Presentation on theme: "Future value Present value Rates of return Amortization Annuities, AND"— Presentation transcript:
1CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics Future valuePresent valueRates of returnAmortizationAnnuities, ANDMany ExamplesSome of later examples change the interest rate from 10% to 12%. I did not do this, since, I then would have to change the excel spreadsheet.
2MINICASE 2 SIMPLE?p. 88Also note financial mathematics problems at end of TAB &Notes on Excel and LOTUS.
3MINICASE 2Why is financial mathematics (time value of money) so important in financial analysis?
4a.Time lines show timing of cash flows. ALWAYS A GOOD IDEA TO DRAW A TIME LINE.123i%CF0CF1CF2CF3Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1, or the beginning of Period 2; and so on.
5Time line for a $100 lump sum due at the end of Year 2. 12 Yearsi%100
6Time line for an ordinary annuity of $100 for 3 years. 123i%100100100
7Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 123i%-501007550
8b(1) What’s the FV of an initial $100 after 3 years if i = 10%? 12310%100FV = ?Finding FVs is compounding.
9b(1) What’s the FV of an initial $100 after 3 years if i = 10%? 12310%100FV = ?110?Finding FVs is compounding.
24Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) Spreadsheet SolutionUse the NPER function: see spreadsheet.= NPER(Rate, Pmt, PV, FV)= NPER(0.20, 0, -1, 2) = 3.8Correction
25ADDITIONAL QUESTIONA FARMER CAN SPEND $60/ACRE TO PLANT PINE TREES ON SOME MARGINAL LAND. THE EXPECTED REAL RATE OF RETURN IS 4%, AND THE EXPECTED INFLATION RATE IS 6%. WHAT IS THE EXPECTED VALUE OF THE TIMBER AFTER 20 YEARS?
26ADDITIONAL QUESTIONBill Veeck once bought the Chicago White Sox for $10 million and then sold it five years later for $20 million. In short, he doubled his money in five years. What compound rate of return did Veeck earn on his investment?
27n * krequired to double = 72 In this case, RULE OF 72A good approximation of the interest rate--or number of years--required to double your money.n * krequired to double = 72In this case,5 * krequired to double = 72k = 14.4Correct answer was 14.87, so for ball-park approximation, use Rule of 72.
28ADDITIONAL QUESTIONJohn Jacob Astor bought an acre of land in Eastside Manhattan in 1790 for $58. If average interest rate is 5%, did he make a good deal?
29d. What’s the difference between an ordinary annuity and an annuity due?
30d. What’s the difference between an ordinary annuity and an annuity due? 123i%PMTPMTPMTAnnuity Due123i%PMTPMTPMT36
31HINTANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT.
32e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%? 12310%100100100FV =
33e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%? 12310%100100100110121FV = 331
34FV Annuity FormulaThe future value of an annuity with n periods and an interest rate of i can be found with the following formula:
35Financial Calculator Formula for AnnuitiesFinancial calculators solve this equation:There are 5 variables. If 4 are known, the calculator will solve for the 5th.Correct but confusing!
36Financial Calculator Solution INPUTS331.00NI/YRPVPMTFVOUTPUTHave payments but no lump sum PV, so enter 0 for present value.
37Spreadsheet SolutionUse the FV function: see spreadsheet.= FV(Rate, Nper, Pmt, Pv)= FV(0.10, 3, -100, 0) =
38e(2). What’s the PV of this ordinary annuity? 12310%100100100_____= PV
39What’s the PV of this ordinary annuity? 12310%10010010090.9182.6475.13= PV
40Have payments but no lump sum FV, so enter 0 for future value. INPUTSNI/YRPVPMTFVOUTPUTHave payments but no lump sum FV,so enter 0 for future value.
41Spreadsheet SolutionUse the PV function: see spreadsheet.= PV(Rate, Nper, Pmt, Fv)= PV(0.10, 3, 100, 0) =
42e(3). Find the FV and PV if the annuity were an annuity due. 12310%100100100
43Could, on the 12C, switch from “End” to “Begin”; i.e. f Begin. Then enter variables to find PVA3 = $INPUTSNI/YRPVPMTFVOUTPUTThen enter PV = 0 and press FV to findFV = $
44Another HINTFV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE FV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k) (slide 30)PV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE PV OF A REGULAR ANNUITY OF n PERIODS TIMES (1+k)
45HINT, illlustratedThe PV of this regular annuity wasMultiply this by ( ), and you get: , the PV of the annuity due.This avoids the necessity of having to switch from end to begin.
46PV and FV of Annuity Duevs. Ordinary AnnuityPV of annuity due:= (PV of ordinary annuity) (1+i)= (248.69) ( ) =FV of annuity due:= (FV of ordinary annuity) (1+i)= (331.00) ( ) = 364.1
47Switch from “End” to “Begin”. Then enter variables to find PVA3 = $INPUTSNI/YRPVPMTFVOUTPUTThen enter PV = 0 and press FV to findFV = $
48Excel Function for Annuities Due Change the formula to:=PV(10%,3,-100,0,1)The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:=FV(10%,3,-100,0,1)
58h.Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why?
59h.Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why?LARGER! If compounding is more frequent than once a year--forexample, semiannually, quarterly,or daily--interest is earned on interest more often.
61Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.Examples:8% quarterly: iPer = 8/4 = 2%.8% daily (365): iPer = 8/365 = %.
62Effective Annual Rate (EAR = EFF%): The annual rate which causes PV to grow to the same FV as under multiperiod compounding.
63An investment with monthly compounding is different from one with quarterly compounding. Must put on EAR% basis to compare rates of return.Banks say “interest paid daily.” Same as compounded daily.
64h(3).How do we find EAR for a nominal rate of 10%, compounded semiannually?Or use a financial calculator (not 12C)
68For multiple years, n (1 + EAR) = (1 + Knom/m)m (1 + EAR)n = (1 + Knom/m)mnTo multiply by a $ of dollars, PRINPRIN * (1 + EAR)n = PRIN * (1 + Knom/m)mn
69i. Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1.If m > 1, EFF% will always be greater than the nominal rate.
70When is each rate used?iNom:Written into contracts, quoted by banks and brokers. May be used in calculations or shown on time lines when compounding is annual.OR USE EAR!
71Can always use EAR! iPer: Used in calculations, shown on time lines. If iNom has annual compounding,then iPer = iNom/1 = iNom.Can always use EAR!
72EAR = EFF%:. Used to compare. returns on investments. with different EAR = EFF%: Used to compare returns on investments with different compounding patterns.Also used for calculations if dealing with annuities where paymentsdon’t match interest compounding periods.
73FV of $100 after 3 years under 10% semiannual compounding. Quarterly FV of $100 after 3 years under 10% semiannual compounding? Quarterly? Daily?mniFV=PV+Nomnm2x30.10FV=$1001+3S2= $100(1.05)6 = $FV3Q = $100(1.025)12 = $
74ALTERNATE SOLUTION USING EAR FOR SEMIANNUAL COMPOUNDING, EAR = 10.25%FOR 3 YEARS: 100*(1.1025)3 = $134.01FOR Quarterly COMPOUNDING and 3 years:100*(1.1038)3 = $134.49
75j(3). What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually?123456 6-mos.periods5%100100100
76Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.
771st Method: Compound Each CF 1234565%100100100.00110.25121.55331.80FVA3 = 100(1.05) (1.05)=
78Could you find FV with a financial calculator? 2nd Method: Treat as an AnnuityI.E. USE EARCould you find FV with a financial calculator?Yes, by following these steps:a. Find the EAR for the quoted rate:EAR = ( ) - 1 = 10.25%.0.1022
79Or, to find EAR with a 17 OR 19b Calculator: NOM% = 10P/YR = 2EFF% = 10.25
80b. The cash flow stream is an annual annuity whose EFF% = 10.25%. INPUTSNI/YRPVPMTFVOUTPUT331.80
81j(2) What’s the PV of this stream? 1235%100100100
82What’s the PV of this stream? 1235%10010010090.7082.2774.62247.59
89Step 2: Find interest charge for Year 1. INTt = Beg balt (i)INT1 = 1,000(0.10) = $100.Step 3: Find repayment of principal inYear 1.Repmt = PMT - INT== $
90Step 4: Find ending balance after Year 1. End bal = Beg bal - Repmt= 1, = $Repeat these steps for Years 2 and 3to complete the amortization table.
91Interest declines. Tax implications. BEG PRIN ENDYR BAL PMT INT REDUCTION BAL1 $1,000 $402 $100 $302 $698TOT 1, ,000Interest declines. Tax implications.
9210% on loan outstanding, which is falling. $402.11Interest302.11Principal Payments123Level payments. Interest declines because outstanding balance declines. Lender earns10% on loan outstanding, which is falling.
93Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, etc. They are very important!Financial calculators (and spreadsheets) are great for setting up amortization tables.
95NEW PROBLEM:On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days).How much will you have on October 1, or after 9 months (273 days)? (Days given.)
96 iPer = 10.0% / 365 = 0.027397% per day. FV = $100 1.00027397 = 12273%-100FV=?273FV=$100273=$100=$Note: % in calculator, decimal in equation.
97Leave data in calculator. iPer = iNom/m= 10.0/365= % per day.INPUTS107.77NI/YRPVPMTFVOUTPUTEnter i in one step.Leave data in calculator.
98Now suppose you leave your money in the bank for 21 months, which is 1 Now suppose you leave your money in the bank for 21 months, which is 1.75 years or = 638 days.How much will be in your account at maturity?Answer: Override N = 273 with N = FV = $
104m*[(1 + EAR)(1/m) - 1] = knom Using the calculator, EAR = 10%, daily compounding. ENTER/X Yx365 MULTIPLY= %
105n. You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of % and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.Should you buy it?
114ADDITIONAL PROBLEM #2IT IS NOW JANUARY 1. YOU PLAN TO MAKE 5 DEPOSITS OF $100 EACH, ONE EVERY 6 MONTHS, WITH THE FIRST PAYMENT BEING MADE TODAY. IF THE BANK PAYS A NOMINAL INTEREST RATE OF 12 PERCENT, SEMIANNUAL COMPOUNDING, HOW MUCH WILL BE IN YOUR ACCOUNT AFTER 10 YEARS?
115ADDITIONAL PROBLEM #3IT IS NOW JANUARY 1, YOU ARE OFFERED A NOTE UNDER WHICH SOMEONE PROMISES TO MAKE 5 PAYMENTS OF $100 EACH, WITH THE FIRST PAYMENT ON JULY 1, 1997 AND SUBSEQUENT PAYMENTS ON EACH JULY 1 THEREAFTER THROUGH JULY 1, 2001, PLUS A FINAL PAYMENT OF $1000 TO BE MADE ON JANUARY 1, WITH A NOMINAL DISCOUNT RATE OF 10 PERCENT, QUARTERLY COMPOUNDING, WHAT IS THE PV OF THE NOTE?