1、Ch04SolutionsManualChapter 4Time Value of MoneyANSWERS TO END-OF-CHAPTER QUESTIONS4-1 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 paramete
2、r iis 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 paymen
3、ts (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
4、iNom 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 paym
5、ent, 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 t
6、he 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
7、 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 e
8、qual 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
9、.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
10、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 yea
11、r. 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 hi
12、gher 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, iNom, is the rate of interest stated in a con
13、tract. 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
14、rate, iPER, 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
15、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 a
16、mounts (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 an
17、d 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.4-3 True. The second series is an uneven payment stream, but it contains an annuity of $
18、400 for 8 years. The series could also be thought of as a $100 annuity for 10 years plus an additional payment of $100 in Year 2, plus additional payments of $300 in Years 3 through 10.4-4 True, because of compounding effects-growth on growth. The following example demonstrates the point. The annual
19、 growth rate is I in the following equation:$1(1 + I)10 = $2.The term (1 + I)10 is the FVIF for I percent, 10 years. We can find I in one of two ways:1. Using a financial calculator input N = 10, PV = -1, PMT = 0, FV = 2, and I/YR = ?. Solving for I/YR you obtain 7.18%.2. Using a financial calculato
20、r, input N = 10, I/YR = 10, PV = -1, PMT = 0, and FV = ?. Solving for FV you obtain $2.59. This formulation recognizes the interest on interest phenomenon.4-5 For the same stated rate, daily compounding is best. You would earn more interest on interest.SOLUTIONS TO END-OF-CHAPTER PROBLEMS10%4-1 0 1
21、2 3 4 5 | | | | | |PV = 10,000 FV5= ?FV5 = $10,000(1.10)5 = $10,000(1.61051) = $16,105.10.Alternatively, with a financial calculator enter the following: N = 5, I/YR = 10, PV = -10000, and PMT = 0. Solve for FV = $16,105.10.7%4-2 0 5 10 15 20 | | | | |PV = ? FV20 = 5,000With a financial calculator e
22、nter the following: N = 20, I/YR = 7, PMT = 0, and FV = 5000. Solve for PV = $1,292.10.I/YR= ?4-3 0 18 | |PV = 250,000 FV18 = 1,000,000With a financial calculator enter the following: N = 18, PV = -250000, PMT = 0, and FV = 1000000. Solve for I/YR = 8.01% 8%.6.5%4-4 0 N = ? | |PV = 1 FVN = 2$2 = $1(
23、1.065)N.With a financial calculator enter the following: I/YR = 6.5, PV = -1, PMT = 0, and FV = 2. Solve for N = 11.01 11 years.12%4-5 0 1 2 N 2 N 1 N | | | | | | PV = 42,180.53 5,000 5,000 5,000 5,000 FV = 250,000Using your financial calculator, enter the following data: I/YR = 12; PV = 42180.53; P
24、MT = 5000; FV = 250000; N = ? Solve for N = 11. It will take 11 years to accumulate $250,000.4-6 Ordinary annuity:7% 0 1 2 3 4 5 | | | | | | 300 300 300 300 300 FVA5 = ?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:7% 0
25、1 2 3 4 5 | | | | | | 300 300 300 300 300 FVA5 = ?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. Dont forget to switch back to “END” mode.8%4-7 0 1 2 3 4 5 6 | | | | | | | 100 100 100 200 300 500 PV = ? FV = ?Us
26、ing a financial calculator, enter the following: CF0 = 0; CF1 = 100; Nj = 3; CF4 = 200 (Note calculator will show CF2 on screen.); CF5 = 300 (Note calculator will show CF3 on screen.); CF6 = 500 (Note calculator will show CF4 on screen.); and I/YR = 8. Solve for NPV = $923.98. To solve for the FV of
27、 the cash flow stream with a calculator that doesnt 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 = 1.0 = (1
28、.01)12 1.0 = 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.6% 4-9 a. 01 | | $500(1.06) = $530.00. -500 FV = ?6%b. 01 2 | | | $500(1.06)2 = $561.80. -500FV = ?6%c. 0 1
29、| | $500(1/1.06) = $471.70. PV = ? 5006%d. 012 | | | $500(1/1.06)2 = $445.00. PV = ? 5006%4-10 a. 0 1 2 3 4 5 6 7 8 9 10 $500(1.06)10 = $895.42. | | | | | | | | | | | -500 FV = ?12% b. 0 1 2 3 4 5 6 7 8 9 10 $500(1.12)10 = $1,552.92. | | | | | | | | | | | -500 FV = ?6% c. 0 1 2 3 4 5 6 7 8 9 10 | |
30、| | | | | | | | | $500(1/1.06)10 = $279.20 PV = ? 50012% d. 0 1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | | $500(1/1.12)10 = $160.99 PV = ? 5007%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 10.10%b. ? | | -2
31、00 400 .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. 18%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 .Wit
32、h 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. 4-12 10%a. 0 1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | | 400 400 400 400 400 400 400 400 400 400 FVA10 = ?With a financial calculator, enter N = 10, I/YR = 10, PV = 0, and PMT = -400
