Bootstrapping OIS Discount Factors from Swap Rates

# How to find OIS discounting factors from OIS swap rates. Please explain with example


# How to find OIS discounting factors from OIS swap rates. Please explain with example












Suppose I have the following OIS Swap rates:

1 year OIS Swap: 0.36% 2 year OIS Swap: 0.37% 3 year OIS Swap: 0.38% 4 year OIS Swap: 0.40%

From these, how do I get the OIS Discounting factors for these respective years?

Could someone please explain with a proper formula that can be applied for this?

## Answer by Jan Stuller (score -1, accepted)

https://quant.stackexchange.com/a/54742

Assuming the frequency of the fixed OIS swap coupons is annual, then the first year swap rate can be converted directly to a discount rate as follows:

$$DF(1y) = \frac{1}{1.0036}\approx0.9964$$

For the second year, the bootstrapping can be done as follows (I assume that swap notional is 100 for illustrative purposes but it can be set to any number without loss of generality):

$$100 = DF(1y)*100*0.0037 + DF(2y)*100*0.0037 + DF(2y)*100 $$

Solving for $DF(2y)$ gives:

$$DF(2y) =\frac{1 - 0.9964* 0.0037}{1.0037} \approx 0.9926$$

For the third year:

$$100=DF(1y)*100*0.0038 + DF(2y)*100*0.0038 + DF(3y)*100*1.0038$$

Solving for $DF(3y)$ gives:

$$DF(3y)=\frac{1-(0.9964*0.0038)-(0.9926*0.0038)}{1.0038}\approx0.9887$$

For the fourth year:

$$100=DF(1y)*100*0.004 + DF(2y)*100*0.004 + DF(3y)*100*0.004+DF(4y)*100*1.004$$

Solving for $DF(4y)$ gives:

$$DF(4y)=\frac{1-(0.9964*0.004)-(0.9926*0.004)-(0.9887*0.004)}{1.004}\approx0.98415$$

Using the OIS swap rates as a proxy to compute the discount rates without bootstrapping: I previously argued that as a proxy, we can just take the OIS swap rates and use them as a proxy for the discount rates without having to bootstrap the curve (for which I got downvoted). I show below that indeed, it is a reasonable proxy:

$$DF(2y)\approx\frac{1}{1.0037^2}\approx0.9926$$

$$DF(3y)\approx\frac{1}{1.0038^3}\approx0.9887$$

$$DF(4y)\approx\frac{1}{1.004^4}\approx0.98416$$

I deliberately used 5 decimal places for the fourth year to show that the approximation gets worse further down the curve, but being off by 0.1 bps for the 4th-year discount factor is still a pretty good approximation. If you ever need a quick "back-of-the-envelop" calculation of discount factors based on the OIS swap curve, just using the swap rates directly works fine.

## Answer by ZelliZello (score 1)

https://quant.stackexchange.com/a/59231

It depends on the discounting index of your OIS swaps : we recently switch from the standard OIS discounting for standard swaps to SOFR discounting. As the basis between OIS and SOFR is small, the effective impact is minimal.

The methodology is the following :

DF(1y) = 1/(1+0.0036) = 0.996413 DF(2y) = DF(1y)*DF(1y1y) DF(3y) = DF(2y)*DF(2y1y) DF(4y) = DF(3y)*DF(3y1y)

and to compute those forwards DFs, if you assume OIS discounting, you would get, by rewriting the definition of the swap fixed rate :

(2Y_fixrate)*(DF(1Y)+DF(2Y)) = (1Y_fixrate)*DF(1Y) + (1Y1Y_fixrate)*DF(2Y) ... DF(1Y1Y) = 0.996214