DASHBOARD LEARN ADAPT MENU
Learn FM 1 1.1 1.1.2 Effective Rate of Interest
Effective Rate of Interest
In this exam, the most important interest rate is the effective rate of interest. To calculate the effective interest rate for period t , divide
the interest earned during period t by the amount invested at the beginning of period t . The t -period effective rate of interest is
denoted as i t :
I (t) A (t) − A (t − 1)
it = = (1.1.2)
A (t − 1) A (t − 1)
Observe we only need the fund value at the beginning and ending period to calculate the effective interest rate.
Consider the following example:
Based on the following amount function graph, calculate each year’s effective interest rate.
Here are the effective interest rates for the first 3 years:
A (1) − A (0) 110 − 100
i1 = = = 10%
A (0) 100
A (2) − A (1) 125 − 110
i2 = = = 13.6364%
A (1) 110
A (3) − A (2) 90 − 125
i3 = = = −28%
A (2) 125
Note an infinite number of amount functions can produce these three effective rates. In this example, a smooth exponential curve
exists in year 1, a stock-market type graph in year 2, and a linear function in year 3. Regardless of what occurs during a year, any
amount function intersecting these points produces the same three effective rates.
Rearrange Formula (1.1.2) to solve for A (t) :
A (t) − A (t − 1)
it =
A (t − 1)
i t A (t − 1) = A (t) − A (t − 1)
i t A (t − 1) + A (t − 1) = A (t)
A (t) = A (t − 1) (1 + i t )
Using the previous effective interest rates, calculate the time-3 amount function as a function of the time-0 amount function.
A (3) = A (2) (1 + i 3 )
= A (1) (1 + i 2 ) (1 + i 3 )
= A (0) (1 + i 1 ) (1 + i 2 ) (1 + i 3 )
= 100 (1.1) (1.1364) (0.72)
= 90
Observe we can calculate the amount function at time t by multiplying the initial amount by (1 + consecutive effective rates of
interest). If the effective interest rate is constant for all periods, the amount function becomes:
A (t) = A (0) (1 + i 1 ) (1 + i 2 ) . . . (1 + i t )
= A (0) (1 + i) (1 + i) . . . (1 + i)
= A (0) (1 + i) t (1.1.3)
where t is the number of periods, and i is the constant effective interest rate for each period.
Note we assumed the initial investment occurs at time 0. Let’s remove this assumption and assume the initial investment occurs at
time k where k ≤ t . Now, the deposit earns interest from time k to t .
t−k
A (t) = A (k) (1 + i)
This equation remains true even if k >t .
EXAMPLE 1.1.1
Sam invested $500 in a bank. After two years, the fund increased to $561.80.
What was the annual effective interest rate offered by the bank?
SOLUTION
Define t in terms of years. $500 accumulates to $561.80 in two years.
A (0) = 500, A (2) = 561.80
A (2) = A (0) (1 + i) 2
561.80 = 500 (1 + i) 2
(1 + i) 2 = 1.1236
i=√− −−−− − 1
1.1236
i = 0.06
The annual effective interest rate was 6%.
EXAMPLE 1.1.2
Beth invests $300 in a fund earning a 1% monthly effective interest rate.
How much is in her fund at the end of eight months?
SOLUTION
Since we are given the monthly effective interest rate, define t in terms of months.
8
A (8) = A (0) ⋅ (1 + i)
= 300 ⋅ (1.01) 8
= 324.8570
Beth has $324.86 at the end of eight months.
EXAMPLE 1.1.3
James invests $100 in a bank. The fund earns an annual effective interest rate of 10%.
Calculate the fund accumulation at the end of six months.
SOLUTION
We are given an annual effective interest rate, but we need to calculate the amount at the end of six months.
Define t in terms of years. Six months is half a year, t = 0.5 .
A (0) = 100, i = 10%, t = 0.5
1/2
A (0.5) = 100 ⋅ (1 + 0.10) = 104.88
The fund accumulation is $104.88 at the end of six months.
Another approach to this question is shown in Example 1.1.11.
EXAMPLE 1.1.4
Toby wants to accumulate $500 in one year.
How much does he need to invest today if the account credits a 7% annual effective interest rate?
SOLUTION
We are not given the initial investment. Instead, we are given the amount at the end of one year.
A (1) = A (0) ⋅ (1 + i)
A (1)
A (0) =
1.07
500
A (0) =
1.07
= 467.2897
Toby needs to invest $467.29 today.
Most exam problems assume a constant effective interest rate. Thus, if a problem states that the annual effective interest rate is 5%,
assume it is constant each year. Accumulate using the factor (1.05) t where t is the number of years the money is invested.
Observe this is an exponential growth, an example of compound interest.
Previous Lesson Next Lesson
Assignment 1.1.1 Amount Function Watch 1.1.2 Effective Rate of Interest
