Interest rates are EQUIVALENT if they provide the same amount of interest on any loan. Consider the two rates:

1. 20.5% compounded semi-annually.
2. 20% compounded quarterly.

If you take any principal and any length of time, you will find that the two rates always result in exactly the same future value – hence the same interest. This is because they are related by the algebraic expression:

To check the equivalence, consider the following example.

$\left(1+\frac{0.205}{2}\right)^2 = \left(1+\frac{0.20}{4}\right)^4$

Example 4.8.1

Suppose $50,000 is invested for seven years at the interest rates noted above. Find the future value of the $50,000 for each interest rate.

You have:

For rate a: 20.5% compounded semi-annually

• n = 7×2 = 14 half-years
• I = $\frac{0.0205}{2}=10.25%$
• PV =$50,0000
• FV = ?

Answer: $196,006.46 for rate b

That exactly the same future value is obtained for both rates bears out the claim that the rates are equivalent. In fact, if two rates produce

the same result for any (non-zero) principal and time, then the rates will do so for any values. Hence they are equivalent.

You  can use calculator functions to find equivalent rates fairly easily, but first we will use the Future Value formula. You can use any size of investment and any length of time, but to illustrate this in the next example, $1 for one year is used.

Example 4.8.2

Suppose you are given the 20% compounded quarterly rate mentioned above and are asked to find the equivalent rate compounded

semi-annually. You are given:

• n = 1× 4 = 4
• $i = \frac{0.20}{4} = 0.05$
• PV = $1
• FV = ?

Using the Future Value Formula, we have;

$FV = PV(1+i)^n = $1(1.05)^4 = $1.21550625$

(Leave this answer in the calculator.)

For the new rate, the only thing that will be different (aside from i) is that it is to be compounded only twice in the year.

So we have

• n = 1× 2 =2
• i = ?
• PV = $1
• FV = $1.21550625

Using the Future Value Formula, we have;

$\begin{align*} PV(1+i)^n &= FV\\ $1(1+i)^2 &= $1.21550625\\ (1+i)^2 &= \frac{$1.21550625}{$1}\\ 1+ i &= \sqrt{$1.21550625} = 1.1025\\ i&= 1.1025-1\\ &=0.1025\\ \end{align*}$

So the nominal rate would be j₂ = 10.25% × 2 = 20.5%.

Effective Rates

Effective Rates with the BAII Plus