Business Finance Final Exam (Chapter 5)

Gerritt wants to buy a car that costs $26,750. The interest rate on his loan is 5.33 percent compounded monthly and the loan is for 7 years. What are his monthly payments?

$26,750 = C[1 − (1/(1 + .0533/12)^84)/(.0533/12)] C = $382.24

ST Trucking just signed a $3.8 million contract. The contract calls for a payment of $1.1 million today, $1.3 million one year from today, and $1.4 million two years from today. What is this contract worth today at a discount rate of 8.7 percent?

$3,480,817.37 PV = $1.1m + ($1.3m/1.087) + ($1.4m/1.0872) = $3,480,817.37

What is the present value of $500 invested each year for 10 years at a rate of 5%?

Financial calculator key strokes: N = 10 I/YR = 5 PMT = $500 FV = $0 PV = ??? = 3,861

What is the future value of $400 invested each year for 15 years at a rate of 6%?

Financial calculator key strokes: N = 15 I/YR = 6 PV = $0 PMT = $400 FV = ??? = $9,310

What is the future value of $500 invested each year for 20 years at a rate of 10%?

Financial calculator key strokes: N = 20 I/YR = 10 PV = $0 PMT = $500 FV = ??? = $28,637

You want to have $2.95 million when you retire in 37 years. You feel that you can save $680 per month until you retire. What APR do you have to earn in order to achieve your goal?

$2,950,000 = $680{[(1 + r)^444 − 1] / r} r = .0081, or .81% r = .81% × 12 = 9.74%

Charlene can afford car payments of $185 a month for 48 months. If the interest rate is 5.65 percent, how much money can she afford to borrow?

$7,931.44 PV = $185 × (1 - {1 / [1 + (.0565 / 12)]^48}) / (.0565 / 12) = $7,931.44

Which statement is true?

All else equal, an increase in the discount rate decreases the present value and increases the future value of an annuity

You are comparing two annuities. Annuity A pays $100 at the end of each month for 10 years. Annuity B pays $100 at the beginning of each month for 10 years. The rate of return on both annuities is 8 percent. Which one of the following statements is correct given this information?

Annuity B has both a higher present value and a higher future value than Annuity A

Which one of the following qualifies as an annuity payment?

Auto loan payment

The value of the following cash flows four years from today is $6,911.44. The interest rate is 3.7 percent. What is the value of the Year 3 cash flow? Year Cash Flow 1 1,420 2 1,532 3 ? 4 2,490

FV = $1,420(1.037)^3 + $1,532(1.037)^2 + $2,490 = $5,720.99 Difference = $6,911.44 − 5,720.99 = $1,190.45 PV = $1,190.45/1.037 = $1,147.97

The variable that you are solving for in a future value of a lump sum problem is:

Future value

Which loan type requires calls for the borrower to pay interest each period and to repay the entire principal at some point in the future?

Interest-only

Which loan type requires the borrower to repay a single lump sum payment at some time in the future with interest? 3.Award: 10 out of 10.00 points Suppose a business takes out a $7,000, five-year loan at 6 percent that will be paid annually with a single, fixed payment each period. How much will be the annual payment? $313.98 $627.88 $1,400.00 $1,661.88 $1,854.88 Total Amount = C × [(1 − 1/(1 + r)t)/r] $7,000 = C × [(1 - 1/(1 + .06)5)/.06] $7,000 = C × [(1 - 1/1.3382)/.06] $7,000 = C × [.2527/.06] C = $7,000 / 4.2121 C = $1,661.88

Pure Discount

The variables in a present value of an annuity problem include all of the following, except:

Risk Profile

The variable that you are solving for in a present value of an annuity problem is:

The Present value

Which one of the following features distinguishes an ordinary annuity from an annuity due?

Timing of the annuity payments

Assume the appropriate discount rate for the following cash flows is 10.59 percent per year. Year Cash Flow 1 2,550 2 2,950 3 5,150 4 5,750 What is the present value of the cash flows?

To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV / (1 + r)^t PV = $2,550 / 1.1059 + $2,950 / 1.10592 + $5,150 / 1.10593 + $5,750 / 1.10594 PV = $12,369.75

he variables in a future value of a lump sum problem include all of the following, except:

Usage

Lee pays 1 percent per month interest on his credit card account. When his monthly rate is multiplied by 12, the resulting answer is referred to as the:

annual percentage rate

Your grandparents would like to establish a trust fund that will pay you and your heirs $195,000 per year forever with the first payment one year from today. If the trust fund earns an annual return of 3.8 percent, how much must your grandparents deposit today?

PV = $195,000/.038 = $5,131,578.95

What is the present value of $3,125 per year, at a discount rate of 10 percent, if the first payment is received 9 years from now and the last payment is received 21 years from now?

Since the first payment is received nine years from today and the last payment is received 21 years from now, there are 13 payments. We can use the present value of an annuity formula, which will give us the present value eight years from now, one period before the first payment. So, the present value of the annuity in eight years is: PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $3,125{[1 - (1 / 1.1013)] / .10} PVA = $22,197.99 And using the present value equation for a lump sum, the present value of the annuity today is: PV = FV / (1 + r)^t PV = $22,197.99 / (1 + .10)^8 PV = $10,355.53

Vandermark Credit Corp. wants to earn an effective annual return on its consumer loans of 17.25 percent per year. The bank uses daily compounding on its loans. What interest rate is the bank required by law to report to potential borrowers?

The reported rate is the APR, so we need to convert the EAR to an APR as follows: EAR = [1 + (APR / m)]^m - 1 APR = m[(1 + EAR)^1/m - 1] APR = 365[(1.1725)^1/365 - 1] = .1592, or 15.92%

Booker, Inc., has identified an investment project with the following cash flows. Year Cash Flow 1 980 2 1,210 3 1,430 4 2,170 If the discount rate is 6 percent, what is the future value of these cash flows in Year 4? What is the future value at an interest rate of 14 percent? What is the future value at an interest rate of 21 percent?

To solve this problem, we must find the FV of each cash flow and sum. To find the FV of a lump sum, we use: FV = PV(1 + r)^t FV@6% = $980(1.06)^3 + $1,210(1.06)^2 + $1,430(1.06) + $2,170 = $6,212.55 FV@14% = $980(1.14)^3 + $1,210(1.14)^2 + $1,430(1.14) + $2,170 = $6,824.63 FV@21% = $980(1.21)^3 + $1,210(1.21)^2 + $1,430(1.21) + $2,170 = $7,407.99

Prepare an amortization schedule for a three-year loan of $63,000. The interest rate is 10 percent per year, and the loan agreement calls for a principal reduction of $21,000 every year. How much total interest is paid over the life of the loan?

Year Beg. Tot. Pay. In.Pay Pri. Pay End. Bal. 1 63,000 27,300 6,300 21,000 42,000 2 42,000 25,200 4,200 21,000 21,000 3 21,000 23,100 2,100 21,000 0 Total interest over life of the loan = $6,300 + 4,200 + 2,100 Total interest over life of the loan = $12,600

Suppose a business takes out a $7,000, five-year loan at 6 percent that will be paid annually with a single, fixed payment each period. How much will be the annual payment?

Total Amount = C × [(1 − 1/(1 + r)^t)/r] $7,000 = C × [(1 - 1/(1 + .06)^5)/.06] $7,000 = C × [(1 - 1/1.3382)/.06] $7,000 = C × [.2527/.06] C = $7,000 / 4.2121 C = $1,661.88

You have just started a new job and plan to save $4,600 per year for 42 years until you retire. You will make your first deposit in one year. How much will you have when you retire if you earn an annual interest rate of 10.38 percent?

V = $4,600[1.103842 − 1)/.1038] = $2,760,838.90

You feel that you will need $1.9 million in your retirement account and when you reach that amount, you plan to retire. You feel you can earn an APR of 9.9 percent compounded monthly and plan to save $260 per month until you reach your goal. How many years will it be until you reach your goal and retire?

$1,900,000 = $260{[1 - 1/(1 + .0990/12)^t]/.0990} t = 500.91 months t = 500.91/12 = 41.74 years

One of your customers has just made a purchase in the amount of $23,200. You have agreed to payments of $445 per month and will charge a monthly interest rate of 1.26 percent. How many months will it take for the account to be paid off?

$23,200 = $445[(1 − 1/1.0126^t) / .0126] t = 85.43 months

Round House Furniture offers credit to its customers at a rate of 1.15 percent per month. What is the effective annual rate of this credit offer?

EAR = (1 + .0115)^12- 1 = .1471, or 14.71 percent

What is the effective annual rate for an APR of 11.10 percent compounded quarterly?

EAR = (1 + .1110/4)^4 = .1157, or 11.57%

What is the effective annual rate of 14.9 percent compounded quarterly?

EAR = [1 + (.149/4)]^4- 1 = .1575, or 15.75 percent

You make $6,000 annual deposits into a retirement account that pays an APR of 10.9 percent compounded monthly. How large will your account balance be in 34 years?

EAR = [1 + (APR / m)]^m - 1 EAR = [1 + (.109 / 12)]^12 - 1 EAR = .1146, or 11.46% Using the FVA equation, we get: FVA = C{[(1 + r)^t - 1] / r} FVA = $6,000[(1.114634 - 1) / .1146] FVA = $2,042,457.36

Assuming an interest rate of 4.9 percent, what is the value of the following cash flows five years from today? Year Cash Flow 1 $ 3,190 2 4,375 3 5,280 4 6,490

FV = $3,190(1.049)^4 + $4,375(1.049)^3 + $5,280(1.049)^2 + $6,490(1.049) = $21,530.99

You plan to save $6,100 per year for the next 9 years. After the last deposit, you will keep the money in the account for 2 more years. The account will earn an interest rate of 6.4 percent. How much will there be in the account 11 years from today?

FV = $6,100[(1.0649 − 1)/.064] = $71,268.13 FV = $71,268.13(1.064)2 = $80,682.37

Bucher Credit Bank is offering 6.8 percent compounded daily on its savings accounts. Assume that you deposit $6,500 today. How much will you have in the account in 3 years? How much will you have in the account in 6 years? How much will you have in the account in 12 years?

FV = PV(1 + r)^t It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so, we get: FV in 3 years = $6,500[1 + (.068 / 365)]^3(365) = $7,970.79 FV in 6 years = $6,500[1 + (.068 / 365)]^6(365) = $9,774.38 FV in 12 years = $6,500[1 + (.068 / 365)]^12(365) = $14,698.22

For each of the following annuities, calculate the annual cash flow: Cash Flow Future Value Years Interest Rate ? 24,650 8 10 ? 1,010,000 41 12 ? 832,000 27 13 ? 137,000 12 9

FVA = C{[(1 + r)^t - 1] / r} $24,650 = C{[(1 + .10)^8 - 1] / .10} C = $24,650 / 11.43589 C = $2,155.50 FVA = C{[(1 + r)^t - 1] / r} $1,010,000 = C{[(1 + .12)^41 - 1] / .12} C = $1,010,000 / 860.14239 C = $1,174.22 FVA = C{[(1 + r)^t - 1] / r} $832,000 = C{[(1 + .13)^27 - 1] / .13} C = $832,000 / 200.84061 C = $4,142.59 FVA = C{[(1 + r)^t - 1] / r} $137,000 = C{[(1 + .09)^12 - 1] / .09} C = $137,000 / 20.14072 C = $6,802.14

For each of the following annuities, calculate the future value: Future Value Annual Payment Years Interest ? 1,790 10 6 ? 8,520 34 7 ? 4,270 9 4 ? 9,050 32 8

FVA = C{[(1 + r)^t - 1] / r} FVA = $1,790{[(1 + .06)^10 - 1] / .06} FVA = $23,593.62 FVA = C{[(1 + r)^t - 1] / r} FVA = $8,520{[(1 + .07)^34 - 1] / .07} FVA = $1,092,764.68 FVA = C{[(1 + r)^t - 1] / r} FVA = $4,270{[(1 + .04)^9 - 1] / .04} FVA = $45,188.54 FVA = C{[(1 + r)^t - 1] / r} FVA = $9,050{[(1 + .08)^32 - 1] / .08} FVA = $1,214,632.51

What is the future value of $1,720 in 14 years assuming an interest rate of 7.25 percent compounded semiannually?

For this problem, we need to find the FV of a lump sum using the equation: FV = PV(1 + r)^t It is important to note that compounding occurs semiannually. To account for this, we will divide the interest rate by two (the number of compounding periods in a year), and multiply the number of periods by two. Doing so, we get: FV = $1,720[1 + (.0725 / 2)]^28 FV = $4,661.61

You have just purchased a new warehouse. To finance the purchase, you've arranged for a 34-year mortgage loan for 80 percent of the $3,240,000 purchase price. The monthly payment on this loan will be $15,700. What is the APR on this loan? What is the EAR on this loan?

Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for the interest rate. First, we need to find the amount borrowed since it is only 80 percent of the building value. So, the amount borrowed is: Amount borrowed = .80($3,240,000) Amount borrowed = $2,592,000 Now we can use the PVA equation: PVA = C({1 - [1 / (1 + r)^t]} / r) $2,592,000 = $15,700[{1 - [1 / (1 + r)^408]} / r] To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial and error, remember that increasing the interest rate decreases the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find: r = .00538, or .538% This is the monthly interest rate. To find the APR with a monthly interest rate, we simply multiply the monthly rate by 12, so the APR is: APR = .00538 × 12 APR = .0645, or 6.45% And the EAR is: EAR = [1 + (APR / m)]^m - 1 EAR = (1 + .00538)^12 - 1 EAR = .0665, or 6.65%

You want to borrow $76,000 from your local bank to buy a new sailboat. You can afford to make monthly payments of $1,350, but no more. Assuming monthly compounding, what is the highest rate you can afford on a 72-month APR loan?

Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for the interest rate. Using the PVA equation: PVA = C({1 - [1 / (1 + r)^t]} / r) $76,000 = $1,350({1 - [1 / (1 + r)^72]} / r) To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial and error, remember that increasing the interest rate decreases the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find: r = .00706, or .706% This is the monthly interest rate. To find the APR with a monthly interest rate, we simply multiply the monthly rate by 12, so the APR is: APR = .00706 × 12 APR = .0847, or 8.47%

A local finance company quotes an interest rate of 15.4 percent on one-year loans. So, if you borrow $37,000, the interest for the year will be $5,698. Because you must repay a total of $42,698 in one year, the finance company requires you to pay $42,698/12, or $3,558.17 per month over the next 12 months. What rate would legally have to be quoted? What is the effective annual rate?

Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for the interest rate. We must be careful to use the cash flows of the loan. Using the present value of an annuity equation, we find: PVA = C({1 - [1 / (1 + r)^t]} / r) $37,000 = $3,558.17({1 - [1 / (1 + r)^12]} / r) To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial and error, remember that increasing the interest rate lowers the PVA, and increasing the interest rate decreases the PVA. Using a spreadsheet, we find: r = .02275, or 2.275% This is the monthly interest rate. To find the APR with a monthly interest rate, we multiply the monthly rate by 12, so the APR is: APR = .02275 × 12 APR = .2731, or 27.31% And the EAR is: EAR = [1 + (APR / m)]^m - 1 EAR = [1 + .02275]^12 - 1 EAR = .3100, or 31.00%

You've just joined the investment banking firm of Dewey, Cheatum, and Howe. They've offered you two different salary arrangements. You can have $7,100 per month for the next three years, or you can have $5,800 per month for the next three years, along with a $31,500 signing bonus today. Assume the interest rate is 5 percent compounded monthly. If you take the first option, $7,100 per month for three years, what is the present value? What is the present value of the second option?

Monthly rate = .05 / 12 Monthly rate = .0042, or .42% The value today of the $7,100 monthly salary is: PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $7,100({1 - [1 / (1.0042)^36]} / .0042) PVA = $236,896.48 To find the value of the second option, we find the present value of the monthly payments and add the bonus. We can add the bonus since it is paid today. So: PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $5,800({1 - [1 / (1.0042)^36]} / .0042) PVA = $193,521.07 So, the total value of the second option is: Value of second option = $193,521.07 + 31,500 Value of second option = $225,021.07 The difference in the value of the two options today is: Difference in value today = $236,896.48 - 225,021.07 Difference in value today = $11,875.41 What if we found the future value of the two cash flows? For the monthly salary, the future value will be: FVA = C{[(1 + r)^t - 1] / r} FVA = $7,100{[(1 + .0042)^36 - 1] / .0042} FVA = $275,148.68 To find the future value of the second option we also need to find the future value of the bonus as well. So, the future value of this option is: FV = C{[(1 + r)^t - 1] / r} + PV(1 + r)^t FV = $5,800{[(1 + .0042)^36 - 1] / .0042} + $31,500(1 + .0042)^36 FV = $261,355.72 So, the first option is still the better choice. The difference between the future value of the two options is: Difference in future value = $275,148.68 - 261,355.72 Difference in future value = $13,792.96

A common error made when solving a future value of an annuity problem is:

Multiplying the annual deposit and the number of years before calculating the problem

Lacey will receive $135,000 a year for 5 years, starting today. If the rate of return is 8.9 percent, what are these payments worth today?

PV = $135,000×({1 - [1 / (1 + .089)^5]} / .089)× (1 + .089) r = $573,323.90

Your crazy uncle left you a trust that will pay you $16,000 per year for the next 23 years with the first payment received one year from today. If the appropriate interest rate is 4.9 percent, what is the value of the payments today?

PV = $16,000[(1 −1/1.049023)/.0490] = $217,866.12

You are in talks to settle a potential lawsuit. The defendant has offered to make annual payments of $26,000, $30,500, $62,000, and $93,000 to you each year over the next four years, respectively. All payments will be made at the end of the year. If the appropriate interest rate is 4.8 percent, what is the value of the settlement offer today?

PV = $26,000/1.048 + $30,500/1.0482 + $62,000/1.0483 + $93,000/1.0484 = $183,541.46

You want to buy a new sports car from Roy's Cars for $51,800. The contract is in the form of a 48-month annuity due at an APR of7.8 percent, compounded monthly. What would be your monthly payment?

PV = $51,800 = C × [(1 - {1 / [1 + (.078 / 12)]^48}) / (.078 / 12)] × [1 + (.078 / 12)] C = $1,251.60

Recently, you needed money and agreed to sell a car you had inherited at a price of $55,000, to be paid in monthly payments of $1,500 for 42 months. What interest rate did you charge for financing the sale?

PV = $55,000 = $1,500 × (1 - {1 / [1 + (r / 12)]^42}) / r r = 7.78 percent

The manager of Gloria's Boutique has approved Carla's application for 24 months of credit with maximum monthly payments of $70.If the APR is 14.2 percent, what is the maximum initial purchase that Carla can buy on credit?

PV = $70 × (1 - {1 / [1 + (.142 / 12)]^24}) / (.142 / 12) = $1,455.08

Maybepay Life Insurance Co. is selling a perpetual annuity contract that pays $2,400 monthly. The contract currently sells for $332,000. What is the monthly return on this investment vehicle? What is the APR? What is the effective annual return?

PV = C / r $332,000 = $2,400 / r We can now solve for the interest rate as follows: r = $2,400 / $332,000 r = .0072, or .72% per month The interest rate is .72 percent per month. To find the APR, we multiply this rate by the number of months in a year, so: APR = (12).72% APR = 8.67% And using the equation to find the EAR, we find: EAR = [1 + (APR / m)]^m - 1 EAR = [1 + .0072]^12 - 1 EAR = .0903, or 9.03%

Curly's Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $27,000 per year forever. A representative for Curly's tells you the policy costs $520,000. At what interest rate would this be a fair deal?

PV = C / r $520,000 = $27,000 / r We can now solve for the interest rate as follows: r = $27,000 / $520,000 r = .0519, or 5.19%

Peter Lynchpin wants to sell you an investment contract that pays equal $12,200 amounts at the end of each year for the next 22 years. If you require an effective annual return of 7 percent on this investment, how much will you pay for the contract today?

PVA = C({1 - [1 / (1 + r)^t]} / r) PVA = $12,200{[1 - (1 / 1.0722)] / .07} PVA = $134,947.13

Prepare an amortization schedule for a three-year loan of $111,000. The interest rate is 10 percent per year, and the loan calls for equal annual payments. How much total interest is paid over the life of the loan?

PVA = C({1 - [1 / (1 + r)t]} / r) $111,000 = C({1 - [1 / (1 + .10)3]} / .10) C = $44,634.74 Year Beginning Balance Total Payment Interest Payment Principal Payment Ending Balance 1 111,000.00 44,634.74 11,100.00 33,534.74 77,465.26 2 77,465.26 44,634.74 7,746.53 36,888.22 40,577.04 3 40,577.04 44,634.74 4,057.70 40,577.04 0.00

Ricky Ripov's Pawn Shop charges an interest rate of 13.75 percent per month on loans to its customers. Like all lenders, Ricky must report an APR to consumers. What rate should the shop report? What is the effective annual rate?

The APR is simply the interest rate per period times the number of periods in a year. In this case, the interest rate is 13.75 percent per month, and there are 12 months in a year, so we get: APR = 12^(13.75%) APR = 165.00% To find the EAR, we use the EAR formula: EAR = [1 + (APR / m)]^m - 1 EAR = (1 + .1375)^12 - 1 EAR = 3.6926, or 369.26%

You're trying to choose between two different investments, both of which have up-front costs of $99,000. Investment G returns $164,000 in 8 years. Investment H returns $284,000 in 15 years. Calculate the rate of return for each of these investments. Rate of Return Investment G Investment H

The investment we should choose is the investment with the higher rate of return. We will use the future value equation to find the interest rate for each option. Doing so, we find the return for Investment G is: FV = PV(1 + r)^t $164,000 = $99,000(1 + r)^8 r = ($164,000 / $99,000)^1/8 - 1 r = .0651, or 6.51% And, the return for Investment H is: FV = PV(1 + r)^t $284,000 = $99,000(1 + r)^15 r = ($284,000 / $99,000)^1/15 - 1 r = .0728, or 7.28%

First Simple Bank pays 9.4 percent simple interest on its investment accounts. First Complex Bank pays interest on its accounts compounded annually. What rate should the bank set if it wants to match First Simple Bank over an investment horizon of 14 years?

The total interest paid by First Simple Bank is the interest rate per period times the number of periods. In other words, the interest by First Simple Bank paid over 14 years will be: .094(14) = 1.316 First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor of $1, or: (1 + r)^14 Setting the two equal, we get: (.094)(14) = (1 + r)^14 - 1 r = 2.3161/14 - 1 r = .0618, or 6.18%

You are looking at a one-year loan of $27,000. The interest rate is quoted as 8 percent plus two points. A point on a loan is simply 1 percent (one percentage point) of the loan amount. Quotes similar to this one are common with home mortgages. The interest rate quotation in this example requires the borrower to pay two points to the lender up front and repay the loan later with 8 percent interest. What rate would you actually be paying here?

To find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be: Loan repayment amount = $27,000(1.08) = $29,160 The amount you will receive today is the principal amount of the loan times one minus the points. Amount received = $27,000(1 - .02) = $26,460 Now, we find the interest rate for this PV and FV. FV = PV(1 + r) $29,160 = $26,460(1 + r) r = ($29,160 / $26,460) - 1 r = .1020, or 10.20%

The variables in a future value of a lump sum problem include all of the following, except:

Volatility

You want to buy a new sports coupe for $75,500, and the finance office at the dealership has quoted you a loan with an APR of 7.9 percent for 72 months to buy the car. What will your monthly payments be? What is the effective annual rate on this loan?

We first need to find the annuity payment. We have the PVA, the length of the annuity, and the interest rate. Using the PVA equation: PVA = C({1 - [1 / (1 + r)^t]} / r) $75,500 = C[1 - {1 / [1 + (.079 / 12)]^72} / (.079 / 12)] Solving for the payment, we get: C = $75,500 / 57.19367 C = $1,320.08 To find the EAR, we use the EAR equation: EAR = [1 + (APR / m)]^m - 1 EAR = [1 + (.079 / 12)]^12 - 1 EAR = .0819, or 8.19%

You want to buy a new sports car from Muscle Motors for $42,100. The contract is in the form of a 48-month annuity due at an APR of 6.45 percent. What will your monthly payment be?

We need to use the PVA due equation, which is: PVAdue = (1 + r)PVA Using this equation: PVAdue = $42,100 = [1 + (.0645 / 12)] × C[{1 - 1 / [1 + (.0645 / 12)]^48} / (.0645 / 12)] $41,874.92 = C[{1 - 1 / [1 + (.0645 / 12)]^48} / (.0645 / 12)] C = $992.10

You are saving to buy a $194,000 house. There are two competing banks in your area, both offering certificates of deposit yielding 7.9 percent. How long will it take your initial $111,000 investment to reach the desired level at First Bank, which pays simple interest? How long will it take your initial $111,000 investment to reach the desired level at Second Bank, which compounds interest monthly?

We will calculate the time we must wait if we deposit in the bank that pays simple interest. The interest amount we will receive each year in this bank will be: Interest = $111,000(.079) Interest = $8,769 per year The deposit will have to increase by the difference between the amount we need and the amount we originally deposit. Dividing this difference by the annual interest earnings, the number of years it will take in the bank that pays simple interest is: Years to wait = ($194,000 - 111,000) / $8,769 Years to wait = 9.47 years To find the number of years it will take in the bank that pays compound interest, we can use the future value equation for a lump sum and solve for the periods. Doing so, we find: FV = PV(1 + r)^t $194,000 = $111,000[1 + (.079 / 12)]^t t = 85.09 months, or 7.09 years