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 Annual
2、 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)nPVIF(in)=p
3、resent 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.your text uses A(n,i)=
4、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 FinanceSlide 710%20%30%01001001001110120130
5、212114416931331732204146207286516124937161772994837195358627821443081692365161060102596191379The 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 five years(t),i=10%pa P=F/(1+i)t or F(1+i)-
6、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%10.92590.91740.90910.90090.892920.85730.84170.82640.81160.797230.79380.77220.75130.731
7、20.711840.73500.70840.68300.65870.635550.68060.64990.62090.59350.567460.63020.59630.56450.53460.506670.58350.54700.51320.48170.452380.54030.50190.46650.43390.403990.50020.46040.42410.39090.3606100.46320.42240.38550.35220.32202010EFN406 Managerial FinanceSlide 10An investment of$100 amounts to$200 in
8、 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=1.10409 i=.10409 or 10.409%to get seventh root use yx button,x=1/7Or use the functionxy
9、2010EFN406 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 effective 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 Financ
10、eSlide 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%12.00%12.20%12.40%12.60%12.80%020406080100120Compounding PeriodsInterest Rate2010EFN406 Managerial FinanceSlide 15 Common sens
11、e 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+CF2012CF2CF12010EFN406 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 i
12、nterest rate is 10%?PV=100/(1.1)+200/(1.1)2=$256.192010EFN406 Managerial FinanceSlide 17012P20010019.2561.12001.11001200110022iiP2010EFN406 Managerial FinanceSlide 18 An annuity is a series of fixed payments at equal intervals of time(eg pension,personal loans,debenture interest payments and home lo
13、an 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 period(interval)we have an annuity due(eg lease payments).A deferred annuity is one which begins sometime in the future.2010EFN406 Managerial FinanceSlide 19
14、012345012345012345100100100100100100100100100NormalDueDeferred2010EFN406 Managerial FinanceSlide 2010001000 1000100012340FV2341000 1000(1.05)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=
15、$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 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
16、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 us
17、ed in reverse to find an annual payment2010EFN406 Managerial FinanceSlide 24 future value of annuity equation 3.28 FVIFA(i,n)your text uses S(n,i)present value of annuity equation 3.19 PVIFA(i,n)your text uses A(n,i)32)05,.4()05.1(1000)05.1(1000)05.1(10001000S432)05,.4(05.1100005.1100005.1100005.110
18、00A2010EFN406 Managerial FinanceSlide 25ExerciseWhat is PV of series of 3 payments of$100?1.First payment end of year one2.First payment end of year three3.First payment nowi=10%2010EFN406 Managerial FinanceSlide 26 PV=A x PVIFA(i,n)Only unknown is A the repayment rearranging A=PV/PVIFA(i,n)A=$10000
19、/6.1446 A=$1627.44 Method also used to calculate annual equivalents2010EFN406 Managerial FinanceSlide 27 A perpetuity is an annuity to infinity,ie a series of equal payments at equal intervals forever PV=CF/i D/k or I/i Where CF is the perpetual cash flow What is the value of$1000 to be received at
20、the end of each year,forever at 10%?PV=1000/.1=$100002010EFN406 Managerial FinanceSlide 28012345DDDDD D=$1000P=D/r =1000/.1=100002010EFN406 Managerial FinanceSlide 29Where:g=constant growth rate of cash flows to infinityr=(cost of capital)g for the model to workCF1=cash flow at end of first time per
21、iod=CF0(1+g)(1grCFPo2010EFN406 Managerial FinanceSlide 30 What is the value of a share which has just paid a dividend(D0)of 50 cents and has dividends expected to grow at 5%pa(g)forever,given a required rate of 15%(r)?Po=D1/(r-g)=.5(1.05)/(.15-.05)=$5.25 Note Do,the current dividend is given but the
22、 equation requires D1.2010EFN406 Managerial FinanceSlide 31012345D1D2D3D4D5 25.51.525.05.15.)05.1(5.10grDPD1.50(1.05)0.525D2.50(1.05)20.551D3.50(1.05)30.579D4.50(1.05)40.608D5.50(1.05)50.6382010EFN406 Managerial FinanceSlide 32 learn how to calculate annual equivalents learn how to decompose repayme
23、nts into principal and interest components,see PBEHP Section 3.7 learn how to do all of these problems on a calculator(will be needed for the exam)Understand continuous compounding(continuous interest rate)Prepare a formula sheet2010EFN406 Managerial FinanceSlide 33 simple versus compound nominal ve
24、rsus effective coupon versus market yield actual(nominal)versus real discrete versus continuous fixed versus variable2010EFN406 Managerial FinanceSlide 34 Chapter Ten2010EFN406 Managerial FinanceSlide 35At what price can I issue a ninety day bank bill with a face value of$100000 if the bill rate is
25、10%pa?97594365901.1100000)1(xiFVPPF=10000090 days2010EFN406 Managerial FinanceSlide 36A vendor offers you goods on credit with the following terms:2%10;net 30What is the nominal annual percentage opportunity cost of not taking the discount and paying the account on day 30?37.24%or 3724.2036598236510
26、0Rate Nominalxtxddd=discountt=number of days after the discount day2010EFN406 Managerial FinanceSlide 37 Section 22.8.4 of PBEHP looks at the same issue but calculates and effective annual rate as opposed to a nominal annual rate.44.6%or 446.1982120365AER2010EFN406 Managerial FinanceSlide 38 What ar
27、e the annual repayments on a loan of$10000 loan over four years at 10%pa compounded annually?R=10000/PVIFA(i,n)R=10000/3.1698=$3155 pa.Prepare a loan repayment schedule over the life of the loan Section 3.7 of text2010EFN406 Managerial FinanceSlide 39YearO/BI10%O/B+IRepayC/B110000100011000(3,155)7,8
28、4527,8457858630(3,155)5,47535,4755486023(3,155)2,86842,8682873155(3,155)0O/B=opening balance:C/B=closing balance:I=interest in dollars:R=repayment.Note:O/B+I=balance just before the repayment is made2010EFN406 Managerial FinanceSlide 40At 10%which is better 150 per year for four years or the followi
29、ng set of cash flows?$173.483.1698$549.89i)PVIFA(n,NPVEAV$549.891.11001.13001.12001.1100432NPV012341002003001002010EFN406 Managerial FinanceSlide 4101234100200300100i=.1173.48173.48173.48173.48See page 142-44,Sometimes referred to as Annual Equivalent AEThis is better than$150 per year for four years.
