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Chapter 05Chapter 05Inputs are in BlueAnswers are in RedNOTE: Some functions used in these spreadsheets may require thatthe "Analysis ToolPak" or "Solver Add-In" be installed in Excel.To install these, click on the Office buttonthen "Excel Options," "Add-Ins" and select"Go." Check "Analysis ToolPak" and"Solver Add-In," then click "OK."

&C2012, The McGraw-Hill Companies&C2012, The McGraw-Hill Companies&C2012, The McGraw-Hill Companies

1Quiz 1Compute the present value of a $100 cash flow for the following combinationsof discount rates and times:a.r =8.00%,t =10.00years.b.r =8.00%,t =20.00years.c.r =4.00%,t =10.00years.d.r =4.00%,t =20.00years.Cash Flow$100.00Solution:a.PV of $100 @ 8% for 10 years=$46.32b.PV of $100 @ 8% for 20 years=$21.45c.PV of $100 @ 4% for 10 years=$67.56d.PV of $100 @ 4% for 20 years=$45.64

&C2012, The McGraw-Hill Companies

2Quiz 2Compute the future value of a $100 cash flow for the following combinationsof discount rates and times:a.r =8.00%,t =10.00years.b.r =8.00%,t =20.00years.c.r =4.00%,t =10.00years.d.r =4.00%,t =20.00years.Cash Flow$100.00Solution:a.FV of $100 @ 8% for 10 years=$215.89b.FV of $100 @ 8% for 20 years=$466.10c.FV of $100 @ 4% for 10 years=$148.02d.FV of $100 @ 4% for 20 years=$219.11


3Quiz 3In 1880 five aboriginal trackers were each promised the equivalent of 100 Australian dollars for helping tocapture the notorious outlaw Ned Kelley. In 1993 the granddaughters of two of the trackers claimed thatthis reward had not been paid. The Victorian prime minister stated that if this was true, the governmentwould be happy to pay the $100. However, the granddaughters also claimed that they were entitled tocompound interest. How much was each entitled to if the interest rate was 4%? What if it was 8%?Years passed113.00yearsAmount promised$100.00Interest rate -14.00%Interest rate -28.00%Solution:Amount using 4% interest rate=$8,409.45Amount using 8% interest rate=$598,252.29


4Quiz 4You deposit $1,000 in your bank account. If the bank pays 4% simple interest, how much will youaccumulate in your account after 10 years? What if the bank pays compound interest? How muchof your earnings will be interest on interest?Deposit$1,000.00Simple interest4.00%Time period10.00yearsSolution:Amount accumulated using simple interest=$1,400.00Amount accumulated using compound interest=$1,480.24Earnings interest on interest=$80.24


5Quiz 5You will require $700 in 5 years. If you earn 5% interest on your funds, how much will you need toinvest today in order to reach your savings goal?Future value$700.00Time period5.00yearsInterest rate5.00%Solution:Present value=$548.47


6Quiz 6Find the interest rate implied by the following combinations of present and future values:Present ValueYearsFuture Value$400.0011.00$684.00183.004.00249.00300.007.00300.00Solution:Present ValueYearsFuture ValueInterest Rate$400.0011.00$684.005.00%183.004.00249.008.00%300.007.00300.000.00%


7Quiz 7Would you rather receive $1,000 a year for 10 years or $800 a year for 15 years ifa.the interest rate is5.00%b.the interest rate is20.00%c.Why do your answers to (a) and (b) differ?Option 1:Amount$1,000.00Time period10.00yearsOption 2:Amount$800.00Time period15.00yearsSolution:a.Present value-1=$7,721.73Present value-2=$8,303.73At 5% interest rate, PV of option 2 is more than option 1, therefore you shouldselect option 1.b.Present value-1=$4,192.47Present value-2=$3,740.38At 20% interest rate, PV of option 1 is more than option 2, therefore you shouldselect option 2.c.When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smallerannuity is more valuable because the impact of discounting on the present valueof future payments is less significant.


8Quiz 8Find the annual interest rate.Present ValueFuture ValueTime Period (years)$100.00$115.763.00200.00262.164.00100.00110.415.00Solution:Present ValueFuture ValueTime Period (years)Interest Rate$100.00$115.763.005.00%200.00$262.164.007.00%100.00$110.415.002.00%


9Quiz 9What is the present value of the following cash-flow stream if the interest rate is 6%?YearCash Flow1$200.002$400.003$300.00Interest rate6.00%Solution:Present Value=$796.56


10Quiz 10How long will it take for $400 to grow to $1,000 at the interest rate specified?a.4.00%b.8.00%c.16.00%Present Value$400.00Desired Future Value$1,000.00Solution:a.Time period @ 4.00%=23.36yearsb.Time period @ 8.00%=11.91yearsc.Time period @ 16.00%=6.17years


11Quiz 11Find the effective annual interest rate for each case.APRCompounding Period12.00%1month8.00%3month10.00%6monthSolution:APRCompounding PeriodEffective Annual Rate12.00%12/year12.68%8.00%4/year8.24%10.00%2/year10.25%


12Quiz 12Find the APR (the stated interest rate) for each case.Effective Annual Interest RateCompounding Period10.00%1month6.09%6month8.24%3monthSolution:Effective Annual Interest RateCompoundingPeriodPer-Period RateAPR10.00%12/ Year0.00809.60%6.09%2/ Year0.03006.00%8.24%4/ Year0.02008.00%


13Quiz 13If you earn 6% per year on your bank account, how long will it take an account with $100 todouble to $200?Interest rate6.00%PV$100.00FV$200.00Solution:Time period=11.9years


14Quiz 14Suppose you can borrow money at 8.6% per year (APR) compounded semiannually or 8.4% peryear (APR) compounded monthly. Which is the better deal?APR compounded semiannually8.60%APR compounded monthly8.40%Solution:APRCompounding periodEffective Annual Rate8.6%2/ Year8.78%8.4%12/ Year8.73%Choose the 8.4% loan for its slightly lower effective rate.


15Quiz 15Lenny Loanshark charges 1 point per week (that is, 1% per week) on his loans. What APRmust he report to consumers? Assume exactly 52 weeks in a year. What is the effectiveannual rate?Loan Charges1.00%per weekWeeks in a year52.00weeksSolution:APR=52.00%EAR=67.77%


16Quiz 16Investments in the stock market have increased at an average compound rate of about5% since 1900. It is now 2012.a.If you invested $1,000 in the stock market in 1900, how much would that investmentbe worth today?b.If your investment in 1900 has grown to $1 million, how much did you invest in 1900?Average compound rate5.00%Time period112.00yearsAmount invested in 1900$1,000.00Investment grown to in 2012 (FV)$1.00millionSolution:a.Present Value=$236,157.40b.Present Value=$4,234.46


17Quiz 17Old Time Savings Bank pays 4% interest on its savings accounts. If you deposit$1,000 in the bank and leave it there, how much interest will you earn in the firstyear? The second year? The tenth year?Interest rate4.00%Deposit$1,000Time period1.00yearTime period2.00yearTime period10.00yearSolution:Interest in first year=$40.00Interest in second year=$41.60Interest in tenth year=$56.93


19Quiz 19A zero-coupon bond that will pay $1,000 in 10 years is selling today for $422.41.What interest rate does the bond offer?Face value$1,000.00Time10.00Market Value$422.41Solution:Interest rate=9.00%


20Quiz 20A famous quarterback just signed a $15 million contract providing $3 million a year for5 years. A less famous receiver signed a $14 million 5-year contract providing $4 millionnow and $2 million a year for 5 years. Who is better paid? The interest rate is 10%.Quarterback Contract:Contract Value$15.00millionCash flow per year$3.00millionTime period5.00yearsInterest rate10.00%Receiver's Contract:Contract Value$14.00millionCash received this year$4.00millionCash flow for next 4 years$2.00millionTime period5.00yearsInterest rate10.00%Solution:Present value of Quarterback's Contract=$11.37millionPresent value of Receiver's Contract=$11.58millionThe receivers contract is worth more than the quarterbacks even though thereceivers undiscounted total payments are less than the quarterbacks.


21Practice Problem 21In mid-2010 a pound of apples cost $1.26, while oranges cost $1.10. Ten years earlier the priceof apples was only $.92 a pound and that of oranges was $.70 a pound. What was the annualcompound rate of growth in the price of the two fruits? If the same rates of growth persist in thefuture, what will be the price of apples in 2030? What about the price of oranges?Apple Cost (per pound)$1.26Oranges Cost (per pound)$1.10Ten years earlier:Apple Cost (per pound)$0.92Oranges Cost (per pound)$0.70Time period (prior)10.00yearsTime period20.00yearsSolution:Rate of growth for apples=3.195%Rate of growth for oranges=4.62%Price of apples in 2030=$2.36Price of oranges in 2030=$2.72


22Practice Problem 22If you take out an $8,000 car loan that calls for 48 monthly payments starting after 1 month at an APRof 10%, what is your monthly payment? What is the effective annual interest rate on the loan?Car loan$8,000.00No. of payments48.00monthlyAPR10.00%Solution:Monthly payment=$202.90The monthly interest rate is=0.8333%Effective annual interest rate=10.47%


23Practice Problem 23a.What is the present value of a 3-year annuity of $100 if the discount rate is 6%?b.What is the present value of the annuity in (a) if you have to wait 2 years insteadof 1 year for the first payment?Time period(1)3.00yearAmount$100.00Discount rate6.00%Time period(2)2.00yearSolution:a.Present Value=$267.30b.Present Value=$252.17


24Practice Problem 24Professors Annuity Corp. offers a lifetime annuity to retiring professors. For a paymentof $80,000 at age 65, the firm will pay the retiring professor $600 a month until death.a.If the professors remaining life expectancy is 20 years, what is the monthly rate onthis annuity? What is the effective annual rate?b.If the monthly interest rate is .5%, what monthly annuity payment can the firm offer tothe retiring professor?Lump sum payment$80,000.00Monthly income$600.00Life expectancy20.00yearsMonthly interest rate0.50%Solution:a.Monthly rate on annuity=0.548%Effective annual rate=6.78%b.Monthly annual payment=$573.14


25Practice Problem 25You want to buy a new car, but you can make an initial payment of only $2,000 and can affordmonthly payments of at most $400.a.If the APR on auto loans is 12% and you finance the purchase over 48 months, what is themaximum price you can pay for the car?b.How much can you afford if you finance the purchase over 60 months?Initial payment$2,000.00Monthly payments$400.00APR12.00%Time period (1)48.00monthsTime period (2)60.00monthsSolution:a.Present Value of loan=$15,189.58Maximum price=$17,189.58b.Present Value of loan=$17,982.02Maximum price=$19,982.02


26Practice Problem 26In a discount interest loan, you pay the interest payment up front. For example, if a 1-year loanis stated as $10,000 and the interest rate is 10%, the borrower pays .10 x $10,000 = $1,000immediately, thereby receiving net funds of $9,000 and repaying $10,000 in a year.a.What is the effective interest rate on this loan?Loan$10,000.00Interest rate10.00%Net funds$9,000.00Solution:a.Effective interest rate=11.11%


28Practice Problem 28If you take out an $8,000 car loan that calls for 48 monthly payments of $240 each, what is the APR ofthe loan? What is the effective annual interest rate on the loan?Car loan$8,000.00No. of monthly payments48.00Monthly payment$240.00Solution:Monthly rate of interest=1.599%APR=19.188%EAR=20.97%


29Practice Problem 29If you take out an $8,000 car loan that calls for 48 monthly payments of $240 each,what is the APR of the loan? What is the effective annual interest rate on the loan?What if the payments are made in four annual year-end installments? What annualpayment would have the same present value as the monthly payment you calculated?Use the same effective annual interest rate. Why is your answer not simply 12times the monthly payment?Car loan$8,000.00No. of monthly payments48.00Monthly payment$240.00No. of yearly payments4.00Monthly payment$240.00Solution:Monthly rate of interest=1.5990%APR=19.188%EAR=20.97%Annual Payment=$3,147.29With monthly payment you would pay $2,880.00 per yearThis value is lower because the monthly payments come beforeyear-end and therefore have a higher PV.


30Practice Problem 30Your landscaping company can lease a truck for $8,000 a year (paid at year end) for 6 years.It can instead buy the truck for $40,000. The truck will be valueless after 6 years. If the interestrate your company can earn on its funds is 7%, is it cheaper to buy or lease?Truck Lease$8,000.00per yearTime period6.00yearsCost of the truck$40,000.00Interest rate7.00%Solution:Present value of annuity=$38,132.32Since $38132.32 < $40000 (the cost of buying a truck),it is less expensive to lease than to buy.


31Practice Problem 31Your landscaping company can lease a truck for $8,000 a year (paid at year end) for 6 years.It can instead buy the truck for $40,000. The truck will be valueless after 6 years. The interestrate your company can earn on its funds is 7%.What if the lease payments are an annuity due, so that the first payment comes immediately?Is it cheaper to buy or lease?Truck Lease$8,000.00Time period6.00yearsCost of the truck$40,000.00Interest rate7.00%Solution:PV of an annuity due=PV of ordinary annuity x (1 + r)PV of ordinary annuity=$38,132.32PV of an annuity due=$40,801.58Since this is greater than $40000 (the cost of buying a truck),we conclude that, if the first payment on the lease is due immediately,it is less expensive to buy the truck than to lease it.


32Practice Problem 32A store offers two payment plans. Under the installment plan, you pay 25% downand 25% of the purchase price in each of the next 3 years. If you pay the entire billimmediately, you can take a 10% discount from the purchase price. Which is abetter deal if you can borrow or lend funds at a 5% interest rate?Down payment25.00%Annuity25.00%n3.00yearsDiscount rate10.00%Interest rate5.00%Solution:Assume the product sells for$100.00Installment plan:Present value=$93.08Pay in full:Payment net of discount=$90.00Choose the second payment plan for its lower present value of payments.


33Practice Problem 33A store offers two payment plans. Under the installment plan, you pay 25% downand 25% of the purchase price in each of the next 3 years. If you pay the entire billimmediately, you can take a 10% discount from the purchase price. Which is abetter deal if you can borrow or lend funds at a 5% interest rate?How will your answer change if the payments on the 4-year installment plan do notstart for a full year?Down Payment25.00%Annuity25.00%n-13.00yearsn-24.00yearsDiscount rate10.00%Interest rate5.00%Solution:Assume the product sells for$100.00Installment plan:Present value=$88.65Pay in full:Payment net of discount=$90.00Now the installment plan offers the lower present value of payments.Therefore choose the first option.


34Practice Problem 34a.If you borrow $1,000 and agree to repay the loan in five equal annual payments at an interestrate of 12%, what will your payment be?b.What if you make the first payment on the loan immediately instead of at the end of the firstyear?Amount borrowed$1,000.00n5.00Interest rate12.00%Solution:a.Annual Payment=$277.41b.PV factor=3.6048Annual Payment=$247.69


35Practice Problem 35Suppose that you will receive annual payments of $10,000 for a period of 10 years. The firstpayment will be made 4 years from now. If the interest rate is 5%, what is the present valueof this stream of payments?Annual payments$10,000.00Time period10.00First payment is made after4.00yearsInterest rate5.00%Solution:PV3=$77,217.35PV0=$66,703.25


36Practice Problem 36Home loans typically involve points, which are fees charged by the lender. Each point chargedmeans that the borrower must pay 1% of the loan amount as a fee.For example, if the loan is for $100,000 and 2 points are charged, the loan repayment scheduleis calculated on a $100,000 loan but the net amount the borrower receives is only $98,000. Whatis the effective annual interest rate charged on such a loan assuming loan repayment occurs over360 months? Assume the interest rate is 1% per month.Fee charges1.00%Loan amount$100,000.00Net amount received$98,000.00Time periods360.00monthsSolution:Payment on the loan=$1,028.61Rate=1.023%Effective Annual Interest Rate=12.99%


37Practice Problem 37You take out a 30-year $100,000 mortgage loan with an APR of 6% and monthly payments. In12 years you decide to sell your house and pay off the mortgage. What is the principal balanceon the loan?Loan amount$100,000.00Time period30.00yearsTime period360.00monthsAPR6.00%Pay off the loan in12.00yearsPay off the loan in144.00monthsSolution:Payment on the mortgage=$599.55After 12 years, 216 months remain on the loan,so the loan balance is:=$79,079.44


38Practice Problem 38Consider a 4-year amortizing loan. You borrow $1,000 initially, and repay it in four equal annualyear-end payments.a.If the interest rate is 8%, show that the annual payment is $301.92.b.Fill in the following table, which shows how much of each payment is interest versus principalrepayment (that is, amortization), and the outstanding balance on the loan at each date.TimeLoanBalanceYear-End InterestDue on BalanceYear-EndPaymentAmortizationof Loan0$1,000.00$80.00$301.92$221.921----------------$301.92--------2----------------$301.92--------3----------------$301.92--------40.00.0----------------c.Show that the loan balance after 1 year is equal to the year-end payment of $301.92 times the3-year annuity factor.No. of installments4.00Amount$1,000.00Interest rate8.00%Annual payment$301.92Solution:a.Annual payment=$301.92b.TimeLoanBalanceYear-End InterestDue on BalanceYear-EndPaymentAmortizationof Loan0$1,000.00$80.00$301.92$221.921$778.08$62.25$301.92$239.672$538.41$43.07$301.92$258.853$279.56$22.36$301.92$279.5640.00.0--c.Loan balance after one year=$778.08


39Practice Problem 39Youve borrowed $4,248.68 and agreed to pay back the loan with monthly paymentsof $200. If the interest rate is 12% stated as an APR, how long will it take you to payback the loan? What is the effective annual rate on the loan?Amount borrowed$4,248.68Monthly payments200.00APR12.00%Solution:Number of months (t)=24.00monthsEffective annual rate=12.68%


40Practice Problem 40The $40 million lottery payment that you just won actually pays $2 million per year for 20 years.If the discount rate is 8% and the first payment comes in 1 year, what is the present value of thewinnings? What if the first payment comes immediately?Lottery amount$40.00millionPer year payment$2.00millionTime period20.00yearsDiscount rate8.00%Solution:Present Value=$19.64millionIf the first payment comes immediately:Present Value=$21.21million


41Practice Problem 41A retiree wants level consumption in real terms over a 30-year retirement. If theinflation rate equals the interest rate she earns on her $450,000 of savings, howmuch can she spend in real terms each year over the rest of her life?Time period30.00yearSavings$450,000.00Solution:Spending each year=$15,000.00


42Practice Problem 42You invest $1,000 at a 6% annual interest rate, stated as an APR. Interest iscompounded monthly. How much will you have in 1 year? In 1.5 years?Amount invested$1,000.00APR6.00%Time period-11.00yearsTime period-21.50yearsSolution:Amount in 1 year=$1,061.68Amount in 1.5 year=$1,093.93


43Practice Problem 43You just borrowed $100,000 to buy a condo. You will repay the loan in equal monthly paymentsof $804.62 over the next 30 years. What monthly interest rate are you paying on the loan?What is the effective annual rate on that loan? What rate is the lender more likely to quoteon the loan?Amount borrowed$100,000.00Monthly payments$804.62Time period30.00yearsSolution:Monthly Interest=0.750%Effective annual rate=9.38%APR=9.00%


44Practice Problem 44If a bank pays 6% interest with continuous compounding, what is the effective annual rate?Interest rate6.00%Solution:EAR=6.18%


45Practice Problem 45You can buy a car that is advertised for $24,000 on the following terms:(a) pay $24,000 and receive a $2,000 rebate from the manufacturer;(b) pay $500 a month for 4 years for total payments of $24,000, implying zero percent financing.Which is the better deal if the interest rate is 1% per month?Car price$24,000.00(a) Rebate$2,000.00(b) Monthly payments$500.00Time4.00yearsInterest rate1.00%per monthSolution:Present Value of Option (a)=$22,000.00Present Value of Option (b)=$18,986.98Option (b) is the better deal.


46Practice Problem 46How much will $100 grow to if invested at a continuously compounded interestrate of 10% for 8 years? What if it is invested for 10 years at 8%?Amount invested$100.00Time period-18.00yearsInterest rate-110.00%Time period-210.00yearsInterest rate-28.00%Solution:Future Value 1=222.54Future Value 2=222.54


47Practice Problem 47I now have $20,000 in the bank earning interest of .5% per month. I need $30,000 tomake a down payment on a house. I can save an additional $100 per month. Howlong will it take me to accumulate the $30,000?Amount in bank$20,000.00Interest rate0.50%per monthAmount required$30,000.00Additional savings$100.00Solution:Time required=44.74months


48Practice Problem 48A local bank advertises the following deal: Pay us $100 a year for 10 years and then wewill pay you (or your beneficiaries) $100 a year forever. Is this a good deal if the interestrate available on other deposits is 6%?Payments$100.00per yearReceipts$100.00per yearTime10.00yearsInterest rate6.00%Solution:The present value of your payments to the bank equals:=$736.01Present value of your receipts=$930.66This is a good deal if you can earn 6% on your other investments.


49Practice Problem 49A local bank will pay you $100 a year for your lifetime if you deposit $2,500 inthe bank today. If you plan to live forever, what interest rate is the bank paying?Receipts per year$100.00Deposit amount$2,500.00Solution:Interest rate=4.00%


50Practice Problem 50A property will provide $10,000 a year forever. If its value is $125,000,what must be the discount rate?Receipts per year$10,000.00Value$125,000.00Solution:Interest rate=8.00%


51Practice Problem 51You can buy property today for $3 million and sell it in 5 years for $4 million.(You earn no rental income on the property.)a.If the interest rate is 8%, what is the present value of the sales price?b.Is the property investment attractive to you? Why or why not?c.Would your answer to (b) change if you also could earn $200,000 per yearrent on the property?Property price$3.00millionTime period5.00yearsProperty price after 5 years$4.00millionInterest rate8.00%Rent per year$200,000.00Solution:a.Present value of the sales price=$2.722millionb.The present value of the sales price is less than the cost of the property,so this would not be an attractive opportunity.c.Present value of the total cash flows=$3.521The property is an attractive investment if you can buy it for $3 million.


52Practice Problem 52A factory costs $400,000. You forecast that it will produce cash inflows of $120,000 in year 1,$180,000 in year 2, and $300,000 in year 3. The discount rate is 12%. Is the factory a goodinvestment? Explain.Factory cost$400,000.00Cash Flows:Year1$120,000.00Year2$180,000.00Year3$300,000.00Discount rate12.00%Solution:Present value of Cash Flow=$464,171.83This exceeds the cost of the factory, so the investment is attractive


53Practice Problem 53You invest $1,000 today and expect to sell your investment for $2,000 in 10 years.a.Is this a good deal if the discount rate is 6%?b.What if the discount rate is 10%?Amount invested$1,000.00Investment worth in 10 years$2,000.00Time period10.00yearsDiscount rate-16.00%Discount rate-210.00%Solution:a.Present value of the future payoff is=$1,116.79This is a good deal: Present value exceeds the initial investmentb.Present value of the future payoff is=$771.09This is now less than the initial investment. Therefore, this is a bad deal.


54Practice Problem 54A store will give you a 3% discount on the cost of your purchase if you pay cash today.Otherwise, you will be billed the full price with payment due in 1 month. What is theimplicit borrowing rate being paid by customers who choose to defer payment for themonth?Discount rate3.00%Solution:Monthly rate=3.09%Effective annual rate=44.12%


55Practice Problem 55Banks sometimes quote interest rates in the form of add-on interest. In this case,if a 1-year loan is quoted with a 20% interest rate and you borrow $1,000, then youpay back $1,200. But you make these payments in monthly installments of $100each. What are the true APR and effective annual rate on this loan? Why shouldyou have known that the true rates must be greater than 20% even before doingany calculations?Time period1.00yearInterest rate20.00%Amount$1,000.00Pay back$1,200.00Monthly installments$100.00Solution:Monthly rate=2.923%APR=35.076%Effective annual rate=41.302%If you borrowed $1,000 today and paid back $1,200 one year from today, the true ratewould be 20%.You should have known that the true rate must be greater than 20%because the twelve $100 payments are made before the end of the year, thusincreasing the true rate above 20%.


56Practice Problem 56Banks sometimes quote interest rates in the form of add-on interest. In this case,if a 1-year loan is quoted with a 20% interest rate and you borrow $1,000, then youpay back $1,200. But you make these payments in monthly installments of $100 each.Suppose you take out a $1,000, 3-year loan using add-on interest with a quotedinterest rate of 20% per year. What will your monthly payments be? (Total paymentsare $1,000 + $1,000 x .20 x 3 = $1,600.) What are the true APR and effective annualrate on this loan? Are they the same as in the previous problem?Time period -11.00yearTime period -23.00yearInterest rate20%Amount$1,000.00Pay back$1,200.00Monthly insatallments-1$100.00Solution:Scenario 1:Monthly rate=2.923%APR=35.076%Effective annual rate=41.302%Scenario 2:Total amount to be repaid=$1,600.00Monthly payments=$44.44Monthly rate=2.799%APR=33.588%Effective annual rate=39.273%APR and EAR of both the scenario are different


57Practice Problem 57What is the effective annual rate on a 1-year loan with an interest rate quoted on a discountbasis of 20%?Discount Rate20.00%Solution:Effective annual rate=25.00%


58Practice Problem 58First National Bank pays 6.2% interest compounded semiannually. Second National Bankpays 6% interest, compounded monthly. Which bank offers the higher effective annual rate?First National Bank Interest6.20%Second National Bank Interest6.00%Solution:After 1 year, each dollar invested at First National will grow to=$1.0630After 1 year, each dollar invested at Second National will grow to=$1.0617First National pays the higher effective annual rate.


59Practice Problem 59You borrow $1,000 from the bank and agree to repay the loan over the next year in 12equal monthly payments of $90. However, the bank also charges you a loan initiationfee of $20, which is taken out of the initial proceeds of the loan. What is the effectiveannual interest rate on the loan taking account of the impact of the initiation fee?Amount borrowed$1,000.00No.of payments12.00Monthly payments$90.00Loan initiation fee$20.00Solution:Monthly rate=1.527%Effective annual rate=19.94%


60Practice Problem 60You believe you will need to have saved $500,000 by the time you retire in 40 yearsin order to live comfortably. If the interest rate is 6% per year, how much must yousave each year to meet your retirement goal?Amount need to be saved$500,000.00Time period40.00yearsInterest rate6.00%Solution:Savings each year=$3,230.77


61Practice Problem 61You believe you will need to have saved $500,000 by the time you retire in 40 yearsin order to live comfortably. If the interest rate is 6% per year, how much would youneed to save if you believe that you will inherit $100,000 in 10 years?Amount need to be saved$500,000.00Time period40.00yearsInterest rate6.00%Amount inherit$100,000.00Time to inherit10.00yearsSolution:Future Value=574,349.12Therefore, you do not need any additional savings; investing the $100,000produces a future value that exceeds your $500,000 requirement.


62Practice Problem 62You believe you will spend $40,000 a year for 20 years once you retire in 40 years.If the interest rate is 6% per year, how much must you save each year untilretirement to meet your retirement goal?Amount to be spent$40,000.00each yearTime period20.00yearsRetirement40.00yearsInterest rate6.00%Solution:Present Value=$458,796.85Savings each year=$2,964.53


63Practice Problem 63A couple thinking about retirement decide to put aside $3,000 each year in a savings plan thatearns 8% interest. In 5 years they will receive a gift of $10,000 that also can be invested.a.How much money will they have accumulated 30 years from now?b.If their goal is to retire with $800,000 of savings, how much extra do they need to saveevery year?Savings each year$3,000.00Interest rate8.00%Time period30.00yearsGift amount$10,000.00Gift (year)5.00yearsSavings goal$800,000.00Solution:a.Accumulated savings=$408,334.38b.Additional accumulations=$391,665.62Extra savings each year=$3,457.40


64Practice Problem 64A couple will retire in 50 years; they plan to spend about $30,000 a year in retirement, whichshould last about 25 years. They believe that they can earn 8% interest on retirement savings.a.If they make annual payments into a savings plan, how much will they need to save eachyear? Assume the first payment comes in 1 year.b.How would the answer to part (a) change if the couple also realize that in 20 years they willneed to spend $60,000 on their childs college education?Retirement50.00yearsInterest rate8.00%Yearly spending$30,000.00Time period25.00yearsChild's Education amount$60,000.00Time period (Education)20.00yearsSolution:a.Present Value=$320,243.29Savings annuity=$558.14PV of consumption=$6,827.98b.Additional savings=$12,872.89Total PV of savings=$19,700.87Savings each year=$1,610.41


65Practice Problem 65An engineer in 1950 was earning $6,000 a year. Today she earns $60,000 a year.However, on average, goods today cost 8.8 times what they did in 1950. What isher real income today in terms of constant 1950 dollars?Earnings in 1950$6,000.00Current earnings$60,000.00Goods today cost8.8timesSolution:Real Income=$6,818.18Real income increased by=$818.18


66Practice Problem 66If investors are to earn a 3% real interest rate, what nominal interest rate mustthey earn if the inflation rate isa.0%?b.4%?c.6%?Real interest rate3.00%Solution:a.Nominal interest rate=3.00%b.Nominal interest rate=7.12%c.Nominal interest rate=9.18%


67Practice Problem 67If investors receive a 6% interest rate on their bank deposits, what real interest ratewill they earn if the inflation rate over the year isa.0%?b.3%?c.6%?Nominal interest rate6.00%Solution:a.Real interest rate=6.00%b.Real interest rate=2.91%c.Real interest rate=0.00%


68Practice Problem 68You will receive $100 from a savings bond in 3 years. The nominal interest rate is 8%.a.What is the present value of the proceeds from the bond?b.If the inflation rate over the next few years is expected to be 3%, what will the real value ofthe $100 payoff be in terms of todays dollars?c.What is the real interest rate?d.Show that the real payoff from the bond [from part (b)] discounted at the real interest rate[from part (c)] gives the same present value for the bond as you found in part (a).Amount$100.00Time period3.00yearsNominal interest rate8.00%Inflation rate3.00%Solution:a.Present Value=$79.38b.Real Value=$91.51c.Real interest rate=4.854%d.Present Value=$79.38


69Practice Problem 69Your consulting firm will produce cash flows of $100,000 this year, and you expectcash flow to keep pace with any increase in the general level of prices. The interestrate currently is 6%, and you anticipate inflation of about 2%.a.What is the present value of your firms cash flows for years 1 through 5?b.How would your answer to (a) change if you anticipated no growth in cash flow?Cash Flow$100,000.00Interest rate6.00%Inflation rate2.00%Time5.00yearsSolution:a.Real interest rate=3.92%Present Value of Cash flow=$446,184.51b.Present Value of Cash flow=$421,236.00


70Challenge Problem 70Good news: You will almost certainly be a millionaire by the time you retire in 50 years.Bad news: The inflation rate over your lifetime will average about 3%.a.What will be the real value of $1 million by the time you retire in terms of todays dollars?b.What real annuity (in todays dollars) will $1 million support if the real interest rate atretirement is 2% and the annuity must last for 20 years?Time period50.00yearsAmount$1.00millionInflation rate3.00%Real interest rate at retirement2.00%Annuity term20.00yearsSolution:a.Real Value of $1 million=$228,107.00b.Real Value of $1 million=$228,107.00Real Annuity=$13,950.00


71Challenge Problem 71If the interest rate is 6% per year, how long will it take for your money to quadruple in value?If the inflation rate is 4% per year, what will be the change in the purchasing power of yourmoney over this period?Interest rate6.00%Inflation rate4.00%Solution:Time required for money to quadruple=23.79yearsReal interest rate=1.92%Purchasing power increases by=57.84%


72Challenge Problem 72In the summer of 2007, Zimbabwes official inflation rate was about 110% per month.What was the annual inflation rateInflation rate110.00%per monthSolution:Prices increased by=778,635%per year.


73Challenge Problem 73British government 4% perpetuities pay 4 interest each year forever. Another bond,2% perpetuities, pays 2.50 a year forever. What is the value of 4% perpetuities ifthe long-term interest rate is 6%? What is the value of 2% perpetuities?Govt. bond interest rate4.00%Perpetuity interest4.00Other bond interest rate2.50%Perpetuity interest2.50Long term interest rate6.00%Solution:The 4% perpetuity will sell for=66.67The 2% perpetuity will sell for=41.67


74Challenge Problem 74a.You plan to retire in 30 years and want to accumulate enough by then to provide yourselfwith $30,000 a year for 15 years. If the interest rate is 10%, how much must you accumulateby the time you retire?b.How much must you save each year until retirement in order to finance your retirementconsumption?c.Now you remember that the annual inflation rate is 4%. If a loaf of bread costs $1 today,what will it cost by the time you retire?d.You really want to consume $30,000 a year in real dollars during retirement and wish to savean equal real amount each year until then. What is the real amount of savings that you needto accumulate by the time you retire?e.Calculate the required preretirement real annual savings necessary to meet your consumptiongoals. Compare with your answer to (b). Why is there a difference?f.What is the nominal value of the amount you need to save during the first year? (Assume thesavings are put aside at the end of each year.) The thirtieth year?Time30.00yearsAnnuity$30,000.00Interest rate10.00%Term15.00yearsAnnual inflation rate4.00%Loaf of bread cost$1.00Solution:a.Present Value=$228,182.39b.Present Value=$228,182.39PV of the retirement goal=$13,076.80You must save=$1,387.18per yearc.Bread loaf future cost=$3.24d.Real interest rate=5.77%The retirement goal in real terms=$295,796.61e.Real interest rate=5.77%The retirement goal in real terms=$295,796.61Annual Savings=3,895.66This value is much higher than the result found in part (b) because therate at which purchasing power grows is less than the nominal interestrate, 10%.f.Annual Savings$3,895.66Nominal value of amount saved in 1st year=$4,051.49Nominal value of amount saved in 30th year=$12,635.17


75Challenge Problem 75A couple will retire in 50 years; they plan to spend about $30,000 a year in retirement, whichshould last about 25 years. They believe that they can earn 8% interest on retirement savings.Assume that the inflation rate over the next 50 years will average 4%.a.What is the real annual savings the couple must set aside?b.How much do they need to save in nominal terms in the first year?c.How much do they need to save in nominal terms in the last year?d.What will be their nominal expenditures in the first year of retirement? The last?Retirement50.00yearsInterest rate8.00%Yearly spending$30,000.00Time25.00yearsInflation rate4.00%Solution:a.Real interest rate=3.85%PV of the required real savings=$476,182.14Real annual savings=$3,266.82b.Real annual savings$3,266.82Nominal savings in year 1=$3,397.49c.Real annual savings$3,266.82Nominal savings in the last year=$23,216.26d.Nominal expenditures in the first year of retirement=$221,728.52Nominal expenditures in the last year of retirement=$568,357.64


76Challenge Problem 76What is the value of a perpetuity that pays $100 every 3 months forever? Thediscount rate quoted on an APR basis is 6%.Perpetuity Pays$100.00n3.00monthsAPR6.00%Solution:Interest rate per 3 months=1.50%Value of the perpetuity=$6,666.67


77Challenge Problem 77If the interest rate this year is 8% and the interest rate next year will be 10%,what is the future value of $1 after 2 years? What is the present value of apayment of $1 to be received in 2 years?Interest rate current year8.00%Interest rate next year10.00%Value$1.00Solution:Future Value=$1.188Present Value=$0.8418


78Challenge Problem 78Your wealthy uncle established a $1,000 bank account for you when you were born.For the first 8 years of your life, the interest rate earned on the account was 6%.Since then, rates have been only 4%. Now you are 21 years old and ready to cash in.How much is in your account?Deposit$1,000Time period8.00yearsInterest rate up to 8 years6.00%Age21.00Interest rate after 8 years4.00%Solution:Your $1,000 has grown to=$2,653.87


79Challenge Problem 79a.It is 2012, youve just graduated college, and you are contemplating your lifetime budget.You think your general living expenses will average around $50,000 a year. For the next 8years, you will rent an apartment for $16,000 a year. After that, you will want to buy a housethat should cost around $250,000. In addition, you will need to buy a new car roughly onceevery 10 years, costing around $30,000 each. In 25 years, you will have to put aside around$150,000 to put a child through college, and in 30 years youll need to do the same foranother child. In 50 years, you will retire, and will need to have accumulated enough savingsto support roughly 20 years of retirement spending of around $35,000 a year on top of yoursocial security benefits. The interest rate is 5% per year. What average salary will you needto earn to support this lifetime consumption plan?b.Whoops! You just realized that the inflation rate over your lifetime is likely to average about3% per year, and you need to redo your calculations. As a rough cut, it seems reasonable toassume that all relevant prices and wages will increase at around the rate of inflation. Whatis your new estimate of the required salary (in todays dollars)?General living expenses$50,000.00per yearTime period 18.00yearsRent$16,000.00Time period 29.00yearsHouse Cost$250,000.00New Car$30,000.00Time period 310.00yearsTime period 425.00yearsChild 1 College$150,000.00Time period 530.00yearsChild 2 College$150,000.00Time period 650.00yearsTime period 720.00yearsSpending$35,000.00Interest Rate5.00%Inflation Rate3.00%Solution:a.Present Values:General living expenses$912,796.00Apartment rental$103,411.00Home purchase$161,152.00Automobile purchases:0years$30,000.0010years$18,417.0020years$11,307.0030years$6,941.0040years$4,261.0050years$2,616.00$73,542.00College education$79,002.00Retirement portfolio$38,036.00All lifetime expenditures$1,367,939.00Average Salary$74,931.00b.Real Rate of Interest1.940%Unrounded interest1.94175%Present Values:General living expenses$1,591,184.00Apartment rental$117,511.00Home purchase$210,300.00Automobile purchases0years$30,000.0010years$24,756.0020years$20,428.0030years$16,857.0040years$13,910.0050years$11,479.00$117,430.00College education$177,070.00Retirement portfolio$220,210.00All lifetime expenditures$2,433,705.00Average Salary$76,475.00


80Challenge Problem 80Suppose you take out a $100,000, 20-year mortgage loan to buy a condo. The interest rateon the loan is 6%, and to keep things simple, we will assume you make payments on theloan annually at the end of each year.a.What is your annual payment on the loan?b.Construct a mortgage amortization table in Excel in which you compute the interestpayment each year, the amortization of the loan, and the loan balance each year.(Allow the interest rate to be an input that the user of the spreadsheet can enter andchange.)c.What fraction of your initial loan payment is interest? What fraction is amortization? Whatabout the last loan payment? What fraction of the loan has been paid off after 10 years(halfway through the life of the loan)?d.If the inflation rate is 2%, what is the real value of the first (year-end) payment? The last?e.Now assume the inflation rate is 8% and the real interest rate on the loan is unchanged.What must be the new nominal interest rate? Recompute the amortization table. Whatis the real value of the first (year-end) payment in this high-inflation scenario? Thereal value of the last payment?Loan amount$100,000.00Time20.00yearsInterest rate6.00%Inflation rate-12.00%Inflation rate-28.00%Solution:a.Annuity factor=11.4699Annual payment on loan=$8,718.47b.Annual payment=$8,718.46YearBeginning BalanceYear End Interest DueYear End PaymentAmortization of loanEnd of year balance1$100,000.00$6,000.00$8,718.46$2,718.46$97,281.54297,281.545,836.898,718.462,881.5694,399.98394,399.985,664.008,718.463,054.4691,345.52491,345.525,480.738,718.463,237.7288,107.80588,107.805,286.478,718.463,431.9984,675.81684,675.815,080.558,718.463,637.9181,037.91781,037.914,862.278,718.463,856.1877,181.72877,181.724,630.908,718.464,087.5573,094.17973,094.174,385.658,718.464,332.8168,761.371068,761.374,125.688,718.464,592.7764,168.591164,168.593,850.128,718.464,868.3459,300.251259,300.253,558.028,718.465,160.4454,139.811354,139.813,248.398,718.465,470.0748,669.751448,669.752,920.188,718.465,798.2742,871.471542,871.472,572.298,718.466,146.1736,725.311636,725.312,203.528,718.466,514.9430,210.371730,210.371,812.628,718.466,905.8323,304.541823,304.541,398.278,718.467,320.1815,984.351915,984.35959.068,718.467,759.398,224.96208,224.96493.508,718.468,224.96(0.00)c.Initial loan payment:Interest fraction=69.00%Amortization fraction=31.00%Last loan payment:Interest fraction=6.00%Fraction of loan paid off after 10 years=36.00%d.Real interest rate=4.00%Real value of the first payment=$8,383.13Real value of the last payment=$3,978.99e.Real interest rate=4.00%Nominal interest rate=12.00%Real value of the first payment=$12,872.96Real value of the last payment=$6,110.05YearBeginning BalanceYear End Interest DueYear End PaymentAmortization of loanEnd of year balance1$100,000.00$12,000.00$13,387.88$1,387.88$98,612.12298,612.1211,833.4513,387.881,554.4297,057.70397,057.7011,646.9213,387.881,740.9595,316.74495,316.7411,438.0113,387.881,949.8793,366.88593,366.8811,204.0313,387.882,183.8591,183.02691,183.0210,941.9613,387.882,445.9288,737.11788,737.1110,648.4513,387.882,739.4385,997.68885,997.6810,319.7213,387.883,068.1682,929.53982,929.539,951.5413,387.883,436.3379,493.191079,493.199,539.1813,387.883,848.7075,644.501175,644.509,077.3413,387.884,310.5471,333.961271,333.968,560.0713,387.884,827.8066,506.161366,506.167,980.7413,387.885,407.1461,099.021461,099.027,331.8813,387.886,056.0055,043.021555,043.026,605.1613,387.886,782.7248,260.301648,260.305,791.2413,387.887,596.6440,663.661740,663.664,879.6413,387.888,508.2432,155.421832,155.423,858.6513,387.889,529.2322,626.201922,626.202,715.1413,387.8810,672.7311,953.462011,953.461,434.4213,387.8811,953.460.00


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