Non‐myopic portfolio choice with unpredictable returns: The jump‐to‐default case

Date01 March 2018
Published date01 March 2018
DOI: 10.1111/eufm.12142
Non-myopic portfolio choice with unpredictable
returns: The jump-to-default case
Anna Battauz
Alessandro Sbuelz
Department of Finance and IGIER,
Bocconi University, Milan, Italy
Department of Mathematical Sciences,
Mathematical Finance and Econometrics,
Catholic University of Milan, Largo
Gemelli 1, 20123 Milan, Italy
If a risky asset is subject to a jump-to-default event, the
investment horizon affects the optimal portfolio rule, even if the
asset returns are unpredictable. The optimal rule solves a non-
linear differential equation that, by not depending on the
investors pre-default value function, allows for its direct
computation. Importantly for financialplanners offering portf olio
advice for the long term, tiny amounts of constant jump-to-
default risk induce marked time variation in the optimal portfolios
of long-run conservative investors. Our results are robust to the
introduction of multiple non-defaultable risky assets.
dynamic asset allocation, irreversible regime change, jump-to-default
risk, return predictability, time-varying hedging portfolio
G01, G11, G12, C61
By the end of 2014, global private financial wealth reached a to tal of $164 trillion and the glo bal
value of professionally mana ged assets grew to $74 trillion.
Financial planners providi ng asset
allocation advice to long-term inv estors benefit greatly from guida nce based on tractable models of
optimal intertemporal por tfolio choice. Understand ing the sources of time varia tion in the optimal
The authors thank two anonymous referees and the Editor for many helpful and constructive comments that greatly
improved earlier versions of this manuscript.
Global Wealth 2015: Winning the Growth Game,Boston Consulting Group, Financial Services Report, June 2015;
Global Asset Management 2015: Sparking Growth with Go-to-Market Excellence,Boston Consulting Group, Financial
Services Report, July 2015.
© 2017 John Wiley & Sons, Ltd. Eur Financ Manag. 2018;24:192208.
portfolio rule is crucial, since optimal portfolio rebalanc ing is a key concern for non-myopic
investors. The seminal work of Mert on (1969, 1971) has determined that ther e is no time variation
in the optimal portfolio rule resul ting from a continuous-time prob lem with purely diffusive risk if
the investment opportunity set is constant. Merton (1971) shows that this is a lso the case in the joint
presence of a risky lognorma l asset and a risklessasset that is subject to an unpredi ctable default
event driven by a Poisson proces s.
Liu, Longstaff, and Pan (2003) broaden the optimal investment analysis by considering value
disruptions for the risky asset, as well as jumps in the stochastic volatility of the risky assets returns. When
focusing on the case of a constant investment opportunityset associated with unpredictable returns, they find
that the optimal portfolio rule remains time invariant while including a constant hedging demand against
asset value jumps. Das and Uppal (2004) prove the same in a multiple-asset jump-diffusion framework.
Importantly, Liu et al. (2003) and Das and Uppal (2004) exclude the possibility of a jump-to-default event.
Our contribution is to show that, if a jump-to-default event for the risky asset is introduced,
unpredictable returns are consistent with a pre-default time-varying optimal portfolio rule. This is
becausethe sudden disappearance of the riskydefaultable asset modifies the dependenceof the investors
value function on theinvestment horizon. Hence, unlike for Liu et al. (2003)and Das and Uppal (2004),
the ratio between the post- and pre-event marginal indirect utility of wealth becomes dependent on the
investment horizon.Importantly, the dependence of theoptimal policy on the investment horizonmakes
it time variantand triggers optimal portfolio rebalancing.The ratio between marginal indirectutilities is a
key component of the hedging demand against the jump-to-default risk, injecting time variation in the
optimal fractional wealth allocation to the riskydefaultable asset. Our analysis of the time-varyingpre-
default hedgingdemand is conducted in a highly tractable model characterizedby different but constant
pre- and post-default investment opportunity sets and no arbitrage.
We show that the optimal portfolio rule follows a first-order non-linear ordinary differential
equation in the investment horizon that does not involve the investors pre-default value function. This
allows for a direct numerical analysis of the investors optimal choice, which we conduct for
investment horizons of up to 15 years. We find that the investment horizon dynamics of the optimal
portfolio rule are particularly conspicuous when the jump-to-default intensity is low and the degree of
relative risk aversion is high. For example, given a frequency of one jump in 25 years, a conservative
15-year investor facing a security exposed to a jump to default with a 30% recovery rate would
optimally allocate to it only 20% of his/her initial capital, whereas pure jump risk severed from default
risk would have, ceteris paribus, implied a Liu et al. (2003) optimal allocation of more than 28%.
Hence, in the sheer absence of asset return predictability, minute amounts of constant jump-to-default
risk cause strong time variation in the optimal portfolios of long-run conservative investors (who are
more risk averse than log utility agents), who will markedly increase the fractional allocation to the
risky defaultable asset as their investment horizon shortens.
This finding is an interesting complement to the well-known investment horizon effect that
predictability in asset returns generates for related classes of investors (e.g., Balvers & Mitchell, 1997;
Brennan, Schwartz, & Lagnado, 1997; Campbell, Chan, & Viceira, 2003; Campbell & Viceira, 1999;
Detemple & Rindisbacher, 2010; Kim & Omberg, 1996; Koijen, Rodríguez, & Sbuelz, 2009; Liu,
2007; McCarthy & Miles, 2013; Merton, 1969, 1971; Wachter, 2002; Wu, 2003). Our results are robust
to the introduction of several non-defaultable securities: optimal multiple-asset portfolio rebalancing
across different horizons is definitely significant for a conservative investor facing a risky asset with a
small probability of sudden immediate default. We perform a careful analysis of the investors total
expected utility with and without taking into account the horizon dependence of the portfolio weights.
We show that the certainty equivalent losses experienced by a non-myopic decision maker who
suboptimally keeps the portfolio composition constant are quantitatively significant.

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