Abstract
decisions on the mean-variance approach. This helped us to develop a model for pricing assets on a financial market, the CAPM. In this chapter we want to generalize this model in that we relax the assumptions on the preferences of the investors.
Toutes les généralisations sont dangereuses, y compris celle-ci. (All generalizations are dangerous, even this one.) Alexandre Dumas
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Notes
- 1.
For the unlikely case that the reader is not familiar with these topics, or in the more likely case that he wants to refresh his memory, the Appendix A.1 gives a quick review on basic linear algebra.
- 2.
Compare, however, the remarks on time-discounting in Sect. 2.7
- 3.
In the case of two commodities the upper contour set is the set that is included by the indifference curve. Hence, the utility function is quasi-concave if the indifference curves are convex.
- 4.
For two vectors \(\boldsymbol{c},\boldsymbol{c}'\) we use the notation \(\boldsymbol{c} >\boldsymbol{ c}'\) to mean that in each component the vector is at least as large as the vector \(\boldsymbol{c}'\) and in at least one \(\boldsymbol{c}\) is strictly greater than \(\boldsymbol{c}'\). We use \(\boldsymbol{c} \gg \boldsymbol{ c}'\) if the vector \(\boldsymbol{c}\) is strictly greater than \(\boldsymbol{c}'\) in all components, see Appendix A.1.
- 5.
The Edgeworth Box can alternatively be used to display intertemporal trade. In that case one would need to assume that there is only one state of the world tomorrow, i.e., S = 1, so all assets are identical to the risk-free asset. In this case, we would of course consider first-period consumption.
- 6.
- 7.
The operator \(\boldsymbol{\varLambda }\) transforms an n-dimensional vector into a n × n diagonal matrix with the vector being the main diagonal. The operator−1 computes the inverse of a matrix. Compare Appendix A.1.
- 8.
See Magill and Quinzii [MQ02] for details.
- 9.
We neglect trading costs!
- 10.
The scalar product is positive (negative) if the angle is smaller (greater) than 90∘. The scalar product of orthogonal vectors is equal to 0 (see Appendix A.1).
- 11.
It is obvious that a representation by state prices satisfies linearity. The converse is a bit harder to see (compare Appendix A.1).
- 12.
Note that we did not assume that all state prices are positive. For this formulation we only need that the sum of the state prices is positive, which holds, for example, with mean-variance utilities.
- 13.
0. 95 ⋅ 95. 06∕number of outstanding 3Com shares.
- 14.
One of the authors was among them.
- 15.
A closed-end fund is a mutual fund with a fixed asset composition.
- 16.
In other words, there exists no asset allocation where nobody is worse off and at least somebody is better off.
- 17.
We will always refer to the normalized state prices as the state price measure. However, as can be seen from the calculations, we do not actually need that all state prices are non-negative. Only the sum of the state prices needs to be positive. Hence, we can accommodate without any special considerations the case of mean-variance preferences for which the positivity of state prices was not guaranteed.
- 18.
See Chap. 4.1.3 for this transformation of the decision problem.
- 19.
Most finance models work right away with a representative investor being in equilibrium with himself. Hence, the market clearing condition is not stated explicitly.
- 20.
In politics, this is sometimes used as an argument against a private pension fund system. The asset melt down, however, is mitigated by the fact that demographic developments are very diverse between different countries and financial markets are global.
- 21.
We assume that the mean-variance utility function \(V ^{i}(\mu,\sigma )\) is quasi-concave so that the first order condition is necessary and sufficient to describe the solution of the maximization problem. This is, for example, the case for the standard mean-variance function \(V ^{i}(\mu,\sigma ):=\mu -\frac{\rho ^{i}} {2}\sigma ^{2}\), since it is even concave.
- 22.
Note that \(\frac{\partial _{\sigma }V ^{i}} {\partial _{\mu }V ^{i}}\) is the slope of the indifference curve in a diagram with the mean as a function of the standard deviation.
- 23.
Note that the linearity of the likelihood ratio process also holds in the CAPM with heterogeneous beliefs (see Sect. 3.3) on expected returns if we define the likelihood ratio process with respect to the average belief of the investors.
- 24.
In the exercise book we derive the CAPM in yet another way. Assume quadratic utility functions and then show that the likelihood ratio process being the marginal rates of substitution becomes proportional to a linear combination of the risk-free asset and the market portfolio.
- 25.
Note that the lower index 1 in the consumption variable denotes the period 1, i.e., \(\boldsymbol{c}_{1}^{i}\) is the vector (c 1 i, …, c s i), which should not be confused with the consumption in state s: c s i, s = 1.
- 26.
Insert \(\boldsymbol{q}' = \boldsymbol{\pi }'\boldsymbol{A}\) from the no-arbitrage condition and substitute to obtain this result.
- 27.
Please don’t be confused: R f denotes the return to factor f while R f denotes the return to the risk-free asset!
- 28.
The likelihood ratio process can always be chosen in the span of \(\boldsymbol{A}\) since any component orthogonal to the span in the sense of \(\mathbb{E}_{p}(\ell A) = 0\) does not change the value of the assets. This is due to the no-arbitrage condition. Moreover, the risk-free asset is the first asset in \(\boldsymbol{A}\).
- 29.
Compare Sect. 2.5 where we have seen that mean-variance preferences can be seen as a special case of EUT with quadratic utility function.
- 30.
See Sect. 4.6 for a justification.
- 31.
Strictly speaking, this is true only if efficient allocations do not lie on the boundary of the Edgeworth Box. Assumptions like that marginal utility is unbounded as consumption converges to the boundary of the Edgeworth Box are necessary here.
- 32.
So far we did never specify the consumption sets. This typically is the set of non-negative vectors in \(\mathbb{R}^{S+1}\), since negative consumption does not have an economic interpretation. The utility functions need only be defined on the consumption sets.
- 33.
Obviously, one could also find a representative consumer with strictly concave utilities since one only needs to satisfy that his marginal rates of substitution at aggregate endowments coincide with the state prices.
- 34.
This last claim is the so called envelope theorem.
- 35.
See Sect. 4.6.3 for empirical studies along this line.
- 36.
- 37.
- 38.
The classical reference is Leland and Genotte [GL90].
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Hens, T., Rieger, M.O. (2016). Two-Period Model: State-Preference Approach. In: Financial Economics. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49688-6_4
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