Abstract
Indeed we will start our journey to financial markets with only one step: the step from one time period (in which we invest into assets) to another time period (in which the assets pay off). To make this two-period model even simpler, we assume in this chapter mean-variance preferences. We will see later that this model is a special case of two-period models with more general preferences (Chap. 4) and that we can extend the model to arbitrarily many time-periods (Chap. 5). Finally we generalize to continuous models, where the time does not any longer consists of discrete steps (Chap. 8). For now, the assumptions of two periods and mean-variance preferences allow us to get some intuition on financial markets without being overwhelmed by an overdose of mathematical formalism. Nevertheless, we want to point out that this simplicity comes at a price: we need to impose strong and not very natural assumptions. In Sect. 2.3, we have seen some of the potential problems of the mean-variance approach. In practical applications, however, this approach is still standard. We will use it to develop a first model of asset pricing, the so-called “Capital Asset Pricing Model” ( CAPM). This model has been praised by many researchers in finance, and in 1990 Markowitz and Sharpe were awarded the Nobel Prize in economics for its development.
“A journey of a thousand miles starts with the first step.”
Chinese proverb
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Notes
- 1.
Expected returns can, for example, be calculated using historical return values adjusted by some market expectations.
- 2.
For information on the historical development of the mean-variance approach and the CAPM see [Var93a] from whom we have taken the above quote.
- 3.
We will see that economically spoken, this portfolio is such that the marginal rate of substitution between the investor’s preferences for risk and return equals the marginal rate of transformation offered by the minimum variance opportunity set.
- 4.
We use in this chapter bold face characters for vectors to increase readability.
- 5.
For the purpose of deriving the Two-Fund Separation Theorem this single utility function is sufficient. Using a more general function like \(V ^{i}(\mu _{\lambda },\sigma _{\lambda })\) would result in expressions similar to those we derive here. In this case we get \(\rho ^{i} = -\frac{\partial _{\sigma }V ^{i}(\mu _{\lambda },\sigma _{\lambda })} {\partial _{\mu }V ^{i}(\mu _{\lambda },\sigma _{\lambda })}\). But as we see below, the point of the Two-Fund Separation Theorem is to show that \(\rho ^{i}\) anyway cancels out from the portfolio of risky assets.
- 6.
The risk aversion concept is often discussed in the expected utility context. Recall, however, that there it is measured by the curvature of a utility function.
- 7.
Note: there is no index i on the Tangent Portfolio \(\boldsymbol{\lambda }^{T}\) since this portfolio is the same for every investor.
- 8.
Note that solving the simplest \((\mu,\sigma )\)-problem is as good as any other \((\mu,\sigma )\)-problem, since by the Two-Fund Separation property all mean-variance utility functions deliver the same Tangent Portfolio.
- 9.
\(\lambda _{0}\) is the first component of \(\boldsymbol{\lambda }\).
- 10.
The market capitalization of a company for example is the market value of total issued shares.
- 11.
Note that this equality is barely supported by empirical evidence, i.e., the Tangent Portfolio does not include all assets. The reason for this mismatch could for example be that not every investor optimizes over risk and return as suggested by Markowitz. For further ideas on this asset allocation puzzle see also [CMW97, BX00]
- 12.
If the j-curve would intersect with the CML then the Sharpe Ratio could still be increased, as can be verified graphically.
- 13.
When prices revert and increase (decrease) in order to reach their fundamental value, the expected returns are decreasing (increasing).
- 14.
Value stocks are for example characterized by high multiples, i.e., book to price ratios, cash flow to price ratio, dividend yield etc.
- 15.
To list some examples: Goldman Sachs offers “Global Alpha”, Merrill Lynch “Absolute Alpha Fund” and UBS “Alpha hedge” and “Alpha select”.
- 16.
Managing Director, Senior Investment Officer, Alternative Investment Solutions at UBS Global Asset Management.
- 17.
There are two tacit assumptions behind this argument that may or may not be true, namely first that the “bad” investors are capable and willing to learn from their mistakes and second that there are not sufficiently many new “bad” investors entering the market to compensate for their dropped out predecessors.
- 18.
See, for example the long run data provided by Robert Shiller on his webpage: http://www.econ.yale.edu/~shiller/data.htm
- 19.
Compare also [Abe89].
- 20.
The factor \((1 -\lambda _{0})\) in the security line appears since we did not normalize the asset allocation in the risky portfolio to sum up to one. Using the notation incorporating this normalization, i.e., the \(\hat{\lambda }_{k}^{i}\), the term would not appear any more.
- 21.
To be more precise: since the optimization problem is concave, we can indeed find the optimal portfolio by optimizing iteratively over the assets, where we improve the portfolio every time we find a positive Alpha, as long as there is one. Concavity is needed in order that this method does not lead to a local optimum, but in fact to the globally optimal portfolio.
- 22.
If the fish and the red wine example doesn’t convince you, you may finally look at a trivial example: imagine a person who only holds a fixed interest rate asset. Would you recommend this person to buy commodities in order to improve the performance through diversification, based on the argument that the efficient frontier will be improved by adding commodities? You probably won’t, since a riskless portfolio can obviously not profit from any diversification effects: its covariance to any other asset is always zero.
- 23.
See Gerber and Hens [GH06] for a generalization towards heterogeneous beliefs on covariances.
- 24.
Note that we have defined the utility on gross returns, i.e., expected returns are larger than one.
- 25.
Note that underdiversified portfolios do not need to be worse than well-diversified portfolios. Based on 78’000 households portfolios observed from 1991 to 1996 Ivković, Sialm and Weisbenner [ISW08] find that the more wealthy have more underdiversified portfolios achieving a positive Alpha to the market.
- 26.
Ex-post means after a state s = 1, …, S of the world has realized. Ex-ante means before the resolution is known.
- 27.
Note that in a model with rational expectations all investors are assumed to know the true market returns that can be expected. In particular they have then homogeneous expectations.
- 28.
This is a consequence of two-fund separation. Every active investor (holding his tangential portfolio) holds the market portfolio, i.e. turns out to be a passive investor in equilibrium.
- 29.
See Gerber and Hens [GH06]
- 30.
That agents may trade actively because they are not at all risk averse but trade for entertainment has recently been observed in Dorn and Segmueller (2006).
- 31.
Applying the SML to the market portfolio itself results in the tautology \(1 \equiv 1\), hence the CAPM is an asset pricing model valuing assets relative to the market portfolio, but it does not evaluate the market portfolio itself.
- 32.
I.e., the covariance between each pair of factors is zero.
- 33.
Skewness is measured by the third moment of a distribution and fat tails are measured by the kurtosis, i.e., the fourth moment, of a distribution in excess of the kurtosis of a normal distribution, see Sect. A.2.
- 34.
The p-values displayed in Table 3.2 mirror the significance level. That is to say a utility value with a p-value not exceeding 0.05 indicates a significance on a 5 % level.
- 35.
For the pros and cons of this approach see Sect. 4.6 of the next chapter.
- 36.
You may include probability weighting by replacing \(\,p_{s}\) with \(w_{s} = w(\,p_{s})\). However, then some care has to be taken in the empirical analysis!
- 37.
In the continuous time version of the financial markets model the likelihood ratio is called the Radon-Nikodym derivative.
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Hens, T., Rieger, M.O. (2016). Two-Period Model: Mean-Variance Approach. In: Financial Economics. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49688-6_3
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