Time Value of Money

Size: px

Start display at page:

Download "Time Value of Money"

창민 서문
6 years ago
Views:

1 CHAPTER 5 Time Value of Money

2 Overview Future value Present value Annuities Perpetuity Uneven cash flow Interest rates Amortization

3 시간가치의위력 Manhattan 면적 : mile 2 (59.47km 2 =180 만평 ) 가격 : 에 450 Park Avenue 의 9,135 평이 $510 million 에매각 ($1,589/feet²= $17,224/m²=$56,000/ 평 ) => $102 billion 역사 : 네덜란드총독 Peter Minuite 이인디언들에게 60 길드 (=$1,000) 에매입 네덜란드총독과인디언, 둘중누가투자의사결정을더잘했나?

4 Time lines I% CF 0 CF 1 CF 2 CF 3 Show the timing of cash flows. Tick marks 각기간말을의미. 즉 Time 0 는오늘 ; Time 1 은첫기간말 ( 년, 월등등 ) or 두번째기간의초를의미.

5 Terminology PV: Present value FV: Future value CF: Cash flow r(i/yr or I): Interest rate Int: Dollar of interest PMT: Payment N: Number of period

6 Future Value Finding the FV of a cash flow or series of cash flows is called compounding. FV can be solved by using the step-by-step (Formula), financial calculator, and table methods. What is the future value (FV) of an initial $100 after 3 years, if r = 10%? % 100 FV =?

7 Solving for FV: The Formula After 1 year: FV 1 = PV (1 + r) = $100 (1.10) = $ After 2 years: FV 2 = PV (1 + r) 2 = $100 (1.10) 2 =$ After 3 years: FV 3 = PV (1 + r) 3 = $100 (1.10) 3 =$ After N years (general case): FV N = PV (1 + r) N = PV (FVIF r,n )

8 이자율수준과미래가치

9 Present Value Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding). The PV shows the value of cash flows in terms of today s purchasing power. What is the present value (PV) of $100 due in 3 years, if r = 10%? % PV =? 100

10 Solving for PV: The formula Solve the general FV equation for PV: PV = FV N / (1 + r) N = FV N (PVIF r,n ) PV = FV 3 / (1 + r) 3 = $100 / (1.10) 3 = $75.13

11 이자율 ( 할인율 ) 과현재가치

12 Interest Rate Solves the general FV equation for r. FV C (1 r) 0 What interest rate would cause $100 to grow to $ in 3 years? = 100(1 + r) 3 T

13 Number of years Solves the general FV equation for T. FV C (1 r) 0 If sales grow at 20% per year, how long before sales double? 2 = 1(1 +0.2) x T

Solving by Spreadsheet Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv)

14 Solving by Spreadsheet Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv) The formula icon is very useful when you can t remember the exact formula Click on the Excel icon to open a spreadsheet containing four different examples.

15 Annuity A series of equal payments at fixed intervals for a specified number of periods Ordinary Annuity i% Annuity Due PMT PMT PMT i% PMT PMT PMT

16 Solving for FVA: The formula 3-year ordinary annuity of $100 at 10% FVA = 100(1.1) (1.1) (1.1) 0 = 331 FVA = PMT(1 + r) N-1 +PMT(1 + r) N-2 + PMT(1 + r) N-3 + PMT(1 + r) N PMT(1 + r) 0 = (1 I ) PMT I N 1 = PMT(FVIFA r,n )

17 Solving for PVA: The formula 3-year ordinary annuity of $100 at 10% PVA = 100/(1.1) /(1.1) /(1.1) 3 = PVA = PMT/(1 + r) 1 +PMT/(1 + r) 2 + PMT/(1 + r) 3 + PMT/(1 + r) PMT/(1 + r) n = n (1 r) -1 PMT n I(1 r) = PMT(PVIFA r,n )

18 Annuity Due: Solving for FVA and PVA Solving for FVA Now, $100 payments occur at the beginning of each period. FVA due = FVA ord (1+r) = $331(1.10) = $ Solving for PVA PVA due = PVA ord (1+r) = $248.69(1.10) = $273.55

19 Perpetuity A stream of equal payments at fixed intervals expected to continue forever What is the present value of a perpetuity $100 at 10%? PV = PMT / r = $100/0.1 = $1,000.

20 Uneven cash flow 0 10% = PV

21 Compounding Definition The arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added Semiannually, quarterly, monthly, daily FV n PV (1 r M NOM N ) M N: number of years, M: periods per year

22 Solving for Compounding FVA % What s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually? Payments occur annually, but compounding occurs every 6 months.

23 재미나는예 하루한잔씩마시는 4 천원짜리카페베네아메리카노커피값을 30 년간복리로저축한다면? 올림픽에서금메달을획득하면월 100 만원씩연금을받는다. 이연금의현재가치는?

24 Loan amortization ( 할부 ) Amortization: A loan that is repaid in equal payments over its life Amortization use: home mortgages, auto loans, business loans, retirement plans, etc. Financial calculators and spreadsheets are great for setting up amortization tables.

25 Amortization cont. 10% 1000 PMT PMT PMT EXAMPLE: Construct an amortization schedule (table) for a $1,000, 10% annual rate loan with 3 equal payments PV PMT1 (1 0.1) 1 PMT2 (1 0.1) 2 PMT3 (1 0.1) 3 3 t 1 PMTt (1 0.1) t

26 Step 1: the annual payment All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan. INPUTS OUTPUT N I/YR PV PMT FV

27 Step 2: the interest and the principal paid in Year 1 The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate. INT t = Beg bal t (r) INT 1 = $1,000 (0.10) = $100 If a payment of $ was made at the end of the first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid. PRIN = PMT INT = $ $100 = $302.11

28 Step 3: the ending balance after Year 1 To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance. END BAL = BEG BAL PRIN = $1,000 - $ = $697.89

29 Constructing an amortization table: Repeat steps 1 3 until end of loan Year BEG BAL PMT INT PRIN END BAL 1 $1,000 $402 $100 $302 $ TOTAL 1, ,000 - Interest paid declines with each payment as the balance declines.

30 Illustrating an amortized payment: Where does the money go? $ Interest Principal Payments Constant payments. Declining interest payments. Declining balance.

31 실제생활에서적용되는예제 3,000 만원짜리차구입시 1,700 만원을선수금으로지급하고나머지를할부로구입하려고한다. 36 개월, 연 9.5% 금리의할부조건이라면월지급해야하는할부액은? 당신은현재 35 세이고연간소득이 5,000 만원이다. 앞으로 25 년동안일을하고 60 세에은퇴할예정이며 90 세까지건강하게살것으로예상한다. 은퇴후에소비하는수준이현재소비하는수준과동일하게되려면연간얼마를저축하고연간얼마를소비해야하는가?

레이아웃 1

레이아웃 1 i g d e d mod, t d e d e d mod, t e,0 e, n s,0 e,n e,0 Division of Workers' Compensation (2009). Iowa workers' compensation manual. Gamber, E. N. & Sorensen, R. L. (1994). Are net discount rates stationary?: