Optimal Investment and Consumption with Default Risk: HARA Utility

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  • Asia-Pacific Finan Markets (2013) 20:261281DOI 10.1007/s10690-013-9167-2

    Optimal Investment and Consumption with DefaultRisk: HARA Utility

    Lijun Bo Xindan Li Yongjin Wang Xuewei Yang

    Published online: 10 March 2013 Springer Science+Business Media New York 2013

    Abstract In this paper, we consider a portfolio optimization problem in a defaultablemarket. The representative investor dynamically allocates his or her wealth among thefollowing securities: a perpetual defaultable bond, a money market account and adefault-free risky asset. The optimal investment and consumption policies that max-imize the infinite horizon expected discounted HARA utility of the consumption areexplicitly derived. Moreover, numerical illustrations are also presented.

    Keywords Optimal control Portfolio optimization Perpetual bond Defaultable market HJB equation

    1 Introduction

    The portfolio optimization problem without default risk is due to the seminal works byMerton (1969, 1971, 1992), in which the author proposed the policy that maximizes thetotal expected discounted utility of the consumption in a market investment problem.So far, the default-free portfolio optimization problems have been extensively studiedin the literature (see, e.g., Bielecki et al. 2005; Callegaro et al. 2010; Fleming and Pang2004; Pang 2006; Pham 2002 and references therein). Among others, a general caseof maximization problem with consumption in a discontinuous default-free market is

    L. BoDepartment of Mathematics, Xidian University, Xian 710071, China

    X. Li X. Yang (B)School of Management and Engineering, Nanjing University, Nanjing 210093, Chinae-mail: xwyangnk@yahoo.com.cn

    Y. WangSchool of Business, Nankai University, Tianjin 300071, China

    123

  • 262 L. Bo et al.

    discussed in Callegaro et al. (2010). Therein the authors used the martingale approachto find an explicit solution for the corresponding HJB equation.

    The study of credit risk has become increasingly attractive in both industry and aca-demic (see, e.g., Moodys Investor Service 20042005, Blanger et al. 2004; Bieleckiand Rutkowski 2002 and the references therein). In particular, portfolio optimiza-tion problems with defaultable securities have been investigated by many scholars. InBielecki and Jang (2006), Jang (2005), Bielecki and Jang discussed an optimal alloca-tion problem associated with a defaultable risky asset. The goal was to maximize theexpected HARA utility of terminal wealth. Lakner and Liang (2008) analyzed the opti-mal investment strategy in a defaultable (corporate) bond and a money market accountin a continuous time model. In Shouda (2006), by considering an optimal investmentproblem for bond holders, the author decomposed credit risk into default-timing risk,recovery risk, and spread risk. Given the tractability of recovery of market value, Houand Jin (2002) considered an optimal allocation problem including a defaultable debt.In Bo et al. (2010), the authors explored an optimal portfolio problem in a default-able market under Log utility. An optimal investment problem under multiple defaultrisk using a BSDE-decomposition approach is presented in Jiao et al. (2012). In Jiaoand Pham (2011), combined duality and dynamic programming principle to optimizeCRRA utility in a defaultable market including a riskless bond and a stock subject tocounterparty risk. These literature share one common characteristic, i.e., the intensity-based reduced-form was adopted to model default risk. However, in Korn and Kraft(2003), the authors used a firm value approach (we also call it structural approach) toexplore optimal portfolios with defaultable securities.

    In this paper, we consider a portfolio optimization problem with a defaultableperpetual bond. Herein we assume that an investor dynamically chooses a consump-tion ratio and allocates the wealth into the following securities: a perpetual default-able bond, a money market account, and a default-free risky asset. The goal is tomaximize the infinite horizon expected discounted HARA utility of consumption.A dynamic programming principle is used to derive the HJB equation. Explicit for-mulas for optimal control strategies and the corresponding value function would bederived

    The outline of this paper is as follows: In the coming section, we describe our modeland derive the price dynamics for defaultable perpetual bond. In Sect. 3, we formulatethe portfolio optimization problem with default risk, and present the optimal controlstrategy and the corresponding value function. Section 4 is devoted to providing theverification theorems. Finally, in Sect. 5, we carry out a sensitivity analysis for theoptimal control strategies and the corresponding value function in non-Log utilitycase. All proofs are arranged in Appendix.

    2 Model Description

    In this section, we present a model with specifications of a reduced-form frameworkfor an intensity-based defaultable market and of the dynamics of financial securi-ties: a defaultable perpetual bond, a money market account and a default-free riskyasset.

    123

  • Optimal Investment and Consumption 263

    2.1 Reduced-Form Framework for Default

    This subsection describes a reduced-form framework for an intensity-based defaultablemarket.

    Let (,F ,P) be a complete real-world probability space and be the (random)first jump time of a Poisson process with constant intensity > 0. For t 0, definea default indicator process Z = (Zt ; t 0) by

    Zt = 1{t}. (1)

    Suppose that W = (Wt ; t 0) is a standard Brownian motion (independent of theaforementioned Poisson process) on the above probability space and F = (Ft ; t 0)is the augmented natural filtration of the Brownian motion. Let Dt =(Zu; 0u t)and Gt = Ft Dt where t 0. Then G = (Gt ; t 0) is the smallest filtrationsuch that the random time is a stopping time. Moreover, the conditional survivalprobability is given by

    St :=P( > t |Ft ) = et , (2)

    and the following process related to default

    Mt :=Zt t

    0

    (1 Zs)ds, t 0 (3)

    is a (P,G)-martingale.

    2.2 Price Dynamics of Financial Securities

    In this subsection, we are concerned with the price dynamics of several financialsecurities mentioned above under the real-world probability measure P.

    As in Bielecki and Jang (2006), Jang (2005), denote by 1/ 1 the default riskpremium and let (0, 1) be the constant loss rate when a default occurs. We makethe assumption that under the chosen risk-neutral measure Q (which is equivalentto the real-world probability measure P) the arrival intensity of default is given by = /. Now we use Y = (Yt ; t 0) to represent the cum-dividend price of adefaultable perpetual bond that pays constant coupon C > 0 per unit time under theconvention of recovery of market value.1

    Recall the default indicator process Z defined by 1 and the martingale M definedby 3. We have the following proposition.

    1 In this paper, we deal with the cum-dividend price process for defaultable bond: the recovery of the bondreceived at default time will be re-invested in the saving account (see 22 below).

    123

  • 264 L. Bo et al.

    Proposition 1 The P-dynamics of the cum-dividend price process Y = (Yt ; t 0)is given by

    dYt = rYt dt + Yt Zt dt C Zt dt YtdMt , (4)

    where the constant r 0 is the spot risk-free interest rate, :=1 1 0 andZt :=1 Zt = 1>t for all positive t .

    Recalling that the spot interest rate is r 0, the corresponding money marketaccount = (t ; t 0) is given by

    t = ert , t 0. (5)

    Finally the P-price dynamics of a default-free risky asset is assumed to obey a geo-metric Brownian motion:

    dt = bt dt + at dWt , 0 > 0, (6)

    where b R is the return rate and a > 0 denotes the volatility.Now let Xt denote the total wealth at time t , and t and t are the proportions at

    time t in the wealth of Y = (Yt ; t 0) and = (t ; t 0) respectively. Accordingly1 t t is the time-t proportion in the wealth of = (t ; t 0). We assume thatthe consumption ratio at time t is ct (ct Xt is the consumption rate at time t). Usingthe self-financing strategy, the P-dynamics of the wealth process is given by

    dXt = [1 t t ]Xtt

    dt + t XtYt dYt +t Xtt

    dt + Ct Xt ZtYt dt ct Xt dt

    = Xt[r ct + (b r)t + t Zt

    ]dt + aXtt dWt t XtdMt , (7)

    X0 = x > 0,

    where x > 0 denotes the initial wealth held by the investor. Moreover, by Its formulawith jump (see Protter 1990, Theorem 37), the total wealth Xt at time t admits thefollowing unique explicit form:

    Xt = x exp

    t

    0

    r cs + (b r)s +

    s Zsds

    exp

    t

    0

    asdWs 12t

    0

    a22s ds

    st[1 sZs] . (8)

    Since Zt {0, 1} for all t 0, we conclude that the solvency constraint Xt > 0holds as long as t < 1/ for all t 0.

    123

  • Optimal Investment and Consumption 265

    3 Optimal Investment and Consumption Under HARA Utility

    The purpose of this section is to find the optimal allocation pair (t , t ; t 0) andthe optimal consumption ratio (ct ; t 0) to maximize the infinite horizon expecteddiscounted HARA utility of consumption. We will provide the analytic expressions foroptimal control policy (t , t , ct ; t 0) and the corresponding value function underHARA utility: Log utility and non-Log utility.

    3.1 Pre-Default and Post-Default Value Functions and HJB Equations

    In this subsection, we introduce the definitions of pre-default and post-default valuefunctions and the corresponding HJB equations.

    Next, we restrict the allocation pair (t , t ; t 0) and the consumption ratio(ct ; t 0) to be in an admissible control space A(G).Definition 2 A r.c.l.l. G-adapted stochastic control policy (t , t , ct ; t 0) is in theadmissible control space A(G), if there is a finite constant K such that:

    t [0, 1/), |t | K , ct > 0, t 0.Remark 3 The condition [0, 1/) in Definition 2 refers to avoiding bankruptcy(see also Jang (2005) Remark 5.2.4). We call each element belonging to A(G) abankruptcy avoiding portfolio. On the other hand, if there exists a control policy(t , t , ct ; t 0) A(G), then the wealth process X = (Xt ; t 0) admits thesolvency constraint condition, i.e., Xt > 0 for all positive t .

    Toward this end, we introduce value functions in term of HARA utility. Let U (x)be a HARA type utility function defined on R+. For an admissible control policy(t , t , ct ; t 0) A(G) and an initial pair (x, z) R+ {0, 1}, define thefollowing objective functional by

    J(x, z; ., ., c.) = E(x,z)

    0

    etU (ct Xt )dt

    := E

    0

    etU (ct Xt )dtX0 = x, Z0 = z

    ,

    where > 0 is the discount factor. The corresponding value function under consid-eration is then

    V(x, z) = max(.,.,c.)A(G)

    J(x, z; ., ., c.). (9)

    Hence the pre-default and post-default value functions are respectively given byV

    0(x) = V(x, 0), (pre-default value function);and

    123

  • 266 L. Bo et al.

    V1(x) = V(x, 1), (post-default value function).

    Using the dynamic programming principle, we have the following pre-default andpost-default HJB equations respectively:

    V0 = r xV0x + max[0, 1

    )

    [

    xV0x + V1(x x) V0(x)

    ]

    + maxR

    [(b r)xV0x +

    12

    a22x2V0xx

    ](10)

    + maxcR+

    [U (cx) cxV0x

    ],

    and

    V1 = r xV1x + maxR

    [(b r)xV1x +

    12

    a22x2V1xx

    ]

    + maxcR+

    [U (cx) cxV1x

    ]. (11)

    Remark 4 Observe the forms of HJB equations 10 and 11. We find that the pre-defaultHJB equation (10) associated to the pre-default value function V0 also depends on thepost-default value function V1. This is different from the HJB equation correspondingto the value function without default risk (see Merton 1969, 1971, 1992).

    3.2 Log Utility Case

    In this subsection, we concentrate on the following Log HARA utility. In other words,the utility function U is given by

    U (x) = log x, where x > 0.

    Under the Log-utility case, it is not hard to verify that the value function V admitsthe property: for all (x, z) R+ {0, 1},

    V(x, z) = 1

    log x + V(1, z),

    where > 0 is the discount rate. Appealing to this property, we can first solve the post-default HJB equation 11 and then solve the pre-default HJB equation 10 explicitly.The corresponding analytic value functions are given by the following lemma.

    Lemma 5 Recall that a > 0 is the volatility rate, b R is the return rate of thedefault-free risky asset, r 0 denotes the risk-free spot interest rate, > 0 is the

    123

  • Optimal Investment and Consumption 267

    arrival intensity of intensity and 1/ 1 denotes the default risk premium. Then thepost-default value function is

    V1(x) = 1

    log x + W1, x > 0,

    where the constant W1 is given by

    W1 = 1

    2

    [(b r)2

    2a2+ r

    ]+ log 1

    . (12)

    Moreover, the pre-default value function has the form:

    V0(x) = 1

    log x + W0, x > 0,

    where the constant W0 is given by

    W0 = W1 +

    ( + )[

    1

    + log 1]

    . (13)

    Remark 6 By virtue of the relationship between W0 and W1 given by 13, we find thatif there is no default, i.e., = 0 or the jump risk premium 1/ = 1, then it holds thatW

    0 = W1 and hence V0(x) = V1(x). This implies that the value function

    V(x, z) = (1 z)V0(x) + zV1(x)= V1(x),

    which reduces to the case without default risk. However in a general case where thedefault risk exists, we have the pre-default value function V0(x) is always greater thanor equal to the post-default value function V1(x) for each fixed initial wealth x > 0.Moreover, the function

    V(x, z):= 1

    log x +[zW1 + (1 z)W0

    ], where x > 0, and z = 0, 1, (14)

    is the nontrivial solution to the HJB equation associated with the value functionV(x, z).

    Finally we give a G-adapted stochastic control policy (Lt , Lt , cLt ; t 0) expressedas

    Lt ={

    1

    , 0 t ,0, t > .

    (15)

    123

  • 268 L. Bo et al.

    where (0, 1) denotes the loss rate given default. The other two controls are givenby

    Lt =b r

    a2, and cLt = , t 0. (16)

    We shall show that the above stochastic control policy is admissible and optimal, andthat the value function is nothing but V(x, z) given by 14.

    3.3 Non-Log Utility Case

    In contrast to the Log utility case, this subsection concerns the following non-LogHARA utility

    U (x) = 1

    x ,

    where x > 0 and (0, 1).The value function V corresponding to the above utility admits the following homo-

    theticity property (see Shreve and Soner 1994, Proposition 3.3 or Fleming and Pang2004, Lemma 2.3):

    V(x, z) = x V(1, z), where x > 0, z = 0, 1.

    Accordingly, we have the following forms concerning the post-default value functionand pre-default value function respectively

    Lemma 7 The post-default value function is given by

    V1(x) = 1

    x W1 , x > 0,

    where the constant W1 is

    W1 =

    [ r1

    (b r)22a2(1 )2

    ]1. (17)

    Moreover, the pre-default value function admits the form

    V0(x) = 1

    x W0 , x > 0,

    123

  • Optimal Investment and Consumption 269

    where the constant W0 is given by

    W0 =

    [W

    1

    ] 11 + ( )

    (1 )

    1 + 1[W1

    ] 11

    1

    . (18)

    By virtue of Lemma 7,

    V(x, z) = 1

    x[zW1 + (1 z)W0

    ], x > 0, z = 0, 1 (19)

    is the nontrivial solution to the HJB equation associated with the value function V inthe case of non-Log utility.

    Finally we give a G-adapted stochastic control policy (t , t , c

    t ; t 0) expressed

    as

    t =

    1

    [1

    [W0W1

    ] 11

    ], 0 t

    0, t > .(20)

    where (0, 1) denotes the loss rate given default. The other two controls are givenby

    t =

    b ra2[1 ] , and c

    t =

    [W

    0

    ] 11

    , 0 t < [W

    1

    ] 11

    , t .(21)

    We shall show that the above stochastic control policy is admissible a...

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