FINAN 450 CHAP 5

All else held constant, the present value of an annuity will decrease if you:

decrease the annuity payment.

You are to make monthly deposits of $500 into a retirement account that earns an APR of 9.5 percent compounded monthly (9.5/12 % per month). If your first deposit will be made one month from now, how large will your retirement account be in 35 years? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

FVA = C{[(1 + r)^t - 1]/r} FVA = $500[{[1 + (.095/12)]^420 - 1}/(.095/12)] FVA = $1,669,731.48

Suppose you just bought a 10-year annuity of $5,200 per year at the current interest rate of 10 percent per year. a. What is the value of the investment at the current interest rate of 10 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b. What happens to the value of your investment if interest rates suddenly drop to 5 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c. What happens to the value of your investment if interest rates suddenly rise to 15 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

A. PVA = C({1 - [1/(1 + r)^t]}/r) PVA = $5,200{[1 - (1/1.10)^10]/.10} PVA = $31,951.75 B. PVA = C({1 - [1/(1 + r)^t]}/r) PVA = $5,200{[1 - (1/1.05)^10]/.05} PVA = $40,153.02 C. PVA = C({1 - [1/(1 + r)^t]}/r) PVA = $5,200{[1 - (1/1.15)^10]/.15} PVA = $26,097.60

Streamsong Credit Bank is offering 4.7 percent compounded daily on its savings accounts. Assume that you deposit $4,750 today. a. How much will you have in the account in 5 years? (Use 365 days in a year. Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b. How much will you have in the account in 10 years? (Use 365 days in a year. Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c. How much will you have in the account in 20 years? (Use 365 days in a year. Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

FV = PV(1 + r)^t a. FV in 5 years = $4,750[1 + (.047/365)]^5(365) = $6,008.23 b. FV in 10 years = $4,750[1 + (.047/365)]^10(365) = $7,599.74 c. FV in 20 years = $4,750[1 + (.047/365)]^20(365) = $12,159.18

Marko, Inc., is considering the purchase of ABC Co. Marko believes that ABC Co. can generate cash flows of $5,300, $10,300, and $16,500 over the next three years, respectively. After that time, they feel the business will be worthless. Marko has determined that a rate of return of 15 percent is applicable to this potential purchase. What is Marko willing to pay today to buy ABC Co.?

PV = $5,300/(1 + .15) + $10,300/(1.15)2 + $16,500/(1.15)3 = $23,245.99

Larry's Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $25,000 per year forever. A representative for Larry's tells you the policy costs $645,000. At what interest rate would this be a fair deal? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

PV = C/r $645,000 = $25,000/r We can now solve for the interest rate as follows: r = $25,000/$645,000 r = .0388, or 3.88%

An investment will pay you $100,000 in 9 years. Assume the appropriate discount rate is 5.5 percent compounded daily. What is the present value? (Use 365 days in a year. Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

PV = FV/(1 + r)^t PV = $100,000/[(1 + .055/365)^9(365)] PV = $60,959.36

You have just purchased a new warehouse. To finance the purchase, you've arranged for a 30-year mortgage loan for 80 percent of the $3,500,000 purchase price. The monthly payment on this loan will be $15,100. a. What is the APR on this loan? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. What is the EAR on this loan? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Amount borrowed = .80($3,500,000) Amount borrowed = $2,800,000 PVA = C({1 - [1/(1 + r)^t]}/r) $3,500,000 = $15,100[{1 - [1/(1 + r)^360]}/r] r = .00420, or .420% APR = .00420 × 12 APR = .0504, or 5.04% b. EAR = [1 + (APR/m)]m - 1 EAR = [1 + .00420]12 - 1 EAR = .0516, or 5.16%

What is the present value of $2,625 per year, at a discount rate of 6.9 percent, if the first payment is received six years from now and the last payment is received 20 years from now? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

PVA = C({1 - [1/(1 + r)^t]}/r) PVA = $2,625({1 - [1/(1.069)^15]}/.069) PVA = $24,060.02 PV = FV/(1 + r)^t PV = $24,060.02/(1 + .069)^5 PV = $17,234.85

You have your choice of two investment accounts. Investment A is a 10-year annuity that features end-of-month $1,525 payments and has an interest rate of 7 percent compounded monthly. Investment B is an annually compounded lump-sum investment with an interest rate of 9 percent, also good for 10 years. How much money would you need to invest in B today for it to be worth as much as Investment A 10 years from now? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

FVA = C{[(1 + r)^t - 1]/r} FVA = $1,525{[(1 + .07/12)^120 - 1]/(.07/12)} FVA = $263,954.33 PV = FV/(1 + r)t PV = $263,954.33/(1 + .09)10 PV = $111,497.16

Although you may know William Shakespeare from his classic literature, what is not well-known is that he was an astute investor. In 1604, when he was 40 and writing King Lear, Shakespeare grew worried about his eventual retirement. Afraid that he would become like King Lear in his retirement and beg hospitality from his children, he purchased grain "tithes," or shares in farm output, for 440 pounds. The tithes paid him 60 pounds per year for 31 years. Even though he died at the age of 52, his children received the remaining payments. What interest rate did the Bard of Avon receive on this investment? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

PVA = C({1 - [1/(1 + r)^t]}/r) £440 = £60[{1 - [1/(1 + r)^31]}/r]

You have arranged for a loan on your new car that will require the first payment today. The loan is for $28,500, and the monthly payments are $525. If the loan will be paid off over the next 60 months, what is the APR of the loan? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

PVAdue = C[{1 - [1/(1 + r)]t}/r](1 + r) $28,500 = $525[{1 - [1/(1 + r)]60}/r](1 + 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 = .00346, or .346% 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 = .00346 × 12 APR = .0415, or 4.15%

Bob has been investing $2,750 in stock at the end of every year for the past 12 years. If the account is currently worth $64,800, what was his annual return on this investment?

$64,800 = $2,750{[(1 + r)12 − 1]/r} r = .1161, or 11.61%

YEAR 1 - 865 YEAR 2 - 1040 YEAR 3 - 1290 YEAR 4 - 1385 a. If the discount rate is 8 percent, what is the future value of these cash flows in Year 4? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b. What is the future value at an interest rate of 11 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c. What is the future value at an interest rate of 24 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

FV = PV(1 + r)^t FV@8% = $865(1.08)^3 + $1,040(1.08)^2 + $1,290(1.08) + $1,385 = $5,080.91 FV@11% = $865(1.11)^3 + $1,040(1.11)^2 + $1,290(1.11) + $1,385 = $5,281.28 FV@24% = $865(1.24)^3 + $1,040(1.24)^2 + $1,290(1.24) + $1,385 = $6,232.93

You're prepared to make monthly payments of $250, beginning at the end of this month, into an account that pays 8 percent interest compounded monthly. How many payments will you have made when your account balance reaches $50,000? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

FVA = C{[(1 + r)^t - 1]/r} $50,000 = $250[{[1 + (.08/12)]^t - 1}/(.08/12)] 200 = {[1 + (.08/12)]^t - 1}/(.08/12) 1.333 = (1 + .00667)^t - 1 2.333 = (1.00667)^t ln 2.333 = t (ln 1.00667) t = ln 2.333/ln 1.00667 t = 127.52 payments

Larry's Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $25,000 per year forever. Assume the required return on this investment is 4 percent. How much will you pay for the policy? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

PV = C/r PV = $25,000/.04 PV = $625,000

Given an interest rate of 6.35 percent per year, what is the value at Year 7 of a perpetual stream of $7,000 payments that begin at Year 20? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

PV = C/r PV = $7,000/.0635 PV = $110,236.22 PV = FV/(1 + r)^t PV = $110,236.22/(1 + .0635)^12 PV = $52,659.22

YEAR 1 - 570 YEAR 2 - 430 YEAR 3 - 840 YEAR 4 - 1230 a. If the discount rate is 10 percent, what is the present value of these cash flows? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b. What is the present value at 18 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c. What is the present value at 24 percent? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

PV = FV/(1 + r)^t PV@10% = $570/1.10 + $430/1.10^2 + $840/1.10^3 + $1,230/1.10^4 = $2,344.76 PV@18% = $570/1.18 + $430/1.18^2 + $840/1.18^3 + $1,230/1.18^4 = $1,937.54 PV@24% = $570/1.24 + $430/1.24^2 + $840/1.24^3 + $1,230/1.24^4 = $1,700.16

One of your customers is delinquent on his accounts payable balance. You've mutually agreed to a repayment schedule of $400 per month. You will charge 1.4 percent per month interest on the overdue balance. If the current balance is $17,320, how long will it take for the account to be paid off? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

PVA = C({1 - [1/(1 + r)^t]}/r) $17,320 = $400{[1 - (1/1.014)^t ]/.014} Now we solve for t: 1/1.014^t = 1 - [($17,320)(.014)/($400)] 1.014^t = 1/.3938 t = ln 2.5394/ln 1.014 t = 67.03 months

Investment X offers to pay you $3,100 per year for 9 years, whereas Investment Y offers to pay you $4,800 per year for 5 years. a. If the discount rate is 6 percent, what is the present value of these cash flows? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b. If the discount rate is 22 percent, what is the present value of these cash flows? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

To find the PVA, we use the equation: PVA = C({1 - [1/(1 + r)^t]}/r) At an interest rate of 6 percent: X@6%: PVA = $3,100{[1 - (1/1.06)^9]/.06} = $21,085.25 Y@6%: PVA = $4,800{[1 - (1/1.06)^5]/.06} = $20,219.35 And at an interest rate of 22 percent: X@22%: PVA = $3,100{[1 - (1/1.22)^9]/.22} = $11,737.48 Y@22%: PVA = $4,800{[1 - (1/1.22)^5]/.22} = $13,745.47