The Time Value of Money

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1 THE TIME VALUE OF MONEY CALCULATING PRESENT AND FUTURE VALUES Fuure Value: FV = PV 0 ( + r) Presen Value: PV 0 = FV ( + r) THE EFFECTS OF COMPOUNDING The effecs/benefis of compounding: Increase wih he ineres rae. Increase wih ime. Increase wih he frequency of compounding. (more on he deails of his laer.)

2 TIPS ON SOLVING PRESENT VALUE AND FUTURE VALUE PROBLEMS For muliple cash flows, jus add up he individual presen (or fuure) values. As, PV and FV As r, PV and FV There are (currenly) only 4 componens: PV, FV,, and r Wih ANY 3 componens, you can solve for he 4h VALUING PERPETUITIES Perpeuiy: A level sream of cash flows which coninue forever (someimes called consols). Presen Value of a Perpeuiy: PV 0 r VALUING PERPETUITIES Assuming ha ineres raes are 0%, wha is he value oday of a perpeuiy paying $500 per year, wih he firs paymen one year from oday?

3 VALUING PERPETUITIES Would you be willing o pay $6,500 for he same perpeuiy if ineres raes were 8%? GROWING PERPETUITIES Presen Value of a Growing Perpeuiy: PV r g 0 GROWING PERPETUITIES Assume a growing perpeuiy jus made a paymen of $20 yeserday. If he cash flow is expeced o grow a 5% and ineres raes are sill 0%, wha is he price of he perpeuiy oday?

4 PRESENT AND FUTURE VALUE OF MULTIPLE CASH FLOWS FV 0 PV 0 0 r r... You jus inheried some money from now dead Uncle Fred. You plan o use he money for a vacaion, bu know you firs need o pu aside some o cover your books and supplies over he nex wo years. You expec o need $4,000 in each of he nex wo years. Ineres raes are 0%, compounded annually. How much of now dead Uncle Fred s money do you need o pu aside oday? r r r PRESENT VALUE OF AN ANNUITY Annuiy: A level sream of cash flows for a fixed period of ime. Presen Value of an Annuiy: PV0 r r PRESENT VALUE OF AN ANNUITY We can rearrange he equaion o he following: Presen Value of an Annuiy: PV 0 r r

5 PRESENT VALUE OF AN ANNUITY Le s reurn o our earlier example: You jus inheried some money from now dead Uncle Fred. You plan o use he money for a vacaion, bu know you firs need o pu aside some o cover your books and supplies over he nex wo years. You expec o need $4,000 in each of he nex wo years. Ineres raes are 0%. How much of now dead Uncle Fred s money do you need o pu aside oday? FUTURE VALUE OF AN ANNUITY Fuure Value of an Annuiy: FV r This, of course, can also be rearranged FV r r r FUTURE VALUE OF AN ANNUITY If you deposi $300 ino a reiremen accoun a he end of each monh, saring nex monh, and he accoun earns 0.75% per monh, how much will you have in 35 years?

6 ANNUITIES: A REAL-LIFE EXAMPLE Books and beer are expensive! You now have a balance of $2,000 on your VISA card. The ineres rae on ha card is 2% per monh. However, you pay only he $50 minimum paymen each monh (saring nex monh) and make no more charges on ha card. How long will i ake you o pay off he balance? GROWING ANNUITIES Presen Value of a Growing Annuiy: g PV0 r g r ANNUITIES DUE Annuiy Due: An annuiy for which he cash flows occur a he beginning of he period. PV Annuiy Due = (PV Ordinary Annuiy) x ( + r) FV Annuiy Due = (FV Ordinary Annuiy) x ( + r)

7 THE EFFECT OF COMPOUNDING Annual Percenage Rae (APR): The nominal, saed annual ineres rae ha ignores he effec of compound ineres wihin he year. The APR is he periodic rae (r) imes he number of compoundings per year (m). THE EFFECT OF COMPOUNDING Effecive Annual Rae (EAR): The effecive annual ineres rae, which akes ino accoun he effec of compound ineres. APR AND EAR Example: A bank loan is quoed as 2% APR, compounded semiannually. Wha is he EAR?

8 APR AND EAR Example: A bank loan is quoed as 2% APR, compounded semiannually. Wha is he EAR? EAR APR m m AMORTIZATION Wha is an amorized loan? You plan o buy a $200,000 house. You will pu 0% down and finance he res wih a 30 year morgage a 6% APR, compounded monhly. Wha are he monhly paymens? SUGGESTED PROBLEMS Quesions 3 hrough 5 Problems, 2, 3, 4, 7, 8, 3, 4, 6, 20, 22, 24, 25, 26, and 28

9 ADDITIONAL PRACTICE Assuming a 0% ineres rae, compounded annually, wha is he value oday of $,000 per year forever, wih he firs paymen saring one year from oday? ADDITIONAL PRACTICE Wha if he firs paymen was in 5 years? ADDITIONAL PRACTICE Given an ineres rae of 0% APR, compounded annually, wha is he value in five years of a perpeual sream of $20 annual paymens saring in nine years?

10 ADDITIONAL PRACTICE You have jus read an adverisemen ha says, Pay us $00 a year for 0 years, saring nex year, and we will pay you (and your heirs) $00 a year hereafer in perpeuiy. A wha range of ineres raes would you accep his deal?

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