A Non-Quant Guide to Constant Maturity Swaps
A Constant Maturity Swap (CMS) is a floating/floating interest rate swap. In many ways, it is similar to a Basis Swap, in which you agree to pay a notional floating rate based on one reference rate (for example, quarterly BBSW) while receiving a floating rate with a different frequency (for example, semi-annual BBSW). While it is possible to have a cross-currency CMS, we’ll keep it simple here and consider only domestic swaps.
The difference with a CMS is that the two rates are on such a different basis, they are not even in the same market. A common CMS would be to swap a quarterly or semi-annual BBSW rate against a two, three or five-year interest rate swap. The BBSW rate would be reset on each swap roll, as would the swap. The difference is in the tenors of the instruments – while the BBSW swap rate applies only until the next rateset date, the swap rate still may have years to run. It is this difference in the underlying rates that turns a very simple concept (from a product point of view) into a reasonably involved modeling exercise.
It is because of this mix of short-term resetting on long-term rates that the CMS is a useful instrument. It gives investors the ability to place bets on the shape of the yield curve over time – if the yield curve steepens, rates on longer swaps will increase, thus increasing the spread to BBSW. Portfolio managers can also use the instruments to hedge a floating rate debt without introducing duration risk from the hedging instrument. This is due to the fact that the fixed ‘coupon’ on a CMS is periodically reset, so the duration on the fixed rate stays constant.
When pricing the swap, we can follow the same high-level approach as with a basis or fixed/floating swap. The fair value will be the value of the short-rate side plus the value of the long-rate side (where one is paying and the other receiving). The only hard part is coming up with the rates on the long-rate side.
Let’s say, for example, we have a three-year CMS where we are are paying semi-annual BBSW plus a margin and receiving the 3 year semi-annual swap rate. Over the 3 year life of this trade, we will have 6 swap rolls, or rate reset dates. For the short-rate (BBSW) side, we can use implied forward rates from a yield curve. For the receiver side, we need a three step process to come up with the rates.
Firstly, we start with par swap rates for each roll – the rate being the fixed swap rate for a theoretical swap starting at the roll date and maturing 3 years after. It would be tempting to think that we could use these rates for the pricing, and indeed, it would not necessarily be a bad approximation. To obtain accuracy, though, we must take into account the different maturities of the respective rates – to compare apples with apples, we must adjust the swap rates to be comparable with the shorter rates.
The first adjustment we need to make is called a ‘convexity adjustment’. A simple interest rate and a swap rate have different sensitivities to underlying curve movements, and therefore we cannot directly compare one rate with another. Consider a six month bill. The relationship between the rate and price of the bill is non-linear and, if you were to plot price according to rate, the shape would be ‘convex’. Similarly, the relationship between a swap rate and price is also non-linear, and also ‘convex’. However, because the rate of curvature in the two relationships differ, we need to make the convexity adjustment. As the adjustment relates to curvature, we need to use a little calculus to find the first and second derivatives of the swap, then also take into account the volatility of the swap rate. Because of these requirements, the pricing of constant maturity swaps becomes a non-trivial exercise.
The second adjustment we need to make is a ‘timing adjustment’. We need to perform this step whenever we are dealing with in-arrears CMS. Again, this adjustment relates to the difference in nature of the two rates. A vanilla swap will take into account paying interest in arrears, as each leg covers the life of the rate for that leg. Calculating a payoff between a short rate and a swap rate, on the other hand, does not adequately reflect the lag in payments throughout the life of the longer rate. In order to make a timing adjustment, we need to know the correlation between the two rates and the volatilities of both the short and long rates.
During the description of the adjustments above, we mentioned the volatility of the rates. Any use of volatility when modelling an instrument will invariably lead to discussions as to the accuracy and appropriateness of the method. Since volatility, like value, is intangible, we cannot expect to find accurate numbers on any vendor screen. Those volatilities that are widely available are subjective, often averaged, and easily manipulated. One way of coming up with the volatility we need is to construct an at-the-money volatility surface based on inputs implied from the prices of exchange traded options, then interpolate from this surface using the tenor and frequency of the swap. This method is very suitable for buy-side and back-office use as it requires no subjective input such as skews or margins. There are of course, many variations and alternatives including incorporating volatility smiles, etc.
While the convexity and timing adjustments obviously depend on a number of factors, including volatility, steepness of the yield curve and correlation, it would not be unreasonable to expect a combined adjustment in the order of 2 – 10 basis points on each rate adjusted. This makes a significant contribution to the pricing of the CMS.
Once we have adjusted the swap rates, we can price the CMS as an ordinary vanilla swap, but using our new adjusted rates in place of BBSW for one of the legs. Typically, for quoting a CMS, you would solve for a margin above BBSW (usually after adding a margin to the swap rate that you are setting from) such that the fixed and floating sides had equal value. This would be the fair-value margin to use in a quote to clients. Buy-side clients who are revaluing a CMS would enter the swap rate and BBSW margins, and solve for a market value using a current yield curve, volatility surface and correlation figures.
The non-quant guide aims to be a cursory overview of financial markets concepts and not an in-depth analysis. This article contains many omissions regarding the mathematical details of pricing constant maturity swaps.
