13.15.1.3 Advanced Historical Volatility Approach

This approach fits relative yield changes on all 2,320 days based on the theoretical relationship between the short rate and longer-term yields by regression analysis and then fits term structure model parameters to the regression coefficients.

For the Canadian market, a regression of par bond coupon yields (as proxies for zero-coupon bond yields) on the one-month Canadian Government bill rate was performed. The results of this regression showed a higher implied mean reversion speed at shorter maturities.

Implied Speed of Mean Reversion by Historical Sensitivity to Movements in the Canadian Treasury Bill Rate (1987 - 1996)

Table 13-12 Implied Speed of Mean Reversion by Historical Sensitivity to Movements in the Canadian Treasury Bill Rate (1987 - 1996)

	3 - Year Bond Yield	10 - Year Bond Yield	25 - Year Bond Yield
Coefficient of Short Rate	0.31430	0.17709	0.13505
Standard Error	0.01960	0.01573	0.01349
t - score	16.03177	11.26116	10.0127
R2	0.09985	0.05189	0.04422
Best Fitting α	1.00921	0.56275	0.29600

The best-fitting α at three years was a very high 1.00921. At 25 years, the α at 0.296 is much more consistent with the historical variances. This chart provides a strong indication that a two-factor model would add value in the Canadian market (assuming other problems, like parameter estimation and the valuation of American options that are strong disadvantages of two-factor models). This is true of most markets where recent interest rate fluctuations have been large and where current rate levels are near historical lows. The Australian market has had similar experiences.