Problem Set CH 5 TVM

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Florida International University *

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Jan 9, 2024

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Problem Set Chapter 5 (TVM) (Please see the syllabus, this is not graded) 1. What is an opportunity cost? How is this concept used in TVM analysis, and where is it shown on a time line? Is a single number used in all situations? Explain. The opportunity cost is the rate of interest one could earn on an alternative investment with a risk equal to the risk of the investment in question. This is the value of I in the TVM equations, and it is shown on the top of a time line, between the first and second tick marks. It is not a single rate—the opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also varies from year to year depending on inflationary expectations (see Chapter 6). 2. Would you rather have a savings account that pays 5% interest compounded semiannually or one that pays 5% interest compounded daily? Explain. For the same stated rate, daily compounding is best. You would earn more “interest on interest.” 3. The present value of a perpetuity is equal to the payment on the annuity, PMT, divided by the interest rate, I: PV=PMT/I. What is the future value of a perpetuity of PMT dollars per year? (Hint: The answer is infinity, but explain why.) The concept of perpetuity implies that payments will be received forever. FV Perpetuity = PV Perpetuity (1 + I)  =  . 4. You want to buy a car, and a local bank will lend you $40,000. The loan will be fully amortized over 5 years (60 months), and the nominal interest rate will be 8% with interest paid monthly. What will be the monthly loan payment? What will be the loan’s EAR? Using a financial calculator, enter the following: N = 60, I/Y = 8/12 = 0.6667, PV = -40000, and FV = 0. Solve for PMT = $811.06. EAR= ( 1 + I m ) m − 1 = (1.00667) 12 – 1.0 = 8.30%. Alternatively, using a financial calculator, enter the following: 2 nd then “2” then ENTER: NOM = 8 and C/Y = 12. Solve for EFF% = 8.30%. 5. Find the present value of $200 due in 10 years at 4%. $200/(1.04) 10 = $135.11. Using a financial calculator, enter N = 10, I/Y = 4, PMT = 0, FV = 200, and PV = ? Solve for PV = $135.11. 6. Sawyer Corporation’s 2020 sales were $5 million. Its 2015 sales were $2.5 million. At what rate have sales been growing? With a calculator, enter N = 5, PV = -6, PMT = 0, FV = 12, and then solve for I/Y = 14.87%.

7. Find the interest rates earned on each of the following: a. You borrow $720 and promise to pay back $792 at the end of 1 year. b. You lend $720, and the borrower promises to pay you $792 at the end of 1 year. c. You borrow $65,000 and promise to pay back $98,319 at the end of 14 years. d. You borrow $15,000 and promise to make payments of $4,058.60 at the end of each year for 5 years. These problems can all be solved using a financial calculator by entering the known values shown on the time lines and then pressing the I/Y button. a. With a financial calculator, enter: N = 1, PV = 720, PMT = 0, and FV = -792. I/Y = 10%. b. With a financial calculator, enter: N = 1, PV = -720, PMT = 0, and FV = 792. I/Y = 10%. c. With a financial calculator, enter: N = 14, PV = 65000, PMT = 0, and FV = -98319. I/Y = 3%. d. With a financial calculator, enter: N = 5, PV = 15000, PMT = -4058.60, and FV = 0. I/Y = 11%. 8. What is the present value of a $600 perpetuity if the interest rate is 5%? If interest rates doubled to 10%, what would its present value be? PV 5% = $600/0.05 = $12,000. PV 10% = $600/0.10 = $6,000. When the interest rate is doubled, the PV of the perpetuity is halved.

9. Jan sold her house on December 31 and took a $10,000 mortgage as part of the payment. The 10-year mortgage has a 10% nominal interest rate, but it calls for semiannual payments beginning next June 30. Next year Jan must report on Schedule B of her IRS Form 1040 the amount of interest that was included in the two payments she received during the year. a. What is the dollar amount of each payment Jan receives? b. How much interest was included in the first payment? How much repayment of principal was included? How do these values change for the second payment? c. How much interest must Jan report on Schedule B for the first year? Will her interest income be the same next year? d. If the payments are constant, why does the amount of interest income change over time? a. This can be done with a calculator by specifying an interest rate of 5% per period for 20 periods with 1 payment per period. N = 10  2 = 20, I/Y = 10/2 = 5, PV = -10000, FV = 0. Solve for PMT = $802.43. b. Set up an amortization table: AMORT: FOR PERIOD 1: P1=1 P2=1 GIVES BAL=9,697.5741 PRN=302.4259 INT=500 FOR PERIOD 2: P1=2 P2=2 GIVES BAL=9,380.0269 PRN=317.5472 INT=484.8787 NOTE : ONE RELATION BETWEEN PRINCIPAL PAYMENT IS : PRN (T+1) =PRN T *(1+i/m) , since a payment towards principal “saves” interest charge of i/m. 317.5472=302.4259 (1+5%) Beginning Payment of Ending Period Balance Payment Interest Principal Balance 1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57 2 9,697.57 802.43 484.88 317.55 9,380.02 $984.88 Because the mortgage balance declines with each payment, the portion of the payment that is applied to interest declines, while the portion of the payment that is applied to principal increases. The total payment remains constant over the life of the mortgage. c. Jan must report interest of $984.88 on Schedule B for the first year. Her interest income will decline in each successive year for the reason explained in part b. d. Interest is calculated on the beginning balance for each period, as this is the amount the lender has loaned and the borrower has borrowed. As the loan is amortized (paid off), the beginning balance, hence the interest charge, declines and the repayment of principal increases.

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10. Find the future values of the following annuities: a. FV of $400 paid each 6 months for 5 years at a nominal rate of 12% compounded semiannually b. FV of $200 paid each 3 months for 5 years at a nominal rate of 12% compounded quarterly c. These annuities receive the same amount of cash during the 5-year period and earn interest at the same nominal rate, yet the annuity in part b ends up larger than the one in part a. Why does this occur? a. Enter N = 5  2 = 10, I/Y = 12/2 = 6, PV = 0, PMT = -400, and then press FV to get FV = $5,272.32. b. Now the number of periods is calculated as N = 5  4 = 20, I/Y = 12/4 = 3, PV = 0, and PMT = -200. The calculator solution is $5,374.07. c. The annuity in part b earns more because the money is on deposit for a longer period of time and thus earns more interest. Also, because compounding is more frequent, more interest is earned on interest. 11. As a jewelry store manager, you want to offer credit, with interest on outstanding balances paid monthly. To carry receivables, you must borrow funds from your bank at a nominal 9%, monthly compounding. To offset your overhead, you want to charge your customers an EAR (or EFF%) that is 3% more than the bank is charging you. What APR rate should you charge your customers? Here you want to have an effective annual rate on the credit extended that is 3% more than what the bank is charging you, so you can cover overhead. First, we must find the EAR = EFF% on the bank loan. Enter NOM = 9, C/Y = 12, and compute EFF to get EAR = 9.38069%. So, to cover overhead you need to charge customers a nominal rate so that the corresponding EAR = 12.38069%. To find this nominal rate, enter EFF= 12.38069, C/Y = 12, and compute NOM to get I NOM = 11.729145%  11.73%. (Customers will be required to pay monthly, so m = 12.) 12. Starting next year, you will need $5,000 annually for 4 years to complete your education. (One year from today you will withdraw the first $5,000.) Your uncle deposits an amount today in a bank paying 6% annual interest, which will provide the needed $5,000 payments. a. How large must the deposit be? b. How much will be in the account immediately after you make the first withdrawal? a. With a calculator, enter N = 4, I/Y = 6, PMT = -5000, and FV = 0. Then compute PV to get PV = $17,325.53. b. At this point, we have a 3-year, 6% annuity whose value is $13,365.06. You can also think of the problem as follows: $17,325.53(1.06) – $5,000 = $13,365.06.

13. A. Set up an amortization schedule for a $19,000 loan to be repaid in equal installments at the end of each of the next 3 years. The interest rate is 8% compounded annually. a. What percentage of the payment represents interest and what percentage represents principal for each of the 3 years? Why do these percentages change over time? a. With a financial calculator, enter N = 3, I/Y = 8, PV = -19000, and FV = 0, and solve for PMT = $7,372.64. Then go through the amortization procedure by entering P1 and P2 values for each of the periods, in this question there are three periods hence enter P1=P2= 1, fill the table then P1=P2= 2, etc. Beginning Repayment Remaining Year Balance Payment Interest of Principal Balance 1 $19,000.00 $ 7,372.64 $1,520.00 $5,852.64 $13,147.36 2 13,147.36 7,372.64 1,051.79 6,320.85 6,826.51 3 6,826.51 7 ,372.64 546.13 6 ,826.51 0 $22 ,117.92 $3 ,117.92 $19 ,000.00 14. a. You plan to make five deposits of $1,000 each, one every 6 months, with the first payment being made in 6 months. You will then make no more deposits. If the bank pays 6% nominal interest, compounded semiannually, how much will be in your account after 3 years? b. One year from today you must make a payment of $4,000. To prepare for this payment, you plan to make two equal quarterly deposits (at the end of Quarters 1 and 2) in a bank that pays 6% nominal interest compounded quarterly. How large must each of the two payments be? a. Since the first payment is made 6 months from today, we have a 5-period ordinary annuity. The applicable interest rate is 6%/2 = 3%. First, we find the FVA of the ordinary annuity in period 5 by entering the following data in the financial calculator: N = 5, I/Y = 6/2 = 3, PV = 0, and PMT = -1000. We find FVA 5 = $5,309.14. Now, we must compound this amount for 1 semiannual period at 3%. $5,309.14(1.03) = $5,468.41. b. Step 1: Discount the $4,000 back 2 quarters to find the required value of the 2-period annuity at the end of Quarter 2, so that its FV at the end of the 4 th quarter is $4,000. Using a financial calculator enter N = 2, I/Y = 1.5, PMT = 0, FV = 4000, and solve for PV = $3,882.65. Step 2: Now you can determine the required payment of the 2-period annuity with a FV of $3,882.65. Using a financial calculator, enter N = 2, I/Y = 1.5, PV = 0, FV = 3882.65, and solve for PMT = $1,926.87.

15. In how many years will weekly deposits of $100 grow to $100,000, if the annual rate of interest is 12 percent? Answer: m=52 FV=100000 I=12/52 N=52*n PMT=-100 CPT N=518.9732 weekly payments, then n=518.9732 /52=9.9803 years 16. In how many years will weekly deposits of $100 grow to $100,000, if the annual rate of interest is 12%, compounded monthly? Answer: m=52 but the interest rate is not compounded weekly, hence find the weekly compounded interest rate first. EAR of 12%, compounded monthly (using formula or ICONV: press 2 nd and 2) NOM=12 C/Y=m=12 EFF CPT=12.6825 now find the weekly compounded interest rate the calculated EAR: EAR of 12%, compounded monthly (using formula or ICONV: press 2 nd and 2) C/Y=m=52 EFF=12.6825 NOM CPT=11.9541 Now solve the annuity problem m=52 I=11.9541/52 N=52*n FV=100000 PMT=-100 CPT N=519.7996 weekly payments, then n=518.9732 /52=9.9961 years 17. If you have $40,000 in an account that earns 10% compounded quarterly, how many years will it take to deplete the account with withdrawals of $1,000 every quarter? Answer: m=4 I=10/4 PMT=1,000 PV=-40,000 CPT N: “Error 1” check periodic interest: 40,000 * (1+10%/4)=1,000. Account will never deplete with a withdrawal of $1,000 18. If you have $40,000 in an account that earns 10% compounded quarterly, how many years will it take to deplete the account with withdrawals of $1,100 every quarter? Answer: m=4 I=10/4 PMT=1,000 PV=-40,000 CPT N: 97.1098 N=97.1098/4=24.2775 years

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19. Consider a mortgage contract with monthly payments of 1,200 for 30 years. If the interest rate is 5%, what is the present value of the mortgage loan payments? m=12 I/Y=5/12 PMT=-1,200 N=30*12 CPT PV=223,537.9405 20. Consider a mortgage loan of 223,537.9405 with monthly payments for 30 years. If the interest rate is 5%, find the monthly payments? Same as the previous question, hence the PMT=1,200. Why? Because in the previous question the present value of all the future payments of $1,200 have a value of 223,537.9405 NOW. Formula enables us to compare present time to future: If you need 223,537.9405 with 5% interest rate, you must pay monthly payments of $1,200 for the next 30 years, because they are the same! That is why the bank is lending you 223,537.9405 now, in exchange for monthly payments of $1,200 in the future. 21. Refer to the previous question, what is the future value of the mortgage payments at the end of the 30th year? (That is, how much would you have at the end of the 30th year if you have invested monthly payments of $1,200 for 30 years at %5 instead of getting the loan? (or how much will the bank have in the account after your payments for 30 years?) Answer: m=12 I/Y=5/12 PMT=-1,200 N=30*12 CPT FV=$998,710.3624 22. If you deposit $223,537.9405 in an account with 5% compounded monthly, how much will you have in the account after 30 years? Answer: Same as previous question! FV=$998,710.3624 Because: The FV of mortgage is $998,710.3624 and its PV was calculated as $223,537.9405. Check the formula: PVA = PMT [ 1 − 1 ( 1 + i / m ) m×n ] i / m and FVA = PMT [ ( 1 + i / m ) m×n − 1 ] i / m PVA find the value of the annuity at the time of one period before the first payment, and FVA finds the value of the SAME annuity at the time when last payment is made. It’s the same annuity but its value at different times and since there is time value of money, PV and FV differ. That is why there is always the link between FV and PV: FV = PV ( 1 + i / m ) m× n

23. Consider a 30-year mortgage loan of $4,028.3971, with an interest rate of 12.00%, assuming monthly payments. The mortgage balance remaining after making 5 payments is ________ and the total interest expense for the 11 th year is _________. Answer: Study the video and the handout for Amortization to understand the steps below: m=12 since monthly payments. First find the payments: Key N I/Y PV PMT FV Enter 30*12 12.00/12 4,028.3971 CPT -41.4366 The mortgage balance remaining after the 5th payment: AMORT P1=5 P2=5 yields MB (t=5)= 4,022.52 Total interest expense in the 11 th year: REMARK: m=12, that is each year has 12 compounding intervals. Therefore, 11 th year has 12 compounding intervals, too. The first payment in the 11 th year is the 121 th payment: Monthly payments are made for (11- 1) years already, that is a total of (11-1)*12 payments. The first payment in year 11 is 121th payment: 120 monthly payments were made and this is the first monthly payment in the 11 th year. AMORT P1=(11-1)*12+1=121 P2=11*12=132 yields TIE in the 11th year is: $ 448.99 24. Mike borrowed some money from a bank to repay the amounts of $1,225, $1,350, $1,500, $1,850, and $2,432 over the next five years. If the interest rate is 6 percent, how much did Mike borrow? Since uneven payments, find the PV of each payment FV = -1,225; N=1; I= 6; CPT PV = 1,155.6604 FV = -1,350; N=2; I= 6; CPT PV = 1,201.4952 FV = -1,500; N=3; I= 6; CPT PV = 1,259.4286 FV = -1,850; N=4; I= 6; CPT PV = 1,465.3733 FV = -2,432; N=5; I= 6; CPT PV = 1,817.3319 Now since all five payments are at time t= 0, add them up: $6,899.2896 25. What is the FV of the loan in the previous question? Answer: we can calculate the FV of each payment or, use the identity FV = PV ( 1 + i m ) m×n PV=6,899.2896; N=5; I/Y=6; CPT FV: $9,232.8059

26. Han is saving for a trip in three years. He will need $5,000 to cover all of his expenses. If he can invest his money in a mutual fund that is expected to earn an average return of 10.3% over the next three years, how much will he have to save every year, starting at the end of this year? N=3 FVA = $5,000 I/Y=10.3 PMT, CPT PMT=1,506.2015 He has to save $1,506.20 every year for the next three years to reach his target of $5,000. 27. Martin is planning to make quarterly deposits of $180 into a fund for 9 years. The fund is expected to pay 12% rate of interest, compounded monthly. Find the present value of the annuity. Answer: m=4 since the deposits are made “quarterly” HOWEVER, The frequency of annuity deposits and the compounding frequency does NOT match: Interest rate is compounded monthly! FIND THE INTEREST RATE “quarterly” compounding, and has the same EAR as the 12% rate of interest, compounded monthly: Find the EAR of the 12% interest rate compounded monthly, then use the EAR to find the nominal interest rate that has “quarterly” compounding. EAR=(1+k/m) m -1 or use ICONV function on calculator. EAR=(1+12%/12) 12 -1=0.1268 STORE then RECALL later: Now find k that has “quarterly” compounding and EAR of 0.1268: 0.1268 =(1+k/4) 4 -1 gives k= 12.1204% is the interest rate compounded quarterly! Now the frequency of annuity per year matches with interest rate compounding frequency. Key: N=m*n I/Y PV PMT FV Enter: 36 12.1204/4 -180 Solve For: 3,912 28. You started depositing $259 in an account every month for 18 months that paid 2% rate of interest compounded monthly. The present value of the annuity is ________. Answer: The frequency of annuity deposits and the compounding frequency match: interval = month Key: N I/Y PV PMT FV Enter: 18 2/12 -259 Solve For: 4,589.00

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29. You will receive $5,917 at year 2089. What is its present value at the beginning of year 2086 if the rate of interest is 13 percent? Answer: Beginning of 2086 is the end of 2085. The FV should be discounted four times, since from year 2085 to 2089, there are four compounding intervals. Key: N I/Y PV PMT FV Enter: 4 13 5,917 Solve For: -3,629.01 or formula: PV=FV/(1+i) n =$3,629.01 30. What is the future value of $685 at the end of 15 years, assuming continuously compounded interest rate of 3.9 percent? Answer: FV=PVe Nk =1,229.57 31. You started depositing annuities of an unknown amount in an account every year that paid 11.5 percent rate of interest per year. After 14 annual deposits your account has $4,224,014. The present value of annuity is Answer: Since the future value of annuity is given but not the payment amount, we can use the relationship between PV and FV to find the PVA: PVA=FVA/(1+i/m) N PVA=4,224,014/(1+11.5%) 14 =920,189 Or, financial calculator: Key: N I/Y PV PMT FV Enter: 14 11.5 4,224,014 Solve For: 920,189 32. If the interest rate is 14.5%, how many years will it take to double your investment? Present value of the investment = PV = $1 FV=$2 Return on investment= i = 14.5% Number of compounding intervals = ? FV= [ PV x (1 + 14.5/100) N ] Key: N I/Y PV PMT FV Enter: 14.5 -1 0 2 Solve For: 5.1191

33. If the simple interest rate is 6.5 percent, how many years will it take to double your investment? Answer: FV=$2, PV=$1, FV=PV(1+nk) 2=1(1+n*6.5/100) n=100*(2-1)/0.035=15.3846 years 34. Julie is planning to invest in a bank account. Bank A offers 4.10% rate of interest, compounded semi- annually. Bank B offers 4.30% rate of interest, compounded daily. Bank C offers 3.50% rate of interest, compounded continuously. Julie, being a Finance major, wisely compares the effective annual rates (EARs) of the three offers and chooses the bank that yields the highest EAR of ________ for her investment. Answer: Bank A : m=2. Since Bank A compounds semi-annually. Using ICONV (press “2 nd ” and then “2”) on calculator: Key: NOM C/Y EFF Enter: 4.10 2 Scroll for EFF then CPT , this is EAR A: 4.14 EAR A: 4.14% Bank B: m=365. Since Bank B compounds daily. Key: NOM C/Y EFF Enter: 4.30 365 Scroll for EFF then CPT, this is EAR B: 4.39 EAR B: 4.39% Bank C compounds continuously, use the formula ( don’t forget to write the formula for continuous compounding on your formula sheet , since financial calculator can’t solve the continuous case) EAR=e annual interest rate -1 EAR C=e 0.0350 -1 Using e x on the calculator: 0.0350 then (press “2 nd ” and then “LN”) - 1 = 0.0356= 3.56% Comparing EAR of Bank A: 4.14% with EAR of Bank B: 4.39%, and EAR of Bank C=3.56%, the highest EAR is 4.4%

35. Han Zolo wants to invest in a bank certificate of deposit that will pay him 15 percent interest, compounded monthly. If he is investing $2,106 today, how many years later will he reach his goal of $586,217? Answer: Remark: PV and FV should have opposite signs, otherwise you’ll get an ERROR 5 code. PV is a cash out flow, so should be negative, FV is a cash in flow, so it’s positive. m=12. Since compounded monthly . I/Y=interest rate/m=15/12 Key: N=m*n I/Y PV PMT FV Enter: 15/12 -2,106 0 586,217 Solve For: 453.1206 N=m*n=453.1206, need to solve for n : Answer in years = n= 453.1206/12=38 years. 36. Aristotle plans to save for a down payment on a house he will buy 7 years later. He will be able to invest $101,410 today in an equity fund that will pay him an interest rate of 7.9 percent, compounded quarterly. If his target down payment, which is due 7 years later, is $175,230, will he be able to meet his target at the end of 7 years? a. No b. Maybe c. Yes ANS: C FV=PV(1+i/m)^m*n), m=4, n=7 Key: N I/Y PV PMT FV Enter: 4* 7 7.9/4 -101,410 Solve For: 175,349.5979 He will have $175,349.5979 in his account 5 years later, when he needs a down payment of $175,230. Yes, since $175,349.5979-$175,230=$120

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Problem Set CH 5 TVM

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