Presentation on theme: "Chapter 5 The Time Value of Money"— Presentation transcript:
1 Chapter 5 The Time Value of Money Laurence Booth, Sean Cleary and Pamela Peterson Drake
2 5.2 Annuities and perpetuities 5.3 Nominal and effective rates Outline of the chapter5.1 Time is moneySimple versus compound interestFuture value of an amountPresent value of an amount5.2 Annuities and perpetuitiesOrdinary annuitiesAnnuity dueDeferred annuitiesPerpetuities5.3 Nominal and effective ratesAPREARSolving for the rate5.4 ApplicationsSavings plansLoans and mortgagesSaving for retirement
4 Simple interestSimple interest is interest that is paid only on the principal amount.Interest = rate × principal amount of loanNote: Some credit unions offer simple interest loansBooth Cleary Drake
5 Simple interest: example A 2-year loan of $1,000 at 6% simple interestAt the end of the first year, interest = 6% × $1,000 = $60At the end of the second year, interest = 6% × $1,000 = $60and loan repayment of $1,000Key: in the second year, only the principal earns interestBooth Cleary Drake
6 Compound interestCompound interest is interest paid on both the principal and any accumulated interest.Interest = rate × principal + accumulated interest
7 CompoundingCompounding is translating a present value into a future value, using compound interest.Future value = Present value × Future value interest factorFuture value interest factor is also referred to as the compound factor.
8 Terminology and notation MeaningFuture valueFVValue at some specified future point in timePresent valuePVValue todayInterestiCompensation for the use of fundsNumber of periodsnNumber of periods between the present value and the future valueCompound factor(1 + i)nTranslates a present value into a future valueWe subscript FV and PV at times when we are dealing with complex situations and need to distinguish the timing of these values.Booth Cleary Drake
9 Compare: simple versus compound Suppose you deposit $5,000 in an account that pays 5% interest per year. What is the balance in the account at the end of four years if interest is:Simple interest?Compound interest?
12 Interest on interestHow much interest on interest? Interest on interest = FVcompound – FVsimple Interest on interest = $6, – 6, = $77.53
13 Comparison Year End of year balance Simple interest Compound interest $5,000.001$5,250.002$5,500.00$5,512.503$5,750.00$5,788.134$6,000.00$6,077.53Bottom line: The value grows faster with compound interest compared to simple interest.Booth Cleary Drake
14 Try it: Simple v. compound Suppose you are comparing two accounts:The Bank A account pays 5.5% simple interest.The Bank B account pays 5.4% compound interest.If you were to deposit $10,000 in each, what balance would you have in each bank at the end of five years?
15 Try it: Answer Bank A: $12,750.00 Bank B: $13,007.78 Booth Cleary Drake
16 A note about interestBecause compound interest is so common, assume that interest is compounded unless otherwise indicated.
17 Short-cutsExample:Consider $1,000 deposited for three years at 6% per year.
20 Input three known values, solve for the one unknown Known: PV, i , nUnknown: FVHP10BBAIIPlusHP12CTI83/841000 +/- PV3 N6 I/YRFV1000 CHS PV3 n6 i[APPS] [Finance][TVM Solver]N =3I%=6PV = -1000FV [Alpha] [Solve}
21 Short-cut: spreadsheet Microsoft Excel or Google Docs =FV(RATE,NPER,PMT,PV,TYPE)TYPE default is 0, end of period=FV(.06,3,0,-1000)orA16%23-10004=FV(A1,A2,0,A3)Type:0 = end of period1 = beginning of periodBasically, this is not relevant until we get to annuities.Booth Cleary Drake
23 Problem 1.1Suppose you deposit $2,000 in an account that pays 3.5% interest annually.How much will be in the account at the end of three years?How much of the account balance is interest on interest?Booth Cleary Drake
24 Problem 1.2If you invest $100 today in an account that pays 7% each year for four years and 3% each year for five years, how much will you have in the account at the end of the nine years?
26 DiscountingDiscounting is translating a future value into a present value.The discount factor is the inverse of the compound factor: 𝑖 𝑛To translate a future value into a present value, PV= FV 1+𝑖 𝑛
27 ExampleSuppose you have a goal of saving $100,000 three years from today. If your funds earn 4% per year, what lump-sum would you have to deposit today to meet your goal?
28 Example, continued Known values: Unknown: PV FV = $100,000 n = 3
34 Frequency of compounding If interest is compounded more than once per year, we need to make an adjustment in our calculation.The stated rate or nominal rate of interest is the annual percentage rate (APR).The rate per period depends on the frequency of compounding.
35 Discrete compounding: Adjustments Adjust the number of periods and the rate per period.Suppose the nominal rate is 10% and compounding is quarterly:The rate per period is 10% 4 = 2.5%The number of periods is number of years × 4
36 Continuous compounding: Adjustments The compound factor is eAPR x n.The discount factor is 1 eAPR x n .Suppose the nominal rate is 10%.For five years, the continuous compounding factor is e0.10 x 5 =The continuous compounding discount factor for five years is 1 ÷ e0.10 x 5 =
37 Try it: Frequency of compounding If you invest $1,000 in an investment that pays a nominal 5% per year, with interest compounded semi-annually, how much will you have at the end of 5 years?
40 Problem 2.1Suppose you set aside an amount today in an account that pays 5% interest per year, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?
41 Problem 2.2Suppose you set aside an amount today in an account that pays 5% interest per year, compounded quarterly, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?
42 Problem 2.3Suppose you set aside an amount today in an account that pays 5% interest per year, compounded continuously, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?
43 5.2 Annuities and Perpetuities 12345|CFPV?FV?5.2 Annuities and Perpetuities
44 What is an annuity? An annuity is a periodic cash flow. Same amount each periodRegular intervals of timeThe different types depend on the timing of the first cash flow.
45 Type of annuities Type First cash flow Examples Ordinary One period from todayMortgageAnnuity dueImmediatelyLottery paymentsRentDeferred annuityBeyond one period from todayRetirement savings
46 Time lines: 4-payment annuity 12345|OrdinaryPVCFFVAnnuity dueDeferred annuity
47 Key to valuing annuities The key to valuing annuities is to get the timing of the cash flows correct.When in doubt, draw a time line.
48 Example: PV of an annuity What is the present value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?1234|$4,000
49 Example: PV of an annuity The long way 1234|$4,000$3,773.583,559.993,358.48$10,692.05
50 Example: PV of an annuity In table form YearCash flowDiscount factorPresent value1$4,000.00$3,773.5823,559.9933,358.48$10,692.05PV = $4, × = $10,692.05
51 Example: PV of an annuity Formula short-cuts PV = $4,000 × 1 − PV = $4,000 × PV = $10,692.05
52 Example: PV of an annuity Calculator short cuts Given:PMT = $4,000i = 6%N = 3Solve for PV
53 Example: PV of an annuity Spreadsheet short-cuts =PV(RATE,NPER,PMT,FV,TYPE) =PV(.06,3,4000,0)Note: Type is important for annuitiesIf Type is left out, it is assumed a 00 is for an ordinary annuity1 is for an annuity dueNote: the default type is ordinary annuity, so this last term can be left off.Booth Cleary Drake
54 Example: FV of an annuity What is the future value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?1234|$4,000
55 Example: FV of an annuity The long way 1234|$4,000.004,240.004,494.40$12,734.40
56 Example: FV of an annuity In table form YearCash flowCompound factorFuture value1$4,000.001.1236$4,494.4021.06004,240.0031.00004,000.003.1836$12,734.40PV = $4, × = $12,734.40
57 Example: FV of an annuity Calculator short cuts Given:PMT = $4,000i = 6%N = 3Solve for FV
58 Example: FV of an annuity Spreadsheet short-cuts =FV(RATE,NPER,PMT,PV,type) =FV(.06,3,4000,0)
59 Annuity dueConsider a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow today.What is the present value of this annuity?What is the future value of this annuity?
60 The time line123|$4,000.00PV?FV?This is an annuity due
61 Present value of cash flow Valuing an annuity duePresent valueFuture valueYearEnd of year cash flowCompound factorPresent value of cash flow$4,000.0013,773.5823,559.99$11,333.57YearEnd of year cash flowFactorFuture value$4,000.00$4,764.0614,494.4024,240.00$13,498.46
62 Valuing an annuity due: Using calculators Present valueFuture valuePMT = 4000 N = 3 I = 6% BEG mode Solve for PVPMT = 4000 N = 3 I = 6% BEG mode Solve for FV
63 Valuing an annuity due: Using spreadsheets Present value =PV(RATE,NPER,PMT,FV,TYPE) =PV(0.06,3,4000,0,1) Future value
64 Any other way?There is one period difference between an ordinary annuity and an annuity due. Therefore: PVannuity due = PVordinary annuity × (1 + i) and FVannuity due = FVordinary annuity × (1 + i)
65 Valuing a deferred annuity A deferred annuity is an annuity that begins beyond one year from today.That means that it could begin 2, 3, 4, … years from today, so each problem is unique.
66 Valuing a deferred annuity 12345|CF4-payment ordinary annuity, then discount value one periodPV0←PV14-payment annuity due, then discount value two periods←PV2
67 Example: Deferred annuity What is the value today of a series of five cash flows of $6,000 each, with the first cash flow received four years from today, if the discount rate is 8%?12345678910|PV?CF
68 Example, cont.Using an ordinary annuity: PV3 = $23, PV0 = $19, Using an annuity due: PV4 = $25,872.76Discount 3 periods at 8%Discount 4 periods at 8%
73 Problem 3.1Which do you prefer if the appropriate discount rate is 6% per year:An annuity of $4,000 for four annual payments starting today.An annuity of $4,100 for four annual payments, starting one year from today.An annuity of $4,200 for four annual payments, starting two years from today.
74 5.3 Nominal and effective rates %𝐴𝑃𝑅=𝑖 ×𝑚𝑬𝑨𝑹= 𝟏+ 𝑨𝑷𝑹 𝒎 𝒎𝑬𝑨𝑹= 𝒆 𝑨𝑷𝑹 −𝟏i5.3 Nominal and effective rates
75 APR & EARThe annual percentage rate (APR) is the nominal or stated annual rate.The APR ignores compounding within a year.The APR understates the true, effective rate.The effective annual rate (EAR) incorporates the effect of compounding within a year.Note to instructors: the post-subprime mortgage rules confuse the APR with the EAR, so the term “APR” is used as both an effective rate and a nominal rate, depending on the part of the paperwork.Booth Cleary Drake
76 APR EAREAR = 1+ 𝐴𝑃𝑅 𝑚 𝑚 −1 Suppose interest is stated as 10% per years, compounded quarterly. EAR = −1= −1 EAR = %
77 EAR with continuous compounding EAR = 𝑒 𝐴𝑃𝑅 −1 Suppose interest is stated as 10% per years, compounded continuously. EAR = e 0.1 −1= −1 EAR = %
78 Frequency of compounding If interest is compounded more frequently than annually, then this is considered in compounding and discounting.There are two approachesAdjust the i and n; orCalculate the EAR and use this
79 Example: EAR & compounding Suppose you invest $2,000 in an investment that pays 5% per year, compounded quarterly. How much will you have at the end of 4 years?
87 Saving for retirementSuppose you estimate that you will need $60,000 per year in retirement. You plan to make your first retirement withdrawal in 40 years, and figure that you will need 30 years of cash flow in retirement. You plan to deposit funds for your retirement starting next year, depositing until the year before retirement. You estimate that you will earn 3% on your funds. How much do you need to deposit each year to satisfy your plans?
88 Deferred annuity time line 12345678…394041424379|DWD = Deposit (39 in total)W = Withdrawal (30 in total)
89 Deferred annuity time line 12345678…394041424379|WPV← Ordinary annuity↓Ordinary annuity →FVD
90 Two steps Step 1: Present value of ordinary annuity N = 30; i = 3%; PMT = $60,000PV39 = $1,176,026.48Step 2: Solve for payment in an ordinary annuityN = 39; i = 3%; FV = $1,176,026.48PMT = $16,280.74
91 What does this mean?If there are 39 annual deposits of $16, each and the account earns 3%, there will be enough to allow for 30 withdrawals of $60,000 each, starting 40 years from today.
94 Problem 1What is the future value of $2,000 invested for five years at 7% per year, with interest compounded annually?Booth Cleary Drake
95 Problem 2What is the value today of €10,000 promised in four years if the discount rate is 4%?Booth Cleary Drake
96 Problem 3What is the present value of a series of five end-of-year cash flows of $1,000 each if the discount rate is 4%?Booth Cleary Drake
97 Problem 4Suppose you plan to save $3,000 each year for ten years. If you earn 5% annual interest on your savings, how much more will you have at the end of ten years if you make your payments at the beginning of the year instead of the end of the year?Booth Cleary Drake
98 Problem 5Sue plans to deposit $5,000 in a savings account each year for thirty years, starting ten years from today. Yan plans to deposit $3,500 in a savings account each year for forty years, starting at the end of this year. If both Sue and Yan earn 3% on their savings, who will have the most saved at the end of forty years?Booth Cleary Drake
99 Problem 6Suppose you have two investment opportunities: Opportunity 1: APR of 12%, compounded monthly Opportunity 2: APR of 11.9%, compounded continuously Which opportunity provides the better return?Booth Cleary Drake
100 Problem 7If you can earn 5% per year, what would you have to deposit in an account today so that you have enough saved to allow withdrawals of $40,000 each year for twenty years, beginning thirty years from today?Booth Cleary Drake
101 Problem 8Suppose you deposit ¥50000 in an account that pays 4% interest, compounded continuously. How much will you have in the account at the end of ten years if you make no withdrawals?Booth Cleary Drake