## Friday, June 1, 2012

### Into the Belly of the Whale: Curve Balls

In our last post, “Into the Belly of the Whale: Hedging and Credit Default Swaps”, we explored what makes a hedge a hedge and then looked at how the ever-so-mysterious Credit Default Swap is priced.
We now take this knowledge and continue exploring the JP Morgan loss of \$2 Billion by their illustrious trader nicknamed “The London Whale”.

What is a “Curve”?
According to Matt Levine and Lisa Pollack, the JPMorgan trade in question was a “curve trade”. While one might think this involves driving a car on mountainous roads, in the world of finance the term “curve” is used differently.
If you put yourself in the shoes of a borrower, you are faced with a choice of when you would like to pay back the money you borrowed. Do we want to issue 5-year debt, 10-year, or 30-year? Homeowners face this same question when they are considering their mortgage.
 Figure A
One of the important factors we consider when evaluating this decision is what the interest rate is going to be, because interest rates will be different depending on when the debt is due. We might be able to issue bonds with a rate of 4% if they are due in 5 years, 4.5% if due in 10, 5% if due in 20, and 5.5% if due in 30.
The length of time until the debt is due is called the “term” of the debt. Figure A plots these rates out by their term (I added a few extra for completeness). This picture, and the concept of different rates for different terms of debt, is called the “yield curve”.
 Figure B
The yield curve will vary over the course of time. Generically, there are three basic types of yield curve  - upward sloping, flat, and inverted. These are shown in Figure B.
For Credit Default Swaps, the curve concept is the same, but instead of interest rates the points on the graph represent the Credit Default Swap Spread (for example, the 102 basis points we calculated in our last post for a 2-year CDS).
Because the shape of a curve can change over the course of time, it is possible to buy and sell securities that will make or lose money should this occur. According to Matt Levine and Lisa Pollack, part of the JPMorgan hedge in question was a “curve flattener”. This means the transactions were set up to pay-off if the curve became flatter than what it was.

Let’s Pull Up Our Bootstraps
 Figure C
Before we think about how curves change, we need to look in particular at one of their properties that become important when thinking about transactions that use them.
Let’s look at a simple probability event like flipping a coin. If the coin is flipped twice, and if heads comes up then there is default, then we know that there is a 50% probability that by the end of period 1 there will be a default and a 75% probability at the end of period 2. This is shown graphically in Figure C.
 Figure D
You probably recall from our prior post’s example that the Credit Default Swap price calculation uses a probability of default for each period of time. Given the above information poses a problem for us if we were going to price a CDS with one annual payment. We know from the above that for the first payment the default rate is 50%.
What we are missing is the rate for the second period. Yes, we have a two-year rate of 75%, but that incorporates the first year as well as the second. We want to get the rate for the second year only. The bracket in Figure D shows the time period where the rate we need is missing.
 Figure E
Fortunately, since we have 2-year rate and a 1-year rate, we can use the difference between the two to come up with the missing rate, which is 25%. This process is known as bootstrapping. This process is used to establish implied default rates per period for Credit Default Swaps, and is also used extensively in interest rate products (we’ll leave that one for later!). Figure E shows the swap valuation with our Heads and Tails data to prove out the result.
 Figure F
Suppose we are faced with a term structure of CDS spreads as shown in Figure F (this curve is from the website onedigit.org). As long as we are willing to “lock down” the loss given default rate, we will be able to work our way through this curve to bootstrap the probabilities.
onedigit.org has been kind enough to provide Visual Basic code so we do not have to go through this process in a tedious, step by step manner.
 Figure G
Figure G shows the cumulative probability for the CDS spreads given the discount rate function used by onedigit.org and a loss given default rate of 60%. It also shows the per period probability.
Figure H shows our valuation for the 10 term swap. The rate of 35.2 basis points for the swap is the correct one in order to get the valuation of the contingent leg equal to the value of the fixed payment leg.
 Figure H

The World is Getting Flatter?
The period probabilities are important because these are the drivers of change in the price of the Credit Default Swap. If the curve is going to flatten, then one of several things are going to occur:
a)      short-term probabilities increase more than long-term ones,
b)      long-term probabilities decrease more than short-term ones, or
c)      some combination of a and b.
 Figure I
This is shown in Figure I.
In order to generate gains when the curve flattens, we need to buy the short-term part of the curve and sell the long-term. The amount we buy vs. what we sell will be different, because for a change in probabilities of default or the CDS rate the change will be different, since one security is much shorter in length than the other, so there is less to impact.
 Figure J
To take an example, say we buy a 2-year CDS and sell the 10-year CDS. To hedge for change in value, it is a ratio of about 4.5 to 1, so Figures J and K show the valuation at the time of our transaction.
 Figure K
We now move the probability of default up by 0.1% in every period (a parallel shift). Figures L and M show the valuation of our securities under that scenario. You will notice that our purchase of the short-term gave us a gain of a little over \$5,000 and our long-term position has a loss of a little more than \$5,000, so we do not make or lose much on a parallel shift.
This is by design, the transaction is supposed to make money when the curve flattens, not when the whole thing shifts up and down.
 Figure L
Now we will increase the probability of default by 1% for the first couple of periods only, and leave the other ones alone (not only did the curve flatten, it inverted). Figures N and O show the valuations. Here we made \$52,000 on our short-term security and lost \$12,000 on the long-term security, for a net gain of \$40,000. Not bad for a day’s work!
We will continue on into the belly with our next post.
 Figure M

Key Takeaways
The term structure of default rates and interest rates can create different impacts depending on the shape of the curve and how it changes. We can execute derivative transactions that take advantage of this.
 Figure N

Questions
·         In your opinion, was JPMorgan speculating or hedging?

 Figure O

1. Stephanie ConnollyJune 07, 2012 11:30 PM

Dave
I worked for you a few years ago as your analyst and stumbled upon this website. It is very impressive, I learned a lot reading through the blogs on here. Hope all is well.

1. Stephanie,

How funny, I just thought about you the other day, someone asked about IRR for an equity buyback, and I had to remember how you did that analysis!

Thanks for the compliments on the blog and reading. Hopefully you continue to find it useful.

Hope all is well with you as well - kindergartner this fall or am I off a year?

Connect with me on LinkedIn if you are there.