<p>I am having issues finding a solid example online that explains how the concepts surrounding the word <em>interest</em> relate to each other. I am after an example that presents a concise mathematical formula that I can ultimately use to write a computer application.</p>

<p>Let us start with nominal versus real interest rates. My understanding is that the simple concept of interest on a loan is called <strong>nominal interest</strong>. If we consider an inflation rate, then their difference is called <strong>real interest</strong>. Mathematically:
<span class="math-container">$$
r_r = r_n - r_i,
$$</span>
where <span class="math-container">$r_r$</span> denotes the real interest rate, <span class="math-container">$r_i$</span> denotes the inflation rate, and <span class="math-container">$r_n$</span> denotes the nominal interest rate.</p>

<p>The so-called "annualized percentage rate (APR)" is an interest rate that accounts for any additional costs the loan may incur in. According to <a href="https://www.investopedia.com/terms/a/apr.asp" rel="nofollow noreferrer">[1]</a> the following is a formula for APR:
<span class="math-container">$$
\begin{aligned} &amp;\text{APR} = \left ( \left ( \frac{ \frac{ \text{Fees} + \text{Interest} }{ \text {Principal} } }{ n } \right ) \times 365 \right ) \times 100 \\ &amp;\textbf{where:} \\ &amp;\text{Interest} = \text{Total interest paid over life of the loan} \\ &amp;\text{Principal} = \text{Loan amount} \\ &amp;n = \text{Number of days in loan term} \\ \end{aligned}
$$</span>
When algebraically manipulating the latter expression, it seems like an equivalent expression can be derived as follows: Let <span class="math-container">$f$</span> denote the sum of all fees incurred by the loan, and let <span class="math-container">$p$</span> denote the amount borrowed (i.e. principal). The APR for a loan period of <span class="math-container">$n$</span> days is then defined as:
<span class="math-container">$$
r_{APR} = 100\times\frac{365}{n}\times\left(\frac{f + p}{p}\right)\times r_n,
$$</span>
where <span class="math-container">$r_n$</span> denotes the nominal rate of interest.</p>

<h1>Examples:</h1>

<p>Let <span class="math-container">$r_i = 0.04$</span> and <span class="math-container">$r_n = 0.1$</span>. Then <span class="math-container">$r_r = 0.06$</span>, or 6%. Then, let <span class="math-container">$p = 200,000$</span> USD and <span class="math-container">$f = 5,000$</span> USD for a <span class="math-container">$n = 365$</span> days loan period, so that <span class="math-container">$r_{APR} = 0.615$</span>, with a total amount to payback of 212,300 USD, meaning the cost of the loan is 12,300 USD. Note, in this example, I’ve used the real interest rate rather than the nominal interest rate to compute the APR.</p>

<h1>Core questions:</h1>

<ol>
<li>Is the definition of real interest rate correct?</li>
<li>Is the second definition for APR correct?</li>
<li>What if <span class="math-container">$n = 365$</span>, thus making the coefficient <span class="math-container">$365/n = 1$</span>? Wouldn't this make this coefficient worthy of the name <strong>annualization coefficient</strong>?</li>
<li>What if instead of using the nominal interest rate to compute the APR, we use the real interest rate?</li>
<li>Isn't APR a misnomer? The word "annualized" may be correct... but "percentage rate" seems like a very general noun phrase. Specifically, the word "percentage" only applies if we take it as a percentage, and the word "rate" is valid, but anywhere in APR the word interest appears.  Wouldn't a better name for this interest rate be "annualized cost-of-borrowing (real) interest rate", in where the word "real" could be used in the case in which the formula used the real interest rate, rather than purely the nominal interest rate?
​   </li>
</ol>

CC BY-SA 4.0
2019-10-20T16:38:05.570Z
Your first equation is only an approximation. See https://en.wikipedia.org/wiki/Real_interest_rate#Fisher_equation
fantastic peace and prosperity
CC BY-SA 4.0
2019-10-20T16:41:12.580Z
As for your other question(s) the def of APR seems to vary by legislation/country: https://en.wikipedia.org/wiki/Annual_percentage_rate#Multiple_definitions_of_effective_APR
fantastic peace and prosperity
CC BY-SA 4.0
2019-10-21T10:55:38.640Z
Along what @Fizz said, different conventions for annualizing an interest rate exist even between different financial markets in the same country. Commercial paper and bill markets, for example, just divide the yield to maturity by 360 to determine the daily accrued interest (no compounding, not even the number of days in a year), and likewise multiply the interest over a period by 360/[number of days] to annualize it for a period. What's important is to be clear about what you're doing, and to not compare differently-expressed rates (convert to a common expression or use cash flows instead).
dismalscience
<interest-rate>
Understanding annualized percentage rate of interest, real interest and nominal interest rate
Eduardo J. Sanchez
# Understanding annualized percentage rate of interest, real interest and nominal interest rate

(Asked by: username_0 on 2019-10-15)

I am having issues finding a solid example online that explains how the concepts surrounding the word *interest* relate to each other. I am after an example that presents a concise mathematical formula that I can ultimately use to write a computer application.

Let us start with nominal versus real interest rates. My understanding is that the simple concept of interest on a loan is called **nominal interest**. If we consider an inflation rate, then their difference is called **real interest**. Mathematically:
$$
r\_r = r\_n - r\_i,
$$
where $r\_r$ denotes the real interest rate, $r\_i$ denotes the inflation rate, and $r\_n$ denotes the nominal interest rate.

The so-called "annualized percentage rate (APR)" is an interest rate that accounts for any additional costs the loan may incur in. According to [[1]](https://www.investopedia.com/terms/a/apr.asp) the following is a formula for APR:
$$
\begin{aligned} &\text{APR} = \left ( \left ( \frac{ \frac{ \text{Fees} + \text{Interest} }{ \text {Principal} } }{ n } \right ) \times 365 \right ) \times 100 \\ &\textbf{where:} \\ &\text{Interest} = \text{Total interest paid over life of the loan} \\ &\text{Principal} = \text{Loan amount} \\ &n = \text{Number of days in loan term} \\ \end{aligned}
$$
When algebraically manipulating the latter expression, it seems like an equivalent expression can be derived as follows: Let $f$ denote the sum of all fees incurred by the loan, and let $p$ denote the amount borrowed (i.e. principal). The APR for a loan period of $n$ days is then defined as:
$$
r\_{APR} = 100\times\frac{365}{n}\times\left(\frac{f + p}{p}\right)\times r\_n,
$$
where $r\_n$ denotes the nominal rate of interest.

# Examples:

Let $r\_i = 0.04$ and $r\_n = 0.1$. Then $r\_r = 0.06$, or 6%. Then, let $p = 200,000$ USD and $f = 5,000$ USD for a $n = 365$ days loan period, so that $r\_{APR} = 0.615$, with a total amount to payback of 212,300 USD, meaning the cost of the loan is 12,300 USD. Note, in this example, I’ve used the real interest rate rather than the nominal interest rate to compute the APR.

# Core questions:

1. Is the definition of real interest rate correct?
2. Is the second definition for APR correct?
3. What if $n = 365$, thus making the coefficient $365/n = 1$? Wouldn't this make this coefficient worthy of the name **annualization coefficient**?
4. What if instead of using the nominal interest rate to compute the APR, we use the real interest rate?
5. Isn't APR a misnomer? The word "annualized" may be correct... but "percentage rate" seems like a very general noun phrase. Specifically, the word "percentage" only applies if we take it as a percentage, and the word "rate" is valid, but anywhere in APR the word interest appears. Wouldn't a better name for this interest rate be "annualized cost-of-borrowing (real) interest rate", in where the word "real" could be used in the case in which the formula used the real interest rate, rather than purely the nominal interest rate?
   ​

**Tags:** interest-rate

> **Comments:**
> **username_1**: Your first equation is only an approximation. See https://en.wikipedia.org/wiki/Real_interest_rate#Fisher_equation  
> **username_1**: As for your other question(s) the def of APR seems to vary by legislation/country: https://en.wikipedia.org/wiki/Annual_percentage_rate#Multiple_definitions_of_effective_APR  
> **username_2**: Along what @Fizz said, different conventions for annualizing an interest rate exist even between different financial markets in the same country. Commercial paper and bill markets, for example, just divide the yield to maturity by 360 to determine the daily accrued interest (no compounding, not even the number of days in a year), and likewise multiply the interest over a period by 360/[number of days] to annualize it for a period. What's important is to be clear about what you're doing, and to not compare differently-expressed rates (convert to a common expression or use cash flows instead).  

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## 0 answers

