Convexity bias

Long-dated yields 144 7.4 Analysing the convexity bias ... 

A common observation in government bond markets is that the longest dated bond trades expensive to the yield curve. It also exhibits other singular features that have been the subject of research, for example, by Pboa (1998), wbicb we review in this chapter. The main feature of long-bond yields is that they reflect a convexity effect. Analysts have attempted to explain the craivexity effects of long-bond yields in a number of ways. These are discussed first. We then consider the volatility and convexity bias that is observed in long-bond yields. 

From Figure 7.3, we see fliat to price a very Iraig-dated bond off the yield of the 30-year government bond would lead to errors. The unbiased expectations hypothesis suggests that 100-year bond yields are essentially identical to 30-year yields however, this is in fact incorrect. The theoretical 100-year yield in fact will be approximately 20-25 basis points lower. This reflects the convexity bias in longer dated yields. In our illustration, we used a hypothetical scenario where only three possible interest-rate states were permitted. Dybvig and Marshall showed that in a more realistic environment, with forward rates calculated using a Monte Carlo simulation, similar observations would result. Therefore, the observations have a practical relevance. 

ANALYSING THE CONVEXITY BIAS IN LONG-BOND YIELDS... 

Phoa (1998) presents an approximation of the convexity bias as follows. Consider a conventional fixed coupon bond, which has a yield at a future time t of r and a price at this time of P r). The convexity bias is estimated using... 

The volatility value used can be estimated in two ways. We can estimate volatility separately and then use this to calculate what the approximate convexity adjustment should be, or we may observe the convexity bias directly and derive a volatility value from this. This would require an examination of market swap rates and bond yields, and use these to estimate the volatility implied by these rates. 

Mean Reversion Rate EsIilltfjM Convexity bias estimation requires an estimate of the mean reversion rate (a) and the standard deviation (a) of the change in short-term interest rates. There are several alternative methodologies for estimating a and o. The first methodology uses historical data to estimate the parameters. 

Cap prices can also be valued analytically using the Hull-White model. The cap prices calculated using the implied volatilities of interest rate caps and the Black-Scholes model serve as the calibrating instruments. After the Hull-White model has been calibrated, the parameters a and o that minimize a goodness-of-fit measure can be used to solve for the convexity bias. 

A third methodology uses a simplified approach to get an approximation of the convexity bias. The mean reversion rate, a, typically varies from O.OOf for negligible effects to O.f for high mean reversion. For example, Bloomberg assumes a constant mean reversion rate of 0.03. Observed volatilities of interest rate caps are multiplied by their corresponding swap rates to get an estimate of o. 

The estimated or assumed values of the parameters a and o can be used in combination with the Hull-White convexity adjustment term to estimate the convexity bias embedded in futures rates. 

Stereoselective allylation of secondary radicals is possible when a suitable steric bias is present. For example, the thiocarbonyl compound 41 reacts to give exclusively the exo allylated product 42, in which allyl tributylstannane approaches from the less-hindered convex face of the cyclic radical (4.40). In acyclic substrates high stereoselectivity can be achieved by chelation with a Lewis acid. For example, allylation of the selenide 43 is much more stereoselective in the presence of trimethylaluminium, in which the aluminium alkoxide chelates to the carbonyl group to give the species 44, such that the approach of the allyl stannane is directed to the less hindered face (4.41). 

Convexity differences between nominals and linkers create a systematic bias in break-even inflation, which is itself hard to quantify reliably (for instance, we have little objective current market-based information about prospective long-term real yield volatility). This makes the isolation of the risk premium even more difficult, if that s possible. The purpose of all of this is not to discourage the investigation of the risk premium, but rather to raise awareness of some important influences that need to be considered before you decide to begin your quest. 

A long position in FRAs or swaps and a short position in futures has net positive convexity. The short futures position has a positive payoff when interest rates rise and lower losses when interest rates fall, as they can be refinanced at a lower rate. This mark to market positive effect of futures contracts creates a bias in favor of short sellers of futures con-... 

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