CHAPTER 4
THE TIME VALUE OF MONEY AND DISCOUNTED CASH FLOW ANALYSIS
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Objectives
To explain the concepts of compounding and discounting, future value and present value. To show how these concepts are applied to making financial decisions.
Outline
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 ? ? ?
Compounding
The Frequency of Compounding Present Value and Discounting
Alternative Discounted Cash Flow Decision Rules Multiple Cash Flows Annuities
Perpetual Annuities Loan Amortization
Exchange Rates and Time Value of Money Inflation and Discounted Cash Flow Analysis Taxes and Investment Decisions
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Summary
Compounding is the process of going from present value (PV) to future value (FV). The future value of $1 earning interest at rate i per period for n periods is (1+i)n.
Discounting is finding the present value of some future amount. The present value of $1 discounted at rate i per period for n periods is 1/(1+i)n.
One can make financial decisions by comparing the present values of streams of expected future cash flows resulting from alternative courses of action. The present value of cash inflows less the present value of cash outflows is called net present value (NPV). If a course of action has a positive NPV, it is worth undertaking.
In any time value of money calculation, the cash flows and the interest rate must be denominated in the same currency.
Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows.
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How to Do TVM Calculations in MS Excel
Assume you have the following cash flows set up in a spreadsheet:
Move the cursor to cell B6 in the spreadsheet. Click the function wizard fx in the tool bar and when a menu appears, select financial and then NPV. Then follow the instructions for inputting the discount rate and cash flows. You can input the column of cash flows by selecting and moving it with your mouse. Ultimately cell B6 should contain the following: =NPV(0.1,B3:B5)+B2
The first variable in parenthesis is the discount rate. Make sure to input the discount rate as a decimal fraction (i.e., 10% is .1). Note that the NPV function in Excel treats the cash flows as occurring at the end of each period, and therefore the initial cash flow of 100 in cell B2 is added after the closing parenthesis. When you hit the ENTER key, the result should be $47.63.
Now move the cursor to cell B7 to compute IRR. This time select IRR from the list of financial functions appearing in the menu. Ultimately cell B7 should contain the following: =IRR(B2:B5)
When you hit the ENTER key, the result should be 34%. Your spreadsheet should look like this when you have finished:
A B t CF 1 0 -100 2 1 50 3 2 60 4 3 70 5 NPV 47.63 6 IRR 34% 7 1 2 3 4 5 6 7 A t 0 1 2 3 NPV IRR B CF -100 50 60 70 I nstructor’s Manual
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Solutions to Problems at End of Chapter
1. If you invest $1000 today at an interest rate of 10% per year, how much will you have 20 years from now,
assuming no withdrawals in the interim?
SOLUTION: n i PV FV PMT Result 20 10 1000 ? 0 FV =6,727.50 2. a. If you invest $100 every year for the next 20 years, starting one year from today and you earn
interest of 10% per year, how much will you have at the end of the 20 years?
b. How much must you invest each year if you want to have $50,000 at the end of the 20 years?
SOLUTION: n i PV FV PMT Result a. 20 10 0 ? 100 FV = 5,727.50 b. 20 10 0 50,000 ? PMT = 872.98
3. What is the present value of the following cash flows at an interest rate of 10% per year? a. $100 received five years from now. b. $100 received 60 years from now.
c. $100 received each year beginning one year from now and ending 10 years from now. d. $100 received each year for 10 years beginning now.
e. $100 each year beginning one year from now and continuing forever.
SOLUTION: n i PV FV PMT Result a. 5 10 ? 100 0 PV = $62.09 b. 60 10 ? 100 0 PV = $.3284 c. 10 10 ? 0 100 ordinary PV = $614.46 d. 10 10 ? 0 100 immediate PV = $675.90 e. Perpetuity 10 ? 0 100 ordinary See below
e. PV = $100 = $1,000 .10
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4. You want to establish a “wasting” fund which will provide you with $1000 per year for four years, at which time the fund will be exhausted. How much must you put in the fund now if you can earn 10% interest per year?
SOLUTION:
n i PV FV PMT Result 4 10 ? 0 1,000 PV =$3,169.87 5. You take a one-year installment loan of $1000 at an interest rate of 12% per year (1% per month) to be repaid in 12 equal monthly payments. a. What is the monthly payment?
b. What is the total amount of interest paid over the 12-month term of the loan?
SOLUTION:
n i PV FV PMT Result 12 1 1,000 0 ? PMT = $88.85 a. PMT = $88.85 b. 12 x $88.85 - $1,000 = $66.20
6. You are taking out a $100,000 mortgage loan to be repaid over 25 years in 300 monthly payments. a. If the interest rate is 16% per year what is the amount of the monthly payment? b. If you can only afford to pay $1000 per month, how large a loan could you take?
c. If you can afford to pay $1500 per month and need to borrow $100,000, how many months would it take
to pay off the mortgage?
d. If you can pay $1500 per month, need to borrow $100,000, and want a 25 year mortgage, what is the
highest interest rate you can pay?
SOLUTION:
n i PV FV PMT Result a. 300 16/12 100,000 0 ? PMT =$1358.89 b. 300 16/12 ? 0 1,000 PV = $73,590 c. ? 16/12 100,000 0 1,500 n = 166 d. 300 ? 100,000 0 1,500 i = 1.482% per month
a. Note: Do not round off the interest rate when computing the monthly rate or you will not get the same answer
reported here. Divide 16 by 12 and then press the i key. b. Note: You must input PMT and PV with opposite signs. c. Note: You must input PMT and PV with opposite signs.
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7. In 1626 Peter Minuit purchased Manhattan Island from the Native Americans for about $24 worth of trinkets. If the tribe had taken cash instead and invested it to earn 6% per year compounded annually, how much would the Indians have had in 1986, 360 years later?
SOLUTION:
n i PV FV PMT Result 360 6 24 ? 0 FV = 3.09 ? 1010 FV = 30,925,930,000
8. You win a $1 million lottery which pays you $50,000 per year for 20 years, beginning one year from now. How much is your prize really worth assuming an interest rate of 8% per year?
SOLUTION:
n i PV FV PMT Result 20 8 ? 0 50,000 PV = $490,907
9. Your great-aunt left you $20,000 when she died. You can invest the money to earn 12% per year. If you spend $3,540 per year out of this inheritance, how long will the money last?
SOLUTION:
n i PV FV PMT Result ? 12 20,000 0 3,540 n = 10 years
10. You borrow $100,000 from a bank for 30 years at an APR of 10.5%. What is the monthly payment? If you must pay two points up front, meaning that you only get $98,000 from the bank, what is the true APR on the mortgage loan?
SOLUTION:
n i PV FV PMT Result 360 .875 100,000 0 ? PMT = $914.74
If you must pay 2 points up front, the bank is in effect lending you only $98,000. Keying in 98000 as PV and computing i, we get:
n i PV FV PMT Result 360 ? 98,000 0 914.74 i = .89575
i =.89575% per month; APR = 12 ? .89575 ? 10.75%
《金融学》答案第四章 货币的时间价值与现金流贴现分析
