EFN406 Lecture 02 2010.ppt

1、2010EFN406 Managerial FinanceSlide 1Financial Mathematics2010EFN406 Managerial FinanceSlide 22010EFN406 Managerial FinanceSlide 3 Time value of money Opportunity cost of funds Simple interest v Compound Interest Present value - Future value Annuities - Perpetuities Effective Interest Rate Growth Ann

2、ual Equivalents Repayment Schedules2010EFN406 Managerial FinanceSlide 40123Cash Flow Map0 represents now - the beginning of the first period1 represents the end of the first period42010EFN406 Managerial FinanceSlide 5FVIF(in) = future value interest factor for t periods at interest rate i. = (1+i)nP

3、VIF(in) = present value interest factor for t periods at interest rate i. = (1+i)-nFVIFA(in) = future value interest factor for annuity of t periods (payments) at interest rate i.your text uses S (n,i) = (1+i)n -1/iPVIFA(in) = present value interest factor for annuity of t periods at interest rate i

4、. your text uses A (n,i) = 1- (1+i)-n/i2010EFN406 Managerial FinanceSlide 6If I invest $1000 (P) for 2 years (t) at 10% pa (i) compounded semi-annually what will it accumulate to?44(1)1000(1 .05)1000(1.05)1000(1.2155)1215.5tFPi012PFFuture ValueAccumulated ValueCompounding2010EFN406 Managerial Financ

5、eSlide 710%20%30%01001001001110120130212114416931331732204146207286516124937161772994837195358627821443081692365161060102596191379The Power of CompoundingWhat a $100 will grow to after 10 years2010EFN406 Managerial FinanceSlide 8What is the present value (P) of $500 (F)to be received at the end of f

6、ive years (t), i = 10% pa P = F/ (1 + i)t or F(1 + i)-t Previous equation rearranged Substituting, P = 500*(1.1)-5 P = 500 x .6209 = $310.45012P = ?F = 500345Present ValueDiscounting“Now” dollars2010EFN406 Managerial FinanceSlide 9Reading a TablePresent Value = (1+r)-n8%9%10%11%12%10.92590.91740.909

7、10.90090.892920.85730.84170.82640.81160.797230.79380.77220.75130.73120.711840.73500.70840.68300.65870.635550.68060.64990.62090.59350.567460.63020.59630.56450.53460.506670.58350.54700.51320.48170.452380.54030.50190.46650.43390.403990.50020.46040.42410.39090.3606100.46320.42240.38550.35220.32202010EFN

8、406 Managerial FinanceSlide 10An investment of $100 amounts to $200 in seven years. What is the compound rate of interest?In this Example we know FV, PV, and the number of periods.Solve for i.FV = PV (1+i)n 200 = 100 (1 + i)7therefore (1+i)7 = 2 - now take 7th root of both sides(1+i)71/7 = 21/71+ i

9、= 1.10409 i = .10409 or 10.409 %to get seventh root use yx button, x = 1/7Or use the functionx y2010EFN406 Managerial FinanceSlide 11 i = (1 + jm/m )m - 1 Where i = effective annual rate jm = nominal interest rate m = frequency of compounding2010EFN406 Managerial FinanceSlide 12- What is the effecti

10、ve annual rate where j4=12%pa ?i = ( 1 + .12/4 )4 - 1= (1.03)4 - 1= 1.12551 - 1= .12551 0r 12.551%2010EFN406 Managerial FinanceSlide 13Continuous Compounding112.00%212.36%412.55%612.62%812.65%912.66%1212.68%2012.71%5012.73%10012.74%Continuous12.75%2010EFN406 Managerial FinanceSlide 14The Limit11.80%

11、12.00%12.20%12.40%12.60%12.80%020406080100120Compounding PeriodsInterest Rate2010EFN406 Managerial FinanceSlide 15 Common sense multiple cash flows - find the PV (or FV) of each cash flow and then take the sum, eg for two cash flows. PV = CF1/(1+r) + CF2/(1+r)2 FV2 = CF1(1+r)1 + CF2012CF2CF12010EFN4

12、06 Managerial FinanceSlide 16 What is the present value of $100 to be received at the end of year one and $200 at the end of year two, if the interest rate is 10% ? PV = 100/(1.1) + 200/(1.1)2 = $256.192010EFN406 Managerial FinanceSlide 17012P20010019.2561.12001.11001200110022iiP2010EFN406 Manageria

13、l FinanceSlide 18 An annuity is a series of fixed payments at equal intervals of time (eg pension, personal loans, debenture interest payments and home loan repayments) If payments are made at the end of the period we have a normal annuity (or ordinary). If payments are made at the beginning of the

14、period (interval) we have an annuity due (eg lease payments). A deferred annuity is one which begins sometime in the future.2010EFN406 Managerial FinanceSlide 19012345012345012345100100100100100100100100100NormalDueDeferred2010EFN406 Managerial FinanceSlide 2010001000 1000100012340FV2341000 1000(1.0

15、5) 1000(1.05)1000(1.05)FV 2010EFN406 Managerial FinanceSlide 21 asking for the future value of an normal annuity. FVIFA(i,n) = (1+i)n - 1/i (Table 3) i = .05, n = 4, annuity = $1000 FV = A x FVIFA(i,n) or A x (1+i)n - 1/i FV4 = 1000 x FVIFA(i,n) 1000 x 4.3101 = 4310.10 Note the FV of Annuity formula

16、 gives the value at the last cash flow2010EFN406 Managerial FinanceSlide 22Use the same example find the present value10001000 1000100012340PV2341 0 0 01 0 0 01 0 0 01 0 0 01 .0 51 .0 51 .0 51 .0 5P V2010EFN406 Managerial FinanceSlide 23 Here you are being asked for the present value of a normal annuity PVIFA(i,n) = 1- (1+i)-n /i (Table 4) PV = A x PVIFA(i,n) = 1000 x 3.546 = $3546 Note all PV formulas give the value at the beginning of the period of the first cash flow this relationship can be