Discount factor

Question

Suppose we have :

$r$ - zero coupon rate, constant over time,

$n$ - a number of years (an integer),

$\theta$ - a fraction of a year $(\theta < 1)$ , calculated with the relevant day count convention.

Which discount factor is the correct one ?

(I'm inclined to think it's $\beta_{2}$ because over a period of time of less than a year, it's a simple interest that is computed).

$\beta_{1} = \frac{1}{(1+r)^{(n+\theta)}}$

$\beta_{2} = \frac{1}{(1+r)^n}\cdot \frac{1}{(1 + r\theta)}$

Answer 1

You have

$\beta_1=\frac{1}{(1+r)^n}\frac{1}{(1+r)^\theta}$

and

$\beta_2=\frac{1}{(1+r)^n}\frac{1}{(1+\theta r)}$.

Both are equal when $\theta=1$. If you consider simple interest then go for $\beta_2$. If you would like compound interest within fraction of year then pick $\beta_1$.

However, because $\theta$ is between 0 and 1 then values $\beta$'s won't be that different.