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

References

Kreps, David. 2018. Notes on the Theory of Choice. Routledge.

Markowitz, Harry M. 1952. “Portfolio Selection.” Journal of Finance 7: 77–91.

Petersen, Kaare Brandt, Michael Syskind Pedersen, et al. 2008. “The Matrix Cookbook.” Technical University of Denmark 7 (15): 510.

Von Neumann, John, and Oskar Morgenstern. 1947. Theory of Games and Economic Behavior. Princeton university press.