Effective Interest Rates

To understand loans it is important to grasp the difference between the the nominal and effective interest rate. The nominal rate is what is standard reporting from banks and is very close to annual percentage rate or APR. The APR is the yearly cost of a loan, including interest, insurance, and the origination fee, expressed as a percentage. The effective interest rate is the rate that will more accurately represent the interest earned over the course of a year [1] [2].

The main point concerning these topics is that there are many different situations involving cash flows that occur in intervals different from the nominal statement. So far, all of our calculations have been based on a yearly rate and nominal interest rates have been used. We use effective interest rates as a means of transforming costs and benefits of different assets/loans into figures with equal units of time. This allows multiple assets with different compounding periods to be placed on an equal comparison plane thereby letting accurate decisions to be made when choosing between them [1].

An Interest Rate is Really Three Numbers

1.) Nominal Statement

The nominal interest rate $i$.

2.) Nominal Period

The number of compounding periods in nominal statement $p$.

3.) Compounding Period

The number of compounding periods in desired effective period $n$.

How to find Effective Interest Rates

The effective interest rate is calculated as if compounded annually. The effective rate is calculated using EIR as the effective annual rate, $i$ as the nominal rate (nominal statement) per nominal period, $p$ as the number of compounding periods per nominal period, and $n$ the number of compounding periods in desired effective period—i.e. the number of compounding periods that will occur in the new period you are calculating.

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

Where the nominal interest rate is $i$, the number of compounding periods in nominal statement is $p$, and the number of compounding periods in desired effective period is $n$.

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 $I$.

Effective ANNUAL Interest Rates

1. 6% per year compounded monthly; e.g. house 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 \end{align}

$\left(1+ \frac {.06}{12} \right)$12 - 1 = .617 = 6.17%

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

(4)
\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 \end{align}

$\left(1+ \frac{.02}{30} \right)$365 - 1 = .2754 = 27.54%

3. 2% per year compounded daily; e.g. savings account

(5)
\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 \end{align}

$\left(1+ \frac {.02}{365} \right)$365 - 1 = .0202 = 2.02%

4. 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 \end{align}

$\left(1+ \frac {.125}{1} \right)$52 - 1 = 456.0158 = 45,601.58%

Why You Need to Find Effective Interest Rates

The effective interest rate is used to make loans and/or investments with different compounding terms (daily, monthly, annually, or other) more comparable.

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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