Effective Interest Rates

Thus far in this class, we have been using nominal interest rates (though we didn't know to call them that). The problems we have been given have asked us to calculate present or future values of money for a time interval ("calculation period") that has units which match the unit of time given in the interest rate. For example, we may have been asked to find the amount of money someone will owe in three years if they take out a $1,000 loan at 5% interest. The calculation period is three years; the calculation period unit is one year. The unit of time given in the interest rate, though unstated, is also one year (5% interest means "5% interest per year"). That unstated interest rate time unit is called the "nominal period."

What we're now going to learn is that there is another unstated unit of time in that problem: the compounding time unit (the "compounding period"). If we were to include the compounding period in the example above, we would say "5% interest per year, compounded annually." Note that all three of the time units in the example match: the calculation period, nominal period , and compounding period units are each one year. However, most real-world loans mix time period units together. For example, how would you calculate the total amount owed after twelve weeks on a $1000 loan with 15% annual interest, compounded daily? So far, we have not been given the tools to solve this problem.

In order to calculate present or future values of money you must match the time period units. Where do we start our conversion? Start by converting the nominal period unit to an "effective period" unit. In the second example given, we start by converting 15% annual interest, which has a nominal period of one year, to some effective unit other than years. What unit shall we choose?

The most convenient effective period unit for the problem above is one week. Why? Because it is the only time unit that is not associated with the interest rate. We can use the effective interest rate (EIR) formula, shown below, to incorporate all of the units associated with the interest rate (the nominal and compounding period units) into any chosen time unit in a single step. Therefore, we choose to convert the two interest rate time units to match the time unit unassociated with the interest rate. Next, we solve the problem using (F|P, i%, n), using the effective interest rate for i%. Again, note that the units of "n" and "i%" in (F|P, i%, n) match.

In summary, we use effective interest rates as a means of transforming costs of loans and benefits of assets into figures with equal units of time. This allows multiple assets with different compounding periods to be placed on an equal basis of comparison. The most convenient way to solve a problem is to convert the nominal interest rate to an effective interest rate which has units of time that match the calculation period unit given in the problem. Accurate decisions cannot be made when choosing between options unless interest units have been transformed to match the calculation period unit [1].

How to find Effective Interest Rates

Three variables are required to calculate effective interest rate (EIR):

  1. $i$, the nominal rate per nominal period,
  2. $p$, the number of compounding periods per nominal period, and
  3. $n$, the number of compounding periods in the desired effective period.

Effective interest is most commonly calculated as if compounded annually ("effective annual interest rate").

(1)
\begin{align} EIR=\left[1+\left(\frac{i}{p}\right)\right]^{n}-1 \end{align}

Another way to look at the effective interest rate equation is to look at it as unit conversion. To do this, separate the EIR equation into the three periods required to solve for effective interest: the nominal period, the compounding period, and the effective period. Use variables to represent the units of the three periods and rearrange the equation above using those variables. If

$nom$ is the nominal period unit (e.g. if the nominal interest rate, $i$, is given as 2% per month, the nominal period unit is one month)
$cmp$ is the compounding period unit (e.g. if the nominal interest rate is compounded daily, the compounding period unit is one day), and
$eff$ is the effective period unit (e.g. if we want to find the effective annual interest rate, the effective period unit is one year),

then by substitution, the number of compounding periods per nominal period, $p$, can be rewritten as $cmp/nom$. The number of compounding periods in the desired effective period, $n$, can be rewritten as $cmp/eff$. The nominal interest rate, $i$, can be thought of as the interest percentage, $I$, per nominal period $nom$. Using these substitutions, the effective interest rate equation can be rewritten as

(2)
\begin{align} EIR=\left[1+\left(\frac{I}{nom}\right)\left(\frac{nom}{cmp}\right)\right]^{{cmp}/{eff}}-1 \end{align}

Note:

  • The nominal period units cancel, and the result inside the brackets is 1 + the nominal interest percentage per compounding period $\left(1+I/cmp\right)$.
  • The numerator unit in the exponent matches the denominator unit inside the brackets.
  • When doing the unit conversion, don't divide through by numbers in the denominator. For example, if the effective period is "biweekly" (i.e. once every two weeks) and the compounding period is daily, do NOT divide 14 days by 2 (weeks) to get 7. In that particular case, the exponent is 14 (not 7).

Remember that in this class, for the purposes of unit conversion, there are always 7 day/wk, 30 day/mo, 365 day/yr, 12 mo/yr, 4 wk/mo, and 52 wk/yr.

Be careful! Always use percentage expressed as a decimal in hundredths for $I$.

Example Calculations

Effective Annual Interest Rate

6% per year compounded monthly; e.g. a home mortgage

(3)
\begin{align} EIR_{annual}=&\left[1+\left(\frac{.06}{1 yr}\right)\left(\frac{1 yr}{12 mo}\right)\right]^{{12 mo}/{1 yr}}-1 \\ =&\left(1+ \frac {.06}{12} \right)^{12} - 1 \\ = &.0617 \\ =&6.17\% \end{align}

2% per year compounded daily; e.g. a savings account

(4)
\begin{align} EIR_{annual}=&\left[1+\left(\frac{.02}{1\,yr}\right)\left(\frac{1\,yr}{365\,day}\right)\right]^{{365\,day}/{1\,yr}}-1 \\ =&\left[1+\left(\frac{.02}{365}\right)\right]^{365}-1 \\ =&.0202 \\ =&2.02\% \end{align}

2% per month compounded daily; e.g. credit card

(5)
\begin{align} EIR_{annual}=&\left[1+\left(\frac{.02}{1\,mo}\right)\left(\frac{1 mo}{30\,day}\right)\right]^{{365\,day}/{1\,yr}}-1 \\ =&\left(1+ \frac{.02}{30} \right)^{365} - 1 \\ =&.2754 \\ =&27.54\% \end{align}

One for Eight weekly; e.g. your friendly neighborhood loan shark.

Essentially this means that for every eight dollars you borrow, you owe one dollar per week for this loan; this translates into 12.5% per week compounded weekly

(6)
\begin{align} EIR_{annual}=&\left[1+\left(\frac{$1/$8}{1\,wk}\right)\left(\frac{1\,wk}{1\,wk}\right)\right]^{{52\,wk}/{1\,yr}}-1 \\ =&(1+.125)^{{52}}-1 \\ =&456.0158 \\ =&45601.58\% \end{align}

Effective Monthly Interest Rate

Note that in the current version (9.5) of the NCEES FE Reference Handbook, the only formula given is for calculating the annual effective interest rate. This means that you must memorize the symbolic forms of the formulas shown on this wiki page.

Why You Need to Find Effective Interest Rates

The effective interest rate is used to compare loans and/or investments with different nominal, compounding, or effective rates. For example, in order to compare a loan with an annual compounding rate to a loan with a monthly compounding rate unless you convert one rate to match the other. It is also used to calculate payments made in increments different from the nominal period. For example, if you want to know what your monthly car payment will be given the nominal annual interest rate, you must calculate the effective monthly interest rate.

Language Matters

One of the things that confuses students about loans are the assumptions that are made about time units, either of the nominal period or compounding periods or the period of the effective interest rate when none is specified. The default is almost always annual. So, if someone tells you that the interest rate on their mortgage is 3% you can safely assume that it is an annual rate. The same rule works for effective interest rates. If you are told, or asked for, “the effective interest rate”, you should just assume that it is the “effective annual rate”.

Questions

Bibliography
1. Park, Chan S. Contemporary Engineering Economics. 4th ed. Saddle River: Pearson, 2007. Print.
2. Rogers, Martin G. Engineering Project Appraisal: The Evaluation of Alternative Development Schemes. Illustrated ed. Hoboken: Wiley-Blackwell, 2001. Print.
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