Chapter 6

A finance company offers to loan you $1,000 today if you will make 48 "low" monthly payments of $32.60. What rate is implicit in the loan?

$1,000 = $32.60 x PVIFA (r, 48) r = 2% x 12 = 24% annually

You are scheduled to receive annual payments of $4,800 for each of the next 7 years. The discount rate is 8 percent. What is the difference in the present value if you receive these payments at the beginning of each year rather than at the end of each year?

$1,999

Your car dealer is willing to lease you a new car for $245 a month for 4 years. Payments are due on the first day of each month starting with the day you sign the lease contract. If your cost of money is 6.5 percent, what is the current value of the lease?

$10,386.99

Given an interest rate of 12% per year, what is the present value of a perpetual stream of $1,000 per month, starting today?

$101,000

Alexa plans on saving $3,000 a quarter and expects to earn an annual rate of 10.25 percent. How much will she have in her account at the end of 45 years?

$11,010,618.72

You are borrowing $17,800 to buy a car. The terms of the loan call for monthly payments for 5 years at 8.6 percent interest. What is the amount of each payment?

$17,800 = C x PVIFA (8.6%/12,60) x C = $366.05

You are planning to save for retirement over the next 30 years. To do this, you will invest $700 a month in a stock account and $300 a month in a bond account. The return of the stock account is expected to be 11%, and the bond account will pay 6%. When you retire, you will combine your money into an account with a 9% return. How much can you withdraw each month from your account assuming a 25-year withdrawal period?

$19,003.76 withdrawal per month

You borrow $165,000 to buy a house. The mortgage rate is 7.5 percent and the loan period is 30 years. Payments are made monthly. If you pay the mortgage according to the loan agreement, how much total interest will you pay?

$250,332.

Samuelson Engines wants to save $750,000 to buy some new equipment six years from now. The plan is to set aside an equal amount of money on the first day of each quarter starting today. The firm can earn 4.75 percent on its savings. How much does the firm have to save each quarter to achieve its goal?

$26,872.94

Given an interest rate of 6.2% per year, what is the value at date t=7 of a perpetual stream of $3,500 payments that begins at date t=15?

$37,051.41

You want to have a $90,000 in your savings account 10 years from now, and you're prepared to make equal monthly deposit into the account at the end of each month. If the account pays 6.8% interest, what amount must you deposit each month?

$525.72

Ellen's son, David, will be entering college in six years. Ellen has not yet put aside any savings. She estimates that she will need $2,000 per month for the next four years David is in school. Any savings she has will earn 9% annually. How much would Ellen have to save each month for the next six years, beginning today, to have enough money to pay for David's education? (Note that the first payment must be made at the end of David's first month of college)

$839.64.

After deciding to buy a new car, you can either lease the car or purchase it on a four-year loan. The car you wish to buy costs $38,000. The dealer has a special leasing arrangement where you pay $299 today and $500 per month for the next four years. If you purchase the car, you will pay it off in monthly payments over the next four years at a 4.8% APR. You believe you will be able to sell the car for $20,000 in four years. Assume that all payments occur at the end of each month. (a) Should you buy or lease the car? (b) What break-even resale price in four years would make you indifferent between buying and leasing?

(a) Should you buy or lease the car? PV of Buying = $38,000 $20,000 PVIF(0.4%,48) = $21,487.54 PV of Leasing = $299 + 500 PVIFA(0.4%,48) = $22,096.12 (b) What break-even resale price in four years would make you indifferent between buying and leasing? $38,000 C PVIF(0.4%,48) = $22,096.12 C = $19,262.88

The Wine Press is considering a project which has an initial cash requirement of $187,000. The project will yield cash flows of $2,832 monthly for 7 years. What is the rate of return on this project?

187,000 = 2,832 x PVIFA(r,84) r = 0.5921% x 12 = 7.105% annually

Today, you are retiring. You have a total of $411,016 in your retirement savings and have the funds invested such that you expect to earn an average of 7.10 percent, compounded monthly, on this money throughout your retirement years. You want to withdraw $2,500 at the end of every month. How long will it be until you run out of money?

411,016 = 2,500 x PVIFA(7.10%/12,t) t = 610.63 months = 50.89 years

Today, you borrowed $6,200 on your credit card to purchase some furniture. The interest rate is 14.9 percent, compounded monthly. How long will it take you to pay off this debt assuming that you do not charge anything else and make regular monthly payments of $120?

6,200 = 120 x PVIFA(14.9%/12,t) T = 83.135 months = 6.928 years.

Your grandmother is gifting you $100 a month for four years while you attend college to earn your bachelor's degree. At a 5.5 percent discount rate, what are these payments worth to you on the day you enter college?

APV = $100 x PVIFA (5.5%/12,48) = $4,299.88

Your employer contributes $75 a week to your retirement plan. Assume that you work for your employer for another 20 years and that the applicable discount rate is 7.5 percent. Given these assumptions, what is this employee benefit worth to you today?

APV = $75 x PVIFA (7.5%/52,5220) = $40,384.69

Consider the following newspaper ad on the passenger car: $1,500 cash rebate or 1.9% financing. Assume that the car price (after negotiation) is $20,000 and he or she can finance the 4-year car loan from a local bank at 8.5%. Which option is better off to the consumer? That is, find the monthly payment for each option. At what interest rate would each option be indifferent?

Cash Rebate option: real price = $20,000 - $1,500 = $18,500. 18,500 = C PVIFA x ( ,) 8 5%. 12 48 x C = $455.99 (b) 1.9% Financing option: 20,000 = C PVIFA x ( ,) 1 9%. 12 48 x C = $433.03 At what interest rate would each option be indifferent? 18,500 = 433.03 x PVIFA(r, 48) Using the financial calculator, we get r = 5.83% annually

Perpetuities

an infinite series of level stream of cash flows, e.g., preferred stock.

What is the present value of $4,000 per year at a discount rate of 10%, if the first payment is received 8 years from now and the last payment is received 25 years from now?

V7 = $4,000 x PVIFA(10%,18) = $32,805.65

Ordinary Annuity

multiple, identical cash flows occurring at the end of each period for some fixed number of periods.