Chapter 5 Optimal individual choices
5.1 The Expected Utility framework
5.1.1 VNM theory
Describing the way economic agent make a choice between two possibilities or asset in an uncertain world is an important part of micro-economics. For a classical survey see Kreps (2018).
Preferences are defined as a relationship between economic outcomes X and Y where either :
- X is preferred to Y, (\(X\succeq Y\)=
- the agent is indifferent between X and Y (\(X \sim Y\))
- the agent doesn’t know
Von Neumann and Morgenstern (1947) show that under reasonable assumptions on preferences such as:
Completeness: \(\forall (X,Y):X\succ Y\quad\mathrm{or}\quad X\prec Y\quad\mathrm{or}\quad X\sim Y\)
Continuity: \(\forall (X,Y,Z):X\succeq Y\succeq Z \Rightarrow \exists p\in[0,1] \; | \; Y\sim pX+(1-p)Z\)
Transitivity: \(\forall (X,Y,Z):X\succeq Y\quad\mathrm{and}\quad Y\succeq Z\quad \Rightarrow X\succeq Z\)
Independence: For any \(N\) and \(p\in (0,1]\), \(L\preceq M\qquad \text{iff}\qquad pL+(1-p)N\preceq pM+(1-p)N\)
then preferences can be represented by the expectation of a utility function u under some probability measure.
Hence, When the set of outcome is in \(\mathbb R\) then \(U\) can be shown to be increasing and concave. Further regularity conditions such as the Inada conditions might be added to ensure that
In a situation where investors want to discriminate among possible portfolio outcomes or different option profile, they should hence choose \(\underset{\tilde V_T}{argsup}\;\mathbb E(U(\tilde V_T))\)
5.1.2 Risk aversion and risk premia,
Let \(\tilde V_T\) the final wealth and \(\bar V_T \equiv \mathbb E (\tilde V_T)\)
Let \[\begin{align*} U\left(\tilde{V_{T}}\right) &\approx U\left(\bar{V_{T}}\right)+\left(\tilde{V_{T}}-\bar{V_{T}}\right)U'\left(\bar{V_{T}}\right)+\frac{1}{2}\left(\tilde{V_{T}}-\bar{V_{T}}\right)^{2}U''\left(\bar{V_{T}}\right)\\ \Rightarrow\mathbb{E}\left(U\left(\tilde{V_{T}}\right)\right) & \approx U\left(\bar{V_{T}}\right)+\frac{\sigma_{V_{T}}^{2}}{2}U''\left(\bar{V_{T}}\right). \end{align*}\]
Since \(U\) is concave one has \[ \mathbb{E}\left(U\left(\tilde{V_{T}}\right)\right)\leq U\left(\bar{V_{T}}\right) \] As a result, when U is continuous, there exists a positive value \(\pi\) such that \[ \mathbb{E}\left(U\left(\tilde{V_{T}}\right)\right)=U\left(\bar{V_{T}}-\pi\right) \]
\(\pi\) is called the risk premia and \(V_{eq}\equiv \bar{V_{T}}-\pi\) is called the certainty equivalent of \(\tilde{V_{T}}\) associated to U.
For small risks, the risk premia can be approximated by a first order expansion of U:
\[ U\left(\bar{V_{T}}-\pi\right)=U\left(\bar{V_{T}}\right)-\pi U'\left(\bar{V_{T}}\right) \]
and hence \[ \pi\approx\frac{\lambda}{2}\sigma_{V}^{2} \]
where \(\lambda_{abs}\equiv-\frac{U''}{U'}\) is called the local Absolute Risk Aversion. The Relative Risk Aversion is defined by \(\lambda_{rel}=\lambda_{abs}\times V\)
Example 5.1 (Constant Absolute Risk Aversion) \(U(x)=-exp(-\lambda x)\) where \(\lambda>0\) is the absolute risk aversion. The corresponding behavior is then described by the fact that \(x_0\) can be seen as some kind of “target wealth”. When \(x<<x_0\), then U is quasi risk neutral and when \[x>>x_O\] the relative risk aversion becomes large. \[ \begin{align} \frac{\Delta U}{U} &=-\lambda\Delta x \\ & = \frac{\Delta x}{x_0} \end{align} \]
::: example name =“Constant Relative Risk Aversion”
\[ U\left(x\right)=\left\{ \begin{array}{c} \frac{x^{1-a}-1}{1-a}\;a\geq0,a\neq1\\ ln\left(x\right)\;a=1 \end{array}\right. \] where \(a\) is the Absolute Risk Aversion. In that case:
\[ \begin{align} \frac{\Delta U}{U} &=-\lambda\Delta x \\ & = \frac{\Delta x}{x_0} \end{align} \]
The corresponding behavior is then described by the fact that \(x_0\) can be seen as some kind of “target wealth”. When \(x<<x_0\), then U is quasi risk neutral and when \[x>>x_O\] the relative risk aversion becomes large.
\[\frac{\Delta U}{U} =(1-a)\frac{\Delta x}{x} \] This corresponds to a behavior with no target satisfactory wealth. Whichever the level of wealth, x% return provides the exact same increase of Utility. :::
5.2 Mean-variance choices
5.2.1 Wealth equation
Consider a situation where, between date 0 and date 1, your initial wealth \(w\) has to be invested into:
- \(n\) risky assets with value \(S^i_t\) such that the random expected return \(\mu^i \quad | \quad S^i_1=S^i_0\times(1+\mu^i)\) is described as a random variable with mean \(\mu\) and covariance matrix \(\Sigma\). \(\mu\) is often referred to as the expected return of that asset,
- a non risky asset which return is known beforehand to be equal to \(R\). \(R\) is also the cost of borrowing money which can be done without constraint.
Let \(\alpha\) be the proportion of \(w\) invested into the risky assets. Let \(\alpha_0=1-\sum_i \alpha_i\) the amount invested or borrowed to the bank. The final (random) wealth yields \[\tilde{W}=w\times(1+R+\alpha'(\tilde{\mu}-R\mathbb I_n))\]
Let us define the excess returns as \(\tilde{\pi_i}\equiv\tilde{\mu_i}-R\) and the excess return of the porfolio as \(\tilde{\pi}(\alpha) \equiv \frac{\tilde{W}}{w}-(1+R)\). The following relationship holds:
\[\begin{equation} \tilde{\pi}(\alpha)=\alpha' \tilde{\pi} \tag{5.1} \end{equation}\]Equation (5.1) shows that the excess return of the portfolio is a linear function of the excess return of the risky asset. That linear relationship between excess return is a key structure to keep in mind. Relation between portfolio moments and risky asset moments follows:
- \(\pi(\alpha)=\mathbb E(\alpha' \tilde{\pi})=\alpha'\bar{\pi}\)
- \(Var(\tilde \pi(\alpha))=\mathbb E (\alpha'\tilde \pi \tilde \pi' \alpha )=\alpha'\Sigma\alpha\)
Definition 5.1 (Sharpe ratio) The Sharpe ratio of an asset with average Excess Return \(\pi\) and volatility \(\sigma\) is defined as the ratio of the two: \(\frac{\pi}{\sigma}\).
5.2.2 Optimal portfolio
The optimal portfolio is the solution of the same maximisation equation
\[\begin{equation} u(w)=\underset{\alpha \in \mathbb R^n}{sup}E(U(\tilde W)) \tag{5.2} \end{equation}\]
For a CARA utility with Gaussian returns, equation (5.2) becomes:
\[\begin{equation} u(w)=\underset{\alpha \in \mathbb R^n}{sup}\alpha'\pi-\frac{\lambda}{2}\alpha'\Sigma\alpha \tag{5.3} \end{equation}\]
The optimisation program (5.3) has to be solved in \(\mathbb R^n\). It can be done line by line as in Markowitz (1952) seminal paper but it makes the result difficult to read.
It is somewhat easier to differentiate directly in \(\mathbb R^n\) even if this requires prior knowledge of some ad-hoc rules. See Petersen, Pedersen, et al. (2008) for instance. One finds that:
\[\begin{equation} \bar \alpha=\frac{1}{\lambda}\Sigma^{-1}\pi \tag{5.4} \end{equation}\]A variational proof:
Define an auxiliary function \[ \phi_{y}^{\alpha}(\epsilon)=f(\alpha+\epsilon y) \] \[ \alpha\;optimal\Longleftrightarrow\left.\frac{d\phi_{y}^{\alpha}(\epsilon)}{d\epsilon}\right|_{\epsilon=0}=0\;\forall y\in\mathbb{R}^{n} \] Practically: \[ \begin{align*} f(\alpha+\epsilon y) & =(\alpha+\epsilon y)'\pi-\frac{\lambda}{2}(\alpha+\epsilon y)'\Sigma(\alpha+\epsilon y)\\ & =cte+\left(y'\pi-\lambda\alpha'\Sigma y\right)\epsilon-\frac{\lambda}{2}y'\Sigma y\epsilon^{2}\\ \left.\frac{d\phi_{y}^{\alpha}(\epsilon)}{d\epsilon}\right|_{\epsilon=0} & =\left(\pi-\lambda\Sigma\alpha\right)'y \end{align*} \]
Since the above has to be equal to zero whatever \(y\), QED.
Comments on equation (5.4):
- The two fund separation theorem
- role of correlation
- exercise in dim 2.
- formula without lending/borrowing
- OSQP solver