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Interest Rate Conversions

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Converting interest rates

There are many situations where an interest rate with a specified compounding frequency must be converted into an equivalent rate with a different compounding frequency. Examples include situations where a need exists to compare alternative interest rates with different compounding frequencies and where the payment frequency does not match the compounding frequency in an annuity problem.
The basic relationship used to convert interest rates from one compounding frequency to another is shown in Figure 1 below.
Figure 1: Formula on Effective Rate
The effective rate is an annually compounded interest rate that is equivalent to the nominal rate compounded more frequently. The nominal rate is the stated rate in a problem, such as 5%, compounded monthly. The number of periods per year is also stated in most problems. An interest rate compounded monthly involves 12 periods per year, for example.
Using the relationship shown in Figure 1 above, any effective annual rate can be converted to a rate compounded more frequently and any rate compounded more frequently than once a year can be converted to an effective annual rate.
Converting an interest rate with continuous compounding uses the formula below. Of course, there is no need for the number of periods per year, which has no meaning under continuous compounding. In the formula below, e is the numeric constant 2.7182818…
Figure 2: Formula

Converting interest rates on the HP 17bII+

The HP 17bII+ calculator has functions that use the relationship shown in Figures 1 and 2 built-in and available to the user by pressing . When pressed, the user must choose between periodic and continuous compounding as shown below in Figure 3.
Figure 3: Selecting between periodic and continuous compounding
If periodic is chosen, the functions are displayed as shown in Figure 4 below.
Figure 4: Displaying the functions in periodic compounding
If continuous is chosen, the functions are displayed as shown in Figure 5 below.
Figure 5: Displaying the functions in continuous compounding
Interest rates are entered as they would be written before a percent sign, i.e., 5% is entered as and not as .

Practice converting interest rates

Example 1

What annual rate is equivalent to 8%, compounded monthly?

Solution

KeystrokeDisplay
Figure 6: Displaying the annual interest rate

Answer

8.30%. Over time, an annual rate of 8.30% would produce the same effects as 8%, compounded monthly.

Example 2

What rate, compounded monthly, is equivalent to an effective annual rate of 8.30%?

Solution

KeystrokeDisplay
Figure 7: Displaying the monthly interest rate

Answer

The result is 8%.

Example 3

Which interest rate would give you better returns as an investor? 4.25%, compounded quarterly or 4.15%, compounded monthly?

Solution

The way to solve problems like these is to convert each rate to an effective annual rate and then compare them.
KeystrokeDisplay
Figure 8: Displaying the annual interest rate
Figure 9: Displaying the annual interest rate

Answer

4.25% compounded quarterly is equivalent to 4.32% compounded annually (or to an effective rate of 4.32%), while 4.15% compounded monthly is equivalent to 4.23% compounded annually.

Example 4

Convert 5%, compounded monthly to an equivalent semiannual rate.

Solution

First, convert the monthly rate to an effective rate.
KeystrokeDisplay
Figure 10: Displaying the effective rate
Then convert this rate to a semiannual rate.
Figure 11: Displaying the semiannual rate

Answer

5%, compounded monthly is equivalent to 5.05%, compounded semiannually (to more decimal places, it is actually 5.05237359091%, compounded semiannually).

Example 5

What effective interest rate is 10% compounded continuously equal to?

Solution

KeystrokeDisplay
Figure 12: Displaying the effective rate

Answer

10% compounded continuously is equivalent to an effective rate of 10.52% (which is 10.52% compounded annually).

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