# Time Value of Money and Discounted Cash Flows

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1 ime Value of Money and Discounted ash Flows ime Value of Money (VM) - the Intuition A cash flow today is worth more than a cash flow in the future since: Individuals prefer present consumption to future consumption. Monetary inflation will cause tomorrow s dollars to be worth less than today s. Any uncertainty associated with future cash flows reduces the value of the cash flow.

2 VM - he One-Period ase: Future Value If you were to invest \$10,000 at 5-percent interest for one year, your investment would grow to \$10,500: \$500 would be interest (\$10, ) \$10,000 is the principal repayment (\$10,000 1) \$10,500 is the total due. It can be calculated as: he total amount due at the end of the investment is called the Future Value (FV). he One-Period ase: Future Value In the one-period case, the formula for FV can be written as: FV 1 0 Where FV 1 is the future value in one period, 0 is the present (date 0) value of cash flow, and r is the appropriate interest rate. 0 \$10,000 0 \$10, FV 1 \$10,500 Year 0 1

3 Some Basic Definitions Present Value earlier money on a time line Future Value later money on a time line Interest rate exchange rate between earlier money and later money. We also use: Discount rate ost of capital Opportunity cost of capital Required return he One-Period ase: Present Value How much do I have to invest today to have some amount in one period? Recall: FV 1 0 Rearrange to solve for 0 FV 1 / When we talk about discounting, we mean finding the present value of some future amount. When we talk about the value of something, we are talking about the present value unless we specifically indicate that we want the future value.

4 he One-Period ase: Present Value You have to pay \$10,000 in tuition next year. Assuming that the annual interest rates is 5-percent, what is the amount that you would need to set aside today to be able to meet this payment in one year? Note that: \$10,000 \$9,53.81 (1.05). he amount that you would need to set aside today to be able to meet the promised payment of \$10,000 in one year is called the Present Value () of \$10,000: 0 he One-Period ase: Present Value In the one-period case, the formula for can be written as: FV r Where 0 is the present value, FV 1 is cash flow at date 1, and r is the appropriate interest rate. 0 \$9,53.81 FV 1 / \$10,000/1.05 FV 1 \$10,000 Year 0 1

5 he Multiperiod ase: Future Value he general formula for the future value of an investment over many periods (VM relationship) can be written as: FV t+ t Where: FV t+ is the value at period t+ (end of the investment), t is the value at period t (beginning of the investment), r is the appropriate interest rate per period, and is the number of periods over which the cash is invested. he Multiperiod ase: Future Value An Example You deposit \$5,000 today in an account paying 1%. How much will you have in 5 years? FV \$ 5,000 \$8,

6 Future Value and ompounding Notice that the value of the investment in year five, \$8,811.71, is considerably higher than the sum of the original investment (principal) plus five increases of 1% on the original \$5,000 principal: \$8, > \$5, [\$5, ] \$8,000 his is due to compounding. Future Value and ompounding \$ 5,000 (1.1) \$ 5,000 (1.1) \$ 5,000 (1.1) \$ 5,000 (1.1) \$ 5,000 (1.1) \$5,000 \$5,600 \$6,7 \$7,04.64 \$7, \$8,

7 Future Value and ompounding How much of your savings is simple interest? How much is compound interest? Use VM to find the FV 5 of \$5,000: \$5000 \$ \$ \$8, hus, the total interest income is: At 1%, the simple interest is 0.1 \$5,000 \$600 per year. After 5 years, this is: the compound interest is thus: \$3, \$3,000 \$ alculator Keys exas Instruments BA-II Plus FV future value present value I/Y period interest rate P/Y must equal 1 for the I/Y to be the period rate Interest is entered as a percent, not a decimal N number of periods Remember to clear the registers (LR VM) after each problem Other calculators are similar in format

8 Example - Future Value Suppose you had a relative deposit \$10 at 5.5% interest 00 years ago. How much would the investment be worth today? FV (1.055) 00 With a calculator 00 N; 5.5 I/Y; 10 ; then P FV What is the effect of compounding? Principal 10 Simple interest 00( ) ompounding added \$447, to the value of the investment Present Value and ompounding How much would an investor have to set aside today in order to have \$0,000 five years from now if the current rate is 15%? Recall: FV t+ t Rearrange to solve for t FV t+ / 0 0 FV5 5 0 \$0,

9 Example - Present Value of a Lump Sum Suppose you are currently 1 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate \$1 million by the time you reach age 65? Set this up as a VM equation and solve for the present value: \$1 million 1 (1.10) 44 Solve for : 1 \$1 million/(1.10) 44 With a calculator: 44 N; 10 I/Y; 1,000,000 FV; then P How Long is the Wait? You want to purchase a new car and you are willing to pay \$0,000. If you can invest at 10% per year and you currently have \$15,000, how long will it be before you have enough money to pay cash for the car? Solve for : FV t+ t 0,000 15,000 (1.10) (1.10) 0,000/15000 With a calculator: I/Y 10; -15,000; FV 0,000 P N 3.0 years

10 Finding the Number of Periods Start with basic VM equation and solve for (remember your logs): FV t+ t > ln(fv t+ / t ) / ln You can use the financial keys on the calculator as well; just remember the sign convention. What Rate Is Enough? Suppose you have a 1-year old son and you want to provide \$75,000 in 17 years towards his college education. You currently have \$5000 to invest. What interest rate must you earn to have the \$75,000 when you need it? Solve for r: FV t+ t 75,000 5, (75,000 / 5,000) (75,000 / 5,000) 1/17 r (75,000 / 5,000) 1/17 1 With a calculator: N 17; -5000; FV 75,000; then P I/Y

11 An Example - Finding the Interest Rate (r): In December 194, the market price of an AB company common stock was \$3.37. According to he Wall Street Journal, the price of an AB company common stock in December 004 is \$7,500. What is the annual rate of increase in the value of the stock? Set this up as a VM problem. FV \$7,500 \$ \$7,500/3.37,5.5 r (,5.5) 1/6-1 With a calculator: N 6; -3.37; FV 7,500; then P I/Y A First Look at the Net Present Value (N) Example for N: You can buy a property today for \$3 million, and sell it in 3 years for \$3.6 million. he annual interest rate is 8%. Qa. Aa. Assuming you can earn no rental income on the property, should you buy the property? he present value of the cash inflow from the sale is: 0 Since this is less than the purchase price of \$3 million - We say that the Net Present Value (N) of this investment is negative: N (Future Fs) - 3,000,000 +,857,796.07

12 Example for N (continued): Qb. Suppose you can earn \$00,000 annual rental income (paid at the end of each year) on the property, should you buy the property now? Ab. he present value of the cash inflow from the sale is: 0 [00,000 /1.08] + [00,000 /1.08 ] + [3,800,000/ ] \$3,373,15.47 Since this is more than the purchase price of \$3 million - We say that the Net Present Value (N) of this investment is positive: N (Future Fs) -3,000,000+ 3,373,15.47 he general formula for calculating N: N /(1+r) + /(1+r) /(1+r) Simplifications Perpetuity A stream of constant cash flows that lasts forever Growing perpetuity A stream of cash flows that grows at a constant rate forever Annuity A stream of constant cash flows that lasts for a fixed number of periods Growing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods

13 Perpetuity A perpetuity is a constant stream of cash flows that lasts forever. 0 1 he present value of a perpetuity is given by: + + ( 1+ r) 3 t t r L A simplified formula for the present value of a perpetuity: Perpetuity: Example What is the value of a British consol that promises to pay 15 each year, every year until the sun turns into a red giant and burns the planet to a crisp? he interest rate is 10-percent

14 More Examples - Present Value for a Perpetuity Q1. AB Life Insurance o. is trying to sell you an investment policy that will pay you and your heirs \$1,000 per year (starting next year) forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy? A1. he most a rational buyer would pay for the promised cash flows is its present value: Q. AB Life Insurance o. tells you that the above policy costs \$9,000. At what interest rate would this be a fair deal? A. Again, the present value of a perpetuity equals /r. Now solve the following equation: Growing Perpetuity A stream of cash flows that grows at a constant rate (g) forever. (1+g) (1+g) he present value of a perpetuity is given by: (1 + g) (1 + g) + + ( 1+ r) 3 t t r g +1 +L A simplified formula for the present value of a growing perpetuity is:

15 Growing Perpetuity: Example Dividend is the portion of a company's profit paid out to shareholders. he expected dividend next year is \$1.30 and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? 0 \$ \$1.30 (1.05) \$1.30 (1.05) 3 0 An Example - Present Value for a Growing Perpetuity Q. Suppose that AB Life Insurance o. modifies the policy, such that it will pay you and your heirs \$1,000 next year, and then increase each payment by 1% forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy? A. he most a rational buyer would pay for the promised cash flows is its present value: Note: Everything else being equal, the value of the growing perpetuity is always higher than the value of the simple perpetuity, as long as g>0.

16 Annuity An annuity is a stream of constant cash flows that lasts for a fixed number of periods. L L 3 A simplified formula for the present value of an annuity is: t r 1 1 of Annuity Intuition An annuity is valued as the difference between two perpetuities: one perpetuity that starts at time 1 less a perpetuity that starts at time + 1 r r ( 1+ r)

17 of Annuity: Example If you can afford a \$400 monthly car payment, how much car can you afford if the monthly interest rate is 0.6% on a 36-month loan? \$400 \$400 \$400 L \$ Annuities and the alculator You can use the PM key on the calculator for the equal payment he sign convention still holds An annuity due is like an ordinary annuity, but its first payment occurs at the beginning of the first period Ordinary annuity versus annuity due: You can switch your calculator between the two types by using the nd BGN nd Set on the I BA-II Plus, or switch between BEG and END on the HP calculator he display of your calculator should indicate the type of annuity it is set for Most problems ainvolve ordinary annuities

18 How to Value Annuities with a alculator N I/Y PM FV , Make sure your calculator is set to 1 payment per period. Present Value of an Annuity - Example 1 Q. A local bank advertises the following: Pay us \$100 at the end of every year for the next 10 years. We will pay you (or your beneficiaries) \$100, starting at the eleventh year forever. Is this a good deal if the effective annual interest rate is 8%? A. We need to compare the of what you pay with the present value of what you get: - he present value of your annuity payments: (1/0.08){1 - [ 1/( )]} - he present value of the bank s perpetuity payments at the end of the tenth year (beginning of the eleventh year): /r (100/0.08) he present value of the bank s perpetuity payments today: 0 10 /(1+r) 10

19 Present Value of an Annuity - Example Q. You take a \$0,000 five-year loan from the bank, carrying a 0.6% monthly interest rate. Assuming that you pay the loan in equal monthly payments, what is your monthly payment on this loan? A. Since payments are made monthly, we have to count our time units in months. We have: monthly time periods in five years, with a monthly interest rate of: r 0.6%, and 0 With the above data we have: 0,000 (1/0.006){1 - [ 1/( )]} Solving for, we get a monthly payment of: Finding the Payment with a alculator N I/Y PM FV , Make sure your calculator is set to 1 payment per period.

20 Annuity Future Value onsider the future value of an ordinary annuity at the time of the last payment (): L FV + (1 + r) + (1 + r) FV t+ r + (1 + r) [( 1+ r) 1] 3 + L(1 + r) 1 A simplified formula for the future value of an annuity is: FV of Annuity: Example Assume that you can earn a 0.8% monthly rate of return on your investment until you retire. You are 5 and you will retire at the age of 65. If you save for your retirement \$500 monthly, how much will you have in your retirement account at the time you retire? \$500 \$500 \$500 \$ L 480 FV 480

21 How to alculate FV of Annuities with a alculator N I/Y PM FV ,801,5.16 Make sure your calculator is set to 1 payment per period. Annuity Due An annuity due is a stream of constant cash flows that is paid at the beginning of each period and lasts for a fixed number of periods (): L L FV (1 + r) + (1 + r) + (1 + r) + L(1 + r)

22 Annuity Due Simplified formulas for the present value and future of an annuity due: t r FV r + 1 [(1 + r) (1 r) ] t + + of Annuity Due : Example A bank offers you a certificate that will make 10 annual payments of \$10,000 at the beginning of each year over the next 10 years. What is the most you should be paying for this certificate if the annual required rate of return is 7%? o

23 alculating of an Annuity Due with a alculator First, switch your calculator to a beginning mode: nd BGN nd Set on the I BA-II Plus, or Set to BEG on the HP calculator N I/Y PM FV , ,000 0 Don t forget to switch your calculator back to end mode FV of Annuity Due : Example You decided to save for your grand children s university education, and you put \$,000 per year in an account paying 8% annually. he first payment is made today. How much will you have at the end of 30 years? FV 30

24 alculating FV of an Annuity Due with a alculator First, switch your calculator to a beginning mode: nd BGN nd Set on the I BA-II Plus, or Set to BEG on the HP calculator N I/Y PM ,000 FV 44, Don t forget to switch your calculator back to end mode Growing Annuity A stream of cash flows that grows at a constant rate (g) over a fixed number of periods: (1+g) (1+g) (1+g) -1 L (1 + g) (1 + g) + + L+ 1 A simplified formula for the present value of a growing annuity: 1+ g t 1 r g

25 of Growing Annuity: Example A defined-benefit retirement plan offers to pay \$0,000 one year after retirement. Subsequent annual payments will grow by three-percent each year. he plan will make a total of 40 annual payments. What is the present value at retirement if the discount rate is 10 percent? \$0, \$0,000 (1.03) \$0,000 (1.03) 39 L 40 0 alculating the of Growing Annuity with a alculator 1+ r Let k 1 and P, then the of growing 1+ g 1+ g annuity formula can be written as: P 1 1 k (1 + k) t N I/Y PM FV , , ,

26 An Example - Present Value of a Growing Annuity Q. Suppose that the bank (from Present Value of an Annuity - Example 1 ) rewords its advertisement to the following: Pay us \$100 next year, and another 9 annual payments such that each payment is 4% lower than the previous payment. We will pay you (or your beneficiaries) \$100, starting at the eleventh year forever. Is this a good deal if the effective annual interest rate is 8%? A. Again, we need to compare the of what you pay with the present value of what you get: - he present value of your annuity payments 0 [1/(r-g)]{1- [(1+g)/(1+r)] } 100[1/(0.08-(-0.04))]{1-[(1+(-0.04))/(1.08)] 10 } [100/0.1]{1-[0.96/1.08] 10 } - he present value of the bank s perpetuity payments today: \$ (see example above) Growing Annuity - Special ases When r < g, we still use the above formula What if r g? onsider two cases: first payment occurs at the end of the first year first payment occurs at the beginning of the first year

27 Growing Annuity Simplification for FV: [( 1+ r) (1 + g ] FV t+ ) r g An example: (r 10%) \$100 \$10 \$ (g %) FV [1/( )]{ }

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