chapter 5, solutions cornett, adair, and nofsinger
TRANSCRIPT
CHAPTER 5 – ANALYSIS OF ANNUITY CASH FLOWSChapter 5, Solutions
Cornett, Adair, and Nofsinger
CHAPTER 5 – Time Value of Money 2: ANALYZING ANNUITY CASH FLOWS
Questions
LG1 5-1 How can you add a cash flow in year two and a cash flow in year four in year seven?
To add cash flows, they need to be moved to the same time period. The cash flows in years two and four should be moved forward with interest to year seven, then they can be added together.
LG2 5-2 People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs). Demonstrate the growth of a $250 monthly contribution for 40 years earning 9 percent APR.
Using equation 5-2, we have:
07.330,170,1$
0.09/12
480
40
LG3 5-3 When you discount multiple cash flows, how does the future period that a cash flow is paid affect its present value and its contribution to the value of all the cash flows?
Discounting reduces a future cash flow to a smaller present value. Cash flows far into the future become very small when discounted to the present. Thus, cash flows in distant future periods have small impacts on present values.
LG4 5-4 How can you use the present value of an annuity concept to determine the price of a house you can afford?
Mortgages are typically for a large enough amount of money that borrowing is required to purchase a home. The amount that one can afford for a home is a function of their current state of wealth. Mortgages allow consumers to spread the expense of a home over a longer period, typically 15 or 30 years. This allows consumers to put a smaller portion of wealth into the home (for example, a 20% down payment) and borrow the balance over the life of the loan. Due to the effect of annuity compounding, the payments for such a long lived debt make the monthly payments of a manageable nature so that they can be paid from current income.
LG5 5-5 Since perpetuity payments continue for ever, how can a present value be computed? Why isn’t the present value infinite?
5-1
Chapter 5, Solutions Cornett, Adair, and Nofsinger
Equation 5-5 is used to calculate the present value of a perpetuity. It is a limiting version of equation 5-4 in which the period N grows infinitely large. As this occurs the expression following the “1” in equation 5-4 drives to the value 0 and the numerator simply become “1.” The present value is not infinite since the terms following the PMT in equation 5-4 converge to a finite limit of 1/i. This also demonstrates how payments far into the future have infinitesimal value today.
LG6 5-6 Explain why you use the same adjustment factor, (1 + i), when you adjust annuity due payments for both future value and present value.
Adjusting an annuity due calculation involves shifting the entire series of payments forward one period. This is accomplished by multiplying by (1 + i) irrespective of whether it is a future value or present value calculation.
LG7 5-7 Use the idea of compound interest to explain why EAR is larger than APR.
The annual percentage rate does not take into account the frequency of interest compounding. Equation 5-8 illustrates the conversion from APR to EAR. The effective annual rate converts the annual percentage rate to a rate that can be compared to other annual rates.
LG8 5-8 Would you rather pay $10,000 for a five year $2,500 annuity or a ten-year $1,250 annuity? Why?
The effective annual rates for these two payment streams are 7.93% and 4.28% respectively. I would rather pay $10,000 for a five year $2,500 annuity as it earns a higher effective annual rate of interest.
LG9 5-9 The interest on your home mortgage is tax deductible. Why are the early years of the mortgage more helpful in reducing taxes than in the later years?
Mortgage payments at the beginning of the amortization schedule are predominantly interest with little principal. In later years, interest payments decline and principal payments make up an ever increasing part of the payments. Thus, the tax deductible part (the interest payment) is larger in the beginning years.
LG10 5-10 How can you use the concepts illustrated in computing the number of payments in an annuity to figure how to pay off a credit card balance? How does the magnitude of the payment impact the number of months?
Utilizing equation 5-2, you can declare the present balance for the credit card and set that equal to the PVA. The interest can be obtained as an APR and converted to and EAR using equation 5-8. This is the value to put into “i” in equation 5-2. You then decide when you want to have the credit card paid off and convert this to a monthly value of N in equation 5-
5-2
Chapter 5, Solutions Cornett, Adair, and Nofsinger
2. Solving for PMT will yield the amount needed to pay the credit card off in the time frame you desire, assuming that no additional charges are made to the credit card and that the interest rate remains level.
Problems Basic Problems 5-1 Future Value Compute the future value in year 8 of a $1,000 deposit in year 1 and LG1 another $1,500 deposit at the end of year 3 using a 10% interest rate.
Use equation 5-1:
FV = $1,000 × (1 + 0.10)7 + $1,500× (1 + 0.10)5 = $1,948.72 + $2,415.77 = $4,364.48
LG1 5-2 Future Value Compute the future value in year 7 of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using a 8% interest rate..
Use equation 5-1:
FV = $2,000 × (1 + 0.08)6 + $2,500 × (1 + 0.08)3 = $3,173.75 + $3,149.28 = $6,323.03
LG2 5-3 Future Value of an Annuity What is the future value of a $500 annuity payment over 5 years if interest rates are 9 percent?
Use equation 5-2:
5
5
LG2 5-4 Future Value of an Annuity What is the future value of a $700 annuity payment over 4 years if interest rates are 10 percent?
Use equation 5-2:
4
4
LG3 5-5 Present Value Compute the present value of a $1,000 deposit in year 1 and another $1,500 deposit at the end of year 3 if interest rates are 10 percent.
Use equation 5-3:
PV = $1,000 ÷ (1 + 0.10)1 + $1,500 ÷ (1 + 0.10)3 = $909.09 + $1,126.97 = $2,036.06
LG3 5-6 Present Value Compute the present value of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using an 8% interest rate.
5-3
Use equation 5-3:
PV = $2,000 ÷ (1 + 0.08)1 + $2,500 ÷ (1 + 0.08)4 = $1,851.85 + $1,837.57 = $3,689.43
LG4 5-7 Present Value of an Annuity What’s the present value of a $500 annuity payment over 5 years if interest rates are 9 percent?
Use equation 5-4:
LG4 5-8 Present Value of an Annuity What’s the present value of a $700 annuity payment over 4 years if interest rates are 10 percent?
Use equation 5-4:
LG5 5-9 Present Value of a Perpetuity What’s the present value, when interest rates are 7.5 percent, of a $50 payment made every year forever?
Use equation 5-5:
50$ perpetuity a of PV
LG5 5-10 Present Value of a Perpetuity What’s the present value, when interest rates are 8.5 percent, of a $75 payment made every year forever?
Use equation 5-5:
75$ perpetuity a of PV
LG6 5-11 Present Value of an Annuity Due If the present value of an ordinary, 7-year annuity is $6,500 and interest rates are 8.5 percent, what’s the present value of the same annuity due?
Use equation 5-7:
PVA7 due = 6,500 × (1 +0.085) = $7,052.50
LG6 5-12 Present Value of an Annuity Due If the present value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what’s the present value of the same annuity due?
Use equation 5-7:
PVA6 due = 8,500 × (1 +0.095) = $9,307.50
LG6 5-13 Future Value of an Annuity Due If the future value of an ordinary, 7-year annuity is $6,500 and interest rates are 8.5 percent, what is the future value of the same annuity due?
Use equation 5-6:
FVA7 due = 6,500 × (1 + 0.085) = $7,052.50
(Note this is the same answer as problem 5-11, as expected)
LG6 5-14 Future Value of an Annuity Due If the future value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what’s the future value of the same annuity due?
Use equation 5-6:
FVA6 due = 8,500 × (1 + 0.095) = $9,307.50
(Note this is the same answer as problem 5-12, as expected)
LG7 5-15 Effective Annual Rate A loan is offered with monthly payments and an 11 percent APR. What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
EAR
LG7 5-16 Effective Annual Rate A loan is offered with monthly payments and a 12 percent APR. What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
EAR
Intermediate Problems 5-17 Future Value Given a 6 percent interest rate, compute the year 6 future value of
deposits made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500. LG1
Use equation 5-1:
FV = $1,000 × (1 + 0.06)5 + $1,200× (1 + 0.06)4 + $1,200 × (1 + 0.06)3 + $1,500× (1 + 0.06)2
5-5
FV = $1,338.23 + $1,514.97 + $1,429.22 + $1,685.40 = $5,967.82
LG1 5-18 Future Value Given a 7 percent interest rate, compute the year 6 future value of deposits made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400.
Use equation 5-1:
FV = $1,000 × (1 + 0.07)5 + $1,300× (1 + 0.07)4 + $1,300 × (1 + 0.07)3 + $1,400× (1 + 0.07)2
FV = $1,402.55 + $1,704.03 + $1,592.56 + $1,602.86 = $6,302
LG2 5-19 Future Value of Multiple Annuities Assume that you contribute $200 per month to a retirement plan for 20 years. Then you are able to increase the contribution to $400 per month for another 20 years. Given a 7 percent interest rate, what is the value of your retirement plan after the 40 years?
Break the annuity streams into a level stream of payments of $200 for 40 years and another level stream of payments of $200 for the last 20 years. Use equation 5-2 for each payment stream and add the results:
01.148,629$9267.520200$2415.796,2200$
0.07/12
240480
2040
LG2 5-20 Future Value of Multiple Annuities Assume that you contribute $150 per month to a retirement plan for 15 years. Then you are able to increase the contribution to $350 per month for the next 25 years. Given an 8 percent interest rate, what is the value of your retirement plan after the 40 years?
Break the annuity streams into a level stream of payments of $150 for 40 years and another level stream of payments of $200 for the last 25 years. Use equation 5-2 for each payment stream and add the results
45.856,713$0264.951200$0078.491,3150$
0.08/12
300480
2540
LG3 5-21 Present Value Given a 6 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500.
Use equation 5-3:
PV = $1,000 ÷ (1 + 0.06)1 + $1,200 ÷ (1 + 0.06)2 + $1,200 ÷ (1 + 0.06)3 + $1,500 ÷ (1 + 0.06)4
PV = $943.40 + $1,068 + $1,007.54 + $1,188.14 = $4,207.08
LG3 5-22 Present Value Given a 7 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400.
Use equation 5-3:
Chapter 5, Solutions Cornett, Adair, and Nofsinger
PV = $1,000 ÷ (1 + 0.07)1 + $1,300 ÷ (1 + 0.07)2 + $1,300 ÷ (1 + 0.07)3 + $1,400 ÷ (1 + 0.07)4
PV = $934.58 + $1,135.47 + $1,061.19 + $1,068.05 = $4,199.29
LG2 5-23 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. The owner states that he can repay a loan at $1,000 per month for the next three years and then $2,000 per month for two years after that. If the bank is charging customers 7.5 percent APR, how much would it be willing to lend the business owner?
Break the annuity streams into a level stream of payments of $2,000 for 5 years and another level stream of payments of $1,000 for the first 3 years. Use equation 5-4 for each payment stream and subtract the results.
70.662,67$92.147,32$62.810,99$
LG2 5-24 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. The owner states that she can repay a loan at $1,500 per month for the next three years and then $500 per month for two years after that. If the bank is charging customers 8.5 percent APR, how much would it be willing to lend the business owner?
Break the annuity into two streams of payments: $500 monthly for five years and $1,000 for three years. Use equation 5-4 for each annuity and add the results.
70.048,56$11.678,31$59.370,24$
LG5 5-25 Present Value of a Perpetuity A perpetuity pays $100 per year and interest rates are 7.5 percent. How much would its value change if interest rates increased to 8.5 percent? Did the value increase or decrease?
Use equation 5-5:
47.176,1$ 085.0
100$ perpetuity a of PV
The difference between these perpetuities is $156.86. The value of the perpetuity decreased with an increase in the interest rate.
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Chapter 5, Solutions Cornett, Adair, and Nofsinger
LG5 5-26 Present Value of a Perpetuity A perpetuity pays $50 per year and interest rates are 9 percent. How much would its value change if interest rates decreased to 8 percent? Did the value increase or decrease?
Use equation 5-5:
00.625$ 08.0
50$ perpetuity a of PV
The difference between these perpetuities is $69.45. The value of the perpetuity increased with a decrease in the interest rate.
LG6 5-27 Future and Present Value of an Annuity Due If you start making $50 monthly contributions today and continue them for 5 years, what’s their future value if the compounding rate is 10 percent APR? What is the present value of this annuity?
Compute the future value using equation 5-2:
98.903,3$00833.14343706824.7750$12/10.01 0.10/12
112/10.01 0$5FVA
88.372,2$00833.106536757.4750$00833.1
LG6 5-28 Future and Present Value of an Annuity Due If you start making $75 monthly contributions today and continue them for 4 years, what is their future value if the compounding rate is 12 percent APR? What is the present value of this annuity?
First calculate the future values and present values, using equations 5-2 and 5-4, respectively. Using these results, the annuity due values can be computed using equations 5-6 and 5-7, respectively.
70.591,4$2226.6175$
0.12/12
5-8
05.848,2$97395951.3775$
PVA48 due = $2,848.05 × (1 +0.12/12) = $2,876.53
LG7 5-29 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $250 today and repay $300 in two weeks. What is the compounded annual rate implied by this 20 percent rate charged for only two weeks?
20% for two weeks needs to be compounded 26 times to form a year.
(1 + i)26 – 1 = (1 + 0.20)26 – 1 = 113.475 = 11,347.5%
LG7 5-30 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $500 today and repay $575 in two weeks. What is the compounded annual rate implied by this 15 percent rate charged for only two weeks?
15% for two weeks needs to be compounded 26 times to form a year.
(1 + i)26 – 1 = (1 + 0.15)26 – 1 = 36.857 = 3,685.7%
LG8 5-31 Annuity Interest Rate What’s the interest rate of a 5-year, annual $5,000 annuity with present value of $20,000?
Use equation 5-2 and solve for i:
%93.7
i
i i
or TVM calculator: N=5, PV=-20,000, PMT=5,000, FV=0, CPT I = 7.93%
LG8 5-32 Annuity Interest Rate What’s the interest rate of a 7-year, annual $4,000 annuity with present value of $20,000?
Use equation 5-2 and solve for i:
%20.9
i
i i
or TVM calculator: N=7, PV=-20,000, PMT=4,000, FV=0, CPT I = 9.20%
LG8 5-33 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $1,000 per month contribution to grow to $75,000 in 5 years?
5-9
%732.
i/12
i i
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields 8.78%.
or TVM calculator: N=60, PV=0, PMT=-1,000, FV=75,000, CPT I = 0.732%
LG8 5-34 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $500 per month contribution to grow to $45,000 in 6 years?
Use equation 5-2 and solve for i:
%60809.
i/12
i i
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields 7.30%.
or TVM calculator: N=72, PV=0, PMT=-500, FV=45,000, CPT I = 0.608%
LG9 5-35 Loan Payments You wish to buy a $25,000 car. The dealer offers you a 4-year loan with a 10 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be?
Use equation 5-9:
If you only paid interest over the length of the loan and your principal balance was repaid at the end of the 48 months, your payment would be $208.33 per month (= $25,000×0.10÷12) for interest only and you would owe $25,000 at the end of the 48 months, too.
LG9 5-36 Loan Payments You wish to buy a $10,000 dining room set. The furniture store offers you a 3-year loan with a 11 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be?
Use equation 5-9:
39.327$03273872.0000,10$
12/11.1
If you only paid interest over the length of the loan and your principal balance was repaid at the end of the 36 months, your payment would be $91.67 per month (= $10,000×0.11÷12) for interest only and you would owe $10,000 at the end of the 36 months, too.
LG10 5-37 Number of Annuity Payments Joey realizes that he has charged too much on his credit card and has racked up $5,000 in debt. If he can pay $150 each month and the card charges 17 percent APR (compounded monthly), how long will it take him to pay off the debt?
Rewrite equation 5-9 in terms of N:
months 4.45
N
LG10 5-38 Number of Annuity Payments Phoebe realizes that she has charged too much on her credit card and has racked up $6,000 in debt. If she can pay $200 each month and the card charges 18 percent APR (compounded monthly), how long will it take her to pay off the debt?
Rewrite equation 5-9 in terms of N:
months 15.40
N
Advanced Problems 5-39 Future Value Given a 8 percent interest rate, compute the year 7 future value if
deposits of $1,000 and LG1 $2,000 are made in years 1, and 3, respectively, and a withdrawal of $500 is made in year 4.
Use equation 5-1:
FV = $1,000 × (1 + 0.08)6 + $2,000× (1 + 0.08)4 - $500× (1 + 0.08)3 = $1,586.87 + $2,720.98 – $629.86 = $3,677.99
LG1 5-40 Future Value Given a 9 percent interest rate, compute the year 6 future value if deposits of $1,500 and $2,500 are made in years 2, and 3, respectively, and a withdrawal of $700 is made in year 5.
5-11
Use equation 5-1:
FV = $1,500 × (1 + 0.09)4 + $2,500× (1 + 0.09)3 - $700× (1 + 0.09)1 = $2,117.37 + $3,237.57 – $763 = $4,591.94
LG4 5-41 Low Financing or Cash Back? A car company is offering a choice of deals. You LG9 can receive $500 cash back on the purchase, or a 3 percent APR, 4-year loan. The price of
the car is $15,000 and you could obtain a 4-year loan from your credit union, at 7 percent APR. Which deal is cheaper?
Compare two cases. The first case is to elect the 3% APR and fully finance $15,000 over 48 months. Using equation 5-9, the payment under this scenario would be:
01.332$
0.03/121
The second case is to take the $500 cash back, apply it to the purchase and finance only $14,500 through your credit union at 7%. The payment under this scenario would be:
22.347$
0.07/121
The lower payment represents the more advantageous scenario that you should choose, electing the 3% financing through the car dealer.
LG4 5-42 Low Financing or Cash Back? A car company is offering a choice of deals. You LG9 can receive $1,000 cash back on the purchase, or a 2 percent APR, 5-year loan. The price of
the car is $20,000 and you could obtain a 5-year loan from your credit union, at 7 percent APR. Which deal is cheaper?
Compare two cases. The first case is to elect the 2% APR and fully finance $20,000 over 60 months. Using equation 5-9, the payment under this scenario would be:
56.350$
0.02/121
The second case is to take the $1,000 cash back, apply it to the purchase and finance only $19,000 through your credit union at 7%. The payment under this scenario would be:
5-12
22.376$
0.07/121
The lower payment represents the more advantageous scenario that you should choose, electing the 2% financing through the car dealer.
LG9 5-43 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid monthly over 3 years using a 9 percent APR.
Month Beginning Balance
Total Payment
Interest Paid
Principal Paid
Ending Balance
1 $15,000.00 $477.00 $112.50 $364.50 $14,635.50 2 14,635.50 477.00 109.77 367.23 14,268.27 3 14,268.27 477.00 107.01 369.98 13,898.29 4 13,898.29 477.00 104.24 372.76 13,525.53 5 13,525.53 477.00 101.44 375.55 13,149.98 6 13,149.98 477.00 98.62 378.37 12,771.61 7 12,771.61 477.00 95.79 381.21 12,390.40 8 12,390.40 477.00 92.93 384.07 12,006.33 9 12,006.33 477.00 90.05 386.95 11,619.38
10 11,619.38 477.00 87.15 389.85 11,229.53 11 11,229.53 477.00 84.22 392.77 10,836.76 12 10,836.76 477.00 81.28 395.72 10,441.03 13 10,441.03 477.00 78.31 398.69 10,042.35 14 10,042.35 477.00 75.32 401.68 9,640.67 15 9,640.67 477.00 72.31 404.69 9,235.98 16 9,235.98 477.00 69.27 407.73 8,828.25 17 8,828.25 477.00 66.21 410.78 8,417.47 18 8,417.47 477.00 63.13 413.86 8,003.60 19 8,003.60 477.00 60.03 416.97 7,586.63 20 7,586.63 477.00 56.90 420.10 7,166.54 21 7,166.54 477.00 53.75 423.25 6,743.29 22 6,743.29 477.00 50.57 426.42 6,316.87 23 6,316.87 477.00 47.38 429.62 5,887.25 24 5,887.25 477.00 44.15 432.84 5,454.41 25 5,454.41 477.00 40.91 436.09 5,018.32 26 5,018.32 477.00 37.64 439.36 4,578.96 27 4,578.96 477.00 34.34 442.65 4,136.31 28 4,136.31 477.00 31.02 445.97 3,690.33 29 3,690.33 477.00 27.68 449.32 3,241.02 30 3,241.02 477.00 24.31 452.69 2,788.33 31 2,788.33 477.00 20.91 456.08 2,332.24 32 2,332.24 477.00 17.49 459.50 1,872.74 33 1,872.74 477.00 14.05 462.95 1,409.79 34 1,409.79 477.00 10.57 466.42 943.37
5-13
Chapter 5, Solutions Cornett, Adair, and Nofsinger
35 943.37 477.00 7.08 469.92 473.45 36 473.45 477.00 3.55 473.45 0.00
LG9 5-44 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid monthly over 2 years using a 8 percent APR.
Month Beginning Balance
Total Payment
Interest Paid
Principal Paid
Ending Balance
1 $5,000.00 $226.14 $33.33 $192.80 $4,807.20 2 4,807.20 $226.14 $32.05 194.09 4,613.11 3 4,613.11 $226.14 $30.75 195.38 4,417.73 4 4,417.73 $226.14 $29.45 196.68 4,221.04 5 4,221.04 $226.14 $28.14 198.00 4,023.04 6 4,023.04 $226.14 $26.82 199.32 3,823.73 7 3,823.73 $226.14 $25.49 200.64 3,623.08 8 3,623.08 $226.14 $24.15 201.98 3,421.10 9 3,421.10 $226.14 $22.81 203.33 3,217.77
10 3,217.77 $226.14 $21.45 204.68 3,013.09 11 3,013.09 $226.14 $20.09 206.05 2,807.04 12 2,807.04 $226.14 $18.71 207.42 2,599.62 13 2,599.62 $226.14 $17.33 208.81 2,390.81 14 2,390.81 $226.14 $15.94 210.20 2,180.61 15 2,180.61 $226.14 $14.54 211.60 1,969.01 16 1,969.01 $226.14 $13.13 213.01 1,756.00 17 1,756.00 $226.14 $11.71 214.43 1,541.57 18 1,541.57 $226.14 $10.28 215.86 1,325.71 19 1,325.71 $226.14 $8.84 217.30 1,108.42 20 1,108.42 $226.14 $7.39 218.75 889.67 21 889.67 $226.14 $5.93 220.21 669.46 22 669.46 $226.14 $4.46 221.67 447.79 23 447.79 $226.14 $2.99 223.15 224.64 24 224.64 $226.14 $1.50 224.64 0.00
LG4 5-45 Investing for Retirement Monica has decided that she wants to build enough LG9 retirement wealth that, if invested at 8 percent per year, will provide her with $3,500 of
monthly income for 30 years. To date, she has saved nothing, but she still has 25 years until she retires. How much money does she need to contribute per month to reach her goal?
First, calculate the amount you would need to have in 25 years time to yield the $3,500 monthly payments for an additional 30 years. Use equation 5-4 to calculate this present value:
5-14
23.992,476$
This amount will become the future value in the next calculation, assuming 8% interest and 300 level monthly payments. Use equation 5-2 and solve for the monthly payment:
56.501$
0.08/12
300
PMT
LG4 5-46 Investing for Retirement Ross has decided that he wants to build enough retirement LG9 wealth that, if invested at 7 percent per year, will provide him with $4,000 of monthly
income for 30 years. To date, he has saved nothing, but he still has 20 years until he retires. How much money does he need to contribute per month to reach his goal?
First, calculate the amount you would need to have in 20 years time to yield the $4,000 monthly payments for an additional 30 years. Use equation 5-4 to calculate this present value:
27.230,601$
This amount will become the future value in the next calculation, assuming 7% interest and 240 level monthly payments. Use equation 5-2 and solve for the monthly payment:
16.154,1$
0.07/12
240
PMT
LG9 5-47 Loan Balance Rachel purchased a $15,000 car two years ago using an 8 percent, 4- year loan. He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What is the minimum price Rachel would need to receive for his car?
First calculate the monthly payment that she has been paying using equation 5-9:
19.366$
0.08/121
The loan balance is the principal amount outstanding. The duration of remaining payments is 24, the interest rate is 8% annual and the monthly payment is $366.19 from the previous calculation. Use these values to calculate the present value of the loan using equation 5-4:
5-15
66.096,8$
This is the minimum price the car needs to be sold for and it represents her break even price.
LG9 5-48 Loan Balance Hank purchased a $20,000 car two years ago using an 9 percent, 5-year loan. He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What’s the minimum price Hank would need to receive for his car?
First calculate the monthly payment that he has been paying using equation 5-9:
17.415$
0.09/121
The loan balance is the principal amount outstanding. The duration of remaining payments is 36, the interest rate is 9% annual and the monthly payment is $415.17 from the previous calculation. Use these values to calculate the present value of the loan using equation 5-4:
77.055,13$
This is the minimum price the car needs to be sold for and it represents his break even price.
LG9 5-49 Teaser Rate Mortgage A mortgage broker is offering a $183,900 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 4 percent APR interest rate. After the second year, the mortgage interest rate charged increases to 7 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year?
Use equation 5-9 to calculate the payment using the teaser rate:
97.877$
12/04.01
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
5-16
62.291,177$
Now use this amount for the present value, the new interest rate of 7% over the remaining 336 payments in equation to calculate the new payment amount after expiration of the teaser rate, using equation 5-9:
89.204,1$
12/07.01
LG9 5-50 Teaser Rate Mortgage A mortgage broker is offering a $279,000 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 4.5 percent APR interest rate. After the second year, the mortgage interest rate charged increases to 7.5 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year?
Use equation 5-9 to calculate the payment using the teaser rate:
65.413,1$
12/045.01
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
04.791,269$
Now use this amount for the present value, the new interest rate of 7.5% over the remaining 336 payments in equation to calculate the new payment amount after expiration of the teaser rate, using equation 5-9:
25.923,1$
12/075.01
5-51 Excel Problem Consider a person who begins contributing to a retirement plan at age 25 and contributes for 40 years until retirement at age 65. For the first ten years, she
5-17
Chapter 5, Solutions Cornett, Adair, and Nofsinger
contributes $3,000 per year. She increases the contribution rate to $5,000 per year in years 11 through 20. This is followed by increases to $10,000 per year in years 21 through 31 and to $15,000 per year for the last ten years. This money earns a 9 percent return, first compute the value of the retirement plan when she turns age 65. Then compute the annual payment she would receive over the next 40 years if the wealth was converted to an annuity payment at 8 percent.
End of Age Contribution Total Wealth 25 $3,000.00 $3,000.00 26 $3,000.00 $6,270.00 27 $3,000.00 $9,834.30 28 $3,000.00 $13,719.39 29 $3,000.00 $17,954.13 30 $3,000.00 $22,570.00 31 $3,000.00 $27,601.30 32 $3,000.00 $33,085.42 33 $3,000.00 $39,063.11 34 $3,000.00 $45,578.79 35 $5,000.00 $54,680.88 36 $5,000.00 $64,602.16 37 $5,000.00 $75,416.35 38 $5,000.00 $87,203.83 39 $5,000.00 $100,052.17 40 $5,000.00 $114,056.87 41 $5,000.00 $129,321.98 42 $5,000.00 $145,960.96 43 $5,000.00 $164,097.45 44 $5,000.00 $183,866.22 45 $10,000.00 $210,414.18 46 $10,000.00 $239,351.45 47 $10,000.00 $270,893.08 48 $10,000.00 $305,273.46 49 $10,000.00 $342,748.07 50 $10,000.00 $383,595.40 51 $10,000.00 $428,118.99 52 $10,000.00 $476,649.70 53 $10,000.00 $529,548.17 54 $10,000.00 $587,207.50 55 $15,000.00 $655,056.18 56 $15,000.00 $729,011.23 57 $15,000.00 $809,622.25 58 $15,000.00 $897,488.25 59 $15,000.00 $993,262.19 60 $15,000.00 $1,097,655.79 61 $15,000.00 $1,211,444.81 62 $15,000.00 $1,335,474.84
5-18
63 $15,000.00 $1,470,667.58 64 $15,000.00 $1,618,027.66
PMT = ($135,688.06)
Research It! Retirement Income Calculators The Internet provides some excellent retirement income calculators. You can find one by Googling “retirement income calculator.” Many of the calculators allow you to determine your predicted annual income from a retirement nest egg under different assumptions. For example, you can spend only the investment income generated from the nest egg. Most retirees try not to touch the principal. Or, you can spend both the income and the nest egg itself. These calculators let you input the size of the retirement wealth and the investment return to be earned. They then make time value computations to determine the annual income the nest egg will provide.
Go to a retirement income calculator like the one at MSN Money. Use the calculator to create a retirement scenario. Use the TVM equations or a financial calculator to check the Internet results.
(http://moneycentral.msn.com/investor/calcs/n_retire/main.asp)
SOLUTION: Assuming the principal amount is exhausted and the amount of savings at my retirement at age 65, here is what the calculator gives.
Summary
Given a certain amount of savings, how much can I spend annually during retirement?
Your annual income is estimated to be $70,000.
Information entered 1. Savings Amount saved $1,000,000 Rate of return 7% Inflation rate 3.8% Average effective tax rate 22.6% 2. Retirement years Retirement age 65 years Life expectancy 85 years Estate amount $0
Chapter 5, Solutions Cornett, Adair, and Nofsinger
Using a financial calculator, the following inputs are needed to determine the projected annual payment:
n = 20 years i = 1.07/1.038 -1 = 3.083% PV = $1,000,000
CPT PMT = $67,732.56 (The calculator rounds this amount to the nearest $10,000 for an estimate of $70,000 gross annually.)
Integrated Mini Case: Paying on your Stafford loan
Consider Gunther, a new freshman who has just received a Stafford student loan and started college. He plans to obtain the maximum loan from Stafford at the beginning of each year. Although Gavin does not have to make any payments while he is in school, the 6.8 percent interest owed (compounded monthly) accrues and is added to the balance of the loan.
Stafford loan limits: Freshman $2,625 Sophomore $3,500 Junior $5,500 Senior $5,500
After graduation, Gavin gets a six month grace period. This means that monthly payments are still not required, but interest is still accruing. After the grace period, the standard repayment plan is to amortize the debt using monthly payments for 10 years.
a. Show a time line of when the loans will be taken. b. What will be the loan balance when Gavin graduates after his fourth year of school? c. What is the loan balance six months after graduation? d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the
monthly payments Gavin owes after the grace period.
SOLUTION: a. Show a time line of when the loans will be taken.
Cash Flows $2,625 $3,500 $5,500 $5,500
Period 0 6.8% 1 6.8% 2 6.8% 3 6.8% 4 6.8% 5 years
5-20
Chapter 5, Solutions Cornett, Adair, and Nofsinger
b. What will be the loan balance when Gavin graduates after his fourth year of school?
Each payment needs to be moved forward with 6.8% interest to the middle of years 4 and 5 to calculate the outstanding accrued loan balance as of the date payments are set to begin:
FV4 = $2,625 × (1.068)4 + $3,500 × (1.068)3 + $5,500 × (1.068)2 + $5,500 × (1.068)1
FV4 = $3,415.19 + $4,263.65 + $6,273.43 + $5,874.00 = $19,826.27
c. What is the loan balance six months after graduation?
Each payment needs to be moved forward with 6.8% interest to the middle of years 4 and 5 to calculate the outstanding accrued loan balance as of the date payments are set to begin:
FV4.5= $2,625 × (1.068)4.5 + $3,500 × (1.068)3.5 + $5,500 × (1.068)2.5 + $5,500 × (1.068)1.5
FV4.5= $3,529.39 + $4,406.23 + $6,483.22 + $6,070.43 = $20,489.27
Note, this can be checked by multiplying $19,826.27 by 1.0680.5.
d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the monthly payments Gavin owes after the grace period.
Using equation 5-9, the monthly payments will be:
79.235$
12/068.01
Combined Chapter 4 and Chapter 5 Problems
4&5-1 Future Value Consider that you are 35 years old and have just changed to a new job. You have $80,000 in the retirement plan from your former employer. You can roll that money into the retirement plan of the new employer. You will also contribute $5,000 each year into your new employer’s plan. If the rolled-over money and the new contributions both earn a 7 percent return, how much should you expect to have when you retire in 30 years?
The future value can be calculated by adding the accumulated value of $80,000 brought forward with interest 30 years to a 30 year annuity of $5,000 per year, both using the 7% interest assumption. Use equations 5-1 and 5-2 and add the results:
33.284,081,1$93.303,472$40.980,608$
0.07
Chapter 5, Solutions Cornett, Adair, and Nofsinger
4&5-2 Future Value Consider that you are 45 years old and have just changed to a new job. You have $150,000 in the retirement plan from your former employer. You can roll that money into the retirement plan of the new employer. You will also contribute $8,000 each year into your new employer’s plan. If the rolled-over money and the new contributions both earn an 8 percent return, how much should you expect to have when you retire in 20 years?
The future value can be calculated by adding the accumulated value of $150,000 brought forward with interest 20 years to a 20 year annuity of $8,000 per year, both using the 8% interest assumption. Use equations 5-1 and 5-2 and add the results:
28.239,065,1$71.095,366$57.143,699$
0.08
20 20
65 Abe
4&5-3 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1 million. He cannot access the money until he turns 65 years old, which is in 25 years. At that time, he can withdrawal $25,000 per month. If the trust fund is invested at a 5.5 percent rate, how many months will it last your client once he starts to withdraw the money?
Using equation 5-1, $1 million will accumulate for 25 more years at 5.5% interest for a future value:
35.392,813,3$055.01000,000,1$FVA 25 65 Abe
months 263
N
4&5-4 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1.5 million. She cannot access the money until she turns 65 years old, which is in 15 years. At that time, she can withdraw $20,000 per month. If the trust fund is invested at a 5 percent rate, how many months will it last your client once she starts to withdraw the money?
Using equation 5-1, $1.5 million will accumulate for 15 more years at 5% interest for a future value:
27.392,118,3$05.01000,500,1$FVA 15 65 Abe
5-22
months 253
N
4&5-5 Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a $3,000 dining room set and do not need to pay for two years (no interest cost is incurred). How much money would you have to deposit now in a savings account earning 5 percent APR, compounded monthly, to pay the $3,000 bill in two years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?
Use equation 5-1 and solve for the lump sum payment:
08.715,2$12/05.01$3,000 PV 24
monthper 11.119$
24
PMT
4&5-6 Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a $5,000 living room set with 3 years before you need to make any payments (no interest cost is incurred). How much money would you have to deposit now in a savings account earning 4 percent APR, compounded monthly, to pay the $5,000 bill in three years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?
Use equation 5-1 and solve for the lump sum payment:
49.435,4$12/04.01$5,000PV 36
95.130$
0.04/12
36
PMT
4&5-7 House Appreciation and Mortgage Payments Say that you purchase a house for $200,000 by getting a mortgage for $180,000 and paying a $20,000 down payment. If you get a 30-year mortgage with a 7 percent interest rate, what are the monthly payments? What would the loan balance be in 10 years? If the house appreciates at 3 percent per year, what will be the value of the house in 10 years? How much of this value is your equity?
Use equation 5-9 to calculate your monthly payment:
5-23
54.197,1$
12/07.01
In ten years, you will have 240 payments of $1,197.54 left to pay. The present value can be calculated using equation 5-4:
71.461,154$
An appreciation of 3% per year will result in a forecast future value of the home using the original purchase price in equation 5-1:
28.783,268$03.1000,200$FV 10 years 10
The amount of equity is the difference between the home’s value and the outstanding balance on the mortgage:
Equity = $268,783.28 - $154,461.17 = $114,321.57
4&5-8 House Appreciation and Mortgage Payments Say that you purchase a house for $150,000 by getting a mortgage for $135,000 and paying a $15,000 down payment. If you get a 15-year mortgage with a 7 percent interest rate, what are the monthly payments? What would the loan balance be in 5 years? If the house appreciates at 4 percent per year, what will be the value of the house in 5 years? How much of this value is your equity?
Use equation 5-9 to calculate your monthly payment:
42.213,1$
12/07.01
In ten years, you will have 120 payments of $1,213.42 left to pay. The present value can be calculated using equation 5-4:
44.507,104$
An appreciation of 4% per year will result in a forecast future value of the home using the original purchase price in equation 5-1:
5-24
94.497,182$04.1000,150$FV 5 5years
The amount of equity is the difference between the home’s value and the outstanding balance on the mortgage:
Equity = $182,497.94-$104,507.44=$77,990.50
4&5-9 Construction Loan You have secured a loan from your bank for 2 years to build your home. The terms of the loan are that you will borrow $100,000 now and an additional $100,000 in one year. Interest of 10 percent APR will be charged on the balance monthly. Since no payments will be made during the two-year loan, the balance will grow at the 10% compounded rate. At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 6 percent interest rate. What will you paying monthly mortgage payments (Principal and Interest only)?
Use equation 5-1 to calculate the capitalized value of your mortgage at the end of year 2:
41.510,232$0.10/121000,100$12/10.01000,100$FV 1224 2
This is the amount that you will need to finance over 30 years at 6%. Use equation 5-9 to compute the monthly payment:
02.394,1$
12/06.01
4&5-10 Construction Loan You have secured a loan from your bank for 2 years to build your home. The terms of the loan are that you will borrow $100,000 now and an additional $50,000 in one year. Interest of 9 percent APR will be charged on the balance monthly. Since no payments will be made during the two-year loan, the balance will grow. At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 7 percent interest rate. What will you pay as monthly mortgage payments (Principal and Interest only)?
Use equation 5-1 to calculate the capitalized value of your mortgage at the end of year 2:
69.331,174$0.09/121000,50$12/09.01000,100$FV 1224 2
This is the amount that you will need to finance over 30 years at 7%. Use equation 5-9 to compute the monthly payment:
5-25
83.159,1$
12/07.01
CHAPTER 5 – Time Value of Money 2: ANALYZING ANNUITY CASH FLOWS
Questions
LG1 5-1 How can you add a cash flow in year two and a cash flow in year four in year seven?
To add cash flows, they need to be moved to the same time period. The cash flows in years two and four should be moved forward with interest to year seven, then they can be added together.
LG2 5-2 People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs). Demonstrate the growth of a $250 monthly contribution for 40 years earning 9 percent APR.
Using equation 5-2, we have:
07.330,170,1$
0.09/12
480
40
LG3 5-3 When you discount multiple cash flows, how does the future period that a cash flow is paid affect its present value and its contribution to the value of all the cash flows?
Discounting reduces a future cash flow to a smaller present value. Cash flows far into the future become very small when discounted to the present. Thus, cash flows in distant future periods have small impacts on present values.
LG4 5-4 How can you use the present value of an annuity concept to determine the price of a house you can afford?
Mortgages are typically for a large enough amount of money that borrowing is required to purchase a home. The amount that one can afford for a home is a function of their current state of wealth. Mortgages allow consumers to spread the expense of a home over a longer period, typically 15 or 30 years. This allows consumers to put a smaller portion of wealth into the home (for example, a 20% down payment) and borrow the balance over the life of the loan. Due to the effect of annuity compounding, the payments for such a long lived debt make the monthly payments of a manageable nature so that they can be paid from current income.
LG5 5-5 Since perpetuity payments continue for ever, how can a present value be computed? Why isn’t the present value infinite?
5-1
Chapter 5, Solutions Cornett, Adair, and Nofsinger
Equation 5-5 is used to calculate the present value of a perpetuity. It is a limiting version of equation 5-4 in which the period N grows infinitely large. As this occurs the expression following the “1” in equation 5-4 drives to the value 0 and the numerator simply become “1.” The present value is not infinite since the terms following the PMT in equation 5-4 converge to a finite limit of 1/i. This also demonstrates how payments far into the future have infinitesimal value today.
LG6 5-6 Explain why you use the same adjustment factor, (1 + i), when you adjust annuity due payments for both future value and present value.
Adjusting an annuity due calculation involves shifting the entire series of payments forward one period. This is accomplished by multiplying by (1 + i) irrespective of whether it is a future value or present value calculation.
LG7 5-7 Use the idea of compound interest to explain why EAR is larger than APR.
The annual percentage rate does not take into account the frequency of interest compounding. Equation 5-8 illustrates the conversion from APR to EAR. The effective annual rate converts the annual percentage rate to a rate that can be compared to other annual rates.
LG8 5-8 Would you rather pay $10,000 for a five year $2,500 annuity or a ten-year $1,250 annuity? Why?
The effective annual rates for these two payment streams are 7.93% and 4.28% respectively. I would rather pay $10,000 for a five year $2,500 annuity as it earns a higher effective annual rate of interest.
LG9 5-9 The interest on your home mortgage is tax deductible. Why are the early years of the mortgage more helpful in reducing taxes than in the later years?
Mortgage payments at the beginning of the amortization schedule are predominantly interest with little principal. In later years, interest payments decline and principal payments make up an ever increasing part of the payments. Thus, the tax deductible part (the interest payment) is larger in the beginning years.
LG10 5-10 How can you use the concepts illustrated in computing the number of payments in an annuity to figure how to pay off a credit card balance? How does the magnitude of the payment impact the number of months?
Utilizing equation 5-2, you can declare the present balance for the credit card and set that equal to the PVA. The interest can be obtained as an APR and converted to and EAR using equation 5-8. This is the value to put into “i” in equation 5-2. You then decide when you want to have the credit card paid off and convert this to a monthly value of N in equation 5-
5-2
Chapter 5, Solutions Cornett, Adair, and Nofsinger
2. Solving for PMT will yield the amount needed to pay the credit card off in the time frame you desire, assuming that no additional charges are made to the credit card and that the interest rate remains level.
Problems Basic Problems 5-1 Future Value Compute the future value in year 8 of a $1,000 deposit in year 1 and LG1 another $1,500 deposit at the end of year 3 using a 10% interest rate.
Use equation 5-1:
FV = $1,000 × (1 + 0.10)7 + $1,500× (1 + 0.10)5 = $1,948.72 + $2,415.77 = $4,364.48
LG1 5-2 Future Value Compute the future value in year 7 of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using a 8% interest rate..
Use equation 5-1:
FV = $2,000 × (1 + 0.08)6 + $2,500 × (1 + 0.08)3 = $3,173.75 + $3,149.28 = $6,323.03
LG2 5-3 Future Value of an Annuity What is the future value of a $500 annuity payment over 5 years if interest rates are 9 percent?
Use equation 5-2:
5
5
LG2 5-4 Future Value of an Annuity What is the future value of a $700 annuity payment over 4 years if interest rates are 10 percent?
Use equation 5-2:
4
4
LG3 5-5 Present Value Compute the present value of a $1,000 deposit in year 1 and another $1,500 deposit at the end of year 3 if interest rates are 10 percent.
Use equation 5-3:
PV = $1,000 ÷ (1 + 0.10)1 + $1,500 ÷ (1 + 0.10)3 = $909.09 + $1,126.97 = $2,036.06
LG3 5-6 Present Value Compute the present value of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using an 8% interest rate.
5-3
Use equation 5-3:
PV = $2,000 ÷ (1 + 0.08)1 + $2,500 ÷ (1 + 0.08)4 = $1,851.85 + $1,837.57 = $3,689.43
LG4 5-7 Present Value of an Annuity What’s the present value of a $500 annuity payment over 5 years if interest rates are 9 percent?
Use equation 5-4:
LG4 5-8 Present Value of an Annuity What’s the present value of a $700 annuity payment over 4 years if interest rates are 10 percent?
Use equation 5-4:
LG5 5-9 Present Value of a Perpetuity What’s the present value, when interest rates are 7.5 percent, of a $50 payment made every year forever?
Use equation 5-5:
50$ perpetuity a of PV
LG5 5-10 Present Value of a Perpetuity What’s the present value, when interest rates are 8.5 percent, of a $75 payment made every year forever?
Use equation 5-5:
75$ perpetuity a of PV
LG6 5-11 Present Value of an Annuity Due If the present value of an ordinary, 7-year annuity is $6,500 and interest rates are 8.5 percent, what’s the present value of the same annuity due?
Use equation 5-7:
PVA7 due = 6,500 × (1 +0.085) = $7,052.50
LG6 5-12 Present Value of an Annuity Due If the present value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what’s the present value of the same annuity due?
Use equation 5-7:
PVA6 due = 8,500 × (1 +0.095) = $9,307.50
LG6 5-13 Future Value of an Annuity Due If the future value of an ordinary, 7-year annuity is $6,500 and interest rates are 8.5 percent, what is the future value of the same annuity due?
Use equation 5-6:
FVA7 due = 6,500 × (1 + 0.085) = $7,052.50
(Note this is the same answer as problem 5-11, as expected)
LG6 5-14 Future Value of an Annuity Due If the future value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what’s the future value of the same annuity due?
Use equation 5-6:
FVA6 due = 8,500 × (1 + 0.095) = $9,307.50
(Note this is the same answer as problem 5-12, as expected)
LG7 5-15 Effective Annual Rate A loan is offered with monthly payments and an 11 percent APR. What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
EAR
LG7 5-16 Effective Annual Rate A loan is offered with monthly payments and a 12 percent APR. What’s the loan’s effective annual rate (EAR)?
Use equation 5-8:
EAR
Intermediate Problems 5-17 Future Value Given a 6 percent interest rate, compute the year 6 future value of
deposits made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500. LG1
Use equation 5-1:
FV = $1,000 × (1 + 0.06)5 + $1,200× (1 + 0.06)4 + $1,200 × (1 + 0.06)3 + $1,500× (1 + 0.06)2
5-5
FV = $1,338.23 + $1,514.97 + $1,429.22 + $1,685.40 = $5,967.82
LG1 5-18 Future Value Given a 7 percent interest rate, compute the year 6 future value of deposits made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400.
Use equation 5-1:
FV = $1,000 × (1 + 0.07)5 + $1,300× (1 + 0.07)4 + $1,300 × (1 + 0.07)3 + $1,400× (1 + 0.07)2
FV = $1,402.55 + $1,704.03 + $1,592.56 + $1,602.86 = $6,302
LG2 5-19 Future Value of Multiple Annuities Assume that you contribute $200 per month to a retirement plan for 20 years. Then you are able to increase the contribution to $400 per month for another 20 years. Given a 7 percent interest rate, what is the value of your retirement plan after the 40 years?
Break the annuity streams into a level stream of payments of $200 for 40 years and another level stream of payments of $200 for the last 20 years. Use equation 5-2 for each payment stream and add the results:
01.148,629$9267.520200$2415.796,2200$
0.07/12
240480
2040
LG2 5-20 Future Value of Multiple Annuities Assume that you contribute $150 per month to a retirement plan for 15 years. Then you are able to increase the contribution to $350 per month for the next 25 years. Given an 8 percent interest rate, what is the value of your retirement plan after the 40 years?
Break the annuity streams into a level stream of payments of $150 for 40 years and another level stream of payments of $200 for the last 25 years. Use equation 5-2 for each payment stream and add the results
45.856,713$0264.951200$0078.491,3150$
0.08/12
300480
2540
LG3 5-21 Present Value Given a 6 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500.
Use equation 5-3:
PV = $1,000 ÷ (1 + 0.06)1 + $1,200 ÷ (1 + 0.06)2 + $1,200 ÷ (1 + 0.06)3 + $1,500 ÷ (1 + 0.06)4
PV = $943.40 + $1,068 + $1,007.54 + $1,188.14 = $4,207.08
LG3 5-22 Present Value Given a 7 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400.
Use equation 5-3:
Chapter 5, Solutions Cornett, Adair, and Nofsinger
PV = $1,000 ÷ (1 + 0.07)1 + $1,300 ÷ (1 + 0.07)2 + $1,300 ÷ (1 + 0.07)3 + $1,400 ÷ (1 + 0.07)4
PV = $934.58 + $1,135.47 + $1,061.19 + $1,068.05 = $4,199.29
LG2 5-23 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. The owner states that he can repay a loan at $1,000 per month for the next three years and then $2,000 per month for two years after that. If the bank is charging customers 7.5 percent APR, how much would it be willing to lend the business owner?
Break the annuity streams into a level stream of payments of $2,000 for 5 years and another level stream of payments of $1,000 for the first 3 years. Use equation 5-4 for each payment stream and subtract the results.
70.662,67$92.147,32$62.810,99$
LG2 5-24 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. The owner states that she can repay a loan at $1,500 per month for the next three years and then $500 per month for two years after that. If the bank is charging customers 8.5 percent APR, how much would it be willing to lend the business owner?
Break the annuity into two streams of payments: $500 monthly for five years and $1,000 for three years. Use equation 5-4 for each annuity and add the results.
70.048,56$11.678,31$59.370,24$
LG5 5-25 Present Value of a Perpetuity A perpetuity pays $100 per year and interest rates are 7.5 percent. How much would its value change if interest rates increased to 8.5 percent? Did the value increase or decrease?
Use equation 5-5:
47.176,1$ 085.0
100$ perpetuity a of PV
The difference between these perpetuities is $156.86. The value of the perpetuity decreased with an increase in the interest rate.
5-7
Chapter 5, Solutions Cornett, Adair, and Nofsinger
LG5 5-26 Present Value of a Perpetuity A perpetuity pays $50 per year and interest rates are 9 percent. How much would its value change if interest rates decreased to 8 percent? Did the value increase or decrease?
Use equation 5-5:
00.625$ 08.0
50$ perpetuity a of PV
The difference between these perpetuities is $69.45. The value of the perpetuity increased with a decrease in the interest rate.
LG6 5-27 Future and Present Value of an Annuity Due If you start making $50 monthly contributions today and continue them for 5 years, what’s their future value if the compounding rate is 10 percent APR? What is the present value of this annuity?
Compute the future value using equation 5-2:
98.903,3$00833.14343706824.7750$12/10.01 0.10/12
112/10.01 0$5FVA
88.372,2$00833.106536757.4750$00833.1
LG6 5-28 Future and Present Value of an Annuity Due If you start making $75 monthly contributions today and continue them for 4 years, what is their future value if the compounding rate is 12 percent APR? What is the present value of this annuity?
First calculate the future values and present values, using equations 5-2 and 5-4, respectively. Using these results, the annuity due values can be computed using equations 5-6 and 5-7, respectively.
70.591,4$2226.6175$
0.12/12
5-8
05.848,2$97395951.3775$
PVA48 due = $2,848.05 × (1 +0.12/12) = $2,876.53
LG7 5-29 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $250 today and repay $300 in two weeks. What is the compounded annual rate implied by this 20 percent rate charged for only two weeks?
20% for two weeks needs to be compounded 26 times to form a year.
(1 + i)26 – 1 = (1 + 0.20)26 – 1 = 113.475 = 11,347.5%
LG7 5-30 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $500 today and repay $575 in two weeks. What is the compounded annual rate implied by this 15 percent rate charged for only two weeks?
15% for two weeks needs to be compounded 26 times to form a year.
(1 + i)26 – 1 = (1 + 0.15)26 – 1 = 36.857 = 3,685.7%
LG8 5-31 Annuity Interest Rate What’s the interest rate of a 5-year, annual $5,000 annuity with present value of $20,000?
Use equation 5-2 and solve for i:
%93.7
i
i i
or TVM calculator: N=5, PV=-20,000, PMT=5,000, FV=0, CPT I = 7.93%
LG8 5-32 Annuity Interest Rate What’s the interest rate of a 7-year, annual $4,000 annuity with present value of $20,000?
Use equation 5-2 and solve for i:
%20.9
i
i i
or TVM calculator: N=7, PV=-20,000, PMT=4,000, FV=0, CPT I = 9.20%
LG8 5-33 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $1,000 per month contribution to grow to $75,000 in 5 years?
5-9
%732.
i/12
i i
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields 8.78%.
or TVM calculator: N=60, PV=0, PMT=-1,000, FV=75,000, CPT I = 0.732%
LG8 5-34 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $500 per month contribution to grow to $45,000 in 6 years?
Use equation 5-2 and solve for i:
%60809.
i/12
i i
Now convert the monthly interest rate to an annual rate by multiplying by 12 which yields 7.30%.
or TVM calculator: N=72, PV=0, PMT=-500, FV=45,000, CPT I = 0.608%
LG9 5-35 Loan Payments You wish to buy a $25,000 car. The dealer offers you a 4-year loan with a 10 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be?
Use equation 5-9:
If you only paid interest over the length of the loan and your principal balance was repaid at the end of the 48 months, your payment would be $208.33 per month (= $25,000×0.10÷12) for interest only and you would owe $25,000 at the end of the 48 months, too.
LG9 5-36 Loan Payments You wish to buy a $10,000 dining room set. The furniture store offers you a 3-year loan with a 11 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be?
Use equation 5-9:
39.327$03273872.0000,10$
12/11.1
If you only paid interest over the length of the loan and your principal balance was repaid at the end of the 36 months, your payment would be $91.67 per month (= $10,000×0.11÷12) for interest only and you would owe $10,000 at the end of the 36 months, too.
LG10 5-37 Number of Annuity Payments Joey realizes that he has charged too much on his credit card and has racked up $5,000 in debt. If he can pay $150 each month and the card charges 17 percent APR (compounded monthly), how long will it take him to pay off the debt?
Rewrite equation 5-9 in terms of N:
months 4.45
N
LG10 5-38 Number of Annuity Payments Phoebe realizes that she has charged too much on her credit card and has racked up $6,000 in debt. If she can pay $200 each month and the card charges 18 percent APR (compounded monthly), how long will it take her to pay off the debt?
Rewrite equation 5-9 in terms of N:
months 15.40
N
Advanced Problems 5-39 Future Value Given a 8 percent interest rate, compute the year 7 future value if
deposits of $1,000 and LG1 $2,000 are made in years 1, and 3, respectively, and a withdrawal of $500 is made in year 4.
Use equation 5-1:
FV = $1,000 × (1 + 0.08)6 + $2,000× (1 + 0.08)4 - $500× (1 + 0.08)3 = $1,586.87 + $2,720.98 – $629.86 = $3,677.99
LG1 5-40 Future Value Given a 9 percent interest rate, compute the year 6 future value if deposits of $1,500 and $2,500 are made in years 2, and 3, respectively, and a withdrawal of $700 is made in year 5.
5-11
Use equation 5-1:
FV = $1,500 × (1 + 0.09)4 + $2,500× (1 + 0.09)3 - $700× (1 + 0.09)1 = $2,117.37 + $3,237.57 – $763 = $4,591.94
LG4 5-41 Low Financing or Cash Back? A car company is offering a choice of deals. You LG9 can receive $500 cash back on the purchase, or a 3 percent APR, 4-year loan. The price of
the car is $15,000 and you could obtain a 4-year loan from your credit union, at 7 percent APR. Which deal is cheaper?
Compare two cases. The first case is to elect the 3% APR and fully finance $15,000 over 48 months. Using equation 5-9, the payment under this scenario would be:
01.332$
0.03/121
The second case is to take the $500 cash back, apply it to the purchase and finance only $14,500 through your credit union at 7%. The payment under this scenario would be:
22.347$
0.07/121
The lower payment represents the more advantageous scenario that you should choose, electing the 3% financing through the car dealer.
LG4 5-42 Low Financing or Cash Back? A car company is offering a choice of deals. You LG9 can receive $1,000 cash back on the purchase, or a 2 percent APR, 5-year loan. The price of
the car is $20,000 and you could obtain a 5-year loan from your credit union, at 7 percent APR. Which deal is cheaper?
Compare two cases. The first case is to elect the 2% APR and fully finance $20,000 over 60 months. Using equation 5-9, the payment under this scenario would be:
56.350$
0.02/121
The second case is to take the $1,000 cash back, apply it to the purchase and finance only $19,000 through your credit union at 7%. The payment under this scenario would be:
5-12
22.376$
0.07/121
The lower payment represents the more advantageous scenario that you should choose, electing the 2% financing through the car dealer.
LG9 5-43 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid monthly over 3 years using a 9 percent APR.
Month Beginning Balance
Total Payment
Interest Paid
Principal Paid
Ending Balance
1 $15,000.00 $477.00 $112.50 $364.50 $14,635.50 2 14,635.50 477.00 109.77 367.23 14,268.27 3 14,268.27 477.00 107.01 369.98 13,898.29 4 13,898.29 477.00 104.24 372.76 13,525.53 5 13,525.53 477.00 101.44 375.55 13,149.98 6 13,149.98 477.00 98.62 378.37 12,771.61 7 12,771.61 477.00 95.79 381.21 12,390.40 8 12,390.40 477.00 92.93 384.07 12,006.33 9 12,006.33 477.00 90.05 386.95 11,619.38
10 11,619.38 477.00 87.15 389.85 11,229.53 11 11,229.53 477.00 84.22 392.77 10,836.76 12 10,836.76 477.00 81.28 395.72 10,441.03 13 10,441.03 477.00 78.31 398.69 10,042.35 14 10,042.35 477.00 75.32 401.68 9,640.67 15 9,640.67 477.00 72.31 404.69 9,235.98 16 9,235.98 477.00 69.27 407.73 8,828.25 17 8,828.25 477.00 66.21 410.78 8,417.47 18 8,417.47 477.00 63.13 413.86 8,003.60 19 8,003.60 477.00 60.03 416.97 7,586.63 20 7,586.63 477.00 56.90 420.10 7,166.54 21 7,166.54 477.00 53.75 423.25 6,743.29 22 6,743.29 477.00 50.57 426.42 6,316.87 23 6,316.87 477.00 47.38 429.62 5,887.25 24 5,887.25 477.00 44.15 432.84 5,454.41 25 5,454.41 477.00 40.91 436.09 5,018.32 26 5,018.32 477.00 37.64 439.36 4,578.96 27 4,578.96 477.00 34.34 442.65 4,136.31 28 4,136.31 477.00 31.02 445.97 3,690.33 29 3,690.33 477.00 27.68 449.32 3,241.02 30 3,241.02 477.00 24.31 452.69 2,788.33 31 2,788.33 477.00 20.91 456.08 2,332.24 32 2,332.24 477.00 17.49 459.50 1,872.74 33 1,872.74 477.00 14.05 462.95 1,409.79 34 1,409.79 477.00 10.57 466.42 943.37
5-13
Chapter 5, Solutions Cornett, Adair, and Nofsinger
35 943.37 477.00 7.08 469.92 473.45 36 473.45 477.00 3.55 473.45 0.00
LG9 5-44 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid monthly over 2 years using a 8 percent APR.
Month Beginning Balance
Total Payment
Interest Paid
Principal Paid
Ending Balance
1 $5,000.00 $226.14 $33.33 $192.80 $4,807.20 2 4,807.20 $226.14 $32.05 194.09 4,613.11 3 4,613.11 $226.14 $30.75 195.38 4,417.73 4 4,417.73 $226.14 $29.45 196.68 4,221.04 5 4,221.04 $226.14 $28.14 198.00 4,023.04 6 4,023.04 $226.14 $26.82 199.32 3,823.73 7 3,823.73 $226.14 $25.49 200.64 3,623.08 8 3,623.08 $226.14 $24.15 201.98 3,421.10 9 3,421.10 $226.14 $22.81 203.33 3,217.77
10 3,217.77 $226.14 $21.45 204.68 3,013.09 11 3,013.09 $226.14 $20.09 206.05 2,807.04 12 2,807.04 $226.14 $18.71 207.42 2,599.62 13 2,599.62 $226.14 $17.33 208.81 2,390.81 14 2,390.81 $226.14 $15.94 210.20 2,180.61 15 2,180.61 $226.14 $14.54 211.60 1,969.01 16 1,969.01 $226.14 $13.13 213.01 1,756.00 17 1,756.00 $226.14 $11.71 214.43 1,541.57 18 1,541.57 $226.14 $10.28 215.86 1,325.71 19 1,325.71 $226.14 $8.84 217.30 1,108.42 20 1,108.42 $226.14 $7.39 218.75 889.67 21 889.67 $226.14 $5.93 220.21 669.46 22 669.46 $226.14 $4.46 221.67 447.79 23 447.79 $226.14 $2.99 223.15 224.64 24 224.64 $226.14 $1.50 224.64 0.00
LG4 5-45 Investing for Retirement Monica has decided that she wants to build enough LG9 retirement wealth that, if invested at 8 percent per year, will provide her with $3,500 of
monthly income for 30 years. To date, she has saved nothing, but she still has 25 years until she retires. How much money does she need to contribute per month to reach her goal?
First, calculate the amount you would need to have in 25 years time to yield the $3,500 monthly payments for an additional 30 years. Use equation 5-4 to calculate this present value:
5-14
23.992,476$
This amount will become the future value in the next calculation, assuming 8% interest and 300 level monthly payments. Use equation 5-2 and solve for the monthly payment:
56.501$
0.08/12
300
PMT
LG4 5-46 Investing for Retirement Ross has decided that he wants to build enough retirement LG9 wealth that, if invested at 7 percent per year, will provide him with $4,000 of monthly
income for 30 years. To date, he has saved nothing, but he still has 20 years until he retires. How much money does he need to contribute per month to reach his goal?
First, calculate the amount you would need to have in 20 years time to yield the $4,000 monthly payments for an additional 30 years. Use equation 5-4 to calculate this present value:
27.230,601$
This amount will become the future value in the next calculation, assuming 7% interest and 240 level monthly payments. Use equation 5-2 and solve for the monthly payment:
16.154,1$
0.07/12
240
PMT
LG9 5-47 Loan Balance Rachel purchased a $15,000 car two years ago using an 8 percent, 4- year loan. He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What is the minimum price Rachel would need to receive for his car?
First calculate the monthly payment that she has been paying using equation 5-9:
19.366$
0.08/121
The loan balance is the principal amount outstanding. The duration of remaining payments is 24, the interest rate is 8% annual and the monthly payment is $366.19 from the previous calculation. Use these values to calculate the present value of the loan using equation 5-4:
5-15
66.096,8$
This is the minimum price the car needs to be sold for and it represents her break even price.
LG9 5-48 Loan Balance Hank purchased a $20,000 car two years ago using an 9 percent, 5-year loan. He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What’s the minimum price Hank would need to receive for his car?
First calculate the monthly payment that he has been paying using equation 5-9:
17.415$
0.09/121
The loan balance is the principal amount outstanding. The duration of remaining payments is 36, the interest rate is 9% annual and the monthly payment is $415.17 from the previous calculation. Use these values to calculate the present value of the loan using equation 5-4:
77.055,13$
This is the minimum price the car needs to be sold for and it represents his break even price.
LG9 5-49 Teaser Rate Mortgage A mortgage broker is offering a $183,900 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 4 percent APR interest rate. After the second year, the mortgage interest rate charged increases to 7 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year?
Use equation 5-9 to calculate the payment using the teaser rate:
97.877$
12/04.01
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
5-16
62.291,177$
Now use this amount for the present value, the new interest rate of 7% over the remaining 336 payments in equation to calculate the new payment amount after expiration of the teaser rate, using equation 5-9:
89.204,1$
12/07.01
LG9 5-50 Teaser Rate Mortgage A mortgage broker is offering a $279,000 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 4.5 percent APR interest rate. After the second year, the mortgage interest rate charged increases to 7.5 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year?
Use equation 5-9 to calculate the payment using the teaser rate:
65.413,1$
12/045.01
Now calculate the outstanding loan balance after the first 24 payments using equation 5-4:
04.791,269$
Now use this amount for the present value, the new interest rate of 7.5% over the remaining 336 payments in equation to calculate the new payment amount after expiration of the teaser rate, using equation 5-9:
25.923,1$
12/075.01
5-51 Excel Problem Consider a person who begins contributing to a retirement plan at age 25 and contributes for 40 years until retirement at age 65. For the first ten years, she
5-17
Chapter 5, Solutions Cornett, Adair, and Nofsinger
contributes $3,000 per year. She increases the contribution rate to $5,000 per year in years 11 through 20. This is followed by increases to $10,000 per year in years 21 through 31 and to $15,000 per year for the last ten years. This money earns a 9 percent return, first compute the value of the retirement plan when she turns age 65. Then compute the annual payment she would receive over the next 40 years if the wealth was converted to an annuity payment at 8 percent.
End of Age Contribution Total Wealth 25 $3,000.00 $3,000.00 26 $3,000.00 $6,270.00 27 $3,000.00 $9,834.30 28 $3,000.00 $13,719.39 29 $3,000.00 $17,954.13 30 $3,000.00 $22,570.00 31 $3,000.00 $27,601.30 32 $3,000.00 $33,085.42 33 $3,000.00 $39,063.11 34 $3,000.00 $45,578.79 35 $5,000.00 $54,680.88 36 $5,000.00 $64,602.16 37 $5,000.00 $75,416.35 38 $5,000.00 $87,203.83 39 $5,000.00 $100,052.17 40 $5,000.00 $114,056.87 41 $5,000.00 $129,321.98 42 $5,000.00 $145,960.96 43 $5,000.00 $164,097.45 44 $5,000.00 $183,866.22 45 $10,000.00 $210,414.18 46 $10,000.00 $239,351.45 47 $10,000.00 $270,893.08 48 $10,000.00 $305,273.46 49 $10,000.00 $342,748.07 50 $10,000.00 $383,595.40 51 $10,000.00 $428,118.99 52 $10,000.00 $476,649.70 53 $10,000.00 $529,548.17 54 $10,000.00 $587,207.50 55 $15,000.00 $655,056.18 56 $15,000.00 $729,011.23 57 $15,000.00 $809,622.25 58 $15,000.00 $897,488.25 59 $15,000.00 $993,262.19 60 $15,000.00 $1,097,655.79 61 $15,000.00 $1,211,444.81 62 $15,000.00 $1,335,474.84
5-18
63 $15,000.00 $1,470,667.58 64 $15,000.00 $1,618,027.66
PMT = ($135,688.06)
Research It! Retirement Income Calculators The Internet provides some excellent retirement income calculators. You can find one by Googling “retirement income calculator.” Many of the calculators allow you to determine your predicted annual income from a retirement nest egg under different assumptions. For example, you can spend only the investment income generated from the nest egg. Most retirees try not to touch the principal. Or, you can spend both the income and the nest egg itself. These calculators let you input the size of the retirement wealth and the investment return to be earned. They then make time value computations to determine the annual income the nest egg will provide.
Go to a retirement income calculator like the one at MSN Money. Use the calculator to create a retirement scenario. Use the TVM equations or a financial calculator to check the Internet results.
(http://moneycentral.msn.com/investor/calcs/n_retire/main.asp)
SOLUTION: Assuming the principal amount is exhausted and the amount of savings at my retirement at age 65, here is what the calculator gives.
Summary
Given a certain amount of savings, how much can I spend annually during retirement?
Your annual income is estimated to be $70,000.
Information entered 1. Savings Amount saved $1,000,000 Rate of return 7% Inflation rate 3.8% Average effective tax rate 22.6% 2. Retirement years Retirement age 65 years Life expectancy 85 years Estate amount $0
Chapter 5, Solutions Cornett, Adair, and Nofsinger
Using a financial calculator, the following inputs are needed to determine the projected annual payment:
n = 20 years i = 1.07/1.038 -1 = 3.083% PV = $1,000,000
CPT PMT = $67,732.56 (The calculator rounds this amount to the nearest $10,000 for an estimate of $70,000 gross annually.)
Integrated Mini Case: Paying on your Stafford loan
Consider Gunther, a new freshman who has just received a Stafford student loan and started college. He plans to obtain the maximum loan from Stafford at the beginning of each year. Although Gavin does not have to make any payments while he is in school, the 6.8 percent interest owed (compounded monthly) accrues and is added to the balance of the loan.
Stafford loan limits: Freshman $2,625 Sophomore $3,500 Junior $5,500 Senior $5,500
After graduation, Gavin gets a six month grace period. This means that monthly payments are still not required, but interest is still accruing. After the grace period, the standard repayment plan is to amortize the debt using monthly payments for 10 years.
a. Show a time line of when the loans will be taken. b. What will be the loan balance when Gavin graduates after his fourth year of school? c. What is the loan balance six months after graduation? d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the
monthly payments Gavin owes after the grace period.
SOLUTION: a. Show a time line of when the loans will be taken.
Cash Flows $2,625 $3,500 $5,500 $5,500
Period 0 6.8% 1 6.8% 2 6.8% 3 6.8% 4 6.8% 5 years
5-20
Chapter 5, Solutions Cornett, Adair, and Nofsinger
b. What will be the loan balance when Gavin graduates after his fourth year of school?
Each payment needs to be moved forward with 6.8% interest to the middle of years 4 and 5 to calculate the outstanding accrued loan balance as of the date payments are set to begin:
FV4 = $2,625 × (1.068)4 + $3,500 × (1.068)3 + $5,500 × (1.068)2 + $5,500 × (1.068)1
FV4 = $3,415.19 + $4,263.65 + $6,273.43 + $5,874.00 = $19,826.27
c. What is the loan balance six months after graduation?
Each payment needs to be moved forward with 6.8% interest to the middle of years 4 and 5 to calculate the outstanding accrued loan balance as of the date payments are set to begin:
FV4.5= $2,625 × (1.068)4.5 + $3,500 × (1.068)3.5 + $5,500 × (1.068)2.5 + $5,500 × (1.068)1.5
FV4.5= $3,529.39 + $4,406.23 + $6,483.22 + $6,070.43 = $20,489.27
Note, this can be checked by multiplying $19,826.27 by 1.0680.5.
d. Using the standard repayment plan and a 6.8 percent APR interest rate, compute the monthly payments Gavin owes after the grace period.
Using equation 5-9, the monthly payments will be:
79.235$
12/068.01
Combined Chapter 4 and Chapter 5 Problems
4&5-1 Future Value Consider that you are 35 years old and have just changed to a new job. You have $80,000 in the retirement plan from your former employer. You can roll that money into the retirement plan of the new employer. You will also contribute $5,000 each year into your new employer’s plan. If the rolled-over money and the new contributions both earn a 7 percent return, how much should you expect to have when you retire in 30 years?
The future value can be calculated by adding the accumulated value of $80,000 brought forward with interest 30 years to a 30 year annuity of $5,000 per year, both using the 7% interest assumption. Use equations 5-1 and 5-2 and add the results:
33.284,081,1$93.303,472$40.980,608$
0.07
Chapter 5, Solutions Cornett, Adair, and Nofsinger
4&5-2 Future Value Consider that you are 45 years old and have just changed to a new job. You have $150,000 in the retirement plan from your former employer. You can roll that money into the retirement plan of the new employer. You will also contribute $8,000 each year into your new employer’s plan. If the rolled-over money and the new contributions both earn an 8 percent return, how much should you expect to have when you retire in 20 years?
The future value can be calculated by adding the accumulated value of $150,000 brought forward with interest 20 years to a 20 year annuity of $8,000 per year, both using the 8% interest assumption. Use equations 5-1 and 5-2 and add the results:
28.239,065,1$71.095,366$57.143,699$
0.08
20 20
65 Abe
4&5-3 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1 million. He cannot access the money until he turns 65 years old, which is in 25 years. At that time, he can withdrawal $25,000 per month. If the trust fund is invested at a 5.5 percent rate, how many months will it last your client once he starts to withdraw the money?
Using equation 5-1, $1 million will accumulate for 25 more years at 5.5% interest for a future value:
35.392,813,3$055.01000,000,1$FVA 25 65 Abe
months 263
N
4&5-4 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1.5 million. She cannot access the money until she turns 65 years old, which is in 15 years. At that time, she can withdraw $20,000 per month. If the trust fund is invested at a 5 percent rate, how many months will it last your client once she starts to withdraw the money?
Using equation 5-1, $1.5 million will accumulate for 15 more years at 5% interest for a future value:
27.392,118,3$05.01000,500,1$FVA 15 65 Abe
5-22
months 253
N
4&5-5 Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a $3,000 dining room set and do not need to pay for two years (no interest cost is incurred). How much money would you have to deposit now in a savings account earning 5 percent APR, compounded monthly, to pay the $3,000 bill in two years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?
Use equation 5-1 and solve for the lump sum payment:
08.715,2$12/05.01$3,000 PV 24
monthper 11.119$
24
PMT
4&5-6 Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a $5,000 living room set with 3 years before you need to make any payments (no interest cost is incurred). How much money would you have to deposit now in a savings account earning 4 percent APR, compounded monthly, to pay the $5,000 bill in three years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?
Use equation 5-1 and solve for the lump sum payment:
49.435,4$12/04.01$5,000PV 36
95.130$
0.04/12
36
PMT
4&5-7 House Appreciation and Mortgage Payments Say that you purchase a house for $200,000 by getting a mortgage for $180,000 and paying a $20,000 down payment. If you get a 30-year mortgage with a 7 percent interest rate, what are the monthly payments? What would the loan balance be in 10 years? If the house appreciates at 3 percent per year, what will be the value of the house in 10 years? How much of this value is your equity?
Use equation 5-9 to calculate your monthly payment:
5-23
54.197,1$
12/07.01
In ten years, you will have 240 payments of $1,197.54 left to pay. The present value can be calculated using equation 5-4:
71.461,154$
An appreciation of 3% per year will result in a forecast future value of the home using the original purchase price in equation 5-1:
28.783,268$03.1000,200$FV 10 years 10
The amount of equity is the difference between the home’s value and the outstanding balance on the mortgage:
Equity = $268,783.28 - $154,461.17 = $114,321.57
4&5-8 House Appreciation and Mortgage Payments Say that you purchase a house for $150,000 by getting a mortgage for $135,000 and paying a $15,000 down payment. If you get a 15-year mortgage with a 7 percent interest rate, what are the monthly payments? What would the loan balance be in 5 years? If the house appreciates at 4 percent per year, what will be the value of the house in 5 years? How much of this value is your equity?
Use equation 5-9 to calculate your monthly payment:
42.213,1$
12/07.01
In ten years, you will have 120 payments of $1,213.42 left to pay. The present value can be calculated using equation 5-4:
44.507,104$
An appreciation of 4% per year will result in a forecast future value of the home using the original purchase price in equation 5-1:
5-24
94.497,182$04.1000,150$FV 5 5years
The amount of equity is the difference between the home’s value and the outstanding balance on the mortgage:
Equity = $182,497.94-$104,507.44=$77,990.50
4&5-9 Construction Loan You have secured a loan from your bank for 2 years to build your home. The terms of the loan are that you will borrow $100,000 now and an additional $100,000 in one year. Interest of 10 percent APR will be charged on the balance monthly. Since no payments will be made during the two-year loan, the balance will grow at the 10% compounded rate. At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 6 percent interest rate. What will you paying monthly mortgage payments (Principal and Interest only)?
Use equation 5-1 to calculate the capitalized value of your mortgage at the end of year 2:
41.510,232$0.10/121000,100$12/10.01000,100$FV 1224 2
This is the amount that you will need to finance over 30 years at 6%. Use equation 5-9 to compute the monthly payment:
02.394,1$
12/06.01
4&5-10 Construction Loan You have secured a loan from your bank for 2 years to build your home. The terms of the loan are that you will borrow $100,000 now and an additional $50,000 in one year. Interest of 9 percent APR will be charged on the balance monthly. Since no payments will be made during the two-year loan, the balance will grow. At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 7 percent interest rate. What will you pay as monthly mortgage payments (Principal and Interest only)?
Use equation 5-1 to calculate the capitalized value of your mortgage at the end of year 2:
69.331,174$0.09/121000,50$12/09.01000,100$FV 1224 2
This is the amount that you will need to finance over 30 years at 7%. Use equation 5-9 to compute the monthly payment:
5-25
83.159,1$
12/07.01