Correlation Breakdown in the Valuation of Collateralized Fund Obligations

UNAI ANSEJO, MARCOS ESCOBAR, AITOR BERGARA, AND LUIS SECO

UNAI ANSEJO ollateralized Debt Obligations have an AAA-rated tranche, an AA-rated is a doctoral student at Uni- (CDO) are structured credit vehi- tranche, a single A tranche, a BBB rated verisdad del Pais Vasco in cles that redistribute credit risk to tranche and an equity tranche. Exhibit 1 shows Bilbao, Spain. [email protected] meet demands for a wide a schematic prototype of a CFO structure. A Crange of rated securities with scheduled interest CFO can be regarded as a financial structure MARCOS ESCOBAR and principal payments. CDOs are securitized with equity and lenders where all the is an assistant professor in by diversified pools of debt instruments. assets, equity and are invested in a port- the Department of Mathe- Recent developments in credit structuring folio of hedge funds. The lenders earn a spread matics at Ryerson Univer- technology include the introduction of Collat- over interest rates and the equity holders, usu- sity in Toronto, Ontario, Canada. eralized Fund Obligations (CFO).The capital ally the manager of the CFO, earn the total [email protected] structure of a Collateralized Fund Obligation return of the fund minus the financing fees. is similar to traditional CDOs, meaning that As pointed out in recent articles on this AITOR BERGARA investors are offered a spectrum of rated debt topic (Mahadevan and Schwartz [2002] and is professor in the Depart- securities and equity interest. Although any Stone and Zissu [2004]), both investors and ment of Physics at Uni- verisdad del Pais Vasco in managed fund can be the source of collateral, issuers find CFO attractive. Bilbao, Spain. the target collateral in these structures tends Investors, because a triple-A-rated bond will [email protected] to be hedge funds, such as relative value hedge have a yield similar to that of a triple-A col- funds, event-driven hedge funds or com- lateralized debt obligation plus a premium, LUIS SECO modity trading advisors (CTAs), along with because with this structure they gain expo- is professor in the Depart- funds that finance the needs of growing com- sure to a diverse collection of hedge funds ment of Mathematics at University of Toronto in panies, such as private equity and mezzanine through a fund of funds manager. And issuers, Toronto, Ontario, Canada. funds. Often, a special purpose entity pur- because is a convenient vehicle [email protected] chases the pool of underlying for raising funds for an otherwise relatively investments, which are then used as collateral illiquid product. Stone and Zissu [2004] pro- to back the notes. vide a detailed overview of the first securiti- These asset-backed notes are also called zation of a fund of funds, the Diversified tranches. The most senior tranche is usually Strategies CFO SA, launched in June 2002. rated AAA and is credit-enhanced due to the This CFO issued US$251 million in five subordination of lower tranches. This means tranches, rated AAA, AA, A, BBB and the that the lowest tranche, which is typically the equity. Also in June of 2002, lawyers from equity tranche, absorbs losses first. When the the Structured Products Group completed the equity tranche is exhausted, the next lowest second Collateralized Fund Obligation backed tranche begins absorbing losses. A CFO may by hedge fund portfolios and assigned ratings

WINTER 2006 THE JOURNAL OF ALTERNATIVE INVESTMENT 1 E XHIBIT 1 Schematic Prototype of a CFO Structure. Typical Maturities 3–7 Years

by Moody’s Investors Service, Inc. and Standard & Poor’s The credit rating of a CFO tranche depends directly Rating Services. The CFO, titled Man Glenwood Alter- on the mark-to- value of the pool of hedge funds. native Strategies I or “MAST I,” issued rated notes Collateralized fund obligations tend to be structured as totaling US$374 million and nonrated notes and prefer- arbitrage market value CDOs, meaning that the fund of ence shares totaling US$176 million. As the first of their funds manager focuses efforts on actively managing the kind, these CFOs were viewed as a cutting edge trans- fund to maximize total return while restraining price action within the industry and attracted considerable volatility within the guidelines of the structure. The diver- attention in the financial press. sity of hedge fund investment strategies and the active As far as valuation is concerned, Moody’s in August management of portfolio positions among strategies ensure 2003, began to use HedgeFund.net data to evaluate the that hedge fund risk and return characteristics are dif- risks of the underlying collateral in order to develop an ferent from those of traditional assets (equities, bonds or accurate Montecarlo-based rating model for CFOs (see ETFs), as illustrated by numerous articles in the literature Moody’s [2003]). Although CDOs have attracted much (e.g., Moody’s [2003] and CISDM [2006]). The unique attention in the academic literature (Hull and White return and risk characteristics of hedge funds are better [2004]; Li [2000], and Laurent and Gregory [2003]), no characterized by more general distributions than the usual effort has been made to develop an analytical rating model multivariate Gaussian approach. Therefore the probability for CFOs. of default of the different tranches will be also more

2 CORRELATION BREAKDOWN IN THE VALUATION OF COLLATERALIZED FUND OBLIGATIONS WINTER 2006 volatile. In addition, lack of transparency within hedge about future cash flows to be generated by the fund may funds in general makes it more difficult to obtain high- be closely guarded. Hedge funds are largely unregulated frequency historical data which leaves investors with no because they are typically limited partnerships with fewer choice but to calibrate rather complex valuation models than 100 investors, which exempts them from the Invest- including the typical non-Gaussian behavior of hedge ment Company Act of 1940. Offshore hedge funds are funds with monthly data. non-U.S. corporations and are not subject to SEC regu- With the ultimate aim of assessing the model and cal- lation. This limited regulation allows hedge funds to be ibration risk incurred by CFOs, we develop in this article extremely flexible in their investment options. Hedge funds an analytical procedure for valuing the different tranches can use selling, leverage, derivatives, and highly con- of a CFO by calculating the probability of default and the centrated investment positions to enhance returns or reduce credit spreads from the bonds, and analyzing the pricing systematic risk. They can also attempt to time the market sensitivities to changes in parameters. When calculating by moving quickly across diverse asset categories. Hedge the probabilities of default and the credit spreads of the funds attract mainly institutions and wealthy individual different tranches of a CFO, one should take into con- investors, with minimum investments typically ranging sideration the non-Gaussian behavior of the returns from from $250,000 to $1 million. Additionally, hedge funds the pool of hedge funds. As we shall show later, those often limit an investor’s liquidity with lock-up periods of credit spreads are directly related to the percentiles of the one year for initial investors and subsequent restrictions underlying portfolio distribution. on withdrawals to certain intervals. Specific investments In our pricing model we will assume that returns made by hedge funds are often carefully guarded secrets. from the pool of hedge funds follow a Covariance- This lack of transparency may make it difficult for a fund Switching (CS) Stochastic Process which is defined in the of hedge funds manager to assess the fund’s aggregate expo- Appendix. This framework allows us to incorporate two sure to a particular investment on a portfolio basis. It also well known characteristics from the return series of hedge presents a challenge to the fund manager attempting to funds: first, the skewed and leptokurtic nature of the mar- monitor a particular hedge fund’s adherence to its adver- ginal distribution functions, and second, the asymmetric tised style or investment approach. However in certain correlation or correlation breakdown phenomenon cases the transparency issues indicated above can be alle- (Longin and Solnik [2001]). As we will later prove, the viated if the manager allows separate accounts. In order to correlation between the different hedge funds depends account for the plethora of hedge fund and CTA strate- on the direction of the market. For instance, correlations gies and the speed at which a given fund’s composition tend to be larger in a bear market than in a bull market. can be changed, we model the alternative assets to exhibit The structure of this article is as follows. Providing more exotic distributions than the Gaussian one, displaying empirical evidence of the existence of correlation regimes, asymmetric and leptokurtic returns. in Section 2 we present the Covariance-Switching process As is the case with most diversified portfolios, fund as the model for the profit and loss function for a pool of of hedge funds reduce the numerous types of risks arising hedge funds and we outline some of their properties. The from individual hedge funds by diversifying both across model for valuing a CFO is presented in Section 3, and strategies as well as the number of funds in each strategy. in Section 4 we illustrate the strong parameter depen- Both types of diversification reduce risk, but because of dence with an example of a hypothetical CFO based on the potential for style drift and a rapid total loss of value the S&P CTA index. This methodology can be applied in a single fund, diversification across funds is especially to other daily hedge fund indexes such as the Dow Jones important. Diversification is most effective for assets that Hedge Fund Strategy Benchmarks and others. Section 5 exhibit low correlations in value over time. Fund of hedge contains our conclusions. funds’ managers pursuing a low-volatility strategy seek to assemble a portfolio of relatively uncorrelated assets. Unfortunately, in times of market distress, such as the POOL OF HEDGE FUNDS MODEL second half of 1998, previously uncorrelated hedge funds Hedge funds in general follow dynamic investment or CTAs can become highly correlated for short periods. strategies using a variety of assets. Typically the investment This phenomenon is often called correlation breakdown. process is not transparent and in most cases knowledge Evidence of this is presented in Exhibit 2, which shows

WINTER 2006 THE JOURNAL OF ALTERNATIVE INVESTMENT 3 E XHIBIT 2 Correlation Switching Phenomenon Each pixel represents a correlation number between the funds, one from each axis. As we can see, it is apparent that in distressed periods the correlations between the different managers tend to be greater in absolute value than in normal or tranquile periods.

the correlation matrices between the return series of hedge infinite number of contributions, any distribution can be fund managers under tranquil and distressed regimes. reconstructed exactly. During tranquil periods correlations are lower, whereas A general CS is defined through its stochastic dif- during periods of market distress, the asset returns become ferential equation as follows: highly correlated, with the magnitudes of off-diagonal Definition: X (t) follow a covariance switching correlation values being close to one in absolute terms. i process with parameters (p,λ,µ, µ0, µ1, σ .,0,σ .,1) if the dif- Therefore, diversifying amongst different assets or markets i i i i fusion process can be represented as: in times of market distress was less effective at reducing risk than many participants had hitherto believed. Arti- dXi()()()() t=+µ i t dtσ i t⋅ dW t (1) cles exploring the contagion phenomenon include Harvey µ()(tJ=+ µ⋅ µ0 + 1− J ). µ1 and Viskanta [1994]; Longin and Solnik [2001], and i t i ti Koedijk and Campbell [2002]. .,01 ., σi()(–)t= J t⋅ σ i+1 J t⋅ σ i In order to capture both the unidimensional lep- σσ.,k ()((t = 1 ,k ttt),… ,σ nk, ( )) tokurtic and asymmetric nature of hedge fund returns i i i and the correlation breakdown phenomenon, we select Where J is a jump process (see appendix for details), the Covariance-Switching Stochastic Process (CS process) t i = 1, ...n, j = 1, ...d, W is a n-dimensional vector of for modeling the returns of a pool of hedge funds from t independent Brownian motion processes, which are inde- the range of parametric alternatives to a Geometric jk, k pendent of J . Moreover, σ, µ are constants (k = 0, 1; Brownian Process. With this election we develop a very t i i i = 1, .., n; j =1, ..., n). tractable model (calculations using this often closely resemble those using a Brownian Process) and allow for In this work the vector of hedgefunds will be a good adjustment to market data. Regime switching assumed CS(p,λ,µ, σ .,0,σ .,1) under the historical measure models have been used before in the field of finance but (P–measure). Notice no jump in the drift is assumed, and mainly in its univariate version (see Yao, Zhang and Zhou then they are CS(p, λ, r, σ .,0, σ .,1) under the risk neutral [2003] for pricing and Choi [2004] for interest measure (Q–measure). Therefore the returns will follow rate modeling). Moreover, regime switching models have CS(p,λ,µ, µ.,0,µ.,1, σ .,0,σ .,1) under P and CS(p,λ, r,µ.,0, the theoretical appeal that by adding together a sufficient µ.,1, σ .,0,σ .,1) under Q (see appendix for details). number of components, any multivariate distribution Notice that the conditional distribution implied may be approximated with reasonable accuracy. With an from a Covariance-Switching process is not closed form

4 CORRELATION BREAKDOWN IN THE VALUATION OF COLLATERALIZED FUND OBLIGATIONS WINTER 2006 but a discrete approximation that leads to a Gaussian mix- typically a year, in which distributions of random vari- ture distribution, providing an intuitive interpretation of ables are sufficient to specify the model. Furthermore, we CS. For example, taking an increment ∆t = 1 then a dis- suppose that returns of the pool of hedge funds follow a crete approximation to the CS density would be the den- Covariance-Switching process CS(p, λ, r, µ.,0, µ.,1, σ .,0, sity function of a mixture of two multivariate Gaussians: σ .,1). Second, we assume that there is no risk of default in the pool of hedge funds, therefore the portfolio’s profit – λ −1 t and loss function will be only market driven. This assump- pe(–)1 –(–)(–)1 xxµµΣ f ()x = e 2 00 0 tion is reasonable insofar as the fund of hedge funds does 2 det π Σ0 not contain a high proportion of speculative grade hedge −λ (2) 1−−−pe()1 − 1(–)(–)xxµµΣ−1 t funds, so the probability of default of each hedge fund + e 2 11 1 2π det Σ is not relevant. Besides, introducing more complexity 1 in the model will result in an increased number of para- meters, and even with the simple model we propose, i .,i . .,i where µi =µ+ µ and Σi =σ σ are the i-th means vector we already have large confidence intervals, as we show and the i-th variance-covariance matrix. If the given later. Gaussian mixture GM distribution were used to describe One could adopt the more realistic but rather the monthly returns of a portfolio, with the first Gaussian abstract view that many hedge funds are nothing but a one could model the tranquil regimes and with the other junior tranche of a massive credit , in recog- the distressed ones. Moreover, the parameter p could be nition of their propensity to default (as was the case of interpreted as the probability of having a tranquil month Long Term Capital Management, Beacon Hill, and more while 1 – p would be the probability of a distressed recently, Amaranth, among others). From this view- month. Unfortunately, GM is not the conditional dis- point, the tranche of a CFO on a fund of hedge funds tribution of a CS process (see appendix), the main dif- will be come a second order , analog of ference between GM and the conditional distributions a CDO squared. The analysis of this, to which this dis- of CS being that for CS, p would be a function of time cussion will provide a frame of reference, is left to a whose path affects the gaussianity of the components in later study. However, this second derivative nature of a the mixture. fund of funds CFO only hightens the need to study Fitting parameters to a Covariance-Switching Process non-gaussian dependence behavior among the hedge has not been explored in great detail in the literature. Few funds, as dependence will not behave linearly in the articles address the unidimensional case (see Choi [2004] tails of the distribution. and Chourdakis [2002]) but none, as far as we know, deals Let us introduce some notation: with multidimensional situations. Therefore we use an ad- hoc algorithm based on a multivariate test of gaussianity. 0 • S i and Si are the initial and end values of the i-th The algorithm focuses on detecting the moments where hedge fund in the pool, 1 < i < n. a jump has occurred, and then uses the samples from tran- • π and π0 are the initial and end values of the port- quil and distress scenarios to estimate standard multidi- folio. mensional Gaussian processes. As a first pass, we define a • Ri is the return of the i-th hedge fund. distressed month as a month where any of the returns of • n is the number of units invested in the i-th the portfolios’ hedge funds is greater in absolute value than i hedge fund and θi is the monetary quantity two standard deviations, and a tranquil month is when all 0 invested, so that θi = niS i. of them are smaller. Then we add or remove sample points • PL = π– π is the profit and loss of the portfolio 0 ππ− in order to create two groups where the hypothesis of and R = 0 is the relative profit and loss. PL π0 multidimensional gaussianity is accepted. • I is the quantity invested by tranche m, 1 ≤ m ≤ m I Σm−1 i M. And Dm + 1= i=1 π0 is the relative cumula- tive quantity invested, D = 0. MODEL M+1 CFO ∗ r∗ • The default rate r is given by Lm = Ime where In our proposed model we make the following Lm is the undertaken future cashflows in one year assumptions. First, we consider a single period model, of tranch m:

WINTER 2006 THE JOURNAL OF ALTERNATIVE INVESTMENT 5 pool of hedge funds. Basically the one-year probability  Im()10+> R PL if R PL    of default is related to the one-year VaR of the collateral I if D<< R 0  m m+1 PL  portfolio. Under the assumption that the returns of the   Lm =   RD–   different hedge funds, Ri, follow a unidimensional CS I PL m if D R D .,0 ., 1 ., 0 ., 1 m   m<< PL m+1 (3) process CS (,,,,,,)prλµµσ σ , using a result pre-   D − D   i i i i  m+1 m  sented in appendix, the profit and loss of the portfolio 0 if R D  PL< m  follows itself a unidimensional CS process with parame- d 0 1 0 0 1 1 ters CS(p, λ, Σi=1 θi µ, θ ⋅ µ , θ ⋅µ , θ ⋅σ ⋅ σ ′ ⋅ θ′, θ⋅ σ ⋅ σ ′⋅ • The recovery rate, f of tranche m, would be θ′). Hence, in order to calculate the probabilities of default RD– m PL m . of the different tranche, we need only calculate the per- DDmm+1– • r is the risk free rate. centiles of a unidimensional CS process. Exhibit 3 pre- •s= r∗– r is the default spread. sents a schematic picture of the model. m Moreover we can obtain the distribution function Consider a portfolio π with underlying hedge funds of the recovery rate of the tranche m, fm, as: S1, ..., Sn, each with ni positions. Its mark-to-market is given by:  RD−  P( f<= x ) P PL m

m−1

π− π=PL < I – π −r ∗ 00∑ i I= L e m i=1 mm

That is, if the mark-to-market of the portfolio at the ∗ end of the period is less than the invested quantity of the smm= r− r m – 1 tranche of less , then the m-th will default.   =InERQ (1 + ) 1 + 1 We can express this inequality in terms of the relative profit   PL{}{} RPL>00 D m+1 < R PL <   and loss and the relative weights in each tranche: (9) RD−  m−1 PL m ππ− I +( 1{}DRD<<  R 0 i – 1 D DD− m PL m+1 PL = <=∑ m mm+1  ππ00i=1

We have to take into account that the yield spread between Note that D1 = 0. Therefore the probability of default a and an otherwise identical bond with from the m-th tranche, PDm, equals: no credit risk reflects the expected actuarial loss, or annual

PDm = P(RPL< Dm+1) (6) expected loss given default, plus a risk premia reflecting, for example, liquidity risk. It is possible to calculate the This equation relates the probability of default to probability of default and the spread of tranche m; details the profit and loss distribution function of the underlying are given in the appendix.

6 CORRELATION BREAKDOWN IN THE VALUATION OF COLLATERALIZED FUND OBLIGATIONS WINTER 2006 E XHIBIT 3 Schematic representation of the CFO valuation model Probabilities of default are given by a percentile of the pool of hedge funds returns distribution. Recovery rates distribution and spread calculations are easy to obtain within this model.

A SAMPLE CFO with the 26% is the equity tranche and the tranche with the 49% is the typically AAA rated tranche. These weights As explained before, CFOs are products which have are in agreement with market conventions and were used not had too much impact among investors. This fact may in the first CFO created, Diversified Strategies CFO SA be explained by the huge variability of the probability of DSCFO1. With these assumptions, the limiting losses default with respect to market conditions. To illustrate would be equal to D6 = 0, D5 = –0.2676, D4 = –0.3326, this feature, we analyze a CFO with the following char- D3 = –0.3724 and D2 = –0.5024. acteristics: There is a total investment of $1 million dol- The pool of Hedge Funds is chosen to be the fol- lars distributed within five tranches with weights 0.4976, lowing six Hedgefunds: BDC Offshore Fund Ltd (A), 0.13, 0.0398, 0.065 and 0.2676 respectively. The tranche Kassirer Market Neutral, Mapleridge Fund LP, New

WINTER 2006 THE JOURNAL OF ALTERNATIVE INVESTMENT 7 Ellington Overseas, Recap Inter and Styx International PARAMETER SENSITIVITIES Ltd. We work with monthly data returns from Jan-2001 to Oct-2005. The investment in each hedge fund is equal Up to now we have calculated the probabilities of to 1/6M$. default of the different tranches from the CFO according First we estimate the parameters of our model using to market conditions during the period of Jan-2001 to the method proposed in the appendix. Notice that hedge Oct-2005. However, how would those probabilities have funds report monthly returns instead of daily, leading to changed if market conditions had been slightly different? a lack of data. Our method is effective because it reduces To answer that question we recalculate the probabilities the number of parameters, allowing for more reliable esti- of default, changing the probability p of a jump in Equa- mations and accurate intervals for the probabilities of tion (1). In this way we measure the effect in default prob- default or for the spread calculations favoring the bid and abilities of a higher probability of being in a distressed offer pricing process. regime. Therefore we valuate the CFO for a grid of prob- In Exhibit 4 we present the estimations of the vari- abilities in the interval from 0 to 1. Exhibit 7 shows that ances for all CTAs returns in the distressed and tranquil the probabilities of default spread over a substantial range regimes and the mean under the historical measure. Vari- when changing the probability of a distress month 1– p ances are greater in distressed than in tranquil regimes. In (market conditions). For example the mezzanine tranche Exhibit 5 we present the estimated correlations in the tran- probability of default could go from 2% to 9%. In Exhibit 8 quil and distressed regime. One can observe the correla- we report the sensitivities of the spread yield to the market tion breakdown phenomenon. Finally the proportion of condition parameter p, which present a similar behavior tranquil months converges to 0.844 as t approaches infinity. to probabilities of default. In both cases a confidence Using these estimations, we can calculate the para- interval for p could be attractive as a reliability measure. meters of the monthly relative profit and loss portfolio A good proxy for the confidence interval of the para- returns under the P-measure. Results are µ= 0.101, σ0 = meter p can be obtained using the binomial distribution, 0.042, σ1 = 0.089, p = 0.867, λ = 1.029. In order to get for example the 95% confidence interval for p ranges from the yearly returns distribution one needs to multiply 0.725 to 0.926, therefore the spread confidence interval monthly means and variances by twelve. With these results, is (1.9%, 3.8%). As a consequence investors could act with using simulations and Equation (2), we are able to calcu- foresight when investing in this type of collateralized fund late the probabilities of default of the different tranches. obligations. In Exhibit 6 we can see that the equity tranche has a one- year probability of default of 19.5% and the most senior CONCLUSION tranche a probability of default of 0.03%. Moreover we report the spread over the risk free rate, supposed to be A covariance switching multidimensional process was 1%. Therefore a mezzanine tranche investor should be proposed and studied for the pricing of collateralized fund expecting a spread of 2.3% and the most senior tranche obligations (CFO). This process presents several desirable investor should be given a spread of 4.85 basis points. We properties, such as closeness under linear transformation. do not report a spread value for the equity tranche because This implies, in particular, that marginals and portfolios usually the yearly coupon on this tranche depends on the will belong to the CS family of processes. Another inter- final portfolio value. esting property of this process is its simplicity for providing

E XHIBIT 4 Mean and Variances

R1 R2 R3 R4 R5 R6

µ 0.012 0.0042 0.00520 0.0095 0.0073 0.00735 2 σt 0.0052 0.0039 0.0136 0.0078 0.0114 0.00313 2 σd 0.0186 0.001 0.0215 0.0127 0.021 0.00654

8 CORRELATION BREAKDOWN IN THE VALUATION OF COLLATERALIZED FUND OBLIGATIONS WINTER 2006 E XHIBIT 5 E XHIBIT 7 Sensitivities of Probabilities of Default to Market Lambdas in Factor analysis Conditions

R1 R2 R3 R4 R5 R6 0 λ 0.564 0.088 0.082 –0.37 0.46 0.458 1 λ 0.862 0.0481 0.0154 –0.697 0.981 0.346

Tranquil Correlation

R1 R2 R3 R4 R5 R6 1 0.0497 0.046 –0.21 0.261 0.26 1 0.0072 –0.033 0.041 0.040 1 –0.031 0.038 0.037 1 –0.172 –0.17 1 0.21 1

Distress Correlation

R1 R2 R3 R4 R5 R6 1 0.0415 0.132 –0.6 0.85 0.30 1 0.0074 –0.033 0.047 0.017 correlation breakdown. We find that because of the lack 1 –0.108 0.15 0.053 of transparency characteristic of hedge funds, as well as the 1 –0.69 –0.24 unavailability of data, confidence intervals in probabilities 1 0.34 of default and credit spreads are high enough to burden 1 CFO proliferation. For example, we find a 95% credit spread interval of 1.9%–3.8%. cumulative distribution probabilities in the unidimensional case, which enables us to compute portfolio cumulative E XHIBIT 8 probabilities by simulating a jump process. A method to Spread Yield Sensitivities to Market Conditions estimate the parameters was also proposed. An empirical analysis shows a better than Gaussian fitting to a time series vector of hedgefunds; the large differences for the covari- ance matrices in tranquil and distress periods not only sup- port the stylized facts described in the literature regarding leptokurtic unidimensional behaviors, but also the reported

E XHIBIT 6 Estimated Probability of Default and Spread Yield for the Tranche m 1 2 3 4 5

PDm 19.56 3.84 2.6 0.65 0.03 sm 4.85 2.3 2.1 1.14 0.1

WINTER 2006 THE JOURNAL OF ALTERNATIVE INVESTMENT 9 dX()()()() t t dt t dW t A PPENDIX i=+µ iσ i ⋅ (12) Covariance-Switching Process µ()(–)t=+ µ J⋅ µ01+1 J ⋅ µ A Covariance-Switching process is presented in this i t i t i .,0 .,1 section. σσi()(–tJ= t⋅ i + 1 Jti)⋅σ We first define a jump process J ∈ {0, 1}. P[∆J ∈ – 1| .,k 1 ,k nk , t t σi (t )= ( σi ( t ),… , σ i ( t )) Jt–=1] =d(t), P[∆Jt ∈ 1| Jt– = 0] = u(t).

Where Jt is a jump process defined previously, i = 1, …n, The law of a jump Jt is described by: j = 1, …d, Wt is a n-dimensional vector of independent Brownian j,k motion processes, which are independent of Jt. Moreover, σi , The intensity of jumps: λ(t)dt k µi are constants (k = 0, 1; i = 1, …, n; j = 1, …, n). The law of the jumps: K(t, dy) = P[∆Jt ∈dy| Jt–]. If P(t) = d a X (t), where X follows a CS Property 1: Σi=1 i ⋅ i i 0 1 0 1 With previous notation, it follows: process with parameters (p, λ, µ, µi , µi , σ i , σ i ) then P(t) fol- lows a covariance-switching process with parameters (p, , d λ Σi=1 K(t, dy) = P[∆J ∈ dy| J ] 0 1 0 0 1 1 t t - aiµ, a ⋅ µ , a ⋅ µ , a ⋅ σ ⋅ σ ′⋅a′, a ⋅ σ ⋅ σ ′ ⋅ a′).

= Jt – [(1 – d(t))δ0+d(t)δ–1](dy)+(1–Jt – ) [(1 Process for the underlying hedgefunds – u(t))δ0 + u(t)δ1](dy)

Si(t) follow a covariance switching process with parame- where δx(dy) denotes the dirac delta. It is known that Jt = J0 + .,0 .,1 t d d ters (p, λ, µi, σi , σi ): ∫0 ∫E yλ(s)K(s, dy)ds + Jt, where Jt is a local martingale (purely d d discontinuous martingale) such that J0 = 0 and d〈 Jt, M〉 = 0 for any continuous local martingale M. dS() t i =+µdtσ ()() t⋅ dWP t Remark: Notice that ∫E yλ(s)K(s, dy) = Js – (–d(s)) + (1 – Js–)u(s). St() ii Then i .,0 .,1 σσi()tJ= t⋅ i + (–)1 Jti⋅σ L .,k 1 ,k nk , d (t ) ( ( t ), , ( t )) JJ J(–()()) sds (–1 J )()() susdsJ σi = σi … σ i t=0 +∫ () s––λλ + s+ t 0 t J J()[()–(() sus dsusJdsJ ()) ] d Where P-historical measure, J is the jump process defined ts=0 +∫ λ +– + t t 0 previously, i = 1, …n, j = 1, …d, Wt is a n-dimensional vector of independent Brownian motion processes, which are inde- j,k Moreover, pendent of Jt. Moreover, σi are constants (k = 0, 1; i = 1, …, t n; j = 1, …, n). EJJ[| ] J ()(()–(() sus dsusEJ ())[ | J ]) dds ts0= 0 +∫ λ + – 0 We assume that the jump process does not change with 0 a change of measure (see Merton [1974] for an explanation of the plausibility of this assumption), therefore if Q is the risk- Let us called q(t) = E [ Jt | J0 ], then: free measure then,

t  s (r )( u ( r ) d ( r )) dr  –t (s )( u ( s ) d ( s )) ds q()()() t=+ Jλ s u s e∫0 λ + ds . e ∫0 λ + (10)  0 ∫  dS() t  0  i Q =+rdtσ i ()() t⋅ dW t Sti ()

Notice that q(t) provides the probability of Jt = 1 given information up to t = 0. For simplicity, we assume the following Property 2: Assumes Xi(t) = ln Si(t) then, by Ito’s lemma, parameters: (t) = , u(t) = p, d(t) = 1 – p, which leads to 1 t t Q λ λ Xi(t) = rt – –2 ∫0 σi(s)⋅ σi(s)′ds + ∫0 σi(s)⋅ dW (s).

–λt –λt q(t) = J0 ⋅ e + p(1 – e ) (11) Remark: The distribution of X(t) = (X1(t), …, Xd(t)) conditional on the history of Js for 0 ≤ s ≤ t, under the Q-mea- Definition: Xi (t) follow a covariance switching process sure, is multivariate Gaussian with mean and volatility as fol- 0 1 .,0 .,1 with parameters (p, λ, µ, µi , µi , σi , σi ) if the diffusion process lows (tj denotes the time of a jump, n-number of jumps, both can be represented as: known under the assumption):

10 CORRELATION BREAKDOWN IN THE VALUATION OF COLLATERALIZED FUND OBLIGATIONS WINTER 2006 n d Remark: The distribution of Π(t) (given Π(0)) does not  2  rt–(–)lk, t t fit into a known family (unless conditional on J ); this is not a µ= ∑∑()σ i ⋅ jj–1  t j=1  l=1  drawback for pricing purposes, expectations can be computed n d   by simulating the jump process Jt. i.e., for CFO pricing, we ll,, k lk (–)tt σσ= ∑∑()()i σ j ⋅ jj–1  can proceed as follows: j=1  l=1    2 π P( ( t ) D ) E [10 | ( )] k = sin  j 0 ΠΠ<={()}Π tD<  2  = EE[ [1{()}Π tD< | JJWJ, (00 )]| ( )] For example:  DJ–()µ   (15) = EΦ Π  J()0  ()J   σ Π   N(,),µ()()11Σ if J=10≤ s ≤ t  ds = EGFJ[ ( )| (0 )]  t N(,)()()22 if J 000, st  dsµ Σ = ≤≤

DJ–()µΠ DABF– +⋅ t ΦΦσ ()J = CEF+⋅ Where G(Ft) = ()Π ()t , A, B, C, E where the i component of (k) (k = 0, 1) is (r – d ( l,k)2) t, µ Σl=1 σi ⋅ were defined in property 3. Then we simulate paths of J (Mon- while the i,j component of Σ(k) is Σd (σ l,k) (σ l,k) ⋅ t. l=1 i j teCarlo) in order to compute G(Ft). These features suggest that a change in Jt can be seen as a change in the market conditions, leading to a change in trend not only for the volatility of the hedgefunds but also the cor- Estimation relation among them. Hedge funds report returns on a monthly basis, leading Assumes (t) = d a X (t), d a = 1, then, to a lack of data for estimation purposes. Therefore it becomes Property 3: Π Σi=1 i ⋅ i Σi=1 i by Ito’s lemma, d (t) = [r – 1 d a (t) (t) ] dt + [ d a (t)] necessary to consider multivariate models with as few parame- Π 2 Σi=1 iσi ⋅ σi ′ Σi=1 iσi ⋅ dW Q(t). Moreover, the distribution of Π(t) conditional on ters as possible. One way to achieve this is by describing the dependence structure of the underlying Multivariate Brownian observing the history of Jt is normal with the following mean and volatility: processes using factor analysis. The parameters of the model are estimated using the fol- d d  2 2  lowing two-step approach: µ()–(–)J= rt a σj,,0 F+ σ j 1 t F  Π ∑ i ∑{}()i ti() t i=1  jj =1  1. Search for the times where a jump has occurred. Here d d  2  = rt– a σ j,1 t we start with an initial guess for distress months described ∑∑i  ()i  (13) i==1  j 1  before, then a sample point is removed if a test rejected the gaussianity hypothesis. We stopped as soon as gaus- d d   2 2  + a  σσj,,0 – j 1 F sianity is accepted. Knowing the distress and the tranquil ∑∑i  ()()()i i  t  i==1  j 1  months allows us to compute the parameters of the jump process (p, λ) by fitting the theoretical probability path =+ABF⋅ t of a tranquil months to the empirical probability path: 2 2 From Equation (8), the theoretical probability of a tran- d   d   d   σ20()(–)J= F⋅ a σj,,+ t F⋅ a σ j 1  quil month at time t given J = 0 is: Π ∑ t ∑ i i  t ∑ i i   0 j=1   i=1   i=1   –λt 2 2 q(t) = p (1 – e )  d d   d  d  j,1   j,0  = a σ ta+  σ t NT() t  ∑∑ ∑ ii  ∑  ii qˆ ,  j 1  i 1    jj 1   i 1  The empirical probability would be: (t) = Σ t  ===   =  i=1 (14) where NT(t) is the number of tranquil months by t. 2  d    Therefore fiting the theoretical to the empirical we get: – aFj,1   ∑ iiσ   t  i 1    =   p = 0.867

=+C EEF⋅ t λ = 1.029

2. Factor analysis is used for estimating each of the covari- t Where Ft = ∫0 Jsds. ance matrices for tranquil and distress events. We use

WINTER 2006 THE JOURNAL OF ALTERNATIVE INVESTMENT 11 both sample data found in step 1 (returns from tranquil Jarrow, J., and S. Turnbull. “The Intersection of Market and months and from distress months). Specifically, we pro- credit risk.” Journal of Banking and Finance, Vol. 24, No. 1–2 pose a 1 factor model as follow: (2000), pp. 271–299.

McNeil, A.J., M. Nyfeler, and R. Frey. “Copulas and Credit cdWt0⋅QQQ()()()=λ 0⋅ dMt+ BdZt 0 ⋅ Models.” RISK (2001), pp. 111–114. c11⋅ dWQ () t=λ ⋅ dM QQ()() t+ B1 ⋅ dZ t Merton, R.C. “On the Pricing of Corporate Debt: The Risk where c i is the correlation matrices under tranquil (i = Structure of Interest Rate.” Journal of Finance, 29 (1974), pp. 0) and distress (i = 1) conditions. λ0, λ1 are constant vec- 449 –470. tors and B i are diagonal matrices with diagonal ()12 Karatzas, I., and S. Shreve. Brownian Motion and Stochastic Cal- 1–().λ j This reduces the total number of parame- ters to be computed for the covariance matrices from n2 + culus. Springer, 1988. n to 4n. Means are computed using standard estimators. Moody’s. “Moody’s Approach to Rating Collateralized Funds of Hedge Funds Obligations.” Moody’s, August 2003. REFERENCES Zangari, P. “An Improved Methodology for Measuring Var.” CISDM. “The Benefits of Hedge Funds.” CISDM Working RiskMetrics Monitor 2Q, 1996. Paper, 2006. Stone, C.A., and A. Zissu. “Fund of Fund Securitizations” Harvey, C.R., and T.E. Viskanta. “Forecasting International Journal of Derivatives (Summer 2004), pp. 62–68. Equity Correlations.” Financial Analysts Journal,1994, pp. 32–45. Schonbucher, P. “Credit Derivatives Pricing Models.” John Wiley Koedijk K., R. Campbell, and P. Kofman. “Increased Corre- and Sons, 2003. lation in Bear Markets.” Financial Analyst Journal, 2002, pp. 87–94. Venkataraman, S. “Value at Risk for a Mixture of Normal Dis- tributions: The Use of Quasibayesian Estimation Techniques.” Laurent, J.P., and J. Gregory. “Basket Default Swaps, CDO’s Economic Perspectives (Mar 1997), pp. 2–13. and Factor Copulas.” Working Paper, ISFA Actuarial School, University of Lyon, 2003. Zhou, Ch. “An Analysis of Default Correlations and Multiple Defaults.” The Review of Financial Studies, Vol. 14, No. 2 Lund, D., and B. Oksendal (eds.). Stochastic Models and Option (2001), pp. 555–576. Values: Applications to Resources, Environment, and Investment Prob- lems. New York, NY: North-Holland. To order reprints of this article, please contact Dewey Palmieri at Li, D.X. “On Default Correlation: A Copula Approach.” Journal [email protected] or 212-224-3675 of , 9 (March 2000), pp. 43–54.

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