FIN 614 Time Value of Money. Professor Robert B.H. Hauswald Kogod School of Business, AU. 1/12/2011 Time Value of Money Robert B.H.

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1 FIN 614 ime Value of Money Professor Robert B.H. Hauswald Kogod School of Business, AU 1/12/2011 ime Value of Money Robert B.H. Hauswald 1 ime Value of Money Business and financial planning revolves around comparing current to past and future cash flows what is it worth? does it make sense? Present value calculus: the price of impatience how much is $1 now worth in the future? how much is $1 in the future worth now? ime value of money: two visions future value and present value from simple to multiple cash flows 1/12/2011 ime Value of Money Robert B.H. Hauswald 2

2 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 ($10,000.05) $10,000 is the principal repayment ($10,000 1) $10,500 is the total due. It can be calculated as: $10,500 = $10,000 (1.05) he total amount due at the end of the investment is call the Future Value (FV). 1/12/2011 ime Value of Money Robert B.H. Hauswald 3 Present Value If you were to be promised $10,000 due in one year when interest rates are at 5-percent, your investment be worth $9, in today s dollars. $10,000 $ 9, = 1.05 he amount that a borrower would need to set aside today to to able to meet the promised payment of $10,000 in one year is call the Present Value (PV) of $10,000. Note that $10,000 = $9, (1.05) 1/12/2011 ime Value of Money Robert B.H. Hauswald 4

3 Net Present Value he Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment. Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy? $10,000 NPV = $9, NPV = $9,500 + $9, NPV = $23.81 Yes! 1/12/2011 ime Value of Money Robert B.H. Hauswald 5 ransferring Money in ime: Exchange Rates Interest rates: the price of money method of exchange or rate of substitution from Future <==> Present incorporates: Maturity, Expected Inflation, erm Structure of Interest Rates Discount rates: adjustment of future cash flow for opportunity cost risk personal preference for cash used in valuation of cash Flows 1/12/2011 ime Value of Money Robert B.H. Hauswald 6

4 he Fundamental VM Principle he Basic ime-value-of-money Relationship: C t+ = C t x (1 + r) r is the interest rate per period is the duration of the investment, stated in the compounding time unit C t (PV: Present Value) is the value at period t (beginning of the investment) C t+ (FV: Future Value) is the value at period t+ (end of the investment) Compounding frequency: how often (how many periods: ) is interest calculated t is now 1/12/2011 ime Value of Money Robert B.H. Hauswald 7 Inputs into ime Value:, r, Cash Flows. C 0 C PV FV $ invested today $ received in future Define some terms C = Future value in th period = FV C 0 = Current value = PV r = Discount rate = Number of future time periods 1/12/2011 ime Value of Money Robert B.H. Hauswald 8

5 Future Value and Compounding Compounding: how much will $1 invested today at 9% be worth in two years? Interest on interest! he ime Line Year C $1 $ Future value (C2) = $1 x = $ /12/2011 ime Value of Money Robert B.H. Hauswald 9 Present Value and Discounting Discounting: how much is $1 that we will receive in two years worth today (r = 9%)? Year C $0.842 $1 Present value (C0) = $1 / = $0.842 he interest rate (9%) is also called the discount rate. 1/12/2011 ime Value of Money Robert B.H. Hauswald 10

6 Future and Present Value oday's Value of a Lump Sum or Stream of Cash Payments Received at a Future Point in ime ( r ) FV = PV 1 + PV = FV ( 1 + r ) 1/12/2011 ime Value of Money Robert B.H. Hauswald 11 Present Value and Compounding How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%? PV $20, $ 9, = $20,000 5 (1.15) 1/12/2011 ime Value of Money Robert B.H. Hauswald 12

7 he Power Of High Discount Rates % % % 15% 20% Periods 1/12/2011 ime Value of Money Robert B.H. Hauswald 13 he Perception of ime So far, we focused on time periods: discrete however, time is a continuous process: gives rise to the notion of continuous compounding 1/12/2011 ime Value of Money Robert B.H. Hauswald 14

8 Continuous Compounding he general formula for the future value of an investment compounded continuously over many periods can be written as: FV = C 0 e r Where C 0 is cash flow at date 0, r is the stated annual interest rate, is the number of periods over which the cash is invested, and e is a transcendental number approximately equal to e x is a key on your calculator. 1/12/2011 ime Value of Money Robert B.H. Hauswald 15 wo Flavors of Compounding Value can be found for any period. Move money around just like exchange rates. FV - Future Value is the Value at time. PV - Present Value is the value at time 0. wo flavors of compounding: beware! Annual Compounding: Continuous Compounding: (almost) equivalent for small r but not for large: watch out for those small discrepancies that make your bankers rich! Focus on VALUAION! 1/12/2011 ime Value of Money Robert B.H. Hauswald 16 PV P V = FV ( 1+ r) F V = e r *

9 PV Practice Problem A zero coupon bond is a bond that does not pay either annual or semi-annual coupons. Instead, it is sold at a discount to its face value. What should be the price an 8 year zero coupon bond with a face of $1,000 if similar bonds are yielding 6%? (Ignore any tax effects. ry both annual and continuous compounding.) 1/12/2011 ime Value of Money Robert B.H. Hauswald 17 Compounding Compounding: interest on interest Benjamin Franklin: Money makes money, and the money that money makes makes more money Which one would you prefer? 1. simple interest: no compounding FV = PV 0 (1 + r*) 2. annual compound interest: your bank account FV = PV 0 ( 1 + r ) 3. continuous compound interest: money growing like trees FV = PV 0 *e r 1/12/2011 ime Value of Money Robert B.H. Hauswald 18

10 Money Makes Money return 10% contribution 100 years Simple Interest Compound Interest $ $ $ $ $ $ $1, Continuous Compound Inter 100 $ $ $ $ $ $ $ $2, /12/2011 ime Value of Money Robert B.H. Hauswald 19 Compounding Simple Interest FV of $ Compound Interest 3. Continuous Compound Interest Years 1/12/2011 ime Value of Money Robert B.H. Hauswald 20

11 Compounding Periods Compounding an investment m times a year for years provides for future value of wealth: FV = C0 1+ m For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to FV 2 3 =.12 $ /12/2011 ime Value of Money Robert B.H. Hauswald 21 r m = $50 (1.06) 6 = $70.93 Effective Annual Interest Rates Effective annual interest rate of an investment? FV = $50 (1 + ) = $50 (1.06) = $ he EAR is the annual rate that yield the same end-of-investment wealth after 3 years: FV = $50 (1 + EAR) 3 = $ $70.93 $70.93 (1 + EAR) 3 = EAR = 1 = $50 $50 Investing at 12.36% compounded annually equals investing at 12% compounded semiannually. 1/12/2011 ime Value of Money Robert B.H. Hauswald 22

12 Generalized Effective Annual Rates (EAR) he formula for multiple compounding periods is (m is the number of times interest is paid annually) Annual basis: 1+r = (1+r/m) m For years: (1+r ) = (1+r/m) m r represents the effective annual rate of interest r represents the stated or simple rate of interest How much will I receive at the end of the year if the 12% annual rate is paid semi-annually? Monthly? Why the difference in value between semi-annual interest and annual interest? What about over the course of 20 years? 1/12/2011 ime Value of Money Robert B.H. Hauswald 23 Making Sound Decisions What are appropriate decision criteria? When do they apply? 1/12/2011 ime Value of Money Robert B.H. Hauswald 24

13 Summary Fundamental building blocks of valuation future value: value attributed to a present cash flow present value: value attributed to a future cash flow Fundamental parameters: the price of money discount factor: the price of future cash flows interest factor: the price of current cash flows time horizon PV and FV allows us to easily switch back and forth between cash flows separated in time 1/12/2011 ime Value of Money Robert B.H. Hauswald 25 EAR eample Appendix: 4 Questions What is the future value with compounding? What is the present value with compounding? How long is the wait? What rate is enough? 1/12/2011 ime Value of Money Robert B.H. Hauswald 26

14 Effective Annual Interest Rates Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly. What we have is a loan with a monthly interest rate rate of 1½ percent. his is equivalent to a loan with an annual interest rate of percent n m r 1 + m.18 = = (1.015) 12 = /12/2011 ime Value of Money Robert B.H. Hauswald 27 EAR on Calculator Hewlett Packard 10B keys: display: description: 12 [gold] [P/YR] Sets 12 P/YR. 18 [gold] [NOM%] Sets 18 APR. [gold] [EFF%] exas Instruments BAII Plus keys: description: [2nd] [ICONV] Opens interest rate conversion menu [ ] [C/Y=] 12 Sets 12 payments per year [ ][NOM=] 18 [ENER] Sets 18 APR. [ ] [EFF=] [CP] /12/2011 ime Value of Money Robert B.H. Hauswald 28

15 Future Value Example Suppose that Jay Ritter invested in the initial public offering of the Modigliani company. Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year for the next five years. What will the dividend be in five years? FV = C 0 (1 + r) $5.92 = $1.10 (1.40) 5 1/12/2011 ime Value of Money Robert B.H. Hauswald 29 Future Value and Compounding Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40- percent on the original $1.10 dividend: $5.92 > $ [$ ] = $3.30 his is due to compounding. 1/12/2011 ime Value of Money Robert B.H. Hauswald 30

16 Future Value and Compounding $ 1.10 (1.40) $ 1.10 (1.40) $ 1.10 (1.40) 2 $ 1.10 (1.40) $ 1.10 (1.40) $1.10 $1.54 $2.16 $3.02 $4.23 $ /12/2011 ime Value of Money Robert B.H. Hauswald 31 Present Value and Compounding How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%? PV $20, $ 9, = $20,000 5 (1.15) 1/12/2011 ime Value of Money Robert B.H. Hauswald 32

17 How Long is the Wait? If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000? FV = C (1 r) $ 10,000 = $5,000 (1.10) 0 + = $10,000 ( 1.10) = = 2 $5,000 ln 2 ln(1.10) ln( 1.10) = = ln = 7.27 years /12/2011 ime Value of Money Robert B.H. Hauswald 33 What Rate Is Enough? Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child s education? About 21.15%. FV ) 12 = C0 (1 + r $ 50,000 = $5,000 (1 + r) (1 + r) 12 = $50,000 $5,000 = 10 ( 1+ r) = r = = = /12/2011 ime Value of Money Robert B.H. Hauswald 34