The Time Value of Money

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1 The Time Value of Money

2 Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment.

3 Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years Interest Earned Value Value at the end of Year 5 = $130

4 Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year s balance. Today Future Years Interest Earned Value Value at the end of Year 5 = $133.82

5 FV $100 ( 1 ) r t Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? FV $100 (1.06) 5 $133.82

6 Interest Rates 0% 5% 10% 15% Number of Years 0 FV of $100

7 Peter Minuit bought Manhattan Island for $24 in Was this a good deal? To answer, determine $24 is worth in the year 2015, compounded at 8%. FV 389 $24 (1.08) $ trillion FYI - The value of Manhattan Island land is well below this figure.

8 Present Value = PV PV = Future Value after t periods (1+r) t

9 Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? PV (1.08) $2,572

10 The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. PV FV 1r t ( 1 )

11 Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? Total Immediate PV PV 1 2 PV payment 4,000 (1.08) 4,000 (1.08) 1 2 8, , , $15,133.06

12 PVs can be added together to evaluate multiple cash flows. PV C C ( 1r ) ( 1r ) 2

13 Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time.

14 PV of Perpetuity Formula C = cash payment r = interest rate PV C r

15 Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? PV 100, 000 $1, 000,

16 Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? PV ,, $751, (. )

17 PV of Annuity Formula PV C 1 1 r r( 1r) t C = cash payment r = interest rate t = Number of years cash payment is received

18 PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. 1 1 ( 1 ) PVAF r r r t

19 Example - Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? PV 4000, ( 1. 10) 3 PV $9,

20 Applications Value of payments Implied interest rate for an annuity Calculation of periodic payments Mortgage payment Annual income from an investment payout Future Value of annual payments FV C PVAF ( 1r) t

21 Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? FV ( 1. 10) 4, 000 ( 1. 10) FV $229, 100

22 NPV = PV(future cash flows) Inv 0 Accept investments that have positive NPV Reject investments with negative NPV Mutual exclusive projects

23 Example Suppose we can invest $50 today and receive $60 in one year. Should we accept the project given a 10% expected return? 60 NPV = -50+ $

24 T T NPV = C C1 C2 C (1 IRR) (1 IRR) (1 IRR) 0 IRR IS THE DISCOUNT RATE FOR WHICH NPV=0

25 FINANCIAL CALCULATOR. TRIAL AND ERROR EXAMPLE: C 0 = - 4,000 C 1 = +2,000 C 3 = +4,000 TRY IRR = 0, NPV = +2,000, IRR > 0 TRY IRR = 50%, NPV = - 889, IRR < 50 TRY IRR = 25%, NPV = +160, IRR >25 TRY IRR = 28%, NPV = 0

26 ACCEPT PROJECT IF IRR IS GREATER THAN THE OPPORTUNITY COST OF CAPITAL LOOKING AT THE NET PRESENT VALUE PROFILE FOR A CONVENTIONAL PROJECT, WE WILL BE ACCEPTING PROJECTS WITH POSITIVE NPV

27 CASH OUTFLOWS FOLLOWED BY CASH INFLOWS NPV DECLINES WITH INCREASING DISCOUNT RATES The optimality of NPV rule

28 NPV Year: 0 1 IRR(%) At 10% ($) A -1,000 +1, B +1,000-1, BOTH PROJECTS HAVE IRR OF 50% NPV PROFILE FOR PROJECT B INCREASES WITH INCREASING DISCOUNT RATES ACCEPT PROJECT B WHEN IRR IS LESS THAN THE OPPORTUNITY COST OF CAPITAL

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