Question

*Round all values to 4 decimal places.

1. What is the present value of $10,000 paid at the end of each of the next 71 years if the interest rate is 8% per​ year?

The present value is $______. ​(Round to the nearest​ cent.)

2. Assume that your parents wanted to have $110,000 saved for college by your 18th birthday and they started saving on your first birthday. They saved the same amount each year on your birthday and earned 5.5% per year on their investments.

a. How much would they have to save each year to reach their​ goal?

b. If they think you will take five years instead of four to graduate and decide to have $150,000 saved just in​ case, how much would they have to save each year to reach their new​ goal?

3. You are trying to decide how much to save for retirement. Assume you plan to save $4,500 per year with the first investment made one year from now. You think you can earn 7.5​% per year on your investments and you plan to retire in 38 years, immediately after making your last $4,500 investment.

a. How much will you have in your retirement account on the day you​ retire?

b.​ If, instead of investing $4,500 per​ year, you wanted to make one​ lump-sum investment today for your retirement that will result in the same retirement​ saving, how much would that lump sum need to​ be?

c. If you hope to live for 29 years in​ retirement, how much can you withdraw every year in retirement​ (starting one year after​ retirement) so that you will just exhaust your savings with the 29th withdrawal​ (assume your savings will continue to earn 7.5​% in​ retirement)?

d.​ If, instead, you decide to withdraw $175,000 per year in retirement​ (again with the first withdrawal one year after​ retiring), how many years will it take until you exhaust your​ savings? (Use​ trial-and-error, a financial​ calculator: solve for​ "N", or​ Excel: function​ NPER)

e. Assuming the most you can afford to save is $900 per​ year, but you want to retire with $1,000,000 in your investment​ account, how high of a return do you need to earn on your​ investments? (Use​ trial-and-error, a financial​ calculator: solve for the interest​ rate, or​ Excel: function​ RATE)

4. A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $2,000.

Each year after​ that, you will receive a payment on the anniversary of the last payment that is 8% larger than the last payment. This pattern of payments will go on forever. Assume that the interest rate is 11% per year.

a. What is​ today's value of the​ bequest?

b. What is the value of the bequest immediately after the first payment is​ made?

5. You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 17 years. You expect that the​ drug's profits will be $1 million in its first year and that this amount will grow at a rate of 2% per year for the next 17 years. Once the patent​ expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is 11% per​ year? (Round to three decimal​ places.)

6. You have an investment account that started with $4,000 10 years ago and which now has grown to ​$12,000.

a. What annual rate of return have you earned​ (you have made no additional contributions to the​ account)?

b. If the investment account earns 16% per year from now​ on, what will the​ account's value be 10 years from​ now?

7. You are thinking of purchasing a house. The house costs $350,000. You have $50,000 in cash that you can use as a down payment on the​ house, but you need to borrow the rest of the purchase price. The bank is offering a 30​-year mortgage that requires annual payments and has an interest rate of 6% per year. What will be your annual payment if you sign this​ mortgage?

8. You are saving for retirement. To live​ comfortably, you decide you will need to save $4 million by the time you are 65. Today is your 33rd ​birthday, and you​ decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 6%​, how much must you set aside each year to make sure that you will have $4 million in the account on your 65th birthday?

9. A local bank is running the following advertisement in the​ newspaper: "For just $2,000 we will pay you $140 ​forever!" The fine print in the ad says that for a $2,000 deposit, the bank will pay $140 every year in​ perpetuity, starting one year after the deposit is made. What interest rate is the bank advertising​ (what is the rate of return of this​ investment)?

Answer 1

Question 1:

Here we have to calculate the present value of the annuity of $10,000 for 71 years at 8% interest

The present value of the annuity is calculated as PV = P*(1-(1+r)^-n)/r where

P is the annuity = 10,000

n = 71

r = 8% = 0.08

Present Value = 10,000*(1-(1+0.08)^-71)/0.08

Present value = 10,000*(1-1.08^-71)/0.08

Present Value =10,000*12.44705519

Present Value =124,470.5519 (Rounded to four decimals)

Present Value = 124,470 .55 (Rounded to the nearest cent)

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