Improving Interest Rate Risk Hedging Strategies through Regularization

Figures & data

Figure 1. Yield Curve’s Level (Left Panel), Slope (Middle) and Curvature (Right Panel) Nelson–Siegel Model Parameters Estimated from Bond Prices, Assuming a Fixed Shape Parameter Equal to $\tau$= 1.37 = 1/(12 × 0.0609) as in Diebold and Li (2006)

Figure 2. Funding Ratio and First-Order Sensitivities to the NS Factors of the Retirement Bond (Continuous Line), the Duration Barbell (Dashed Line), and the Duration-Convexity (Dotted Line) strategies implemented with k + 1 Bonds, Where k Is the Number of Factor Sensitivities Matched by the Strategy (2 and 3 for the Duration and Duration-Convexity Strategies, Respectively)

The upper-left panel presents the time series of the funding ratios, the upper-right panel presents the first-order sensitivities to the level factor (D0), the bottom-left panel the first-order sensitivities to the slope factor (D1), and the bottom-right panel presents the first-order sensitivities to the curvature factor (D2).

Figure 3. Second-Order Sensitivities to the NS Factors of the Retirement Bond (Continuous Line), the Duration Barbell (Dashed Line) and the Duration-Convexity (Dotted Line) Strategies Implemented with k + 1 Bonds, Where k Is the Number of Factor Sensitivities Matched by the Strategy (2 and 3 for the Duration and Duration-Convexity Strategies, Respectively)

The left panel presents the time series of the second-order sensitivities to the level factor (D0D0), the middle panel the second-order sensitivities to the slope factor (D1D1), and the bottom-right panel presents the first-order sensitivities to the curvature factor (D2D2).

Table 1. Number of Constituents (NC), Median and Maximum Leverage (Med Lev, Max Lev), Final Funding Ratio (FR.Final), Funding Ratio Median Absolute Deviation (FR.MEAD) and Maximum Absolute Deviation (FR.MAAD), and Average One-Way Monthly Turnover (TO) of Hedging Strategies Matching the Retirement Bond Yield Curve Factor Sensitivities Rebalanced Monthly

	NC	Med Lev	Max Lev	FR.Final	FR.MEAD	FR.MAAD	TO
A. Unrestricted leverage
Duration Barbell	2	1	1.13	78.93	11.73	22.46	68.1
DurConv k + 1	3	1.77	3.48	112.17	11.27	14.28	159.93
NS k + 1	4	31.44	6703.9	8.28	161.24	351.86	20737.88
B. Restricted leverage—L3
Duration lasso	127	1	1.32	96.21	2.05	3.80	8.58
DurConv lasso	87	1.02	1.64	102.04	2.37	3.77	12.6
NS lasso	83	1	1.53	100.75	1.74	3.39	17.89
C. Long-only
Duration lasso	127	1	1	95.25	1.48	4.76	9.16
DurConv lasso	84	1	1	101.06	1.54	2.90	10.67
NS lasso	82	1	1	99.46	0.61	2.92	16.27

Panel A corresponds to the strategies without regularization/leverage constraints. Panel B presents the regularized lasso strategies with a (gross) leverage limit of 3. Panel C presents the long-only lasso strategies.

Table 2. Summary of Deviations in NS Factor Sensitivities of Hedging Strategies Relative to the Retirement Bond

	Duration Barbell	DurConv k + 1	NS k + 1	Duration lasso	DurConv lasso	NS lasso
D0.MEAD	0.019	0.006	0.018	0.006	0.006	0.007
D0.MAAD	0.180	0.052	3,646.864	0.055	0.051	0.087
D1.MEAD	0.470	0.388	0.009	0.051	0.041	0.009
D1.MAAD	0.711	1.024	1,635.570	0.092	0.224	0.168
D2.MEAD	0.461	0.329	0.008	0.106	0.042	0.008
D2.MAAD	0.660	0.701	1,703.917	0.166	0.163	0.110
D0D0.MEAD	1.471	0.013	1.426	0.408	0.007	0.004
D0D0.MAAD	2.159	0.180	12,129.316	0.873	0.117	0.138
D1D1.MEAD	0.477	0.412	0.013	0.083	0.049	0.008
D1D1.MAAD	0.701	0.945	1,747.534	0.139	0.203	0.145
D2D2.MEAD	0.431	0.123	0.045	0.151	0.026	0.007
D2D2.MAAD	0.624	0.279	1,614.618	0.214	0.088	0.046
D0D1.MEAD	0.029	0.028	0.018	0.010	0.006	0.006
D0D1.MAAD	0.164	0.144	3,715.280	0.054	0.035	0.074
D0D2.MEAD	0.043	0.108	0.031	0.014	0.009	0.006
D0D2.MAAD	0.218	0.359	3,805.218	0.065	0.058	0.056
D1D2.MEAD	0.451	0.287	0.009	0.124	0.039	0.007
D1D2.MAAD	0.649	0.559	1,678.741	0.187	0.137	0.087

MEAD stands for median absolute deviation and MAAD for maximum absolute deviation. For instance, D0.MEAD stands for median absolute deviation in sensitivity to ${\beta }_0$ (the level factor) between each strategy and the corresponding sensitivity of the retirement bond over time. D1.MAAD is the maximum absolute deviation in first-order sensitivity with respect to ${\beta }_1$ (slope) from the sensitivity of the retirement bond, and D1D1.MAAD is the maximum absolute deviation in the second-order sensitivity with respect to ${\beta }_1.$ All MEAD and MAAD series are divided by the median value of the corresponding retirement bond’s factor sensitivity, to express them as percentages.

Figure 4. Funding Ratio and First-Order Sensitivities to the NS Factors of the Retirement Bond (Continuous Line), and the Hedging Strategies Implemented with All the Bonds Available and with a Lasso Regularization that Restricts the Gross Leverage below Three at All Times

Strategies matching Duration, Duration and Convexity, and the strategy matching the first- and second-order sensitivities to the three NS factors. The upper-left panel presents the time series of the funding ratios, the upper-right panel presents the first-order sensitivities to the level factor (D0), the bottom-left panel the first-order sensitivities to the slope factor (D1), and the bottom-right panel presents the first-order sensitivities to the curvature factor (D2).

Figure 5. Second-Order Sensitivities to the Three NS Factors of the Retirement Bond, and of the Hedging Strategies Implemented with All the Bonds Available and with a Lasso Regularization that Restricts the Gross Leverage below Three at All Times

Strategies matching Duration, Duration and Convexity, and the strategy matching the first- and second-order sensitivities to the three NS factors. The left panel presents the time series of the second-order sensitivities to the level factor (D0D0), the middle panel the second-order sensitivities to the slope factor (D1D1), and the right panel presents the second-order sensitivities to the curvature factor (D2D2).

Table 3. Summary of Deviations in NS Factor Sensitivities of Long-Only Hedging Strategies Relative to the Retirement Bond

	Duration Lasso	DurConv Lasso	NS Lasso
D0.MEAD	0.006	0.006	0.007
D0.MAAD	0.165	0.283	0.289
D1.MEAD	0.054	0.027	0.010
D1.MAAD	0.107	0.099	0.099
D2.MEAD	0.111	0.033	0.008
D2.MAAD	0.207	0.201	0.201
D0D0.MEAD	0.408	0.007	0.005
D0D0.MAAD	1.406	0.120	0.104
D1D1.MEAD	0.087	0.034	0.009
D1D1.MAAD	0.159	0.154	0.154
D2D2.MEAD	0.153	0.027	0.007
D2D2.MAAD	0.296	0.291	0.291
D0D1.MEAD	0.010	0.007	0.007
D0D1.MAAD	0.185	0.308	0.313
D0D2.MEAD	0.014	0.010	0.007
D0D2.MAAD	0.218	0.348	0.354
D1D2.MEAD	0.128	0.035	0.008
D1D2.MAAD	0.239	0.233	0.233

MEAD stands for median absolute deviation and MAAD for maximum absolute deviation. Hence, for instance, D0.MEAD stands for median absolute deviation in sensitivity to ${\beta }_0$ (the level factor) between each strategy and the corresponding sensitivity of the retirement bond over time. D1.MAAD is the maximum absolute deviation in first-order sensitivity with respect to ${\beta }_1$ (slope) from the sensitivity of the retirement bond, and D1D1.MAAD is the maximum absolute deviation in the second-order sensitivity with respect to ${\beta }_1.$ All MEAD and MAAD series are divided by the median value of the corresponding retirement bond’s factor sensitivity, to express them as percentages.

Diebold, F. X., and C. Li. 2006. “Forecasting the Term Structure of Government Bond Yields.” Journal of Econometrics 130 (2): 337–364. doi:10.1016/j.jeconom.2005.03.005.

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