5,000 to be received in 4 years discounted at 9% is $3,542.15 ($5,000 × .70843).
2
n = 4
Present Value of an Annuity
The preceding discussion involved the discounting of only a single future amount. Businesses and individuals frequently engage in transactions in which a series of equal dollar amounts are to be received or paid periodically. Examples of a series of periodic receipts or payments are loan agreements, installment sales, mortgage notes, lease (rental) contracts, and pension obligations. As discussed earlier, these periodic receipts or payments are annuities.
The present value of an annuityis the value now of a series of future re-ceipts or payments, discounted assuming compound interest. In computing the present value of an annuity, it is necessary to know (1) the discount rate, (2) the number of discount periods, and (3) the amount of the periodic receipts or pay-ments. To illustrate the computation of the present value of an annuity, assume
that you will receive $1,000 cash annually for three years at a time when the dis-count rate is 10%. This situation is depicted in the time diagram in Illustration C-14. Illustration C-15 shows computation of the present value in this situation.
Illustration C-14 Time diagram for a three-year annuity
Illustration C-15 Present value of a series of future amounts computation i = 10% 2 Today 3 years PV = ? $1,000 $1,000 $1,000 1 n = 3 Present Value of 1
Future Amount ⴛ Factor at 10% ⴝ Present Value
$1,000 (one year away) .90909 $ 909.09
1,000 (two years away) .82645 826.45
1,000 (three years away) . 75132 751.32
2.48686 $2,486.86
This method of calculation is required when the periodic cash flows are not uniform in each period. However, when the future receipts are the same in each period, an annuity table can be used. As illustrated in Table 4 below, an annuity table shows the present value of 1 to be received periodically for a given number of periods.
TABLE 4 Present Value of an Annuity of 1 (n) Periods 4% 5% 6% 8% 9% 10% 11% 12% 15% 1 .96154 .95238 .94340 .92593 .91743 .90909 .90090 .89286 .86957 2 1.88609 1.85941 1.83339 1.78326 1.75911 1.73554 1.71252 1.69005 1.62571 3 2.77509 2.72325 2.67301 2.57710 2.53130 2.48685 2.44371 2.40183 2.28323 4 3.62990 3.54595 3.46511 3.31213 3.23972 3.16986 3.10245 3.03735 2.85498 5 4.45182 4.32948 4.21236 3.99271 3.88965 3.79079 3.69590 3.60478 3.35216 6 5.24214 5.07569 4.91732 4.62288 4.48592 4.35526 4.23054 4.11141 3.78448 7 6.00205 5.78637 5.58238 5.20637 5.03295 4.86842 4.71220 4.56376 4.16042 8 6.73274 6.46321 6.20979 5.74664 5.53482 5.33493 5.14612 4.96764 4.48732 9 7.43533 7.10782 6.80169 6.24689 5.99525 5.75902 5.53705 5.32825 4.77158 10 8.11090 7.72173 7.36009 6.71008 6.41766 6.14457 5.88923 5.65022 5.01877 11 8.76048 8.30641 7.88687 7.13896 6.80519 6.49506 6.20652 5.93770 5.23371 12 9.38507 8.86325 8.38384 7.53608 7.16073 6.81369 6.49236 6.19437 5.42062 13 9.98565 9.39357 8.85268 7.90378 7.48690 7.10336 6.74987 6.42355 5.58315 14 10.56312 9.89864 9.29498 8.24424 7.78615 7.36669 6.98187 6.62817 5.72448 15 11.11839 10.37966 9.71225 8.55948 8.06069 7.60608 7.19087 6.81086 5.84737 16 11.65230 10.83777 10.10590 8.85137 8.31256 7.82371 7.37916 6.97399 5.95424 17 12.16567 11.27407 10.47726 9.12164 8.54363 8.02155 7.54879 7.11963 6.04716 18 12.65930 11.68959 10.82760 9.37189 8.75563 8.20141 7.70162 7.24967 6.12797 19 13.13394 12.08532 11.15812 9.60360 8.95012 8.36492 7.83929 7.36578 6.19823 20 13.59033 12.46221 11.46992 9.81815 9.12855 8.51356 7.96333 7.46944 6.25933
Table 4 shows that the present value of an annuity of 1 factor for three periods at 10% is 2.48685.1 This present value factor is the total of the three individual present value factors, as shown in Illustration C-15. Applying this amount to the annual cash flow of $1,000 produces a present value of $2,486.85. The following demonstration problem (Illustration C-16) illustrates how to use Table 4.
Computing the Present Value of a Long-Term Note or Bond C-11
Illustration C-16 Demonstration problem— Using Table 4 for PV of an annuity of 1
Compute the present value of notes and bonds.
7
i = 12% 4 PV = ? Today $6,000 5 years 1Kildare Company has just signed a capitalizable lease contract for equip-ment that requires rental payequip-ments of $6,000 each, to be paid at the end of each of the next 5 years. The appropriate discount rate is 12%. What is the present value of the rental payments—that is, the amount used to capitalize the leased equipment?
Answer: The present value factor from Table 4 is 3.60478 (5 periods at 12%). The present value of 5 payments of $6,000 each discounted at 12% is $21,628.68 ($6,000 × 3.60478).
$6,000 $6,000
2 3
$6,000 $6,000
n = 5
Time Periods and Discounting
In the preceding calculations, the discounting was done on an annual basis us-ing an annual interest rate. Discountus-ing may also be done over shorter periods of time such as monthly, quarterly, or semiannually.
When the time frame is less than one year, it is necessary to convert the an-nual interest rate to the applicable time frame. Assume, for example, that the investor in Illustration C-14 received $500 semiannually for three years instead of $1,000 annually. In this case, the number of periods becomes six (3 2), the discount rate is 5% (10% 2), the present value factor from Table 4 is 5.07569 (6 periods at 5%), and the present value of the future cash flows is $2,537.85 (5.07569 $500). This amount is slightly higher than the $2,486.86 computed in Illustration C-15 because interest is computed twice during the same year. That is, during the second half of the year, interest is earned on the first half-year’s interest.
Computing the Present Value
of a Long-Term Note or Bond
The present value (or market price) of a long-term note or bond is a function of three variables: (1) the payment amounts, (2) the length of time until the amounts are paid, and (3) the discount rate. Our illustration (on the next page) uses a five-year bond issue.
1The difference of .00001 between 2.48686 and 2.48685 is due to rounding.
The first variable (dollars to be paid) is made up of two elements: (1) a series of interest payments (an annuity) and (2) the principal amount (a single sum). To compute the present value of the bond, both the interest payments and the principal amount must be discounted—two different computations. The time diagrams for a bond due in five years are shown in Illustration C-17.
Illustration C-17 Present value of a bond time diagram
Illustration C-18 Time diagram for present value of a 10%, five-year bond paying interest semiannually Interest Rate (i) 1 yr. Present Value (?) Today Principal Amount 5 yr. Diagram for Principal
2 yr. 3 yr. 4 yr.
Annuity 1 yr. Present Value (?) Today 5 yr. Diagram for Interest
2 yr. 3 yr. 4 yr. Interest Rate (i)
Annuity Annuity Annuity Annuity n = 5
n = 5
When the investor’s market interest rate is equal to the bond’s contractual interest rate, the present value of the bonds will equal the face value of the bonds. To illustrate, assume a bond issue of 10%, five-year bonds with a face value of $100,000 with interest payable semiannually on January 1 and July 1. If the discount rate is the same as the contractual rate, the bonds will sell at face value. In this case, the investor will receive (1) $100,000 at maturity and (2) a series of ten $5,000 interest payments [($100,000 10%) 2] over the term of the bonds. The length of time is expressed in terms of interest periods—in this case—10, and the discount rate per interest period, 5%. The following time diagram (Illustration C-18) depicts the variables involved in this discounting situation.
i = 5% 1 Present Value (?) Today Principal Amount $100,000 10 Diagram for Principal 5 6 1 Present Value (?) Today 10 Diagram for Interest 5 6 i = 5% $5,000 2 2 3 3 4 4 7 7 8 8 9 9 $5,000 $5,000 $5,000 $5,000 $5,000 $5,000 $5,000 $5,000 n = 10 n = 10 $5,000 Interest Payments
Illustration C-19 shows the computation of the present value of these bonds.
Computing the Present Value of a Long-Term Note or Bond C-13
Illustration C-19 Present value of principal and interest—face value
Illustration C-20 Present value of principal and interest—discount
Illustration C-21 Present value of principal and interest—premium 10% Contractual Rate—10% Discount Rate
Present value of principal to be received at maturity $100,000 PV of 1 due in 10 periods at 5%
$100,000 .61391 (Table 3) $ 61,391
Present value of interest to be received periodically over the term of the bonds
$5,000 PV of 1 due periodically for 10 periods at 5%
$5,000 7.72173 (Table 4) 38,609*
Present value of bonds $100,000
*Rounded
Now assume that the investor’s required rate of return is 12%, not 10%. The future amounts are again $100,000 and $5,000, respectively, but now a discount rate of 6% (12% 2) must be used. The present value of the bonds is $92,639, as computed in Illustration C-20.
10% Contractual Rate—12% Discount Rate Present value of principal to be received at maturity
$100,000 .55839 (Table 3) $ 55,839
Present value of interest to be received periodically over the term of the bonds
$5,000 7.36009 (Table 4) 36,800
Present value of bonds $92,639
Conversely, if the discount rate is 8% and the contractual rate is 10%, the present value of the bonds is $108,111, computed as shown in Illustration C-21.
10% Contractual Rate—8% Discount Rate Present value of principal to be received at maturity
$100,000 .67556 (Table 3) $ 67,556
Present value of interest to be received periodically over the term of the bonds
$5,000 8.11090 (Table 4) 40,555
Present value of bonds $108,111
The above discussion relied on present value tables in solving present value problems. Electronic hand-held calculators may also be used to compute present values without the use of these tables. Many calculators, especially “financial” calculators, have present value (PV) functions that allow you to calculate present values by merely inputting the proper amount, discount rate, periods, and press-ing the PV key. We discuss the use of financial calculators in the next section.
Business professionals, once they have mastered the underlying concepts in sections 1 and 2, often use a financial calculator to solve time value of money problems. In many cases, they must use calculators if interest rates or time periods do not correspond with the information provided in the compound interest tables.
To use financial calculators, you enter the time value of money variables into the calculator. Illustration C-22 shows the five most common keys used to solve time value of money problems.2
section three
Using Financial Calculators
Use a financial calculator to solve time value of money problems.
8
Illustration C-22 Financial calculator keys
N I PV PMT FV
where
N number of periods
I interest rate per period (some calculators use I/YR or i) PV present value (occurs at the beginning of the first period) PMT payment (all payments are equal, and none are skipped) FV future value (occurs at the end of the last period)
In solving time value of money problems in this appendix, you will gener-ally be given three of four variables and will have to solve for the remaining vari-able. The fifth key (the key not used) is given a value of zero to ensure that this variable is not used in the computation.
Present Value of A Single Sum
To illustrate how to solve a present value problem using a financial calculator, assume that you want to know the present value of $84,253 to be received in five years, discounted at 11% compounded annually. Illustration C-23 depicts this problem.
Illustration C-23 Calculator solution for present value of a single sum
2
On many calculators, these keys are actual buttons on the face of the calculator; on others they appear on the display after the user accesses a present value menu.
? 0 84,253 –50,000 Inputs: 5 Answer: 11 N I PV PMT FV study objective
Illustration C-23 shows you the information (inputs) to enter into the calculator: N ⫽ 5, I ⫽ 11, PMT ⫽ 0, and FV ⫽ 84,253. You then press PV for the answer: ⫺$50,000. As indicated, the PMT key was given a value of zero because a series of payments did not occur in this problem.
PLUS AND MINUS
The use of plus and minus signs in time value of money problems with a finan-cial calculator can be confusing. Most finanfinan-cial calculators are programmed so that the positive and negative cash flows in any problem offset each other. In the present value problem above, we identified the $84,253 future value initial investment as a positive (inflow); the answer ⫺$50,000 was shown as a negative amount, reflecting a cash outflow. If the 84,253 were entered as a negative, then the final answer would have been reported as a positive 50,000.
Hopefully, the sign convention will not cause confusion. If you understand what is required in a problem, you should be able to interpret a positive or neg-ative amount in determining the solution to a problem.
COMPOUNDING PERIODS
In the problem above, we assumed that compounding occurs once a year. Some financial calculators have a default setting, which assumes that compounding occurs 12 times a year. You must determine what default period has been pro-grammed into your calculator and change it as necessary to arrive at the proper compounding period.
ROUNDING
Most financial calculators store and calculate using 12 decimal places. As a result, because compound interest tables generally have factors only up to five decimal places, a slight difference in the final answer can result. In most time value of money problems, the final answer will not include more than two decimal places.
Present Value of an Annuity
To illustrate how to solve a present value of an annuity problem using a finan-cial calculator, assume that you are asked to determine the present value of rental receipts of $6,000 each to be received at the end of each of the next five years, when discounted at 12%, as pictured in Illustration C-24.
Present Value of an Annuity C-15
Illustration C-24 Calculator solution for present value of an annuity
Inputs: 5 12 ? 6,000 0
Answer: –21,628.66
N I PV PMT FV
In this case, you enter N ⫽ 5, I ⫽ 12, PMT ⫽ 6,000, FV ⫽ 0, and then press PV to arrive at the answer of ⫺$21,628.66.
Useful Applications of
the Financial Calculator
With a financial calculator you can solve for any interest rate or for any num-ber of periods in a time value of money problem. Here are some examples of these applications.
AUTO LOAN
Assume you are financing a car with a three-year loan. The loan has a 9.5% stated annual interest rate, compounded monthly. The price of the car is $6,000, and you want to determine the monthly payments, assuming that the payments start one month after the purchase. This problem is pictured in Illustration C-25.
Illustration C-25 Calculator solution for
auto loan payments Inputs: 36 9.5 6,000 ? 0
Answer:
N I PV FV
–192.20 PMT
To solve this problem, you enter N 36 (12 3), I 9.5, PV 6,000, FV 0, and then press PMT. You will find that the monthly payments will be $192.20. Note that the payment key is usually programmed for 12 payments per year. Thus, you must change the default (compounding period) if the payments are other than monthly.
MORTGAGE LOAN AMOUNT
Let’s say you are evaluating financing options for a loan on a house. You decide that the maximum mortgage payment you can afford is $700 per month. The annual interest rate is 8.4%. If you get a mortgage that requires you to make monthly payments over a 15-year period, what is the maximum home loan you can afford? Illustration C-26 depicts this problem.
Illustration C-26 Calculator solution for
mortgage amount Inputs: 180 8.4 ? 0
Answer:
N I PV PMT FV
–700
71,509.81
You enter N 180 (12 15 years), I 8.4, PMT 700, FV 0, and press PV. With the payments-per-year key set at 12, you find a present value of $71,509.81— the maximum home loan you can afford, given that you want to keep your mort-gage payments at $700. Note that by changing any of the variables, you can quickly conduct “what-if” analyses for different situations.
Summary of Study Objectives
Brief Exercises C-17
1
Distinguish between simple and compound interest. Simple interest is computed on the principal only, while compound interest is computed on the princi-pal and any interest earned that has not been with-drawn.2
Solve for future value of a single amount. Prepare a time diagram of the problem. Identify the principal amount, the number of compounding periods, and the interest rate. Using the future value of 1 table, multi-ply the principal amount by the future value factor specified at the intersection of the number of periods and the interest rate.3
Solve for future value of an annuity. Prepare a time diagram of the problem. Identify the amount of the periodic payments, the number of compounding pe-riods, and the interest rate. Using the future value of an annuity of 1 table, multiply the amount of the pay-ments by the future value factor specified at the inter-section of the number of periods and the interest rate.4
Identify the variables fundamental to solving present value problems.The following three variables are fun-damental to solving present value problems: (1) the future amount, (2) the number of periods, and (3) the interest rate (the discount rate).5
Solve for present value of a single amount.Prepare a time diagram of the problem. Identify the future amount, the number of discounting periods, and the discount (interest) rate. Using the present value of a single amount table, multiply the future amount by the present value factor specified at the intersection of the number of periods and the discount rate.6
Solve for present value of an annuity.Prepare a time diagram of the problem. Identify the amount of future periodic receipts or payment (annuities), the number of discounting periods, and the discount (interest) rate. Using the present value of an annuity of 1 table, mul-tiply the amount of the annuity by the present value factor specified at the intersection of the number of periods and the interest rate.7
Compute the present value of notes and bonds. Deter-mine the present value of the principal amount: Mul-tiply the principal amount (a single future amount) by the present value factor (from the present value of 1 table) intersecting at the number of periods (number of interest payments) and the discount rate. Determine the present value of the series of interest payments: Multiply the amount of the interest payment by the present value factor (from the present value of an an-nuity of 1 table) intersecting at the number of periods (number of interest payments) and the discount rate. Add the present value of the principal amount to the present value of the interest payments to arrive at the present value of the note or bond.8
Use a financial calculator to solve time value of money problems.Financial calculators can be used to solve the same and additional problems as those solved with time value of money tables. One enters into the finan-cial calculator the amounts for all of the known ele-ments of a time value of money problem (periods, in-terest rate, payments, future or present value) and solves for the unknown element. Particularly useful situations involve interest rates and compounding periods not presented in the tables.Glossary
Annuity (p. C-4) A series of equal dollar amounts to be paid or received periodically.
Compound interest (p. C-2) The interest computed on the principal and any interest earned that has not been paid or withdrawn.
Discounting the future amount(s) (p. C-7) The process of determining present value.
Future value of a single amount (p. C-2) The value at a future date of a given amount invested, assuming com-pound interest.
Future value of an annuity (p. C-5) The sum of all the payments or receipts plus the accumulated compound interest on them.
Interest (p. C-1) Payment for the use of another per-son’s money.
Present value (p. C-7) The value now of a given amount to be paid or received in the future assuming compound interest.
Present value of an annuity (p. C-9) The value now of a series of future receipts or payments, discounted as-suming compound interest.
Principal (p. C-1) The amount borrowed or invested. Simple interest (p. C-1) The interest computed on the principal only.
Brief Exercises
(Use tables to solve exercises BEC-1 to BEC-23.)
BEC-1 Danny Reid invested $6,000 at 5% annual interest, and left the money invested without withdrawing any of the interest for 12 years. At the end of the 12 years, Danny withdrew the accumulated amount of money. (a) What amount did Danny withdraw, suming the investment earns simple interest? (b) What amount did Danny withdraw, as-suming the investment earns interest compounded annually?
Compute the future value of a single amount.
BEC-2 For each of the following cases, indicate (a) to what interest rate columns and (b) to what number of periods you would refer in looking up the future value factor. (1) In Table 1 (future value of 1):
Annual Number of
Rate Years Invested Compounded
Case A 6% 3 Annually
Case B 8% 4 Semiannually
(2) In Table 2 (future value of an annuity of 1):
Annual Number of
Rate Years Invested Compounded
Case A 5% 8 Annually
Case B 6% 6 Semiannually
BEC-3 Piper Company signed a lease for an office building for a period of 12 years. Under the lease agreement, a security deposit of $8,000 is made. The deposit will be re-turned at the expiration of the lease with interest compounded at 4% per year. What amount will Piper receive at the time the lease expires?
BEC-4 Weisman Company issued $1,000,000, 10-year bonds and agreed to make an-nual sinking fund deposits of $75,000. The deposits are made at the end of each year into an account paying 5% annual interest. What amount will be in the sinking fund at the end of 10 years?
BEC-5 Jack and Susan Stine invested $5,000 in a savings account paying 4% annual interest when their daughter, Regina, was born. They also deposited $1,000 on each of her birthdays until she was 18 (including her 18th birthday). How much was in the sav-ings account on her 18th birthday (after the last deposit)?
BEC-6 Kurt Heflin borrowed $30,000 on July 1, 2010. This amount plus accrued inter-est at 9% compounded annually is to be repaid on July 1, 2015. How much will Kurt have to repay on July 1, 2015?
BEC-7 For each of the following cases, indicate (a) to what interest rate columns and (b) to what number of periods you would refer in looking up the discount rate.
(1) In Table 3 (present value of 1):
Annual Number of Discounts
Rate Years Involved per Year
Case A 12% 6 Annually
Case B 10% 11 Annually
Case C 6% 9 Semiannually
(2) In Table 4 (present value of an annuity of 1):
Annual Number of Number of Frequency of
Rate Years Involved Payments Involved Payments
Case A 12% 20 20 Annually
Case B 10% 5 5 Annually
Case C 8% 4 8 Semiannually
BEC-8 (a) What is the present value of $30,000 due 9 periods from now, discounted at 10%?
(b) What is the present value of $30,000 to be received at the end of each of 6 periods, discounted at 9%?
Use future value tables.
(SO 2, 3)
Compute the future value of a single amount.
(SO 2)
Compute the future value of an annuity.
(SO 3)
Compute the future value of a single amount and of an annuity.
(SO 2, 3)
Compute the future value of a single amount.
(SO 2)
Use present value tables.
(SO 5, 6)
Determine present values.
BEC-9 Concord Company is considering an investment which will return a lump sum of $800,000 five years from now. What amount should Concord Company pay for this in-vestment to earn a 9% return?
BEC-10 Cunningham Company earns 10% on an investment that will return $525,000 eight years from now. What is the amount Cunningham should invest now to earn this rate of return?
BEC-11 Shaw Company is considering investing in an annuity contract that will return $40,000 annually at the end of each year for 15 years. What amount should Shaw Com-pany pay for this investment if it earns a 5% return?
BEC-12 Koehn Enterprises earns 8% on an investment that pays back $110,000 at the end of each of the next 6 years. What is the amount Koehn Enterprises invested to earn the 8% rate of return?
BEC-13 Agler Railroad Co. is about to issue $400,000 of 10-year bonds paying a 9% interest rate, with interest payable semiannually. The discount rate for such securities is 8%. How much can Agler expect to receive for the sale of these bonds?
BEC-14 Assume the same information as BEC-13 except that the discount rate was 10% instead of 8%. In this case, how much can Agler expect to receive from the sale of these bonds?
BEC-15 Molina Taco Company receives a $70,000, 6-year note bearing interest of 6% (paid annually) from a customer at a time when the discount rate is 8%. What is the pre-sent value of the note received by Molina?
BEC-16 Henderson Enterprises issued 9%, 8-year, $3,000,000 par value bonds that pay interest semiannually on October 1 and April 1. The bonds are dated April 1, 2010, and are issued on that date. The discount rate of interest for such bonds on April 1, 2010, is 10%. What cash proceeds did Henderson receive from issuance of the bonds?
BEC-17 George Basler owns a garage and is contemplating purchasing a tire retread-ing machine for $16,100. After estimatretread-ing costs and revenues, George projects a net cash flow from the retreading machine of $2,900 annually for 8 years. George hopes to earn a return of 10 percent on such investments. What is the present value of the retreading operation? Should George purchase the retreading machine?
BEC-18 Englehart Company issues an 8%, 5-year mortgage note on January 1, 2010, to obtain financing for new equipment. Land is used as collateral for the note. The terms provide for semiannual installment payments of $90,260. What were the cash proceeds received from the issuance of the note?
BEC-19 Popper Company is considering purchasing equipment. The equipment will produce the following cash flows: Year 1, $30,000; Year 2, $45,000; Year 3, $55,000. Popper requires a minimum rate of return of 10%. What is the maximum price Popper should pay for this equipment?
BEC-20 If Andrea Costello invests $3,152.40 now and she will receive $10,000 at the end of 15 years, what annual rate of interest will Andrea earn on her investment? [Hint: Use Table 3.]
BEC-21 Robert Wilk has been offered the opportunity of investing $40,388 now. The investment will earn 12% per year and at the end of that time will return Robert $100,000. How many years must Robert wait to receive $100,000? [Hint: Use Table 3.]
BEC-22 Jane Duncan made an investment of $9,818.15. From this investment, she will receive $1,000 annually for the next 20 years starting one year from now. What rate of interest will Jane’s investment be earning for her? [Hint: Use Table 4.]
BEC-23 Jessica Bakely invests $7,536.08 now for a series of $1,000 annual returns be-ginning one year from now. Jessica will earn a return of 8% on the initial investment. How many annual payments of $1,000 will Jessica receive? [Hint: Use Table 4.]
BEC-24 Reba McEntire wishes to invest $19,000 on July 1, 2010, and have it accumu-late to $49,000 by July 1, 2020.
Brief Exercises C-19
Compute the present value of a single amount investment.
(SO 5)
Compute the present value of a single amount investment.
(SO 5)
Compute the present value of an annuity investment.
(SO 6)
Compute the present value of an annuity investment.
(SO 6)
Compute the present value of bonds.
(SO 5, 6, 7)
Compute the present value of bonds.
(SO 5, 6, 7)
Compute the present value of a note.
(SO 5, 6, 7)
Compute the present value of bonds.
(SO 5, 6, 7)
Compute the present value of a machine for purposes of making a purchase decision.
(SO 6, 7)
Compute the present value of a note.
(SO 6)
Compute the maximum price to pay for a machine.
(SO 6, 7)
Compute the interest rate on a single amount.
(SO 5)
Compute the number of periods of a single amount.
(SO 5)
Compute the interest rate on an annuity.
(SO 6)
Compute the number of periods of an annuity.
(SO 6)
Determine interest rate.
Instructions
Use a financial calculator to determine at what exact annual rate of interest Reba must invest the $19,000.
BEC-25 On July 17, 2010, Tim McGraw borrowed $42,000 from his grandfather to open a clothing store. Starting July 17, 2011, Tim has to make 10 equal annual payments of $6,500 each to repay the loan.
Instructions
Use a financial calculator to determine what interest rate Tim is paying.
BEC-26 As the purchaser of a new house, Patty Loveless has signed a mortgage note to pay the Memphis National Bank and Trust Co. $14,000 every 6 months for 20 years, at the end of which time she will own the house. At the date the mortgage is signed the purchase price was $198,000, and Loveless made a down payment of $20,000. The first payment will be made 6 months after the date the mortgage is signed.
Instructions
Using a financial calculator, compute the exact rate of interest earned on the mortgage by the bank.
BEC-27 Using a financial calculator, solve for the unknowns in each of the following situations.
(a) On June 1, 2010, Shelley Long purchases lakefront property from her neighbor, Joey Brenner, and agrees to pay the purchase price in seven payments of $16,000 each, the first payment to be payable June 1, 2011. (Assume that interest compounded at an annual rate of 7.35% is implicit in the payments.) What is the purchase price of the property?
(b) On January 1, 2010, Cooke Corporation purchased 200 of the $1,000 face value, 8% coupon, 10-year bonds of Howe Inc. The bonds mature on January 1, 2018, and pay interest annually beginning January 1, 2011. Cooke purchased the bonds to yield 10.65%. How much did Cooke pay for the bonds?
BEC-28 Using a financial calculator, provide a solution to each of the following situations.
(a) Bill Schroeder owes a debt of $35,000 from the purchase of his new sport utility vehicle. The debt bears annual interest of 9.1% compounded monthly. Bill wishes to pay the debt and interest in equal monthly payments over 8 years, beginning one month hence. What equal monthly payments will pay off the debt and interest? (b) On January 1, 2010, Sammy Sosa offers to buy Mark Grace’s used snowmobile for
$8,000, payable in five equal annual installments, which are to include 8.25% inter-est on the unpaid balance and a portion of the principal. If the first payment is to be made on December 31, 2010, how much will each payment be?
Determine interest rate.
(SO 8)
Determine interest rate.
(SO 8)
Various time value of money situations.
(SO 8)
Various time value of money situations.
