# Basis risk, spreads and discounting

There is a lot of information to be read on basis risk, spreads and discounting. After reading some information, I have an idea about what basis risk is about and why this type of risks should be considered, but I don't know very well how one accounts for this risk in practice, say in discounting cash flows in a swap. How is basis risk incorporated here? Is this simply an additional term that is added to the floating rate? Can someone provide me with a clear example?

Secondly, after the financial crisis, one has moved from LIBOR to OIS for discounting purposes. How is the OIS spread (which I assume is the difference between LIBOR and OIS) linked to basis?

Perhaps I am mixing up these things, but I hope someone can explain with a clear example.

Thanks!

First lets clear one thing up, 'basis' and 'spreads' are the same thing. Often this is called the 'basis spread'. This represents the difference between curves at different points in time.

For example if from 1st Jan 2019: 6M (from 6M LIBOR cv) is 1.20% and 6M (from 3M LIBOR cv) is 1.10% then the 6M/3M basis on 1st Jan 2019 is 10bps. You have equivalent basis numbers between any curves you choose, 1M/3M or OIS/3M or OIS/6M etc. The specific basis you are interested in referencing discounting is the LIBOR/OIS basis.

Now lets give you a practical example of what happens in a hypothetical scenario:

Say 6M Libor are forecast to be 2% every day for the next five years, i.e. you have a perfectly flat curve. It is not hard to see that the 5Y IRS fixed rate is 2% in that case, and that every cash flow (assuming equal payment frequencies on the fixed and floating legs) would be zero for an IRS struck at 2% (this is irrespective of whatever discount method you choose).

But instead suppose that your fixed rate is struck at 2.10% in 10mm USD. Now if you are the receiver you are ITM by 5,000 USD every 6M cashflow. The total value of the swap is roughly 5K x 10periods x some discounting = say 45K.

Your profit has been derived from the discounting methodology applied, if you have higher discount factors the swap is worth more to you. And that means you need a lower discount (OIS) curve, so if the basis widens this is favourable to you (6M LIBOR remains the same so your floating and fixed cashflows are stable but your valuations of them changes). So you have exposure to basis moves, i.e. you have basis risk.

In this particular case you would have roughly 11 USD pv01 exposure to the 5Y 6M/OIS basis, so if the 6M/OIS basis widened by 1bp you would make 11 USD on mark-to-market. Note this was calculated with the approximation: $$Discounting Basis Risk = \frac{PV}{10,000} * \frac{Tenor}{2}$$

• Thanks for the clear example, this clears my doubt about the basis concept. I assume it is very likely to say OIS spread for LIBOR-OIS? But I guess one generally means something else when referring to LIBOR spread? Btw, is it common to construct discount curves using LIBOR spreads and OIS spreads? Any documentation on this is always welcome. – user39039 Feb 20 '18 at 14:29
• Academia teaches you specific formal terms. Finance is much more flexible. People use many terms, combine terms or simply use the wrong terms and people are expected to know what is meant in context. OIS spread or OIS basis is usually relative to LIBOR (6M or 3M), yes. Libor spread might mean against a bonds curve but thats a bit fishy. Best documentation by far is book Darbyshire: Pricing and Trading Interest Rate Derivs (check out the Table of Contents on Amazon). If the answer helped please accept it. – Attack68 Feb 20 '18 at 15:25
• Thanks, the contents of this book look promising. I was just wondering what you exactly mean with "And that means you need a lower discount (OIS curve)." Furthermore, if one would discount the cash flows using LIBOR rates as well, then the risk is rather movements in LIBOR from one day to another, so it is then just common to speak about IR risk or? – user39039 Feb 21 '18 at 13:28
• Higher curves mean lower discount factors and vice versa, so for positively valued swap (i.e. and asset) you desire low curves and high DFs. Any swap has two components of risk; forecasting basis risk (the floating leg payment dependencies) and discounting basis risk (how you discount cashflows), the sum of the two is the total market delta. I.e if rates go up and all curves go up (forecasting and discounting) then both risks are impacted. If you forecast and discount with the same (LIBOR) curve it can still be useful to consider two copies of the curve for different purposes. – Attack68 Feb 21 '18 at 15:10

Hope that give you an overview and helps you.

In the financial crisis it can be seen on the difference between IBOR spot rates and OIS rates, which we will further refer to as the IBOR-OIS spread. As IBOR market quotes now involve the average credit and liquidity risk of the interbank money market, OIS rates have become the new proxy for the risk-free rate.

The new risk-free curve for discounting is built from OIS rates. We can still decompose the swap into two legs. However, the floating payments estimated by implied forward rates change under OIS discounting.

• Thanks for the answer. Can you perhaps explain me how the floating payment implied by the LIBOR forward rates change under OIS discounting? Is the idea to take the LIBOR and simply add some kind of OIS spread as basis? I'm just looking for a simple, intuitive way on how the basis risk is captured in the discount rate – user39039 Feb 20 '18 at 9:10