Diversification benefits for bond portfolios

Abstract

Finance research has focused primarily on the diversification of stock portfolios. Various metrics are used herein to assess the diversification benefits, and the optimal bond portfolio sizes (PSs) for investment opportunity (IO) sets differentiated by issuer type, credit ratings and term-to-maturity. While PSs of 25–40 bonds appear optimal for the marginal reduction of dispersion with increasing PS, larger (smaller) PSs are optimal if the investor is concerned about left tail weight (positive skewness or reward-to-downside risk). Although the marginal reduction of dispersion is less than 1% beyond these optimal PSs, much potential diversification benefits still remain unrealized for many of the IO sets studied herein.

Keywords:

• diversification benefits
• portfolio size
• derived and realized dispersion
• skewness
• Sortino ratio
• tail shape

JEL Classification :

• G11
• G23
• C15
• D81

Acknowledgements

Financial support from the Concordia University Research Chair in Finance, IFM2, SSHRC and SSQRC-CIRPÉE are gratefully acknowledged. The usual disclaimer applies.

Notes

Since the foreign bonds are issued in US dollars by non-US companies from all sectors, all of our returns are based on US dollars. Clearly, the number of foreign bonds is much smaller than the number of US bonds, which means that our study has an investor home bias. Nevertheless, it accurately reflects the home-bias selection followed by investors that are referred to as ‘the home bias puzzle’ in the literature, since international diversification not subject to his bias could achieve higher diversification benefits.

The decision to require a minimum of 27 monthly returns to compute the correlations is based on a number of considerations. Requiring fewer monthly returns leads to a bias in the correlations due to the increase in missing observations, whereas requiring no missing values would result in the elimination of too many bonds from our sample. Having 75% of the observations as nonmissing, in our opinion, is an appropriate balance between the negative consequences of requiring too many or too few returns in order to include more bonds (i.e., a more representative sample of bonds) in our correlation computations.

When the number of bonds for a specific month is less than 101, the PS of ‘All’ represented by a portfolio of N−1 bonds is lower in size than a PS of 100. Consequently, we eliminate months from our metric calculations where the number of bonds available for selection is less than 101. This results in the elimination of four months for the foreign (short maturities) IO set, one month for financial (long maturities) IO set, one month for foreign (long maturities) IO set, and 32 months for speculative grade (long maturities) IO set.

We also examine the normalized portfolio variance metric of Goetzmann and Kumar (2008) that is equal to the ratio of the two variances in the MDD metric, and not to the difference of their standard deviations. Based on untabulated results, a SMB-determined minimum PS of around 20–30 bonds captures a high percentage of potential diversification benefits of 91–98% (except for a capture of only 83% for the foreign IO set for both short and long maturities). Similar results are found for most IO sets differentiated by credit rating. Their SMB-determined minimum PSs are 20–40 bonds that correspond to the capture of 87–98% of the potential overall benefits of diversification.

Equal weights are used in forming the portfolios given the findings reported in the literature for equities that no sample-based mean–variance portfolio formation strategy is consistently better in terms of out-of-sample performance than using equal weights. To illustrate, DeMiguel et al. (2007) find that none of the 14 models that they evaluate across seven empirical data sets is consistently better than the 1/N rule in terms of Sharpe ratio, certainty-equivalent return, or turnover. They conclude that this indicates that the out-of-sample gain from optimal diversification is more than offset by estimation error.

The SMB-determined minimum PSs for the Sharpe ratio range from 30 (Utility) to 50 (industrial) bonds, and the relative increase in the Sharpe ratios range from 80 (Tr./Ag.) to 95% (foreign) for IO sets differentiated by issuer type. The SMB-determined minimum PSs range from 25 (Aa) to 60 (Aaa) bonds with associated relative overall increases of 67% (Aaa) to 92% (Aa-A) for IO sets differentiated by credit rating.

By selling out-of-the-money put options on the S&P 500, Lo (2001) obtains a Sharpe ratio of 1.94 for the period from January 1992 to December 1999. This is higher than the corresponding Sharpe ratio of 0.98 for the S&P 500. In Lo's example, the maximum loss for his fund is −18.3% compared with −8.9% for the S&P 500.

Unlike the other metrics, diversification benefits are captured by the decrease in kurtosis between a PS of two, and the PS under investigation since measuring the potential diversification benefits as the difference in the kurtosises between a PS of two and PS of ‘All’ bonds is not applicable due to the very high kurtosis for a PS of ‘All’ bonds.

According to Ruppert (1987), kurtosis measures both peakedness and tail weight, because if probability mass is moved from the flanks to the center of a distribution, then mass has to be moved from the flanks to the tail to keep the scale fixed. As a result, Brys et al. (2006) conclude that, since no agreement exists on what kurtosis really estimates, its use is often restricted to symmetric distributions. They also note that the kurtosis coefficient is very sensitive to outliers in the data.

For a continuous univariate distribution F, and in which 0<p<1/2 and 1/2<q<1and where is the quantile function. The results of the right tail weight are not reported to conserve valuable journal space.

We also test the left medcouple and right medcouple robust measures of tail weight, and the results emit similar implications as those discussed herein (for further details about these tests refer to Brys et al. 2006). We also test the tail behavior for up to 20,000 randomly selected portfolios and again the results have the same patterns. The right tail weight tables are not reported to conserve valuable journal space.

We also examine semi-variance with the risk-free rate as the target return given the evidence that bond return distributions are not symmetric and investors dislike negative returns. Based on untabulated results, the SMB minimum PS range is 20–25 bonds, and the overall reductions in the semi-variances range between 93 and 96% for IO sets differentiated by issuer type. Similar results are observed for further differentiation by maturities and for the various IO sets differentiated by credit ratings (un)differentiated by maturities.

For each month, the cumulative holding-period return is first calculated. Then, the probability is calculated based on the number of times that the holding-period returns for the specific PS underperforms the holding-period return on the market (the target return).

The results reported in this section tend to have the same pattern for the IO sets differentiated by rating category.