# OIS Discount Factor Bootstrapping - Do we assume simple interest?

When I am reading papers (ie here and here) on bootstrapping discount curves they refer to obtaining discount factors from rates for swaps maturing less than a year with:

$$D(t, T_i) = \frac{1}{1+s_i(T_i-t)}$$

(where $s_i$ is the swap rate, $T_i-t$ is the time to maturity in years and $D$ is the discount factor)

In short my question is why not this?:

$$D(t, T_i) = \frac{1}{(1+s_i)^{(T_i-t)}}$$

Is the difference because we are assuming there is no compounding? Is this a correct assumption? It's quite a key question because the denominator in the bootstrapping formula for tenors > 1 year (because the ois swaps pay annually) depends on this:

$$D(0,T_i) = \frac{1-s_i\sum_{j=1}^{i-1}(T_j-T_{j-1})D(0,T_j)}{1+s_i(T_i-T_{i-1})}$$

• Yes, this is the convention. For less than 1 year, use simple interest. That is how everyone quotes and calculates rates. Unless otherwise specified. – noob2 Oct 11 '17 at 15:49
• @noob2 So if we are converting these discount factors back into rates, do we use simple interest for all tenors? Even those greater than a year? ie. r = (1/s - 1)/T – JSharm Oct 18 '17 at 16:43

An OIS interest rate swap rate with annual-annual freq is determined under one year by: $$1 + d_i s_i = \prod_{j=1}^{n(i)}(1+ d_j r_j) \; , \quad \text{where} \quad d_i = \sum_{j=0}^{n(i)} d_j \;.$$ Each $r_j$ is a forecast overnight OIS rate which as you can see are compounded in the floating side. Therefore a discount factor in the future, for maturity $m_i<1Y$, which would be represented by: $$\text{discount factor at }m_i = \frac{1}{\prod_{j=1}^{n(i)}(1+d_jr_j)}=\frac{1}{1+d_is_i}\;.$$