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2 - 1 CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics Future value Present value Rates of return Amortization Annuities, AND Many Examples

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

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2 - 3 MINICASE 2 Why is financial mathematics (time value of money) so important in financial analysis?

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2 - 4 a.Time lines show timing of cash flows. ALWAYS A GOOD IDEA TO DRAW A TIME LINE. CF 0 CF 1 CF 3 CF i% 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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2 - 5 Time line for a $100 lump sum due at the end of Year Years i%

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2 - 6 Time line for an ordinary annuity of $100 for 3 years i%

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2 - 7 Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through i% -50

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2 - 8 b(1) Whats the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding.

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2 - 9 b(1) Whats the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding. 110 ?

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After 1 year FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $ After 2 years FV 2 = FV 1 (1 + i) = PV(1 + i) 2 = $100(1.10) 2 = $

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

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

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USING TABLES: See handout 3 PERIODS 10 % = times 100 = $ 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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NI/YR PV PMTFV Heres the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END INPUTS OUTPUT

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b(2) Whats the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and its the reverse of compounding % PV = ?

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PV = Solve FV n = PV(1 + i ) n for PV: PV = $100 / ( ) = = $100(0.7513) = $ FV n (1 + i) n n

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Financial Calculator Solution INPUTS OUTPUT NI/YR PV PMTFV 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 FUNCTIONS 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 ? FV n = 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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INPUTS OUTPUT Graphical Illustration: FV 3.8 Years NI/YR PV PMTFV 3.8 Beware: Some Calculators round up.

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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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RULE OF 72 A good approximation of the interest rate--or number of years--required to double your money. n * k required to double = 72 In this case, 5 * k required 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. Whats the difference between an ordinary annuity and an annuity due?

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d. Whats the difference between an ordinary annuity and an annuity due? PMT 0123 i% PMT 0123 i% PMT Annuity Due Ordinary Annuity 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). Whats the FV of a 3-year ordinary annuity of $100 at 10%? % FV =

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e(1). Whats the FV of a 3-year ordinary annuity of $100 at 10%? % 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 calculators solve this equation: There are 5 variables. If 4 are known, the calculator will solve for the 5th. Financial Calculator Formula for Annuities Correct but confusing!

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NI/YRPVPMTFV Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT

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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). Whats the PV of this ordinary annuity? % _____ = PV

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Whats the PV of this ordinary annuity? % = PV

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INPUTS OUTPUT N I/YR PVPMTFV 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 % 100

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INPUTS OUTPUT NI/YRPV PMTFV Could, on the 12C, switch from End to Begin; i.e. f Begin. Then enter variables to find PVA 3 = $ 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 (1 +.10), 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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NI/YRPVPMTFV Switch from End to Begin. Then enter variables to find PVA 3 = $ Then enter PV = 0 and press FV to find FV = $ INPUTS OUTPUT

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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? % ______ = PV

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

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Input in CFLO register: CF 0 =0 CF 1 =100 CF 2 =300 CF 3 =300 CF 4 =-50 Enter I = 10%, then press NPV button to get NPV =

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CALCULATOR SOLUTION 0 g CF g CF j 300 g CF j 2 g N j 50 CHS g CF j 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 Excel Formula in cell A3: =NPV(10%,B2:E2) ABCDE

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g. What interest rate would cause $100 to grow to $ in 3 years? INPUTS OUTPUT NI/YR PVFVPMT 8.00% $100 (1 + i ) 3 = $

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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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% % Annually: FV 3 = 100(1.10) 3 = Semiannually: FV 6 = 100(1.05) 6 =

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Periodic rate = i Per = i Nom /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: i Per = 8/4 = 2%. 8% daily (365): i Per = 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+k nom /m) m ENT 2 divide Y x 1 - =.1025 (1 + EAR) = (1+k nom /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% EAR Annual = 10%. EAR Q =( /4) 4 - 1= 10.38%. EAR M =( /12) = 10.47%. EAR D(360) =( /360) = 10.52%.

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For multiple years, n (1 + EAR) = (1 + K nom /m) m (1 + EAR) n = (1 + K nom /m) mn To multiply by a $ of dollars, PRIN PRIN * (1 + EAR) n = PRIN * (1 + K nom /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? i Nom : 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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Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom. Can always use EAR! i Per :

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EAR = EFF%:Used to compare returns on investments with different compounding patterns. Also used for calculations if dealing with annuities where payments dont match interest compounding periods.

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FV of $100 after 3 years under 10% semiannual compounding? Quarterly? Daily? = $100(1.05) 6 = $ FV 3Q = $100(1.025) 12 = $ FV = PV1.+ i m n Nom mn FV = $ S 2x3

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

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j(3). Whats the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually? % mos. periods 100

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Payments occur annually, but compounding occurs each 6 months. So we cant use normal annuity valuation techniques.

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st Method: Compound Each CF % FVA 3 = 100(1.05) (1.05) =

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

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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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INPUTS OUTPUT NI/YRPVFV PMT b. The cash flow stream is an annual annuity whose EFF% = 10.25%. c.

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j(2) Whats the PV of this stream? %

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Whats the PV of this stream? %

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Calculator solution 100 PMT 3 n i f NPV =

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Whats the FV of this stream under quarterly compouning?

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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. PMT % INPUTS OUTPUT NI/YRPVFV PMT

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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 = 1000/ [1/(1+k) + 1/(1+k) 2 + 1/(1+k) 3 ]

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Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = 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. BEGPRINEND YRBALPMTINT REDUCTION BAL 1$1,000$402$100$302$ TOT1, ,000

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

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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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i Per = 10.0% / 365 = % per day. FV=? % -100 Note: % in calculator, decimal in equation. FV = $ = $ = $

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INPUTS OUTPUT N I/YRPVFV PMT i Per =i Nom /m =10.0/365 = % per day. 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 1.75 years or = 638 days. How much will be in your account at maturity? Answer:Override N = 273 with N = 638. FV = $

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i Per = % per day. FV = days -100 FV=$100( /365) 638 =$100( ) 638 =$100(1.1910) =$

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ALTERNATIVE SOLUTION USING EAR FIND EAR.10 ENTER 365 DIVIDE Y x [= (1 + EAR)].75 Y x [= (1 + EAR).75 ] 100 MULTIPLY = $107.79

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MORE PRECISELY, instead of.75 exponent: Calculate [1+EAR] as above, then 273 ENTER 365 DIVIDE [=(1+EAR)] Y x [=(1+EAR) (273/365)] 100MULTIPLY = $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 + k nom /m) m (1 + EAR) (1/m) = (1 + k nom /m) (1 + EAR) (1/m) - 1 = k nom /m m*[(1 + EAR) (1/m) - 1] = k nom

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m*[(1 + EAR) (1/m) - 1] = k nom Using the calculator, EAR = 10%, daily compounding. 1.1 ENTER 365 1/X Y x MULTIPLY = 9.53%

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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 dont buy the note. The note is riskless. Should you buy it?

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

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Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with notes FV = $1000. FV Bank =$850( ) 456 =$ in bank. Buy the note: $1000 > $

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INPUTS OUTPUT NI/YRPVFV PMT Calculator Solution to FV: i Per =i Nom /m =7.0/365 = % per day. Enter i Per in one step.

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

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

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

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% per day INPUTS OUTPUT NI/YRPV FV PMT Convert % to decimal: Decimal = /100 = EAR = EFF%= ( ) = 13.89%.

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Using interest conversion: P/YR=365 NOM%= (365)= 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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