1 Lecture 2: Time Value of Money

2 Why learn this future value & present value analysis Personal finance application Home mortgage calculation car loan amortization Business applications Project investment analysis stock and bond valuation firm valuation

3 Future value of a cash flow Assuming the interest rate for a period is r, then the future value (FV) of C dollars one period from today is C(1+r), two periods from today is C(1+r) 2, and n periods from today is C(1+r) n FV=C(1+r) n

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6 Present value (PV) of a cash flow Assuming the interest rate for a period is r, then the PV of C dollars one period from today is C/(1+r), two periods from today is C/(1+r)2, and n periods from today is C/(1+r) n

7 The present value of cash flows {C t } is

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9 如果現金流量中的數額不等, 如何計算 現值 ?

10 FVIF & PVIF FVIF: future value interest factor = (1+r) t r: interest rate, rate of return, discount rate, or discount factor PVIF: present value interest factor = 1/(1+r) t Basic present value equation:

11 Perpetuities A perpetuity is a constant payment of C dollars every period forever.

12 The present value of a perpetuity is

13 Example An asset is promised to pay $500 every year forever. If we want to earn 10 percent on our money. How much would we pay for this asset?

14 Growth perpetuities Assuming, the payment on a growth perpetuity grows at the rate of g:

15 The present value of a growth perpetuity is:

16 Annuities An annuity is a constant payment of C dollars every period until t=T.

17 The present value of an annuity is:

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19 =FV(0.08,10,-1000,0)

20 =PV(0.10,15,1500,10000)

21 =PMT(0.10,15,-23579,0)

22 =IRR(A2:E2)

23 =NPV(0.12,A2:E2)-1523=166.14

24 Example: How much would your annual end-of-year payments have to be on a $12,000 loan with a 15% interest rate that must be repaid in three years?

25 Exercise: Dick Tracy just turned 40. He has decided that he would like to retire when he is 65. He thinks that he will need $1,500,000 in his special retirement account at age 65 to maintain his current lifestyle. For the next 15 years he can afford to put $12,000 per year into the account. At age 55 he will need to withdraw $40,000 to purchase membership in the local country club. If his retirement account earns 11% compounded annually, how much will Dick need to deposit into it each year for the last ten years of his career to attain the $1,500,000 goal?