Present Value of an Annuity

Problem 4-13
Present Value of an Annuity

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Find the present value of the following ordinary annuities. Round your answers to the nearest cent. (Notes:If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can “override” the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in many situations, to see how changes in input variables affect the output variable. Also, note that you can leave values in the TVM register, switch to “BEG,” press FV, and find the FV of the annuity due.)

a. $800 per year for 10 years at 8%.
$

b. $400 per year for 5 years at 4%.
$

c. $800 per year for 5 years at 0%.
$

Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

d. $800 per year for 10 years at 8%.
$

e. $400 per year for 5 years at 4%.
$

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Problem 4-11
Time for a Lump Sum to Double

To the next whole year, how long will it take $200 to double if it is deposited and earns the following rates? Round your answers up to the next highest year. [Notes:(1) If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can “override” the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in parts b and d, and in many other situations, to see how changes in input variables affect the output variable.) (2) This problem cannot be solved exactly with some financial calculators. For example, if you enter PV = -200, PMT = 0, FV = 400, and I = 7 in an HP-12C, and then press the N key, you will get 11 years. The correct answer is 10.2448 years, which rounds to 10, but the calculator rounds up. However, the HP-10B gives the correct answer.]

a. 6%.
year(s)

b. 10.1%.
year(s)

c. 16%.
year(s)

d. 100%.
year(s)

Problem 4-4
Number of Periods of a Single Payment

If you deposit money today in an account that pays 5.2% annual interest, how long will it take to double your money? Round your answer to the nearest whole.

years

Problem 4-7
Present and Future Value of an Uneven Cash Flow Stream

An investment will pay $200 at the end of each of the next 3 years, $400 at the end of Year 4, $600 at the end of Year 5, and $800 at the end of Year 6. If other investments of equal risk earn 4% annually, what is its present value? Round your answer to the nearest cent.
$

What is its future value? Round your answer to the nearest cent.
$
A $50,000 loan is to be amortized over 7 years, with annual end-of-year payments. Which of these statements is CORRECT?

Q8

A $150,000 loan is to be amortized over 7 years, with annual end-of-year payments. Which of these statements is CORRECT?

Q9

Your bank account pays a 6% nominal rate of interest. The interest is compounded quarterly. Which of the following statements is CORRECT?

Problem 4-28
PV and Effective Annual Rate

Assume that you inherited some money. A friend of yours is working as an unpaid intern at a local brokerage firm, and her boss is selling securities that call for 4 payments of $50 (1 payment at the end of each of the next 4 years) plus an extra payment of $1,000 at the end of Year 4. Your friend says she can get you some of these securities at a cost of $875 each. Your money is now invested in a bank that pays an 6% nominal (quoted) interest rate but with quarterly compounding. You regard the securities as being just as safe, and as liquid, as your bank deposit, so your required effective annual rate of return on the securities is the same as that on your bank deposit. You must calculate the value of the securities to decide whether they are a good investment. What is their present value to you? Round your answer to the nearest cent.

$

Problem 4-30
Loan Amortization

Your company is planning to borrow $2,000,000 on a 7-year, 11%, annual payment, fully amortized term loan. What fraction of the payment made at the end of the second year will represent repayment of principal? Round your answer to two decimal places.

%

Problem 4-15
Effective Rate of Interest

Find the interest rate (or rates of return) for each of the following situations. Round your answers to two decimal places.

1. You borrow $700 and promise to pay back $728 at the end of 1 year.
   %
2. You lend $700 and receive a promise to be paid $728 at the end of 1 year.
   %
3. You borrow $95,000 and promise to pay back $156,788 at the end of 11 years.
   %
4. You borrow $8,000 and promise to make payments of $2,444.9 at the end of each year for 4 years.
   %

Problem 4-19
Effective versus Nominal Interest Rates

Universal Bank pays 4% interest, compounded annually, on time deposits. Regional Bank pays 3%, compounded quarterly.

1. Based on effective interest rates, in which bank would you prefer to deposit your money?

   I. You are indifferent between the banks and your decision will be based upon which one offers you a gift for opening an account.
   II. You would choose Universal Bank because its EAR (or EFF%) is higher.
   III.You would choose Regional Bank because its nominal interest rate is higher.
   IV.You would choose Universal Bank because its nominal interest rate is higher.
   V. You would choose Regional Bank because its EAR (or EFF%) is higher.

2. Could your choice of banks be influenced by the fact that you might want to withdraw your funds during the year as opposed to at the end of the year? In answering this question, assume that funds must be left on deposit during the entire compounding period in order for you to receive any interest.

   I. If funds must be left on deposit until the end of the compounding period (1 year for Universal Bank and 3 months for Regional Bank), and you have no intentions of making a withdrawal during the year, then Regional Bank might be preferable.
   II. If funds must be left on deposit until the end of the compounding period (1 year for Universal Bank and 3 months for Regional Bank), and you think there is a high probability that you will make a withdrawal during the year, then Universal Bank might be preferable.
   III.If funds must be left on deposit until the end of the compounding period (3 months for Universal Bank and 1 year for Regional Bank), and you think there is a high probability that you will make a withdrawal during the year, then Universal Bank might be preferable.
   IV.If funds must be left on deposit until the end of the compounding period (1 year for Universal Bank and 3 months for Regional Bank), and you think there is a high probability that you will make a withdrawal during the year, then Regional Bank might be preferable.
   V. If funds must be left on deposit until the end of the compounding period (3 months for Universal Bank and 1 year for Regional Bank), and you think there is a high probability that you will make a withdrawal during the year, then Regional Bank might be preferable.

Problem 4-24
Required Lump-Sum Payment

To complete your last year in business school and then go through law school, you will need $20,000 per year for 4 years, starting next year (that is, you will need to withdraw the first $20,000 one year from today). Your rich uncle offers to put you through school, and he will deposit in a bank paying 6.44% interest a sum of money that is sufficient to provide the 4 payments of $20,000 each. His deposit will be made today.

1. How large must the deposit be? Round your answer to the nearest cent.
   $
2. How much will be in the account immediately after you make the first withdrawal? Round your answer to the nearest cent.
   $
   How much will be in the account immediately after you make the last withdrawal? Round your answer to the nearest cent.
   $