Federico Thibaud - Capital Structure Arbitrage

MIT SLOAN SCHOOL OF MANAGEMENT

An update on Capital Structure Arbitrage

15.961: Independent Study

Federico Thibaud 12/10/2015

Abstract:

This paper offers a review of existing literature on Capital Structure Arbitrage, a relative value strategy that exploits differences between actual spreads on Debt securities and those implied in Equity prices. Taking as a starting point the model presented in Merton (1974), we provide an introductory overview of the main issues in the field. We then update the performance of the CSA strategy proposed in Yu (2006) and propose and test modifications to the original Strategy.

15.961 Independent Study in Management (Fall 2015) Federico Thibaud – MIT ID: 917456011

Contents Introduction: ................................................................................................................................................. 2

Merton Model: .......................................................................................................................................... 3

CreditGrades Model: ................................................................................................................................. 4

Diffusion processes and low-risk underestimation: ................................................................................. 6

Yu (2006) – Using structural models in a trading strategy: .................................................................... 10

Trading Strategy Implementation: .............................................................................................................. 12

Data: ........................................................................................................................................................ 12

Trading Strategy Description: ................................................................................................................. 14

Results – An update of Yu for 2004-2014 ............................................................................................... 15

Original Strategy – Static Hedge Trade Return Distribution ............................................................... 15

Dynamic Hedge Trade Return Distribution ......................................................................................... 18

Monthly Returns ................................................................................................................................. 21

Explaining portfolio returns ................................................................................................................ 22

Periodic Recalibration ............................................................................................................................. 25

Results ................................................................................................................................................. 25

Conclusions: ................................................................................................................................................ 28

Appendix: .................................................................................................................................................... 30

Merton model: ........................................................................................................................................ 30

Market spreads table: ............................................................................................................................. 33

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Introduction:

Capital Structure Arbitrage (CSA) attempts to exploit mispricings between different securities on the same underlying capital structure. The logic of the strategy is to price both instruments according to a single variable and then determine whether any of them is trading at a premium compared to the other.

The first analysis of the returns of a CSA strategy is described in Duarte, Longstaff and Yu (2005) and later extended in Yu (2006). The paper finds the strategy to produce significantly positive returns over the 2001-2004 period.

Since then, several other papers have extended results following a similar methodology with mixed results. Bajlum and Larsen (2008) repeat the test from October 2002 to September 2004 and find the strategy tends to have positive returns (depending on parameter specifications). Avino and Lazar (2013) find that the strategy generates slightly negative returns in the 2005-2009 periods. For practically the same period Visockis (2011) finds positive returns, with a large unexplained outlier in 2009.

The typical starting point in a CSA strategy is a structural model (most of these are derived from Merton (1974)), which prices all instruments as claims on a firm’s assets. Since the value and volatility of assets are not directly observable in the market, the model will make specific assumptions in estimating them.

For implementing our trading strategy, we will use a particular structural model (we take the CreditGrades model) to first infer an equity-implied CDS spread which we compare against the actual 5 year CDS spread. If a significant discrepancy is identified we build a convergence portfolio: if the market spread trades above our model-implied value we short the 5 year CDSs and hedge with a short equity position; if it trades below our model, we buy CDSs and hedge with a long equity position.

It is important to note that, despite its name, CSA is not an arbitrage. There is no mechanism which forces convergence within a given period between the equity-implied value of the debt and its market value. Even in the case when the two do converge, this does not guarantee a profit, since convergence can come from non-equity related factors, such a change in the face value of the debt or the equity’s volatility.

The strategy is more akin to a “statistical arbitrage” in which a strong fundamental reasoning allows us to expect the two securities to move in conjunction and assume divergences come from market noise, such as short term liquidity issues. The key is, then, understanding how empirically reliable the strategy is as well as what we can expect from its convergence.

In this paper we will replicate a well-known CSA strategy and attempt to determine some of its characteristics that will allow us to best exploit it in a trading strategy. One of the issues we will deal with is the empirically observed fact that structural models tend to underestimate the spreads for low-risk obligors.

The original strategy based on Yu (2006) attempts to salvage this gap by “calibrating” one of the parameters in the model to match observed spreads over the first data points. This, however,

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introduces unwanted assumptions into the model and going forward does not produce spreads comparable to the market.

We first replicate the strategy presented in Yu (2006) using implied volatility as proposed in Bajlum and Larsen (2008) and provide results for 2004 to 2014. We then propose an extension of Yu’s model in which rather than using a fixed period for calibration we use a periodic recalibration to match model spreads to levels comparable to the market.

In the final conclusion we analyze the extent to which the examined models can be used as a basis for a trading strategy and what possible extensions could be examined to improve its performance.

Merton Model:

As mentioned above, most structural models are based on the original ideas proposed in Merton (1974). The key insight behind the model is the notion that equity and debt can be modelled as derivative claims on the underlying firm’s assets and can be priced accordingly using the Black-Scholes formulas.

In his original paper, Merton proposes a simplified framework in which the firm is assumed to have only issued non-dividend paying equity and a single zero-coupon bond with face value B maturing at time T. Both securities are priced as claims on the firm’s assets V, assumed to follow a diffusion process with constant volatility:

1. 𝑑𝑑𝑑𝑑 = (𝛼𝛼𝑑𝑑 − 𝐶𝐶) 𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑑𝑑𝑑𝑑𝜎𝜎

Where α is the expected return on the firm, 𝐶𝐶 are the total payouts to equity and debt holders, 𝜎𝜎2 is the variance of the return on the firm and 𝑑𝑑𝜎𝜎 is a standard Wiener process.

At maturity T, the bond’s face value will either be repaid in full, with any residual assets going to equity holders or, if assets are insufficient to cover it, bondholders will take control of any assets held by the firm. These payoffs can thus be described as:

2. 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸: 𝑓𝑓(𝑑𝑑, 𝑇𝑇) = 𝑀𝑀𝑀𝑀𝑀𝑀 (𝑑𝑑 − 𝐵𝐵, 0) 3. 𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑: 𝐹𝐹(𝑑𝑑, 𝑇𝑇) = 𝑀𝑀𝐸𝐸𝑀𝑀 (𝑑𝑑, 𝐵𝐵)

The payoffs to debt and equity are thus comparable to a call and a combination of a short put on the firm’s assets and a risk-free bond and they can be priced accordingly using the Black-Scholes equation.

Since we are interested in calculating an implied CDS spread we will focus on the debt. At each point in time t, we can value this security according to its time to maturity (𝜏𝜏):

4. 𝐹𝐹(𝑑𝑑, 𝜏𝜏) = 𝐵𝐵𝐷𝐷−𝑟𝑟𝑟𝑟 �𝜙𝜙(ℎ2) + 1𝑑𝑑

ϕ(ℎ1)�

Where

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5. ℎ1 ≡ −𝑀𝑀1 = −�1

2𝜎𝜎2𝑟𝑟−ln (𝑑𝑑)�

𝜎𝜎√𝑟𝑟

6. ℎ2 ≡ 𝑀𝑀2 = −�1

2𝜎𝜎2𝑟𝑟+ln (𝑑𝑑)�

𝜎𝜎√𝑟𝑟

7. 𝑑𝑑 = 𝐵𝐵𝑒𝑒−𝑟𝑟𝑟𝑟

𝑉𝑉

The above formula describes the value of the firm’s risky bond in terms of its price. Since we are interested in calculating a CDS spread, we express the bond price in terms of its yield and then subtract the risk-free rate:

8. 𝐸𝐸 − 𝑟𝑟 = − 1𝑟𝑟

𝑙𝑙𝑀𝑀 �𝜙𝜙(ℎ2) + 1𝑑𝑑

ϕ(ℎ1)�

It is important to note that, for a given term (𝜏𝜏) the spread will thus only depend on the firm’s debt-to-asset ratio (𝑑𝑑) and its asset volatility (𝜎𝜎).

The asset volatility can be derived from the observable equity volatility using the equation:

9. 𝜎𝜎𝐸𝐸𝜎𝜎

= 𝜕𝜕𝜕𝜕𝜕𝜕𝑉𝑉

∗ 𝑉𝑉𝜕𝜕

Given our model specification this will be:

10. 𝜎𝜎𝜕𝜕 = 𝜎𝜎𝜎𝜎(𝑥𝑥1)𝜎𝜎(𝑥𝑥1)−𝑑𝑑𝜎𝜎(𝑥𝑥2)

In order to estimate these values, we take the balance-sheet debt-per-share (𝐵𝐵�) as the debt’s face value B and take the equity share price and equity historic volatility as 𝑓𝑓(𝑑𝑑, 𝑇𝑇) and 𝜎𝜎𝜕𝜕 respectively. It is important to note here that, similar to Black-Scholes the relevant number is the expected future volatility, so there is some discretion (as we will see when we implement our model) in choosing what number to plug in here.

Using these inputs we can solve simultaneously the equations to get the asset value (𝑑𝑑, required to calculate the debt-to-asset ratio (𝑑𝑑)) and the asset volatility (𝜎𝜎).

CreditGrades Model:

The CreditGrades model builds on the original Merton framework by alleviating some of its assumptions and simplifying its implementation.

The model uses the same diffusion process for the evolution of the asset prices but incorporates the idea of a default barrier based on Black and Cox (1976). In this framework default can be triggered before maturity if the firm’s assets fall below a certain threshold. Intuitively, this can be thought of as a covenant which allows bondholders to demand the liquidation of the firm if assets hit a certain level.

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To correct for the empirically low predicted spreads predicted by structural models in the short term, the CreditGrades model incorporates a random barrier. This is modeled using a lognormal random variable defined as:

11. 𝐷𝐷𝐷𝐷𝑓𝑓𝑀𝑀𝐸𝐸𝑙𝑙𝑑𝑑 𝐵𝐵𝑀𝑀𝑟𝑟𝑟𝑟𝐸𝐸𝐷𝐷𝑟𝑟: 𝐿𝐿𝐷𝐷 = 𝐿𝐿�𝐷𝐷𝐷𝐷𝜆𝜆𝜆𝜆−𝜆𝜆2 2⁄

Where 𝐿𝐿� is the average recovery rate, 𝐷𝐷 is the observed level of debt, 𝜎𝜎 is a standard normal random variable and 𝜆𝜆 is the volatility of the barrier.

𝐿𝐿� and 𝜆𝜆 are set to 0.5 and 0.3 respectively, based on an analysis of the observed mean and volatility in recovery rates between 1987 and 1997 compiled in Hu and Lawrence (2000).

The intuition behind using a random barrier is that it allows for some uncertainty in the actual value of the firm’s debt and has the beneficial effect of increasing short term spreads.

Default will thus occur whenever assets fall below the barrier level:

12. 𝐷𝐷𝐷𝐷𝑓𝑓𝑀𝑀𝐸𝐸𝑙𝑙𝑑𝑑 𝑇𝑇𝑟𝑟𝐸𝐸𝑇𝑇𝑇𝑇𝐷𝐷𝑟𝑟: 𝑑𝑑0𝐷𝐷𝜎𝜎𝑊𝑊𝑡𝑡−𝜎𝜎2𝑡𝑡/2 > 𝐿𝐿�𝐷𝐷𝐷𝐷𝜆𝜆𝜆𝜆−𝜆𝜆2/21

The probability of the first hitting time of the barrier will be:

13. 𝑃𝑃(𝑑𝑑) = 𝜙𝜙 �− 𝐴𝐴𝑡𝑡2

+ ln(𝑑𝑑)𝐴𝐴𝑡𝑡

� − 𝑑𝑑 𝜙𝜙 �− 𝐴𝐴𝑡𝑡2

− ln(𝑑𝑑)𝐴𝐴𝑡𝑡

�

Where

14. 𝑑𝑑 = 𝑉𝑉0𝑒𝑒𝜆𝜆2

𝐿𝐿𝐿𝐿

15. 𝐴𝐴𝑡𝑡

2 = 𝜎𝜎2𝑑𝑑 + 𝜆𝜆2

Intuitively, default will depend on the initial distance to default (𝑑𝑑) and the combined variance of the firm assets and the barrier (𝐴𝐴𝑡𝑡

2).

The second major difference between the CreditGrades model and the Merton model is in the estimation of the asset value and variance.

Rather than deriving the asset value by solving the simultaneous equations in the Merton model, CreditGrades simply assumes a linear relationship between the firm value and the share price:

16. 𝑑𝑑0 = 𝐸𝐸0 + 𝐿𝐿�𝐷𝐷

Where 𝑑𝑑0 is the initial asset value, 𝐸𝐸0 is the observed share price and 𝐿𝐿�𝐷𝐷 is the expected value of the default barrier, based on observed debt levels.

1 One detail here is what happens to the asset drift in the firm’s assets evolution process. In the case of the Merton model, this disappeared when we considered the construction of the no-arbitrage portfolios. In the case of the CreditGrades model, it is assumed to be zero relative to the default barrier. Meaning that both assets and the default barrier grow at a rate μ. The logic here is assuming a constant leverage ratio.

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While this may seem an over simplification, it is based on the fact that at the two extremes, namely a firm sufficiently far from the default barrier or a firm at default it will produce the correct estimation. Empirical studies such as Bharat and Shumway (2004) have found that the additional complexity in the Merton model does not increase CDS spread prediction power over a naïve linear alternative.

Once we have found the asset value, asset volatility can be inferred from equity volatility following (9). In this case, however, since we are assuming a linear relationship between assets and the share price this will simply be:

17. 𝜎𝜎 = 𝜎𝜎𝜕𝜕 ∗ 𝜕𝜕𝜕𝜕+𝐿𝐿𝐿𝐿

Where 𝜎𝜎 is the unobservable asset volatility and 𝜎𝜎𝜕𝜕 is the historic stock volatility.

In order to calculate the Hedge Ratio between the CDS and equity position, we use a linear approximation of the derivative of the predicted CDS spread against the equity price. The equity price is increased marginally and compared against the resulting increase in CDS Price. This is calculated as:

18. 𝐻𝐻𝐷𝐷𝑑𝑑𝑇𝑇𝐷𝐷 𝑅𝑅𝑀𝑀𝑑𝑑𝐸𝐸𝑎𝑎 = (𝜕𝜕−(𝜕𝜕+𝜀𝜀))

𝜕𝜕(𝑃𝑃−(𝑃𝑃′))

𝑃𝑃

�

Where, E is the equity price and P is the CDS price (calculated from its spread). ε represents a 1% increment on equity price and 𝑃𝑃′ represents the CDS price resulting from the modified Equity base.

Diffusion processes and low-risk underestimation:

An observed issue with these and similar models based on a diffusion-growing assets is that they seem to systematically underestimate spreads for low-risk obligors, a shortcoming already noted by the authors of the original CreditGrades paper.

Bajlum and Larsen (2008) test the impact of using alternative model specifications, both in structural form and inputs. They test several alternative structural models based on the same underlying diffusion process and find little dispersion in results. In particular they test the CreditGrades implementation against the more complex model proposed in Leland and Toft (1996). Their findings suggest that the choice of the particular model actually has little impact on results.

To assess the model specification’s importance from a more general perspective, we could first ask how different estimates for the CreditGrades and the Merton model are for a given firm.

To show the potential the models have for producing different results, we first calculate the spreads for the Merton and the CreditGrades model based theoretical levels of Equity-to-Debt and equity volatility. We choose these two variables because they fully describe all we need to calculate a firm’s spread (in

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the Merton model we use these values to find the corresponding debt-to-asset ratio (𝑑𝑑) and asset volatility (𝜎𝜎)).

Below are the tables of implied 5 year CDS spreads (expressed in basis points) for each model, calculated for each level of Equity-to-Debt and equity volatility:

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Table 1: Merton model - implied CDS spreads for a variety of combinations of stock-to-debt ratios and 𝜎𝜎𝜕𝜕. The vertical axis marks the stock-to-debt ratio (share price over debt-per-share) and the horizontal axis shows expected equity volatility (in percentage). The values in the table are the model-predicted CDS spreads.

Table 2: CreditGrades - implied CDS spreads for a variety of combinations of stock-to-debt ratios and 𝜎𝜎𝜕𝜕. The vertical axis marks the stock-to-debt ratio (calculated as share price over debt-per-share) and the horizontal axis shows expected equity volatility (in percentage). The values in the table are the model-predicted CDS spreads.

The first observation we can make is that both models predict essentially zero spreads for low-risk obligors (the bottom left corner of the table). This makes sense when we consider we are using a diffusion process for both models. Assuming a constant continuous volatility it is essentially impossible for assets to fall to the levels which would trigger a default.

A second, observation is the fact that the CreditGrades model predicts much higher spreads for obligors near the default barrier. This is explained by the use of a random default barrier. When we remove the stochastic component in the default barrier spreads for these securities become much closer to those predicted by the Merton model

Merton 20 25 30 35 40 45 50 55 60 650.5 0 3 9 24 49 87 142 216 317 4311 0 2 7 21 45 83 136 207 296 407

1.5 0 1 5 16 37 70 119 183 264 3652 0 1 4 12 30 59 103 161 237 329

2.5 0 0 3 9 24 50 89 143 213 2973 0 0 2 7 20 42 78 127 191 272

3.5 0 0 1 6 16 37 68 115 173 2504 0 0 1 4 14 32 60 103 159 230

4.5 0 0 1 4 11 27 54 93 146 2145 0 0 0 3 10 24 48 84 134 199

5.5 0 0 0 2 8 21 43 78 124 1866 0 0 0 2 7 18 39 71 116 174

CreditGrades 20 25 30 35 40 45 50 55 60 650.5 56 86 127 177 235 301 372 447 527 6101 8 22 47 84 132 190 257 330 409 492

1.5 2 8 22 48 86 136 196 263 338 4182 1 3 12 30 60 102 155 217 287 363

2.5 0 2 7 20 44 79 126 183 248 3203 0 1 4 14 33 63 104 156 217 285

3.5 0 0 3 10 25 51 88 135 192 2574 0 0 2 7 19 41 74 118 172 234

4.5 0 0 1 5 15 34 64 104 155 2135 0 0 1 4 12 29 55 93 140 196

5.5 0 0 1 3 10 24 48 83 127 1816 0 0 0 2 8 21 43 75 117 168

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In order to gauge the significance of these differences, we now contrast these values against actual spreads for companies as observed on a recent date. As a universe we take all securities in the CDX North American Investment Grade and High Yield Indices and the iTraxx Europe Main and Crossover Indices on 12/01/2014:

Table 3: Market - CDS spreads observed on 12/01/2014 for issuers falling into the corresponding stock-to-debt ratios and 𝜎𝜎𝜕𝜕 bins. The issuer universe comprises the constituents of the CDX North America Investment and High Yield indices and the iTraxx Main and Crossover. The vertical axis marks the stock-to-debt ratio (calculated as share price over debt-per-share) and the horizontal axis shows expected equity volatility (in percentage). The values in the table represent the model-predicted CDS spread.

Although we have a limited number of companies that do not span the entire Equity-to-Debt -equity volatility space (included in the Appendix is a table with the number of firms in each quadrant) we can see that observed spreads are significantly above implied spreads for both models, particularly for low-risk obligors.

This phenomenon (as noted in the “CreditGrades Technical Document”) can be associated with the shortcomings of using a diffusion process. Intuitively, we can link this underestimation of spreads to the underestimation of OTM option prices as priced by the Black-Scholes model using historic stock volatility.

In order to fit the discrepancy between the observed and the model-predicted prices of options, the convention is to fit different implied volatility for individual strike options, giving rise to the “volatility smile” phenomenon. It is important to note how counter-intuitive this compromise is: all options on the same underlying should be based on the same expected future volatility. In our base model, taken from Yu (2006) we take a similar approach and apply it to model-predicted CDS spreads.

Market 20 25 30 35 40 45 50 55 60 650.5 80 165 187 401 172 114 350 3041 68 134 189 270 274 380

1.5 97 125 147 796 157 3042 71 103 114 216 164

2.5 52 69 161 189 2563 46 83 82 229

3.5 37 47 57 4054 69 45 115

4.5 39 112 1615 78 88

5.5 31 1046

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Yu (2006) – Using structural models in a trading strategy:

As noted in the introduction, Duarte, Longstaff and Yu (2005), followed in more detail by Yu (2006) are the first papers to describe and quantify the returns of a CSA strategy. To identify trading opportunities, the model takes as a starting point the CreditGrades model but, rather than using the average recovery parameter (𝐿𝐿� = 0.5) specified in the original paper as a default barrier, he leaves it as a floating parameter and calibrates for the barrier that best fits the first 10 days of data.

In his original paper, Yu cites as motivation for this decision literature (in particular Leland (1994) and Leland and Toft (1996)) suggesting the expected default barrier should endogenously be determined by firm-specific factors.

In practice, however, deriving an “implied” default barrier from market spreads (rather than the underlying firm’s fundamental characteristics) is analogous to the calibration of individual underlying volatilities to the prices of each option (applied to each obligor, rather than each individual contract).

Note here the change in the approach to the trading strategy: by using a structural model we attempt to determine an expectation for the fair value of the instrument’s payoff and trade on price divergences from it. By using this calibration we are going much closer to reduced form models, which derive an implied default frequency from market spreads and use it to value other instruments.

Like the implied volatility calibration, this default barrier fitting can also lead to highly counter-intuitive results. In particular, for low-risk obligors where the need for adjustment is largest, the implied barriers can be well above one, meaning that, if the market were indeed using a CreditGrades-like model to price CDSs, it would be assuming that the firm would default at asset levels above the face value of their total debt.

Table 4: Calibrated default barrier – the values below show statistics for the fitted values for the default barrier, calibrated over the first 10 data points for each obligor. Results represent the percentage of debt at which the company would be expected to default.

Yu makes the observation that CDS-implied recovery values often require “unreasonable” levels assumed. Bajlum and Larsen report similar results in their obtained calibration.

One shortcoming of this firm-specific calibration is that rather than updating it as new information about the firm is incorporated into prices, the calibrated values are static for each obligor. Intuitively, we

Calibrated Recovery ValuesMean 0.94Median 0.60Max 5.50Min (Constrained to >= 0.1) 0.10

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would assume that the under-over estimation of implied CDSs would depends on time varying factors such as the firm’s distance to default or its debt structure.

By optimizing once per obligor we are basing the adjustment solely on the initial difference between model and market spreads and applying it for the future history of the firm regardless of the convergence of the unadjusted model and market spreads.

Furthermore, it will introduce an arbitrary bias into our trading strategy, since by taking the beginning of the test as the starting point for the optimization, future trades will depend on the starting date we choose.

While this issue is not as significant for a short, stable test period such as the 2002-2004 date range of the original paper, it becomes a significant problem when considering a more extended period in which firm conditions vary significantly.

In the following section we will present results for our version of the original Yu and a version of the strategy intended to alleviate this issue.

As an example of the difference introduced by calibration we show below graphs using predicted vs market spreads for a given security (21st Century Fox):

Figure 1: Model vs Market Spreads for FOXA (2006-2014) – predicted spreads using the uncalibrated (default barrier 𝐿𝐿� = 0.5) version of CreditGrades plotted against market spreads.

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Figure 2: Model vs Market Spreads for FOXA (2006-2014) – predicted spreads using the calibrated (default barrier 𝐿𝐿� = 2.28) version of CreditGrades plotted against market spreads.

As we can see in these graphs, by calibrating we get a much closer match between model and actual spreads during normal periods. During the 2008 crisis, however, as volatilities spike up we see the model actually overshooting (with the difference being only slightly larger for the calibrated version.

Trading Strategy Implementation:

In our implementation of the CSA model we mostly follow you but also incorporate results from Bajlum and Larsen (2008). In their paper they show that the performance of the model can be significantly improved by using option-implied volatility rather than historic volatility. Based on this result, rather than using historic volatility as the equity volatility input as in Yu (2006) we will instead use ATM option-implied volatility.

In order to reduce the impact of using an out-dated calibration, we will then test the impact of periodically (every two years) recalibrating the default barrier to match market spreads.

Data:

Our universe consists of 25 Investment Grade and 10 High Yield obligors. These companies are randomly chosen from the current constituents of the CDX IG and CDX HY indices. The tables below show the Bloomberg CDS Ticker, Name and Sector for the companies used:

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Table 4: Investment Grade obligors – Ticker, Name and Sector for entities in our sample

Table 5: High Yield obligors – Ticker, Name and Sector for entities in our sample

CDS Ticker Name SectorCACE1U5 TWENTY-FIRST CENTURY FOX-A Consumer DiscretionaryCAET1U5 ACE LTD FinancialsCMO1U5 AETNA INC Health CareCAEP1U5 ALTRIA GROUP INC Consumer StaplesCAXP1U5 AMERICAN ELECTRIC POWER UtilitiesCAIG1U5 AMERICAN EXPRESS CO FinancialsCAPC1U5 AMERICAN INTERNATIONAL GROUP FinancialsCARW1U5 AMGEN INC Health CareCFSAI1U5 ANADARKO PETROLEUM CORP EnergyCSBC1U5 ARROW ELECTRONICS INC Information TechnologyCAZO1U5 ASSURED GUARANTY LTD FinancialsCAVT1U5 AT&T INC Telecommunication ServicesCAVP1U5 AUTOZONE INC Consumer DiscretionaryCBAX1U5 AVNET INC Information TechnologyCFO1U5 AVON PRODUCTS INC Consumer StaplesCHRB1U5 BARRICK GOLD CORP MaterialsCBMY1U5 BAXTER INTERNATIONAL INC Health CareCCPB1U5 BEAM SUNTORY INC Consumer StaplesCCOF1U5 H&R BLOCK INC Consumer DiscretionaryCCAH1U5 BOSTON SCIENTIFIC CORP Health CareCNCP1U5 BRISTOL-MYERS SQUIBB CO Health CareCAMG1U5 CAMPBELL SOUP CO Consumer StaplesCABX1U5 CAPITAL ONE FINANCIAL CORP FinancialsCBSX1U5 CARDINAL HEALTH INC Health Care

CDS Ticker Name SectorCCZN1U5 FRONTIER COMMUNICATIONS CORP Telecommunication ServicesCCTL1U5 CENTURYLINK INC Telecommunication ServicesCARM1U5 MERITOR INC IndustrialsCGT1U5 GOODYEAR TIRE & RUBBER CO Consumer DiscretionaryCBZH1U5 BEAZER HOMES USA INC Consumer DiscretionaryCCD1U5 AVIS BUDGET GROUP INC IndustrialsCT370292 AMERICAN AXLE & MFG HOLDINGS Consumer DiscretionaryCRCL1U5 ROYAL CARIBBEAN CRUISES LTD Consumer DiscretionaryCILFC1U5 IRON MOUNTAIN INC FinancialsCGCI1U5 GANNETT CO Consumer Discretionary

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We get CDS spreads, equity prices and fundamental data for the debt calculation from Bloomberg. While this is an admittedly small sample (and one suffering from survivorship bias) it will allow us to quickly test and develop an intuition for the working of the model.

To test the integrity of Bloomberg’s CDS spreads we check using CMA data taken from Datastream (available until 2010) with very similar results).

In order to transform CDSs into prices and calculate mark-to-market performance, we also take 6 month, 1 year, 2 years, 5 years and 10 years Treasury rates from Bloomberg.

Trading Strategy Description:

We first replicate a CSA strategy identical to that implemented in Yu (2006), with the exception of using ATM implied volatility instead of historic.

As in the original strategy, we will use different differences between the model-implied and the observed spreads to initiate trades:

If: 𝑐𝑐𝑀𝑀𝑀𝑀𝑑𝑑𝑒𝑒𝑀𝑀 > 𝑐𝑐𝑀𝑀𝑀𝑀𝑟𝑟𝑀𝑀𝑒𝑒𝑡𝑡 (1 + 𝛼𝛼) ⇒ we buy CDS and hedge with a long equity position

If: 𝑐𝑐𝑀𝑀𝑀𝑀𝑑𝑑𝑒𝑒𝑀𝑀 (1 + 𝛼𝛼) < 𝑐𝑐𝑀𝑀𝑀𝑀𝑟𝑟𝑀𝑀𝑒𝑒𝑡𝑡 ⇒ we sell CDS and hedge with a short equity position

As a convention we will call a trade which sells CDS and shorts equity “Long” (from “long credit risk”) and one which buys CDS and buys equity “Short” (from “short credit risk”).

In order to determine the hedge ratios we will simply calculate the model-implied change in the value of a CDS position resulting from a movement in the stock price.

Trades are held until the earliest of convergence (𝑐𝑐𝑀𝑀𝑀𝑀𝑑𝑑𝑒𝑒𝑀𝑀 <= 𝑐𝑐𝑀𝑀𝑀𝑀𝑟𝑟𝑀𝑀𝑒𝑒𝑡𝑡 for Shorts and 𝑐𝑐𝑀𝑀𝑀𝑀𝑑𝑑𝑒𝑒𝑀𝑀 >= 𝑐𝑐𝑀𝑀𝑀𝑀𝑟𝑟𝑀𝑀𝑒𝑒𝑡𝑡 for Longs) or a time-based stop at 180 days (this is again based on the trading strategy proposed in Yu.

Positions are sized assuming we trade protection for 10% of the trade’s capital. That is, when we look at individual trade results they will show the return on an initial capital of $100, affected by buying/selling protection for a notional of $10 and hedging based on the appropriate hedge ratio. This is significantly smaller than the leverage used in Yu (he used 2-to-1 rather than 0.1-to-1) but it is simply a scaling issue.

We will present results for the strategy using the three trigger levels used in the original paper (0.5, 1 and 2) as well as show the results using a static and a dynamic hedge.

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Results – An update of Yu for 2004-2014

Original Strategy – Static Hedge Trade Return Distribution In the tables below we show results for the original strategy. Yu notes that since correlation between CDS spreads and equity is low, dynamic hedging is not worth the additional trading costs and thus chooses a static hedge.

Table 6: Static hedge IG - summary statistics of holding period returns for Investment Grade obligors. N is the number of trades. The “Long” panel shows trades which are short CDS (=Long Credit Risk) and short stocks for the hedge. The “Short” panel shows trades which are long CDS (=Short Credit Risk) and long stocks for the hedge. The “All” panel shows both long and short trades. % Positive is the percentage of trade with positive returns. Returns are in percentage. Holding Period is in days.

Trigger N Mean Median Std Dev T-Stat Max Min % Positive Median Holding Period0.5 39,973 0.06% 0.05% 0.21% 51.4 4.36% -1.89% 76.0% 180

1 37,005 0.06% 0.05% 0.22% 50.6 4.36% -1.89% 76.9% 1802 32,898 0.06% 0.05% 0.20% 51.2 4.36% -1.89% 77.9% 180


1 8,218 0.22% 0.08% 0.73% 26.9 7.07% -3.44% 59.8% 1802 5,045 0.36% 0.19% 0.82% 30.9 7.07% -2.20% 65.4% 180


1 45,223 0.09% 0.05% 0.37% 48.9 7.07% -3.44% 73.8% 1802 37,943 0.10% 0.05% 0.37% 51.2 7.07% -2.20% 76.3% 180

Long

Short

All

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Table 7: Static hedge HY - summary statistics of holding period returns for High Yield obligors. N is the number of trades. The “Long” panel shows trades which are short CDS (=Long Credit Risk) and short stocks for the hedge. The “Short” panel shows trades which are long CDS (=Short Credit Risk) and long stocks for the hedge. The “All” panel shows both long and short trades. % Positive is the percentage of trade with positive returns. Returns are in percentage. Holding Period is in days.

The first observation we can make is that the strategy still seems to produce positive returns over its holding period. As noted in the “All” section of the above tables, returns are significantly positive for both Investment Grade and High Yield firms for all entry triggers.

Furthermore, when we segregate between Long and Short trades we see that both are significantly profitable (with the exception of Short trades using a trigger of 2 for High Yield firms).

As for the distribution of trade returns, the max and min seem to be relatively symmetric (with a small positive skew). The positive performance of the strategy seems to mostly come from a higher proportion of profitable trades.

Another interesting feature we can observe is the distribution of trades as we vary the different entry triggers. While we see the number of trades in the High Yield portfolio more than halving (falls from 18,424 to 8,351 trades) as we go from a trigger of 0.5 to 2, the Investment Grade portfolio seems to remain a lot more constant, falling only 25% (from 51,091 to 37,943 trades). This would indicate the existence of larger spreads on the Investment Grade side, consistent with the underpricing we had seen before.

The histograms below provide additional information regarding the returns distribution.


1 8,539 0.21% 0.21% 0.53% 37.4 3.67% -2.44% 74.9% 1802 7,207 0.22% 0.22% 0.55% 34.8 3.67% -2.43% 75.1% 180


1 4,279 0.15% -0.03% 0.83% 11.6 3.22% -2.20% 48.0% 1802 1,144 -0.03% -0.28% 0.73% -1.5 2.46% -1.29% 33.1% 180


1 12,818 0.19% 0.16% 0.64% 33.6 3.67% -2.44% 65.9% 1802 8,351 0.19% 0.20% 0.58% 29.7 3.67% -2.43% 69.4% 180

Long

Short

All

16


Figure 3: Holding period return distribution- histogram depicting holding period returns for triggers of 0.5, 1 and 2 for Investment Grade and High Yield obligors. Returns are in percentage and N denotes the number of trades in each bin.

Returns for both Investment Grade and High Yield seem to be symmetrical. Volatility for returns for investment grades seem to be a lot smaller. This, however masks the fact that although a very high mass of the distribution is concentrated very near the mean, we still get tails at almost the level of the High Yield debt.

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Dynamic Hedge Trade Return Distribution We next test the impact of using dynamic hedging in setting up the equity hedge ratios. The tables below show the holding period returns for the delta-hedged strategy:

Table 8: Dynamic hedge IG - summary statistics of holding period returns for Investment Grade obligors. N is the number of trades. The “Long” panel shows trades which are short CDS (=Long Credit Risk) and short stocks for the hedge. The “Short” panel shows trades which are long CDS (=Short Credit Risk) and long stocks for the hedge. The “All” panel shows both long and short trades. % Positive is the percentage of trade with positive returns. Returns are in percentage. Holding Period is in days.

Trigger N Mean Median Std Dev Max Min % Positive Median Holding Period0.5 39,973 0.06% 0.05% 0.23% 4.71% -1.73% 76.0% 180

1 37,005 0.06% 0.05% 0.22% 4.71% -1.73% 76.6% 1802 32,898 0.05% 0.05% 0.20% 4.33% -1.73% 77.5% 180


1 8,218 0.24% 0.14% 0.66% 5.02% -3.45% 67.2% 1802 5,045 0.37% 0.26% 0.70% 5.02% -3.18% 74.6% 180


1 45,223 0.09% 0.05% 0.35% 5.02% -3.45% 74.9% 1802 37,943 0.10% 0.05% 0.33% 5.02% -3.18% 77.1% 180

Long

Short

All

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Table 9: Dynamic hedge HY - summary statistics of holding period returns for High Yield obligors. N is the number of trades. The “Long” panel shows trades which are short CDS (=Long Credit Risk) and short stocks for the hedge. The “Short” panel shows trades which are long CDS (=Short Credit Risk) and long stocks for the hedge. The “All” panel shows both long and short trades. % Positive is the percentage of trade with positive returns. Returns are in percentage. Holding Period is in days.

Interestingly the hedge not only does not reduce trade volatility, but for some cases it actually increases it. This observation merits further analysis since it suggest that rather than hedging it, the equity position compounds on the CDS. In our model this would mean that the equity and debt securities are moving in opposite directions.


1 8,539 0.20% 0.21% 0.53% 35.4 3.65% -2.43% 75.1% 1802 7,207 0.21% 0.22% 0.55% 32.3 3.65% -2.37% 75.0% 180


1 4,279 0.14% 0.05% 0.82% 11.4 3.09% -2.60% 53.7% 1802 1,144 -0.09% -0.24% 0.80% -3.8 2.25% -2.60% 36.4% 180


1 12,818 0.18% 0.18% 0.65% 32.3 3.65% -2.60% 68.0% 1802 8,351 0.17% 0.20% 0.60% 25.7 3.65% -2.60% 69.7% 180

Long

Short

All

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Figure 4: Holding period return distribution Delta Hedge - histogram depicting holding period returns for triggers of 0.5, 1 and 2 for Investment Grade and High Yield obligors. Returns are in percentage and N denotes the number of trades in each bin.

The graphs of the Delta Hedged trades are practically identical to the ones we find for static hedge.

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Monthly Returns We now look at monthly returns for the strategy. On any given day we aggregate the returns for all open trades. The result is that (particularly for lower triggers where the amount of trades increases) volatility is very high. In practice we would further scale down leverage or cap the amount of possible simultaneous open trades.

Table 10: Static hedge HY & IG monthly returns- summary statistics of monthly returns for the portfolio of Investment Grade and High Yield issuers (taken separately) for each trigger. N is the number of months in the sample. All numbers are daily returns aggregated over the month (not annualized) returns for the sum of all open trades

Table 11: Dynamic hedge HY & IG monthly returns- summary statistics of monthly returns for the portfolio of Investment Grade and High Yield issuers (taken separately) for each trigger. N is the number of months in the sample. All numbers are daily returns aggregated over the month (not annualized) returns for the sum of all open trades

Monthly returns yield a similar picture as our analysis of trades. Returns are significantly positive (no risk-free rate is subtracted in calculating this) across all triggers and both Investment Grade and High Yield issuers.

One interesting aspect is that results for delta hedging seem to reduce volatility for Investment Grade issuers but not so for those in the High Yield portfolio.

We next analyze the time series of returns:

Trigger N Mean Median Min Max Std T-Stat Sharpe0.5 132 2.7% 1.7% -39.0% 115.1% 13.0% 2.4 0.21

1 132 2.9% 1.6% -38.8% 116.2% 13.3% 2.5 0.212 132 2.5% 1.7% -28.6% 73.6% 9.5% 3.0 0.27

0.5 132 3.7% 2.4% -37.7% 62.1% 14.6% 2.9 0.251 132 3.4% 2.9% -38.3% 62.4% 14.1% 2.8 0.242 132 3.1% 1.6% -32.0% 53.7% 13.7% 2.6 0.22

Inv Grade

High Yield


1 132 2.7% 1.4% -36.8% 112.5% 12.7% 2.5 0.222 132 2.4% 1.5% -24.9% 64.6% 8.2% 3.3 0.29

0.5 132 4.2% 2.6% -59.5% 110.0% 18.4% 2.6 0.231 132 3.6% 2.6% -39.0% 67.7% 15.1% 2.7 0.232 132 2.7% 2.4% -20.4% 28.1% 8.1% 3.9 0.34

Inv Grade

High Yield

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Figure 5: Time series of monthly returns IG vs HY – evolution of monthly returns (in percentage) for Investment Grade and High Yield portfolios.

When we look at the returns series for both portfolios we can see a markedly different skewness. The High Yield portfolio exhibits high and relatively constant volatility throughout most of the period, diminishing towards the end of the sample. The Investment Grade portfolio however, exhibits very low volatility throughout most of the period with a marked spike in the 2008 financial crisis.

Explaining portfolio returns

A feature of the table which is a cause for concern in tables 6 and 7 is that the median holding period is the same as the time-stop, indicating that at least half of the trades are not converging. This raises an issue towards what explains the models returns.

In a statistical arbitrage in which we assumed we are using the same pricing model as the market, we would expect returns to come from a short-term mispricing caused by the market impact of a large participant. With a typical time to convergence above six months, however, it appears that the model is not targeting these short-term liquidity based trades but rather making a directional bet based on a different model.

The issue then centers around whether the model is profitable because it prices credit risk more accurately than the market or because it is making a directional bet that happened to pay off during the period.

Another way of looking at this is how balanced the portfolio is in its trading. If the strategy is exploiting short term mispricing, effectively acting as a market maker, we would expect it to be fairly balanced in its taking the long and short sides of trades.

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To measure this we will particularly look into the Investment Grade portfolio, where we had seen the divergence between model and market spreads seems to be largest.

Table 12: Trade details for long and shorts – the table shows the number of trades long and short in the Investment Grade portfolio (taking the 0.5 trigger portfolio). A large percentage of returns coming from either side would indicate we are taking a directional position in the overall CDS market.

When we go to results we see that the portfolio is fairly balanced in its sourcing of returns. Although long trades are more than three times more frequent, short trades compensated during the period by having a larger average return.

Another danger, however, is that the portfolio is consistently making directional bets in single securities:

Trades Long Trades ShortAverage 0.06% Average 0.16%Median 0.05% Median 0.06%Std Dev 0.21% Std Dev 0.67%Count 39973 Count 11118Std Error 0.001% Std Error 0.006%T-Stat 51.4 T-Stat 25.3Max 4.36% Max 7.07%Min -1.89% Min -3.44%Total Return 2204% Total Return 1791%

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Table 13: Number of long and short trades for each issuer – the table shows the number of trades long and short for each of the issuers in the Investment Grade (taking the 0.5 trigger portfolio) and the percentage of trades concentrated in the most heavily traded side. A large concentration on either side would indicate we are taking a directional position in the instrument.

In the table above we can see that the trades the system is taking are highly directional, with an average weight of the most traded side of 84%. This indicates that the system is consistently taking one side of the trades.

What this seems to indicate is that the spreads we see between market and model are so pervasive that the model is making one-sided bets in each security. Although results indicate that it did fairly well during the period, this is a drift from the model’s original objective.

Long ShortCACE1U5 2688 148 95% LongCAET1U5 1252 427 75% LongCMO1U5 2501 187 93% LongCAEP1U5 1777 432 80% LongCAXP1U5 2564 214 92% LongCAIG1U5 1080 930 54% LongCAPC1U5 1834 534 77% LongCARW1U5 2478 153 94% LongCFSAI1U5 1414 101 93% LongCSBC1U5 2725 66 98% LongCAZO1U5 1992 310 87% LongCAVT1U5 2668 14 99% LongCAVP1U5 607 449 57% LongCBAX1U5 2705 2 100% LongCFO1U5 795 414 66% LongCHRB1U5 1111 116 91% LongCBMY1U5 2396 133 95% LongCCPB1U5 2699 12 100% LongCCOF1U5 2166 402 84% LongCCAH1U5 1150 451 72% LongCNCP1U5 108 1303 92% ShortCAMG1U5 430 2093 83% ShortCABX1U5 183 993 84% ShortCBSX1U5 626 1210 66% Short

% Directional

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Periodic Recalibration

By fitting the default barrier to market spreads more frequently we intend to reduce the drift away from the initial conditions in which we calibrate and thus reduce a systematic divergence from market levels. What we expect to see here is shorter trades with a better long-short balance as the short term noise creates trades opportunities around the market mean.

To test this assumption we increase the rebalance frequency to once every two years (matching Yu’s original sample size).

Results When we go to the results of the trading strategy we can see that it does worse both in terms of holding period returns and monthly returns:

Table 14: Static Hedge IG with recalibration - summary statistics of holding period returns for Investment Grade obligors. N is the number of trades. The “Long” panel shows trades which are short CDS (=Long Credit Risk) and short stocks for the hedge. The “Short” panel shows trades which are long CDS (=Short Credit Risk) and long stocks for the hedge. The “All” panel shows both long and short trades. % Positive is the percentage of trade with positive returns. Returns are in percentage. Holding Period is in days.


1 23,681 0.03% 0.03% 0.19% 28.3 3.88% -1.82% 70.5% 178


1 7,677 0.23% 0.07% 0.81% 24.8 7.57% -2.80% 64.2% 146


1 31,358 0.08% 0.03% 0.44% 33.0 7.57% -2.80% 69.0% 167

Long

Short

All

25


Table 15: Static Hedge HY with recalibration - summary statistics of holding period returns for Investment Grade obligors. N is the number of trades. The “Long” panel shows trades which are short CDS (=Long Credit Risk) and short stocks for the hedge. The “Short” panel shows trades which are long CDS (=Short Credit Risk) and long stocks for the hedge. The “All” panel shows both long and short trades. % Positive is the percentage of trade with positive returns. Returns are in percentage. Holding Period is in days.

When we look at holding period returns we see that both portfolios are still highly significant, although returns have diminished as compared to the single-calibration portfolio. In particular, median returns for the High Yield portfolio are now negative.

We do see, however, median holding period falling for the Investment Grade portfolio.

Table 16: Holding period returns for periodic recalibration – summary statistics of monthly returns for the portfolio of Investment Grade and High Yield issuers (taken separately) for each trigger. N is the number of months in the sample. All numbers are daily returns aggregated over the month (not annualized) returns for the sum of all open trades

In the above table we can see that Investment Grade returns are no longer significantly positive and returns for the High Yield portfolio are only slightly above the significance threshold.

One caveat to be taken into account, however, is that with such a small sample size a significant event in a single security can have a very large impact on results. In particular, a trade on American Axle (part of the High Yield portfolio) explains a big part of the decay in results. In this trade the stock rose 150% on a single day following a favorable earnings report (on May 2009) but the movement was not followed by


1 6,693 0.14% 0.18% 0.82% 14.2 3.67% -15.71% 72.4% 180

Trigger N Mean Median Std Dev T-Stat Max Min % Positive Median Holding Period0.5 5,602 0.08% -0.02% 0.87% 6.8 6.36% -5.51% 48.0% 180

1 2,890 0.04% -0.13% 0.84% 2.8 3.22% -4.43% 41.0% 180


1 9,583 0.11% 0.12% 0.82% 13.3 3.67% -15.71% 62.9% 180

Long

Short

All


1 132 2.7% 0.9% -55.3% 212.4% 20.3% 1.5 0.130.5 132 3.2% 1.0% -37.4% 56.8% 14.4% 2.5 0.22

1 132 4.1% 1.3% -64.8% 140.2% 23.2% 2.0 0.18High Yield

Inv Grade

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the CDS. The result was that our portfolio, heavily loaded on the short side took a very large loss (the 15.7% observed above). Being a highly idiosyncratic risk, its impact could be significantly reduced as we add more securities to our portfolios.

As a check on whether the portfolio is more balanced, we repeat the test on the percentage of long and short trades in each security.

Table 17: Number of long and short trades for each issuer – the table shows the number of trades long and short for each of the issuers in the Investment Grade (taking the 0.5 trigger portfolio) and the percentage of trades concentrated in the most heavily traded side. A large concentration on either side would indicate we are taking a directional position in the instrument.

We see that after we incorporate more frequent rebalancing the portfolio becomes a lot more balanced, with the average directional concentration in the portfolio falling to about 73%

CDS Ticker Long ShortCACE1U5 1818 272 87% LongCAET1U5 821 547 60% LongCMO1U5 1294 352 79% LongCAEP1U5 1252 389 76% LongCAXP1U5 1006 738 58% LongCAIG1U5 830 1060 56% ShortCAPC1U5 1078 883 55% LongCARW1U5 1794 276 87% LongCFSAI1U5 1211 148 89% LongCSBC1U5 1903 175 92% LongCAZO1U5 1061 483 69% LongCAVT1U5 1126 275 80% LongCAVP1U5 729 340 68% LongCBAX1U5 1701 247 87% LongCFO1U5 965 373 72% LongCHRB1U5 1005 121 89% LongCBMY1U5 1264 712 64% LongCCPB1U5 1316 424 76% LongCCOF1U5 1483 678 69% LongCCAH1U5 1163 423 73% LongCNCP1U5 1174 525 69% LongCAMG1U5 1178 533 69% LongCABX1U5 421 739 64% ShortCBSX1U5 1162 448 72% Long

% Directional

27


Conclusions: Although returns for the original CSA strategy were significantly positive during the period, several factors lead us to question whether the explanation behind the strategy’s profitability is what we had initially assumed.

Taking as a starting the notion that the market is in fact using a structural model as the one proposed in the Yu-CreditGrades framework, we would expect spreads for individual securities to be centered on any given day around this “expected” spread and short-term supply and demand to cause short-term fluctuations around it. Under these circumstances we would expect small magnitudes of divergence, with a corresponding quick time to convergence. We would also expect the model to be fairly neutral, as we have no reason to believe supply and demand to act in any single direction (making market spreads rich or cheap randomly). Given this model’s objective of trading on supply-demand imbalances around the market’s spread expectation, we could call this a “market maker” or “liquidity providing” model

The picture we see, however, seems to be quite different. The time to convergence seems to be too long (with more than 50% of trades not reaching convergence during the 180 day trading period). We also see that, for individual securities, long and short trades seem to occur in a heavily uneven distribution (with an average of 86% of trades occurring in a single direction). Furthermore, issues such as the apparent “reversing” in the relationship between the CDS and the hedge returns indicate a less reliable relationship.

The picture described above does not seem to correspond to what we would expect from simply providing liquidity, but rather taking a directional view on spreads of particular securities. The question remains though as to whether the model’s assessment of credit risk, different as it may be from the market’s, is not the right one. Significantly positive results would seem to indicate this, with results robust to changing parameter specifications and changing the sample (from Investment Grade to High Yield obligors). The issue, though, is how representative is this sample of the overall market. Particularly dangerous is the “survivorship bias” inherent in the sample, caused by exclusively taking firms which are part of the index today. One caveat about this danger is that results do not seem to depend exclusively on either the long or short side so, if we have inadvertently introduced a sample with a certain spread drift, the model does not seem to be picking up on it.

The re-calibrating version of the strategy seems to be a step in the direction of the “market maker” model. Here we are periodically removing the effect of our “directional view” by matching the market’s spreads. As we would expect, we see holding periods decrease and long/short trades being more balanced for individual securities. We also see returns becoming less significant as compared to the original model’s but, given that we do not yet have a clear comparison of the trades each strategy takes we cannot yet say how redundant or complimentary both strategies are.

In terms of the next steps for research in the model there seem to be three main possible directions.

The first is to extend the analysis on the current path. Going down this route would require learning more about what causes the positive performance of the “directional view” model, starting by using a

28


larger sample with a historically-consistent method for portfolio selection (possibly using index constituents as of each date). A sample of trades on this sample using random entry signals with similar holding periods could also be gathered to test for any skew and isolate the model’s timing abilities. As for the “market maker” portfolio, further steps typically associated with a market neutral strategy could be taken: exploring the impact of more frequent calibrations or forming long-short portfolios based on the level of deviation.

A second alternative is moving to hedging with assets in which convergence within a certain time-frame is more likely to occur, such as trading CDSs against options with matching maturities. A particularly interesting comparison is that of CDSs and very OTM puts, since these securities would profit under similar events. The intuition here is that even if we are not using the same model as market participants to value these extreme events, the models used to price these related security should be consistent with each other.

The third option would involve moving to a different model which does better in matching the market expectations. Insights into what sort of models could work can be taken from the options world (Huang and Huang (2003), for example incorporate a jump-diffusion model in estimating the evolution of the value of assets).

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Appendix:

Merton model: Taking as a starting point the diffusion process assumed for the evolution of the firm’s assets, the paper then defines a security whose value 𝑌𝑌 = 𝐹𝐹(𝑑𝑑, 𝑑𝑑) depends on the value of the firm’s assets and time following a similar process:

𝑑𝑑𝑌𝑌 = ( 𝛼𝛼𝑌𝑌𝑑𝑑 − 𝐶𝐶𝑌𝑌) 𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑌𝑌𝑑𝑑𝑑𝑑𝜎𝜎𝑌𝑌

By Ito’s Lemma and combining with the asset evolution process defined above we can express return and volatility of the price of the derivative security as:

𝛼𝛼𝑌𝑌𝐹𝐹 =12

𝜎𝜎2𝑑𝑑2 𝜕𝜕2𝐹𝐹𝜕𝜕𝑑𝑑2 + (𝛼𝛼𝑑𝑑 − 𝐶𝐶)

𝜕𝜕𝐹𝐹𝜕𝜕𝑑𝑑

+𝜕𝜕𝐹𝐹𝜕𝜕𝑑𝑑

+ 𝐶𝐶𝑌𝑌

𝜎𝜎𝑌𝑌𝐹𝐹 = 𝜎𝜎𝑑𝑑𝜕𝜕𝐹𝐹𝜕𝜕𝑑𝑑

We will also say the stochastic term in the derivative’s pricing process corresponds to the same Wiener process we followed by the firm’s assets:

𝑑𝑑𝜎𝜎𝑦𝑦 ≡ 𝑑𝑑𝜎𝜎

Similar to the Merton’s no-arbitrage derivation of the option pricing formulas, assuming we can trade in continuous time, we can build a zero net investment portfolio 𝑋𝑋 consisting of an investment of 𝑊𝑊1dollars in the firm’s assets, 𝑊𝑊2dollars in the derivative security and 𝑊𝑊3dollars in a risk-free security.

We choose 𝑊𝑊3 = −(𝑊𝑊1 + 𝑊𝑊2) to satisfy the zero net investment condition. The return on this portfolio will thus be:

𝑑𝑑𝑋𝑋 = 𝑊𝑊1(𝑑𝑑𝑑𝑑 + 𝐶𝐶𝑑𝑑𝑑𝑑)

𝑑𝑑+ 𝑊𝑊2

(𝑑𝑑𝑌𝑌 + 𝐶𝐶𝑌𝑌𝑑𝑑𝑑𝑑)𝑌𝑌

− (𝑊𝑊1 + 𝑊𝑊2) 𝑟𝑟 𝑑𝑑𝑑𝑑

We then replace the original equations for the evolution of 𝑑𝑑𝑑𝑑 and 𝑑𝑑𝑌𝑌:

𝑑𝑑𝑋𝑋 = 𝑊𝑊1(𝛼𝛼𝑑𝑑𝑑𝑑𝑑𝑑 − 𝐶𝐶𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑑𝑑𝑑𝑑𝜎𝜎 + 𝐶𝐶𝑑𝑑𝑑𝑑)

𝑑𝑑+ 𝑊𝑊2

(𝛼𝛼𝑌𝑌𝑌𝑌𝑑𝑑𝑑𝑑 − 𝐶𝐶𝑌𝑌𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑌𝑌𝑌𝑌𝑑𝑑𝜎𝜎 + 𝐶𝐶𝑌𝑌𝑑𝑑𝑑𝑑)𝑌𝑌

− (𝑊𝑊1 + 𝑊𝑊2) 𝑟𝑟 𝑑𝑑𝑑𝑑

𝑑𝑑𝑋𝑋 = 𝑊𝑊1(𝛼𝛼𝑑𝑑𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑑𝑑𝑑𝑑𝜎𝜎)

𝑑𝑑− 𝑊𝑊1𝑟𝑟𝑑𝑑𝑑𝑑 + 𝑊𝑊2

(𝛼𝛼𝑌𝑌𝑌𝑌𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑌𝑌𝑌𝑌𝑑𝑑𝜎𝜎)𝑌𝑌

− 𝑊𝑊2𝑟𝑟𝑑𝑑𝑑𝑑

𝑑𝑑𝑋𝑋 = 𝑊𝑊1(𝛼𝛼 − 𝑟𝑟)𝑑𝑑𝑑𝑑 + 𝑊𝑊2(𝛼𝛼𝑌𝑌 − 𝑟𝑟)𝑑𝑑𝑑𝑑 + 𝑊𝑊1 𝜎𝜎𝑑𝑑𝜎𝜎 + 𝑊𝑊2 𝜎𝜎𝑌𝑌𝑑𝑑𝜎𝜎

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We choose weights so that the portfolio is risk-free and its expected return is zero (to satisfy no-arbitrage):

𝑁𝑁𝑎𝑎 𝑅𝑅𝐸𝐸𝑅𝑅𝑅𝑅: 𝑊𝑊1∗𝜎𝜎 + 𝑊𝑊2

∗𝜎𝜎𝑌𝑌 = 0

𝑁𝑁𝑎𝑎 𝐴𝐴𝑟𝑟𝐷𝐷𝐸𝐸𝑑𝑑𝑟𝑟𝑀𝑀𝑇𝑇𝐷𝐷: 𝑊𝑊1∗(𝛼𝛼 − 𝑟𝑟) + 𝑊𝑊2

∗(𝛼𝛼𝑌𝑌 − 𝑟𝑟) = 0

A non-trivial solution to the above equations exists if and only if the return on the derivative is simply a leveraged return on the firm’s assets:

𝛼𝛼 − 𝑟𝑟𝜎𝜎

=𝛼𝛼𝑌𝑌 − 𝑟𝑟

𝜎𝜎𝑌𝑌

By replacing the formulas we obtained for the return and volatility of the derivative (𝛼𝛼𝑌𝑌 and 𝜎𝜎𝑌𝑌 respectively) we get:

𝛼𝛼 − 𝑟𝑟𝜎𝜎

=12 𝜎𝜎2𝑑𝑑2 𝜕𝜕2𝐹𝐹

𝜕𝜕𝑑𝑑2 + (𝛼𝛼𝑑𝑑 − 𝐶𝐶) 𝜕𝜕𝐹𝐹𝜕𝜕𝑑𝑑 + 𝜕𝜕𝐹𝐹

𝜕𝜕𝑑𝑑 + 𝐶𝐶𝑌𝑌

𝜎𝜎𝑑𝑑 𝜕𝜕𝐹𝐹𝜕𝜕𝑑𝑑

We then rearrange the terms to get:

0 =12

𝜎𝜎2𝑑𝑑2 𝜕𝜕2𝐹𝐹𝜕𝜕𝑑𝑑2 + (𝑟𝑟𝑑𝑑 − 𝐶𝐶)


− 𝑟𝑟𝐹𝐹 +𝜕𝜕𝐹𝐹𝜕𝜕𝑑𝑑

+ 𝐶𝐶𝑌𝑌

The key observation in this result is that the value of any derivative security on the firm’s asset does not depend on the asset’s expected return.

Since the above equation would be satisfied by any derivative security, we then specify further conditions to define the value of the debt and equity.

As mentioned above, the firm will only issue a single form of debt and equity. The debt will further be assumed to be a zero-coupon bond with face value B and maturity T. At maturity, the bond will either be repaid in full or bondholders will assume control of the company, making equity worthless. Further covenants on the debt state that the firm cannot pay dividends or issue new debt prior to maturity.

We next define τ as the time to maturity and, given the zero-coupon and no-dividend assumptions simplify the above equation as:

0 =12

𝜎𝜎2𝑑𝑑2 𝜕𝜕2𝐹𝐹𝜕𝜕𝑑𝑑2 + 𝑟𝑟𝑑𝑑


− 𝑟𝑟𝐹𝐹 −𝜕𝜕𝐹𝐹𝜕𝜕𝜏𝜏

Since we have only two types of claims on the firm’s assets, they will be completely divided between debt (F) and equity (f):

𝑑𝑑 ≡ 𝐹𝐹(𝑑𝑑, 𝜏𝜏) + 𝑓𝑓(𝑑𝑑, 𝜏𝜏)

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Given our definition of the debt security, at maturity it will either be repaid in full (paying the face value B) or, if assets are not sufficient to cover the face value, the firm will default and all of the firm’s assets (V) will go to bondholders. Thus:

𝐹𝐹(𝑑𝑑, 0) = min (𝑑𝑑, 𝐵𝐵) or 𝐹𝐹(𝑑𝑑, 0) = 𝐵𝐵 − min (𝑑𝑑 − 𝐵𝐵, 0)

At maturity, equity, defined as the residual claim on the assets, will then be:

𝑓𝑓(𝑑𝑑, 0) = max (0, 𝑑𝑑 − 𝐵𝐵)

The value of equity is thus identical to a call option on the firm’s assets with strike price equal to the debt’s face value (𝐵𝐵) and can be valued according to Black-Scholes as:

𝑓𝑓(𝑑𝑑, 𝜏𝜏) = Vϕ(𝑀𝑀1) − 𝐵𝐵𝐷𝐷−𝑟𝑟𝑟𝑟𝜙𝜙(𝑀𝑀2)

Where

𝑀𝑀1 ≡ �ln �𝑑𝑑

𝐵𝐵� + (𝑟𝑟 + 12 𝜎𝜎2)𝜏𝜏�

𝜎𝜎√𝜏𝜏

𝑀𝑀2 ≡ 𝑀𝑀1 − 𝜎𝜎√𝜏𝜏

Before writing the price of the debt security 𝐹𝐹(𝑑𝑑, 𝜏𝜏) we define 𝑑𝑑, the debt-to-firm value as:

𝑑𝑑 =𝐵𝐵𝐷𝐷−𝑟𝑟𝑟𝑟

𝑑𝑑

We then substitute the value we found for 𝑓𝑓(𝑑𝑑, 𝜏𝜏) into our definition 𝑑𝑑 ≡ 𝐹𝐹(𝑑𝑑, 𝜏𝜏) + 𝑓𝑓(𝑑𝑑, 𝜏𝜏) and rearrange to find 𝐹𝐹(𝑑𝑑, 𝜏𝜏):

𝐹𝐹(𝑑𝑑, 𝜏𝜏) = 𝐵𝐵𝐷𝐷−𝑟𝑟𝑟𝑟 �𝜙𝜙(ℎ2) +1𝑑𝑑

ϕ(ℎ1)�

Where

ℎ1 ≡ −𝑀𝑀1 = −�12 𝜎𝜎2𝜏𝜏 − ln (𝑑𝑑)�

𝜎𝜎√𝜏𝜏

ℎ2 ≡ 𝑀𝑀2 = −�12 𝜎𝜎2𝜏𝜏 + ln (𝑑𝑑)�

𝜎𝜎√𝜏𝜏

The CDS spread implied in the risky bond’s price can be calculated as the difference between its yield and the risk-free rate. We first define the bond’s yield as:

𝐹𝐹(𝑑𝑑, 𝜏𝜏) = 𝐵𝐵−𝑦𝑦𝑟𝑟

We then rewrite the above equation to find 𝐸𝐸 − 𝑟𝑟:

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𝐸𝐸 − 𝑟𝑟 = −1𝜏𝜏

𝑙𝑙𝑀𝑀 �𝜙𝜙(ℎ2) +1𝑑𝑑

ϕ(ℎ1)�

The firm’s asset volatility can be inferred from Equity volatility using the formula:

𝜎𝜎𝜕𝜕 =𝜎𝜎𝐴𝐴𝜙𝜙(𝑀𝑀1)

𝜙𝜙(𝑀𝑀1) − 𝑑𝑑𝜙𝜙(𝑀𝑀2)

Market spreads table:

The table below provides additional information to Table 3 which gave a snapshot of average market spreads on for

Table 18: Number of obligors per stock-to debt/equity volatility bins - the table below shows the number of entities falling into each quadrant used in table 3. The issuer universe comprises the constituents of the CDX North America Investment and High Yield indices and the iTraxx Main and Crossover. The vertical axis marks the stock-to-debt ratio (calculated as share price over debt-per-share) and the horizontal axis shows expected equity volatility (in percentage). The values in the table are the number of entities falling into each quadrant as of the observation date.

Market 20 25 30 35 40 45 50 55 60 650.5 11 9 9 7 1 1 1 11 16 16 8 4 3 5

1.5 12 5 6 3 2 12 14 3 6 2 2

2.5 12 8 3 2 13 11 2 2 1

3.5 5 2 2 14 4 1 1

4.5 7 1 15 1 1

5.5 1 16

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Bibliography:

Yu, Fan (2006) “How Profitable is Capital Structure Arbitrage” Financial Analysts Journal 62(5), 47-62

Merton, Robert (1974) “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates” The Journal of Finance 29, 449-470.

Black, Fisher and Cox, John (1976) “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions” The Journal of Finance 31, 351-367

Bharath, Sreedhar and Shumway, Tyler “Forecasting Default with the KMV-Merton Model” AFA 2006 Boston Meetings Paper (2004)

Finger, Christopher “CreditGrades Technical Document” RiskMetrics Group (2002)

Sundaresan, Suresh (2013) “A Review of Merton’s Model of the Firm’s Capital Structure with its Wide Applications” The Annual Review of Financial Economics

Bajlum, Claus and Larsen, Peter Tind (2007) “Capital Structure Arbitrage: Model Choice and Volatility Calibration” Copenhagen Business School, Department of Finance, Working Papers

Leland, Hayne and Toft, Klaus Bjerre (1996) “Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads” The Journal of Finance 51(3),987-1019

Sepp, Artur (2006) “Extended CreditGrades Model with Stochastic Volatility and Jumps” Wilmott Magazine

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Federico Thibaud - Capital Structure Arbitrage