15.5: Interest Rates
- Page ID
- 185462
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Learn the differences between the following interest rates: the annual percentage rate (APR), the actuarial rate, and the effective rate.
- Learn how to compute the APR, the actuarial rate, and the effective rate on alternative loans.
- Learn how to use the compound interest formula to find a loan’s effective interest rate.
In this section, we identify the various types of interest rates. This is an important exercise because it highlights the difference between stated interest rates and effective interest rates—the actual rate paid on the loan funds.
Interest rates on loans are similar to opportunity costs. The loan amount is like an investment, and loan payments are comparable to an investment's cash flows. Therefore, we can use the compound amount formula to find the effective interest rate for a loan (set \(P = \$1) and \(t = 1\) year). This section focuses on different definitions of interest rates.
Actuarial Rate, Annual Percentage Rate (APR), and Effective Interest Rate
Loans charge interest rates, of which there are at least three types that are closely related to each other. These rates and their commonly used synonyms are listed below. They are: (1) actuarial rate, compound rate, true rate, or periodic rate; (2) Annual Percentage Rate (APR), annual rate, or nominal rate; and (3) effective interest rate or effective annual rate.
In financial transactions, interest may be computed and charged more than once a year. For example, interest on savings deposits is usually calculated on a daily basis, while many corporate bonds pay interest on a semiannual basis. The interest rate used for computations over periods of less than one year is called the actuarial interest rate.
The actuarial rate is defined as the interest rate per compounding period, or the interest rate per period of conversion. It is the actuarial rate used to charge interest on the principal sum during each successive conversion period.
For example, consider a 1% actuarial rate charged monthly on a $1,000 loan. In the first month, 1% of $1,000, or $10, is charged as interest. In the second month, interest is charged on $1,010, which equals $10.10, and so on.
The Annual Percentage Rate (APR) represents the true yearly cost of borrowing money (or the real return on investment), expressed as a percentage. It includes not only the nominal interest rate but also any fees or additional costs associated with the loan or credit.
Let \(r\) represent the APR, and let \(n\) stand for the number of times during the year that interest is calculated or charged. Thus, \(n\) equals the number of compounding periods per year. The ratio of \(r/n\) is the actuarial rate, compound rate, true rate, or periodic rate.
We find the APR from actuarial rates by expressing the actuarial rate on an annual basis. To convert the actuarial rate to an APR, we multiply the actuarial rate by \(n\). In the previous example, we multiply the actuarial rate of 1% per month by 12 to yield an APR of 12%. When the compounding period or conversion period is one year in length, the actuarial rate and the APR are equal.
Consider two savings institutions, both offering the same APR. The only difference is that institution A offers monthly compounding of interest, while institution B offers annual compounding. Which one should the saver prefer? Obviously, monthly compounding is preferred because the saver earns interest on the interest earned during the same year. With institution B, interest is earned only on the principal saved and on interest earned in previous years.
Effective interest rates are the actual interest charged, measured on an annual basis. When APRs have different numbers of compounding periods per year, the different actuarial rates should be converted to their effective interest rates for comparison. The effective rate is obtained by compounding the actuarial rate for a period of one year. As the number of compounding periods per year increases, the difference between the APR and the effective rate also increases.
Relationship between interest rates
The relationships between the actuarial rate, the APR, and the effective rate can be easily summarized. Let \(n\) be the number of compounding periods per year, let \(r\) be the APR, let \(r_a\) be the actuarial rate, and let \(r_e\) be the effective rate. The relationship between the effective rate \(r_e\), the APR \(r\), and the actuarial rate \(r_a\) can be expressed as follows:
\[ r = r_a\times n, \quad \quad r_a= \frac{r}{n}\]
\[r_e=\left(1+\frac{r}{n}\right)^{n}-1 \label{16.1}\]
Note that when \(n = 1\), the effective interest rate, the APR, and the actuarial rate are equal. However, when \(n\) is not 1, the rates are no longer equal. For example, suppose we wish to find the \(r_e\) assuming \(r\) were compounded quarterly. To solve this problem let \(n = 4\), and \(r=0.12\). Substituting 0.12 for \(r\) and 4 for \(n\) in Equation \ref{16.1},
\[r_e=\left(1+\frac{0.12}{4}\right)^{4}-1 \approx 0.1255 = 12.55\% \nonumber\]
Thus, the effective interest rate is about 12.55%.
If \(n\) is increased to 12, or monthly compounding periods, the effective rate is found as before. Let \(n= 12\), and \(r= 12\) in Equation \ref{16.1}. Then
\[ r_e=\left(1+\frac{0.12}{12}\right)^{12}-1 \approx 0.1268 = 12.68\%\nonumber\]
We find the effective interest rate to equal 12.68%.
Comparison Table: APR and EIR
| Feature | APR | Effective Interest Rate |
|---|---|---|
| Includes fees (e.g., loan origination, processing fees) | ✔️ Yes | ❌ No (unless added manually) |
| Includes compounding frequency (monthly, quarterly, daily, etc.) | ❌ No | ✔️ Yes |
| Best for comparing | Loan offers with fees | Investment or loan growth over time |
| Affected by compounding (how often interest is added) | ❌ No | ✔️ Yes |
Our national debt in 1992 was about $4 trillion. If the annual interest rate was 7% at that time, what was the daily interest on the national debt? Round your answer to the nearest dollar.
Solution
The interest rate per day \(\displaystyle = \frac{r}{n} = \frac{0.07}{365}\). Therefore, the daily interest \(\displaystyle = 4,000,000,000,000\times \frac{0.07}{365} \approx \$767,123,288.\) It is more than $767 million.
Estimating the APR for a Simple Interest Installment Loan
The annual percentage rate (APR) represents the interest rate on a loan, calculated based on the actual amount owed and the length of time it remains outstanding. For a loan with an add-on (simple) interest rate \(r\) and \(N\) payments, the APR can be approximated using the following formula:
\[ APR \approx \frac{2Nr}{N+1} \]
Consider a loan for the purchase of a computer priced at $2,500, to be repaid in monthly installments over three years at a simple interest rate of 15%. Approximate the APR to the nearest tenth of a percent.
Solution
\[ N = 3\times 12 =36. \nonumber\]
\[APR \approx \frac{2\times 36 \times 15}{36+1} = \frac{1080}{37}\approx 29.2\% \nonumber\]
Consider an autobike priced at $1,899, advertised with no down payment, a 5% annual simple interest rate, and 60 monthly payments. What is the APR, rounded to the nearest tenth of a percent?
- Answer
-
9.8%
Disguised Interest Rates and Effective Interest Rates
One of the challenges financial managers face when making borrowing decisions is understanding the actual cost of borrowing—or, in other words, determining the effective interest rate. Sometimes lenders offer loans that are designed to disguise the real cost of borrowing. These are known as disguised interest rate loans. Disguised interest rate loans have effective interest rates that are higher than the stated interest rate, achieved through methods other than simply increasing the loan’s interest rate.
For example, interest costs can be subtracted in the initial period, reducing the actual loan amount received by the borrower (a discount loan). Interest can also be charged as if the original loan balance were outstanding throughout the life of the loan (an add-on loan). Alternatively, the lender may charge a loan closing fee, reducing the actual loan balance received by the borrower. Additionally, the interest may compound more frequently than the loan payments occur. Each of these methods increases the effective interest rate above the stated rate. There are also several other types of loans that disguise the true interest rate.
Exercises
- Which would you prefer to earn on your savings? An APR of 12.5% or a 1% actuarial rate compounded monthly?
- Consider a loan of $80,000 at an APR of 13%. What is the loan payment that would retire the loan if repaid in monthly payments for 10 years? If repaid in monthly installments for nine years? Compare the percentage change in the term versus the percentage change in the loan payment.
- Assume a loan of $54,000 with a remaining term of 21 years. The existing loan requires monthly payments at an APR of 11.25%. For a 3% closing fee, the borrowers could refinance their loan at an APR of 10% for the same term. What is the effective interest rate on the new loan? What is the total interest paid on the two loans?
- A consumer obtains an installment loan of $12,000 from which $2,700 is deducted for interest costs. The loan is to be repaid over two years with monthly payments equal to $500 (=$12,000/24). Please determine the effective interest rate on this loan.