Abstract
Between 2005 and 2009, we document evident time-varying credit risk price discovery between the equity and credit default swap (CDS) markets for 174 US non-financial investment-grade firms. We test the economic significance of a simple portfolio strategy that utilizes fluctuation in CDS spreads as a trading signal to set stock positions, conditional on the CDS price discovery status of the reference entities. We show that a conditional portfolio strategy which updates the list of CDS-influenced firms over time, yields a substantively larger realized return net of transaction cost over the unconditional strategy. Furthermore, the conditional strategy’s Sharpe ratio outperforms a series of benchmark portfolios over the same trading period, including buy-and-hold, momentum and dividend yield strategies.
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Notes
Equity-holders are able to retire debt at maturity and claim firm ownership. This is akin to holding a call option against debt-holders on firm assets, with the face-value of debt as the strike price. Accordingly, the probability of non-exercise is analogous to the probability of default. Any information that affects a firm’s credit worthiness will affect the value of equity-holders’ embedded call options, hence its stock price.
The list of companies and their details are available upon request.
The CreditGrades model is jointly developed by four leading financial institutions in the credit market. They are RiskMetrics, JP-Morgan, Goldman Sachs and Deutsche Bank. In Finger et al. (2002, p. 5), the purpose of the CreditGrades model is to establish a robust but simple framework linking the credit and equity market”.
Since the CDS contracts are denominated into US dollar, we use share code 10 and 11 to pull out those firms with common shares traded on US exchanges.
Cao et al. (2010) find that option-implied volatility gives a more accurate measure of \({\text{ICDS}}_{it}\) compared to historical volatility. Our main objective is to analyze credit-risk information flow between the equity and CDS markets. By using implied volatility, the information content of \({\text{ICDS}}_{it}\) spans both equity and option markets. This would contaminate the interpretation of our main results.
In that regard, there are various problems with calculating the liability figures for banking and financial service firms. Furthermore, our sample period encompasses the GFC, which has an anomalous impact on financial firms. That is why we focus on high quality investment-grade non-financial firms.
On the stock-split day, the share price will drop by around 50 %, but the quarterly number of shares remains unchanged. This will cause the per-share debt/equity (D/E) ratio to double. The D/E ratio is a key input parameter to determine probability of default and back out \({\text{ICDS}}_{it}\).
This is based on an industry rule-of-thumb that is consistent with Moodys statistics on historical corporate bond recovery rate. See Moodys Investor’s Service historical Default Rates of Corporate Bond Issuers, 1920–1999” (January 2000).
See Bakshi et al. (1997) for a discussion on the benefits of frequent calibration on option pricing model.
For example, in Blanco et al. (2005) Table IV for Daimler-Chrysler, \({\lambda }_1=-0.03\) and \({\lambda }_2=0.07\) with t-statistics of \(-\)1.4 and 2.8 respectively. The GG measure for Daimler-Chrysler’s CDS market is 0.71. This makes it a C1 firm, even though its price discovery is not per se dominated by the CDS market. We thank an anonymous referee for raising the potential concern to us.
We have also implemented TS4 and TS5, where the candidate list for TS4 (TS5) is mutually exclusive to TS2 (TS3). TS4 and TS5 are designed as control strategies that consider only non-C1 and non-C2 firms. TS4 and TS5 evoke very few trades. For example, for strategies with 1 day (2days) holding period, the number of trades for TS1, TS2, TS3 are 569 (477), 474 (391), and 439 (363) respectively. In contrast, the corresponding number for TS4 and TS5 are only 47 and 83 for 1 day holding period, and 44 and 72 for 2 days holding period. Not surprisingly, TS2 and TS3 outperform TS4 and TS5 respectively. But given the large difference in the frequency of trading, we believe that the two pairs of strategies are not directly comparable. The results for TS4 and TS5 are available upon request.
Firms with CDS contracts tend to be more liquid stocks. Furthermore, hedge funds and investment bank trading desks tend to face lower trading costs, which makes 0.5 % a conservative measure of transaction cost.
Since we consider mainly high-quality investment-grade firms, the short-sale ban applies to only 5.75 % of our firm sample.
Our trading algorithm invests the same capital on each executed trade. Since any gains we realize over time are not reinvested, we report simple rather than compound returns. The choice of return calculation does not affect the relative ranking among portfolio strategies.
The other trading results base on 100 bps transaction cost are available upon request.
For weekly re-balancing, we sort firms every Tuesday based on dividend yield. The next trading day, we form an equally-weighted portfolio in the top ten dividend yielding firms. This portfolio is liquidated next Tuesday. For monthly re-balancing, we sort firms on the first trading day of each month. The portfolio is formed the next trading day, and is subsequently liquidated on the last trading day of the month. For quarterly re-balancing, we sort firms on the first trading day of each of the March, June, September and December quarter. We form a long portfolio the next trading day, which is subsequently liquidated on the last trading day of each quarter.
Conceptually, \(\tau _0=0\). Since our risk-free rate is a proxy, hence we allow for \(\tau _0=c\).
Define a time-shift Brownian motion \(\hat{W}_t\) that starts at \(\hat{t}_0\). Then \(\frac{d\hat{X}_t}{\hat{X}_t}=-\frac{{\sigma }^2}{2}d\hat{t}+{\sigma }d\hat{W}_t\) also follows time-shift Brownian motion with \(\hat{X}_{\hat{t}_0}=0; E(\hat{X})=-\frac{{\sigma }^2}{2}(t+\frac{{\lambda }^2}{{\sigma }^2}); Var(\hat{X})={\sigma }^2(t+\frac{{\lambda }^2}{{\sigma }^2})\).
The general formula for the probability of a Brownian motion is \(Y_t=at+bW_t>y, \forall s<t\) is \(P[Y_s>y]=\phi (\frac{at-y}{b\sqrt{t}}-e^{\frac{2ay}{b^2}\phi (\frac{at+y}{b\sqrt{t}})})\), where \(\phi (\cdot )\) is the cumulative probability distribution function. The CreditGrades model focuses on the Brownian motion probability \(\hat{X}_t=-\frac{{\sigma }^2}{\hat{t}}+{\sigma }\hat{W}_t\) exceeding the fixed level of \(log(\frac{\bar{L} D}{V_0})-{\lambda }^2\).
R is different from \(\bar{L}\). R is the expected recovery rate for specific debt covered by the CDS contract, whereas \(\bar{L}\) is the expected global recovery rate i.e. expected average recovery rate for all debt of the firm. Base on \(P_{(t)}\), the risk neutral probability of the default density function can be defined as \(f(t)=-\frac{dP_{(t)}}{dt}\).
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Acknowledgments
We thank Tarun Chordia, Ning Gong and Yuelan Chen for helpful comments. We acknowledge comments from seminar participants at the UNSW, University of Adelaide, Fudan University and National Taiwan University of Science and Technology. We retain full property rights to all existing errors.
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Appendix: Technical notes on the CreditGrades model
Appendix: Technical notes on the CreditGrades model
The CreditGrades model assumes that a firm’s asset value \(V_t\) follows a geometric Brownian motion without drift \(\frac{dV_t}{V_t}={\sigma }dW_t\). A zero-drift assumption is consistent with evidence of stationary leverage ratios documented by Collin-Dufresne and Goldstein (2001). In the event of default, debt-holders receive a recovery amount LD, where L is the global average recovery rate and D is debt per share. Denote \(E(L)=\bar{L}\). In CreditGrades, the recovery amount upon default LD is defined as the default barrier and is assumed to follow a stochastic process. The model assumes that L follows a log-normal distribution with \(Log(L)\sim N(\mu ,{\lambda }^2)\), such that LD can be expressed as Eq. (4), where \(Z\sim N(0,1)\). Since \(\bar{L}=e^{(\mu +\frac{{\lambda }^2}{2})}\), hence \(\bar{L} \cdot e^{({\lambda }Z-\frac{{\lambda }^2}{2})}=e^{(\mu +{\lambda }Z)}\). As \(Z\sim N(0,1), log(e^{\mu +{\lambda }Z})\sim N(\mu ,{\lambda }^2)\). Hence \(LD=e^{(\mu +{\lambda }Z)}D=\bar{L} D \cdot e^{({\lambda }Z-\frac{{\lambda }^2}{2})}\).
A default event is triggered by \(V_t<LD\). Using Ito’s Lemma and given the initial asset value \(V_0\), the firm will exist as long as Eq. (5) is satisfied.
Denote \(X_t={\sigma }W_t-\frac{1}{2}{\sigma }^2t-{\lambda }Z+\frac{{\lambda }^2}{2}\). It can be shown that \(X_t\sim N[(-\frac{{\sigma }^2}{2}(t+\frac{{\lambda }^2}{{\sigma }^2})), {\sigma }^2(t+\frac{{\lambda }^2}{{\sigma }^2})]\). Then \(X_t\) can be approximated by a time-shift Brownian motion \(\hat{X}_t\) that starts at \(t_0=-\frac{{\lambda }^2}{{\sigma }^2}\).Footnote 19 Default is triggered by \(\hat{X}_t\le (log(\frac{\bar{L} D}{V_0})-{\lambda }^2)\). The survival probability is the cumulative probability before \(\hat{X}_t\) hits and falls below a certain level of \((log(\frac{\bar{L} D}{V_0})-{\lambda }^2)\) for the first time. Applying distributions for the first-time hitting of \(\hat{X}_t\), the CreditGrades model provides a closed-form solution in Eq. (6) to calculate the survival probability \(P_{(t)}\) up to time t.Footnote 20
\(P_{(t)}\) allows us to specify the implied credit default spread ICDS. Denote R as recovery rate for underlying debt,Footnote 21 f(t) as the default density function and r as the risk-free rate. The present values of expected compensation and expected CDS spread payments due to a default event are given by Eqs. (7) and (8) respectively.
The day \(\tau\) value of a CDS contract \(M_\tau\) for the protection buyer is the difference between present values of expected compensation and expected spreads payments in Eq. (9).
Since \(\int _0^t P_{(s)}\cdot e^{-rs} ds=\frac{1}{r}(P_{(0)}-P_{(t)}\cdot e^{-rt})-\frac{1}{r}\int _0^t f(s)\cdot e^{-rs}ds\), then Eq. (9) can be re-expressed as Eq. (10).
Using equation(11), we rewrite equation(10) as equation(12), where \(\xi =\frac{{\lambda }^2}{{\sigma }^2}; z=\sqrt{\frac{1}{4}+\frac{2r}{{\sigma }^2}}\).
Finally, by setting \(M_\tau =0\), we obtain the close-form solution for ICDS in Eq. (13). Since it uses only stock prices and balance sheet information, ICDS is the CDS spread implied by the equity market.
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Xiang, V., Chng, M.T. & Fang, V. The economic significance of CDS price discovery. Rev Quant Finan Acc 48, 1–30 (2017). https://doi.org/10.1007/s11156-015-0540-2
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DOI: https://doi.org/10.1007/s11156-015-0540-2