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LECTURE 3 PROBLEMS

3.1 Calculating Present values. Suppose you have just celebrated your 19th birthday. A rich uncle has set up a trust fund for you that will pay you $150, 000 when you turn 30. If the relevant discount rate is 9%, how much is this fund worth today?

3.2 Calculating rates of return. You’ve been offered an investment that will double your money in 10 years. What rate of return are you being offered?

3.3 Calculating the number of periods. You’ve been offered an investment that will pay you 9% per year. If you invest $15,000, how long until you have $30,000? How long until you have $45,000?

3.4 Calculating interest rates and future values. In 1985, the first US Open Golf Championship was held. The winner’s prize money was $150. In 2001, the winner’s check was $900,000. What was the percentage increase in the winner’s check over this period? If the winner’s prize increases at the same rate, what will it be in 2040?

3.5 Calculating Present values. Suppose you are committed to owning a $120,000 Ferrari. If you believe your mutual fund can achieve an 11% annual rate of return and you want to buy the car in 10 years on the day you turn 30. How much must you invest today?

3.6 Calculating the number of periods. You expect to receive $10,000 at graduation in 2 years. You plan on investing it at 12% until you have $120,000. How long will you wait from now?

3.7 Annuity Present value. You are looking into an investment that will pay you $12,000 per year for the next 10 years. If you require a 15% return, what is the most you would pay for this investment?

3.8 APR vs EAR. The going rate on student loan is quoted as 8% APR. The terms of loans call for monthly payments. What is the effective annual rate (EAR) on such a student loan?

3.9 It’s the Principal that matters. Suppose you borrow $10,000. You are going to repay the loan by making equal annual payments for 5 years. The interest rate on the loan is 14% per year. Prepare an amortization schedule for the loan. How much interest rate will you pay over the life of loan?

3.10 Just a little bit each month. You’ve recently finished your MBA at the Darnit School. Naturally, you must purchase a new BMW immediately. The car costs about $21,000. The bank quotes an interest rate of 15% APR for 72- month loan with a 10% down payment. You plan on trading the car in for a new one in two years. What will your monthly payment be? What is the effective interest rate on the loan? What will the loan balance be when you trade the car in?

3.11 Calculating Annuity values. You want to have $50,000 in your savings account five years from now and you‘re prepared to make equal annual deposits into the account at the end of each year. If the account pays 6.2% interest, what amount must you deposit each year?

3.12 Calculating perpetuity values. The perpetual life insurance Corp. is trying to sell you an investment policy that will pay you and your heirs $5,000 per year forever. If the required return on this investment is 9%, how much will you pay for the policy?

3.13 Calculating loan payments. You want to buy new sport car for $48,250 and the finance office at the dealership has quoted you a 9.8% APR loan for 60 months to buy the car. What will your monthly payments be? What is the EAR on this loan?

3.14 Calculating the Number of payments. You’ve prepared to make monthly payments of $95, beginning at the end of this month, into an account that pays 10% interest compounded monthly. How many payments will you have made when your account balance reaches $18,000?

3.15 Calculating Annuities Due. An ordinary annuity assumes equal payments at the end of each period over the life of the annuity. An annuity due is the same thing except the payments occur at the beginning of each period instead.

a. At a 10.5% annual discount rate, find the present value of a six- year ordinary annuity contract of $475 payments.

b. Find the present value of the same contract if it is an annuity due.

3.16 EAR vs. APR. There are two banks in the area that offer 30-year, $150,000 mortgages at 8.5% and charge a $1,000 loan application fee. However, the application fee charged by Insecurity Bank and Trust is refundable if the loan application is denied, whereas that charged by I.M Greedy and Sons Mortgage Bank is not. The current disclosure law requires that any fees that will be refunded if the application is rejected be included in calculating the APR, but this is not requires with nonrefundable fees (presumably because refundable fees are part of the loan rather than a fee)

What are the EARs on these loans? What are the APRs?

3.17 Calculating a Balloon payment. You have just arranged for a $300,000 mortgage to finance the purchase of a large tract of land. The mortgage has a 9% APR, and it calls for monthly payments over the next 15 years. However the loan has a five- year balloon payment, meaning that the loan must be paid off then. How big will the balloon payment be?

3.18 Break- Even Investment Returns. Your financial planner offers you two different investment plans. Plan X is an $8,000 annual perpetuity. Plan Y is a 10-year, $20,000 annual annuity. Both plans will make their first payment one year from today. At what discount rate would you be indifferent between these two plans?

3.19 Calculating Annuity payments. This is a classic retirement problem. Your friend is celebrating her 35th birthday today and want to start saving for her anticipated retirement at age 65. She wants to be able to withdraw $80,000 from her savings account on each birthday for 15 years following her retirement; the first withdrawal will be on her 66th birthday. Your friend intends to invest her money in the local credit union, which offers 9% interest per year. She wants to make equal annual payments on each birthday into the account established at the credit union for her retirement fund.

a. If she starts making these deposits in her 36th birthday and continues to make deposits until she is 65 (the last deposit will be in her 65th birthday), what amount must she deposit annually to be able to make the desired withdrawals at retirement?

b. Suppose your friend has just inherited a large sum of money. Rather than making equal annual payments, she has decided to make one lump- sum payment on her 35th birthday to cover her retirement needs. What amount does she have to deposit?

c. Suppose your friend’s employer will contribute $1,500 to the account every year as part of the company’s profit- sharing plan. In addition, your friend expects a $30,000 distribution from a family trust fund on her 55th birthday, which she will also put into the retirement account. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?

3.20 Future Values and Multiple Cash Flows. An insurance company is offering a new policy to its customers. Typically, the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: The purchasers makes the following six payments to the insurance company:

First birthday: $750

Second birthday: $750

Third birthday: $850

Fourth birthday: $850

Fifth birthday: $950

Sixth birthday: $950

After the child’s sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $175,000. If the relevant interest rate is 10% for the first six years and 6% for all subsequent years, is the policy worth buying?

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