The method of bond trading based on the convexity characteristics of the yield curve (hereinafter referred to as “convex strategy”) is one of the most commonly used active fixed income strategies for overseas institutional investors, but this strategy has not received sufficient attention and application in the domestic market. . This article will analyze and explore this.

Introduction to convex strategy

The convexity strategy is an important means of trading the yield curve. It constructs a long-term neutral butterfly trading position to ensure that the portfolio is immune to changes in the yield curve and puts the risk exposure on the yield curve. Morphological characteristics change.

The convexity characteristic of the yield curve is visually expressed as the degree of flatness/steepness of the curve. Specifically, when the yield curve is too flat and the expected medium-term bond yields will rise significantly relative to the short-term and long-term, the butterfly trading can be carried out by means of air-time bonds and holding long-term and short-term bonds; Butterfly trading in the form of medium-term bonds and short-term short-term bonds. If the curve shape changes consistently with expectations, excess returns can be obtained after the yield curve is fixed.

Mathematical Derivation of Convexity Strategy

By mathematically deriving the convexity strategy and combining the morphological characteristics of the typical yield curve, the profit source of the strategy can be better understood.

Suppose that at the initial moment, it is judged that the yield curve is too flat, so the short-term one-year yield is the short-term treasury bond, and the buying rate is the same as the annual treasury bond, which is the duration of the national debt when it is closed. The short selling amount is respectively. Since the convexity strategy needs to maintain a neutral position on the held position, there are:

There is a spread in the above positions, which is expressed as the combined cost of construction (revenue), namely:

When the yield curve is expected to be closed after the repair, the position holding period is the year, and the capital gains of each position include the gains from the riding effect, namely:

Among them, the income from the holding of the national debt of each term (negative is the expenditure) is the excess return of the entire transaction.

In the above formula, the change in the rate of return can be divided into two parts: one is the term spread under the current rate of return curve, and the other is the rate of change in the same period. for example:

Since the investment portfolio satisfies the duration of neutrality, the formula (1) is brought into the formula (8) (9), and both ends are simultaneously divided, and the order is:

In view of the graph, Figure 1 shows a yield curve with typical characteristics, the yield is tilted to the upper right, and the slope is gradually reduced, and the corresponding period spread curve is inclined to the lower right. With the change, the value moves on the line segment in the figure, and the key to determining the positive and negative of the spread protection is the concave-convex feature of the term spread curve. Since the term spread curve in the figure is upward and concave, in the vast majority of cases. In addition, the value is related to the holding amount of the short-end position, which depends on the specific barbell trading strategy. Generally, the 50-50 strategy, the maturity date weighting strategy, etc. are roughly balanced for the long-short-end risk exposure, and will run until even The middle end of the line is near point A, and its specific impact will be further analyzed in later evidence.

The ideal return conditions for butterfly trading include two points. One is that the spread protection of the end is as large as possible; the other is that the convexity change of the yield curve is in line with expectations. Therefore, there are two different opening logics for the corresponding butterfly transaction. The first is the spread protection strategy: the two-way transaction locks the spread protection, that is, the guaranteed holding income is always positive regardless of the change of the yield curve shape. At this time, as long as the convexity adjustment of the yield curve is insufficient to completely offset the You can get excess returns by paying off spreads. The second is the convexity level strategy: actively judge the convexity change of the yield curve, and earn the excess return of the convexity change. In addition, it is also possible to comprehensively consider the above two strategies for opening positions, taking into account spread protection and convex changes.

The empirical effect of convex strategy in the Chinese market

After the theoretical analysis, the following attempts to solve the following problems through empirical research: (1) How effective is the different opening strategy of convex trading in the Chinese bond market, and how should it be chosen? (2) How should the long and short-end bond weights be configured? (3) What is the impact of the holding period on the excess returns of the convex trading? (4) Is it possible to further enhance the performance of convex transactions?

(1) Sample and data compilation

This paper uses the monthly data of China Bond Debt Yield from January 2007 to August 2018 to conduct a backtest of the convex strategy. The data is taken from Wind. Figure 2 shows the treasury yield curve and the term spread curve at a certain point in time. It can be found that the fluctuation of the term spread curve is more obvious at the short end, indicating that there is a certain profit margin in the spread protection strategy at this position. Figure 3 shows the convex variation of the portfolio of different maturity bonds in the sample interval. The convexity is defined as, here, taken. It can be seen from Figure 3 that the convexity variation of the term combination 1-7-10, 3-7-10, 1-5-7, 1-5-10 in years is larger, which is more conducive to the convexity level strategy. Implementation.

(2) Empirical evidence on the effectiveness of convexity strategies

Based on the government bond yield data of the Chinese bond market, this paper back-tests the spread protection strategy, the convex level strategy and the comprehensive strategy of the former two.

The spread protection strategy: Calculate the value of the corresponding combination according to formula (11). If it is steep, it is to do the air-period bond and hold the long-term and short-term bonds; otherwise, it is flat. Therefore, the protection of the spread of the butterfly transaction is always positive, and the excess return of the butterfly transaction is further calculated.

Convex level strategy: Calculate the convexity of the corresponding combination, use the representation, and flatten when its value is greater than the historical mean, and vice versa.

Comprehensive strategy: if and only when the butterfly trading direction indicated by the spread protection and convex historical level information is the same, the position is established, and the combined convexity level is lower than the historical average, and the combined convexity level is higher than the history. The mean is flat.

The above backtest results are shown in Figure 4.

It can be seen from Figure 4 that except for the term combination 3-7-10, the average excess return obtained by the spread protection strategy is negative, and the average excess return rate of the butterfly transactions of other time combinations is significantly greater than 0, and the corresponding winning rate is generally higher. high. The partial excess return on the partial portfolio can even reach around 150BP. Overall, the convex strategy has a lot of room for implementation in the Chinese bond market.

In terms of horizontal comparison, the effect of the convex level strategy is generally better than the spread protection strategy, especially for combinations with large convexity variation and average regression trend, such as term combination 1-5-7, 1-5-10. 3-7-10, 1-7-10, the average excess return under the convexity level strategy is 104BP, 137BP, 152BP and 251BP, respectively, compared with the spread protection strategy, and the winning rate is also increased by about 10%. The profit margin provided by spread protection is more limited than the predicted convexity change.

In addition, the comprehensive strategy to establish positions has the best return, and the spread protection strategy and the convex level strategy have synergistic contributions to the income. Among them, the comprehensive strategy of term combination 1-3-5, 1-3-7, 1-3-10 has better performance, and the average excess return and winning rate are greatly improved compared with the single strategy; The benefit contribution of the spread protection strategy of 1-5-10, 1-7-10, and 3-7-10 is relatively small, and the profit status of the comprehensive strategy is not much different from the strategy of relying solely on the convexity level.

(3) The weight ratio of the long and short ends of the barbell

Observing formula (11) (12), we can find that the position of the short-end bond is increased, and the coefficient value is gradually increased. For the spread protection strategy, when it is steep, it will increase the safety pad of the butterfly transaction. On the contrary, the interest margin protection will be reduced when doing the usual. For a convex level strategy, an increase means that the bet on short-term yield changes is increasing.

The empirical results shown in Figures 5 through 7 can be obtained by dynamically adjusting the values in the range [0, 1] and repeating the previous backtesting steps.

Looking at the left graph in Figure 5, under the spread protection strategy, increasing the matching weight of the barbell trading short-end bond can not effectively improve the average excess return rate of the butterfly transaction; observe the left graph in Figure 6 and Figure 7, Under the convex level strategy and the comprehensive strategy, with the increase, the average excess return of the portfolio shows a trend increase, and the greater the convexity variation of the combination, the greater the slope of the average excess return rise. This is mainly due to the fact that short-term bonds are more volatile than the long-term, and more short-term bonds can significantly increase the average return when the strategy win rate is basically stable. In practice, in the case of risk control, in order to obtain higher returns, the allocation ratio of short-term bonds can be appropriately increased.

Looking at the right graphs in Figures 5 to 7, it can be seen that the variation of the Sharp Index of each strategy shows a considerable degree of consistency: first rising and then falling, there is a local best. This shows that the ideal butterfly strategy needs to balance the ratio of long- and short-term bonds.

In addition, in traditional convex transactions, there are configuration strategies such as 50-50 strategy, maturity date weighting, etc. Are the theoretical barbell proportions of these strategies consistent with the actual empirical optimal values?

According to the 50-50 strategy:

Weighting strategy based on due date:

The k values corresponding to the above two strategies can be calculated as the sum, and compared with the empirical empirically obtained optimal experience values, and Figure 8 can be obtained. It can be seen that the empirical optimal values corresponding to the spread protection strategy, the convex level strategy and the comprehensive strategy are not much different, and the short-end matching weights given by the maturity weighting strategy are basically the same. This indicates that the short-end optimal ratio of barbell trading is related to the magnitude of the change in long- and short-end yields, and is less related to the specific holding strategy. Considering that the empirical optimal value may have an over-fitting problem, the applicability of the data outside the sample is poor. It is recommended that the butterfly transaction be configured based on the maturity weighting strategy (ie, let k=).

(4) Impact on the excess income during the holding period

According to formula (11)(12), there are, and. Excess income, we can know that the impact on the average excess return during the holding period mainly includes two parts. First, as the holding period increases, the coefficient term will fall, thus dragging down the final gain of the butterfly trading. Second, as the holding period increases, the spread curve will be more stable, and the over-concave features will be alleviated. For the spread protection strategy, the increase in the holding period has a downward trend on the overall security pad; for the convex level strategy, the increase in the holding period makes the prediction of the yield curve more difficult, and the convexity is not expected to change. The risk is increasing.

As shown in the left graph of Figures 9 to 11, as the holding period increases, the average excess returns under the three opening strategies all show a significant decline, and the influence of the coefficient term is quite significant. It should be noted that in the right graph of Figure 10, the Sharpe index generally decreases with the increase of the holding period under the convexity level strategy. When the holding period is one year, the excess returns of some combinations are even absent. This indicates that the increase in holding period makes it difficult to predict the convexity of the yield curve, and it has a great interference with the implementation of the convex level strategy, especially for the period of 1-3-5, 1-3-7, 1- In the combination of 3-10, it is a combination of a small change in convexity, and the difficulty of related prediction is greatly increased.

On the contrary, as shown in the right figure of Figure 11, under the comprehensive strategy, the Sharpe ratio has increased significantly. This is because as the holding period increases, the same opportunities for double-standard positions in the comprehensive strategy decrease, but the certainty increases.

Therefore, for the spread protection strategy and the convexity level strategy, lowering the holding period is beneficial to controlling risks and increasing average income. For the comprehensive strategy, a modest increase in the holding period means that the threshold for opening a position increases, which helps to increase the portfolio’s return-to-risk ratio.

Exploring the efficiency of convex strategy

(1) Construction of threshold indicators

After comparing the effects of different opening strategies, the weight of the long and short ends of the barbell transaction, and the length of the holding period on the excess return, this paper attempts to construct indicators to further optimize the benefits of the convexity strategy.

According to the previous analysis, the performance of the comprehensive strategy is better than any single strategy as a whole. Here, we try to further improve the threshold of building a warehouse based on the comprehensive strategy, and screen out the trading opportunities that are not significant enough, in order to obtain more certain excess returns.

On the spread protection side, the greater the spread protection, the stronger the ability to resist the unexpected change of the convexity; at the horizontal end of the convexity, the more extreme the convexity level is, the greater the probability of the mean return of the convexity in the future. Therefore, the spread protection threshold coefficient and the convexity horizontal threshold coefficient can be set, respectively corresponding to the threshold value, wherein the average value of the convexity of the yield curve is the historical volatility of the absolute value of the spread protection, which is the historical volatility of the convexity.

(2) Threshold indicator empirical

Select 1, 3, 5, 7 and 10 years of national debt to construct a butterfly transaction, set the holding period, and if and only if the following conditions are met:

Investigate the threshold coefficient and the impact on excess returns. Let, 0, 0.2, and 0.4 respectively, obtain Table 1, and let, 0, 0.2, and 0.4 respectively, and Table 2 can be obtained.

It can be seen from Tables 1 and 2 that after increasing the threshold for building a warehouse, the average excess return and the Sharpe index of the butterfly transaction are generally significantly higher than before, but at the same time there are fewer opportunities for trading. From the vertical comparison point of view, the impact of raising the margin protection threshold is relatively weak, and the partial excess return and the Sharp index have a short-term correction. The effect of raising the threshold of convexity is more significant. In addition, from the perspective of improving efficiency (increased average earnings or Sharpe ratio/reduced trading frequency), raising the threshold of convexity is also better than raising the threshold of spread protection.

(III) Further comparison between convex level strategy and comprehensive strategy

We noticed that the improvement of the comprehensive strategy’s income status is brought about by the double standard to increase the threshold of building a warehouse. Then, can we increase the threshold coefficient in the convexity level strategy to achieve an excess or higher level of excess return with the comprehensive strategy?

To this end, the convexity level threshold can be adjusted, which is divided into the calculation of the convexity level strategy and the average excess return and the Sharpe index when the comprehensive strategy transaction frequency is 10%, that is, the final return status of the 10% ideal opening position of the two strategies.

It can be seen from Figure 12 that increasing the threshold coefficient in the convexity level strategy can achieve an investment return similar to the comprehensive strategy. Some term combinations such as 1-5-10 and 1-7-10 perform even better than the comprehensive strategy. It is related to the magnitude of the combined convexity variation and the contribution of the spread protection. However, the stability of the comprehensive strategy is better, especially for the mid-end short-span combination with a duration of 3-5-7 and 3-7-10. The strategy to improve the threshold variation of the threshold coefficient will be invalid. The performance of the strategy is far superior to the convex level strategy.

main conclusion

The profit margin provided by the spread protection strategy is more limited than the convex level strategy. Especially for the combination of large convexity variation and average regression trend, the convexity level strategy performs better. The overall strategy’s revenue situation is better than any single strategy.

Under the convex level strategy and the comprehensive strategy, increasing the matching weight of the barbell trading short-end bond can significantly increase the average excess return of the butterfly transaction, and the spread protection strategy has no such effect. From the perspective of Sharp Index, the long-term and short-term bond ratios of each strategy have approximate empirical optimal values, and the relative level is basically the same as the maturity weighting strategy. It is recommended to use the maturity date as the benchmark. Adjust after configuration.

As the holding period increases, the average excess returns under each of the opening strategies have shown a significant decline, and the convex level strategy will even fail under the longer holding period. For the spread protection strategy and the convex level strategy, lowering the holding period is beneficial to controlling risks and increasing average income. For the comprehensive strategy, a modest increase in the holding period means that the threshold for opening a position increases, which helps to increase the portfolio’s return-to-risk ratio.

After increasing the threshold for building a warehouse, the average excess return of the butterfly trade and the Sharp Index generally increased significantly. From the perspective of improving efficiency (increased average earnings or Sharpe ratio/reduced trading frequency), the threshold for raising the level of convexity is better than the threshold for improving the spread protection strategy. The strategy of improving the post-contour level of the Jianceng threshold can indeed achieve an investment return similar to the comprehensive strategy, but the stability is still weaker than the comprehensive strategy.