Skip to main content
Log in

Portfolio allocation problems between risky and ambiguous assets

  • Original Research
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 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.

  2. 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.

  3. 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).

  4. Borgonovo et al. (2018) study and provide a method to connect operational risk management with the theoretical background of decision theory.

  5. Particularly, we adopt Neilson’s (2010) model that is a special case of Klibanoff et al. (2005). Neilson’s (2010) model is popular in applications.

  6. We acknowledge an anonymous reviewer who points out these works and provides an idea of an elementary proof of Theorem 1.

  7. In the literature, for example, Gollier (2011) considers one safe asset and one ambiguous asset.

  8. See Arrow (1965) and Pratt (1964).

  9. See “Appendix” in detail.

  10. For the definition of the convolution property, see “Appendix A”.

  11. For example, see Rothschild and Stiglitz (1971), Fishburn and Porter (1976), and Hadar and Seo (1990).

  12. In Eeckhoudt and Gollier (1995), RHRD is referred to as monotone probability ratio order.

  13. See Eeckhoudt and Gollier (1995, Lemma 2). Eeckhoudt and Gollier (1995, Lemma 1) also show that RHRD is stronger than FSD.

  14. For a survey of the home bias puzzle, see Lewis (1999). For recent studies of the home bias puzzle, see Solnik and Zuo (2012, 2017).

  15. Epstein and Miao (2003) explain the home bias puzzle under ambiguity within the framework of MEU.

  16. 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.

References

  • Anantanasuwong, K., Kouwenberg, R., Mitchell, O. S., & Peijnenburg, K. (2019). Ambiguity attitudes about investments: Evidence from the field. Wharton Pension Research Counsil Working Papers, University of Pennsylvania.

  • Arrow, K. J. (1965). Aspects of the theory of risk-bearing. Helsinki: Yrjo Jahnsonin Saatio.

    Google Scholar 

  • Bianchi, M., & Tallon, J.-M. (2018). Ambiguity preferences and portfolio choices: Evidence from the field. Management Science. https://doi.org/10.1287/mnsc.2017.3006.

    Google Scholar 

  • Billingsley, P. (1995). Probability and measure (3rd ed.). Hoboken: Wiley.

    Google Scholar 

  • Borgonovo, E., Cappelli, V., Maccheroni, F., & Marinacci, M. (2018). Risk analysis and decision theory: A bridge. European Journal of Operational Research, 264, 280–293.

    Google Scholar 

  • Boyle, P., Garlappi, L., Uppal, R., & Wang, T. (2012). Keynes meets Markowitz: The trade-off between familiarity and diversification. Management Science, 58, 253–272.

    Google Scholar 

  • Chateauneuf, A. (1994). Modeling attitudes towards uncertainty and risk through the use of choquet integral. Annals of Operations Research, 52, 3–20.

    Google Scholar 

  • Chiu, W. H., Eeckhoudt, L., & Rey, B. (2012). On relative and partial risk attitudes: Theory and implications. Economic Theory, 50, 151–167.

    Google Scholar 

  • Clark, E., & Jokung, O. (1999). A note on asset proportions, stochastic dominance, and the 50% rule. Management Science, 45, 1724–1727.

    Google Scholar 

  • Driouchi, T., Trigeorgis, L., & So, R. H. Y. (2018). Option implied ambiguity and its information content: Evidence from the subprime crisis. Annals of Operations Research, 262, 463–491.

    Google Scholar 

  • Eeckhoudt, L., Fiori, A. M., & Gianin, E. R. (2016). Loss-averse preferences and portfolio choices: An extension. European Journal of Operational Research, 249, 224–230.

    Google Scholar 

  • Eeckhoudt, L., & Gollier, C. (1995). Demand for risky assets and the monotone probability ratio order. Journal of Risk and Uncertainty, 11, 113–122.

    Google Scholar 

  • Eeckhoudt, L., & Schlesinger, H. (2006). Putting risk in its proper place. American Economic Review, 96, 280–289.

    Google Scholar 

  • Ekern, S. (1980). Increasing \(N\)th degree risk. Economics Letters, 6, 329–333.

    Google Scholar 

  • Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics, 75, 643–669.

    Google Scholar 

  • Epstein, L. G., & Miao, J. (2003). A two-person dynamic equilibrium under ambiguity. Journal of Economic Dynamics and Control, 27, 1253–1288.

    Google Scholar 

  • Epstein, L. G., & Schneider, M. (2008). Ambiguity, information quality, and asset pricing. Journal of Finance, 63, 197–228.

    Google Scholar 

  • Fang, Y., & Post, T. (2017). Higher-degree stochastic dominance optimality and efficiency. European Journal of Operational Research, 261, 984–993.

    Google Scholar 

  • Fishburn, P. C., & Porter, R. B. (1976). Optimal portfolios with one safe and one risky asset: Effects of changes in rate of return and risk. Management Science, 22, 1064–1073.

    Google Scholar 

  • French, K. R., & Poterba, J. M. (1991). Investor diversification and international equity markets. American Economic Review, 81, 221–226.

    Google Scholar 

  • Ghirardato, P., & Marinacci, M. (2001). Risk, ambiguity, and the separation of utility and beliefs. Mathematics of Operations Research, 26, 864–890.

    Google Scholar 

  • Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique priors. Journal of Mathematical Economics, 18, 141–153.

    Google Scholar 

  • Gilboa, I., & Schmeidler, D. (1994). Additive representations of non-additive measures and the choquet integral. Annals of Operations Research, 52, 43–65.

    Google Scholar 

  • Gilboa, I., & Schmeidler, D. (1995). Canonical representation of set functions. Mathematics of Operations Research, 20, 197–212.

    Google Scholar 

  • Gollier, C. (2011). Portfolio choice and asset prices: The comparative statics of ambiguity aversion. Review of Economic Studies, 78, 1329–1344.

    Google Scholar 

  • Hadar, J., & Seo, T. K. (1988). Asset propotions in optimal portfolios. Review of Economic Studies, 55, 459–468.

    Google Scholar 

  • Hadar, J., & Seo, T. K. (1990). The effects of shifts in a return distribution on optimal portfolios. International Economic Review, 31, 721–736.

    Google Scholar 

  • Huang, Y.-C., & Tzeng, L. Y. (2018). A mean-preserving increase in ambiguity and portfolio choices. Journal of Risk and Insurance, 85, 993–1012.

    Google Scholar 

  • Jewitt, I., & Mukerji, S. (2017). Ordering ambiguous acts. Journal of Economic Theory, 171, 213–267.

    Google Scholar 

  • Jindapon, P., & Neilson, W. S. (2007). Higher-order generalizations of Arrow–Pratt and ross risk aversion: A comparative statics approach. Journal of Economic Theory, 136, 719–728.

    Google Scholar 

  • Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263–291.

    Google Scholar 

  • Kelsey, D., Kozhan, R., & Pang, W. (2010). Asymmetric momentum effects under uncertainty. Review of Finance, 15, 603–631.

    Google Scholar 

  • Keynes, J. M. (1921). A treatise on probability. London: MacMillan.

    Google Scholar 

  • Kijima, M., & Ohnishi, M. (1996). Portfolio selection problems via the bivariate characterization of stochastic dominance relations. Mathematical Finance, 6, 237–277.

    Google Scholar 

  • Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73, 1849–1892.

    Google Scholar 

  • Knight, F. H. (1921). Risk, uncertainty and profit. Boston: Houghton Mifflin.

    Google Scholar 

  • Landsberger, M., & Meilijson, I. (1990). Demand for risky financial assets: A portfolio analysis. Journal of Economic Theory, 50, 204–213.

    Google Scholar 

  • Lehmann, E. L. (2005). Testing statistical hypotheses (3rd ed.). Berlin: Springer.

    Google Scholar 

  • Levy, H. (1992). Stochastic dominance and expected utility: Survey and analysis. Management Science, 38, 555–593.

    Google Scholar 

  • Lewis, K. K. (1999). Trying to explain home bias in equities and consumption. Journal of Economic Literature, 37, 571–608.

    Google Scholar 

  • Menezes, C., Geiss, C., & Tressler, J. (1980). Increasing downside risk. American Economic Review, 70, 921–932.

    Google Scholar 

  • Meyer, D. J., & Meyer, J. (2005). Relative risk aversion: What do we know? Journal of Risk and Uncertainty, 31, 243–262.

    Google Scholar 

  • Neilson, W. (2010). A simplified axiomatic approach to ambiguity aversion. Journal of Risk and Uncertainty, 41, 113–124.

    Google Scholar 

  • Osaki, Y., & Schlesinger, H. (2014). Portfolio choice and ambiguous background risk. Working Paper, University of Alabama. Available at http://hschlesinger.people.ua.edu/uploads/2/6/8/4/26840405/ambiguousbgr.pdf.

  • Peter, R. (2019). Revisiting precautionary saving under ambiguity. Economics Letters, 174, 123–127.

    Google Scholar 

  • Peter, R., & Ying, J. (2018). Do you trust your insurer? Ambiguity about contract nonperformance and optimal insurance demand. Journal of Economic Behavior and Organization. https://doi.org/10.1016/j.jebo.2019.01.002.

  • Post, T., & Kopa, M. (2013). General linear formulations of stochastic dominance criteria. European Journal of Operational Research, 230, 321–332.

    Google Scholar 

  • Pratt, J. W. (1964). Risk aversion in the small and the large. Econometrica, 32, 122–136.

    Google Scholar 

  • Roman, D., Mitra, G., & Zverovich, V. (2013). Enhanced indexation based on second-order stochastic dominance. European Journal of Operational Research, 228, 273–281.

    Google Scholar 

  • Rothschild, M., & Stiglitz, J. E. (1970). Increasing risk: I. A definition. Journal of Economic Theory, 2, 225–243.

    Google Scholar 

  • Rothschild, M., & Stiglitz, J. E. (1971). Increasing risk: II. Its economic consequences. Journal of Economic Theory, 3, 66–84.

    Google Scholar 

  • Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.

    Google Scholar 

  • Segal, U. (1987). The Ellsberg paradox and risk aversion: An anticipated utility approach. International Economic Review, 28, 175–202.

    Google Scholar 

  • Solnik, B., & Zuo, L. (2012). A global equilibrium asset pricing model with home preference. Management Science, 58, 273–292.

    Google Scholar 

  • Solnik, B., & Zuo, L. (2017). Relative optimism and the home bias puzzle. Review of Finance, 21, 2045–2074.

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takao Asano.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

$$\begin{aligned} H(z) \equiv \int _a ^b Q(z-x) dP(x). \end{aligned}$$
(8)

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,

$$\begin{aligned} E[u((w-k) {\tilde{x}} + k {\tilde{y}}_i)] \le E[u((w-k) {\tilde{x}} + k {\tilde{y}}_j)]. \end{aligned}$$

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

$$\begin{aligned} V_B '(k^B) = \sum _{\theta =1}^n q_{\theta } \phi '_B (U(k^B,\theta )) g(\theta ,k^B)=0. \end{aligned}$$

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:

$$\begin{aligned} V'_A (k^B) = \sum _{\theta =1}^n q_{\theta } t' (\phi _B (U(k^B,\theta ))) \phi '_B (U(k^B,\theta ))g(\theta ,k^B) \end{aligned}$$

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 \(k^i \le k^B \le k^i+1\),

$$\begin{aligned} {\left\{ \begin{array}{ll} g(\theta ,k^B) \le 0 &{} \text{ for }~\theta \in \{1,\ldots ,i \} \\ g(\theta ,k^B) \ge 0 &{} \text{ for }~\theta \in \{i+1,\ldots ,n \}. \end{array}\right. } \end{aligned}$$

With this decomposition in mind, and noting that \(t'\) is decreasing in \(\theta \), we obtain the following:

$$\begin{aligned}&V_A'(k^B)\\&\quad =\sum _{\theta =1}^i q_{\theta } t' (\phi _B (U(k^B,\theta ))) \phi '_B(U(k^B,\theta ))g(\theta ,k^B) +\sum _{\theta =i+1}^n q_{\theta } t' (\phi _B (U(k^B,\theta ))) \phi '_B(U(k^B,\theta ))g(\theta ,k^B)\\&\quad \le t' (\phi _B (U(k^B,i))) \sum _{\theta =1}^i q_{\theta } \phi '_B(U(k^B,\theta )) g(\theta ,k^B)+ t' (\phi _B (U(k^B,i))) \sum _{\theta =i+1}^n q_{\theta } \phi '_B(U(k^B,\theta )) g(\theta ,k^B)\\&\quad = t'(\phi _B(U(k^B,i))) V_B'(k^B)=0. \end{aligned}$$

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

$$\begin{aligned}&q_1 \phi '_A ( U(k^A,1)) + q_2 \phi '_A ( U(k^A,2)) \\&\quad = q_1 t' (\phi _B (U(k^A,1))) \phi '_B ( U(k^A,1)) + q_2 t' (\phi _B (U(k^A,2)) \phi '_B (U(k^A,2)). \end{aligned}$$

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

$$\begin{aligned} t' (\phi _B ( U(k^A,1))) \ge t' (\phi _B ( U(k^A,2))). \end{aligned}$$

Because \(\phi _i\) is unique up to a positive affine transformation for \(i=A,B\), we can obtain the following normalization,

$$\begin{aligned} q_1 \phi '_A ( U(k^A,1)) + q_2 \phi '_A ( U(k^A,2)) = q_1 \phi '_B (U(k^A,1)) + q_2 \phi '_B ( U(k^A,2)), \end{aligned}$$
(9)

which implies that the following inequalities must be satisfied:

$$\begin{aligned} t' (\phi _B ( U(k^A,1))) \ge 1 \ge t' (\phi _B (U(k^A,2))). \end{aligned}$$

From the first inequality,

$$\begin{aligned} \phi '_B (U(k^A,1)) \le t' (\phi _B ( U(k^A,1))) \phi '_B ( U(k^A,1)) = \phi '_A (U(k^A,1)) \end{aligned}$$

holds. Now, we obtain that

$$\begin{aligned}&\phi '_B (U(k^A,1)) \le \phi '_A (U(k^A,1)) \\&\quad \Leftrightarrow \frac{\phi '_B (U(k^A,1))}{q_1 \phi '_B (U(k^A,1)) + q_2 \phi '_B ( U(k^A,2))} \\&\quad \le \frac{\phi '_A (U(k^A,1)) }{q_1 \phi '_A ( U(k^A,1)) + q_2 \phi '_A ( U(k^A,2))} \\&\quad \Leftrightarrow {\hat{q}}^A_1(k^A) \ge {\hat{q}}^B_1 (k^A), \end{aligned}$$

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

$$\begin{aligned} H_\theta (z) = \int _a^b G_\theta (z-x) dF (x), \end{aligned}$$

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

$$\begin{aligned} H_\theta ^n (z) = \int _a^b G_\theta ^n (z -x) d F(x). \end{aligned}$$

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-019-03206-1

Keywords

JEL Classification

Navigation