Fixed Income Portfolio Management

Fixed-Income Portfolio Selection
Kay Giesecke∗ and Jack Kim† Stanford University June 29, 2009; this draft January 11, 2012‡

Abstract The equity portfolio selection problem is the subject of a substantial literature. Though equally important in practice, the selection problem for a ﬁxed-income portfolio of corporate and government bonds, industrial loans and credit derivatives, is less well-understood. The ﬁxed-income portfolio problem presents unique challenges: the risk of issuer default induces skewed return distributions, the correlation of defaults inﬂuences the tail of the portfolio return distribution, and credit derivative positions have complex risk/return implications. This paper addresses the static selection problem for a ﬁxed-income portfolio. We optimize the total mark-to-market value of the portfolio at the investment horizon. This value incorporates the intermediate premium and default cash ﬂows of long and short cash and derivative positions, and the survival-contingent market value of these positions at the horizon. The selection problem is cast as a polynomial goal program that involves a two-stage constrained optimization of preference weighted moments of the portfolio mark-to-market. The decision variable is the vector of contract notionals. A capital constraint guarantees the solvency of the investor. The multi-moment formulation addresses the non-Gaussian distribution of the portfolio mark-tomarket. It is also computationally tractable, because we obtain analytical expressions for the moments of the portfolio mark-to-market, which are given in terms of nested expectations under risk-neutral and actual probability measures. The expressions are valid for a broad class of intensity-based, doubly-stochastic models of correlated default timing that are widely used in portfolio credit risk and derivatives pricing. Numerical results illustrate the features of optimal portfolios of credit swaps.

Department of Management Science & Engineering, Stanford University, Stanford, CA 94305-4026, USA, Phone (650) 723 9265, Fax (650) 723 1614, email: giesecke@stanford.edu, web: www.stanford.edu/∼giesecke. † Department of Management Science & Engineering, Stanford University, Stanford, CA 94305-4026, USA, Fax (650) 723 1614, email: jackkim@stanford.edu, web: www.stanford.edu/∼jackkim. ‡ We thank Anil Bangia, Vineer Bhansali, John Birge, Agostino Capponi, Andreas Eckner, Taosong He, Pete Meindl, Hideyuki Takada and seminar participants at the 2009 American Mathematical Society Joint Mathematics Meeting, the 2008 INFORMS Annual Meeting, the Stanford Financial Mathematics Seminar, JP Morgan, and PIMCO for comments.

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Introduction

The equity portfolio selection problem is treated in a substantial literature pioneered by Markowitz (1952).1 Despite its signiﬁcance in practice, the problem of selecting a portfolio of corporate bonds and other credit instruments, is less well-understood. This paper formulates and solves the static selection problem for a “ﬁxed-income” portfolio of corporate and government bonds, risky loans, and various credit derivatives such as credit and standard index swaps. The ﬁxed-income problem has unique features. Due to the risk of default, the position return distributions are often skewed. The correlation of defaults, i.e., the tendency of defaults to cluster, inﬂuences the tail of the portfolio return distribution. Additional complexity comes from credit derivative positions. A manager can, for instance, hedge a corporate bond position by buying protection through a credit default swap. In this case, the manager pays a premium and receives compensation if the bond issuer defaults. This position does not suﬀer from the notorious problems associated with shorting bonds. The manager may also act as the seller of protection. Then, the manager receives a premium for promising to cover the loss due to default by the bond issuer. Unlike a long bond position, the swap position does not require a payout at contract inception. There is yet another aspect. Suppose the credit quality of the bond issuer improves after contract inception. The manager, who has sold protection for a ﬁxed premium, can now realize a mark-to-market proﬁt by buying protection on the issuer over the remaining term. Then, the swap is eﬀectively closed out at a mark-to-market proﬁt proportional to the diﬀerence between the initial premium and the available market premium. This proﬁt represents the market value of the swap. It is income in addition to the premiums already received, and inﬂuences the distribution of the portfolio return. To address the implications of long and short credit derivative positions, we propose to optimize the distribution of the total mark-to-market value of a ﬁxed-income portfolio at the investment horizon. The portfolio value is given by the sum of the mark-to-market values of the constituent positions. For example, the mark-to-market value of a credit swap protection selling position is given by the swap premiums received, the loss paid out at default, and the mark-to-market value of the swap realized at the horizon contingent on the survival of the reference issuer. The mark-to-market value of the swap is equal to the expected present value of the premium income over the remaining term of the swap less the expected present value of the potential default payments. We develop generic, model-free expressions for the mark-to-market values of typical constituent positions, treating the timing and magnitude of most contractual cash ﬂows exactly. The distribution of the portfolio mark-to-market value takes a complicated form: it is a nested expectation under diﬀerent probability measures. The inner expectation represents the mark-tomarket value of the portfolio at the horizon, and is taken under a risk-neutral pricing measure. The outer expectation is taken under the statistical probability measure that describes the empirical distribution of the state variables determining the portfolio mark-to-market value at the horizon. A
Important developments include the treatment of parameter estimation error as in Goldfarb & Iyengar (2003), DeMiguel & Nogales (2009) and DeMiguel, Garlappi, Nogales & Uppal (2009), transaction costs as in Perold (1984) and Lobo, Fazel & Boyd (2007), trading constraints as in Bonami & Lejeune (2009), higher moments as in Athayde & Flores (2004), and tail risk measures such as the lower partial moment (Jarrow & Zhao (2006)) or the value at risk (El Ghaoui, Oks & Oustry (2002)).
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dynamic model of default timing under actual and risk-neutral probabilities is required to calculate the distribution of the portfolio mark-to-market. We formulate the optimization of the distribution of the portfolio mark-to-market as a polynomial goal program (PGP). The PGP involves a two-stage constrained optimization of preference weighted moments of the portfolio mark-to-market. The decision variable is the vector of contract notionals, i.e., the principal value of a bond or loan position, or the protection notional of a credit derivative position. The sign of a notional value indicates the nature of a position (long or short). In the initial stage we perform the individual odd moment maximizations and even moment minimizations. In the second stage, we balance the multiple objectives by optimizing the exponentially weighted sum of distances from the individual optima. We stipulate a capital constraint that guarantees the solvency of the investor while exposing the entire investment capital to risk. The PGP formulation of the ﬁxed-income portfolio problem has an important advantage over an alternative expected utility formulation. Not only can it address the non-Gaussian distribution of the portfolio mark-to-market, but the multi-moment objective is also computationally tractable. Although the full distribution of the portfolio mark-to-market is generally analytically intractable, the moments turn out to be less so. We obtain analytical expressions for the moments that are valid for a broad class of intensity-based, doubly-stochastic models of default timing under actual and risk-neutral probabilities. Such models are widely used for the pricing of credit derivatives, see Duﬃe & Garleanu (2001), Eckner (2009), Feldh¨tter (2007), Kou & Peng (2009), Mortensen (2006), u Papageorgiou & Sircar (2007) and others, and for default prediction, see Chava & Jarrow (2004), Duﬃe, Saita & Wang (2006), Eckner (2008), and others. Our numerical results for a portfolio of credit swaps have important implications. The large negative skewness of mean-variance optimal portfolios highlights the importance of including higher moments in the analysis. Multiple horizon analysis shows that signiﬁcant skewness is present for both short and long horizons. Optimal portfolio behavior is also examined under elevated default risk, and default correlation scenarios. These analyses show that an overall elevation in default risk, default correlation or the introduction of a risky obligor all have momentous implications in the portfolio variance-skewness trade oﬀ. Finally the performance, and robustness of the optimization methodology with respect to model parameters, are examined. Prior research has analyzed ﬁxed-income portfolio management. Akutsu, Kijima & Komoribayashi (2004), Kraft & Steﬀensen (2008), Meindl & Primbs (2006), and Wise & Bhansali (2002) focus on corporate bond portfolios, using expected utility or other objectives. They do not consider credit derivative positions and the implications of mark-to-market valuation. Lai (1991), Chunhachinda, Dandapani, Hamid & Prakash (1997), Sun & Yan (2003) and Davies, Kat & Lu (2009) formulate PGPs for the selection of stock portfolios. The PGP we formulate for the ﬁxed-income problem is mathematically diﬀerent from these PGPs, however. Because of the diﬀerences in cash outlays for swap and cash positions, and contingent payments, the typical capital constraint of equity portfolios is no longer applicable in the ﬁxed-income setting. The altered constraint no longer makes the optimal notional vector scalable. The rest of this paper is organized as follows. Section 2 characterizes the portfolio mark-to-market value. Section 3 formulates the selection problem as a PGP over multiple moments of the portfolio

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mark-to-market. Section 4 demonstrates the application of this approach to intensity-based, doublystochastic models of default timing under actual and risk-neutral measures, and develops analytical expressions for the moments of the portfolio mark-to-market. Section 5 provides numerical results that illustrate the properties of the optimal portfolios, as well as an analysis of robustness, performance, and horizon eﬀects. Section 6 discusses several extensions. Section 7 concludes. There are three technical appendices.

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Portfolio mark-to-market

This section characterizes the mark-to-market value of a ﬁxed-income portfolio. For clarity in the exposition, we make simplifying assumptions regarding the portfolio composition, payment of premium accruals, and interest rates. Section 6 relaxes each of these assumptions.

2.1

Portfolio

Consider a portfolio of credit swaps referenced on a collection of names i = 1, 2, . . . , n. Sections 6.1 and 6.2 discuss extensions to a mixed portfolio of corporate loans and bonds, credit swaps, and index swaps. Section 6.4 discusses the inclusion of a non-defaultable government bond. A credit swap is a bilateral ﬁnancial contract. One party, called the protection seller, provides default insurance on a reference name i for a speciﬁed notional amount. At default, the protection seller makes a payment equal to the product of the notional and the loss rate , which we assume is constant. The other party, called the protection buyer, pays a premium for this coverage. The premium, or credit swap spread, is negotiated at contract inception and is expressed as a fraction Si of the notional. It is paid at dates (tm ), where the ﬁnal premium date is equal to the swap maturity date T . The spread Si is stated on a per-period basis. We ignore premium accruals; Section 6.3 discusses the general case. We focus on a static “buy and hold” strategy for a ﬁxed investment horizon H < T . The decision variable is a vector δ = (δ1 , . . . , δn ), where δi ∈ R is the contract notional for the credit swap referenced on name i. A positive δi indicates a protection selling position, through which the investor assumes the credit risk of reference name i. A negative δi indicates a protection buying position, through which the investor hedges the credit risk of name i.2 There is a constant risk-free rate r. All cash ﬂows generated by the investment strategy are reinvested in a money market account earning the compound interest r. This is a realistic assumption as it corresponds to standard duration strategies. It also ensures consistency in both protection selling and buying positions. When there is a surplus of funds received, this scheme implies a netting eﬀect in which the payments received can be used in paying out obligations, and that payouts are otherwise funded from the money market. Section 6.4 extends the analysis to a stochastic interest rate.
The practical diﬃculties of shorting bonds are well known. However in the case of credit swaps, long and short positions are symmetric counterparty transactions. Therefore we do not impose a restriction on the sign of δi . However we do require a constraint on δ that is consistent with the capital available to the investor, see Section 3.1.
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2.2

Mark-to-market values

To describe the cash ﬂows to the portfolio positions, suppose reference name i defaults at a stopping time τi > 0, relative to a complete probability space (Ω, F , P) and an information ﬁltration F = (Ft )t≥0 satisfying the usual conditions (see Protter (2004)). Here P denotes the actual (statistical) probability measure. The credit swap market is free of arbitrage opportunities, so under mild technical conditions there exists a risk-neutral pricing measure equivalent to P. We ﬁx a risk-neutral measure P∗ with respect to the risk-free rate r. Consider a unit notional, protection selling position in a credit swap on name i, initiated at the reference time 0 at a (per-period) swap spread Si . Two diﬀerent scenarios can unfold during the investment period [0, H]. If name i defaults before H, then the cash ﬂows to the protection seller consist of the premiums received minus the loss paid out at default. With the loss payment at τi the swap expires. Hence the discounted cumulative payments to the protection seller take the form
H

Di (H) = tm ≤H

e−rtm Si (1 − Ntim ) −
0

i e−rs dNs

(1)

where Nti = 1{τi ≤t} is the default indicator for name i, which is zero before default and one after default. The ﬁrst term on the right side of (1) represents the cumulative discounted premium payments to be received before H. A premium payment is contingent upon the survival of the name to the payment date, hence the factor 1{τi >tm } = (1 − Ntim ). The default cash ﬂow consists of a one-time i payment of the loss at the time of default. The discounted default payment is therefore e−rτi NH , which corresponds to the second term on the right side of (1) in integral form. If the reference name survives to the investment horizon H, then the protection seller receives all premium payments scheduled during (0, H]. The contract is still active at H, and may have a nonzero value due to the evolution of the credit quality of the reference name, as measured by the realized market spread. The protection seller realizes a mark-to-market proﬁt or loss depending on whether the spread at H has tightened or widened. In addition to the premium income, this markto-market value is a matter of concern for the investor (here the protection seller) because at H the investor is free to close out the position. If the spread tightens, the investor can buy protection on the reference name, covering the remaining term (H, T ]. Ignoring counter-party risks, if the reference name defaults, the loss compensation that is received can be used to pay out the investor’s initial obligation as a protection seller, canceling all future obligations while locking in the diﬀerence between Si and the time H market spread during (H, T ∧ τi ]. Similarly, if the spread widens, the protection seller incurs a mark-to-market loss. Hence from the protection seller’s point of view, if the reference name survives to H, the discounted cumulative cash ﬂow takes the form
T

Si (H, T ) = tm ≤H

e

−rtm

Si + E

∗ tm >H

e

−rtm

Si (1 −

Ntim )

−
H

i e−rs dNs FH

(2)

where E∗ denotes the expectation under the risk-neutral measure P∗ . The conditional P∗ -expectation on the right side of (2) represents the mark-to-market value of the position at H, discounted to time

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0. The mark-to-market value at H is given by the diﬀerence between the value at H of the premium payments during (H, T ∧ τi ] and the value at H of a potential default payment during (H, T ]. From the protection seller’s perspective, the mark-to-market value at time H of a unit notional credit swap position referenced on name i is given by the random variable Pi = erH 1{τi ≤H} Di (H) + 1{τi >H} Si (H, T ) . (3)

The event indicators separate the two possible scenarios, and the factor erH represents the reinvestment policy of rolling over net payments to H at the risk-free rate r. The mark-to-market value of the portfolio at H is given by δ P = δ1 P1 + · · · + δn Pn , where δi is the notional for the credit swap referenced on ﬁrm i. The sign of δi indicates the nature of a position (long or short).

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Optimizing the portfolio mark-to-market

The investor optimizes the portfolio mark-to-market value δ P at the investment horizon H. The decision variable is the vector of contract notionals δ. The complex features of δ P impose restrictions on the choice of the optimization objective. The optimization of the expected utility of δ P appears infeasible, because the full distribution of δ P required for this optimization is computationally intractable unless one signiﬁcantly relaxes the assumptions regarding default timing and position cash ﬂows made in Section 2 above. To avoid such an over-simpliﬁcation, we propose to cast the selection problem as a polynomial goal program that involves a two-stage constrained optimization of preference weighted moments of δ P. Unlike the distribution, the moments of δ P can be calculated in closed form for a broad family of default timing models, in the general setting of Section 2. This motivates a moment-based formulation of the portfolio problem.

3.1

Capital constraint

We begin by deﬁning a capital constraint that represents a minimal restriction on how much capital the investor can use for covering cash ﬂow obligations arising from a strategy δ. To this end, it is essential to recognize a key diﬀerence between a credit swap position and a bond position. When entering into a long bond position, the investor has a cash outﬂow equal to the price or principal of the bond. When entering into a short bond position, the investor has a cash inﬂow. A credit swap position does not involve initial cash ﬂows. Hence, while the capital constraint can be imposed as a simple bound on the sum of net initial investments in the bond portfolio case,3 we need to modify the notion of a capital constraint when dealing with credit swaps. To this end, we assume that there is a ﬁxed amount of time H capital C that the investor can use to cover obligations generated by the credit swap positions. We impose a constraint so that in the worst possible scenario, the investor’s obligations, measured in terms of time H capital, are exactly C. The worst case scenario is the situation where the names referenced by the short protection positions (positive δi ) default
For coupon bonds the sum would also have to account for the coupon payments to be paid out or received. Given the variable timing of these payments this would require an assumption on the reinvestment policy as we have done so here.
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immediately, while the names referenced by the long protection positions (negative δi ) survive to H. In this case, the obligations consist of the default payments on the short protection positions, and premium payments on the long protection positions up to H. Thus, we require that the strategy δ satisﬁes the convex constraint g(δ) = C, where n g(δ) = e

rH i=1

max(δi , 0) − min(δi , 0) tm ≤H

Si e−rtm .

(4)

We adopt an equality constraint as opposed to an inequality constraint so that all the capital C committed to investment is fully exposed to risk. The capital is initially invested in the money market account; the strategy governs the subsequent withdrawals and infusions. The balance at the horizon H will be zero only if the worst case scenario occurs. Note that this constraint is the least restrictive bound that guarantees solvency in face of the cash ﬂow obligations associated with the strategy. It does not limit the generality of the formulation. In practice, portfolio managers may incorporate additional constraints that account for margin requirements, pledged collateral, ﬁxed moment constraints etc., customized to the particular problem at hand. On a more technical note, the equality constraint also precludes the second moment objective in Section 3 from attaining the trivial no risky investment, zero variance optimum – a case that is of no interest.

3.2

Polynomial goal program

Polynomial goal programming is a method by which a decision maker can resolve problems that involve multiple objectives or goals.4 The often conﬂicting nature of simultaneous objectives results in solution methods that oﬀer a compromise between the optimality of individual goals, depending on the decision maker’s preference for each objective. In the context of the ﬁxed-income portfolio markto-market problem, the investor wishes to maximize E(δ P) while minimizing risk as measured by the variance of δ P, given the capital constraint. At the same time, as noted in Kraus & Litzenberger (1983), Simkowitz & Beedles (1978), and Briec, Kerstens & Jokung (2007), investors prefer positive skewness because it implies a low probability of a large negative return.5 On the other hand, large portfolio kurtosis is something that investors wish to avoid as it implies large tail risk i.e. a large contribution of extreme events to the variance. Generalizing, the investor wishes to maximize odd moments and minimize even moments, within the capital constraint.6 This leads to a constrained multi-objective optimization problem: max δ Z1 (δ) = δ M1 Z2 (δ) = δ M2 δ (−1)k Zk (δ), g(δ) = C k≥3

min δ δ

min such that
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See Deckro & Hebert (1977) and Deckro & Hebert (1988) for an early introduction to the method. Arditti (1967) shows that a preference for positive skewness is consistent with decreasing absolute risk aversion. 6 For a formal justiﬁcation, see Scott & Horvath (1980).

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where Mk is the kth central moment tensor7 of P and Zk (δ) = (δ Mk (δ ⊗ δ ⊗ · · · ⊗ δ ))Z2 (δ)−k/2 , k−1 k≥3

(5)

is the kth standardized central moment of the portfolio mark-to-market δ P. In general, this multiobjective problem does not have a solution that satisﬁes all optimizations simultaneously.8 Therefore, we balance the objectives through a 2-step polynomial goal program (PGP). Given moment preference parameters γk ≥ 0, we consider the problem
K

min δ Z(δ) = k=1 (1 + dk (δ))γk k = 1, 2, . . . , K

(6) (7)

such that

∗ dk (δ) = (−1)k (Zk (δ) − Zk ),

∗ ∗ where (−1)k Zk = min{(−1)k Zk (δ) : g(δ) = C}, so that Zk is the optimal kth moment of δ P ∗ under the capital constraint, and dk (δ) is the diﬀerence between Zk and the kth moment evaluated ∗ at δ. The optimization is a 2-step process in which we ﬁrst ﬁnd the optimal Zk . Then we substitute these values into equation (7) and solve the minimization problem (6) according to the preference proﬁle (γ1 , γ2 , . . . , γK ). The minimization problem (6) is simply a weighted sum of the deviational variables from each optimal objective, where the exponential weight of each term corresponds to the moment preference parameter. The maximum order of the problem is speciﬁed by K, so for instance K = 4 represents the 4-moment problem.9 The PGP formulation (6)–(7) decomposes the overall portfolio problem into simpler solvable problems and then iteratively attempts to ﬁnd solutions that preserve, as closely as possible, the individual goals while also reﬂecting the investor’s priority over the diﬀerent goals.

3.3

Discussion

The traditional mean-variance formulation is a special case of the multi-moment PGP (6)–(7), where γ1 , γ2 > 0 and γk = 0, k > 2. The 2-moment approach is only consistent with expected utility when either δ P is normally distributed or the investor has a quadratic utility function. Given the rational investor’s preference for high odd moments and low even moments, a literal interpretation of quadratic investor utility is an unrealistic postulation. The K-moment approach can be seen as a Kth-order polynomial approximation of expected utility that addresses the non-normality of δ P. The relevance of higher-order moments in the mark-to-market credit swap setting is easily recognized. Consider a high-quality reference name i with a swap spread Si at contract inception of 10 basis points, or 10−3 . As explained in Section 2.2, the protection seller realizes a mark-to-market proﬁt only if the spread tightens over the investment period. Hence the maximum mark-to-market
Mk is the mean vector of P for k = 1, the covariance matrix for k = 2, and the k-dimensional kth moment tensor for k ≥ 3. 8 The mean optimal portfolio, for instance, is likely to include large notionals on names that have high expected default payments and/or large spread movements. This composition is unlikely to be optimal for the portfolio variance. 9 Individual moment objectives can be deactivated by setting the corresponding preference parameter to zero.
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proﬁt attainable for a protection seller is bounded within a narrow range: the spread can only drop by 10 basis points. On the other hand, there is potential for a much larger loss if the spread widens, since the spread has no upper bound. Thus, the distribution of the position mark-to-market value δi Pi is asymmetric. This asymmetry may increase with default correlation, i.e. with the tendency of portfolio positions to move in the same direction. The ﬁrst two portfolio moments fail to capture these higher order features of asymmetry and tail risk of the portfolio mark-to-market value δ P, and this motivates a multi-moment formulation. Lai (1991), Chunhachinda et al. (1997), Sun & Yan (2003) and Davies et al. (2009) formulate PGPs to select a portfolio of stocks. However there are key technical distinctions of the PGP (6)–(7), which make it a diﬀerent mathematical problem. We can no longer subsume the second moment objective as a constraint in the individual optimizations. In the aforementioned papers, the decision variables are the relative portfolio weights. Hence the unit variance constraint can be subsumed in the individual optimizations at the ﬁrst stage and the weights re-scaled after the second stage to satisfy the capital constraint. However, in our problem, the decision variables are the contract notionals restricted by a nonlinear capital constraint. Therefore the second moment is explicitly incorporated in the two stages (as an objective in the ﬁrst and a constraint in the second). Both stages are governed by the capital constraint. The PGP approach provides advantages over alternative formulations of the multi-moment portfolio selection problem in terms of transparency, economic interpretability of the marginal rates of moment-substitution, and computational tractability. Comparative advantages against the constantskewness constrained mean-variance locus method of Arditti & Levy (1975) and the 1-step, 3-moment optimization of Athayde & Flores (2004) are well documented in Lai (1991) and Davies et al. (2009). Briec et al. (2007) and Jurczenko, Maillet & Merlin (2006) deﬁne mean-variance-skewness and meanvariance-skewness-kurtosis utility functions as third and fourth order polynomial approximations, respectively, of expected utility. They deﬁne a shortage function relative to a convexiﬁed multimoment frontier to establish a duality result. This approach addresses some of the shortcomings of the PGP approach in that it conforms to the theoretical notion of a frontier portfolio,10 and provides a direct link to expected utility, as well as a precise characterization of the distance from the frontier as a performance measure. However the speciﬁcation of the necessary parameters is less transparent than the direct characterization of moment preferences, and the dual interpretation becomes ambiguous in fourth and higher moments. Also cubic utility functions do not exhibit increasing absolute risk aversion for all wealth levels (Levy (1969)).

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Applying the PGP

This section illustrates the application of the PGP (6)–(7). To calculate the required moments of the portfolio mark-to-market δ P, we formulate a standard model of default timing in Section 4.1. Then, in Section 4.2, we develop analytical expressions for the moments under this model.
The optimal portfolio in the PGP approach is still eﬃcient in the sense that it is a nondominated portfolio, for which no goal can be improved upon without the change of subjective preference.
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4.1

Default timing model

We model the default time τi of reference name i as the ﬁrst jump time of a non-explosive counting t process with intensity λi relative to the actual measure P. The intensity satisﬁes 0 λi (s)ds < ∞ almost surely, for each t. Prior to default, the intensity represents the conditional default rate of a reference name relative to P. There is also an intensity λ∗ relative to the risk-neutral pricing measure i ∗ P , see Artzner & Delbaen (1995). The risk-neutral intensity λ∗ represents the conditional default i ∗ rate under P , and governs the mark-to-market valuation of a swap position. We must specify the processes followed by λi and λ∗ under both P and P∗ because the moments of δ P involve nested i expectations under actual and risk-neutral measures. 4.1.1 Actual intensity

We adopt a doubly-stochastic model of ﬁrm intensities under P. Similar models appear in Chava & Jarrow (2004), Duﬃe & Garleanu (2001), Duﬃe et al. (2006), Eckner (2009), Eckner (2008), Feldh¨tter (2007), Mortensen (2006), Papageorgiou & Sircar (2007) and others. In this formulation, u the P-intensity λi of reference name i = 1, 2, . . . , n is a speciﬁed function Λi of an idiosyncratic risk factor Xi and a systematic risk factor X0 . The factors X0 , X1 , . . . , Xn are independent of one another. Conditional on a realization of (Xi , X0 ), the default time τi is the ﬁrst jump time of an inhomogeneous Poisson process with (conditionally deterministic) intensity λi = Λi (Xi , X0 ). Given a realization of the systematic factor X0 , which drives the intensities of all reference names, the default times are P-independent of one another. This property facilitates the analytical calculations of the moments of δ P. It also renders the parameter estimation problem computationally tractable. As we will see in Section 4.2, the only requirement on the risk factor dynamics is that we have a computationally tractable characterization of the Laplace transforms of cumulative intensities. Examples of models that meet this requirement include vector-valued aﬃne jump-diﬀusions and quadratic diﬀusions. Here, for purposes of illustration, we assume that the risk factors follow P-Feller diﬀusions (Feller (1951)), which are also known as Cox-Ingersoll-Ross (CIR) processes: dXi (t) = κXi [θXi − Xi (t)]dt + σXi Xi (t)dWi (t), i = 0, 1, 2, . . . , n (8)

2 where the Wi s are independent P-Brownian motions. We assume that 2κXi θXi > σXi so that Xi (t) > 0 almost surely, and from hereon we always assume that this condition is met when a new Feller diﬀusion is introduced. The P-intensities are

λi = Xi + ωi X0 ,

i = 1, 2, . . . , n

(9)

where the ωi s are the factor loadings of the common risk factor.11 The ωi s control and diﬀerentiate the level of exposure that a ﬁrm has to the common risk factor, and therefore they modulate the correlation structure of the default times. As shown by Duﬃe & Garleanu (2001), certain restrictions apply to the parameters for the resulting ﬁrm intensity process to remain a P-Feller diﬀusion. De11 It should be noted that the ωi s are relative in that if they are all scaled by a common constant, equivalent intensity dynamics can be obtained by scaling the common risk factor parameters.

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noting the parameter triplets of the idiosyncratic and common factors by (κXi , θXi , σXi ) and (κ, θ, σ) respectively, we require that κXi = κ √ σXi = ωi σ. (10) (11)

There are no restrictions on the mean level parameters θXi . The resulting P-Feller diﬀusion parameter triplet for the ﬁrm intensity λi is given by (κi , θi , σi ) = (κ, θXi + ωi θ, √ ωi σ). (12)

The parameter triplet (12) can be estimated from historical default experience and risk factor observations by maximum likelihood, as in Chava & Jarrow (2004) or Duﬃe et al. (2006). The doublystochastic property implies that the likelihood function is the product of the risk factor-conditional likelihood functions of the ﬁrms’ survival events. 4.1.2 Risk-neutral intensity

The models of the actual intensities under the actual measure do not themselves intrinsically prescribe any a priori conditions that must be satisﬁed by the risk-neutral model or the relationship between the two measures. In the presence of a premium for default timing risk, the risk-neutral intensity λ∗ will be diﬀerent from the actual intensity λi , and there is no innate restriction preventing λ∗ i i from having dynamics that are diﬀerent from the dynamics of λi . The risk-neutral intensity may no longer be a Feller diﬀusion or it may even be the case that the doubly-stochastic property no longer holds under P∗ . The only real endogenous restriction (by the no arbitrage principle) is that the two measures be equivalent. However based on the empirical ﬁndings of Berndt, Douglas, Duﬃe, Ferguson & Schranz (2005) and Eckner (2008), we see that it is reasonable for the joint model to preserve certain process characteristics under their respective measures. Here we propose a measure change that incorporates the premium for default timing risk as adjustments to the intensities λi , and accounts for the premium for diﬀusive mark-to-market ﬂuctuations through adjustments to the Brownian motions driving the λi . Berndt et al. (2005) provide regression results and an empirical time-series ﬁtting in which the risk-neutral intensity is modeled as an aﬃne function of the actual intensity plus a stochastic noise term. Motivated by this formulation, we model the relationship between λ∗ and λi as i λ∗ (t) = αi + βi λi (t) + ui (t), ˆ ˆ i i = 1, 2, . . . , n (13)

where αi and βi are constant parameters and the ui s are stochastic noise terms. With (9), ˆ ˆ λ∗ (t) = αi + βi Xi (t) + ωi βi X0 (t) + ui (t). ˆ ˆ i (14)

Since Xi and X0 are P-Feller diﬀusions, λ∗ will also also be a P-Feller diﬀusion as long as the noise i term ui is a P-Feller diﬀusion, and parameter restrictions are met. ˆ 11

The diﬀerence between our formulation and that of Berndt et al. (2005) is that we incorporate correlation between ﬁrm intensities via (9). In the P-dynamics of λ∗ given by (14), the λ∗ s remain i i correlated through the common risk factor X0 . Given the structure of the transformation (14), we will decompose (13) into two independent transformations – a component attributed to the transformation ∗ of the idiosyncratic factor process Xi → Xi∗ , and another one describing X0 → X0 . We achieve (13) by an explicit speciﬁcation of these two transformations. This is consistent with the empirical results of Eckner (2008), who ﬁnds that the default timing risk premium is composed of an idiosyncratic component and a component ascribed to the correlation risk. As we will see, this decomposition leads to a natural formulation for the diﬀusive premium as well, and is convenient for the moment computations that follow in Section 4.2. To this end, we assume that there are 2n+2 state variables – n idiosyncratic risk factors, 1 common factor, and n + 1 factors that modulate the relationship between the actual and risk-neutral factors. The modulator terms, denoted by ui , are P-Feller diﬀusions that track the stochastic evolution of the default timing risk premia. We posit the following relationship between the actual factor processes Xi and their risk-neutral counterparts Xi∗ : Xi∗ (t) = αi + βi Xi (t) + ui (t), i = 0, 1, 2, . . . , n (15)

with β0 = 1. Here the αi s and βi s are constant parameters and the ui s are again independent P-Feller diﬀusions with dynamics given by dui (t) = κui [θui − ui (t)]dt + σui ui (t)dBi (t), i = 0, 1, 2, . . . , n. (16)

The Bi s are independent Brownian motions under P, also independent of the Wi s of (8). From (14), λ∗ (t) = αi + βi Xi (t) + ui (t) + ωi βi (α0 + X0 + u0 (t)) i = Xi∗ (t) +
∗ ωi βi X0 (t)

(17) (18)

which gives us a factor decomposition of the risk-neutral intensity λ∗ , analogous to (9) of the actual i ∗ intensity λi . Just as λi is driven by (Xi , X0 ), the transformed factors (Xi∗ , X0 ) now drive λ∗ . Equation i (17) gives us our intended formulation of (13) where αi = αi + ωi βi α0 , and ui = ui + ωi βi u0 . There ˆ ˆ is now a correlation structure that is also embedded in the stochastic noise term ui , which captures ˆ 12 ﬂuctuations in the default timing risk premium. The process ui is a P-Feller diﬀusion if ˆ κui = κ σui = ωi βi σ, i = 0, 1, 2, . . . , n (19) (20)

with ω0 = β0 = 1. These restrictions ensure that the risk-neutral factors and intensities remain
While Eckner (2008) also decomposes the risk premia in terms of idiosyncratic and common inﬂuences, he chooses to model the measure change as a scaling of the factors and a modiﬁcation of the factor loading. Hence, unlike our formulation, ﬂuctuations in the default timing risk premia are driven solely by the scaled common factor. In our formulation the ﬂuctuations inherit the correlation structure of the intensities.
12

12

P-Feller diﬀusions. The resulting P-Feller parameter triplet of the risk-neutral intensity λ∗ is i αi + ωi βi α0 + βi θXi + ωi βi θ + θui + ωi βi θu0 , ωi βi σ). (¯ ∗ , θi , σi ) = (κ, κi ¯∗ ¯ ∗ (21) κ √ For the individual risk factors, (¯ ∗ i , θXi , σXi ) = (κ, αi + βi θXi + θui , ωi βi σ), i = 0, 1, 2, . . . , n. While κX ¯∗ ¯ ∗ κ our model speciﬁcation allows the ﬂexibility of having ﬁrm-speciﬁc scale (α) and shift (β) parameters, in practice they may be estimated per sector or industry for model parsimony. Equation (21) speciﬁes the P-dynamics of λ∗ . To obtain the P∗ -dynamics of λ∗ , we only need to i i specify how the Brownian motions driving (X0 , X1 , . . . , Xn , u0 , u1 , . . . , un ) change under P∗ . Let η be a parameter. Given the actual measure P, we deﬁne the risk-neutral measure P∗ to be the measure under which, along with the previous intensity transformations for the default point processes, ∗ ∗ ∗ ∗ ∗ ∗ (W0 , W1 , . . . , Wn , B0 , B1 , . . . , Bn ) is an (2n + 2)-dimensional standard Brownian motion where t η Wi (t) = Wi∗ (t) − √ Xi (s)ds ωi 0 t η ui (s)ds, Bi (t) = Bi∗ (t) − √ ωi βi 0

(22) (23)

for ωi , βi > 0 and i = 0, 1, 2, . . . , n. The parameter η governs the risk premium for the diﬀusive volatility of mark-to-market values. It is appropriately scaled to adjust the Brownian motions driving the individual factors and modulators. This synchronization preserves model parsimony and enforces the κ restrictions that ensure that the overall risk-neutral intensity processes remain P∗ -Feller diﬀusions. Now we substitute (22) and (23) into the P-dynamics of Xi∗ and λ∗ , to rewrite them in terms of i ∗ ∗ the P -Brownian motions. This results in P -Feller diﬀusion dynamics for Xi∗ and λ∗ speciﬁed by i
∗ ∗ (κ∗ i , θXi , σXi ) = (κ + ησ, X

αi + κ(βi θXi + θui ) , ωi βi σ) κ + ησ αi + ωi βi α0 + κ(βi θXi + ωi βi θ + θui + ωi βi θu0 ) ∗ ∗ , (κ∗ , θi , σi ) = (κ + ησ, i κ + ησ

(24) ωi βi σ). (25)

The overall measure change that governs both the point process and diﬀusion transformations can be made precise by the corresponding Radon-Nikodym derivative, which is a product of the exponential martingale terms for the independent transforms. The default process, by construction, remains doubly-stochastic in the risk-neutral measure. Technical details are in Appendix A. Given the parameters (12) of the actual intensity, the parameters (21) and (25) of the risk-neutral intensity can be estimated from time series of single-name, index and tranche swap market rates, as in Eckner (2008). Once again the doubly-stochastic property generates signiﬁcant computational advantages for the estimation problem.

4.2

Analytical moment calculations

Given the Markov doubly-stochastic property of the default processes under P and P∗ , and the aﬃneparametric relationship between the risk-neutral and actual intensities, we can express the moments of 13

the portfolio mark-to-market δ P in terms of certain Laplace transforms of the cumulative intensities under actual and risk-neutral measures. This characterization, described in Sections 4.2.1 and 4.2.2, does not depend on the speciﬁc dynamics of the risk factors. The risk factor dynamics only enter in the explicit computation of these transforms in Section 4.2.3. The Feller diﬀusion formulation is not essential; we can extend our explicit calculations to other risk factor dynamics. 4.2.1 First moment

The ﬁrst moment M1 = (µi ) of the mark-to-market vector P is given by µi = E(Pi ). From equation (3), we have µi e−rH = E(1{τi ≤H} Di (H)) + E(1{τi >H} Si (H, T )). (26)

Formula (1) indicates that the ﬁrst term in equation (26) is a standard P-expectation, which is governed by the P-dynamics of the actual intensity λi speciﬁed in Section 4.1.1. Formula (2) shows that the second term is given by a nested expectation under diﬀerent measures: for some function f E(1{τi >H} Si (H, T )) = E(1{τi >H} E∗ (f (Nti : H < t ≤ T ) | FH )). (27)

On the set {τi > H}, the inner expectation takes the form g(λ∗ (H)) for a function g that is determined i ∗ ∗ by the P -dynamics of the risk-neutral intensity λi . The outer expectation is governed by the joint P-dynamics of λ∗ and λi .13 If the outer expectation were taken under P∗ instead of P, then the nested i expectation would be analogous to the undiscounted value of a forward-starting credit derivative that is knocked out at a default before the forward start date H. The doubly-stochastic property states that conditional on a realization of the risk factor processes driving the intensity, the default time τi is the ﬁrst jump time of a nonhomogenous Poisson process. Applying this property under P, iterated expectation yields the formula i P(τi > s | Ft ) = (1 − Nti )Lt,s

(28)

where s ≥ t and where s i i i Lt,s = E(Et,s | Ft ) for Et,s = exp − t

λi (v)dv

(29)

is the Ft -conditional P-Laplace transform of the cumulative P-intensity evaluated at 1. Applying the doubly-stochastic property relative to P∗ , we obtain the risk-neutral counterpart to formula i∗ (28), which is expressed in terms of the Ft -conditional P∗ -Laplace transform Lt,s of the cumulative P∗ -intensity evaluated at 1. These transforms can be calculated explicitly for many intensity speciﬁcations, including the model in Section 4.1. Therefore, our goal is to express µi in terms of these and related transforms. Equation (26) indicates that we can decompose e−rH µi into the sum of ﬁve expectations. However
13 ∗ ∗ The intensities λi , and λ∗ can be further decomposed into functions of (X0 , Xi ), and (X0 , Xi ) respectively. i

14

since an obligor’s survival at the horizon also implies survival at each previous premium date, we can combine the ﬁrst terms of the default and survival components of (26) as ai = 1{τi ≤H} tm ≤H

e−rtm Si (1 − Ntim ) + 1{τi >H} tm ≤H

Si e−rtm = tm ≤H

e−rtm Si (1 − Ntim ).

This leaves us with four additive components: e−rH µi = E(ai − bi + ci − di ) where
H

bi = 1{τi ≤H}
0

i e−rs dNs

ci = 1{τi >H} tm >H

Si e−rtm E∗ (1 − Ntim | FH )
T ∗ H i e−rs dNs FH

di = 1{τi >H} E

We treat each expectation separately. Formula (28) then leads to E(ai ) = E tm ≤H

e−rtm Si (1 − Ntim )

= Si tm ≤H

i e−rtm L0,tm .

(30)

Similarly, by Stieltjes integration by parts
H H i e−rs dNs 0 i = e−rH (1 − L0,H ) + r 0 i e−rs (1 − L0,s )ds.

E(bi ) = E 1{τi ≤H}

(31)

The parametrization of the measure change in Section 4.1 facilitates the calculation of the mixed measure terms E(ci ) and E(di ). The risk-neutral intensity (13) is an aﬃne function of the actual intensity plus a stochastic noise term. Therefore, by exploiting the risk-neutral counterpart to formula i∗ (28), recognizing that the P∗ -Laplace transform Lt,s is a function of λ∗ (t) by the P∗ -Markov property i ∗ of λi , and conditioning on the paths of the intensity and the noise term over [0, t], we have i∗ i i∗ E(1{τi >t} P∗ (τi > s | Ft )) = E(1{τi >t} Lt,s ) = E(E0,t Lt,s )

(32)

for s ≥ t. This equation implies that E(ci ) = Si tm >H i i∗ e−rtm E(E0,H LH,tm )

and similarly, by Stieltjes integration by parts,
T i i i∗ E(di ) = e−rT (L0,H − E(E0,H LH,T )) + r H i i∗ We deem component terms such as E(E0,H LH,s ) “compound Laplace transforms.” In Section 4.2.3, we explicitly compute the relevant compound Laplace transforms and the more complex compound i i i∗ e−rs (L0,H − E(E0,H LH,s ))ds.

15

terms that arise in the higher moment calculations. 4.2.2 Higher order moments

The covariance matrix M2 = (µij ) of P takes the form µij = E(Pi Pj ) − µi µj . The skewness tensor M3 = (µijk ) of P is given by µijk = E(Pi Pj Pk ) − µi E(Pj Pk ) − µj E(Pi Pk ) − µk E(Pi Pj ) + 2µi µj µk . (33)

All higher-order moment tensors can be expanded similarly. To compute these tensors, we must compute product terms of the form e−krH E( j=1,...,k Pij ) = E( j=1,...,k (aij − bij + cij − dij )), for k ≥ 2. Proposition 4.1 develops an expression for these terms. We note that the expression does not depend on the speciﬁc dynamics of the risk factor processes. Proposition 4.1. Under the Markov doubly-stochastic assumption and measure change parametrization of Section 4.1, the expectation of a kth order product term can be expressed in the form E j=1,...,k (aij − bij + cij − dij )

= m Cm
[0,T ]k

Dm (t)I m (t)Lm (t)dt,

(34)

where t = (t1 , . . . , tk ) is the k-dimensional time vector, C m is a constant, Dm (t) is a discount function, I m (t) is an indicator function that limits the domain of integration, Lm (t) is an unconditional compound Laplace transform, and m is an index for the additive terms that arise in the product. A proof is provided in Appendix B. To provide some insight into the arguments leading to formula (34), we give a representative example. We calculate E(di di bj ), a component of the cross-skewness of P. By using Stieltjes integration by parts on both product terms we have
T 2

E(di di bj ) =

3

E 1{τi >H} e
T

−2rT

1−

i∗ 2LH,T

+

i∗ LH,T 2

+r

2 H

e

−rs

(1 −

i∗ LH,s )ds H j e−rs Ns ds 0

+ 2re−rT
H

i∗ i∗ i i∗ e−rs 1 − LH,s − LH,T + LH,s LH,T ds

j e−rH NH + r

.

The last equality follows from the risk-neutral counterpart to formula (28). The apparent nonlinearity of the squared integral term can be circumvented by re-writing it as
T 2 T T i∗ i∗ e−rs (1 − LH,s )e−ru (1 − LH,u )dsdu H H H

e
H

−rs

(1 −

i∗ LH,s )ds

=

(35)

j j by which, after multiplying and expanding through the factor (e−rH NH + r 0 e−rs Ns ds), we can now pass the outer expectation via Fubini’s theorem, and linearity. As we are trying to formulate the expression in terms of the Laplace transforms of the cumulative intensities, which requires the indicator arguments to be of the form {τi > t, τj > s}, we further decompose the expression using the fact that 1{τi >s,τj ≤u} = 1{τi >s} − 1{τi >s,τj >u} . The doubly-stochastic property, and in particular

16

the conditional independence property relative to P lead to a generalization of formula (28): ij P(τi > s, τj > u | Ft ) = (1 − Nti )(1 − Ntj )Lt,s,u

(36)

where t ≤ s ≤ u and14 s ij Lt,s,u u

=

ij E(Et,s,u

| Ft ) for

ij Et,s,u

= exp − t λi (v)dv − t λj (v)dv .

(37)

Using a conditioning argument similar to the one used to derive equation (32), we conclude that ji i∗ i∗ i∗ i∗ E(1{τi >H,τj >v} LH,s LH,u ) = E(E0,v,H LH,s LH,u ).

(38)

With these calculations we obtain the full characterization of E(di di bj ) in terms of the integrals of compound Laplace transforms; see Appendix B for further details. Appendix B also treats the case where all indices are distinct, which is similar to the above but requires further generalizations of (28), and (37), and a more intricate decomposition of the indicator terms. 4.2.3 Explicit computation of the compound Laplace transforms

We ﬁnally compute the compound Laplace transforms. Feller diﬀusions are examples of aﬃne jump diﬀusion processes. The key result we use here is due to Duﬃe, Pan & Singleton (2000). Under mild technical conditions, the extended transform with respect to P of a P-aﬃne jump diﬀusion process X takes an exponentially aﬃne form E e−u
Rs
t

X(v)dv+zX(s)

Ft = exp(µX (t, s; u, z) + νX (t, s; u, z)X(t)),

(39)

for u, z ∈ R, where µX and νX are solutions to a system of Riccati ordinary diﬀerential equations. In the case where X is a Feller diﬀusion, the system has closed-form solutions. By formula (39), the aﬃne form of the measure change, and conditioning, we can compute the compound Laplace transforms explicitly. Although we encounter a variety of diﬀerent forms of compound Laplace transforms, the computations of these transforms are similar. Consider the computation of ij i∗ i∗ E(E0,s,v Lv,w Lv,p ) appearing in (38). By an application of formula (39) with respect to P∗ and the ∗ ∗ P -Feller diﬀusions Xi∗ and X0 with parameter triplets given by (24), we have i∗ Lv,w = E∗ e− Rw v ∗ ∗ Xi (q)+ωi βi X0 (q)dq ∗ Xi (q)dq

Fv
Rw
v ∗ X0 (q)dq

= E∗ e−

Rw v Fv E∗ e−ωi βi

Fv

i∗ 0∗ ∗ = exp(µi∗ (v, w; 1, 0) + νX (v, w; 1, 0)Xi∗ (v) + µ0∗ (v, w; ωi βi , 0) + νX (v, w; ωi βi , 0)X0 (v)) (40) X X

where the second equality follows from the P∗ -independence of the risk factors and the ∗ superscript indicates that we are dealing with the extended transform of the risk-neutral factors under P∗ . Now we
The order of the indices is signiﬁcant. The subscripts are written in ascending order, and the superscripts are ordered so that they align with the upper limit of the corresponding integral in the exponential transform.
14

17

∗ can make the appropriate substitutions of Xi∗ (q) = αi +βi Xi (q)+ui (q), and X0 (q) = α0 +X0 (q)+u0 (q), using the measure change parametrization described in Section 4.1. The independence of all risk factors and modulator terms permits a multiplicative separation of the individual exponential transforms. Each transform can then be further decomposed by iterative conditioning on the appropriate break points of the time integrals in the transforms. For details of the calculation, as well as the ij i∗ i∗ explicit characterization of E(E0,s,v Lv,w Lv,p ), see Appendix C. Clearly, with the same technique we can compute all the other compound Laplace transform terms, and this allows us to replace the Lm (t) term in Proposition 4.1, with an exponentially aﬃne function of the initial factor values, with deterministic coeﬃcients. These coeﬃcients can be computed explicitly via the sequence of Riccati ODEs that arise from successive conditioning arguments. Of course for alternative risk factor models (e.g. quadratic diﬀusions), the characterization of the speciﬁc function that replaces Lm (t) will diﬀer, but the sequence of arguments remains the same.

5

Numerical results

To illustrate the features of the PGP, this section provides numerical results for the model formulation in Section 4. These results indicate, in particular, the implications for optimal portfolios of ﬁrm idiosyncratic default risk and default correlation, and the signiﬁcance of skewness over both long and short horizons. They also illustrate robustness of the optimal portfolios to parameter estimation errors, and initialization of the optimization.

5.1

Base case with alternative moment preferences

We consider a portfolio of n = 15 reference names. The names are ordered according to their credit quality, as measured by their swap spreads Si , ascending. Thus, name 1 is the least risky name, while name 15 is the most risky. The swap spreads, actual and risk-neutral risk factor parameters, factor loadings, aﬃne measure change parameters, and noise means are given in Table 1. Other parameters include = 0.6, r = 0.02, κi = κ = 0.414, κ∗ = κ∗ = 0.357, η = −0.5. These parameters represent i a realistic range of low, medium, and high risk names based on the estimates of Duﬃe et al. (2006) and Eckner (2008), whereas the parameters regarding the change of measure are consistent with the ﬁtted sector data of Berndt et al. (2005).15 The investment horizon and contract tenure are H = 0.5 and T = 1.75, respectively. We compute the moments of the portfolio mark-to-market using the methods developed in Section 4.2. The time integrals in the moment expressions are discretized on a quarterly grid, consistent with credit derivatives market convention. Since none of the expressions require any simulation or highcomplexity numerical approximation techniques, the moments can be computed and re-computed for alternative parameter sets easily. We substitute the computed moments into the PGP (6)–(7), which can be solved by standard nonlinear optimization routines. Here we carried out the higher order optimizations via nonlinear interior-point methods over randomized initial values. The feasible
15

The parameter values were randomly sampled from a uniform distribution centered at the empirical estimates.

18

Name 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Spread (bp) N/A 21.33 77.93 107.65 199.92 277.50 311.85 317.60 327.76 691.13 802.73 907.92 1130.49 1198.85 1359.57 1948.28

ωi 1 0.09 0.16 0.27 0.24 0.56 1.32 0.54 0.48 0.09 1.11 0.74 0.03 1.91 1.06 2.55

θi 0.0228 0.0003 0.0029 0.0027 0.0089 0.0042 0.0549 0.0109 0.0338 0.0621 0.0947 0.0984 0.0685 0.0530 0.0872 0.1591

σi 0.1140 0.0342 0.0456 0.0592 0.0558 0.0853 0.1310 0.0838 0.0790 0.0342 0.1201 0.0981 0.0197 0.1576 0.1174 0.1820

∗ θi 0.1592 0.0016 0.0101 0.0111 0.0270 0.0292 0.0341 0.0306 0.0469 0.1105 0.1010 0.1143 0.1825 0.0961 0.1645 0.2254

∗ σi 0.1140 0.0210 0.0253 0.0392 0.0373 0.0617 0.0631 0.0707 0.0407 0.0254 0.0839 0.0890 0.0170 0.1523 0.1146 0.1444

αi 0.0300 0.0003 0.0029 0.0027 0.0073 0.0076 0.0049 0.0052 0.0122 0.0249 0.0134 0.0033 0.0442 0.0022 0.0177 0.0286

βi 1 0.376 0.309 0.437 0.445 0.523 0.232 0.712 0.265 0.553 0.488 0.823 0.733 0.934 0.954 0.629

θui 0.0420 0.0005 0.0008 0.0019 0.0017 0.0046 0.0048 0.0060 0.0020 0.0008 0.0085 0.0096 0.0004 0.0280 0.0159 0.0252

Table 1: Annualized swap spreads (1bp=10−4 ) and risk factor parameters for reference names. set for (6) is always nonempty.16 To deal with the existence of local optima we use randomization and perturbations of the boundaries. In conjunction with the capital constraint, the premium and loss payments deﬁne eﬀective ranges for each name over which we generate 1000 random points used as initial values. We keep track of the running minimum of the objective value over the iterations. After each randomized session we also perturbed the constraint C by small increments.17 Table 2 reports the results for the 3-moment problem with preference triplets (γ1 , γ2 , γ3 ) and capital constraint C = 100.18 The ﬁrst ﬁve columns of Table 2 provide an interesting comparison between a full ﬂedged 3moment optimal portfolio and simpler alternative investment strategies. Note that the capital constraint of Section 3.1 corresponds to a minimal risk protocol that the investor must abide by. The (1, 0, 0) or maximum expected mark-to-market proﬁt portfolio represents the strategy in which the investor only considers the expected mark-to-market gains of the positions and decides upon an allocation that will maximize this gain. This is a naive but realistic strategy that roughly corresponds to the investor looking for undervalued positions that will appreciate or overvalued positions that will depreciate the most over the horizon, while complying with the risk protocol on maximum loss. Since the governing capital constraint is nonlinear, the allocation under this criterion is still nontrivial – one does not simply align the expected mark-to-market gains and concentrate the entire allocation on a single name. The maximum expected portfolio value is 33.09. While this is much higher than any other single or multi-moment strategy, the overall risk proﬁle is highly unattractive if we consider the other moments. The portfolio has an extremely large variance of 100.00 and a signiﬁcantly negative skew of −44.95. The (0, 1, 0) portfolio of Table 2 corresponds to a minimum variance strategy. This strategy has
C ˜ ˜ The capital constraint always contains the element δ where δi = n . It is a standard neighborhood search technique in nonlinear programming to go through C ± ∆ for small values of ∆ to check for signiﬁcant turning points. 18 The 0.00s in Table 2 are not identically zero but 0.0000 rounded to the 4th decimal. 16 17

19

(γ1 , γ2 , γ3 ) Mean Var Skew δ1 δ2 δ3 δ4 δ5 δ6 δ7 δ8 δ9 δ10 δ11 δ12 δ13 δ14 δ15

(1, 0, 0) 33.09 100.00 −44.95 −1.30 0.00 0.00 114.60 32.09 −17.94 −40.66 42.56 −13.37 −19.82 0.00 −24.25 −10.42 −20.53 0.00

(0, 1, 0) 3.33 5.18 −143.24 76.87 24.06 18.83 7.73 2.85 2.30 10.85 5.41 3.02 1.76 1.71 3.10 2.44 2.77 1.32

(0, 0, 1) 3.17 5.27 8.26 86.14 21.53 16.47 7.28 2.74 1.86 9.44 5.01 2.17 1.06 1.82 2.41 2.91 2.58 1.59

(1, 1, 0) 10.29 5.23 −2.48 53.65 33.15 18.86 26.67 6.93 −1.44 3.89 15.46 0.81 −1.68 2.90 −2.19 0.084 −1.33 1.61

(0, 1, 1) 3.37 5.28 8.23 84.13 22.54 17.37 7.51 2.81 1.98 9.97 5.16 2.36 −0.64 1.76 2.52 2.78 2.58 1.50

(1, 1, 1) 11.50 12.61 2.23 −13.67 51.67 33.08 28.17 8.10 2.24 12.96 16.62 2.64 −2.29 3.23 −2.90 2.48 1.73 1.88

(3, 3, 1) 16.02 16.90 1.44 23.63 38.53 17.37 43.73 10.40 −2.28 −6.02 24.75 −1.54 −2.41 4.14 −3.49 −1.43 −2.60 1.84

(3, 2, 1) 24.38 40.15 0.39 0.00 30.49 0.00 73.09 15.28 −5.56 −13.51 38.60 −4.62 −6.33 4.73 −8.45 −3.57 −6.79 1.27

(1, 1, 3) 4.19 5.32 8.15 82.09 23.51 17.32 9.51 3.25 1.70 9.17 6.15 2.03 −0.91 1.84 2.08 2.59 2.24 1.50

(1, 3, 1) 4.25 5.31 8.07 83.49 23.10 16.46 10.31 3.40 1.42 8.26 6.50 1.64 0.53 1.92 1.71 2.64 2.04 1.60

Table 2: Optimization results: optimal mean, variance, and skewness of the portfolio mark-to-market value, and the corresponding optimal position notional values. The capital constraint C = 100. recently been emphasized in the equity portfolio selection literature, see DeMiguel & Nogales (2009) and others. It is motivated by the fact that mean equity returns are hard to estimate accurately. In view of these estimation errors and the resulting instability of the portfolio policies, the investor may be better oﬀ ignoring the mean return as a performance metric altogether. In our credit swap setting, the (0, 1, 0) portfolio represents the strategy in which the investor minimizes portfolio variance while fully utilizing the assigned credit exposure. The minimum variance is 5.18. However this is accompanied by a low expected mark-to-market value of 3.33, and a portfolio skewness of −143.24 that is hard to ignore. On the other hand, we may also seek to maximize the skewness of the portfolio subject to the capital constraint. This corresponds to the (0, 0, 1) strategy in Table 2. The maximum portfolio skewness is 8.26. While the portfolio variance remains a modest 5.27, the expected markto-market value of the portfolio is the lowest of all three test cases. In light of their full 3-moment proﬁles, none of these single moment strategies oﬀer attractive portfolios. The next two columns of Table 2 represent mean-variance optimal (γ3 = 0) and skewness-variance optimal (γ1 = 0) portfolios. As expected, mean-variance optimality enjoys a high expected value of 10.29, and a relatively low variance of 5.23. However, the portfolio has a signiﬁcant negative skewness. Skewness-variance optimality produces a high positive skewness of 8.23, but this comes at the expense of a signiﬁcantly reduced expected value. Altering the ratio between γ1 and γ2 in the former, and γ2 and γ3 in the latter case traces out mean-variance and skewness-variance eﬃcient portfolios, respectively. The signiﬁcant negative skewness (−2.48) in the mean-variance optimal portfolio indicates the signiﬁcance of including the third moment into the optimization. See Figure 1. The mixed preference cases strike a balance between these extreme cases. Looking at the 5 columns of Table 2 where all three preference parameters are non-zero, we see that raising one parameter while keeping the other ﬁxed increases the corresponding moment at the expense of the 20

Figure 1: Optimal moments of portfolio mark-to-market for diﬀerent preferences.

others. This is especially noticeable in the case of (1, 3, 1) → (3, 3, 1) in which the expected value rises from 4.25 to 16.02 but the portfolio variance increases from 5.31 to 16.90 and the skewness drops from 8.07 to 1.44. Whether this is a worthy trade-oﬀ is a matter that depends entirely on the investor’s risk appetite. Altering the ratio between the three preference parameters would trace out the mean-variance-skewness optimal surface, whereas holding one of the parameters ﬁxed while altering the other two would trace out iso-mean, iso-skewness, and iso-variance curves. An interesting observation is that the optimal allocations vary drastically over the diﬀerent risk proﬁles. For instance, in the (1, 1, 1) portfolio we sell protection on reference name 7 for a notional value of 12.96, while in the (3, 3, 1) portfolio we buy protection for a notional of 6.02. Skewnessvariance optimality recommends that we sell protection for a notional value of 84.13 on the least risky name 1, whereas in the (1, 1, 1) portfolio we buy protection with a notional value of 13.67 and in the (3, 2, 1) portfolio we exclude this name entirely. This could be indicative of the fact that this low risk name has an extremely skewed mark-to-market distribution but contributes little towards mean-sensitive objectives because of its low spread and low risk of default. With the exception of the mean-optimal portfolio, all preference triplets recommend a small positive notional for the riskiest name 15. Because of the pronounced skewness and co-skewness that exist in the mark-to-market gains, the mixed preference cases appear to oﬀer more attractive proﬁles, sacriﬁcing a relatively small portion of the expected value while gaining signiﬁcantly in portfolio skewness, and allowing the investor to avoid negatively skewed distributions.

5.2

Scenario analysis

To understand the implications on optimal portfolios of idiosyncratic default risk and default correlation, we analyze alternative scenarios. The optimization results for these scenarios are given in Table 3, and Figure 2 visualizes the corresponding optimal moments. First we consider a moderately elevated state of common risk by increasing the equilibrium level of the common risk factor to θ0 = 0.0858. With preferences (1, 1, 1), the optimal expected value becomes 11.82 with a portfolio variance of 18.57, and skewness of 1.25. This shows that we can achieve a higher expected value in this elevated state of common risk. However this expected value is overshadowed by

21

Scenario Mean Var Skew δ1 δ2 δ3 δ4 δ5 δ6 δ7 δ8 δ9 δ10 δ11 δ12 δ13 δ14 δ15

Base 11.50 12.61 2.23 −13.67 51.67 33.08 28.17 8.10 2.24 12.96 16.62 2.64 −2.29 3.23 −2.90 2.48 1.73 1.88

MG 11.82 18.57 1.25 53.41 30.55 18.68 19.09 7.15 7.97 −1.09 9.36 7.42 −0.72 8.83 −1.22 −0.49 −2.93 1.78

DC 16.70 14.97 1.29 19.54 36.39 16.71 42.75 29.98 −1.45 −7.95 11.79 1.01 −1.36 4.17 −1.41 −4.03 −2.37 1.48

CS 8.03 37.91 0.42 31.47 24.80 20.98 16.56 10.27 6.30 15.76 13.72 8.47 −4.52 5.03 −7.22 4.85 5.26 −1.24

EXT 31.70 20.44 1.08 2.01 64.19 33.19 16.82 25.76 3.51 7.17 −2.76 5.20 −3.66 −4.03 5.37 −4.66 −1.08 −3.57

Table 3: Optimization results for the alternative scenarios, with preferences (1, 1, 1). a larger portfolio variance, and a signiﬁcantly reduced positive portfolio skewness. The high portfolio variance is a result of the larger univariate variances and an increase in the co-movements of the positions. This ﬁrst scenario is denoted MG in Table 3 and Figure 3. In order to isolate the eﬀect of an increase in default correlation, we examine a scenario denoted DC in Table 3 and Figure 3. Here we increase default correlation among the names by 50% while maintaining the overall level of risk for each name. This is achieved by increasing the factor loadings √ by a multiplicative factor of 1.5, while simultaneously scaling σ0 to σ0 / 1.5, and subtracting 0.5ωi θ0 from each θi . This leaves the actual intensity parameters κi , θi , and σi for each obligor unchanged but it shifts the contribution from the idiosyncratic factors to the common factor. The directionality of the changes in the optimal portfolio moments over the base case are identical to the previous scenario MG. This can be attributed to the fact that in MG, due to the increase in the equilibrium level of the common risk factor alone, the common factor assumes a more prominent role, eﬀectively increasing default correlation. The diﬀerence is that in MG the overall level of risk is also elevated so it was not possible to isolate the eﬀects of one from the other. In the DC scenario, the expected portfolio value rises to 16.70 but the increase in portfolio variance is more moderate at 14.97. The portfolio skewness drops to a similar level of 1.29. Interestingly, the directionality of the positions (long/short) are identical to MG, apart from the 6th obligor which is a short protection position in DC. This indicates that for comparable risk levels, the correlation structure among the reference names is more important than the absolute risk levels in deciding optimal compositions. Next we analyze a “crisis state,” denoted CS in Table 3 and Figure 3, in which increase the equilibrium level of the common risk factor to θ0 = 0.0628, and also increase all factor loadings by 50%. Not only does this induce higher default correlation but, as speciﬁed in Section 4.1, this change im√ plies an increase in the volatilies of the idiosyncratic risk factors via the restriction σi = ωi σ. Hence this represents a state in which we have both high systematic and idiosyncratic risk across the board. 22

8 7

6.5 b113 b131 6 5.5 b321 CS

6 5 4 3 2 1 5 standard deviation

5 4.5 4 3.5 b111 3 MG b331 DC EXT

skewness

b111 b331 DC EXT CS 10 MG 15 20 expected P&L b321 25 30

2.5

b113 10 15 20 expected P&L 25 30

b131 2 5

Figure 2: Optimal moments for alternative scenarios (MG – increase in long-term global intensity factor mean; DC – increased default correlation; CS – crisis state; EXT – introduction of high default risk name). The label b(γ1 , γ2 , γ3 ) refers to the base case with preferences (γ1 , γ2 , γ3 ). Left panel : Mean-skewness cross section. Right panel : Mean-standard deviation cross section. Under this regime, and with preference parameters (1, 1, 1), the optimal expected value becomes 8.04 with a portfolio variance of 37.91 and portfolio skewness of 0.42. The expected value shows a large decrease over both the base case (1, 1, 1) and the previous case, and the portfolio variance has also grown signiﬁcantly. The skewness drops signiﬁcantly from both the previous case of moderately elevated common risk and the base case. The interactions between the mark-to-market values of the 15 credit swaps are complex, and a number of observations warrant discussion. Greater risk can act in both directions by simultaneously increasing the spread and the expected loss payment. We can capitalize on large movements in either direction by selling protection when the expected value is positive and buying protection otherwise. Therefore the directionality of change in expected portfolio value in greater risk scenarios is not clear-cut. In this crisis state, both the larger idiosyncratic intensity volatilities and higher default correlation contribute to a much larger portfolio variance. As a result, the variance objective places a signiﬁcantly greater strain on the other two objectives, contributing to both lower expected portfolio value and lower portfolio skewness. Finally we examine the case where the least risky name name 1 is replaced by a high risk name. We raise the equilibrium mean level parameter of the idiosyncratic factor of name 1 in Table 1 from 0.0003 to 0.1883, and also increase the common factor loading to 1.39. The results are listed under the label EXT in Table 3 and Figure 3. The optimal allocation dictates that we sell protection on name 1, though on a much smaller notional than in the 2 other scenarios. However, the introduction of this high risk name drastically changes the optimal holdings in other positions as well, and consequently the overall risk/return proﬁle of the portfolio. The most notable diﬀerence is that now we buy protection for all of the 3 riskiest obligors (apart from the newly introduced obligor). This demonstrates the complex way in which the polynomial goal program achieves attractive risk/return 23

Figure 3: Optimal moments of portfolio mark-to-market for diﬀerent scenarios.

proﬁles via trading oﬀ the intricate interactions between the individual mark-to-market changes. Only a small portion of the large gain in expected portfolio value can be attributed to actual changes in the weight of the ﬁrst obligor. The rest is due to the changes that are now more favorable in the other positions because of the diﬀerent overall coskewness and covariance structure.

5.3

Robustness of optimal portfolios

We consider the sensitivity of optimal portfolios to changes in the model parameters. Understanding this sensitivity is important, for several reasons. First, if the selection problem is solved sequentially over shorter horizons, then changes in allocations incur costs that impact overall returns. The costs increase with the discrepancies in adjacent allocations, which are governed by the changes in the ﬁtted model parameters. Second, in any single one-period setting, parameter estimation errors may play a role. While the maximum likelihood estimation problem for doubly-stochastic models of correlated default timing are relatively well-understood (Berndt et al. (2005), Duﬃe et al. (2006), Eckner (2009) and Eckner (2008)), not all parameters can be estimated with high accuracy. It would be undesirable if estimation errors were to drastically change the optimal investment policies. Hence, we examine the eﬀect of perturbing some of the parameters that are diﬃcult to estimate. We start from the base parameters of Table 1, and focus on the (1, 1, 3)-portfolio of Table 2. The parameters that are most diﬃcult to estimate are those related to the measure change, η, αi and βi , and the long-term equilibrium levels θi of the P-intensities. Since all factors are structurally symmetric, for clarity we perturb the global measure change parameter η, the common risk factor P-equilibrium level θ, and common risk factor measure change adjustment α0 , each over a range of ±10%. Figure 4 contains boxplots of the notionals for the four names with the largest range of deviation. For the ﬁrst name, the optimal notional ranges between [79.98, 82.78], [79.16, 83.39] and [79.77, 82.23], for η, θ and α0 perturbations, respectively. Given that the total notional committed for the base (1, 1, 3)-portfolio is 165.89, a turnover range of less than 4.23 only corresponds to 2.5% of the committed notional. For the second name, the respective ranges are even narrower at [23.33, 24.24], [23.04, 24.49] and [23.35, 24.19]. For the subsequent names the deviations are further reduced, and even for the most volatile θ perturbations, the maximum total deviation of the 15 names amounts

24

24.5 83 notional (1) notional (2) 1 2 3 parameter (eta, theta, alpha) 82 81 80 79 23 1 2 parameter 3 24

23.5

18 17.8 notional (3) 17.6 17.4 17.2 1 2 parameter 3 notional (4)

10 9.8 9.6 9.4 9.2 1 2 parameter 3

Figure 4: Boxplots of the (1, 1, 3)-portfolio notionals of the ﬁrst four names with the η, θ, and α0 parameters being perturbed over a range of ±10%, with step size of 1%. The box indicates the interquartile (25-75%) range, the end points indicate the minimum and maximum, and the horizontal line indicates the median of the notional after perturbation.

to less than 16% of the committed notional. Thus the optimal portfolios are remarkably robust to univariate parameter perturbations. A deﬁnite trend across the names is that θ perturbations have a more pronounced eﬀect on the optimal allocations while the solutions are much more stable under perturbations to the measure change parameters. This trend is also indicated in Figure 5, which displays the optimal mean, variance, and skewness of the (1, 1, 3)-portfolio with η, θ and α0 perturbations. Since the parameters relate to the moments in a highly nonlinear way and the moments are components of a nonlinear optimization problem, not much can be said from an analytical point of view about monotonicity trends. However, what is apparent is that θ perturbations again exhibit the most salient eﬀects. The optimal variance and skewness are constricted to extremely narrow bands for the η and α0 perturbations, while the band for the mean is somewhat larger. The θ perturbations prompt relatively greater but not excessive changes to all optimal moments.

25

4.5 4.4 mean 4.3 4.2 4.1 4 −10 variance

5.8 5.6 5.4 5.2 5 eta theta alpha

−5

0

5

10

−10

−5

0

5

10

9.5 9 skewness 8.5 8 7.5 7 −10

−5 0 5 10 percentage deviation of parameters

Figure 5: Changes to the optimal mean (left), variance (center), and skewness (right) of the (1, 1, 3)portfolio with the η, θ and α0 parameters being perturbed over a range of ±10%.

5.4

Performance

The computational complexity of computing the skewness tensor is O(n3 ), where n denotes the number of products in the investment universe. In the 15 name example of Section 5.1, the total runtime for this computation was 153.38 seconds, implemented on Matlab 7.9.0.529, with an Intel Xeon E5620 2.4GHz processor with 3.5GB of RAM. Once the normalized mean vector, covariance matrix, and skewness tensor are computed, they can be cached and reused in the goal function revaluations. Optimizations were performed over 1, 000 randomized initial vectors. Table 4 provides the total runtime for optimization problems with diﬀerent preference parameters over 1000 randomizations. (γ1 , γ2 , γ3 ) (1, 1, 1) (3, 3, 1) Runtime (seconds) 2464.74 2389.91 (3, 2, 1) (1, 1, 3) (1, 3, 1) 4281.19 2357.06 2435.54

Table 4: Optimization runtime

26

Portfolio Moment 1 Week Mean 0.60 (1,1,0) Var 1.95 Skew −8.17 Mean 0.41 (1,1,0) Var 4.42 Skew 5.48

1 Month 1 Quarter 2 Quarters 1 Year 2.21 5.50 10.29 16.33 2.59 3.23 5.23 8.01 −4.32 −3.83 −2.48 −2.12 2.04 5.65 11.5 19.52 7.34 10.96 12.61 23.56 3.79 3.53 2.23 2.05

Table 5: Moments of (1, 1, 0), and (1, 1, 1) optimal portfolios over multiple horizons.

5.5

Skewness over multiple horizons

The H = 0.5 horizon is illustrative of medium to long-term credit investments, less liquid loan or bond positions, and corresponding hedges. It is also relevant to risk capital frameworks where conservative assumptions must be made with respect to liquidity horizons.19 However one may also consider the context of more active credit portfolio management, and the qualitative implications of shorter horizons. Table 5 shows the moments of the (1, 1, 0), and (1, 1, 1)-optimal portfolios for horizons of 1 week, 1 month, 1 quarter, 2 quarters,20 and 1 year. The portfolio skewness plots are shown in Figure 6. As we can see, neither the large negative skewness of the mean-variance optimal portfolios nor the positive skewness attainable in mean-variance-skewness optimal portfolios vanish at either end of the investment horizon spectrum. Signiﬁcant skewness is present across all horizons. This is a key observation that demonstrates that the pronounced skewness in the portfolios of previous sections are not an artifact of the particular choice of horizon. Furthermore, the asymmetries of the distributions increase with shorter horizons. Note that the shortest horizon we look at is 1-week, and the distributions are model-implied. Hence, this eﬀect is distinct from the microstructure noise of higher frequency return data mentioned in empirical equity portfolio literature. As we move to shorter horizons we are changing the structure of the payoﬀs. Less of the payoﬀ is attributable to premium cash ﬂows or default-contingent payments while the mark-to-market contribution at the end of the horizon dominates the proﬁt and loss. As discussed previously, mark-to-market ﬂuctuations for CDS positions have naturally asymmetric distributions because the spreads cannot tighten below zero. The mean and variance ﬁgures of Table 5 show that the skewness objective places considerable pressure on the other two objectives at shorter horizons. Rather than being immune to higher moment considerations, an active portfolio manager, managing short-term positions, may be unknowingly exposed to large negative skewness or alternatively unaware of the positive portfolio skewness that he or she can take advantage of.
In Basel II Incremental Risk Charge calculations, the minimum liquidity horizon for most credit instruments are at least 1 quarter. See Basel Committee on Banking Supervision (2009). 20 This corresponds to H = 0.5 of previous sections.
19

27

6

0 1 Week

0.00 -1

0.20

0.40

0.60

0.80

1.00

1.20

5 -2 4 1 Month 1 Quarter 3 -3 2 Quarters 1 Year

Skewness

Skewness

-4 -5 -6 -7 -8

1 Month

1 Quarter

2 Quarters
2

1 Year

1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 -9

1 Week

Horizon

Horizon

Figure 6: Skewness of (1,1,0), and (1,1,1)-optimal portfolios over diﬀerent horizons.

6

Mixed portfolios

This section discusses the treatment of mixed portfolios of credit swaps, corporate or government bonds or loans, and standard index swaps, as well as premium accruals, and stochastic interest rates.

6.1

Corporate bonds and loans

A long corporate bond position is similar to a credit swap protection selling position. The diﬀerence is that there is an initial cash outﬂow, the premium stream is replaced by a coupon stream, and the default loss paid out becomes a recovery payment received. Also if the ﬁrm survives to the maturity T > H, then the bond investor receives the principal 1. Thus, if the bond issuer defaults before H, then the cumulative discounted payments are
H

Dib (H)

=

−Pib

+ tm ≤H

e

−rtm

ci (1 −

Ntim )

+
0

i (1 − )e−rs dNs

where ci is the per-period coupon rate and Pib is the initial price of the bond. If the bond issuer survives to H, then the investor receives the scheduled coupons, and the mark-to-market value of the bond at H. Hence the cumulative discounted cash ﬂows are21 Sib (H, T ) = −Pib + tm ≤H
21

ci e−rtm

This formulation does not account for the liquidity of the corporate bond market, which may be limited in practice. A liquidity discount could be included in the mark-to-market formulation to account for this.

28

T

+ E∗ tm >H

e−rtm ci (1 − Ntim ) +
H

i i (1 − )e−rs dNs + e−rT (1 − NT ) FH .

Risky loans can be modeled as corporate zero coupon bonds, while risky loans with ﬁxed prepayment schedules can be modeled as corporate coupon bonds. From the bond investor’s perspective, the mark-to-market value at time H on a unit notional bond position is given by Pib = erH 1{τi ≤H} Dib (H) + 1{τi >H} Sib (H, T ) . (41)

Modulo scaling and additive constants, (41) shares the same basic components as our original formulation (3) and thus the analytical methods of Section 4.2 can be applied directly. In addition to the regular payments and loss payments that can be mapped directly to their credit swap counterparts, the capital constraint must now be adjusted to account for the initial cash outﬂow for long bond positions and cash inﬂow for short bond positions. We can also incorporate short sale restrictions by placing lower bound constraints on the bond notionals in addition to the capital constraints at both stages of the goal program.

6.2

Standard index swaps

Next we show how to incorporate a standard index swap into the portfolio problem. An index swap is referenced on a pool of k single-name credit swaps with notional 1/k, premium payment dates tm and maturity T > H, which is also the maturity of the index swap. The index reference pool may overlap with the set of single-name swaps that are available on a stand-alone basis for inclusion in the portfolio. We must expand the model of default timing in Section 4.1 to the superset of all reference names. The index reference pool is a subset {l1 , l2 , . . . , lk } of this superset of reference names. The index swap protection seller covers default losses in the underlying pool as they occur; the swap is not canceled at a default. The premium at date tm is a fraction of the notional of the names that have survived to tm . It is given by Six (1 − Ntm /k), where Six is the per-period index swap spread and N = N l1 + · · · + N lk is the default count in the index portfolio. Thus Dix (H), the discounted stream of premium payments received during (0, H] less the discounted stream of default payments made during that period, is given by Dix (H) = tm ≤H

e−rtm Six 1 −

Ntm k

H

−

k

e−rs dNs .
0

Further Six (H, T ), the mark-to-market value of the index swap at H, discounted to time 0, is Six (H, T ) =E
∗ tm >H

e

−rtm

Six

Nt 1− m k

T

−

k

e−rs dNs FH .
H

The mark-to-market value at H of a unit-notional index swap protection selling position is then Pix = erH Dix (H) + Six (H, T ) . 29

Note that the index default process N is simply a summation of default indicators of individual reference names. A review of the techniques of Section 4.2 reveals that the analytical framework can be adopted seamlessly for any derivative product whose payoﬀ can be decomposed into a linear combination of indicator terms of the form 1{τ1 >s1 ,...,τn >sn } and Stieltjes integrals of deterministic functions with respect to the default processes. This includes both the bonds and loans of Section 6.1 and the index swaps of this section. Although we have thus far dealt with diﬀerent credit sensitive instruments separately, there is nothing in our analytical framework that prevents us from expanding the security pool to include all of the aforementioned securities together. Each individual mark-to-market value stream is a linear combination of indicator terms of the form 1{τ1 >s1 ,...,τn >sn } and Stieltjes integrals with respect to the default indicators. Hence the cross terms between diﬀerent securities that appear in the higher-order moment computations are not diﬀerent from those of the credit swaps. Their additive decompositions will involve the same compound Laplace transforms as in Section 4.2.3. After the moments are fully decomposed, the default and survival scenarios of the obligors are propagated through the calculations by the Laplace transform terms. Thus the problem of overlapping names is addressed naturally.

6.3

Accrued premium payments

In practice, if default occurs between payment dates the credit swap protection buyer must pay the protection seller the premium that has accrued from the last payment date to the default time. Incorporating this feature into our framework only requires a minor modiﬁcation. We must add Si tm ≤H

e−rτi

τi − tm−1 1{tm−1 H T

e

−rtm

(1 −

Ntim )

1 + ∆m

tm i e−rs (s − tm−1 )dNs H∨tm−1

FH

− E∗
H

i e−rs dNs FH

respectively. The extra terms are Stieltjes integrals of simple deterministic functions with respect to the default process. They can be treated by integration by parts in the moment computations.

30

6.4

Stochastic interest rates and government bonds

There is a natural extension of our formulation to a stochastic risk-free rate of interest. We can model the short rate as an additional risk factor following Feller diﬀusion dynamics or an aﬃne function of multiple factors that are all Feller diﬀusions, for a more elaborate model. Positive correlation with default intensities can also be accommodated by adding the common risk factor X0 , with an appropriate factor loading, to the independent interest rate factor(s). The same paramterization of the measure change of Section 4.1 should govern the interest rate factors. Because of the mean reversion and nonnegativity characteristics of Feller diﬀusions, they have long been benchmark processes for short rate models. This modiﬁcation is unobtrusive to our overall formulation, as the interest rate factors simply act as additional risk factors that add coeﬃcient terms to the compound Laplace transforms in our moment computations. The introduction of stochastic interest rates leads to a meaningful extension of the framework to include non-defaultable government bonds in the investment pool. Defaultable sovereign bonds can be dealt with analogously to the corporate bonds of Section 6.1. The cash ﬂow stream of a default risk-free government bond consists of ﬁxed discounted payments of coupons and principal. With the aforementioned Feller diﬀusion factor model for the short rate, the discount factors will simply be exponentially aﬃne functions of Feller risk factors. Hence, the computation of the related moments and compound Laplace transforms can be seamlessly incorporated into our existing analytical framework of Section 4.2.

7

Conclusion

We formulate and solve the static optimization problem for a ﬁxed-income portfolio that may contain corporate or government bonds, industrial loans or positions in credit derivatives such as credit and index swaps. We propose to optimize the portfolio mark-to-market proﬁt or loss at the investment horizon, which incorporates the premium and default cash ﬂows of long and short cash and derivative positions, and the survival-contingent market value of these positions at the horizon. The selection problem is cast as a polynomial goal program that involves a two-stage constrained optimization of preference weighted moments of the portfolio mark-to-market. The decision variable is the vector of contract notionals. A capital constraint guarantees the solvency of the portfolio investor during the investment period. Our multi-moment formulation takes account of the non-normal distribution of the portfolio markto-market, in particular the skewness due to the risk of issuer default. It is also computationally tractable for the large portfolios common in practice. This is because we succeed in developing analytical expressions for the moments of the portfolio mark-to-market, which are valid for a class of doubly-stochastic models of correlated default timing. These models are widely used for the analysis of portfolio credit risk and the valuation of credit derivatives, and their estimation is well-understood. These features facilitate a practical application of our optimization methodology. Numerical experiments illustrate the properties of the optimal portfolios, and document the computational advantages of the moment-based formulation. They indicate the economic and statistical importance of including higher moments in the optimization, and highlight the implications for op31

timal portfolios of ﬁrm idiosyncratic default risk and default correlation. The optimal policies are found to be robust with respect to errors in the estimation of the parameters of the default timing model.

A

Measure change

This appendix details the measure change in Section 4.1. Let {λi , λ∗ , Xj , uj } be the (almost surely i strictly positive) P-Feller diﬀusions speciﬁed in Sections 4.1.1 and 4.1.2. The default indicator process (N 1 , . . . , N n ) is a doubly-stochastic Poisson process driven by X = (X0 , X1 , . . . , Xn , u0 , u1 , . . . , un ) √ with P-intensity (λ1 , . . . , λn ). Given these dynamics, for any η ∈ R and ﬁnite horizon T > 0, √η i Xi ω T T η √ and √ωi βi ui satisfy Novikov’s condition for T , and 0 λi (s)ds, and 0 λ∗ (s)ds are both ﬁnite Pi almost surely for each i. Let n t

ZtN ZtW

= i=1 n

exp
0

log

λ∗ (s) i λi (s) t 0 t

t i dNs + 0 t

(λi (s) − λ∗ (s))ds i Xi (s)ds
0 t

η = exp − √ ωi i=0 n η2 Xi (s)dWs − 2ωi ui (s)dBs −
0

ZtB = i=0 exp − √

η ωi βi

η2 2ωi βi

ui (s)ds
0

for ω0 = β0 = 1. Then, a strictly positive P-martingale Z = (Zt )t≤T is given by Zt = ZtN × ZtW × ZtB . √ To show that Z is a martingale we argue as follows. The Novikov conditions for √η i Xi and ω √ √η ui are suﬃcient for ZtW and ZtB to deﬁne martingales. A suﬃcient condition for ZtN to deﬁne ωi β i a martingale is that 0 ( λs −1)2 λs ds < ∞ almost surely for any ﬁnite horizon T (Liptser & Shiryayev λs (1989)). We have that both λ∗ < ∞ and λt < ∞ almost surely for every t, therefore, upon expansion t λ∗2 of the integrand, we only need to verify that the quotient λtt is ﬁnite almost surely. However Feller diﬀusions are ﬁnite variance processes so λ∗2 < ∞ almost surely, and the Feller condition bounds the t denominator λt away from 0 almost surely. Hence the given quotient is indeed almost surely ﬁnite. The process Z deﬁnes an equivalent probability measure P∗ on FT via dP∗ = ZT dP. On [0, T ], the default indicator process (N 1 , . . . , N n ) is a doubly-stochastic Poisson process driven by {X(t) : t ∈ [0, T ]}, with intensity (λ∗ , . . . , λ∗ ) relative to P∗ and {Ft : 0 ≤ t ≤ T }. The fact that λ∗ is 1 n i T adapted to the ﬁltration generated by X, along with the fact that 0 λ∗ (s)ds is ﬁnite P∗ -almost i surely, guarantees that the doubly-stochastic property is preserved. Furthermore, the processes Wi∗ , and Bi∗ , i = 0, 1, 2, . . . , n deﬁned by (22)–(23) are P∗ -standard Brownian motions on [0, T ] relative to {Ft : 0 ≤ t ≤ T }.
T
∗

32

B

Moment computations and technical results

Proposition 4.1 is a direct result of the following lemma, which in turn is proved inductively. Lemma B.1. Under the Markov doubly stochastic assumption of Section 4.1, the kth order product term can be expressed in the form (aij − bij + cij − dij ) = j=1,...,k m

Cm
[0,T ]k

Dm (t)I m (t)1m (t)Lm∗ (t)dt, τ H

(42)

where t = (t1 , . . . , tk ) is the k-dimensional time vector, C m is a constant, Dm (t) is a discount function, I m (t) an indicator function that limits the domain of integration, and m is an index for the additive terms that arise in the product. The expression Lm∗ (t) represents a product of k FH -conditional H P∗ -Laplace transforms, and 1m (t) is a random indicator function of the form 1{τi1 >s1 ,...,τik >sk } where τ sj is either a constant or an element of the time vector of integration. Proof. The proof is by induction. For k = 1, Stieltjes integration by parts and the doubly stochastic property under P∗ yields
H

ai − b i + c i − d i = tm ≤H

e−rtm 1{τi >tm } − e−rH 1 − 1{τi >H} − r
0 i∗ 1{τi >H} LH,tm

e−rs (1 − 1{τi >s} )ds
T i∗ e−rs 1{τi >H} 1 − LH,s ds. H

+ Si tm >H

e

−rtm

− e

−rT

1{τi >H} 1 −

i∗ LH,T

− r

The non-integral components can be expressed as [0,T ] · dt integrals with the indicator I m (t) = 1{0≤t≤T } and the multiplicative constant scaled by a factor of 1/T . The time integrals can be expressed with the indicators I m (t) = 1{0≤t≤H} , and I m (t) = 1{H≤t≤T } respectively. Also, by assumpi∗ tion, 1{τi >0} = LH,H = 1. Hence, this expression clearly conforms to (42). Note that I m (t) is a deterministic indicator for the time variable while 1m (t) is a random indicator function. Next we τ assume that (42) holds for k = l. Then for k = l + 1 we have (aij − bij + cij − dij ) j=1,...,l+1 = m [0,T ]l

Dm (t)I m (t)1m (t)Lm∗ (t)dt τ H p [0,T ]

Dp (u)I p (u)1p (u)Lp∗ (u)du τ H

= m,p [0,T ] [0,T ]l

Dm (t)Dp (tl+1 )I m (t)I p (tl+1 )1m (t)1p (tl+1 )Lm∗ (t)Lp∗ (tl+1 )dtdtl+1 τ τ H H

= q [0,T ]l+1

Dq (t)I q (t)1q (t)Lq∗ (t)dt. τ H

The ﬁrst equality is due to the k = 1 case that we have veriﬁed and the induction hypothesis. Clearly the products Dm (t)Dp (tl+1 ), I m (t)I p (tl+1 ), and Lm∗ (t)Lp∗ (tl+1 ) can be subsumed H H into their equivalent expressions in the augmented dimension and 1{τi1 >s1 ,...,τil >sl } 1{τi1+1 >sl+1 } = 33

1{τi1 >s1 ,...,τil >sl ,τi1+1 >sl+1 } . As in the case for k = 1, if il+1 ∈ (i1 , . . . , il ) so that there is an overlap in the names, a redundant integration dimension and appropriate scaling allows us to remain conformed to (42). Therefore, with a relabeling of the indices the last equality follows and we conclude the proof by induction. The integrands of (42) are bounded. Hence, in taking an unconditional expectation we may apply Fubini’s theorem to interchange the integration and expectation and then take out the deterministic factors C m Dm (t)I m (t). We can now condition on the path of the intensities and noise terms over [0, H], and apply the doubly stochastic property under P to arrive at Proposition 4.1. The demonstrative computations of Section 4.2.2 are completed here. First we have E(di di bj ) =
3 ij ij i i i∗ i∗ e−r(H+2T ) L0,H − L0,H,H − 2E E0,H LH,T + 2E E0,H,H LH,T i i∗ + E E0,H LH,T T T ij ij i i i∗ i∗ i i∗ e−r(s+u+H) L0,H − L0,H,H − E E0,H LH,s + E E0,H,H LH,s − E E0,H LH,u H H ij ij i∗ i i∗ i∗ i∗ i∗ + E E0,H,H LH,u + E E0,H LH,s LH,u − E E0,H,H LH,s LH,u T −r(H+T ) H ij i∗ i i∗ i∗ i i∗ i∗ + E E0,H,H LH,T + E E0,H LH,s LH,T − E E0,H,H LH,s LH,T H ji ji i i i∗ i∗ e−r(s+2T ) L0,H − L0,s,H − 2E E0,H LH,T + 2E E0,s,H LH,T 0 i i∗ + E E0,H LH,T H T H T ji ji i i i∗ i∗ i i∗ e−r(s+u+v) L0,H − L0,v,H − E E0,H LH,s + E E0,v,H LH,s − E E0,H LH,u 0 H ji ji i∗ i i∗ i∗ i∗ i∗ + E E0,v,H LH,u + E E0,H LH,s LH,u − E E0,v,H LH,s LH,u H T ji ji i∗ i∗ i i i i∗ e−r(s+u+T ) L0,H − L0,u,H − E E0,H LH,s + E E0,u,H LH,s − E E0,H LH,T 0 H ji ji i i∗ i∗ i∗ i∗ i∗ + E E0,u,H LH,T + E E0,H LH,s LH,T − E E0,u,H LH,s LH,T 2 ji i∗ − E E0,s,H LH,T 2 2 ij i∗ − E E0,H,H LH,T 2

+ r2

dsdu

+ 2re

ij ij i i i∗ i∗ i i∗ e−rs L0,H − L0,H,H − E E0,H LH,s + E E0,H,H LH,s − E E0,H LH,T

ds

+r

ds

+ r3

dsdudv

+ 2r2

dsdu .

For components in which all indices are distinct, we require a natural extension of the notation in (37): s ijk Lt,s,u,w u w

=

ijk E(Et,s,u,w

| Ft ) for

ijk Et,s,u,w

=

exp − t λi (v)dv − t λj (v)dv − t λk (v)dv ,

34

as well as the further generalization of formula (28): ijk P(τi > s, τj > u, τk > w | Ft ) = (1 − Nti )(1 − Ntj )(1 − Ntk )Lt,s,u,w

(43)

with an appropriate re-alignment of the indices to accommodate an ascending order of the cut-oﬀ times. Consider E(ai bj bk ), a component of the third moment in which all indices are distinct. By Stieltjes integration by parts we have
H

E(ai bj bk ) =

2

T S0 (i)E tm ≤H

e−r(tm +H) 1{τi >tm ,τj ≤H} + r
0 H

e−r(tm +s) 1{τi >tm ,τj ≤s} ds

× e−rH 1{τk ≤H} + r
0

e−rs 1{τk ≤s} ds
H

=

2

T S0 (i)E tm ≤H H

e−r(tm +2H) 1{τi >tm ,τj ≤H,τk ≤H} + r
0 H H

e−r(tm +s+H) 1{τi >tm ,τj ≤s,τk ≤H} ds e−r(tm +s+u) 1{τi >tm ,τj ≤s,τk ≤u} dsdu .

+r
0

e−r(tm +s+H) 1{τi >tm ,τj ≤H,τk ≤s} ds + r2
0 0

Now we can use the fact that 1{τi >t,τj ≤s,τk ≤w} = 1{τi >t} − 1{τi >t,τj >s} − 1{τi >t,τk >w} + 1{τi >t,τj >s,τk >w} , and property (43) to arrive at E(ai bj bk ) =
2 T S0 (i) tm ≤H H i,j i,k i,j,k i e−r(tm +2H) L0,tm − L0,tm ,H − L0,tm ,H + L0,tm ,H,H

+r
0 H

i,j i,k i,j,k i e−r(tm +s+H) L0,tm − L0,tm ,s − L0,tm ,H + L0,tm ,s,H ds i,j i,k i,j,k i e−r(tm +s+H) L0,tm − L0,tm ,H − L0,tm ,s + L0,tm ,H,s ds H H i,j i,k i,j,k i e−r(tm +s+u) L0,tm − L0,tm ,s − L0,tm ,u + L0,tm ,s,u dsdu . 0 0

+r
0

+r

2

C

Transform calculations

ij i∗ i∗ Here we provide the details of the computation of E E0,s,v Lv,w Lv,p from Section 4.2.3. With the ∗ appropriate substitutions of Xi∗ (q) = αi + βi Xi (q) + ui (q), and X0 (q) = α0 + X0 (q) + u0 (q), we obtain ij i∗ i∗ E E0,s,v Lv,w Lv,p s v

= E exp

−
0

Xi (q) + ωi X0 (q) + Xj (q) + ωj X0 (q)dq − s Xj (q) + ωj X0 (q)dq

35

i∗ 0∗ ∗ + µi∗ (v, w; 1, 0) + νX (v, w; 1, 0)Xi∗ (v) + µ0∗ (v, w; ωi βi , 0) + νX (v, w; ωi βi , 0)X0 (v) X X i∗ 0∗ ∗ + µi∗ (v, p; 1, 0) + νX (v, p; 1, 0)Xi∗ (v) + µ0∗ (v, p; ωi βi , 0) + νX (v, p; ωi βi , 0)X0 (v) X X s v

= E exp

−
0

Xi (q) + ωi X0 (q) + Xj (q) + ωj X0 (q)dq − s Xj (q) + ωj X0 (q)dq

i∗ + µi∗ (v, w; 1, 0) + νX (v, w; 1, 0)(αi + βi Xi (v) + ui (v)) X 0∗ + µ0∗ (v, w; ωi βi , 0) + νX (v, w; ωi βi , 0)(αi + X0 (v) + u0 (v)) X i∗ + µi∗ (v, p; 1, 0) + νX (v, p; 1, 0)(αi + βi Xi (v) + ui (v)) X 0∗ + µ0∗ (v, p; ωi βi , 0) + νX (v, p; ωi βi , 0)(αi + X0 (v) + u0 (v)) X i∗ i∗ = exp µi∗ (v, w; 1, 0) + µi∗ (v, p; 1, 0) + αi νX (v, w; 1, 0) + νX (v, p; 1, 0) X X s v

×E exp −
0

Xi (q)dq + s i∗ βi νX (v, w; 1, 0) v

+

i∗ βi νX (v, p; 1, 0)

Xi (v)

E exp − s Xj (q)dq

×E exp −(ωi + ωj )
0

X0 (q)dq − ωj s 0∗ 0∗ X0 (q)dq + νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0) X0 (v)

×E exp

i∗ i∗ νX (v, w; 1, 0) + νX (v, p; 1, 0) ui (v)

E exp

0∗ 0∗ νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0) u0 (v)

.

The last equality is a result of the independence of the risk factors and the modulator terms. Next by iterative conditioning we have s v

E exp −(ωi + ωj )
0

X0 (q)dq − ωj s s

0∗ 0∗ X0 (q)dq + νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0) X0 (v)

= E exp

− (ωi + ωj )
0

0∗ 0∗ X0 (q)dq + µ0 (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)) X

0 0∗ 0∗ + νX (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0))X0 (t) 0∗ 0∗ = exp µ0 (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)) X 0 0∗ 0∗ + µ0 (0, s; ωi + ωj , νX (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0))) X 0 0 0∗ 0∗ + νX (0, s; ωi + ωj , νX (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)))X0 (0) .

After applying similar conditioning arguments to the other product terms, we arrive at the following explicit characterization ij i∗ i∗ E E0,s,v Lv,w Lv,p i∗ i∗ = exp µi∗ (v, w; 1, 0) + µi∗ (v, p; 1, 0) + αi νX (v, w; 1, 0) + αi νX (v, p; 1, 0) X X 0∗ 0∗ + µ0∗ (v, w; ωi βi , 0) + µ0∗ (v, p; ωi βi , 0) + α0 νX (v, w; ωi βi , 0) + α0 νX (v, p; ωi βi , 0) X X i∗ i∗ + µi (s, v; 0, βi (νX (v, w; 1, 0) + νX (v, p; 1, 0))) X 0∗ 0∗ + µ0 (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)) X i∗ i∗ + µi (s, v; , 0, νX (v, w; 1, 0) + νX (v, p; 1, 0)) u 0∗ 0∗ + µ0 (0, v; 0, νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)) u i i∗ i∗ + µi (0, s; 1, νX (s, v; 0, βi (νX (v, w; 1, 0) + νX (v, p; 1, 0)))) X

36

i i i∗ i∗ + νX (0, s; 1, νX (s, v; 0, βi (νX (v, w; 1, 0) + νX (v, p; 1, 0))))Xi (0)

+ µj (0, v; 1, 0) + µj (0, v; 1, 0)Xj (0) X X
0∗ 0∗ 0 + µ0 (0, s; ωi + ωj , νX (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0))) X 0 0 0∗ 0∗ + νX (0, s; ωi + ωj , νX (s, v; ωj , νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)))X0 (0) i i∗ i∗ + µi (0, s; 0, νu (s, v; 0, νX (v, w; 1, 0) + νX (v, p; 1, 0))) u i i i∗ i∗ + νu (0, s; 0, νu (s, v; 0, νX (v, w; 1, 0) + νX (v, p; 1, 0)))ui (0) 0∗ 0∗ g + µ0 (0, s; 0, νu (s, v; 0, νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0))) u 0 g 0∗ 0∗ + νu (0, s; 0, νu (s, v; 0, νX (v, w; ωi βi , 0) + νX (v, p; ωi βi , 0)))u0 (0) .

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39

Fixed Income Portfolio Management

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