1. Basics of Interest Theory | Exam FM
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Problem 1.1
You invest $3,200 in a savings account on January 1, 2004. On December 31,
2004, the account has accumulated to $3,294.08. What is the annual interest
rate?
Problem 1.2
You borrow $12,000 from a bank. The loan is to be repaid in full in one
year’s time with a payment due of $12,780.
(a) What is the interest amount paid on the loan?
(b) What is the annual interest rate?
Problem 1.3
The current interest rate quoted by a bank on its savings accounts is 9% per
year. You open an account with a deposit of $1,000. Assuming there are no
transactions on the account such as depositing or withdrawing during one
full year, what will be the amount value in the account at the end of the
year?
Problem 1.4
The simplest example of interest is a loan agreement two children might
make:“I will lend you a dollar, but every day you keep it, you owe me one
more penny.” Write down a formula expressing the amount value after t days.
Problem 1.5
When interest is calculated on the original principal only it is called simple
interest. Accumulated interest from prior periods is not used in calculations
for the following periods. In this case, the amount value A, the principal P,
the period of investment t, and the annual interest rate i are related by the
formula A = P(1 + it). At what rate will $500 accumulate to $615 in 2.5
years?
Problem 1.6
Using the formula of the previous problem, in how many years will 500 accumulate to 630 if the annual interest rate is 7.8%?
Problem 1.7
Compounding is the process of adding accumulated interest back to the
principal, so that interest is earned on interest from that moment on. In this
case, we have the formula A = P(1 + i)
t and we call i a yearly compound
interest. You can think of compound interest as a series of back-to-back
simple interest contracts. The interest earned in each period is added to the
principal of the previous period to become the principal for the next period.
You borrow $10,000 for three years at 5% annual interest compounded annually. What is the amount value at the end of three years?
Problem 1.8
Using compound interest formula, what principal does Andrew need to invest
at 15% compounding annually so that he ends up with $10,000 at the end of
five years?
Problem 1.9
Using compound interest formula, what annual interest rate would cause an
investment of $5,000 to increase to $7,000 in 5 years?
Problem 1.10
Using compound interest formula, how long would it take for an investment
of $15,000 to increase to $45,000 if the annual compound interest rate is 2%?
Problem 1.11
You have $10,000 to invest now and are being offered $22,500 after ten years
as the return from the investment. The market rate is 10% compound interest. Ignoring complications such as the effect of taxation, the reliability of
the company offering the contract, etc., do you accept the investment?
Problem 1.12
Suppose that annual interest rate changes from one year to the next. Let
i1 be the interest rate for the first year, i2 the interest rate for the second
year,· · · , in the interest rate for the nth year. What will be the amount value
of an investment of P at the end of the nth year?
Problem 1.13
Discounting is the process of finding the present value of an amount of
cash at some future date. By the present value we mean the principal that
must be invested now in order to achieve a desired accumulated value over a
specified period of time. Find the present value of $100 in five years time if
the annual compound interest is 12%.
Problem 1.14
Suppose you deposit $1000 into a savings account that pays annual interest
rate of 0.4% compounded quarterly (see the discussion at the end of page
11.)
(a) What is the balance in the account at the end of year.
(b) What is the interest earned over the year period?
(c) What is the effective interest rate?
Problem 1.15
The process of finding the present value P of an amount A, due at the end of
t years, is called discounting A. The difference A−P is called the discount
on A. Notice that the discount on A is also the interest on P. For example,
if $1150 is the discounted value of $1250, due at the end of 7 months, the
discount on the $1250 is $100. What is the interest on $1150 for the same
period of time?
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