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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

It is quite strange to me because the rf(T) should be rf(T)/(1/T) following many sources online, e.g. WallStreetMojo, they should divide the rf(T) for number of periods per year

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

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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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