Math 325
Chapter 1 – The Measurement of Interest
Interest: the compensation that a borrower of capital pays to a lender of capital for its use. It can be
viewed as a form of rent that the borrower pays to the lender to compensate for the loss of use of
capital by the lender.
Principal: the initial amount of money invested.
Accumulated Value: the total amount received after a period of time, t
Interest = Accumulated Value – Principal
Period: The unit in which time is measured. Unless otherwise stated, assume the period to be one year.
Accumulation function, a(t): the accumulated value at time t ³ 0 of an original investment of 1.
Properties:
a) a(0) = 1
b) For positive interest rates, a(t) is a non-decreasing function.
c) If effective interest accrues continuously, a(t) is continuous. Otherwise, a(t) will have
discontinuities.
Amount Function, A(t): the accumulated value at time t ³ 0 of an original investment of k.
Properties:
a) A(t) = k · a(t)
b) A(0) = k
c) For positive interest rates, A(t) is a non-decreasing function.
d) If effective interest accrues continuously, A(t) is continuous. Otherwise, A(t) will have
discontinuities.
Define In = amount of interest earned during the nth period. It's given by:
In = A(n) – A(n – 1)
Examples of Amount functions:
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Math 325
Chapter 1 – The Measurement of Interest
Effective rate of interest, i: the amount of money that 1 unit invested at the beginning of a period will
earn during the period, where interest is paid at the end of the period.
i = a(1) – a(0), or a(1) = 1 + i
Observations about the definition
a) “effective” is used for rates of interest in which interest is paid once per measurement period.
b) Effective rate of interest is often expressed as a percentage, e.g. i= 8% is equivalent to 0.08
earned per unit of principal.
c) The amount of principal remains constant throughout the period.
d) The interest is paid at the end of the period.
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Math 325
Chapter 1 – The Measurement of Interest
Example 1. An investment of $10,000 is made into a fund at time t = 0. The fund develops the
following balances over the next 4 years:
t
A(t)
0
10,000
1
10,600
2
11,130
3
11,575
4
12,153
a) Find the effective rate of interest for each of the four years.
b) If $5000 is invested at time t = 2, under the same interest environment, find the accumulated value
of the $5000 at time t = 4.
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Math 325
Chapter 1 – The Measurement of Interest
Example 2. It is known that a(t) is of the form ae0.09t + b. If $250 invested at time 0 accumulates to
$257.10` at time 5, find the accumulated value at time 12 of $250 invested at time 3.
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Math 325
Chapter 1 – The Measurement of Interest
Simple Interest
With simple interest, the amount of interest earned during each period is constant.
a(t) = 1 + it
for t = 1, 2, 3,...
A constant rate of simple interest does not imply a constant effective rate of interest.
Amount function A(t) for Simple Interest
Example 3. Find the accumulated value of $3500 invested for 10 years if the simple interest rate is 4%
per annum.
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Math 325
Chapter 1 – The Measurement of Interest
Compound Interest: interest is automatically reinvested to earn additional interest.
a(t) = (1 + i)t
for t = 1, 2, 3,...
A constant rate of simple interest implies a constant effective rate of interest and that the two are equal.
Note: Unless stated otherwise, we will assume that interest is accrued over functional periods according
to a(t) = 1 + it for simple interest, and a(t) = (1 + i)t for compound interest.
Example 4. Find the accumulated value of $100 at the end of 3 years and 8 months invested at 7% per
annum.
Example 5. Rework example 4 assuming simple interest during the final fractional period.
Example 7. Which of the two accumulated values is higher – one in example 4 or example 5. Explain.
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Math 325
Chapter 1 – The Measurement of Interest
Present Value:
We have seen that an investment of 1 will accumulate to 1 + i at the end of one period.
1 + i is known as the accumulation factor.
Now consider: How much should be invested initially, so that the balance will be 1 at the end of one
period?
Let that initial amount be x.
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v is called the discount factor, since it “discounts” the value of an investment at the end of the
period to its value at the beginning of the period.
•
How much should be invested initially such that the balance is 1 at the end of t periods?
The discount factor is denoted by a– 1(t). We get, for t ³ 0,
For simple interest : a– 1(t) =
For compound interest : a– 1(t) =
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Math 325
Chapter 1 – The Measurement of Interest
Example 8. If an investment of $1000 will grow to $6000 after 20 years, find the sum of present values
of two payments of $9000 each, which will occur at the end of 30 and 50 years, assuming the same
interest rate.
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