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**CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics**

Future value Present value Rates of return Amortization Annuities, AND Many Examples Some 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.

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MINICASE 2 SIMPLE? p. 88 Also note financial mathematics problems at end of TAB & Notes on Excel and LOTUS.

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MINICASE 2 Why is financial mathematics (time value of money) so important in financial analysis?

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**a.Time lines show timing of cash flows.**

ALWAYS A GOOD IDEA TO DRAW A TIME LINE. 1 2 3 i% CF0 CF1 CF2 CF3 Tick 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.

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**Time line for a $100 lump sum due at the end of Year 2.**

1 2 Years i% 100

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**Time line for an ordinary annuity of $100 for 3 years.**

1 2 3 i% 100 100 100

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**Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.**

1 2 3 i% -50 100 75 50

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**b(1) What’s the FV of an initial $100 after 3 years if i = 10%?**

1 2 3 10% 100 FV = ? Finding FVs is compounding.

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**b(1) What’s the FV of an initial $100 after 3 years if i = 10%?**

1 2 3 10% 100 FV = ? 110 ? Finding FVs is compounding.

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After 1 year FV1 = PV + INT1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $ After 2 years FV2 = FV1(1 + i) = PV(1 + i)2 = $100(1.10)2 = $

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After 3 years FV3 = PV(1 + i)3 = 100(1.10)3 = $ In general, FVn = PV(1 + i)n.

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Four Ways to Find FVs Solve the equation with a regular calculator Use tables Use a financial calculator Use a spreadsheet

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**USING TABLES: See handout**

3 PERIODS 10 % = times 100 = $133.10 SAY GOOD-BYE TO USING TABLES!

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**Financial Calculator Solution**

Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th.

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**Here’s the setup to find FV:**

N I/YR PV PMT FV INPUTS OUTPUT 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END

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**b(2) What’s the PV of $100 due in 3 years if i = 10%?**

Finding PVs is discounting, and it’s the reverse of compounding. 1 2 3 10% PV = ? 100

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**Solve FVn = PV(1 + i )n for PV:**

= $100(0.7513) = $75.13. 3 1.10

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**Financial Calculator Solution**

N I/YR PV PMT FV -75.13 INPUTS OUTPUT Either PV or FV must be negative. Here PV = Put in $75.13 today, take out $100 after 3 years.

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EXCEL SOLUTION LOOK AT FUNCTION’S PAGE FOR EXCEL/LOTUS.

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Spreadsheet Solution Use the FV function: see spreadsheet in Ch 02 Mini Case.xls. = FV(Rate, Nper, Pmt, PV) = FV(0.10, 3, 0, -100) =

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Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, 100) =

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**c. If sales grow at 20% per year, how long before sales double?**

Solve for n: Time line ? FVn = PV(1 + i)n 2 = 1(1.20)n (1.20)n = 2 n ln (1.20) = ln 2 n(0.1823) = n = / = 3.8 years.

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**Graphical Illustration:**

N I/YR PV PMT FV 3.8 Beware:Some Calculators round up. INPUTS OUTPUT Graphical Illustration: FV 2 3.8 1 Years 1 2 3 4

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**Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) **

Spreadsheet Solution Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) = NPER(0.20, 0, -1, 2) = 3.8 Correction

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ADDITIONAL QUESTION A 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?

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ADDITIONAL QUESTION Bill 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?

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**n * krequired to double = 72 In this case, **

RULE OF 72 A good approximation of the interest rate--or number of years--required to double your money. n * krequired to double = 72 In this case, 5 * krequired to double = 72 k = 14.4 Correct answer was 14.87, so for ball-park approximation, use Rule of 72.

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ADDITIONAL QUESTION John 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?

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**d. What’s the difference between an ordinary annuity and an annuity due?**

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**d. What’s the difference between an ordinary annuity and an annuity due?**

1 2 3 i% PMT PMT PMT Annuity Due 1 2 3 i% PMT PMT PMT 36

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HINT ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR ANNUITY OF (n-1) PERIODS PLUS THE PMT.

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**e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%?**

1 2 3 10% 100 100 100 FV =

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**e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%?**

1 2 3 10% 100 100 100 110 121 FV = 331

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FV Annuity Formula The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

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**Financial Calculator Formula**

for Annuities Financial calculators solve this equation: There are 5 variables. If 4 are known, the calculator will solve for the 5th. Correct but confusing!

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**Financial Calculator Solution**

INPUTS 331.00 N I/YR PV PMT FV OUTPUT Have payments but no lump sum PV, so enter 0 for present value.

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Spreadsheet Solution Use the FV function: see spreadsheet. = FV(Rate, Nper, Pmt, Pv) = FV(0.10, 3, -100, 0) =

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**e(2). What’s the PV of this ordinary annuity?**

1 2 3 10% 100 100 100 _____ = PV

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**What’s the PV of this ordinary annuity?**

1 2 3 10% 100 100 100 90.91 82.64 75.13 = PV

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**Have payments but no lump sum FV, so enter 0 for future value.**

INPUTS N I/YR PV PMT FV OUTPUT Have payments but no lump sum FV, so enter 0 for future value.

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Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, Fv) = PV(0.10, 3, 100, 0) =

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**e(3). Find the FV and PV if the annuity were an annuity due.**

1 2 3 10% 100 100 100

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**Could, on the 12C, switch from “End” to “Begin”; i.e. f Begin. **

Then enter variables to find PVA3 = $ INPUTS N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $

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Another HINT FV 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)

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HINT, illlustrated The PV of this regular annuity was Multiply this by ( ), and you get: , the PV of the annuity due. This avoids the necessity of having to switch from end to begin.

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PV and FV of Annuity Due vs. Ordinary Annuity PV 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

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**Switch from “End” to “Begin”. **

Then enter variables to find PVA3 = $ INPUTS N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = $

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**Excel 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)

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EXCEL SOLUTION

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**(f) What is the PV of this uneven cash flow stream?**

1 2 3 4 10% 100 300 300 -50 ______ = PV

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**(f) What is the PV of this uneven cash flow stream?**

1 2 3 4 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 = PV

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**Input in “CFLO” register:**

Enter I = 10%, then press NPV button to get NPV =

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CALCULATOR SOLUTION g CF0 100 g CFj 300 g CFj g Nj 50 CHS g CFj 10 i f NPV

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EXCEL SOLUTION REMEMBER EXCEL/LOTUS READS THE 1ST CASH FLOW AS OCCURING ONE PERIOD HENCE.

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**Spreadsheet Solution A B C D E 1 0 1 2 3 4 2 100 300 300 -50 3 530.09**

Excel Formula in cell A3: =NPV(10%,B2:E2)

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**g. What interest rate would cause $100 to grow to $125.97 in 3 years?**

INPUTS N I/YR PV PMT FV OUTPUT 8.00%

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EXCEL SOLUTION

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h.Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? Why?

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h.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--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

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1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 1 2 3 1 2 3 4 5 6 5% 100 134.01 Semiannually: FV6 = 100(1.05)6 =

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Periodic 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 = %.

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**Effective Annual Rate (EAR = EFF%):**

The annual rate which causes PV to grow to the same FV as under multiperiod compounding.

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An 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.

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**h(3).How do we find EAR for a nominal rate of 10%, compounded**

semiannually? Or use a financial calculator (not 12C)

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EAR = (1+knom/m)m - 1 ENT divide Yx = .1025 (1 + EAR) = (1+knom/m)m

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CALCULATOR WHAT IS EAR IF COMPOUNDING QUARTERLY? COMPOUNDING DAILY? COMPOUNDING CONTINUOUSLY?

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**EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 - 1 = 10.38%.**

EARM = ( /12) = %. EARD(360) = ( /360) = %.

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**For multiple years, n (1 + EAR) = (1 + Knom/m)m**

(1 + EAR)n = (1 + Knom/m)mn To multiply by a $ of dollars, PRIN PRIN * (1 + EAR)n = PRIN * (1 + Knom/m)mn

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**i. 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.

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When 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!

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**Can 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!

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**EAR = 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 payments don’t match interest compounding periods.

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**FV of $100 after 3 years under 10% semiannual compounding. Quarterly**

FV of $100 after 3 years under 10% semiannual compounding? Quarterly? Daily? mn i FV = PV + Nom n m 2x3 0.10 FV = $100 1 + 3S 2 = $100(1.05)6 = $ FV3Q = $100(1.025)12 = $

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**ALTERNATE SOLUTION USING EAR**

FOR SEMIANNUAL COMPOUNDING, EAR = 10.25% FOR 3 YEARS: 100*(1.1025)3 = $134.01 FOR Quarterly COMPOUNDING and 3 years: 100*(1.1038)3 = $134.49

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j(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? 1 2 3 4 5 6 6-mos. periods 5% 100 100 100

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**Payments occur annually, but compounding occurs each 6 months.**

So we can’t use normal annuity valuation techniques.

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**1st Method: Compound Each CF**

1 2 3 4 5 6 5% 100 100 100.00 110.25 121.55 331.80 FVA3 = 100(1.05) (1.05) =

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**Could you find FV with a financial calculator?**

2nd Method: Treat as an Annuity I.E. USE EAR Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = ( ) - 1 = 10.25%. 0.10 2 2

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**Or, to find EAR with a 17 OR 19b Calculator:**

NOM% = 10 P/YR = 2 EFF% = 10.25

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**b. The cash flow stream is an annual annuity whose EFF% = 10.25%.**

INPUTS N I/YR PV PMT FV OUTPUT 331.80

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**j(2) What’s the PV of this stream?**

1 2 3 5% 100 100 100

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**What’s the PV of this stream?**

1 2 3 5% 100 100 100 90.70 82.27 74.62 247.59

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Calculator solution PMT n i f NPV = 247.59

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**What’s the FV of this stream under quarterly compouning?**

1 2 3 100 100 100

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EAR WORKSHEET

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k. Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

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**Step 1: Find the required payments.**

1 2 3 10% -1000 PMT PMT PMT INPUTS N I/YR PV PMT FV OUTPUT 402.11

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ALGEBRA PMT/(1+k) + PMT/(1+k)2 + PMT/(1+k)3 = 1000 PMT [1/(1+k) + 1/(1+k)2 + 1/(1+k)3] = 1000, or PMT = / [1/(1+k) + 1/(1+k)2 + 1/(1+k)3]

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**Step 2: Find interest charge for Year 1.**

INTt = Beg balt (i) INT1 = 1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = = $

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**Step 4: Find ending balance after Year 1.**

End bal = Beg bal - Repmt = 1, = $ Repeat these steps for Years 2 and 3 to complete the amortization table.

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**Interest declines. Tax implications.**

BEG PRIN END YR BAL PMT INT REDUCTION BAL 1 $1,000 $402 $100 $302 $698 TOT 1, ,000 Interest declines. Tax implications.

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**10% on loan outstanding, which is falling.**

$ 402.11 Interest 302.11 Principal Payments 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.

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Amortization 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.

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EXCEL SOLUTION

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NEW 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.)

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** iPer = 10.0% / 365 = 0.027397% per day. FV = $100 1.00027397 =**

1 2 273 % -100 FV=? 273 FV = $100 273 = $100 = $ Note: % in calculator, decimal in equation.

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**Leave data in calculator.**

iPer = iNom/m = 10.0/365 = % per day. INPUTS 107.77 N I/YR PV PMT FV OUTPUT Enter i in one step. Leave data in calculator.

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**Now 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 = $

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**iPer = 0.027397% per day. FV = $100(1 + 0.10/365)638**

365 638 days -100 FV = FV = $100( /365)638 = $100( )638 = $100(1.1910) = $

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**ALTERNATIVE SOLUTION USING EAR**

FIND EAR ENTER DIVIDE Yx [= (1 + EAR)] Yx [= (1 + EAR).75] MULTIPLY = $107.79

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**MORE PRECISELY, instead of .75 exponent:**

Calculate [1+EAR] as above, then ENTER DIVIDE [=(1+EAR)] Yx [=(1+EAR)(273/365)] 100 MULTIPLY = $107.77

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PROBLEM SUPPOSE THAT YOU WERE TOLD THAT THE EFFECTIVE ANNUAL RATE WERE 10%, WITH DAILY COMPOUNDING. WHAT THE STATED, OR NOMINAL RATE BE IN THIS CASE?

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ALGEBRA (1 + EAR) = (1 + knom /m)m (1 + EAR)(1/m) = (1 + knom /m) (1 + EAR)(1/m) - 1 = knom /m m*[(1 + EAR)(1/m) - 1] = knom

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**m*[(1 + EAR)(1/m) - 1] = knom Using the calculator, EAR = 10%, daily compounding.**

ENTER /X Yx 365 MULTIPLY = %

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n. 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?

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**1. Greatest future wealth: FV 2. Greatest wealth today: PV **

iPer = % per day. 365 456 days -850 1,000 3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF%

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**1. Greatest Future Wealth**

Find FV of $850 left in bank for 15 months and compare with note’s FV = $1000. FVBank = $850( )456 = $ in bank. Buy the note: $1000 > $

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**Calculator Solution to FV:**

iPer = iNom/m = 7.0/365 = % per day. INPUTS 927.67 N I/YR PV PMT FV OUTPUT Enter iPer in one step.

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**2. Greatest Present Wealth**

Find PV of note, and compare with its $850 cost: PV = $1000/[( )456] = $

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7/365 = INPUTS N I/YR PV PMT FV OUTPUT PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

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3. Rate of Return Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital: FVn = PV(1 + i)n 1000 = $850(1 + i)456 Now we must solve for i.

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**Convert % to decimal: Decimal = 0.035646/100 = 0.00035646.**

INPUTS % per day N I/YR PV PMT FV OUTPUT Convert % to decimal: Decimal = /100 = EAR = EFF% = ( ) = 13.89%.

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**Using interest conversion:**

P/YR = 365 NOM% = (365) = 13.01 EFF% = 13.89 Since 13.89% > 7.25% opportunity cost, buy the note.

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ADDITIONAL PROBLEM #2 IT 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?

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ADDITIONAL PROBLEM #3 IT 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?

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JOHNM PROBLEMS

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