Source: China Investment Anxin Futures Research Institute

The ratio of hedging determines the overall risk exposure of the portfolio, and the net position affects portfolio volatility and return performance. Due to the convexity of treasury bond futures and spot prices, when the yield changes, futures and spot prices do not change linearly, hedging will inevitably lead to a mismatch of futures spot prices, which will affect the hedging effect. In the previous study of commodities and stock indexesAcesandeightspokerWe have begun to use statistical models to calculate hedging ratios in order to smooth the portfolio curve. Because the "nominal standard coupon" corresponding to treasury bond futures refers to the virtual coupon with standardized coupon interest rate and fixed term, so the traditional calculation method is not suitable, we imagine that the traditional bond value conversion method is based on the base point value method and then superimposed the statistical model for calculation.

As the linkage reflection of futures is constantly changing, the results of static hedging include biased estimates, and dynamic hedging has gradually become the focus of research. Theoretically, the autoregressive conditional heteroscedasticity model (ARCH) solves the influence of data heteroscedasticity on the hedging ratio, while the generalized autoregressive conditional heteroscedasticity model (GARCH) is improved on this basis. Based on the historical trading data of ten-year treasury bond futures contracts, domestic scholars build a volatility correlation model. Through the relevant methods of VaR measurement, it is found that there is a certain positive correlation between treasury bond futures and the spot market, and there is a certain extreme risk in the treasury bond futures market, and the risk value can be better reflected by the GARCH model. Based on the Granger test based on information share model and vector autoregressive VAR model, this paper studies the price discovery mechanism between treasury bond spot market, treasury bond futures market and interest rate swap market. It is found that interest rate swap has information advantage over treasury bond futures and treasury bond spot, while treasury bond futures has information advantage over treasury bond spot. The price discovery ability of treasury bond futures is significantly stronger than that of the other two markets over time. Therefore, the above methods have a certain practical basis for treasury bond futures hedging.

Based on the empirical analysis of 2-year, 5-year and 10-year treasury bond futures, 30-year treasury bond futures are not included in the calculation because of their short listing time and less data. Firstly, the current series is balanced by the base point value method, and then the correction coefficient of the hedging ratio is calculated by the OLS, VAR, ECM and GARCH models; according to the spot market, futures market and hedging ratio, the net value curve is evaluated, and several indicators are selected to evaluate the performance of the net value curve, and then find the optimal model of different varieties.

From the empirical test, OLS and GARCH methods are better than VAR and ECM models as a whole, in which the gap between them is small for five-year contracts, better for two-year contracts GARCH model, and relatively better for ten-year contracts OLS. As the hedging ratio is originally the adjustment of risk exposure, the change of the overall portfolio yield is not significant, but it still plays a certain role in the reduction of volatility, especially the reduction of losses caused by position shift.

Statistical model-hedging ratio

The main assumption of the base point value method of treasury bond futures is that the change of futures yield is consistent with that of spot yield. As the term structure of deliverable coupons may be different from that of current coupons, the non-parallel movement of the yield curve will lead to non-synchronous changes in yields between them, which will lead to incomplete hedging. For the systematic error caused by term mismatch, the hedging ratio can be modified by adjusting the coefficient. The historical correlation between the base point value of current coupons and CTD coupons is analyzed by statistical model, and the hedging ratio obtained by the base point value method is multiplied by the coefficient β.

In this paper, the β coefficient is obtained by OLS, VAR, ECM and GARCH statistical models to modify the hedging ratio for comparison.

oneAcesandeightspoker.1 OLS regression model

OLS is the most basic statistical regression model, and its premise is that the error series has the same variance and no correlation, that is:

When the above conditions are not satisfied, the result of model regression is biased.

The OLS regression model of the base point value of current coupons and the rate of change of CTD voucher base point value (DVBP) is as follows:

Among them, Rs,t is the DVBP change of t-time coupons, and Rf,t is the DVBP change of t-time CTD coupons.

In this model, the regression coefficient β is the β coefficient.

1.2 VAR model

For economic variables, the sequence often has autocorrelation. The vector autoregressive model solves the problem of sequence autocorrelation. The VAR model for the DVBP change of current coupons and CTD coupons is shown as follows:

The error series of DVBP change rate regression of current coupons and CTD coupons respectively are ε _ s _ ~ ~ t and ε _ f _ ~ t, and all items obey normal distribution, and n is the order of autoregressive lag. In the VAR model, the β coefficient is:

According to the change of the AIC value of the regression result of the VAR model, we set the maximum lag order to 3.

1.3 ECM model

For non-stationary series, OLS and VAR models are no longer suitable. In order to explain the sequence with cointegration relationship, that is, there is a long-term equilibrium relationship between variables, we introduce an error correction term on the basis of the VAR model to establish the ECM model.

Where:

St-1 is the DVBP,Ft-1 of the current coupon at the time of tMui 1 and the DVBP of the CTD coupon at the time of tMui 1.

The β coefficients obtained by the model are as follows:

1.4 GARCH model

For the residual term, the above three models all assume that the residual term is the same variance, but a large number of empirical studies show that there is a problem of residual heteroscedasticity in financial time series, so the GARCH model is proposed. The specific models are as follows:

The optimal β coefficient is:

2. Empirical hedging

We select the historical data from March 2019 to March 2024 as the research object, select the most active cash coupons as the current positions on the transfer date, the CTD coupons corresponding to the following contract as the reference, and use the base point value data of 120 days before the transfer date to determine the β coefficient. Under the condition that five days before the expiration of the contract is selected as the transfer date of treasury bond futures, and the leverage of the futures contract is 1, the β coefficient of the selected treasury bond futures under the OLS, VAR, ECM and GARCH models is shown in the following figure.

Next, we evaluate the net performance of the hedging portfolio through the following five indicators: the maximum loss on a single day of position transfer, the cumulative gain on position transfer, the annualized rate of return, the maximum retracement rate, and the Sharp rate. For the underlying two-year, five-year and ten-year treasury bond futures, the net performance of the hedging portfolio under the four models was compared with the benchmark strategy, namely the basis point value method.

By comparing the calculation results of the beta coefficient, the difference in CTD coupons when moving positions will bring about obvious changes in the beta coefficients, and in most cases, the difference between the coefficient and the benchmark case is large. Since in most cases, the market is more uncertain about forward fundamentals and macro policy strength, and requires a higher implied rate of return to compensate, long-month treasury bonds are usually discounted compared with recent months. A small β coefficient will reduce the loss when moving positions, so the portfolio's return performance after adjusting for the hedging coefficient is better than the benchmark case. Because the liquidity of delivery bonds is worse than that of active bonds, interest rate changes between the two may be decoupled, resulting in extremely low correlation results.

Judging from the hedging effects of the three major treasury bond futures contracts, the OLS and GARCH methods are generally better than the VAR and ECM models. Among them, the gap between the two is small for five-year contracts, the GARCH model is better for two-year contracts, and the OLS for ten-year contracts is relatively better. Since the hedging ratio is originally an adjustment in terms of risk exposure, it does not change significantly the overall portfolio yield, but it still has a certain effect on the reduction of volatility, especially the reduction of losses caused by shifting positions.