1、1.Understand what is meant by the time value of money.2.Understand the relationship between present and future value.3.Describe how the interest rate can be used to adjust the value of cash flows both forward and backward to a single point in time.4.Calculate both the future and present value of:(a)
2、an amount invested today;(b)a stream of equal cash flows(an annuity);and(c)a stream of mixed cash flows.5.Distinguish between an“ordinary annuity”and an“annuity due.”6.Use interest factor tables and understand how they provide a shortcut to calculating present and future values.7.Use interest factor
3、 tables to find an unknown interest rate or growth rate when the number of time periods and future and present values are known.8.Build an“amortization schedule”for an installment-style loan.The Interest Rate Simple Interest Compound Interest Amortizing a LoanCompounding More Than Once per YearObvio
4、usly,.You already recognize that there is!Which would you prefer or?allows you the opportunity to postpone consumption and earn.Why is such an important element in your decision?Interest paid(earned)on any previous interest earned,as well as on the principal borrowed(lent).Interest paid(earned)on on
5、ly the original amount,or principal,borrowed(lent).SI=P0(i)(n)SI:Simple InterestP0:Deposit today(t=0)i:Interest Rate per Periodn:Number of Time PeriodsSI=P0(i)(n)=$1,000(0.07)(2)=Assume that you deposit$1,000 in an account earning 7%simple interest for 2 years.What is the accumulated interest at the
6、 end of the 2nd year?=P0+SI=$1,000+$140=is the value at some future time of a present amount of money,or a series of payments,evaluated at a given interest rate.What is the()of the deposit?The Present Value is simply the$1,000 you originally deposited.That is the value today!is the current value of
7、a future amount of money,or a series of payments,evaluated at a given interest rate.What is the()of the previous problem?050001000015000200001st Year 10thYear20thYear30thYearFuture Value of a Single$1,000 Deposit10%SimpleInterest7%CompoundInterest10%CompoundInterestFuture Value(U.S.Dollars)Assume th
8、at you deposit at a compound interest rate of 7%for.0 1 7%=(1+i)1=(1.07)=Compound InterestYou earned$70 interest on your$1,000 deposit over the first year.This is the same amount of interest you would earn under simple interest.=(1+i)1 =(1.07)=FV1(1+i)1=(1+i)(1+i)=(1.07)(1.07)=(1+i)2 =(1.07)2 =You e
9、arned an EXTRA in Year 2 with compound over simple interest.=P0(1+i)1=P0(1+i)2General Formula:=P0(1+i)n or =P0(i,n)etc.i,n is found on Table I at the end of the book.Period6%7%8%11.0601.0701.08021.1241.1451.16631.1911.2251.26041.2621.3111.36051.3381.4031.469=$1,000(7%,2)=$1,000(1.145)=Due to Roundin
10、gPeriod6%7%8%11.0601.0701.08021.1241.1451.16631.1911.2251.26041.2621.3111.36051.3381.4031.469Julie Miller wants to know how large her deposit of today will become at a compound annual interest rate of 10%for.0 1 2 3 4 10%Calculation based on Table I:=$10,000(10%,5)=$10,000(1.611)=Due to RoundingCalc
11、ulation based on general formula:=P0(1+i)n =$10,000(1+0.10)5=We will use the Quick!How long does it take to double$5,000 at a compound rate of 12%per year(approx.)?Approx.Years to Double=/i%/12%=Actual Time is 6.12 YearsQuick!How long does it take to double$5,000 at a compound rate of 12%per year(ap
12、prox.)?Assume that you need in Lets examine the process to determine how much you need to deposit today at a discount rate of 7%compounded annually.0 1 7%PV1 =/(1+i)2=/(1.07)2 =/(1+i)2=0 1 7%=/(1+i)1=/(1+i)2General Formula:=/(1+i)n or =(i,n)etc.i,n is found on Table II at the end of the book.Period
13、6%7%8%1 0.943 0.935 0.926 2 0.890 0.873 0.857 3 0.840 0.816 0.794 4 0.792 0.763 0.735 5 0.747 0.713 0.681 =(PVIF7%,2)=(.873)=Due to RoundingPeriod 6%7%8%1 0.943 0.935 0.926 2 0.890 0.873 0.857 3 0.840 0.816 0.794 4 0.792 0.763 0.735 5 0.747 0.713 0.681 Julie Miller wants to know how large of a depos
14、it to make so that the money will grow to in at a discount rate of 10%.0 1 2 3 4 10%Calculation based on general formula:=/(1+i)n =/(1+0.10)5=Calculation based on Table I:=(10%,5)=(0.621)=Due to Rounding:Payments or receipts occur at the end of each period.:Payments or receipts occur at the beginnin
15、g of each period.represents a series of equal payments(or receipts)occurring over a specified number of equidistant periods.Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings0 1 2 3$100$100$100(Ordinary Annuity)ofPeriod 1 ofPeriod 2Today Cash Flows Each 1
16、 Period Apart ofPeriod 30 1 2 3$100$100$100(Annuity Due)ofPeriod 1 ofPeriod 2Today Cash Flows Each 1 Period Apart ofPeriod 3=R(1+i)n-1+R(1+i)n-2+.+R(1+i)1+R(1+i)0 R R R0 1 2 n+1R=Periodic Cash FlowCash flows occur at the end of the periodi%.=$1,000(1.07)2+$1,000(1.07)1+$1,000(1.07)0 =$1,145+$1,070+$
17、1,000 =$1,000$1,000$1,0000 1 2 47%$1,070$1,145Cash flows occur at the end of the periodThe future value of an ordinary annuity can be viewed as occurring at the of the last cash flow period,whereas the future value of an annuity due can be viewed as occurring at the of the last cash flow period.=R(F
18、VIFAi%,n)=$1,000(FVIFA7%,3)=$1,000(3.215)=Period6%7%8%11.0001.0001.00022.0602.0702.08033.1843.2153.24644.3754.4404.50655.6375.7515.867 =R(1+i)n+R(1+i)n-1+.+R(1+i)2+R(1+i)1 =(1+i)R R R R R0 1 2 3 ni%.Cash flows occur at the beginning of the period=$1,000(1.07)3+$1,000(1.07)2+$1,000(1.07)1 =$1,225+$1,
19、145+$1,070 =$1,000$1,000$1,000$1,0700 1 2 47%$1,225$1,145Cash flows occur at the beginning of the period=R(FVIFAi%,n)(1+i)=$1,000(FVIFA7%,3)(1.07)=$1,000(3.215)(1.07)=Period6%7%8%11.0001.0001.00022.0602.0702.08033.1843.2153.24644.3754.4404.50655.6375.7515.867=R/(1+i)1+R/(1+i)2 +.+R/(1+i)n R R R0 1 2
20、 n+1R=Periodic Cash Flowi%.Cash flows occur at the end of the period=$1,000/(1.07)1+$1,000/(1.07)2+$1,000/(1.07)3 =$934.58+$873.44+$816.30 =$1,000$1,000$1,0000 1 2 47%$934.58$873.44$816.30Cash flows occur at the end of the periodThe present value of an ordinary annuity can be viewed as occurring at
21、the of the first cash flow period,whereas the future value of an annuity due can be viewed as occurring at the of the first cash flow period.=R(PVIFAi%,n)=$1,000(PVIFA7%,3)=$1,000(2.624)=Period6%7%8%10.9430.9350.92621.8331.8081.78332.6732.6242.57743.4653.3873.31254.2124.1003.993 =R/(1+i)0+R/(1+i)1+.
22、+R/(1+i)n1 =(1+i)R R R R0 1 2 nR:Periodic Cash Flowi%.Cash flows occur at the beginning of the period=$1,000/(1.07)0+$1,000/(1.07)1+$1,000/(1.07)2 =$1,000.00$1,000$1,0000 1 2 4=7%$934.58$873.44Cash flows occur at the beginning of the period =R(PVIFAi%,n)(1+i)=$1,000(PVIFA7%,3)(1.07)=$1,000(2.624)(1.
23、07)=Period6%7%8%10.9430.9350.92621.8331.8081.78332.6732.6242.57743.4653.3873.31254.2124.1003.9931.Read problem thoroughly2.Create a time line3.Put cash flows and arrows on time line4.Determine if it is a PV or FV problem5.Determine if solution involves a single CF,annuity stream(s),or mixed flow6.So
24、lve the problem7.Check with financial calculator(optional)Julie Miller will receive the set of cash flows below.What is the at a discount rate of.0 1 2 3 4 0 1 2 3 4 10%0 1 2 3 4 10%$600(PVIFA10%,2)=$600(1.736)=$1,041.60$400(PVIFA10%,2)(PVIF10%,2)=$400(1.736)(0.826)=$573.57$100(PVIF10%,5)=$100(0.621
25、)=$62.10General Formula:FVn=(1+i/m)mnn:Number of Yearsm:Compounding Periods per Yeari:Annual Interest RateFVn,m:FV at the end of Year n:PV of the Cash Flow todayJulie Miller has to invest for 2 Years at an annual interest rate of 12%.Annual FV2=(1+0.12/1)(1)(2)=Semi FV2=(1+0.12/2)(2)(2)=Qrtly FV2=(1
26、+0.12/4)(4)(2)=Monthly FV2=(1+0.12/12)(12)(2)=Daily FV2=(1+0.12/365)(365)(2)=Effective Annual Interest RateThe actual rate of interest earned(paid)after adjusting the nominal rate for factors such as the number of compounding periods per year.(1+i/m )m 1Basket Wonders(BW)has a$1,000 CD at the bank.T
27、he interest rate is 6%compounded quarterly for 1 year.What is the Effective Annual Interest Rate()?=(1+0.06/4)4 1=1.0614-1=0.0614 or 1.Calculate the payment per period.2.Determine the interest in Period t.(Loan Balance at t 1)x(i%/m)3.Computein Period t.(Payment-Interest from Step 2)4.Determine endi
28、ng balance in Period t.(Balance-from Step 3)5.Start again at Step 2 and repeat.Julie Miller is borrowing at a compound annual interest rate of 12%.Amortize the loan if annual payments are made for 5 years.Step 1:Payment =R(PVIFA i%,n)=R(PVIFA 12%,5)=R(3.605)=/3.605=End ofYearPaymentInterestPrincipal
29、EndingBalance0$10,0001$2,774$1,200$1,5748,42622,7741,0111,7636,66332,7748001,9744,68942,7745632,2112,47852,7752972,4780$13,871$3,871$10,000Last Payment Slightly Higher Due to Rounding The quantity of outstanding debt may be used in financing the day-to-day activities of the firm.Interest expenses may reduce taxable income of the firm.
