New research being carried out jointly at Dublin City University Business School and University College Cork provides important findings for those considering an investment in convertible bond arbitrage. Evidence is provided that in severe market downturns convertible arbitrage exhibits negative returns. This negative return is due to increases in credit spreads. Evidence is also found that in severe market upturns the daily returns from a convertible bond arbitrage portfolio are negatively related to equities. In effect the returns to convertible bond arbitrage are akin to writing naked out of the money put and call options. Although this is not the first study to document the put option like feature in convertible arbitrage returns ^{1} , it is the first to document the negative correlation between daily convertible bond arbitrage and equity market returns in extreme up markets. This negative correlation is explained by the long volatility nature of convertible bond arbitrage. In extreme up markets implied volatility generally decreases having a negative effect on portfolio returns. This is an important finding for any investor considering adding a convertible bond arbitrage fund to an existing buy and hold long only equity portfolio.

While the overall market for convertible bonds has been growing to an estimated $351.9 billion by the end of December 2003 (BIS, 2004), the hedge fund industry has also been growing at a phenomenal rate. Initially investors were interested in large global/macro hedge funds and the majority of the funds went into these strategies. Fung and Hsieh (2000) estimate that in 1997 27 large hedge funds accounted for at least one third of the assets managed by the industry. However, since the burst of the dotcom bubble, perhaps due to a reduction in appetite for risk, investors have been increasingly interested in lower volatility non-directional arbitrage strategies. According to Tremont Advisors, convertible arbitrage total market value grew from just $768 million in 1994 to $25.6 billion in 2002, an astonishing growth rate of 50% on average per annum.

Rather than using combinations of derivatives which you would expect to intuitively share the characteristics of a trading strategy's returns, this study innovates by creating a convertible arbitrage portfolio by combining financial instruments in a manner akin to that ascribed to practitioners who operate the strategy. The portfolio is created by matching long positions in convertible bonds, with short positions in the issuer's equity to create a delta neutral hedged convertible bond position which captures income and volatility. Returns on each individual position are calculated daily as follows.

Daily returns are then compounded to produce a position value index for each hedged convertible bond over the entire sample period. The delta neutral hedged positions are then combined into two convertible bond arbitrage portfolios, one equally weighted, the other weighted by market capitalisation of the convertible issuers' equity. To confirm that the portfolios have the characteristics of a convertible bond arbitrageur, the returns of the convertible bond arbitrage portfolio and the returns from two indices of convertible arbitrage hedge funds are examined in a variety of market conditions.

Table 1, taken from Hutchison and Gallagher (2005), presents the correlation coefficients between the monthly returns on the equally weighted convertible bond arbitrage portfolio (Equal Portfolio), the market capitalisation weighted portfolio (MC Portfolio), the CSFB Tremont Convertible Arbitrage Index (CSFB Tremont Convertible), the HFRI Convertible Arbitrage Index (HFRI Convertible), the Russell 3000, the Merrill Lynch Convertible Securities Index (ML Convertible Securities) and the VIX Index (VIX). The VIX index is an equity volatility index calculated by the Chicago Board Option Exchange. It is calculated by taking a weighted average of the implied volatilities of 8 30-day call and put options to provide an estimate of equity market volatility. As the CSFB Tremont data is unavailable prior to 1994, the correlation coefficients cover returns from January 1994 to December 2002 ^{2}.

The equal weighted portfolio, the market capitalisation weighted portfolio, the CSFB Tremont index and the HFRI index are all positively correlated with the Merrill Lynch convertible index. With the exception of the CSFB Tremont index they are also all positively correlated with equities. The equal weighted portfolio is positively correlated with the market capitalisation weighted portfolio, the CSFB Tremont index and the HFRI index over the entire sample period. Surprisingly, the market capitalisation weighted portfolio is not correlated with the CSFB Tremont index, although it is positively correlated with the HFRI index. Monthly returns on the VIX are negatively correlated with both the equal weighted portfolio and the market capitalization weighted portfolio indicating that they are both negatively correlated with implied volatility. Neither of the hedge fund indices has any correlation with the VIX. This is surprising as convertible bond arbitrage is a long volatility strategy.

**Table 1 - Hutchinson and Gallagher (2005)
Correlation between monthly convertible bond arbitrage returns and market factors**

This table presents correlation coefficients for monthly returns on the equally weighted (Equal Portfolio) and market capitalisation weighted (MC Portfolio) convertible bond arbitrage portfolios, the CSFB Tremont Convertible Arbitrage Index, the HFRI Convertible Arbitrage Index and market factor returns. The Russell 3000 is a broad based index of US equities. The Merrill Lynch Convertible Securities Index is an index of US convertible securities and the VIX is an equity volatility index calculated by the Chicago Board Option Exchange. It is calculated by taking a weighted average of the implied volatilities of 8 30-day call and put options to provide an estimate of equity market volatility.

A further analysis of correlations in different states of equity market return reported in Hutchinson and Gallagher (2005) indicates that the relationship between convertible arbitrage and equity market returns is non-linear. As discussed previously we are not the first authors to come to this conclusion. However, studies to date have been restricted to analysing relatively low frequency monthly returns data. To provide a closer examination of this non-linear effect we look in Table 2, taken from Hutchinson and Gallagher (2005), at the estimation of the market model using the equally weighted portfolio limiting the sample to those days when the equity risk premium is more than two and a half standard deviations form its mean. This represents a relatively infrequent seven trading days per year but from an investors perspective these may be the most important. Panel A looks at those days when the equity risk premium is at least two and a half standard deviations less than its mean. The explanatory power of the regression is higher than for the entire sample (adjusted R2 of 13.7% v. 1.9%) and the convertible bond arbitrage beta has again increased, to 0.137 from 0.048. Panel B of table 7 looks at those days when the equity risk premium is at least two and a half standard deviations greater than its mean and the results are striking. The explanatory power of the regression is high with an adjusted R2 of 11.9%, and the beta is -0.267, providing evidence of the negative relationship between convertible bond arbitrage and equity returns in extremely positive equity markets.

**Table 2 - Hutchinson and Gallagher (2005)
Regression of daily equally weighted convertible bond arbitrage returns
at market extremes**

This table presents results from the following regression of convertible bond arbitrage returns.

Where is the daily return on the equal weighted convertible bond arbitrage portfolio, is the daily return on the Russell 3000 stock index and is the daily yield on a three month treasury bill.

Panel A of the table presents results after restricting the sample to those days with excess market returns at least two and a half standard deviations less than their mean. Panel B presents results after restricting the sample to those days with excess market returns at least two and a half standard deviations greater than their mean.

The analysis of convertible bond arbitrage conducted in this research provides some useful evidence on the characteristics of this dynamic trading strategy. This evidence should be particularly interesting to any institution or individual considering an investment in convertible bond arbitrage.

**References**

*Agarwal, V. and Naik, N.Y., 2004. Risks and portfolio decisions involving hedge funds. Review of Financial Studies 17, 63-98.*

*BIS, 2004. BIS Quarterly Review. Bank of International Settlement Publication, March.*

*Fung, W. and Hsieh, D.A., 2000a. Measuring the market impact of hedge funds. Journal of Empirical Finance 7, 1-36.*

*Hutchinson, M. and Gallagher, L., 2005. Convertible Bond Arbitrage, Working Paper.*

**Footnote**

^{1}
Agarwal and Naik (2004) also document this feature of convertible arbitrage using monthly hedge fund asset values.

^{2}
Correlation coefficients were estimated for the entire sample period 1990 to 2002 for all variables excluding the CSFB Tremont data. There was no change in the sign or significance of any of the coefficients.