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Decisions in Economics and Finance

Erschienen in:

01.11.2017

Reaching nirvana with a defaultable asset?

verfasst von: Anna Battauz, Marzia De Donno, Alessandro Sbuelz

Erschienen in: Decisions in Economics and Finance | Ausgabe 1-2/2017

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Abstract

We study the optimal dynamic portfolio exposure to predictable default risk, taking inspiration from the search for yield by means of defaultable assets observed before the 2007–2008 crisis and in its aftermath. Under no arbitrage, default risk is compensated by an ‘yield pickup’ that can strongly attract aggressive investors via an investment-horizon effect in their optimal non-myopic portfolios. We show it by stating the optimal dynamic portfolio problem of Kim and Omberg (Rev Financ Stud 9:141–161, 1996) for a defaultable risky asset and by rigorously proving the existence of nirvana-type solutions. We achieve such a contribution to the portfolio optimization literature by means of a careful, closed-form-yielding adaptation to our defaultable asset setting of the general convex duality approach of Kramkov and Schachermayer (Ann Appl Probab 9(3):904–950, 1999; Ann Appl Probab 13(4):1504–1516, 2003).

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Cox and Huang (1989) require the global Lipschitz continuity of the diffusive coefficient for the risky asset-value process [see Conditions A and B at p. 46 in Cox and Huang (1989)]. By contrast, the diffusive coefficient in our setting is the square root of the risky asset value.

See, for instance, Davydov and Linetsky (2001), p. 952, first paragraph, with \(S_{t}=P_{t}\) and \(p=\frac{1}{2}.\)

The objective probability of the asset defaulting within the date \(T>0\) is \({\mathbb {P}}\left[ P_{h}=0,\,0\le h\le T\quad P_{0}=p\,\right] =\Gamma \left( \frac{2\left( r+\xi \right) p}{1-e^{-\left( r+\xi \right) T}},1\right) ,\) where \(\Gamma \left( k,l\right) =\int _{k}^{+\infty }u^{l-1}e^{-u}du,\) \(k\ge 0\) is the incomplete gamma function [see, e.g., the Proposition 1 in Campi and Sbuelz (2005)].

The nonnegativity requirement is innocuous in our setting as the utility function U is \(-\infty \) for negative wealth levels.

Battauz et al. (2015) apply the Kramkov and Schachermayer (1999, 2003) approach to the standard Kim and Omberg (1996) portfolio problem with a non-defaultable risky asset.

There are parameter values that make the investor with \(\phi >1\) take a net short position in the risky defaultable asset.

Similar 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 constant volatility [see, for example, Koijen et al. (2009)].

The functions U and V are conjugate if and only if \(U(w)=\inf _{y>0}(V(y)+wy)\) and \(V(y)=\sup _{w>0}(U(w)-wy)\).

The superscript \(\left( \cdot \right) ^{DS}\) refers to the notations of Delbaen and Shirakawa (2002), whereas a, b, c are defined in the statement of Proposition 3.2.

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Titel: Reaching nirvana with a defaultable asset?
verfasst von: Anna Battauz
Marzia De Donno
Alessandro Sbuelz
Publikationsdatum: 01.11.2017
Verlag: Springer Milan
Erschienen in: Decisions in Economics and Finance / Ausgabe 1-2/2017
Print ISSN: 1593-8883
Elektronische ISSN: 1129-6569
DOI: https://doi.org/10.1007/s10203-017-0192-x

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