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PV = \$100(2.4759) = \$247.59 AT 10.25%.

TO USE A FINANCIAL CALCULATOR, INPUT N = 3, I = 10.25, PMT = 100, FV = 0, AND THEN PRESS THE PV KEY TO FIND PV = \$247.59.

J. 3. IS THE STREAM AN ANNUITY?

ANSWER: THE PAYMENT STREAM IS AN ANNUITY IN THE SENSE OF CONSTANT AMOUNTS AT REGULAR INTERVALS, BUT THE INTERVALS DO NOT CORRESPOND WITH THE COMPOUNDING PERIODS. THIS KIND OF SITUATION OCCURS OFTEN. IN THIS SITUATION THE INTEREST IS COMPOUNDED SEMIANNUALLY, SO WITH A QUOTED RATE OF 10%, THE EAR WILL BE 10.25%. HERE WE COULD FIND THE EFFECTIVE RATE AND THEN TREAT IT AS AN ANNUITY. ENTER N = 3, I = 10.25, PMT = 100, AND FV = 0. NOW PRESS PV TO GET \$247.59.
J. 4. AN IMPORTANT RULE IS THAT YOU SHOULD NEVER SHOW A NOMINAL RATE ON A TIME LINE OR USE IT IN CALCULATIONS UNLESS WHAT CONDITION HOLDS? (HINT: THINK OF ANNUAL COMPOUNDING, WHEN iNom = EAR = iPer.) WHAT WOULD BE WRONG WITH YOUR ANSWER TO QUESTIONS J(1) AND J(2) IF YOU USED THE NOMINAL RATE (10%) RATHER THAN THE PERIODIC RATE (iNom /2 = 10%/2 = 5%)?

ANSWER: iNom CAN ONLY BE USED IN THE CALCULATIONS WHEN ANNUAL COMPOUNDING OCCURS. IF THE NOMINAL RATE OF 10% WAS USED TO DISCOUNT THE PAYMENT STREAM THE PRESENT VALUE WOULD BE OVERSTATED BY \$272.32 - \$247.59 = \$24.73.

K. 1. CONSTRUCT AN AMORTIZATION SCHEDULE FOR A \$1,000, 10 PERCENT ANNUAL RATE LOAN WITH 3 EQUAL INSTALLMENTS.

2. WHAT IS THE ANNUAL INTEREST EXPENSE FOR THE BORROWER, AND THE ANNUAL INTEREST INCOME FOR THE LENDER, DURING YEAR 2?

ANSWER: TO BEGIN, NOTE THAT THE FACE AMOUNT OF THE LOAN, \$1,000, IS THE PRESENT VALUE OF A 3-YEAR ANNUITY AT A 10 PERCENT RATE:

0 1 2 3
| | | |
-1,000 PMT PMT PMT

PVA3 = PMT + PMT + PMT
\$1,000 = PMT(1 + i)-1 + PMT(1 + i)-2 + PMT(1 + i)-3
= PMT(1.10)-1 + PMT(1.10)-2 + PMT(1.10)-3.

WE HAVE AN EQUATION WITH ONLY ONE UNKNOWN, SO WE CAN SOLVE IT TO FIND PMT. THE EASY WAY IS WITH A FINANCIAL CALCULATOR. INPUT N = 3, I = 10, PV = -1,000, FV = 0, AND THEN PRESS THE PMT BUTTON TO GET PMT = 402.1148036, ROUNDED TO \$402.11.
NOW MAKE THE FOLLOWING POINTS REGARDING THE AMORTIZATION SCHEDULE:

THE \$402.11 ANNUAL PAYMENT INCLUDES BOTH INTEREST AND PRINCIPAL. INTEREST IN THE FIRST YEAR IS CALCULATED AS FOLLOWS:

1ST YEAR INTEREST = i ? BEGINNING BALANCE = 0.1 ? \$1,000 = \$100.

THE REPAYMENT OF PRINCIPAL IS THE DIFFERENCE BETWEEN THE \$402.11 ANNUAL PAYMENT AND THE INTEREST PAYMENT:

1ST YEAR PRINCIPAL REPAYMENT = \$402.11 - \$100 = \$302.11.

THE LOAN BALANCE AT THE END OF THE FIRST YEAR IS:

1ST YEAR ENDING BALANCE = BEGINNING BALANCE – PRINCIPAL REPAYMENT
= \$1,000 - \$302.11 = \$697.89.

WE WOULD CONTINUE THESE STEPS IN THE FOLLOWING YEARS.

NOTICE THAT THE INTEREST EACH YEAR DECLINES BECAUSE THE BEGINNING LOAN BALANCE IS DECLINING. SINCE THE PAYMENT IS CONSTANT, BUT THE INTEREST COMPONENT IS DECLINING, THE PRINCIPAL REPAYMENT PORTION IS INCREASING EACH YEAR.

THE INTEREST COMPONENT IS AN EXPENSE WHICH IS DEDUCTIBLE TO A BUSINESS OR A HOMEOWNER, AND IT IS TAXABLE INCOME TO THE LENDER. IF YOU BUY A HOUSE, YOU WILL GET A SCHEDULE CONSTRUCTED LIKE OURS, BUT LONGER, WITH 30 ? 12 = 360 MONTHLY PAYMENTS IF YOU GET A 30-YEAR, FIXED RATE MORTGAGE.

THE PAYMENT MAY HAVE TO BE INCREASED BY A FEW CENTS IN THE FINAL YEAR TO TAKE CARE OF ROUNDING ERRORS AND MAKE THE FINAL PAYMENT PRODUCE A ZERO ENDING BALANCE.

THE LENDER RECEIVED A 10% RATE OF INTEREST ON THE AVERAGE AMOUNT OF MONEY THAT WAS INVESTED EACH YEAR, AND THE \$1,000 LOAN WAS PAID OFF. THIS IS WHAT AMORTIZATION SCHEDULES ARE DESIGNED TO DO.

MOST FINANCIAL CALCULATORS HAVE AMORTIZATION FUNCTIONS BUILT IN.

FRACTIONAL TIME PERIODS

THUS FAR ALL OF OUR EXAMPLES HAVE DEALT WITH FULL YEARS. NOW WE ARE GOING TO LOOK AT THE SITUATION WHEN WE ARE DEALING WITH FRACTIONAL YEARS, SUCH AS 9 MONTHS, OR 0.75 YEARS. IN THESE SITUATIONS, PROCEED AS FOLLOWS:

AS ALWAYS, START BY DRAWING A TIME LINE SO YOU CAN VISUALIZE THE SITUATION.

THEN THINK ABOUT THE INTEREST RATE--THE NOMINAL RATE, THE COMPOUNDING PERIODS PER YEAR, AND THE EFFECTIVE ANNUAL RATE. IF YOU HAVE BEEN GIVEN A NOMINAL RATE, YOU MAY HAVE TO CONVERT TO THE EAR, USING THIS FORMULA:

EAR = .

IF YOU HAVE THE EFFECTIVE ANNUAL RATE--EITHER BECAUSE IT WAS GIVEN TO YOU OR AFTER YOU CALCULATED IT WITH THE FORMULA--THEN YOU CAN FIND THE PV OF A LUMP SUM BY APPLYING THIS EQUATION:

PV = FVt .

HERE T CAN BE A FRACTION OF A YEAR, SUCH AS 0.75 IF YOU NEED TO FIND THE PV OF \$1,000 DUE IN 9 MONTHS, OR 450/365 = 1.2328767 IF THE PAYMENT IS DUE IN 450 DAYS.

IF YOU HAVE AN ANNUITY WITH PAYMENTS DIFFERENT FROM ONCE A YEAR, SAY EVERY MONTH, YOU CAN ALWAYS WORK IT OUT AS A SERIES OF LUMP SUMS. THAT PROCEDURE ALWAYS WORKS. WE CAN ALSO USE ANNUITY FORMULAS AND CALCULATOR FUNCTIONS, BUT YOU HAVE TO BE CAREFUL.

L. SUPPOSE ON JANUARY 1, 2002, YOU DEPOSIT \$100 IN AN ACCOUNT THAT PAYS A NOMINAL, OR QUOTED, INTEREST RATE OF 11.33463 PERCENT, WITH INTEREST ADDED (COMPOUNDED) DAILY. HOW MUCH WILL YOU HAVE IN YOUR ACCOUNT ON OCTOBER 1, OR AFTER 9 MONTHS?

ANSWER: THE DAILY PERIODIC INTEREST RATE IS kPer = 11.3346%/365 = 0.031054%. THERE ARE 273 DAYS BETWEEN JANUARY 1 AND OCTOBER 1. CALCULATE FV AS FOLLOWS:

FV273 = \$100(1.00031054)273
= \$108.85.

USING A FINANCIAL CALCULATOR, INPUT N = 273, I = 0.031054, PV = -100, AND PMT = 0. PRESSING FV GIVES \$108.85.
AN ALTERNATIVE APPROACH WOULD BE TO FIRST DETERMINE THE EFFECTIVE ANNUAL RATE OF INTEREST, WITH DAILY COMPOUNDING, USING THE FORMULA:

EAR = - 1 = 0.12 = 12.0%.

(SOME CALCULATORS, E.G., THE HP 10B AND 17B, HAVE THIS EQUATION BUILT IN UNDER THE ICNV [INTEREST CONVERSION] FUNCTION.)
THUS, IF YOU LEFT YOUR MONEY ON DEPOSIT FOR AN ENTIRE YEAR, YOU WOULD EARN \$12 OF INTEREST, AND YOU WOULD END UP WITH \$112. THE QUESTION, THOUGH, IS THIS: HOW MUCH WILL BE IN YOUR ACCOUNT ON OCTOBER 1, 2002?
HERE YOU WILL BE LEAVING THE MONEY ON DEPOSIT FOR 9/12 = 3/4 = 0.75 OF A YEAR.

0 0.75 1
| | |
-100 FV = ? 112

YOU WOULD USE THE REGULAR SET-UP, BUT WITH A FRACTIONAL EXPONENT:

FV0.75 = \$100(1.12)0.75 = \$100(1.088713) = \$108.87.

THIS IS SLIGHTLY DIFFERENT FROM OUR EARLIER ANSWER, BECAUSE N IS ACTUALLY 273/365 = 0.7479 RATHER THAN 0.75.

M. NOW SUPPOSE YOU LEAVE YOUR MONEY IN THE BANK FOR 21 MONTHS. THUS, ON JANUARY 1, 2002, YOU DEPOSIT \$100 IN AN ACCOUNT THAT PAYS A 12 PERCENT EFFECTIVE ANNUAL INTEREST RATE. HOW MUCH WILL BE IN YOUR ACCOUNT ON OCTOBER 1, 2003?

ANSWER: USING A FINANCIAL CALCULATOR INPUT THE FOLLOWING:

N = 365 + 273 = 638, I = 11.3346/365 = 0.031054, PV = -100, PMT = 0. PRESSING FV GIVES A RESULT OF FV = \$121.91.

ALTERNATIVELY, HERE YOU WILL BE LEAVING THE MONEY ON DEPOSIT FOR
1 + 9/12 = 1 + 3/4 = 1.75 OF A YEAR.

0 1 1.75 2
| | | |
-100 112 FV = ? 125.44

YOU WOULD USE THE REGULAR SET-UP, BUT WITH A FRACTIONAL EXPONENT:

FV = \$100(1.12)1.75 = \$100(1.219359) = \$121.94.
N IS ACTUALLY 638/365 = 1.7479, WHICH ACCOUNTS FOR THE SLIGHT DIFFERENCE IN THE ANSWERS.

N. SUPPOSE SOMEONE OFFERED TO SELL YOU A NOTE CALLING FOR THE PAYMENT OF \$1,000 15 MONTHS FROM TODAY. THEY OFFER TO SELL IT TO YOU FOR \$850. YOU HAVE \$850 IN A BANK TIME DEPOSIT WHICH PAYS A 6.76649 PERCENT NOMINAL RATE WITH DAILY COMPOUNDING, WHICH IS A 7 PERCENT EFFECTIVE ANNUAL INTEREST RATE, AND YOU PLAN TO LEAVE THE MONEY IN THE BANK UNLESS YOU BUY THE NOTE. THE NOTE IS NOT RISKY--YOU ARE SURE IT WILL BE PAID ON SCHEDULE. SHOULD YOU BUY THE NOTE? CHECK THE DECISION IN THREE WAYS: (1) BY COMPARING YOUR FUTURE VALUE IF YOU BUY THE NOTE VERSUS LEAVING YOUR MONEY IN THE BANK, (2) BY COMPARING THE PV OF THE NOTE WITH YOUR CURRENT BANK ACCOUNT, AND (3) BY COMPARING THE EAR ON THE NOTE VERSUS THAT OF THE BANK ACCOUNT.

ANSWER: YOU CAN SOLVE THIS PROBLEM IN THREE WAYS--(1) BY COMPOUNDING THE \$850 NOW IN THE BANK FOR 15 MONTHS AND COMPARING THAT FV WITH THE \$1,000 THE NOTE WILL PAY, (2) BY FINDING THE PV OF THE NOTE AND THEN COMPARING IT WITH THE \$850 COST, AND (3) FINDING THE EFFECTIVE ANNUAL RATE OF RETURN ON THE NOTE AND COMPARING THAT RATE WITH THE 7% YOU ARE NOW EARNING, WHICH IS YOUR OPPORTUNITY COST OF CAPITAL. ALL THREE PROCEDURES LEAD TO THE SAME CONCLUSION. HERE IS THE TIME LINE:

0 1 1.25
| | |
-850 1,000

(1) FV = \$850(1.07)1.25 = \$925.01 = AMOUNT IN BANK AFTER 15 MONTHS VERSUS \$1,000 IF YOU BUY THE NOTE. (AGAIN, YOU CAN FIND THIS VALUE WITH A FINANCIAL CALCULATOR. NOTE THAT CERTAIN CALCULATORS LIKE THE HP 12C PERFORM A STRAIGHT-LINE INTERPOLATION FOR VALUES IN A FRACTIONAL TIME PERIOD ANALYSIS RATHER THAN AN EFFECTIVE INTEREST RATE INTERPOLATION. THE VALUE THAT THE HP 12C CALCULATES IS \$925.42.) THIS PROCEDURE INDICATES THAT YOU SHOULD BUY THE NOTE.
ALTERNATIVELY, 15 MONTHS = (1.25 YEARS)(365 DAYS PER YEAR) = 456.25 ˜ 456 DAYS.
FV456 = \$850(1.00018538)456
= \$924.97.

THE SLIGHT DIFFERENCE IS DUE TO USING N = 456 RATHER THAN N = 456.25.

(2) PV = \$1,000/(1.07)-1.25 = \$918.90. SINCE THE PRESENT VALUE OF THE NOTE IS GREATER THAN THE \$850 COST, IT IS A GOOD DEAL. YOU SHOULD BUY IT.
ALTERNATIVELY, PV = \$1000/(1.00018538)456 = \$918.95.

(3) FVn = PV(1 + i)n, SO \$1,000 = \$850(1 + i)1.25 = \$1,000. SINCE WE HAVE AN EQUATION WITH ONE UNKNOWN, WE CAN SOLVE IT FOR i. YOU WILL GET A VALUE OF i = 13.88%. THE EASY WAY IS TO PLUG VALUES INTO YOUR CALCULATOR. SINCE THIS RETURN IS GREATER THAN YOUR 7% OPPORTUNITY COST, YOU SHOULD BUY THE NOTE. THIS ACTION WILL RAISE THE RATE OF RETURN ON YOUR ASSET PORTFOLIO.
ALTERNATIVELY, WE COULD SOLVE THE FOLLOWING EQUATION:

\$1,000 = \$850(1 + i)456 FOR A DAILY i = 0.00035646,

WITH A RESULT OF EAR = EFF% = (1.00035646)365 - 1 = 13.89%.

O. SUPPOSE THE NOTE DISCUSSED IN PART N HAD A COST OF \$850, BUT CALLED FOR 5 QUARTERLY PAYMENTS OF \$190 EACH, WITH THE FIRST PAYMENT DUE IN 3 MONTHS RATHER THAN \$1,000 AT THE END OF 15 MONTHS. WOULD IT BE A GOOD INVESTMENT FOR YOU?

ANSWER: HERE IS THE TIME LINE:

YEAR 0 0.25 0.5 0.75 1.0 1.25 USE EAR = 7.0%
QUARTER 0 1 2 3 4 5 USE iNom/4 = 1.70585%
| | | | | |
-850 190 190 190 190 190

AGAIN, WE CAN SOLVE THE PROBLEM IN SEVERAL WAYS, AS ABOVE:

FIND THE PVA:

PVA = \$190(1.07)-0.25 + \$190(1.07)-0.5 ... + \$190(1.07)-1.25
= \$190(0.9832) + ... = \$903.25.
WE COULD FIND THE iNom FOR EAR = 7.0%, QUARTERLY COMPOUNDING, WHICH IS iNom = 6.8234%, OR 6.8234/4 = 1.70585% PER QUARTER, AND USE WHOLE NUMBER EXPONENTS: OR PVA = \$190(1.017...)-1 + \$190(1.017...)-2 ... + \$190(1.017...)-5 = \$903.25. SINCE THE PVA IS GREATER THAN THE COST, THE NOTE IS A GOOD DEAL. NOTE THAT IN THE SECOND EQUATION ABOVE WE COULD USE THE ANNUITY FUNCTION, BUT WE HAD TO FIND iNom AND THEN DIVIDE BY 4 TO GET THE QUARTERLY RATE FOR USE IN THE ANNUITY EQUATION.

FIND THE INTEREST RATE BUILT INTO THE NOTE:

PVA = \$850 = \$190(1 + i)-1 + \$190(1 + i)-2 ... + \$190(1 + i)-5.

HERE WE HAVE AN EQUATION WITH ONE UNKNOWN, SO WE CAN SOLVE IT FOR i, THE RATE OF RETURN. WITH A FINANCIAL CALCULATOR, WE FIND i = 3.8258658%, BUT THIS IS THE QUARTERLY RATE. WE CONVERT TO THE EFFECTIVE ANNUAL RATE AS FOLLOWS:

EAR = (1 + 0.038258658)4 - 1 = 16.2%.

SINCE THIS RATE IS GREATER THAN YOU COULD EARN ON THE BANK DEPOSIT, WE AGAIN SEE THAT THE NOTE IS A GOOD BUY FOR YOU.

YOU COULD ALSO COMPOUND THE \$190 PAYMENTS AND THE BANK DEPOSIT. HERE, THIS QUESTION WOULD ARISE: WHEN YOU RECEIVE THE \$190 PAYMENTS, WHAT WOULD YOU DO WITH THEM? THE ANSWER TO THAT QUESTION INVOLVES THE "REINVESTMENT RATE." IN THIS INSTANCE, YOU WOULD PROBABLY DEPOSIT THE PAYMENTS IN THE BANK AND EARN 7% ON THEM. THAT WOULD PRODUCE THIS SITUATION:

0 0.25 0.5 0.75 1.0 1.25
| | | | | |
190 190 190 190 190

FVA1.25 = \$190(1.07)1 + \$190(1.07)0.75 + \$190(1.07)0.5 + \$190(1.07)0.25 + \$190
= \$982.97.

AS WE SAW EARLIER, THE BANK DEPOSIT WOULD GROW TO:

FV = \$850(1.07)1.25 = \$925.01.

0 1 1.25 YEARS
| | |
-850 FV = 925.01

OR FV = \$850 = \$850 = \$925.02.

0 1 2 3 4 5 QUARTERS
| | | | | |
-850 925.01

SINCE THE NOTE WOULD GROW TO A LARGER FINAL AMOUNT, THIS ANALYSIS ALSO INDICATES THAT THE NOTE IS A GOOD BUY. NOTE ALSO THAT IF YOU COULD KEEP FINDING OTHER NOTES LIKE THIS ONE, YOU COULD REINVEST THE CASH FLOWS--THE \$190 PAYMENTS--AT A HIGHER RATE THAN 7%, SO YOU WOULD END UP WITH EVEN MORE. WE WILL LOOK AT THE REINVESTMENT RATE ASSUMPTION IN GREATER DEPTH LATER IN THE COURSE, AND IN LATER FINANCE COURSES. THE REINVESTMENT RATE ASSUMPTION IS IMPORTANT IN DECISIONS INVOLVING ALTERNATIVE INVESTMENT OPPORTUNITIES.

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