Download Risks in Hedge Fund Strategies: Case of Convertible Arbitrage

Risks in Hedge Fund Strategies: Case of Convertible Arbitrage
Vikas Agarwal
Georgia State University
William Fung
London Business School
Yee Cheng Loon
Georgia State University
and
Narayan Y. Naik
London Business School
JEL Classification: G10, G19
This version: May 04, 2004
____________________________________________
Vikas Agarwal and Yee Cheng Loon are from Georgia State University, Robinson College of Business, 35, Broad
Street, Suite 1221, Atlanta GA 30303, USA: e-mail: [email protected] (Vikas) and [email protected]
(Yee Cheng) Tel: +1-404-651-2699 (Vikas) +1-404-651-2628 (Yee Cheng) Fax: +1-404-651-2630. William Fung
and Narayan Y. Naik are from London Business School, Sussex Place, Regent's Park, London NW1 4SA, United
Kingdom: e-mail: [email protected] (William) and [email protected] (Narayan) Tel: +44-207-262-5050,
extension 3579 (Narayan) Fax: +44-207-724-3317. We gratefully acknowledge the support of Center for Hedge
Fund Research and Education at the London Business School. We are thankful to Otgontsetseg Erhemjamts and
Kari Sigurdsson for excellent research assistance. We are responsible for all errors.
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Risks in Hedge Fund Strategies: Case of Convertible Arbitrage
Abstract
Using data on Japanese convertible bonds and underlying stocks, we analyze the risk-return
characteristics of convertible arbitrage strategy. We conduct an asset-based style (ABS) analysis
to extract the common risk factors that are related to observable market prices. Our asset-based
style factors provide important insights into the nature of risks associated with investing in
convertible arbitrage hedge funds. We hypothesize that there are three primitive trading
strategies that serve as ABS factors to explain convertible arbitrage strategy – positive carry,
volatility arbitrage, and credit arbitrage. Using our Japanese convertible bond sample, we
compute the returns to these three primitive trading strategies to explain the basic risk-return
characteristics of convertible arbitrage strategy. Our results show that these ABS factors can
explain a significant proportion of the return variation of four popular convertible arbitrage
indexes. Furthermore, the positive carry and volatility arbitrage factors have greater statistical
significance than the credit arbitrage factor. This suggests that, in the Japanese convertible
market, explo itable pockets of market inefficiency were more likely to be found via the interest
rate and volatility channels. Overall, our findings have important implications for risk
management, portfolio construction, and benchmark design in the hedge fund industry. We are in
the process of (1) studying the explanatory power of these ABS factors for individual convertible
arbitrage hedge funds and (2) applying the same analysis to a sample of U.S. convertible bonds.
2
Risks in Hedge Fund Strategies: Case of Convertible Arbitrage
It is now well-established that hedge funds employ dynamic trading strategies with nonlinear option- like returns that cannot be captured by standard linear factor models as in Sharpe
(1992). 1 For some hedge fund strategies, researchers have directly modeled the non- linear
relationship between hedge fund returns and conventional asset markets in which hedge funds
operate. For example, Fung and Hsieh (2001) stud ied the risks of trend-followers using lookback
options on publicly traded stocks, bonds and commodity markets to capture the path-dependent,
long- gamma orientation of trend- following strategies Mitchell and Pulvino (2001) used naked
put options on equity index to mimic the deal risk inherent in diversified portfolios of announced
mergers of publicly traded companies.
Other researchers have also used alternative
combinations (long-short and nonlinear combinations as opposed to conventional long-only) of
conventional asset-class benchmarks to study the risks of different hedge fund strategies. These
include Gatev, Goetzmann, and Rouwenhorst (1999) who investigated the risks of pairs trading
strategy and Fung and Hsieh (2002a, 2004) who studied the risks of fixed- income and equity
long/short hedge funds using nonlinear combinations of the underlying assets.
Our paper is in the spirit of this stream of research. The objective of our paper is to
characterize the risk-return characteristics of convertible arbitrage (“CA” for short) strategies.
Industry sources estimate the global convertible bond market to be $434 billion in 2000 (Brown,
2000).2 This highlights the growing importance of studying convertible arbitrage strategy as well
as the growth in opportunities for the CA hedge fund managers.
1
See Fung and Hsieh (1997) and Agarwal and Naik (2004).
The paper also documents that the estimated number of convertible arbitrage hedge funds grew from less than 50
in 1995 to over 120 by the end of 2000. Further, as of June 2000, out of a total manager universe of about 2,200
hedge funds managing approximately $205 billion, about $10 billion (or 5%) is invested in convertible arbitrage
2
3
The nonlinear technology used in this study follows the models proposed in Fung and
Hsieh (2001, 2002b). The idea is to construct Asset-Based Style (“ABS” for short) factors that
would capture the return characteristics of convertible arbitrage hedge funds.
The main
advantage of an ABS approach is that it provides an intuitive explanation of risks (because it is
expressed in terms of familiar conventional asset classes such as equities and bonds) associated
with a particular hedge fund strategy. Further, the ABS factors have attractive properties of being
investable and transparent (Fung and Hsieh, 2002b). An ABS factor is a portfolio of
conventional assets defined by a simplified proxy of a particular class of hedge fund strategies or
style. One that captures the strategy’s essence but not its details—Fung and Hsieh (2001)
referred to these as primitive trading strategies. This conception is founded on the desire to
extend Sharpe’s (1992) style analysis, which was originally designed for long-only trading
strategies, to hedge funds. It is intuitively obvious that these ABS factors are likely to be
dynamic, nonlinear combinations of conventional assets mimicking hedge funds investment
styles.
The construction of primitive trading strategies begins with the identification of the risks
characteristics of the underlying asset markets and the ways they can be managed. The starting
point here is to recognize the three basic sources of risk (and therefore potential sources of value)
of a convertible bond—the interest rate component, the credit component, and the equity option
component. 3 Next, we need to hypothesize on the way a CA goes about neutralizing parts of the
funds. Over this period, out of total estimated asset flow of approximately $57.5 billion, convertible arbitrage
strategies attracted $5.25 billion (or 9%) of the flows.
3
A convertible bond is a fixed-income instrument which allows the holder to exchange the instrument for the
common stock of the issuing firm. The exchange is governed by provisions stipulated in the convertible security’s
indenture (Dialynas, Durn, and Ritchie, 2000). For example, we can think of the convertible bond as an option-free
bond plus a call option on the issuer’s stock. Typically, convertible securities have other embedded options. The
security may be callable by the issuer and/or putable by the security holder. For details on the valuation of
convertible securities, see Ingersoll (1977), Brennan and Schwartz (1977), and Brennan and Schwartz (1980).
4
convertible bond’s risks that are fairly priced leaving himself with the highest expected reward
for placing capital at risk. 4
A convertible arbitrage strategy (“C AS” for short) usually involves buying a portfolio of
convertible securities and hedging equity risk by short-selling the underlying stock. The amount
of stocks sold short is a function of the conversion ratio, delta of the embedded call option, and
the sensitivity of delta to changes in stock price, i.e., gamma. In addition to hedging equity risk,
some fund managers may choose to hedge (in part or fully) other risks associated with investing
in a convertible bond (“CB” for short) such as credit risk, interest rate risk, and volatility risk.
Lhabitant (2002) attributes the rationale for following a CAS to the fact that CBs may be
mispriced due to illiquidity, small issue size, and complexities associated with the valuation of
these hybrid securities, whose characteristics keep changing over time. While the precise nature
of potential mispricing of a CB may arise from a myriad of sources, in general they manifest
themselves in how the risk factors of a CB are priced. In other words, if market friction is such
that a CB becomes mispriced, it is likely that mispricing will manifest itself in one or more of the
three key components—the implied interest rate, the implied credit spread, and the implied
option price. Consequently, we conjecture that there are three primitive trading strategies in
CA—positive carry, credit arbitrage, and volatility arbitrage. We label these as PCASi, PCASc,
and PCASv respectively. These form the ABS factors that we use to explain the returns of
convertible arbitrage strategy.
The positive carry strategy, PCASi, is designed to create a delta neutral portfolio with
positive interest income comprising a long position in the CB, with an appropriate hedge against
4
We are implicitly assuming that there is no free lunch.
Although some hedge funds, typically the larger ones, segregate exposures to each of the two CB markets into
separate funds, this is not always the case and can run into inefficiencies in terms of allocating risk capital.
8
5
equity risk. The credit arbitrage strategy, PCASc, is designed to create a long credit spread
position while retaining neutrality to interest rate and equity risks. It is designed to capture value
from over/under priced credit risk inherent in the CBs. The volatility arbitrage strategy, PCASv,
seeks to exp loit underpricing in the embedded option in CBs by actively managing a deltaneutral, but long gamma, position in the underlying equity whilst maintaining a neutral position
to interest rate and credit risks.
All three primitive trading strategies are variations from a basic theme where CAs are
essentially short-term providers of liquidity to the CB market. In doing so, CAs seek to capture a
liquidity premium for managing the risk inherent in these PCASs. The concept is predicated on
an illiquid, incomplete CB market where CAs are rewarded for managing an inventory of CBs
and indirectly acting as market makers to issuers of CBs. As the paper develops, it will also
become apparent that the construction of these primitive trading strategies implicitly assumes an
incomplete securities lending market for CBs.
We begin our investigation using daily data on 593 Japanese convertible bonds and the
underlying stocks. We are currently in the process of repeating our analysis for US CBs as some
of the convertible hedge funds invest in both the US and Japanese CB markets and can
dynamically allocate capital from one market to another. 8 Extending our analysis to both markets
allows us to investigate important microstructure differences of the two markets.
To validate the PCASs, we compare the returns of the PCASs to the monthly returns on
CA hedge fund index from Hedge Fund Research (HFR), TASS, Morgan Stanley Capital
International (MSCI), and Center for International Securities and Derivatives Markets (CISDM).9
The results will be extended to include individual convertible arbitrage hedge funds utilizing a
9
For details on index construction methodology, please refer www.hfr.com (HFR), www.hedgeindex.com (TASS),
http://www.msci.com/hfi/index.html (MSCI) and http://cisdm.som.umass.edu/indices/activeindices.shtml (CISDM).
6
comprehensive hedge fund database created by merging HFR, TASS, MSCI, and CISDM
databases.
Using our Japanese convertible bond sample, we create ABS factors to capture the basic
risk-return characteristics of the convertible arbitrage strategy. Our results show that the ABS
factors can explain between 9% and 24% of the return variation of the four CA hedge fund
indexes. Furthermore, the positive carry and volatility arbitrage factors have greater statistical
significance than the credit arbitrage factor. Thus, to the extent that convertible arbitrage hedge
funds operate in the Japanese market, exploitable pockets of market inefficiency were more
likely to be found via the interest rate and volatility channels. The credit arbitrage factor, PCASc,
does not appear to apply to the Japanese CB market.
Why doesn’t credit risk offer much exploitable arbitrage opportunities in Japan? We
believe there are three possible reasons. The first one relates to the credit quality of convertible
issuers. Japanese CB issues tend to be larger and are often issued by blue-chip companies. In
contrast, US convertibles are often issued by smaller companies of medium credit quality
(mostly B-grade paper). 10 Hence, it is likely that credit risk is less critical in valuing Japanese
CBs. We are in the process of repeating our analysis with US convertibles that will allow us to
compare and contrast the risk characteristics for these two biggest convertible markets. 11 The
second reason for insignificant exposure to conventional credit risk measures such as yield
spreads between corporate and comparable default- free bonds may be a systematic consequence
of the corporate loan market in Japan. Corporate borrowing in Japan is dominated by bank
lending and consequently has a much less developed corporate bond market. Consequently, the
10
See Soldofsky (1971), Altman (1989), and Guedos and Opler (1994) for empirical risk characteristics of
convertibles and their issuers. Stein (1992) discusses this issue where a separating equilibrium corresponds to only
medium-quality firms issuing convertible debt.
11
Japanese domestic convertible bond market is the largest convertible market in the world with a $160 billion face
amount compared to $130 billion for the US, representing about 38% and 31% respectively of the global convertible
bond market size of $421 billion as of October 1999 (Source: http://www.gabelli.com/news/ahw_102299.html).
7
treatment of nonperforming loans revolves around bilateral agreement between the bank lenders
and the corporate borrower.
With a publicly traded corporate debt market, the need for
standardization in the treatment of default has led to defined legal procedures dealing with
bankruptcy such as Chapter 11 filings (and its variants) in the US. Koshi, Kashyap, and
Scharfstein (1990) present evidence that the bank-oriented financial structure along with crossshare ownership in Japan results in lower costs of financial distress. This, together with a lack of
a price discovery mechanism like a public market, implies that CA funds may find it more
efficient to hedge credit risk by holding a short position in the stock. This may result in a larger
short stock position to cope with.
12
The rest of the paper is organized as follows. Section I describes the data. Section II
provides a description of our methodology. Section III reviews the results from validating our
ABS factors and finally, we conclude the paper in Section IV.
I.
Data
Our sample consists of daily closing prices of the convertible bonds and their underlying
stocks for 593 convertible bonds issued by Japanese companies. All bonds are denominated in
Japanese Yen. We also obtain contractual information of each bond including the conversion
price, maturity date, call provision, and dividend yield on the underlying stock. Table I provides
descriptive statistics of our sample of Japanese convertible bonds. In Panel A of Table I, we
observe that the number of issuers decreases as the number of issues per issuer increases. This
suggests that there are more firms selling a single issue of convertible bonds than there are firms
12
Hedging equity risk by shorting the stock does result in hedging a portion of the credit risk because as credit
spreads widen, stock prices generally decline.
8
selling multiple issues of bonds. In total, there are 462 Japanese firms with 593 separate issues of
convertible bonds.
In Panel B of Table I, we provide summary statistics on issue size. The average
convertible issue is close to 48 billion Yen while the median issue size is 15 billion Yen. The
distribution of issue size is positively skewed, suggesting that our sample consists mostly of
small and medium size issues coupled with a few relatively large issues. This is consistent with
the state of the Japanese convertible bond market (Lhabitant, 2002). In the following section, we
describe the methodology for creating our ABS factors i.e., three primitive convertible arbitrage
strategies – positive carry, credit arbitrage, and volatility arbitrage.
II. Methodology
To construct the first ABS factor, the positive carry strategy (PCASi), we define a simple
convertible bond trading rule and implement it for our convertible bond sample. In other words,
we simulate the positive carry strategy on a daily basis for our bond sample. This simulation
produces a monthly return series which embodies the risk-return characteristics of the positive
carry strategy. To create the other two ABS factors, credit arbitrage and volatility arbitrage
strategies, we adopt a slightly different approach. We first select bonds for the credit and
volatility arbitrage strategies based on their moneyness, which we measure using the bond’s
parity/conversion value. Parity is simply the market value of the shares obtained if we convert
the bond immediately. Once the bonds are identified, we create an equally-weighted bond
portfolio and use the equally-weighted portfolio returns as the basis for the ABS factor. Although
the specifics differ, there is a common theme underlying the creation of all the ABS factors. For
each strategy (PCASi, PCASc, PCASv), we have to specify a rule to identify bonds appropriate
for that strategy.
9
A.
Positive carry strategy, PCASi
PCASi entails creating a delta neutral portfolio when the portfolio has a positive carry,
holding the portfolio as long as carry is positive and liquidating the portfolio when the carry
becomes negative. The delta neutral position is effected by short selling an appropriate amount
of the underlying stock (usually based on the delta of the embedded call option). As discussed
above, we adopt a simulation approach in creating the ABS factor for PCASi. This means that
we define a simple bond trading rule based on the positive carry condition and apply the rule to
each individual bond on a daily basis.
The best way to describe this approach is to put ourselves in the shoes of a convertible
arbitrage hedge fund manager who wishes to implement PCASi. We assume that this manager
has sufficient capital to buy the convertible bond. This implies that he is self- financing his
position. Hence, the borrowing cost acceptable to him would simply be the discount rate, the rate
at which he would be able to invest in an alternative scenario. With the discount rate as the
opportunity cost of capital, we can compute carry on date t as:
Carry t
= current income on delta neutral position − financing cost on delta netural position
(1)
= ( Bt × cyt ) + ( deltat × Bt × ( DISCt − s − dyt ) ) − ( Bt × DISCt )
where
Bt is the value of the convertible bond using the closing price of day t,

1

Bt =  midpricet +  × spread t   × 0.01 × F .
2


10
(2)
Midpricet is the average of the bid and ask prices13 ,
Spreadt is the average (or maximum) of all ava ilable positive bid-ask spreads for day t,14
F = unit face value of the bond, 15
cy t is the convertible bond’s current yield on date t,
deltat is the convertible bond’s delta on date t,
DISCt is the daily discount rate on Japanese Yen on date t,
dy t is the compensation to the stock lender for dividend paid on the stock, as a proportion of the
bond price, Bt , and
s is the lending spread.
The first term of equation (1), ( Bt × cy t ) , represents the income from the coupon interest
while the second term,
( delta × B × ( DISC
t
t
proceeds from the short-stock sale.
t
−s − dyt ) ) , represents interest income on the
( DISCt −s − dyt )
is the lending rate enjoyed by the
arbitrageur, which is lower than the discount rate. Finally, ( Bt × DISC t ) represents the financing
cost of the convertible bond position. This financing cost is dynamic because the bond price, Bt ,
changes from day to day.
The condition for starting the carry strategy is Carryt > 0 , which is equivalent to the
condition: 16
cyt + ( deltat × ( DISCt −s − dyt ) ) − DISCt > 0
(3)
13
If only the bid (ask) price is available, we set midpricet equal to bid (ask) price. The bid and ask prices are full
(dirty) prices, i.e., prices which incorporate accrued coupon income.
14
In our data, midpricet and spreadt are stated as percentages of par value (e.g., 105 means 105.0% of par value).
1

× spreadt  , is included to
2


Thus, we need to multiply by 0.01 to restate the bond value in Yen. The half-spread, 
account for the transaction costs incurred in purchasing the bond.
15
F denotes the unit value (par value) of the bond. If the face value of a bond is 1 million Yen, then F = ¥ 1,000,000.
Thus, Bt gives the Yen value of the bond position at the close of day t.
16
Obtained by setting (1) to be greater than 0 and factoring out Bt from both sides of the inequality.
11
We will refer to (3) as the boundary condition. The position is maintained as long as the
boundary condition holds. We end the position when the boundary condition is violated, i.e.,
when
cyt + ( deltat × ( DISCt −s − dyt ) ) − DISCt ≤ 0
(4)
To recap, equations (3) and (4) constitute our simple convertible bond trading rule to capture the
essence of the positive carry strategy. Next, we discuss the computation of returns on our
simulated strategy.
A.1.
Computation of marked-to-market portfolio value and total return
The first issue is the definition of capital base for the carry trade. This is an essential task
because a wrong capital base assumption can result in misleading inferences about the strategy’s
profitability. For the positive carry strategy, the capital base depends on the short stock position
and the convertible bond position.
The value of the short stock position is,
SSt = deltat × Bt
(5)
where deltat is computed assuming the stock pays a continuous dividend yield [see Merton
(1973)). 17 Hence, SSt represents the Yen value of the short stock position at the close of day t.
Combining (2) and (5) gives the marked-to-market portfolio value for day t,
Vt = Bt - SSt
17
(6)
The volatility estimate used in computing delta t is the historical volatility estimated over the past 10 trading days.
We acknowledge that delta t can be computed using a host of alternative option pricing models. We are in the
process of designing a Monte Carlo simulation to check whether our results are robust to different delta values.
12
We assume that the hedge fund manager collateralizes the trade by paying for the bond
and using the bond value as collateral for the short stock position. Now, the next issue the
manager faces is how to raise this capital on a daily basis. We consider two scenarios. In the first
scenario, the manager has perfect foresight in terms of capital requirements and can raise capital
equal to the maximum bond value during the period of the trade. An example might be the best
way to illustrate this treatment of the capital base. Suppose the convertible arbitrageur starts the
carry strategy at the end of day 0, i.e., carry0 > 0. Further suppose that the trade is terminated at
the end of day T, i.e., carryT = 0. Changes in the capital requirement are determined based on
prices observed at the close of day 1, day 2 through to day T-1 (The capital available at the end
of day T-1 is used to support the position during day T, the last day of the trade). During the time
between the end of day 0 and the end of day T-1, the bond price fluctuates and the capital
requirement changes. ‘Perfect foresight’ means that at the inception of the trade (end of day 0),
the manager obtains capital equal to the maximum bond price occurring between the end of day
0 and the end of day T-1. In this scenario, the manager only raises capital once, but he raises
sufficient capital to support the trade throughout its entire life. This also means that the capital
base remains unchanged throughout the life of trade. Therefore, for a positive carry trade
initiated at the end of day 0 and terminated at the end of day T, the capital base, Cmax is
Cmax = Max ( Bt | t = 0,1,2,..., T − 1) = Max ( B0 , B1 ,..., BT −1 )
(7)
where B0 refers to the bond price paid by the arbitrageur to establish the position, B1 is the bond
price at the end of the first day of the trade and so on.
13
In this scenario, the manager will have excess capital on days when Bt is less than Cmax.
We assume that the manager is able to invest this excess capital at the discount rate. Thus, R1,t,
the total return for day t is
marked - to - market return + interest on short sale proceeds + interest on excess capital
capital base
(V − Vt−1 ) + ( SSt −1 × (DISC t − s − dy t )) + max( 0, Cmax − Bt−1 ) × DISCt
= t
Cmax
R1, t =
(8)
In the second scenario, the manager is able to raise enough capital on a daily basis. This
means that the manager can raise capital “just- in-time” and does not have to “store” excess
capital. In this “just- in-time” scenario, the manager only needs the bond value on day t as the
required capital. This implies that
Ct = Bt
(9)
In this scenario, the manager will have no excess capital and hence, R2,t , the total return
for day t is
marked - to - market return + interest on short sale proceeds
capital base
(V − Vt−1 ) + ( SSt −1 × ( DISCt − s − dy t ))
= t
Ct−1
R2,t =
(10)
R1,t and R2,t represents the daily return on a positive carry trade in the two scenarios.
Since we implement the positive carry simulation for all eligible bonds in our sample, we create
and maintain an equally-weighted portfolio of positive carry trades over the sample period. The
composition of the portfolio changes as old trades are liquidated and new trades are put on. Our
14
positive carry portfolio produces two monthly return series: Carry1 and Carry2. Carry1 is the
return series derived from the “perfect foresight” capital base assumption of scenario 1 while
Carry2 is derived from the “just- in-time” capital base assumptio n of scenario 2. These two return
series are the ABS factors for PCASc. In the following section, we describe the construction of
the ABS factor for PCASc.
B.
Credit arbitrage strategy, PCASc
The credit arbitrage strategy is designed to capture value from the mispricing of credit
risk inherent in CBs. We conjecture that the mispricing of credit risk is most likely for an out-ofthe-money (OTM) convertible, which we define as a bond whose parity is equal to or less than
20% of par value. On each day, we form an equally-weighted portfolio of bonds having parity
values equal to or less than 20% of par (henceforth the “credit bond portfolio”). Because we
apply the selection rule on a daily basis, the bond portfolio is reformed every day. The formation
method implicitly rebalances the portfolio so that it retains the same basic characteristic with
respect to moneyness. Since a bond’s parity value can change with firm-specific and market
conditions, the bond portfolio’s composition do change over time.
In computing the daily parity of a bond, we adjust the underlying stock price for the
effect of dividends. The adjustment is based on Merton (1973) and basically reduces the stock
price by the present value of the stock’s expected dividends. Merton’s formulation uses a
continuous dividend yield. To estimate such a yield, we take the time series average of the
underlying stock’s daily dividend yield and convert the average into a continuous dividend yield.
Thus, the parity value of a bond is computed as:
15
 conversion ratio × St −1 × exp( −d × t ) 
Parity (conversion value) as a % of par = 
 ×100 (11)
par value


where conversion ratio is the number of shares into which the bond can be converted, St-1 is the
previous day’s closing price for the underlying stock, d is our estimate of the continuous
dividend yield (stated on an annual basis), t is the time to maturity of the convertible bond (in
years) and par value is the convertible bond’s face value. For each bond, we have a single
estimate of d. If the conversion ratio and par value remain constant during our sample period,
then parity will change due to (a) a shortening of the bond’s time to maturity and (b) changes in
the underlying stock price. Parity is used to describe the “moneyness” of the convertible bond.
As the underlying (dividend-adjusted) stock price increases, the bond’s conversion option
becomes more valuable and parity increases.
Since each convertible bond is associated with an underlying stock, once the bond
portfolio is formed, we are also able to form the equally-weighted portfolio of underlying stocks.
We then compute the daily returns of both the bond and stock portfolios. Recall that PCASc
hedges against both equity and interest rate risk. Thus, to construct the ABS factor PCASc, we
need to identify proxies for both equity and interest rate risks. For the equity risk proxy, we use
the daily returns of the equally- weighted stock portfolio. For the interest rate risk proxy, we use
the daily returns of the Japanese 3-5 year Gove rnment Bond index. We use a regression-based
approach to estimate the hedge ratios for equity and interest rate risks. Specifically, we estimate
the following OLS regression:
BRET tc = γ 0 + γ 1 EQ t + γ 2 IR t + η t
16
(12)
where BRET tc is the day t return on the credit bond portfolio, EQt is the day t return on the
equally- weighted portfolio of underlying stocks, IRt is the Japanese 3-5 year Government Bond
total return index (from Datastream) and η t is the random error term.
We present the results from the regression in equation (12) in the second column (labeled
“Credit”) of Table II. The equity factor is positive and highly significant. This is consistent with
the fact that a convertible bond ’s value depends on the underlying stock. The interest rate factor
is inversely related to the bond portfolio return. This reflects the inverse relationship between
bond values and interest rates. However the interest rate factor is not statistically significant.
With the results from Table II, we proceed to construct our second ABS factor - credit arbitrage
strategy.
The ABS factor, PCASc, is simply the time series of daily returns obtained from the
following equation:
Credit t = BRET tc − γˆ1 EQ t − γˆ 2 IR t
(13)
where Credit t is the ABS factor, PCASc, on day t, BRET tc , EQt , and IRt are as defined above.
γˆ1 and γˆ 2 are the OLS estimates from estimating equation (12). γˆ1 and γˆ 2 serve as the “hedge
ratios” for equity and interest rate risks respectively. We compound the daily series of Credit t to
obtain a monthly series of our ABS factor, PCASc.
C.
Volatility arbitrage strategy, PCASv
17
The creation of our third ABS factor, PCASv, follows essentially the same steps as the
creation of the second ABS factor, PCASc. For the sake of brevity, we only discuss construction
details specific to PCASv. Volatility arbitrage requires a convertible bond with a large positive
gamma. A convertible bond’s gamma is the highest when it is at-the- money (ATM) [see
Woodson (2002, p.130) and Calamos (2003, Chapter 5)]. Thus, ATM convertibles are the most
likely candidates for volatility arbitrage trades. Accordingly, the equally- weighted bond portfolio
for PCASv (henceforth “gamma bond portfolio”) is made up of convertible bonds having parity
values within 90% and 110% of par value. It should be noted that with a change in the bond
selection rule, the equally-weighted portfolio of underlying stocks also changes. As a result, we
are using two different equity risk factors in estimating the equity hedge ratios for the credit and
gamma bond portfolios.
The difference between PCASc and PCASv is that the latter hedges against credit risk, in
addition to equity and interest rate risks. For the credit risk proxy, we use the daily change in the
Japanese corporate bond credit spread. The hedge ratios for PCASv are estimated via the
following OLS regression:
BRET tv = γ 0 + γ 1 EQt + γ 2 IR t + γ 3CRt + η t
(14)
where BRET tv is the day t return on the gamma bond portfolio, EQt is the day t return on the
equally- weighted portfolio of underlying stocks, IRt is the Japanese 3-5 year Government Bond
total return index (from Datastream), CRt is proxy for credit risk, measured as the daily change
in the spread between the Japanese corporate bond yield and the Japanese long term government
bond yield (from Datastream) and η t is the random error term.
18
We present the results from regression in equation (14) in the third column (labeled
“Gamma”) of Table II. The equity factor is positive and highly significant, as in the credit bond
regression. The interest rate and credit factors are also positively related to the gamma bond
portfo lio. However, both factors are not statistically significant. The results suggest that ATM
bonds are behaving more like the underlying stocks. As a result, the interest rate and credit risks
have lesser impact on the values of the convertible bonds. With the results from Table II, we
proceed to construct our third ABS factor of volatility arbitrage.
The ABS factor, PCASv, is simply the time series of daily returns obtained from the
following equation:
Gammat = BRET t v − γˆ1 EQ t − γˆ 2 IR t − γˆ 3 CRt
(15)
where Gammat is the ABS factor, PCASv, on day t, BRET tv , EQ,
t IR t and CRt are as defined
above. γˆ1 , γˆ 2 and γˆ 3 are the OLS estimates from the estimating equation (14). γˆ1 , γˆ 2 and γˆ 3
serve as the “hedge ratios” for equity, interest rate, and credit risks respectively. We compound
the daily series of Gammat to obtain a monthly series of our ABS factor, PCASv.
Before we validate the ABS factors, it’s instructive to study the characteristics of these
factors. Table III shows that, on an average day, the positive carry portfolio has about 10 times as
many bonds as the gamma and credit bond portfolios. The carry portfolio has 233 bonds on an
average day while the gamma and credit portfolios have 26 and 23 bonds respectively. This
implies that the positive carry strategy can be implemented for a wide group of bonds while the
credit and volatility arbitrage strategies tend to be very selective. The carry and credit portfolios
have higher current yields than the gamma portfolio. This makes sense because current yield is
19
positively related to carry, thus it’s not surprising that the positive carry portfolio has a relatively
high current yield. For the credit portfolio, the current yield is high because of the low bond
prices associated with out-of-the- money convertible bonds. The average parity for the gamma
portfolio is 98% while the average parity for the credit portfolio is 9.2%. These figures are
consistent with our bond selection criteria. The gamma portfolio has parity values ranging
between 90% and 110% of the par value while the credit portfolio has parity values not
exceeding 20% of the par value. The credit portfolio’s average daily delta is 0.13, consistent with
the fact that it consists of out-of-the- money convertible bonds. By the same token, the gamma
portfolio has the highest average daily delta among the three portfolios. This is because it
consists of at-the- money convertible bonds, which behave more like stocks.
The monthly return characteristics of the ABS factors are provided in Table IV. Looking
at the first two columns, we see that Carry2 has a higher mean return than Carry1. This implies
that the “just- in-time” capital base assumption of Carry2 results in a smaller capital base and
hence higher average monthly return. However, the “just- in-time” assumption also makes Carry2
a more volatile series than Carry1. Carry2 has a higher standard deviation and experience larger
swings than Carry1. It’s interesting to note that Gamma (PCASv ABS factor) loses money on
average. This reminds us that convertible arbitrage strategies are no t riskless. Another interesting
point is that Gamma has the lowest volatility among the three ABS factors. Turing to the credit
ABS factor, we see that Credit has the highest mean monthly return but is not as volatile as either
Carry1 or Carry2. Finally, the distributions of Carry1 and Carry2 are roughly symmetrical while
the distributions of Gamma and Credit are skewed to the right.
Having constructed our three ABS factors, namely positive carry, credit arbitrage, and
volatility arbitrage (or gamma), we next examine how well these ABS factors explain the returns
of the various convertible arbitrage indexes.
20
III. Validating the ABS factors: Convertible Arbitrage Hedge Fund Indexes
It is possible that the convertible arbitrage manager may choose to use one or more of the
three primitive trading strategies (or ABS factors) that we identify earlier. Hence, in order to
examine how well each ABS factor explains the returns of the convertible arbitrage indexes, we
allow for the possibility of one or more of these factors to explain the returns of the different
indexes and estimate the following OLS regressions:
Model 1: CAIt = α + β 1Carry t + ε t
(16)
Model 2: CAIt = α + β1Carry t + β 2 Gammat + ε t
(17)
Model 3: CAIt = α + β 1Carry t + β 2 Gammat + β 3 Credit t + ε t
(18)
where CAIt is the convertible arbitrage index’s month t return, Carryt is month t return for the
PCASi ABS factor, Gammat is the month t return for the PCASv ABS factor, and Credit t is the
month t return for the PCASc ABS factor. α , β1 , β 2 , and β3 are the regression parameters and ε t
is the error term. For CAIt, we use the convertible arbitrage indexes from four different sources:
TASS, HFR, MSCI, and CISDM.
Recall from section II that we have two versions of Carryt : Carry1 t is the month t return
corresponding to the assumption of “perfect foresight” capital base while Carry2 t is the month t
return corresponding to the assumption of “just-in-time” capital base. Thus, Models 1, 2, and 3
are estimated twice: once for Carry1 t and once for Carry2 t . Table V (VI) presents the regression
results when Carry1 t (Carry2 t ) is the ABS factor for PCASi.
21
The results in Tables V and VI are very similar. For brevity, we will focus our discussion
on Table V, the regression results with Carry1 t as the carry ABS factor.Carry1 t is positively and
significantly related to all four indexes. However, on its own, Carry1t has low explanatory power.
The adjusted R2 for Model 1 ranges from 1.7% (HFR) to 5.9% (CISDM). These two
observations imply that convertible arbitrageurs seem to pursue PCASi, in conjuction with other
trading strategies. The results for Models 2 and 3 support this view.
Model 2 uses both the positive carry and volatility arbitrage strategies to explain
variations in the index returns. Carry1 t continues to be positive and significant for TASS, MSCI,
and CISDM while Gammat is positively and significantly related to HFR, MSCI, and CISDM.
More importantly, Model 2 achieves higher explanatory power than Model 1 for all four indexes.
The adjusted R2 for Model 2 ranges from 6.3% (TASS) to 23.5% (CISDM).
Finally, Model 3 employs all three ABS factors as regressors. Moving from Model 2 to
Model 3, the signs and statistical significance of Carry1 t and Gammat remain intact. Carry1 t
remains positively and significantly related to TASS, MSCI, and CISDM; Gammat is still
positively and significantly related to HFR, MSCI, and CISDM. The values of the estimated
coefficients remain largely similar. These relatively stable patterns suggest that the positive carry
and volatility arbitrage strategies capture at least some of the risk-return characteristics of
convertible arbitrage. Turning to the credit ABS factor, we note that Creditt is significant only for
HFR and TASS. Compared to Model 2, Model 3 yields higher adjusted R2 for TASS, HFR, and
MSCI. However, there is a slight decrease in the adjusted R2 value for CISDM. This indicates
that adding the credit ABS factor does not help in understanding the CISDM convertible
arbitrage index.
In all three models, the intercept term remains positive and highly significant. Equally
apparent is that fact that our models can explain at most 24% of return variations at the index
22
level. These results should be interpreted with a couple of caveats. First, our ABS factors are
constructed using only Japanese convertible bonds. Thus, these factors represent only one piece
of the convertible arbitrage puzzle. We are currently extending our study to construct similar
ABS factors using U.S. convertible bonds. If convertible arbitrage managers dynamically
allocate capital in these two major convertible bond markets, one would hope that a combination
of ABS factors from the US and Japanese markets will be better in expla ining the risk-return
characteristics of convertible arbitrageurs. Second, convertible arbitrage indexes are an
amalgamation of different convertible arbitrage funds. To gauge the explanatory power of our
ABS factors, we need to test our models on the returns of individual convertible arbitrage funds.
We are in the process of conducting our analysis at the individual fund level.
IV. Concluding Remarks
In this paper, we analyze the risk-return characteristics of convertible arbitrage strategy
by using daily data on Japanese convertible bonds and the underlying stocks. We contribute to
the extant literature by conducting an asset-based style (ABS) analysis to extract the common
risk exposures of convertible arbitrage strategy followed by large number of hedge funds. Our
asset-based style factors provide important insights into the nature of risks associated with
investing in convertible arbitrage hedge funds.
We hypothesize that there are three primitive trading strategies or ABS factors associated
with convertible arbitrage strategy – positive carry, volatility arbitrage, and credit arbitrage.
Using our sample of Japanese convertible bonds and underlying stocks, we construct these three
ABS factors to capture the basic risk-return characteristics of the convertible arbitrage strategy.
Our results show that these ABS factors can explain between 9% and 24% of the return variation
23
of four popular convertible arbitrage indexes. Furthermore, the positive carry and volatility
arbitrage factors have greater statistical significance than the credit arbitrage factor. This
suggests that, in the Japanese convertible market, exploitable pockets of market inefficiency
were more likely to be found via the interest rate and volatility channels.
To the extent that convertible arbitrage funds operate globally, ABS factors based on a
single market will have limited explanatory power. Thus, we are in the process of extending our
analysis to a sample of U.S. convertible bonds. We hope that the ABS factors from the Japanese
and U.S. convertible markets will provide a more complete picture of the risk-return
characteristics of convertible arbitrage. Further, we intend examining how well these ABS
factors explain the returns of individual convertible arbitrage hedge funds. Such an analysis will
shed light on the importance and validity of asset-based style analysis of convertible arbitrage
strategy and its implications for extending to other hedge fund strategies.
*** *** ***
24
References
Agarwal, Vikas, and Narayan Y. Naik, 2004, “Risks and Portfolio Decisions involving Hedge
Funds,” Review of Financial Studies 17(1) , 63-98.
Altman, Edward I., 1989, “The Convertible Debt Market: Are Returns Worth the Risk?,” Financial
Analysts Journal, 45.
Brennan, Michael J., and Eduardo S. Schwartz, 1977, “Convertible Bonds: Valuation and Optimal
Strategies for Call and Conversion,” Journal of Finance 32, 1699-1715.
Brennan, Michael J., and Eduardo S. Schwartz, 1980, “Analyzing Convertible Bonds,” Journal of
Financial and Quantitative Analysis 15, 907-929.
Brown, William Anthony, 2000, “Convertible Arbitrage: Opportunity & Risk,” Tremont White Paper,
November 2000.
Buchan, Jane, 1997, “Convertible Bond Pricing: Theory and Evidence”, Doctoral Dissertation,
Harvard University.
Calamos, Nick, 2003. Convertible arbitrage : insights and techniques for successful hedging (John
Wiley, Hoboken, N.J.).
Carayannopoulos, Peter, and Madhu Kalimipalli, 2003, “Convertible Bond Prices and Inherent
Biases,” forthcoming Journal of Fixed Income.
Dialynas, Chris P., Sandra Durn, and John C. Jr. Ritchie, 2000, “Convertible securities and their
investment characteristics,” in Frank J. Fabozzi, ed.: The Handbook of Fixed Income Securities
(Mc-Graw Hill, New York).
Fung, W., and D.A. Hsieh, 1997, “Empirical characteristics of dynamic trading strategies: The case
of hedge funds,” Review of Financial Studies, 10, 275-302.
Fung, W., and D.A. Hsieh, 2001, “The Risk in Hedge Fund Strategies: Theory and Evidence from
Trend Followers,” Review of Financial Studies, 14(2), 313-341.
Fung, W., and D.A. Hsieh, 2002a, “Risk in Fixed-Income Hedge Fund Styles,” Journal of Fixed
Income, 12(2) , 6-27.
Fung, W., and D. A. Hsieh, 2002b, “Asset-Based Style Factors for Hedge Funds,” Financial Analysts
Journal, 58(5), 16-27.
Fung, W., and D.A. Hsieh, 2004, “Extracting Portable Alphas from Equity Long-Short Hedge
Funds,” Working Paper, Duke University and London Business School.
Gatev, Evan G., William N. Goetzmann, and K. Geert Rouwenhorst, 1999, “Pairs Trading:
Performance of a Relative Value Arbitrage Rule,” NBER Working Paper Series w7032.
Getmansky, Mila, Andrew W. Lo, and Igor Makarov, 2003, “An Econometric Model of Serial
Correlation and Illiquidity in Hedge Fund Returns,” forthcoming Journal of Financial Economics.
Guedos, Jose, and Tim Opler, 1994, “Determinants of the Maturity of New Corporate Debt Issues,”
Working Paper.
Ingersoll, Jr., Jonathan E., 1977, “A contingent-claims valuation of convertible securities,” Journal of
Financial Economics 4, 289-321.
Kang, Jun-Koo, and Yul W. Lee., 1996, “The Pricing of Convertible Debt Offerings”, Journal of
Financial Economics, 41(2), 231-248.
King, Raymon Dean,1980, “An Empirical Examination of a Contingent Claims Valuation Model
Applied to Convertible Bonds”, Doctoral Dissertation, University of Oregon.
Koshi, T., Anil Kashyap, and David Scharfstein, 1990, “The role of banks in reducing the costs of
financial distress in Japan,” Journal of Financial Economics 27(1) , 67-88.
Lhabitant, François-Serge, 2002, “Hedge funds: myths and limits,” Wiley, Chichester; New York.
Merton, Robert C., 1973, “Theory of Rational Option Pricing,” Bell Journal of Economics and
Management Science 4, 141-183.
25
Mitchell, Mark, and Todd Pulvino, 2001, “Characteristics of risk in risk arbitrage,” Journal of
Finance 56, 2135-2175.
Newey, W., and K. West, 1987, “A simple , positive semi-definite , heteroskedasticity and
autocorrelation consistent covariance matrix,” Econometrica, 55, 703-708.
Sharpe, William, 1992, “Asset Allocation: Management Style and Performance Measurement,”
Journal of Portfolio Management, 18, 7-19.
Soldofsky, Robert M. , 1971, “Yield-Risk Performance of Convertible Securities,” Financial Analysts
Journal, 27.
Stein, Jeremy C., 1992, “Convertible Bonds as Backdoor Equit y Financing,” Journal of Financial
Economics, 32(1), 3-21.
Woodson, Hart, 2002. Global convertible investing : the Gabelli way (Wiley, New York).
26
Table II: Hedge Ratios for Constructing Credit Arbitrage and Volatility Arbitrage
(or Gamma) ABS Factors
This table provides the estimates of slope coefficients or hedge ratios for constructing the Credit Arbitrage
and Volatility Arbitrage (or Gamma ) ABS factors. The hedge ratios for the Credit Arbitrage ABS factor are
the OLS betas obtained from the regression of the daily returns of the Credit bond portfolio (consisting of
bonds whose parity ≤20% of the par value) on two factors: Equity and Interest rate. Equity refers to the daily
returns on the equally-weighted portfolio of underlying stocks and Interest rate refers to the Japanese 3-5 year
Government Bond total return index. The hedge ratios for the Volatility Arbitrage (or Gamma) ABS factor
are the OLS betas obtained from the regression of the daily returns of the Gamma bond portfolio (consisting
of bonds whose parity lies between 90% and 110% of the par value) on three factors: Equity, Interest rate,
and Credit spread. Equity and Interest rate have been defined already. Credit spread is the daily change in the
spread between the Japanese Corporate Bond yield and Japanese long-term Government Bond yield. * , ** , and
***
indicate that the coefficient is significantly different from zero at the 10, 5, and 1% levels respectively. pvalues are computed using Newey-West (1987) standard errors.
Credit
Gamma
Intercept
0.00
0.00
Equity
0.05***
0.44***
Interest rate#
-3.7
7.7
Credit
0.16
No. of obs
2262
2109
F
52.0
1218.7
p-value
0
0.0
Adj. R2
4.3%
63.4%
# coefficient has been multiplied by 107
28
Table III: Characteristics of Portfolios generating Carry, Volatility Arbitrage (or
Gamma), and Credit Arbitrage Asset-Based Style Factors
This table provides the daily average current yield, parity and delta of the portfolios generating the Carry,
Volatility Arbitrage (or Gamma), and Credit Arbitrage Asset-Based Style (ABS) factors, where the daily
average is taken across all bonds in each of the portfolios. Current yield is computed using a full year's
coupon divided by the bond's mid-price (which incorporates accrued interest). Parity is the bond's
conversion value as a percentage of the par value. Delta is computed based on Merton's (1973) option
pricing model. The start and end dates mark the start and end of the respective portfolios.
Carry
Gamma
Credit
Size of portfolio
233
26
23
Current yield (%)
1.6
1.0
1.9
Parity (%)
59
98
9.2
Delta
0.39
0.7
0.13
Start date
3/19/1993
3/17/1993
1/5/1993
End date
2/1/2002
2/1/2002
2/1/2002
29
Table IV: Descriptive Statistics of the Carry, Volatility Arbitrage (or Gamma), and
Credit Arbitrage Asset-Based Style Factors
This table provides the descriptive statistics of the monthly returns of the Carry, Volatility Arbitrage (or
Gamma ), and Credit Arbitrage Asset-Based Style (ABS) factors. Carry1 (Carry2) is the Carry ABS factor
computed using a capital base equal to the maximum (previous day’s) bond value during the period for
which the carry is positive. Gamma is the ABS factor for volatility arbitrage. Credit is the ABS factor for
credit arbitrage.
Carry1
Carry2
Gamma
Credit
No. of observations
108
108
104
110
Mean
0.7%
0.9%
-1.7%
1.5%
Median
0.3%
0.3%
-1.6%
1.2%
Standard deviation
6.3%
8.1%
1.7%
3.7%
Minimum
-22.5%
-25.5%
-5.7%
-11.5%
Maximum
17.3%
21.5%
6.0%
16.5%
Skewness
-0.14
0.05
0.90
0.85
Kurtosis
1.41
0.94
4.18
4.98
30
Table V: Regression of Convertible Arbitrage Indices on Asset-Based Style Factors
This table provides the results of OLS regression for the returns on the convertible arbitrage hedge fund
index as the dependent variable and the three Asset-Based Style (ABS) factors namely Carry, Volatility
Arbitrage, and Credit Arbitrage as the independent variables. Carry is the ABS factor for the positive carry
strategy, computed using a capital base equal to the maximum bond value during the period for which the
carry is positive. Gamma is the ABS factor for volatility arbitrage. Credit is the ABS factor for credit
arbitrage.* , ** , and *** indicate that the coefficient is significantly different from zero at the 10, 5, and 1%
levels respectively. p-values are computed using Newey-West (1987) standard errors.
Model 1
Model 2
Model 3
Intercept
0.01***
0.01***
0.01***
Carry
0.05**
0.04**
0.04**
0.18
0.13
Panel A:TASS
Gamma
Credit
0.07*
2
Adj. R
3.1%
6.3%
8.7%
Intercept
0.01***
0.01***
0.01***
Carry
0.03*
0.02
0.01
0.18**
0.14**
Panel B: HFR
Gamma
Credit
0.07**
2
Adj. R
1.7%
9.6%
14.5%
Intercept
0.01***
0.01***
0.01***
Carry
0.04**
0.03**
0.03**
0.22**
0.23**
Panel C: MSCI
Gamma
Credit
0.00
2
Adj. R
4.4%
13.1%
11.6%
Intercept
0.01**
0.02***
0.01***
Carry
0.06**
0.04**
0.04*
0.44**
0.37*
Panel D: CISDM
Gamma
Credit
0.04
2
Adj. R
5.9%
23.5%
31
22.7%
Table VI: Regression of Convertible Arbitrage Indexes on Asset-Based Style Factors
This table provides the results of OLS regression for the returns on the convertible arbitrage hedge fund index as the
dependent variable and the three Asset-Based Style (ABS) factors namely Carry, Volatility Arbitrage, and Credit
Arbitrage as the independent variables. Carry is the ABS factor for the positive carry strategy, computed using a
capital base equal to the previous day’s bond value during the period for which carry is positive. Gamma is the ABS
factor for volatility arbitrage. Credit is the ABS factor for credit arbitrage.* , ** , and *** indicate that the coefficient is
significantly different from zero at the 10, 5, and 1% levels respectively. p-values are computed using Newey-West
(1987) standard errors.
Model 1
Model 2
Model 3
Intercept
0.01***
0.01***
0.01***
Carry
0.04**
0.03***
0.03**
0.17
0.13
Panel A:TASS
Gamma
Credit
0.07*
2
Adj. R
3.7%
6.7%
9.0%
Intercept
0.01***
0.01***
0.01***
Carry
0.02**
0.02*
0.01
0.18**
0.14**
Panel B: HFR
Gamma
Credit
0.07**
Adj. R2
2.2%
9.8%
14.6%
Intercept
0.01***
0.01***
0.01***
Carry
0.03**
0.02**
0.02**
0.22**
0.22**
Panel C: MSCI
Gamma
Credit
0.00
2
Adj. R
5.2%
13.5%
12.0%
Intercept
0.01**
0.02***
0.01***
Carry
0.05**
0.03**
0.03**
0.43**
0.36*
Panel D: CISDM
Gamma
Credit
0.04
2
Adj. R
7.3%
24.0%
32
23.3%