Posted by Peter Orr on Sun, Aug 09, 2009 @ 07:54 AM
"Our lives improve only when we take chances - and the first and most difficult risk we can take is to be honest with ourselves."
- Walter Anderson
Although it may not look related out of the gate, this post is a continuation of the prior post on LIBOR swaps over-hedging tax-exempt variable rate bonds. I want to start by looking at how one might build a reasonable interest rate model that would facilitate calculating this % LIBOR correctly, so that we expect to minimize the volatility of our synthetic fixed rate structure. Let's say your job is to build an interest rate model that captures the uncertainty inherent in SIFMA and LIBOR. This would be an unusual task for "quants" in public finance whose primary responsibility is coming up with accurate and often elaborate variations of present value ideas. The "model" used most frequently among front-line in investment bankers/advisors in this sector and in part due to an overreliance on spreadsheets looks something like this:
Some historic average over a selected time-period is used to create a static, flat, deterministic assumption for short rates over the time horizon of the analysis. This IS a type of interest rate model no doubt though one whose strength is not in capturing uncertainty/variability.
If an analyst were trying to create a SIFMA and LIBOR market model using two risk factors, perhaps un-intuitively s/he would not want to use "SIFMA" and "LIBOR" as the risk factors themselves. A detailed reason why is beyond the scope of this post (though you can find an outstanding thorough treatment here), but to put it simply, too much of the variability in SIFMA is also present in LIBOR. Let's face it, as US$ denominated short term interest rates, both SIFMA and LIBOR will be driven largely by changes in US monetary policy.
The better choice for a 2 factor model is LIBOR and SIFMA/LIBOR ratios. SIFMA/LIBOR ratios better reflect the unique component of risk in SIFMA itself i.e. the taxable/tax-exempt relationship. But how does this relate to the correlation impact on swap structure mentioned in the first post? It turns out that historically and on average, as LIBOR falls SIFMA/LIBOR ratios tend to go up and vice versa. In the industry vernacular bankers call this "yield compression" and it has a number of reasonable economic and technical explanations.
How do we capture this in a two-factor interest rate model that doesn't take a PhD to understand? For details on that you can read this and/or get a spreadsheet example. Suffice it to say, it really isn't so bad. To ultimately answer the original question, does this inverse relationship between rates (LIBOR) and ratios (SIFMA/LIBOR) impact the *right* percentage of LIBOR to use when hedging tax-exempt variable rate bonds? Absolutely. The graph below shows the LIBOR swap % that minimizes debt service volatility at different levels of expected correlation between LIBOR and SIFMA/LIBOR ratios.
The bottom line is that using simple averages for this LIBOR swap hedge calculation does 2 things: a) ignores the fact that these structures are not equivalent to fixed rate bonds, a fact that's been sometimes painfully understood over the last 18 months and b) implicitly assumes a correlation of zero between rates and ratios which leads to a hedge ratio that is too high, and ultimately more LIBOR swap than is necessary. What are the cash flow and mark to market ramifications of this over-hedging? Stay tuned for the 3rd and final installment on this topic. In the meantime and if you're involved in the biz, how do YOU do this calculation?
Posted by Peter Orr on Thu, Jun 25, 2009 @ 08:59 AM
"Opportunity is missed by most people because it is dressed in overalls and looks like work."
- Thomas Edison
Over the last decade, many tax-exempt issuers have executed interest rate swaps based upon a percentage of the 1M or 3M London Interbank Bank Offered Rate in order to hedge the interest rate risk inherent in tax-exempt variable rate, or (perhaps unfortunately) auction rate securities. Whether or not these "synthetic fixed-rate" structures are appropriate for all or any tax-exempt issuers is a hot topic these days in light of expected regulatory changes, and one I'm not touching here. What I am going to discuss is something more fundamental to implementation: how does one arrive at the *best* structure to minimize overall debt service volatility, have historical practices led to chronic over-hedging and if so, what are the ramifications.
The usual methodology employed to determine the "right" percentage of LIBOR for the floating leg of the swap is usually calculated based upon some historic average of the ratio of the SIFMA swap index, a weekly tax-exempt floating index, to 1month or 3 month LIBOR. Ignoring tax and accounting issues for the moment, if the goal is to minimize the expected variability of overall net debt service payments this simple averaging method is incorrect in all cases save one: the expected correlation between SIFMA and LIBOR is precisely 1. My personal assessment is that most issuer/advisor/banker participants in this particular market are not accustomed to thinking explicitly about correlation as it relates to their risk management decisions. However, they're making implicit assessments of correlation quite often, sometimes with regrettable consequences.
Let me explain. Any recently test-taking CFA candidate (congratulations by the way) will tell you that the variance minimizing hedge ratio is calculated using the following simple formula:
Volhedged item / Volhedging item * Correlationbetween the 2
This formula has intuitive appeal. If the volatility of my hedging item is far greater than the item hedged, my hedge ratio should fall, which you can see it does. As it relates to correlation and in the canonical edge case, if I have a 0 expected correlation between the item I'm hedging and the item I'm hedging with, we'd expect our hedge ratio to be zero as well; you can't hedge something with something else if you expect no co-movement between the two items.
But is it our best judgment that (changes in) SIFMA and LIBOR will be perfectly correlated going forward? What does history tell us? If we look at the levels themselves, it's clear their correlation isn't perfect. Below are 5 year rolling correlations of one month averaged SIFMA and 1M LIBOR.
What does all this mean? If we expect that SIFMA LIBOR ratios will be 68% going forward (a common and frequently used historically calculated average ratio), then the right hedge ratio would scale this by our expected correlation which likely does not equal 1. A reasonable expected correlation between 85-90% would yield an optimal hedge ratio of between 57.8 and 61.2%. Some banks have gotten to numbers like this by performing regression calculations, which of course are just alternative ways of determining correlation.
So what and who cares? Well, let's compare an issuer with a 70% LIBOR swap paying a 3.5% fixed rate versus one with a 60% LIBOR swap with a 3% fixed rate. The latter has used roughly 14% less swap to do its hedging, which means the 60% LIBOR swap's value is less sensitive to changes in interest rates. This will have important ramifications for mark to markets and collateral posting. For more detail on that and how to capture these effects within a market model, stay tuned for Part2.