<p>Example 5.2 is the risk-free return for the given horizon (half-year here, meaning it is NOT an annualized - it would be 5.42 following the logic of WallStreetMojo that uses annualized rates). Therefore, as written, <span class="math-container">$T = 1/2$</span> and <span class="math-container">$1/T = 2$</span>. This turns the half-yearly return into an annualized return (2.71% -&gt; 5.49%).</p>
<p>The thing to consider is that almost all interest rate quotes (and many other metrics like implied volatilities etc.) are quoted annualized by convention. The next section defines the annual percentage rates, which is what you would usually look at:
<span class="math-container">$$ APR = n \times r_f(T)$$</span>
and the per period rate (used in the example you highlighted) is given by
<span class="math-container">$$r_f(T) = T \times APR$$</span>.</p>
<p>It also uses example 5.2 again to show that it is 5.42% (as I wrote above).</p>

CC BY-SA 4.0
2023-04-24T18:56:30.310Z
2023-04-24T18:56:30.310Z
AKdemy
<p>When reading the seminal textbook of Bodie, Kane and Marcus, 2014, chapter 5, page 123 they have the equation for Effective Annual Rate is
<a href="https://i.sstatic.net/n1oJM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/n1oJM.png" alt="enter image description here" /></a></p>
<p>It is quite strange to me because the rf(T) should be rf(T)/(1/T) following many sources online, <a href="https://www.wallstreetmojo.com/effective-annual-rate/" rel="nofollow noreferrer">e.g. WallStreetMojo</a>, they should divide the rf(T) for number of periods per year</p>
<p><a href="https://i.sstatic.net/nnqiK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nnqiK.png" alt="enter image description here" /></a></p>
<p>I am curious they are conflict in that point or I may miss something important</p>

<financial-economics>
Concern about Effective Annual Rate in textbook and reality
Phil Nguyen
# Concern about Effective Annual Rate in textbook and reality

(Asked by: username_0 on 2023-04-24)

When reading the seminal textbook of Bodie, Kane and Marcus, 2014, chapter 5, page 123 they have the equation for Effective Annual Rate is
[![enter image description here](https://i.sstatic.net/n1oJM.png)](https://i.sstatic.net/n1oJM.png)

It is quite strange to me because the rf(T) should be rf(T)/(1/T) following many sources online, [e.g. WallStreetMojo](https://www.wallstreetmojo.com/effective-annual-rate/), they should divide the rf(T) for number of periods per year

[![enter image description here](https://i.sstatic.net/nnqiK.png)](https://i.sstatic.net/nnqiK.png)

I am curious they are conflict in that point or I may miss something important

**Tags:** financial-economics

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## 1 answers

### Answer by username_1 on 2023-04-24. Score: 2 | Accepted answer

Example 5.2 is the risk-free return for the given horizon (half-year here, meaning it is NOT an annualized - it would be 5.42 following the logic of WallStreetMojo that uses annualized rates). Therefore, as written, $T = 1/2$ and $1/T = 2$. This turns the half-yearly return into an annualized return (2.71% -> 5.49%).

The thing to consider is that almost all interest rate quotes (and many other metrics like implied volatilities etc.) are quoted annualized by convention. The next section defines the annual percentage rates, which is what you would usually look at:
$$ APR = n \times r\_f(T)$$
and the per period rate (used in the example you highlighted) is given by
$$r\_f(T) = T \times APR$$.

It also uses example 5.2 again to show that it is 5.42% (as I wrote above).


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