CAREFIN Centre for Applied Research in Finance

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11/2010 Anna Battauz Alessandro Sbuelz Reaching Nirvana With A Defaultable Asset?

Reaching Nirvana With A Defaultable Asset?

by Anna Battauz Alessandro Sbuelz

n. 11/10 Milan, June 2010

Copyright Carefin, Università Bocconi

Further research materials can be downloaded at

www.carefin.unibocconi.eu

CAREFIN WORKING PAPER I

INDEX Abstract I 1. THE DEFAULTABLE ASSET 8 2. DYNAMIC ASSET ALLOCATION AND NIRVANA 12 3. CONCLUSIONS 20 References 17 A. APPENDIX 25

CAREFIN WORKING PAPER II

Abstract*

We contribute a new insight into the pre-crisis massive levered exposures to default risk by formulating a parsimonious, closed-form analysis of conspicuous risk taking in default-prone assets. Under no arbitrage, default risk is compensated by an ‘yield pickup’ that can strongly attract aggressive investors via an investmenthorizon effect in their optimal non-myopic portfolios. We show that default risk decoupled from event risk does not discourage the formation of markedly geared portfolios. Our results add a new portfolio-based perspective to a mainstream model of default risk: even if arbitrage-free, the model may not find unconstrained support in general equilibrium.

JEL: G11, G12, C61. Keywords: dynamic asset allocation, defaultable asset, Sharpe-ratio uncertainty, levered

non-myopic speculation, the Constant Elasticity of Variance (CEV) model.

*Anna Battauz Bocconi University, Department of Finance (anna.battauz@unibocconi.it), Milan, Italy. Alessandro Sbuelz Catholic University of Milan, Department of Quantitative Finance and Econometrics (alessandro.sbuelz@unicatt.it)Milan, Italy. Corresponding author and CAREFIN Research Fellow. The authors thank Viral Acharya, Giovanni Barone-Adesi, Claudio Albanese, Flavia Barsotti, Emilio Barucci, Jonathan Berk, Patrick Bolton, Francesco Corielli, Jaksa Cvitanic,Mikhail Chernov, Bernard Dumas, Francesco Franzoni, Patrick Gagliardini, Francesco Garzarelli, Jens Jackwerth, Semyon Malamud, Christine Parlour, Francesco Sangiorgi, Christian Schlag, Peter Schotman, Claudio Tebaldi, Fabio Trojani, TizianoVargiolu, Luis Viceira, S. Vish Viswanathan, and the participants at the evening sessions of the 2009 European Summer Symposium in Financial Markets (Gerzensee), the 2009-10 Finance Seminar Series of the University of Italian Switzerland (Lugano), and the 11th Workshop on Quantitative Finance (Palermo, 2010). Any remaining error is our sole responsability.

The massive risk taking observed before the 2007-2009 �nancial crisis did

not shun defaultable assets. A chief example is offered by the banking industry's

course of action during the bullish years up to early 2007 as it can be gauged

from the accounts of the run-up to the crisis-among the many brilliant studies see

Brunnermeier (2009), Diamond and Rajan (2009), Duf�e (2008), and Hellwig

(2009). In essence, banks became with growing gusto originators as well as ac-

tive buyers of defaultable mortgage-backed securities. A large quantity of assets

exposed to mortgage default risk did �nd its way into commercial and investment

banks' balance sheets, the corresponding escalation of which was mainly backed

by short-term wholesale funding. Diamond and Rajan (2009), p. 607, state:

... [I]t is surprising that [banks] held on to so many of the mortgage-

backed securities (MBS) in their own portfolios. ... The real answer

seems to be that bankers thought these securities were worthwhile

investments, despite their risk.

While banks' course of action may have been inspired by a variety of motives1,

the statement is evocative of how sophisticated investors with possibly long invest-

ment horizons were not insensitive to the glamor then surrounding defaultable as-

sets. Two elements behind such a glamor were notable. (1) Defaultable assets

seemed to bring relief to professional investors' tremendous hunger for yield at a

time of surging markets2. (2) Event risk (the chance of abrupt changes in asset1For instance, Diamond and Rajan (2009) re�ect on warped incentives for bank bosses and

�awed internal compensation and control for subordinates.2Among others, King (2006) emphasized how an `unusual' �ow of funds from countries with a

CAREFIN WORKING PAPER 3

values) seemed to be incompatible with the astonishingly benign credit prospects3.

Our key contribution is to show how, under a parsimonious no-arbitrage model

of optimal dynamic portfolio choice, the ability of a default-prone asset to pro-

vide an `yield pickup' (an excess expected return that endures at times of sub-

dued volatility) and the absence of event risk can lead an aggressive investor to

take vast long geared positions in the asset through an investment-horizon effect.

The investment-horizon effect is associated with the so-called nirvana solution for

the optimal portfolio problem, that is a solution-and an investor's value function-

growing to extreme levels at long-enough horizons (see Kim and Omberg (1996)).

The intuition is as follows. In our model, no-arbitrage comes from the balance be-

tween `yield-pickup' and default risk. Such a balance is rooted in the inverse

relationship between asset returns and their subsequent volatility: upward asset-

value paths that enjoy tremendous Sharpe ratios (paths characterized by shrinking

volatility) are counteracted by downward asset-value paths that end up with de-

fault (paths characterized by swelling volatility). The conservative investor non-

myopically hedges against Sharpe-ratio uncertainty by selling the asset (the in-

vestor seeks more wealth in the states coupled with dismal investment opportuni-

ties). In contrast, the aggressive investor non-myopically speculates on Sharpe-

ratio uncertainty by going long the asset (the investor seeks more wealth in the

high return on physical capital to countries with a low one was channelled into Western traditionalhigh-grade assets and pushed yield-seeking professional investors into an eager search for returnsamong alternative assets.

3Diamond and Rajan (2009) call attention to how investors' belief that any troubles were faraway was fuelled by global savings pouring into the markets of mature economies and by theFederal Reserve being actually inclined to ride to the rescue whenever �nancial crises threatened(the so-called �Greenspan Put�). The last golden years of the �Great Moderation�-a period ofsustained decline in the volatility of real aggregates, see Bernanke (2004)-did help.

CAREFIN WORKING PAPER 4

states coupled with a high productivity of wealth). Since absence of event risk

makes deleveraging painless, the aggressive investor places a non-myopic (possi-

bly geared) bet on the upward asset-value paths to unfold before her investment

maturity. A clear investment-horizon effect emerges. The longer the maturity

is, the more muscular and geared non-myopic speculation becomes: the bet has

more time to make good. An `yield pickup' intensity effect also emerges. The

stronger the inverse link between asset returns and their subsequent volatility is,

the stronger non-myopic speculation is: coeteris paribus, the same upward asset-

value path offers juicier Sharpe ratios.

We assume a Constant Elasticity of Variance (CEV) diffusion for the value

process of the defaultable asset, so that our portfolio-choice model involves only

�ve primitives. Three describe the investables and are the riskfree rate, the excess

expected return on the defaultable asset, and the elasticity of asset returns' volatil-

ity with respect to the current asset value. Importantly, the elasticity is assumed to

be negative to engender the `yield pickup' as well as the possibility of default. The

other two primitives describe the investor and are the level of constant relative risk

aversion and the investment horizon. Parsimony brings about tractability by em-

powering a closed-form analysis of how the �rst three parameters shape the asset's

`yield-pickup' and default probability and how all �ve parameters determine opti-

mal dynamic portfolio choice. Optimal portfolios turn out to have the well-known

closed form studied by Kim and Omberg (1996). Following the seminal work

of Merton (1971), Kim and Omberg (1996) have provided the standard analytic

framework for many dynamic portfolio problems with a stochastic opportunity set

CAREFIN WORKING PAPER 5

(see Liu (2007) for a thorough discussion and Jurek and Yang (2007) for an appli-

cation to time-varying arbitrage opportunities). While being fully agnostic about

the source of default risk (see Korn and Kraft (2002) on the peculiar issues im-

plied by dynamic investment in a corporation's capital structure), we are the �rst

to use the Kim and Omberg (1996) framework to explicitly calculate non-myopic

wealth allocation to a defaultable asset under no arbitrage.

Delbaen and Shirakawa (2002) have shown that no arbitrage in the CEVmodel

with negative elasticity comes from the balance between upward asset-value paths

with huge Sharpe ratios and downward asset-value paths to default. This result

bears a resemblance to the principle-explored by Weitzman (1998, 2009), Gollier

(2002), and Martin (2009)-that the no-arbitrage value of a long-dated asset may be

dictated by extreme outcomes. In our model such extreme outcomes take the form

of exploding Sharpe ratios and cannot be related to asset-value bubbles. (Hardly

exploitable) asset-value bubbles arise in the context of the CEV model only if the

elasticity is assumed to be positive, which rules out the possibility of default (see

Heston, Loewenstein, and Willard (2007)). At any rate, we show that the fair

pricing of those extreme outcomes does not deter rational aggressive investors

with a suf�ciently long investment horizon from massively gambling on them.

Our results contribute a new portfolio-based angle on a well-known model: albeit

arbitrage-free and mainstream in the default-risk literature4, the CEV model with

negative elasticity may not have unrestricted general-equilibrium foundations.

The massive gambling associated with nirvana-type portfolios, an articulate4Carr and Linetsky (2006) and Campi, Polbennikov, and Sbuelz (2009) thoroughly review the

CEV-based literature on credit risk and extend it to consider default-type event risk.

CAREFIN WORKING PAPER 6

Sharpe-ratio-based discussion of the non-myopic demands, and the role played

by investors' preferences in resolving the non-myopic tradeoff between `yield-

pickup' and default risk are subtle but momentous issues that are utterly omitted

in Gao's (2009) study of wealth allocation under the CEVmodel-Gao (2009) over-

looks the very presence of default risk and the wealth-allocation analysis of Kim

and Omberg (1996), which may explain the omissions. Those issues are at the

core of our work. Their handling contributes results that bring fresh evidence on

the importance of event risk for investment strategies. Liu, Longstaff, and Pan

(2003) show how the presence of event risk5 makes rational investors behave as

if they faced borrowing constraints albeit none are imposed. They consider non-

defaultable investables. We show that, even with a defaultable zero-recovery asset,

de�ciency of event risk is conducive to extreme levered positions.

The paper is organized as follows. Section 1 details the features of the CEV-

type defaultable asset. Section 2 shows that huge long geared positions can be the

rational outcome of a dynamic portfolio problem with the CEV-type defaultable

asset. Section 3 draws the conclusions and an Appendix collects the proofs of the

propositions in Section 3 and other technical material.5The literature on dynamic portfolio choice with jump-diffusion settings typically avoids con-

sidering defaultable zero-recovery assets and by and large �nds penalizations for investors whohold levered positions. Such a literature includes the contributions of Aase (1984), Jeanblanc-Picqué and Pontier (1990), Wu (2003), Das and Uppal (2004), and is reviewed in Kurmann (2009).Cvitanic, Polimenis, and Zapatero (2008) focus on optimal portfolio allocation in the presence ofpure-jump risk with similar results. Merton (1971) studies a constant-opportunity-set portfolioproblem with a non-defaultable risky asset and a money-market account with a jump to default (asexpected, optimal allocation is never concentrated in such an account).

CAREFIN WORKING PAPER 7

1 The defaultable asset

There are essentially three properties we want the value process fV g of the risky

asset to have: (i) it must render default possible and the probability of its occur-

rence analytic; (ii) it must be arbitrage-free; (iii) it must support an `yield pickup'

(a positive excess expected return that endures at times of subdued volatility).

Property (i) is meant to empower a tractable study of the asset's default. Tractabil-

ity and parsimony also suggest the assumption of zero recovery at default. Prop-

erty (ii) is meant to rule out the possible emergence of extreme portfolios due to

the existence of free lunches. Property (iii) is meant to bestow the asset with a

glamor similar to the one defaultable assets seemed to possess before the crisis.

The Constant-Elasticity-of-Variance (CEV) value process with negative elas-

ticity parameter � has been introduced in �nancial economics by Cox (1975)6.

The CEV value process is frugally parametrized and meets the three properties of

interest. Its dynamics is

dVtVt

D .r C X t� t/ dt C � tdZ t ; � t � Vt�; � 1 < � < 0; V0 D v � 0;

(1)

X t ��

� t(Sharpe ratio), (2)

where the excess expected return on the asset is the constant � > 0 and the riskfree

rate is r > 0. fZg is a Wiener process under the objective probability measure P.6The geometric Brownian motion value process of Black and Scholes (1973) is obtained with

the special case � D 0 and the square-root process of Cox and Ross (1976) is obtained with� D � 12 (see Davidov and Linetsky (2001) for an extensive review of the CEV model).

CAREFIN WORKING PAPER 8

From the boundary classi�cation, the point 0 is an attainable state for the process

fV g only if � < 0. As soon as the process fV g falls down to zero, we con�ne it

there so that the level 0 becomes the absorbing state for the asset value process

(consistently with zero recovery at default). Default becomes possible only if asset

returns' local volatility in�ates as the asset value deteriorates (� < 0), which is

quite realistic. Importantly, what makes default possible also engenders what we

refer to as the asset's `yield pickup' (the excess expected return � is positive and

constant no matter how small, along upward asset-value paths that rise above 1,

the local volatility V � is). Thus, the CEV value process with negative elasticity

naturally possesses property (iii). j�j measures the `yield pickup' intensity: along

upward asset-value paths that mount above 1, the higher j�j, the larger the Sharpe

ratio.

The CEV value process with negative elasticity substantiates property (i) with

a well-known analytic expression for the probability of default.

Proposition 1 If the asset value process follows a CEV value process with � 2

.�1; 0/, then the objective probability of the asset defaulting within the date T >

0 admits the following closed form:

P [Vh D 0; 0 � h � T j V0 ] D0�� .rC�/v�2��[1�e2.rC�/�T ] ;�

12�

�0�� 12�

� ;

CAREFIN WORKING PAPER 9

where

0 .p; l/ D

Z C1

pul�1e�udu; p � 0 (Incomplete Gamma function),

0 .l/ D

Z C1

0ul�1e�udu (Gamma function):

Proof. Proposition 1 in Campi and Sbuelz (2006) holds.

Figure 1 plots the probability of default against the current value v and the date

T . As v approaches 0, the probability converges to 1 for any date T . Given � < 0

and the positive drift r C � , a higher j�j causes lower probabilities of default for

distant dates as it tends to curb volatility along the surviving trajectories, which

include the asset-value paths that rise above 1. Of course, the lower the drift rC� ,

the higher the probability of default for any date T .

The CEV value process with negative elasticity does enjoy property (ii), as the

following proposition maintains.

Proposition 2 The CEV value process with � 2 .�1; 0/ complies with the no-

arbitrage assumption.

Proof. Theorem 2.3 in Delbaen and Shirakawa (2002) holds.

Under no arbitrage, default risk balances the presence of bullish asset-value

paths along which the Sharpe ratio X t bloats. If no free lunches are to emerge,

default risk must counteract the `yield pickup' offered by the defaultable asset.

This is borne out by the following proposition.

CAREFIN WORKING PAPER 10

1030

20

T

0.0

0.5prob.

0

1.0

1

initial value v2

34 0

30

T

2010

0.0

0.5

1.0

0

prob.

1 2

initial value v3

04

� D �0:2, r C � D 7% � D �0:5, r C � D 7%

1030

20

T

0.5

0.0

1.0

0

prob.

1

initial value v2

34 0 T

2030

10

0.0

0.5

0

1.0

prob.

1

initial value v2

4 03

� D �0:2, r C � D 1% � D �0:5, r C � D 1%

Figure 1: Probability of asset default within date T (years)

CAREFIN WORKING PAPER 11

Proposition 3 Assume � 2 .�1; 0/ and consider the positive value process of the

CEV model, that is fVt I 0 � t � T g under the conditional objective probability

measure P [ � j VT > 0 ]. For the positive value process of the CEV model there

always exist arbitrage opportunities.

Proof. Theorem 4.2 of Delbaen and Shirakawa (2002) holds.

The next section shows that the allure of the CEV asset-value paths along

which the Sharpe ratio X t in�ates can become supreme for aggressive investors.

Easy deleveraging can make them `rationally forget' about the balancing force

represented by default risk.

2 Dynamic asset allocation and nirvana

The CEV asset value process enables a closed-form dynamic portfolio analysis

along the lines set by Kim and Omberg (1996). The investor allocates her wealth

Wt to two assets, the risk-free asset and the defaultable asset. There is no con-

sumption or income during the investment horizon. The investor has Constant-

Relative-Risk-Aversion (CRRA) utility from terminal wealth:

U .z/ D

8<: z1��= .1� �/ for z > 0 ,

�1 for z � 0 ,

where the level of relative risk aversion equals the parameter � > 0.

The defaultable asset value Vt has the CEV dynamics speci�ed in (1) and the

point 0 is its cemetery state. By construction (see De�nition 2), the Sharpe ratio

CAREFIN WORKING PAPER 12

X t is always non-negative, admits a cemetery state at 0, and has dynamics totally

deprived of mean-reversion7,

dX t D ���X t .r C �/�

12.� C 1/

�2

X t

�dt � ��dZ t , X0 D x � 0 . (3)

The correlation between its innovations and defaultable-asset value innovations

is 1. The current Sharpe ratio x supplies all currently available information on

current and future investment opportunities. Indeed, Nielsen and Vassalou (2006)

show that, in typical continuous-time portfolio problems, the only time-variation

that matters for portfolio choice is the time-variation in the slope (the Sharpe ratio)

and the intercept (the riskfree rate) of the instantaneous capital market line.

Let W0 D w and � be the investor's current wealth and investment horizon

and y� .w; x; � / be the optimal monetary investment in the defaultable asset. The

optimal portfolio problem is:

maxfyt g�tD0

E [ U .W� / j W0; X0 ] s.t. (3) and dWt D WtrdtCyt�dVtVt

� rdt�, w > 0 .

(4)

The following proposition and corollary qualify optimal dynamic portfolios.

7The process that �ows according to (3) is known as the Rayleigh process (see for exampleGiorno, Ricciardi, Nobile, and Sacerdote (1986)). Whereas C1 is unattainable but attracting(�� .r C �/ > 0), the level 0 is always an attainable boundary for � 2 .�1; 0/ and we im-pose absorption there. Kim and Omberg (1996) assume the Sharpe-ratio process fXg to be amean-reverting Ornstein-Uhlenbeck process. It can be shown that, in the Kim and Omberg (1996)setting, �imsy mean-reversion for the process fXg stokes nirvana-type optimal portfolios (the tech-nical details are at the end of the Appendix). We argue that, given a frail mean-reversion, the un-folding of paths with swelling Sharpe ratios becomes a plausible event to punt on for non-myopicaggressive investors (see the discussion of our Proposition 7).

CAREFIN WORKING PAPER 13

Proposition 4 The optimal fraction of wealth allocated to the defaultable asset

is:

y� .w; x; � /w

D1�

x2

�C

1�.��/C .� / x2 ;

C .� / D2a�1� exp

��pq�

��2pq �

�b Cpq

� �1� exp

��pq�

�� ;

a D1�� 1;

b D 2��1�

1�

��� � � .r C �/

�;

c D1��2�2;

q D b2 � 4ac D 4�2�r2 C

1�

�.r C �/2 � r2

��:

Proof. See the Appendix.

Corollary 5 The optimal fraction of wealth allocated to the defaultable asset van-

ishes as the Sharpe ratio dies out:

limx!0C

y� .w; x; � /w

D 0 :

Proposition 4 states that the optimal fraction of wealth invested in the default-

CAREFIN WORKING PAPER 14

able asset contains two components. The �rst component,

1�

x2

�,

is the myopic demand for the defaultable asset. It is the allocation that an investor

optimally holds if the investment horizon � shrinks to zero-the investor does not

care about future investment opportunities. The second component,

�1��C .� / x2 ,

is the intertemporal non-myopic demand (see for instance Merton (1971)). The

reason the investor forms non-myopic demands is to deal optimally with changes

in future investment opportunities.

Both components of the optimal investment in the defaultable asset are con-

tingent on the current squared Sharpe ratio, which implies myopic as well as non-

myopic market timing. The zero level is a cemetery state for the Sharpe ratio and

for the defaultable-asset value. Corollary 5 emphasizes the investment implication

of that. A rational risk-averse investor does move out of the defaultable asset as

its value plunges to zero.

The absence of intermediate consumption hampers the use of arguments based

on the substitution/income effects on consumption to gauge the sign of the non-

myopic demand for the defaultable asset. However, since an unexpected drop in

the asset value implies a deterioration in the investment opportunities offered by

the asset (the Sharpe ratio drops), one conjectures that a non-log-utility and suf�-

CAREFIN WORKING PAPER 15

ciently risk-averse investor will hedge against such an adverse effect by shorting

the asset to pro�t from the unexpected drop in the asset value. The �rst part

of the following proposition con�rms the conjecture and highlights a standard

investment-horizon effect-the longer � is, the more energetic the hedging act be-

comes8.

Proposition 6 Given a conservative investor (� > 1), the non-myopic component

of y� .w; x; � / in Proposition 4 is negative and monotonically decreasing in the

investment horizon:

C .� / < 0 and C 0 .� / < 0 for any � > 0 .

Furthermore, y� .w; x; � / is a non-nirvana solution to the optimal portfolio

problem (4) in the sense of Kim and Omberg (1996):

��y� .w; x; � /�� <1 for any � � 0 .

Proof. See the Appendix.

The second part of Proposition 6 states that conservative non-myopic investors

never take extreme positions in the defaultable asset, no matter how long the in-

vestment horizon � is. In contrast, aggressive non-myopic investors do take ex-

treme positions if their investment horizon is long enough, as the following propo-8Similar investment-horizon effects have been found in the literature on dynamic portfolio

choice with a risky non-defaultable asset characterized by a mean-reverting drift and a constantvolatility (see for example Campbell and Viceira (1999), Koijen, Rodriguez, and Sbuelz (2009),and Wachter (2002)).

CAREFIN WORKING PAPER 16

sition shows.

Proposition 7 Given an aggressive investor (� < 1), y� .w; x; � / in Proposition

4 is a nirvana solution to the optimal portfolio problem (4) in the sense of Kim

and Omberg (1996), i.e., there is a critical horizon9 � c such that

lim�!��c

y� .w; x; � / D lim�!��c

C .� / D 1 with � c D1pq

ln�b Cpqb �pq

�> 0 .

Furthermore, the optimal non-myopic demand for the defaultable asset is positive

and monotonically increasing in the investment horizon:

C .� / > 0 and C 0 .� / > 0 for any � 2 .0; � c/ .

Proof. See the Appendix.

Proposition 7 shows that the aggressive investor non-myopically speculates on

Sharpe-ratio uncertainty by buying the defaultable asset, in contrast with the con-

servative investor who non-myopically hedges against Sharpe-ratio uncertainty by

shorting the asset. Essentially, the aggressive investor uses the defaultable asset to

seek pro�ts in the states that come with a high productivity of wealth, that is, with

high Sharpe ratios. In the assumed absence of abrupt changes in the asset value, it

is optimal for the aggressive investor to make a non-myopic gamble on the upward

asset-value paths to unfold before her investment maturity � (upward asset-value

paths go along with swelling Sharpe ratios). If � stretches out toward the critical9As Kim and Omberg (1996) point out, the nirvana solutions are not de�ned for longer horizons

than the critical horizon � c. Investors with a longer horizon can pursue any policy up to the criticalhorizon and attain nirvana by initiating the optimal policy at that time.

CAREFIN WORKING PAPER 17

50 100

­0.6

­0.4

­0.2

0.0

0.2

tau (years)

y*/w

0 10 20 3001234

tau (years)

y*/w

(� D 2 black; � D 5 red) (� D 0:5 black; � D 0:75 red)

Figure 2: Allocation to the defaultable asset versus the investment horizon

horizon � c, non-myopic gambling becomes increasingly hefty and levered: the

bet has more time to succeed.

Given r D 2%, � D 5%, � D �0:7, and x D 0:5, Figure 2 visualizes the

results in Propositions 6 and 7. In the left-hand panel of Figure 2, conservative

investors do take short positions in the defaultable asset for non-myopic hedging

purposes. In the right-hand panel of Figure 2, the critical horizon � c is about

17 years for � D 0:5 and about 29 years for � D 0:75. An aggressive investor

(� D 0:5) with an horizon just over 10 years borrows to invest in the defaultable

asset twice as much as her current wealth. The same will do a more cautious but

still aggressive investor (� D 0:75) with a 23-year horizon.

The impact of market conditions (changes in the riskfree rate r , the excess

expected return � , and the `yield pickup' intensity j�j) on optimal portfolio choice

is visualized in Figures 4, 5, and 6. Figures 4, 5, and 6 are based on the reference

levels � D 0:5, r D 2%, � D 5%, � D �0:7, and x D 0:5.

Interestingly, an increase in the riskfree rate slightly reduces the critical hori-

zon (see Figure 3). There is a drift effect and an opportunity-cost effect and they

CAREFIN WORKING PAPER 18

0 20 400

5

10

tau (years)

y*/w

0 20 400

5

10

tau (years)

y*/w

(� D 0:5) (� D 0:75)

Figure 3: The allocation impact of the riskfree rateThe allocation with r D 4% is the solid black line; The allocation with r D 0% isthe solid red line.

0 20 400

5

10

tau (years)

y*/w

0 20 400

5

10

tau (years)

y*/w

(� D 0:5) (� D 0:75)

Figure 4: The allocation impact of the excess expected returnThe allocation with � D 7% is the solid black line; The allocation with � D 3% isthe solid red line.

go in contrasting directions. The drift effect is the drop in the probability of de-

fault caused by a higher drift r C � . The opportunity-cost effect is the allocation

shift, justi�ed by a higher r , from the defaultable asset to the riskfree asset. The

drift effect results stronger and non-myopic risk taking is mildly encouraged.

An increase in the excess expected return decidedly lowers the critical horizon

(see Figure 4). There is a drift effect and a level effect and they both go in the

same direction of strengthening the incentive of placing the non-myopic bet. The

CAREFIN WORKING PAPER 19

0 20 400

5

10

tau (years)

y*/w

0 20 400

5

10

tau (years)

y*/w

(� D 0:5) (� D 0:75)

Figure 5: The allocation impact of the `yield pickup' intensityThe allocation with j�j D 0:9 is the solid black line; The allocation with j�j D 0:5is the solid red line.

drift effect is the decrease in the probability of default implied by a higher drift

r C � . The level effect is an upward shift of every Sharpe-ratio path implied by a

higher � .

Chie�y, an increase in the `yield pickup' intensity brings down the critical

horizon (see Figure 5). There is a tilt effect that bolsters non-myopic risk taking.

A higher j�j implies an upward tilt of the pivotal Sharpe ratio paths (the upward

asset-value paths that soar above 1). The upward tilt of the pivotal paths is asso-

ciated with a decrease of the probability of default for distant dates, as seen in the

analysis of Figure 1.

3 Conclusions

Defaultable assets became de�nitely fashionable among sophisticated investors

before the 2007-2009 crisis. We use a frugal no-arbitrage model of dynamic

portfolio choice to show why non-naive investors may �nd a defaultable asset

CAREFIN WORKING PAPER 20

extremely appealing.

Our economic insight is clear: under no arbitrage, default risk is compensated

by an `yield pickup' that can strongly attract non-myopic aggressive investors.

The balance between possible default at the end of downward asset-value paths

and alluring Sharpe ratios along upward asset-value paths implies no arbitrage.

The allure of large Sharpe ratios in states in which the asset performs well fosters

an optimal non-myopic long exposure to the asset among aggressive investors-

they seek more pro�ts in states associated with a high productivity of wealth.

Such exposure balloons as the investment horizon increases-the non-myopic bet

on upward asset-value paths has more time to succeed.

We employ the CEVmodel with negative elasticity to parsimoniously describe

the value process of the defaultable asset. The model is standard in the default-

risk literature and serves us well by enabling closed-form default probabilities,

absence of arbitrage, and `yield pickup'. We show that the CEV model also en-

ables optimal dynamic portfolios in the closed form studied by Kim and Omberg

(1996). Our portfolio results offer a new angle on such a popularly used model:

by encouraging extreme positions among certain market participants, it may not

have unchecked backing in general equilibrium despite being arbitrage-free.

Finally, our results contribute fresh evidence on the importance of event risk

(possible discontinuities in the asset value) for investment strategies. With non-

defaultable investables, event risk is known to prompt endogenous borrowing con-

straints. We show that, even with a defaultable zero-recovery asset, de�ciency of

event risk can lead to extreme geared portfolios.

CAREFIN WORKING PAPER 21

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A Appendix

A.1 Proof od Proposition 4

We closely follow the steps in Kim and Omberg (1996). Let J .w; x; � / be the

optimal expected utility for the portfolio problem (4). Given X 's dynamics (3),

CAREFIN WORKING PAPER 25

the Hamilton-Jacobi-Bellman conditions result in the optimal investment function

y� .w; x; � / D�Jw

�Jww

�x�

C

�Jwx�Jww

�����

�

�; (5)

the partial differential equation

�J�CJwrw�12Jww

�y��2� 2�Jx�

�x .r C �/�

12.� C 1/

�2

x

�C12Jxx�2�2 D 0 ;

(6)

the terminal condition,

J .w; x; 0/ D U .w/ ;

and the second-order condition for a maximum Jww < 0. The optimal investment

function for the logarithmic case (� D 1) is the usual myopic investment function,x� . For � 6D 1, assume trial solutions of the form

J .w; x; � / D U .w/ exp�A .� /C

12C .� / x2

�;

A .0/ D C .0/ D 0 :

Solutions of this form automatically satisfy both the terminal condition and the

second-order condition for a maximum. Substitution of the trial solutions into

equation (5) yields the investment function

y� .w; x; � / Dw

�

x�

Cw

�C .� / x

����

�

�:

CAREFIN WORKING PAPER 26

Further substitution of the trial solutions into equation (6) produces a quadratic

equation for x , but deprived of the term with the degree of 1. The two coef�cients

of such an equation must be zero, yielding the following system of �rst-order

nonlinear ordinary differential equations:

C 0 D C2c C Cb C a;

A0 D C f C g;

A .0/ D C .0/ D 0;

a D1�� 1;

b D 2��1�

1�

��� � � .r C �/

�;

c D1��2�2;

f D12��.� C 1/ �2 C ��2

�;

g D .1� �/ r:

Given the assumptions that � > 0, r > 0, and � > 0, the quantity

q D b2 � 4ac D 4�2�r2 C

1�

�.r C �/2 � r2

��

is always positive. Hence, the results in Kim and Omberg (1996), p. 158, apply

CAREFIN WORKING PAPER 27

and the solutions have this form:

C .� / D2a�1� exp

��pq�

��2pq �

�b Cpq

� �1� exp

��pq�

�� ;A .� / D f

Z �

0C .u/ du C g� :

This completes the proof.�

A.2 Proof of Proposition 6

The condition for non-nirvana solutions boils down to the impossibility of having

zeros for the denominator in C .� /:

2pq > b C

pq ()

pq > b (see Kim and Omberg (1996), p. 158).

Such a condition is equivalent to considering conservative investors only:

pq > b()

s4�2

�r2 C

1�

�.r C �/2 � r2

��> �2�

�r C

1��

�() � 2 .1;1/ :

Focusing on conservative investors only implies a < 0. Hence, we have

C .� / < 0 and C .� /0 D4aq exp

��pq�

��2pq �

�b Cpq

� �1� exp

��pq�

���2 < 0 .This completes the proof.�

CAREFIN WORKING PAPER 28

A.3 Proof of Proposition 7

The condition for nirvana solutions coincides with the possibility of having a zero

for the denominator in C .� / at some critical horizon � c > 0 :

pq < b :

Such a condition is equivalent to considering aggressive investors only:

s4�2

�r2 C

1�

�.r C �/2 � r2

��< �2�

�r C

1��

�() � 2 .0; 1/ :

The expression for the critical horizon � c comes from equating to zero the denom-

inator in C .� / :

2pq �

�b C

pq� �1� exp

��pq���D 0() � c D

1pq

ln�b Cpqb �pq

�> 0 :

Focusing on aggressive investors only implies a > 0. Hence, we have

C .� / � 0 and C .� /0 D4aq exp

��pq�

��2pq �

�b Cpq

� �1� exp

��pq�

���2 > 0 for any � 2 [0; � c/ .

This completes the proof.�

CAREFIN WORKING PAPER 29

A.4 Nirvana solutions andmean-reversion in Kim andOmberg

(1996)

We show that, in the Kim and Omberg (1996) setting, feeble mean-reversion

for the Sharpe-ratio process fXg fosters nirvana-type dynamic optimal portfolios.

Kim and Omberg (1996) consider a risky asset with a mean-reverting Sharpe ratio:

dPt=Pt D .r C X t�m/ dt C �mdZ t , dX t D ��X�X t � X

�dt C � XdBt ,

with �X ; X ; � X > 0 and dBtdZ t D �mXdt . Weak mean-reversion (large levels

of the ratio � X=�X ) does lead aggressive investors (0 < � < 1) to implement

nirvana-type optimal portfolios, the condition for which is:

4�2X

1�

�1�� 1

�"�� X

�X

�2C 2�mX

�� X

�X

�#!< 0 (see p. 159 of their paper).

CAREFIN WORKING PAPER 30

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