Abstract
This paper considers a portfolio allocation problem between a risky asset and an ambiguous asset, and investigates how greater ambiguity aversion influences the optimal proportion invested in the two assets. We derive several sufficient conditions under which greater ambiguity aversion decreases the optimal proportion invested in the ambiguous asset. Furthermore, we consider an international diversification problem as an application and show that ambiguity aversion partially resolves the home bias puzzle.
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Notes
Throughout this paper, to avoid confusion, we say that an asset whose return is known with certainty is safe, rather than riskless or risk-free.
Throughout this paper, we say that an asset whose return is captured by a unique probability measure is risky and an asset whose return is not captured by a unique probability measure is ambiguous.
For a survey of stochastic dominance, see Levy (1992). For applications of stochastic dominance to portfolio strategies, in particular, second-order stochastic dominance, see Roman et al. (2013). Recent studies of stochastic dominance in operations research and management science include Post and Kopa (2013), Eeckhoudt et al. (2016), and Fang and Post (2017).
Borgonovo et al. (2018) study and provide a method to connect operational risk management with the theoretical background of decision theory.
We acknowledge an anonymous reviewer who points out these works and provides an idea of an elementary proof of Theorem 1.
In the literature, for example, Gollier (2011) considers one safe asset and one ambiguous asset.
See “Appendix” in detail.
For the definition of the convolution property, see “Appendix A”.
In Eeckhoudt and Gollier (1995), RHRD is referred to as monotone probability ratio order.
Epstein and Miao (2003) explain the home bias puzzle under ambiguity within the framework of MEU.
The previous studies examine conditions under which the optimal portfolio allocation for one asset is greater than 50%, \(k \ge 0.5\). Because it is essentially identical, their results are restated as \(k \le 0.5\), to agree with the settings in this paper.
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Acknowledgements
We acknowledge an anonymous reviewer whose comments improved this paper substantially. We are grateful to Masamitsu Ohnishi and participants at Paris Financial Management Conference 2017 and the 2017 Annual Meeting of the Nippon Finance Association. Needless to say, we are responsible for any remaining errors. This research is financially supported by the JSPS KAKENHI Grant Nos. 26380240, 26380411, 26705004, 16H02026, 16H03619, 16K03558, 17K03806, and the Joint Research Program of KIER.
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Appendices
Appendix A
We provide the definition of convolution in probability theory based on Billingsley (1995, p. 266) and Lehmann (2005, p. 103).
Definition 4
Let \({\tilde{x}}\) and \({\tilde{y}}\) be independent random variables with probabilities \(\mu \) and v, respectively, and let P and Q be the corresponding probability distribution functions. The convolution of P and Q is defined by
It can be shown that H is a probability distribution function. It can also be shown that if two random variables \({\tilde{x}}\) and \({\tilde{y}}\) with probability distribution functions P and Q are independent, then \({\tilde{x}}+{\tilde{y}}\) has the probability distribution function H defined by (8). Next, we introduce the convolution property.
Definition 5
A stochastic order \(\lesssim _{\mathrm {st}}\) satisfies the convolution property if \({\tilde{x}}+{\tilde{y}}_i \lesssim _{\mathrm {st}} {\tilde{x}}+{\tilde{y}}_j\) for any random variable \({\tilde{x}}\) such that \({\tilde{x}}\) and \({\tilde{y}}_i\) are independent and \({\tilde{x}}\) and \({\tilde{y}}_j\) are independent.
Appendix B. Proof of Theorem 1
Before providing the proof of Theorem 1, we present the following two lemmas. As in Kijima and Ohnishi (1996, Proposition 3.3), the convolution property holds for FSD, which shows the following lemma.
Lemma 1
Let \({\tilde{x}}\) and \({\tilde{y}}_i\) be independent, and \({\tilde{x}}\) and \({\tilde{y}}_j\) be independent for \(i,j \in \Theta \) with \(i < j\). Let \(\{{\tilde{y}}_i,\ldots {\tilde{y}}_n \}\) be ranked by FSD. Let \(k \in [0,w]\). Then,
Lemma 2
(Hadar and Seo 1990) Suppose that (a) \(u'>0\), \(u'' \le 0\), (b) \({\tilde{x}}^i\) and \({\tilde{y}}\) are independent for \(i=1,2\), and (c) \(E[u((w-k_i) {\tilde{x}}^i+ k_i {\tilde{y}})]\) is maximized at \(k_i ^*\). Then, \(k_1 ^* \le k_2 ^*\) for any \({\tilde{x}}^2 \lesssim _{\mathrm {FSD}} {\tilde{x}}^1\) if and only if \(u'(z)z\) is non-decreasing if and only if \(R(z) \le 1\).
The following lemma follows from Lemma 2.
Lemma 3
Let \({\tilde{x}}\) and \({\tilde{y}}_i\) be independent, and \({\tilde{x}}\) and \({\tilde{y}}_j\) be independent for \(i,j \in \Theta \) with \(i < j\). Let \({\tilde{y}}_i \lesssim _{\mathrm {FSD}} {\tilde{y}}_j\) for \(i,j \in \Theta \) with \(i <j\). Then, \(k^i \le k^j\) if \(R(z) \le 1\).
Now, we are in a position to show Theorem 1.
Proof of Theorem 1
Let \(V_i(k)=\sum _{\theta =1}^n q_{\theta } \phi _i (E[u((w-k){\tilde{x}}+k {\tilde{y}}_{\theta })])\) be the objective functions for \(i=A,B\). Let \(\phi _A=t\circ \phi _B\) where t is an increasing and concave function. Define \(U(k,\theta )= E[u((w-k) {\tilde{x}} + k {\tilde{y}}_\theta )]\) and \(g(\theta ,k)=E[({\tilde{y}}_{\theta }-{\tilde{x}})u'((w-k) {\tilde{x}} + k {\tilde{y}}_\theta )]\). The optimal portfolio allocation for investor B must satisfy
By the concavity of the objective function, it suffices to show that the sign of \(V'_A (k^B)=\sum _{\theta =1}^n q_{\theta } \phi '_A (U(k^B,\theta ))g(\theta ,k^B)\) is negative. Because \(\phi _A = t \circ \phi _B\), \(V'_A (k^B)\) can be rewritten as follows:
Now, \(t'(\phi _B(U(k^B,\theta )))\) is decreasing in \(\theta \) because, as \(\theta \) increases, \((w-k){\tilde{x}}+k{\tilde{y}}_{\theta }\) improves in the sense of FSD by Lemma 1, so that \(U(k^A,\theta )\) increases in \(\theta \), and \(\phi \) is increasing in \(\theta \) because \(\phi \) is increasing by assumption, but the concavity of t implies that \(t' (\phi _B (U(k^B,\theta )))\) is decreasing in \(\theta \). From Lemma 3, \(k^\theta \) is increasing in \(\theta \) if \(R(z) \le 1\). Thus, we obtain that, for \(k^i \le k^B \le k^i+1\),
With this decomposition in mind, and noting that \(t'\) is decreasing in \(\theta \), we obtain the following:
Because we show that \(V'_A (k^B)=\sum _{\theta =1}^n q_{\theta } \phi '_A (U(k^B,\theta ))g(\theta ,k^B)\) is negative, the proof is completed. \(\square \)
Appendix C. Derivation of (4)
Let \(\phi _A = t \circ \phi _B\), where t is increasing and concave. Then, we can rewrite
Because \(U(k^A,1) \le U(k^A,2)\), \(\phi _i\) is increasing for \(i=A,B\), and \(t'\) is decreasing by t’s concavity, it holds that
Because \(\phi _i\) is unique up to a positive affine transformation for \(i=A,B\), we can obtain the following normalization,
which implies that the following inequalities must be satisfied:
From the first inequality,
holds. Now, we obtain that
where the first equivalence follows from (9). Therefore, we complete the proof.
Appendix D. Proofs of Propositions 1 and 2
We can show Propositions 1 and 2 based on the Proof of Theorem 1. For that purpose, it suffices to show that the results corresponding to Lemmas 1 and 3 hold for MLRD and RHRD.
First, Lemma 1 holds for MLRD and RHRD because both MLRD and RHRD are stronger than FSD.
Second, as in the main text, the result corresponding to Lemma 3 can be shown by Landsberger and Meilijson (1990, Proposition 2) for MLRD, and the result corresponding to Lemma 3 can be shown by Kijima and Ohnishi (1996, Theorem 4. 12 and its Corollary 4.7) for RHRD. Thus, the proofs of Propositions 1 and 2 are completed.
Appendix E. Proof of Proposition 3
Similar to Propositions 1 and 2 , we can show Proposition 3 based on the proof of Theorem 1. For that purpose, it suffices to show that the results corresponding to Lemmas 1 and 3 hold for N-th degree risk.
First, we show the result corresponding to Lemma 3. Because \({\tilde{x}}\) and \({\tilde{y}}_\theta \) are independent for any \(\theta \in \Theta \), the convolution of F and \(G_{\theta }\) is
where F and \(G_{\theta }\) denote the probability distribution functions of \({\tilde{x}}\) and \({\tilde{y}}_{\theta }\), respectively. It can be shown that the convolution H is also a probability distribution function. Let us define \(G_\theta ^n (y-x) = \int _a^y G_\theta ^{n-1} (t-x) dt\). By Fubini’s theorem, we can rewrite the probability distribution function as
Note that \({\tilde{y}}_j \lesssim _{\mathrm {N-risk}} {\tilde{y}}_i\) is equivalent to \(H_i^n (z) = \int _a^b G_i^n (z -x) d F(x) \ge \int _a^b G_j^n (z -x) d F(x) =H_j^n (z)\), that is, \({\tilde{x}} + {\tilde{y}}_j \lesssim _{\mathrm {N-risk}} {\tilde{x}} + {\tilde{y}}_i\). From the convolution property, the claim is proved.
Second, the result corresponding to Lemma 3 can be shown by Chiu et al. (2012) for N-th degree risk. Thus, the proof of Proposition 3 is completed.
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Asano, T., Osaki, Y. Portfolio allocation problems between risky and ambiguous assets. Ann Oper Res 284, 63–79 (2020). https://doi.org/10.1007/s10479-019-03206-1
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DOI: https://doi.org/10.1007/s10479-019-03206-1
Keywords
- Uncertainty modelling
- Home bias puzzle
- Portfolio allocation problem
- Smooth ambiguity model
- Greater ambiguity aversion