When Correlations Go To One

Frequent readers know that I am always reading something.

Recently a co-worker lent me Nate Silver's The Signal and the Noise. In the first chapter the author discussed the financial crisis and how mortgage pools change based on the correlation of their risks.

In prior posts (see Mei's Monte Carlo Adventure for example) we have modeled investment portfolios assuming that the correlation between securities was constant. We have always cautioned, however, that this assumption may not be correct.

In his book, Mr. Silver provided some data on how default rates in mortgage pools change under both an uncorrelated assumption and a perfect correlation assumption, and I thought it would be fun to replicate this and to take a deeper dive into what happens when correlations change.

The Uncorrelated Pool

Say we have bought 5 mortgages, and the risk of default on these is 5%. If this rate is too risky for some investors, we can create an investment that they can invest in by using them to create a different type of security.

We place the 5 mortgages together, creating a "pool". We then sell 5 securities to the market, A through E, whose payout will be based on order of default. Securities A through E are called "tranches" of the pool.

If all mortgages pay out, all 5 of our created securities pay investors. If one defaults,then investors in A through D get paid but E does not. If two default, investors in A through C get paid but D and E do not, and so on.

Figure A

Mortgage Pools

Five mortgages are pooled and five new securities are created. Payout for each security is based on the number of defaults.

Figure A depicts this structure.

One way to think about this is to imagine we have large soup pot on the stove. Each of the payments for the five mortgages are the ingredients that go into the soup.

After we stir a little bit, we ladle out the soup, only we do so in a certain order. Person A always gets the first ladle full, Person B the second, and so on. If one of the mortgages defaults, so we only have 4 ladles full of ingredients going into the soup, then there will not be enough for person E when they show up holding out their bowl.

Figure B

Uncorrelated Default Rates

First column is occurrence of the specific event - answering the question 'what percent of the time was there exactly x defaults?'. The second column is cumulative occurrence, answering the question 'what percent of the time was there at least x defaults?'

If we assume that the mortgage defaults are uncorrelated - meaning that if one defaults it in no way indicates that any of the other four default, then the probabilities of default occurrence for each of our newly created securities will be close to that of the results in Figure B, which is R output for a simulation of 10 million runs. Sometimes we will hear the phrase 'the variables are independent', which is stat-speak for uncorrelated.

These results match (with rounding) those produced in Mr. Silver's book.

Back to our mortgage situation, we had the case where an investor would not be able to purchase the 5% default risk mortgage securities because of their risk.

By creating our pool structure, we have created mortgage securities where three of them (A, B and C) have less than 0.12% probability of default, a much less risky proposition for these investors.

Of course, these results only hold if the securities are in fact uncorrelated.

The Correlated Pool

In the prior example the mortgages going into the pot were uncorrelated, meaning that the behavior of one does not impact the other.

In our simulation every now and then a mortgage would default and person E did not get paid. In a few instances 2 mortgages defaulted at the same time, but the other ones did not. In well over 99% of the simulation cases no more than 2 defaulted at any one time, so persons A through C would almost always get paid.

However, if the mortgages are correlated, meaning they move more "together" rather than "independently", this dynamic changes.

At the extreme, if the correlation is 1 (the highest a correlation can go), then when one defaults all five of them default. This means either there is soup for everyone or soup for no one - nothing in between.

In this case, the benefit of carving out different tranches of the pool is of no benefit whatsoever. The investor may as well have invested in one mortgage with a 5% chance of default, because that is the same probability as their perfectly correlated tranche security offers.

Figure C

Bond Prices By Tranche

Bond prices are calculated based on (loss given default * probability of default)/risk-free rate

This brings us to an interesting point. By creating a structure based on one assumption, should that assumption change it creates both winners and losers. Figure C shows theoretical bond prices based on the probability of default and 100% loss under that scenario.

Under the uncorrelated scenario, tranche E is worth about 73.69 cents on the dollar, since it is the tranche that gets hit with the vast majority of defaults.

However, under the perfect correlation assumption it is worth a little over 90 cents on the dollar. All the other tranches experience decreases in value from the uncorrelated to perfectly correlated scenarios.

In total, however, the entire pool of bonds is worth the same amount. This has to be so, as we have not changed the inherent risk of the mortgages (i.e. 5% default probability) in either case, only the correlation.

The Spaces In Between

Between the perfectly correlated and the perfectly uncorrelated, we have the partially correlated.

In Cholesky to the Rescue, we discussed how to create partial correlation between different simulated events according to a target correlation matrix.

But in that situation, we were using the normal distribution. In the mortgage default case, we are using the uniform distribution, one that has a range between 0% and 100%. There is no normally distributed mean or standard deviation to use like there is in the former case.

How do we go about creating a simulation that creates correlation between variables while staying true to the original variables' uniform distribution parameters?

For one answer to this, we can turn to the work done by Enrico Schuman in his post Generating Correlated Uniform Variates.

In this approach, we perform 4 steps.

::Create our correlation matrix

::Convert the matrix to Spearman correlation

::Simulate variables using the Cholesky decomposition of this matrix using normal distribution

::Convert the normal variables to the uniform distribution by using their p-values (probability of occurrence)

Under this method, we can use the well-developed normal distribution processes and simply convert the results to the uniform distribution in a manner that maintains the correlation.

Figure D

Default Probability at Different Correlation Levels

As correlation trends towards 1 default probabilities converge to .05

Figure D shows the default rates for each tranche of our pool resulting from one thousand simulations performed at each level of correlation (to the second decimal point, or for each .01). During each simulation the variables were run 100 times, so the total for each correlation level is 10,000.

As the correlation approaches 1, each of the security's default rates converge to 5%. The change in the price of these securities shown in Figure C, will change as the lines in Figure D change (opposite the sign of the change, so increases in Figure D mean decreases in bond prices).

The impact to each tranche varies. Tranche D's probability of default almost immediately starts to increase (thus it's value is decreasing) as things become more correlated, while Tranche A's holds out almost to the very end, but then increases rapidly.

Tranche D also has the interesting dynamic that it's probability of default increases and then decreases - the only security to do so. All the others only increase or only decrease.

What You Can Do

Correlation is a condition often stipulated for a model but not always completely thought out. Yet its' impacts in certain situations can be quite significant. Some questions to ask when analyzing our models or the results from them are:

What happens if our correlation assumption changes? - using Figure C as our example, asking this question allows us to consider the fact that our correlation assumption will drive a 5% change in our investments value.

Figure E

Average and 100-Day Rolling Correlation

Average correlation for entire time period is shown by the colored dashed line, and the rolling 100 day correlation by the solid lines

How strong is our correlation assumption? - For whatever period of time we calculate a correlation, we are inherently assuming that it will apply to the time period going forward. This is almost never the case. Figure E shows the correlation between Short and Long Term Fixed Income funds and US and Global Equity Funds (original data from Should You Rebalance Your Investment Portfolio?) from 1996 to 2012. The entire period correlation is almost never equal to the 100 day correlations, and the 100 day correlations can change dramatically in relatively short periods of time. In addition, there are distinct correlation differences in different time periods, such as equity correlations being slightly higher than .5 (and volatile) for much of the 90's compared to being close to 1 during the past few years (and not so volatile).

Insist on Scenario Analysis - one way to achieve the objectives in our first two "What Can You Do" items, we can insist on reviewing results of scenarios where the correlation conditions have been altered. For example, looking at the Equity correlations in Figure E, some possible scenarios are "Maintain recent high correlation", "Regress to the average", "Intermittent periods between high and low", "Return to the 90's", and "Go to Zero". Comparing the results of these scenarios will help us to understand the risk potential, though not necessarily the probability of each.

Learn From the Traders - a book I read once (though the name of it escapes me) made the claim that in security trading "every trader's position will eventually get wiped out", implying that trading is fundamentally a race to cash out before that eventuality occurs. Long Term Capital Management did great...until it didn't. Mortgage Bond traders and banks did great...until they didn't. So it will go with correlation, at some point the uncorrelated will go to 1, or the strongly correlated will go to 0. We need to make sure we are clear what we will do when that occurs.

Key Takeaways

The assumptions that go into a model are essential to the results it generates. Investigating what can occur if a) the assumptions change, or b) are simply incorrect, is an essential component of building an actionable analytical framework.

You May Also Like

Simulations Using Correlation

Mei's Monte Carlo Adventure

Cholesky to the Rescue

Uniform Distribution

Generating Correlated Uniform Variates

Questions

::What assumptions have you challenged that significantly impacted the results of a model's output?

::What is the best way to proceed if correlation is uncertain?

::Do you agree that eventually "all correlations will go to 1" or "all correlations will go to 0"? Why or why not?

Add to the discussion with your thoughts, comments, questions and feedback! Please share Treasury Café with others. Thank you!

Should You Rebalance Your Investment Portfolio?

Investors are often advised to rebalance their portfolios periodically in order to maintain their target asset allocations.

Translated, this means the following:

·         Suppose we want to have 20% of our investments in global equity, 40% in domestic equity, and 40% in fixed income,

·         A year of time goes by,

·         Each of our investment classes has changed in value. Some went up, others went down.

·         Because of this, our last investment statement shows our current portfolio is 25% global equity, 42% domestic equity and 33% fixed income.

·         We sell 5% of our global equity holdings, sell 2% of our domestic equity holdings, and use those proceeds to buy more fixed income assets, so that in the end we are back to a 20%/40%/40% mix of assets.

Undertaking these transactions to get back to our target levels is called rebalancing.

But what happens if we don’t? How much performance difference does the act of rebalancing create versus buying and holding? If we are making periodic contributions to the portfolio, such as funding a pension plan or payroll deductions into a 401k, won’t the fact that we are investing periodically at different prices take care of this and eliminate the need to rebalance?

Monte-Carlo simulation can give us a quantitative perspective to these questions.

 

Portfolio Math…Briefly

Figure A

Portfolio Returns: Mean and Variance

Figure A shows the equations we will need to use to calculate our portfolio data.

Equation 1 calculates the return to our portfolio. For each of the securities in our portfolio (n securities in total), we multiply the weight of that security by the return for that security and simply add them all together.

Either Equation 2 or 3 calculates the variance of our portfolio. This equation is a little more complicated because the variation between securities can be correlated. If security a goes up every time security b goes up, then our variance will be magnified. If security a goes down every time security b goes up, then our variance will be a lot smaller.

Variance and its companion metric – standard deviation (which is just the square root of variance) - is the measure of our portfolio risk. We have calculated this measure in prior posts (see "Your Regression Results Step by Step" ), but stated in words variance is a measure of how close or far each of the returns are from the average.

For example, look at the following 2 data sets:

A          {4,5,6}

B          {0,5,10}

Each data set has an average of 5. A’s first element is 4, which is 1 away from 5. B’s first element is 0, which is 5 away from 5. Thus, B will have a much higher variance and standard deviation because its data elements are “farther away” from the average than A’s.

 

The Efficient Frontier

A portfolio can be constructed in an almost infinite number of combinations, however very few are “best in their class”.

Figure B

Portfolio Efficient Frontier

Figure B shows a graph with Return on the y-axis and Risk on the x-axis. Each of the three letters A, B, and C represents three different portfolio combinations. Starting from A, we can follow the line to B. Since this line moves straight up, Portfolio B is superior to A because it has higher returns for the same level of risk.

Starting from A again, and this time moving to C, we see that Portfolio C is also superior to A because it generates the same returns for lower levels of risk.

The dashed orange line represents those portfolios where it is not possible to increase returns without also incurring greater risk, and vice versa. In other words, there is no B or C that beats A. No higher returns for the same risk, and no same returns for lower levels of risk.

This line is called the “Efficient Frontier”, and as investors it is where we want to be.

 

Setting-Up the Simulation: Asset Specific Information

In order to perform our portfolio Monte Carlo analysis, we need to define the asset classes that will comprise our investment portfolio. For this post we will use the following:

·         Short-term fixed income,

·         Long-term fixed income,

·         Domestic (US) equity and

·         Global equity.

In addition, to perform the simulation we need to provide the following inputs:

·         Expected returns for each asset class,

·         Expected standard deviation for each asset class, and

·         Correlations for all the assets classes involved

Finally, we need to establish the methods around which these values will vary during each simulation run.

The inputs above can come from a number of sources. For example, we may have a proprietary estimation methodology or use various forecasts provided by those we regard as experts. Some will use historical results under the assumption that they are the best proxy for future values (though any investment firm will tell you that past results do not predict future performance!).

Figure C

Historical Asset Class Performance

Mean and Standard Deviation Annual Basis, Min and Max Monthly Basis

Otherwise, we can simply use our own intuition and judgment, perhaps informed by a number of the other methods.

In order to calculate the historical values of these asset classes I used 4 index funds from Vanguard covering the period from January, 1996 to January, 2013. These funds’ return performance is summarized in Figure C, which was calculated using the open-source statistical software R.

Figure D

Asset Class Data

In addition, estimates of expected future asset portfolio performance were reviewed such as those provided by JP Morgan.

Based on this activity, Figure D shows the expected annual rates of return and standard deviations that we will use in the simulation, and Figure E shows the R output of the correlation matrix between these asset classes. For purposes of this simulation we will assume that returns will vary according to the normal distribution.

Figure E

Asset Correlation Matrix

st = Short-Term Fixed Income

lt = Long-Term Fixed Income

de = Domestic Equity

gl = Global Equity

 

Setting-Up the Simulation: Portfolio Decisions

As discussed in the last section, we want to target a point on the efficient frontier for our portfolio’s performance. There are equations that can do this, but the drawback to them is that they allow negative values in the target weights, which implies “shorting” those asset classes (to short a security you sell first, and buy back later, thus making money if the price falls while your trade is in effect). While shorting transactions occur and are perfectly legitimate, in my experience individual investors and conservative investment managers such as defined benefit trust funds do not undertake these even if they are possible.

Lacking the ability to use these equations, we can still set our targets using what amounts to “computational brute force” - calculate every possible combination of portfolio weight and then determine which of those lie on the frontier.

The left panel of Figure F shows the return and risk of the entire universe of portfolio combination using 5% intervals and our simulation assumptions. At this point we need to make a decision: what is going to be our target?

Figure F

Efficient Frontier

By taking a “slice” of this universe close to one of our target values, we can find the portfolio combination that lies on the efficient frontier. This is shown in the right panel of Figure F. All portfolio combination lying within a range of 9.5% and 9.6% are shown, with the red dot signifying the one that delivers the highest return for that level of variation.

This efficient frontier portfolio is weighted as follows:

            Short-Term Fixed Income       10%

            Long-Term Fixed Income        60%

            Domestic Equity                      20%

            Global Equity                           10%

 

Using Figure A - Equation 1, we can

Figure G

Portfolio Variance Calculation

calculate that the expected return on this portfolio should be 5.95% (.1 * .035 + .6 * .05 + .2 * .08 + .1 * .1), and using Equation 2 its standard deviation to be 9.57%. Figure G shows the standard deviation calculation in Excel.

The final input is the investment amount. For purposes of this simulation, we will assume that the portfolio begins with ²100 (the symbol ²  stands for Treasury Café Monetary Units, freely convertible into any other currency at any exchange rate of your choosing), and that each month an additional ²100 is contributed, representing conditions one might expect to find in a trust funding situation.

 

Simulation Results

Each simulation performed produced a 360 month progression (30 years) of asset and portfolio values. The average return and standard deviation for each 30 year run was then stored. For each scenario – Rebalancing and No Rebalancing - this simulation was repeated 10,000 times. For the Rebalancing scenario, we executed transactions once per year to buy or sell in order to get back to the target percentages.

We targeted a

Figure H

Portfolio Standard Deviation Comparison

standard deviation of about 9.5% as our target portfolio variance. Figure H shows the density curves (probabilities of each outcome) of each scenario’s 10,000 results. Clearly rebalancing keeps the portfolio a lot closer to the target standard deviation, as indicated by the fact that the rebalanced portfolio’s standard deviation density is “taller and skinnier” than the portfolio that was not rebalanced. This means that we are much more likely to attain our targets by rebalancing.

For the portfolios that were not rebalanced, the average standard deviation varies much more widely from our targets. For example, there were a significant number of outcomes where the portfolios that were not rebalanced would average standard deviations

Figure I

Portfolio Return Comparison

higher than 15%, or over 50% from where we wanted it to be.

Figure I shows the average portfolio return over the 30 year time period for each scenario. As was the case in our look at standard deviation, rebalanced portfolios were closer to the target and did not vary as much as non-rebalanced ones. The portfolios that were not rebalanced had slightly higher returns, as you can notice by the gap between the two around the 10% mark.

Figure J

Figure J

Portfolio Ending Value

Summary Table Values in 1,000's

shows the final values of the portfolios after 30 years. While the ending values were slightly higher for the non-rebalanced portfolios, the average “improvement” was only about ²8,000, less than 10% of the total values. This does not seem efficient given the up to 50% increases in standard deviation required to achieve them. Thus, while we might earn higher returns, it is almost a certainty that the non-rebalanced portfolio does not lie on the efficient frontier.

 

Key Takeaways

Investors aiming to achieve specific objectives with their investment portfolios need to rebalance their portfolios periodically in order to keep performance in line with targets.

Questions

·         How often do you rebalance your portfolio?

·         How would you improve the Monte Carlo methodology employed here?

 

Add to the discussion with your thoughts, comments, questions and feedback! Please share Treasury Café with others. Thank you!

Into the Belly of the Whale: Basis Risk

In our last post, “Into the Belly of the Whale: Curve Balls”, we looked at the term structure of the probability of default and looked at how a “curve flattener” trade pays off when the term structure flattens.

We now apply this knowledge to some of what has been proposed with respect to JP Morgan’s derivative activities and derive some lessons that we can take with us.

It’s a Hedge After All

According to Matt Levine’s assessment, JPMorgan had a significant amount invested in debt securities of other firms.

We have already examined how to price a Credit Default Swap in “Into the Belly of the Whale: Hedging and Credit Default Swaps”.

Figure A

In theory, we can price a bond using the risk free rate if we know the same factors that go into the Credit Default Swap calculation – default probabilities and recovery/loss rates. Figure A shows this expected value calculation in “tree form”.

Using this methodology, the upper leg of the tree is worth $999,519 ($1,050,000 x (1 - 1%) x 0.962) and the lower leg is worth $4,038 ($1,050,000 x 1% x 0.962 x 40% recovery rate), for a total expected value of $1,003,558. This compares to a risk-free bond (i.e. a bond that will pay out with 100% certainty) valued at $1,009,615 ($1,050,000 x 0.962).



Figure B

The difference between the risky bond and the risk free bond price is $6,058. Figure B shows our Credit Default Swap pricing calculation, which comes to the same amount - ah, all is right with the world!

But It Ain’t Perfect

Figure C

If we extend the term of the risk free and risky bonds to 2 years, assuming an annual payment, then the risky bond will be priced according to Figure C. Using this methodology, the risky bond is valued at $983,177, and the risk free bond is valued at $1,000,458. This is a difference of $17,280.



Figure D

Running our Credit Default Swap pricing (Figure D), we come up with a slightly different value of $17,486, for a difference of $206 versus our Bond Pricing methodology. This difference is attributable to a couple of different factors. If there is a default in time 1, then there will be no interest payment nor recovery of it in time 2, whereas for the risk-free bond this would not occur. If there is a default in time 2, we have already earned the time 1 interest payment on both sets of bonds, while the Credit Default Pricing deals only with one set amount which does not vary from period to period.

The valuation difference is a little over 1% of the total value, so there is still a fairly good tracking of the Credit Default Swap pricing versus the Bond Pricing, but there is a difference. In hedging terminology, the fact that there is some fundamental difference between the hedge instrument pricing and the pricing of the item being hedged is called “basis risk”.

Figure E

Let’s see what happens if the curve flattens. If we increase the year 1 probability of default to 5% (thereby inverting the probability of default curve) from 1%, the Credit Default Swap pricing jumps from $17,486 to $41,717 (see Figure E). The Bond Pricing for the risky bond is now $959,770, which creates a difference to the risk-free bond of $40,688. The difference between the two methods is now 2.5% of the Bond Pricing valuation, up from 1% previously. Thus, because of the basis risk the hedge can become less effective as things change through time.

Slight Imperfections Can Add Up!

JP Morgan described the hedge they established as a “macroeconmic hedge”, which would suggest that we can interpret it as follows – “we have investments in some bonds, and in general these things will lose value if the default curve flattens, so we will use CDS index to hedge this, but there is an element of basis risk in the hedge”.

Matt Levine suggested that JPMorgan had up to $1 Trillion in corporate loan exposure. If this were hedged using just our simple example worked out above, the movement of the basis risk between 1% to 2.5% would create a $15 Billion shift in value. So maybe they should be happy if the losses are only $3 Billion!

The Moral of the Story

My purpose in this series of posts was to explore the pricing mechanics of Credit Default Swaps and how they might be used to hedge a debt security investment, and to determine whether a sensational story can be made out of a seemingly prudent undertaking. It seems that it can.

This is not meant to expunge JPMorgan from any wrongdoing. There are a lot of other factors that have been reported about JPMorgan’s hedge which can contribute to the loss other than what we have explored to this point, and not being privy to the gory details leaves me in a position where I cannot fairly judge.

However, we have seen that a simple element such as basis risk can have a large impact on the bottom line if we are dealing in large numbers, and it is therefore important to consider. Many of the spectacular blow-ups in the past, such as Long-Term Capital Management, have basis risk as part of the underlying story as to why they occurred.

For this reason, risk management is something that needs to be executed with an understanding of what may or may not occur under a number of different scenarios. When markets are moving fast, it may be too late to correct for changes in the basis risk inherent in your risk management program.

Key Takeaways

Risk Management is not a precise and exact science. There are often differences between what a business needs to hedge and the securities available to execute it, leading to basis risk. In order to understand potential risks related to the risk management effort, scenario analysis of extreme, sudden, and unexpected market movements should be performed in order to keep our firms from being surprised by the results of our risk management efforts.

Questions

·         Do you think JPMorgan’s losses were the result of speculation or the realities of imperfect hedging solutions?

Add to the discussion with your thoughts, comments, questions and feedback! Please share Treasury Café with others. Thank you!

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

Add to the discussion with your thoughts, comments, questions and feedback! Please share Treasury Café with others. Thank you!