Download financial management chapter solution and more Exercises Financial Management in PDF only on Docsity!
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4
Time Value of Money
ANSWERS TO END-OF-CHAPTER QUESTIONS
4-1 a. PV (present value) is the value today of a future payment, or stream of payments, discounted at the appropriate rate of interest. PV is also the beginning amount that will grow to some future value. The parameter i is the periodic interest rate that an account pays. The parameter INT is the dollars of interest earned each period. FVn (future value) is the ending amount in an account, where n is the number of periods the money is left in the account. PVAn is the value today of a future stream of equal payments (an annuity) and FVAn is the ending value of a stream of equal payments, where n is the number of payments of the annuity. PMT is equal to the dollar amount of an equal, or constant cash flow (an annuity). In the EAR equation, m is used to denote the number of compounding periods per year, while i (^) Nom is the nominal, or quoted, interest rate.
b. The opportunity cost rate (i) of an investment is the rate of return available on the best alternative investment of similar risk.
c. An annuity is a series of payments of a fixed amount for a specified number of periods. A single sum, or lump sum payment, as opposed to an annuity, consists of one payment occurring now or at some future time. A cash flow can be an inflow (a receipt) or an outflow (a deposit, a cost, or an amount paid). We distinguish between the terms cash flow and PMT. We use the term cash flow for uneven streams, while we use the term PMT for annuities, or constant payment amounts. An uneven cash flow stream is a series of cash flows in which the amount varies from one period to the next. The PV (or FVn ) of an uneven payment stream is merely the sum of the present values (or future values) of each individual payment.
d. An ordinary annuity has payments occurring at the end of each period. A deferred annuity is just another name for an ordinary annuity. An annuity due has payments occurring at the beginning of each period. Most financial calculators will accommodate either type of annuity. The payment period must be equal to the compounding period.
e. A perpetuity is a series of payments of a fixed amount that last indefinitely. In other words, a perpetuity is an annuity where n equals infinity. Consol is another term for perpetuity. Consols were originally bonds issued by England in 1815 to consolidate past debt.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
f. An outflow is a deposit, a cost, or an amount paid, while an inflow is a receipt. A time line is an important tool used in time value of money analysis; it is a graphical representation which is used to show the timing of cash flows. The terminal value is the future value of an uneven cash flow stream.
g. Compounding is the process of finding the future value of a single payment or series of payments. Discounting is the process of finding the present value of a single payment or series of payments; it is the reverse of compounding.
h. Annual compounding means that interest is paid once a year. In semiannual, quarterly, monthly, and daily compounding, interest is paid 2, 4, 12, and 365 times per year respectively. When compounding occurs more frequently than once a year, you earn interest on interest more often, thus increasing the future value. The more frequent the compounding, the higher the future value.
i. The effective annual rate is the rate that, under annual compounding, would have produced the same future value at the end of 1 year as was produced by more frequent compounding, say quarterly. The nominal (quoted) interest rate, i (^) Nom, is the rate of interest stated in a contract. If the compounding occurs annually, the effective annual rate and the nominal rate are the same. If compounding occurs more frequently, the effective annual rate is greater than the nominal rate. The nominal annual interest rate is also called the annual percentage rate, or APR. The periodic rate, i (^) PER , is the rate charged by a lender or paid by a borrower each period. It can be a rate per year, per 6-month period, per quarter, per month, per day, or per any other time interval (usually one year or less).
j. An amortization schedule is a table that breaks down the periodic fixed payment of an installment loan into its principal and interest components. The principal component of each payment reduces the remaining principal balance. The interest component is the interest payment on the beginning-of-period principal balance. An amortized loan is one that is repaid in equal periodic amounts (or "killed off" over time).
4-2 The opportunity cost rate is the rate of interest one could earn on an alternative investment with a risk equal to the risk of the investment in question. This is the value of i in the TVM equations, and it is shown on the top of a time line, between the first and second tick marks. It is not a single rate--the opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also varies from year to year depending on inflationary expectations.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
PV = 10,000 FV 5 =?
FV 5 = $10,000(1.10) 5
Alternatively, with a financial calculator enter the following: N = 5, I/YR = 10, PV = - 10000, and PMT = 0. Solve for FV = $16,105.10.
PV =? FV 20 = 5,
With a financial calculator enter the following: N = 20, I/YR = 7, PMT = 0, and FV =
- Solve for PV = $1,292.10.
PV = 250,000 FV 18 = 1,000,
With a financial calculator enter the following: N = 18, PV = -250000, PMT = 0, and FV = 1000000. Solve for I/YR = 8.01% ≈ 8%.
4-4 0 N =?
PV = 1 FVN = 2
$2 = $1(1.065) N.
With a financial calculator enter the following: I/YR = 6.5, PV = -1, PMT = 0, and FV =
- Solve for N = 11.01 ≈ 11 years.
4-5 0 1 2 N – 2 N – 1 N
PV = 42,180.53 5,000 5,000 5,000 5,000 FV = 250,
Using your financial calculator, enter the following data: I/YR = 12; PV = -42180.53; PMT = -5000; FV = 250000; N =? Solve for N = 11. It will take 11 years to accumulate $250,000.
7%
6.5%
12%
I/YR =?
10%
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-6 Ordinary annuity:
0 1 2 3 4 5 | | | | | | 300 300 300 300 300 FVA 5 =?
With a financial calculator enter the following: N = 5, I/YR = 7, PV = 0, and PMT = 300. Solve for FV = $1,725.22. Annuity due: 0 1 2 3 4 5 | | | | | | 300 300 300 300 300 FVA 5 =?
With a financial calculator, switch to “BEG” and enter the following: N = 5, I/YR = 7, PV = 0, and PMT = 300. Solve for FV = $1,845.99. Don’t forget to switch back to “END” mode.
PV =? FV =?
Using a financial calculator, enter the following: CF 0 = 0; CF 1 = 100; Nj = 3; CF 4 = 200 (Note calculator will show CF 2 on screen.); CF 5 = 300 (Note calculator will show CF 3 on screen.); CF 6 = 500 (Note calculator will show CF 4 on screen.); and I/YR = 8. Solve for NPV = $923.98.
To solve for the FV of the cash flow stream with a calculator that doesn’t have the NFV key, do the following: Enter N = 6, I/YR = 8, PV = -923.98, and PMT = 0. Solve for FV = $1,466.24.
4-8 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = -20000, and FV = 0. Solve for PMT = $444.89.
EAR =
M NOM M
I (^1)
= (1.01) 12 – 1. = 12.68%.
Alternatively, using a financial calculator, enter the following: NOM% = 12 and P/YR =12. Solve for EFF% = 12.6825%. Remember to change back to P/YR = 1 on your calculator.
7%
8%
7%
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-11 a.? | | -200 400
With a financial calculator, enter I/YR = 7, PV = -200, PMT = 0, and FV = 400. Then press the N key to find N = 10.24 ≈ 10.
b.? | | -200 400.
With a financial calculator, enter I/YR = 107, PV = -200, PMT = 0, and FV = 400. Then press the N key to find N = 7.27 ≈ 7.
c.? | | -200 400.
With a financial calculator, enter I/YR = 18, PV = -200, PMT = 0, and FV = 400. Then press the N key to find N = 4.19 ≈ 4.
d. 100%? | | -200 400.
With a financial calculator, enter I/YR = 100, PV = -200, PMT = 0, and FV = 400. Then press the N key to find N = 1.00 ≈ 1.
a. 0 1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | | 400 400 400 400 400 400 400 400 400 400 FVA 10 =?
With a financial calculator, enter N = 10, I/YR = 10, PV = 0, and PMT = -400. Then press the FV key to find FV = $6,374.97.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
b. 0 1 2 3 4 5 | | | | | | 200 200 200 200 200 FVA 5 =?
With a financial calculator, enter N = 5, I/YR = 5, PV = 0, and PMT = -200. Then press the FV key to find FV = $1,105.13.
c. 0 1 2 3 4 5 | | | | | | 400 400 400 400 400 FVA 5 =?
With a financial calculator, enter N = 5, I/YR = 0, PV = 0, and PMT = -400. Then press the FV key to find FV = $2,000.
d. To solve Part d using a financial calculator, repeat the procedures discussed in Parts a, b, and c, but first switch the calculator to "BEG" mode. Make sure you switch the calculator back to "END" mode after working the problem.
(1) 0 1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | | 400 400 400 400 400 400 400 400 400 400 FVA 10 =?
With a financial calculator set to “BEG” mode, enter N = 10, I/YR = 10, PV = 0, and PMT = -400. Then press the FV key to find FV = $7,012.46.
(2) 0 1 2 3 4 5 | | | | | | 200 200 200 200 200 FVA 5 =?
With a financial calculator set to “BEG” mode, enter N = 5, I/YR = 5, PV = 0, and PMT = -200. Then press the FV key to find FV = $1,160.38.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
| | | | | | 200 200 200 200 200 PV =?
With a financial calculator set to “BEG” mode, enter N = 5, I/YR = 5, PMT = - 200, and FV = 0. Then press the PV key to find PV = $909.19.
| | | | | | 400 400 400 400 400 PV =?
With a financial calculator set to “BEG” mode, enter N = 5, I/YR = 0, PMT = - 400, and FV = 0. Then press the PV key to find PV = $2000.00.
4-14 a. Cash Stream A Cash Stream B 0 1 2 3 4 5 0 1 2 3 4 5 | | | | | | | | | | | | PV =? 100 400 400 400 300 PV =? 300 400 400 400 100
With a financial calculator, simply enter the cash flows (be sure to enter CF 0 = 0), enter I/YR = 8, and press the NPV key to find NPV = PV = $1,251.25 for the first problem. Override I = 8 with I = 0 to find the next PV for Cash Stream A. Repeat for Cash Stream B to get NPV = PV = $1,300.32.
b. PVA = $100 + $400 + $400 + $400 + $300 = $1,600. PVB = $300 + $400 + $400 + $400 + $100 = $1,
4-15 These problems can all be solved using a financial calculator by entering the known values shown on the time lines and then pressing the I/YR button.
a. 0 1 | | +700 -
With a financial calculator, enter N = 1, PV = 700, PMT = 0, and FV = -749. Then press the I/YR key to find I/YR = 7%.
0%
8% 8%
I =?
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
b. 0 1 | | -700 +
With a financial calculator, enter N = 1, PV = -700, PMT = 0, and FV = 749. Then press the I/YR key to find I/YR = 7%.
c. 0 10 | | +85,000 -201,
With a financial calculator, enter N = 10, PV = 85,000, PMT = 0, and FV = -201,229. Then press the I/YR key to find I/YR = 9%.
d. 0 1 2 3 4 5 | | | | | | +9,000 -2,684.80 -2,684.80 -2,684.80 -2,684.80 -2,684.
With a financial calculator, enter N = 5, PV = 9,000, PMT = -2,684.8, and FV = 0. Then press the I/YR key to find I/YR = 15%.
I =?
I =?
I =?
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-17 a. 0 2 4 6 8 10 | | | | | | PV =? 500
With a financial calculator, enter N = 10, I/YR = 6, PMT = 0, and FV = -500. Then press the PV key to find PV = $279.20. Alternatively,
PV = FVn
mn
m
1 i
1
2 ( 5 )
2
- 12 1
1
10
- 06
1
= $500(PVIF
b. 0 4 8 12 16 20 | | | | | | PV =? 500
With a financial calculator, enter N = 20, I/YR = 3, PMT = 0, and FV = -500. Then press the PV key to find PV = $276.84, or
PV = $
4 ( 5 )
4
1 0.^12
1
20
- 03
1
c. 0 1 2 12 | | | • • • | PV =? 500
With a financial calculator, enter N = 12, I/YR = 1, PMT = 0, and FV = -500. Then press the PV key to find PV = $443.72, or
PV = $
12 ( 1 )
12
1 0.^12
1
12
- 01
1
6%
3%
1%
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-18 a. 0 1 2 3 9 10 | | | | • • • | | 400 400 400 400 400 FVA 10 =?
Enter N = 5 × 2 = 10, I/YR = 12/2 = 6, PV = 0, PMT = -400, and then press FV to get FV = $5,272.32.
b. Now the number of periods is calculated as N = 5 x 4 = 20, I/YR = 12/4 = 3, PV = 0, and PMT = -200. The calculator solution is $5,374.07.
Note that the solution assumes that the nominal interest rate is compounded at the annuity period.
c. The annuity in Part b earns more because some of the money is on deposit for a longer period of time and thus earns more interest. Also, because compounding is more frequent, more interest is earned on interest.
4-19 a. Universal Bank: Effective rate = 7%.
Regional Bank:
Effective rate =
4 4
1 0.^06
With a financial calculator, you can use the interest rate conversion feature to obtain the same answer. You would choose the Universal Bank.
b. If funds must be left on deposit until the end of the compounding period (1 year for Universal and 1 quarter for Regional), and you think there is a high probability that you will make a withdrawal during the year, the Regional account might be preferable. For example, if the withdrawal is made after 10 months, you would earn nothing on the Universal account but (1.015) 3 - 1.0 = 4.57% on the Regional account. Ten or more years ago, most banks and S&Ls were set up as described above, but now virtually all are computerized and pay interest from the day of deposit to the day of withdrawal, provided at least $1 is in the account at the end of the period.
6%
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-21 a. 0 I=? 1 2 3 4 5 | | | | | | -6 12 (in millions)
With a calculator, enter N = 5, PV = -6, PMT = 0, FV = 12, and then solve for I/YR = 14.87% ≈ 15%.
b. The calculation described in the quotation fails to take account of the compounding effect. It can be demonstrated to be incorrect as follows:
$6,000,000(1.20) 5 = $6,000,000(2.4883) = $14,929,800,
which is greater than $12 million. Thus, the annual growth rate is less than 20 percent; in fact, it is about 15 percent, as shown in Part a.
| | | | | | | | | | | -4 8 (in millions)
With a calculator, enter N = 10, PV = -4, PMT = 0, FV = 8, and then solve for I/YR = 7.18%.
| | | | | • • • | 85,000 -8,273.59 -8,273.59 -8,273.59 -8,273.59 -8,273.
With a calculator, enter N = 30, PV = 85000, PMT = -8273.59, FV = 0, and then solve for I/YR = 9%.
I =?
I =?
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-24 a. 0 1 2 3 4 | | | | | PV =? -10,000 -10,000 -10,000 -10,
With a calculator, enter N = 4, I/YR = 7, PMT = -10000, and FV = 0. Then press PV to get PV = $33,872.11.
b. (1) At this point, we have a 3-year, 7% annuity of $10,000 whose present value is $26,243.16: N = 3, I/YR = 7, PMT = -10000, and FV = 0. Then press PV to get PV = $26,243.16. You can also think of the problem as follows:
(Beginning balance)(1+I) – PMT = Ending balance $33,872.11 (1.07) ─ $10,000 = $26,243.16.
(2) Zero after the last withdrawal.
| | | • • • | 12,000 -1,500 -1,500 -1,
With a calculator, enter I/YR = 9, PV = 12000, PMT = -1500, and FV = 0. Press N to get N = 14.77 ≈ 15 years. Therefore, it will take approximately 15 years to pay back the loan.
4-26 0 1 2 3 4 5 6 | | | | | | | 1,250 1,250 1,250 1,250 1,250? FV = 10,
With a financial calculator, get a "ballpark" estimate of the years by entering I/YR = 12, PV = 0, PMT = -1250, and FV = 10000, and then pressing the N key to find N = 5. years. This answer assumes that a payment of $1,250 will be made 94/100th of the way through Year 5. Now find the FV of $1,250 for 5 years at 12%; N = 5, I/YR = 12, PV = 0, and PMT = -1250. Press FV to get FV = $7,941.06. Compound this value for 1 year at 12% to obtain the value in the account after 6 years and before the last payment is made; it is $7,941.06(1.12) = $8,893.99. Thus, you will have to make a payment of $10,000 - $8,893.99 = $1,106.01 at Year 6, so the answer is: it will take 6 years, and $1,106.01 is the amount of the last payment.
7%
9%
12%
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-31 a. Begin with a time line:
6 -mos. 0 1 2 3 4 5 6 8 10 12 14 16 18 20 Years 0 6% 1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | | | | | | | | | | | | 100 100 100 100 100 FVA
Since the first payment is made today, we have a 5-period annuity due. The applicable interest rate is I = 12/2 = 6 per period, N = 5, PV = 0, and PMT = -100. Setting the calculator on "BEG," we find FVA (Annuity due) = $597.53. That will be the value at the 5th^ 6-month period, which is t = 2.5. Now we must compound out to t = 10, or for 7.5 years at an EAR of 12.36%, or 15 semiannual periods at 6%.
$597.53 → 20 - 5 = 15 periods @ 6% → $1,432.02,
or $597.53 → 10 - 2.5 = 7.5 years @ 12.36% → $1,432.02.
b. 1 10 years 0 1 2 3 4 5 40 quarters | | | | | | • • • | PMT PMT PMT PMT PMT FV = 1,432.
The time line depicting the problem is shown above. Because the payments only occur for 5 periods throughout the 40 quarters, this problem cannot be immediately solved as an annuity problem. The problem can be solved in two steps:
(1) Discount the $1,432.02 back to the end of Quarter 5 to obtain the PV of that future amount at Quarter 5.
(2) Then solve for PMT using the value solved in Step 1 as the FV of the five- period annuity due.
Step 1: Input the following into your calculator: N = 35, I/YR = 3, PMT = 0, FV = 1432.02, and solve for PV at Quarter 5. PV = $508.92.
Step 2: The PV found in Step 1 is now the FV for the calculations in this step. Change your calculator to the BEGIN mode. Input the following into your calculator: N = 5, I/YR = 3, PV = 0, FV = 508.92, and solve for PMT = $93.07.
3%
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4-32 Here we want to have the same effective annual rate on the credit extended as on the bank loan that will be used to finance the credit extension. First, we must find the EAR = EFF% on the bank loan. Enter NOM% = 15, N = P/YR = 12, and press EFF% to get EAR = 16.08%. Now recognize that giving 3 months of credit is equivalent to quarterly compounding- -interest is earned at the end of the quarter, so it is available to earn interest during the next quarter. Therefore, enter P/YR = 4, EFF% = EAR = 16.08%, and press NOM% to find the nominal rate of 15.19 percent. Therefore, if a 15.19 percent nominal rate is charged and credit is given for 3 months, the cost of the bank loan will be covered. Alternative solution: We need to find the effective annual rate (EAR) the bank is charging first. Then, we can use this EAR to calculate the nominal rate that should be quoted to the customers.
Bank EAR: EAR = (1 + I (^) NOM/M) M^ - 1 = (1 + 0.15/12) 12 - 1 = 16.08%.
Nominal rate that should be quoted to customers:
16.08% = (1 + I (^) NOM/4) 4 - 1 1.1608 = (1 + I (^) NOM/4) 4 1.0380 = 1 + I (^) NOM/ I (^) NOM = 0.0380(4) = 15.19%.
