Pedagogy dictates that students learn about (and teachers teach) the mathematical approach to portfolio risk long before the art. A quick historical recap typically starts from Bachelier's thesis written in 1900 presenting what is known as the Normal model for asset price dynamics, through to the mean-variance framework presented by Markowitz, to (multi-) factor models, statistical models and models based around numerical simulation.

To understand the art, we typically start by critically analysing the weaknesses of such models, under what conditions they fail, and what stylistic features of portfolio (or market) variance they fail to capture - and why. For example, Taleb's discourse on anti-fragility is a good example of the art of risk measurement. While both the science and art of risk measurement are important, mastery and understanding of the latter takes longer and is more difficult than the former.

Part of the art of risk measurement is knowing when a philosophical deep-dive is needed (ala Taleb); when assumptions need to be revisited (as happened during and after the financial crisis) or when a simple tweak to governing equations is required (as we show below). In other words, delineating between the science and art is crucial.

In this short note, we take a critical look at convexity (see image right/above showing convexity for a par bond) within a fixed-income factor model for portfolio risk and consider how vanilla short expiry rate swaptions are represented. It is assumed the reader has a basic familiarity with the concepts of duration and convexity in the fixed income markets, and these are not covered here. It is further assumed that the factor model, mentioned above, uses key rate durations (KRDs) and key rate bucket convexities (KRBCs) as exposure vectors in a standard covariance framework.

The convexity calculation is defined with respect to two yield curve shifts at tenor points . Subsequently, Vertical Key Rate Bucket Convexity (VKRBC) denoted aggregates the KRBCs vertically when the KRBC are arranged in matrix form. Finally, effective convexity is the sum over tenors of VKRBC. Mathematically, we have the following

The finite difference approximation used for can be derived by informally defining the price of an asset to be a function of the tenor points used in the shift, . Using partial cross derivatives, its possible to derive the approximation:

Post the 2007-2008 financial crisis, with attention focussed on long-dated exposure to counterparts and their credit worthiness, it is common to trade swaptions 'forward premium', meaning unlike a traditional asset purchase via an upfront premium paid at spot. The exact date on which a forward premium can be paid can, in theory, be on any date from spot through to expiry - although at expiry would be a reasonably common choice. Premiums can also be split over several dates, although this can complicate valuation and accounting.

In the case where the security is a forward premium (at expiry) swaption, we can decompose the price, as , where is a spot swaption with the same terms and conditions and is a hypothetical zero coupon bond maturing on the premium payment date. Here, is the swaption (forward) premium price in % of face of the swaption position.

Applying the derivation in the case of a forward premium swaption

For swaptions with expiry of 2yrs or less, the zero coupon bond term tends to zero (for all except ) due to insensitivity to longer tenor rates. For example, with and

Ensuring

When this happens, the KRBC for a forward premium swaption reduces to a multiple of the KRBC for a spot swaption

Because the price, , for a forward premium swaption can pass through, or equal, when this happens very large KRBC values can be obtained, leading to very large VKRBC exposure values and in turn risk estimates.

If such convexity exposures are left uncorrected, estimates of portfolio risk can be exaggerated, and in the case of portfolios consisting largely of such positions, significantly erroneous.

Portfolio risk: swaption convexity