In a previous learning curve (DW, 1/5), we introduced a notion of convexity cost in the option adjusted valuation. That discussion referred to an OAS compenent caused by the nonlinear (convex or concave) relationship between a volatile market factor (the interest rate) and the price of an embedded-option asset. The interest rate factor is a traded factor as the market assesses it and compensates when trading the asset.
However, returns on dynamic assets are affected by non-interest-rate factors as well. Such factors as potential errors in underlying modeling assumptions (prepayment model, index model, interest rate volatility, etc.) can be simulated as standard Brownian motions disturbing the asset's price. If the factor-price relationship is nonlinear, each factor contributes some convexity cost into OAS. In many cases, the market does not recognize these contributions thus making the factors non-traded; convexity costs due to non-traded factors become a hidden source of an additional return reward (or penalty) for holding the asset. If the price depends on several independent non-traded factors, the corresponding convexity costs are simply additive, each of them computed (see figure).
Note that, unlike for a traded factor, we do not generally subtract benchmark convexity because there is no benchmark. The drift for a non-traded factor is eliminated from the analysis being a part of the traded model (for example, the error for an unbiased prepayment model can be considered a zero-mean factor).
If a factor is non-traded, computations of the corresponding convexity cost are rather simple. First, traded OAS is sought for the model that involves traded factors only. A non-traded factor is then shocked up and down and new prices are computed. Since the factor is non-traded, the market does not change traded OAS. Using thus calculated 3 prices, we estimate factor convexity (and, if needed, factor duration). Below, we present a brief coverage of major non-traded factors that could concern sophisticated MBS, whole loan and floating bond investors and traders.
PREPAYMENT MODEL ERROR
There exists no ideal prepayment model. The deviations of actual prepayment speeds from those computed by a model can be treated as a factor. A. Sparks and F. Feiken Sung (March 1995) conclude that a positive (negative) prepayment convexity will improve (reduce) OAS if this factor is randomized in an OAS model, but, unlike this article, they provide no analytical tool to quantify this addition.
It is a simple, but notable, finding that the prepayment convexity grows with price (in a very simplified form, this conclusion can be explained by a generalized Gordon's stock pricing formula). Thus, given everything else even and assuming 20% of unexplained annual prepayment volatility, premium MBS' or whole loans provide as much as 15 basis points of additional expected return hidden from the market, whereas discount assets may cause 3-5 basis points of expected loss.
INDEX MODEL ERROR
If a floater or an ARM is indexed to a non-Treasury and non-LIBOR index (COFI, NMCR, PRIME, etc.) having no established term structure, it needs to be modeled. An error arising in such a model can be treated as an additional non-traded factor. Due to the nature of the prepayment option, convexity cost with respect to the ARM index model error is always positive (ignoring the reset cap's role). Indeed, should the index be above the modeled values, potential interest income benefits for investors will be limited by a rising prepayment activity. If the index is below the model's projections, the loss of interest will be extended. Assuming that an autoregressive COFI model allows 15 basis points of annualized residual volatility, a COFI ARM investor is expected to lose 2-3 basis points in return due to this factor, while being exposed to additional risk. Of course, a regular bond floater has no prepayment option, but reset caps may cause some convexity cost even under symmetrical deviations of the index around its model.
OTHER NON-TRADED FACTORS
Practically all parameters in a pricing model are known with certain accuracy and could be treated as factors themselves. Interest rate volatility is an obvious candidate as heteroskedasticity is a known market phenomenon. For the sake of simplicity, we propose to treat it as a non-traded factor and employ our general method to find its convexity cost. Our analysis indicates some positive convexity cost (return loss) not exceeding 1-2 basis points for most MBS.
Loans and MBS are traded at an option-adjusted spread above the Treasury market. This spread can vary over the time thus becoming a pricing factor itself, its volatility is easily observed. Since the spread convexity is always positive, investors enjoy a slight (within a 1 basis point range) additional return.
This week's Learning Curve was written by Alexander Levin, a senior quantitative developer, and James Daras, the treasurer of The Dime Savings Bank of New York.
