Exam (2018)

This is the exam for the 2018 ensae course.

Exercise 1

The investment universe is composed of $N$ risky assets (indices $i=1,\ldots,N$) with expected returns $\mu=(\mu_{1},\ldots,\mu_{N})'$ and covariance matrix $\Sigma$ (dimension $(N,N)$) which is assumed non singular. A portfolio is a vector $\pi$ of size $N$ which records the proportions invested on the assets. The proportions sum to $1$, i.e. $\pi'e=1$ where $e$ is the vector of size $N$ where each component is equal to $1$.

1. Given a return objective $\bar{\mu}$, we are looking for the portfolio with the lowest possible level of risk. Give the corresponding optimization problem. Derive the Lagrangian, noting $\gamma$ the multiplier of the return constraint and $\delta$ the multiplier of the full investment constraint.

2. What is the first order condition corresponding to the optimum ? For which portfolio do we get $\gamma=0$ ?

3. We consider an optimal portfolio $\bar{\pi}$ with $\gamma \neq 0$ and return $\bar{r}$. Show that the first order condition for this portfolio can be written as: $$\text{cov}(r_{i},\bar{r})=\gamma \mu_{i}+\delta.$$

4. Show that: $$\text{var}(\bar{r})=\gamma \bar{\mu}+\delta,$$ and: $$\beta_{i}(\bar{\mu}+\frac{\delta}{\gamma})=\mu_{i}+\frac{\delta}{\gamma},$$ where $\beta_{i}$ is the beta of asset $i$ against portfolio $\bar{\pi}$.

5. We now consider portfolio $\tilde{\pi}$ with return $\tilde{r}$ satisfying: $\text{cov}(\tilde{r},\bar{r})=0$. Let $\tilde{\mu}$ be its expected return. Show that for asset $i$: $$\mu_{i}-\tilde{\mu}=\beta_{i}(\bar{\mu}-\tilde{\mu}).$$ This is called a one factor representation of expected returns.

6. Only one efficient portfolio does not give rise to a one factor representation of expected returns. Which one ?

7. Inversely, assume that portfolio $\bar{\pi}$ with return $\bar{r}$ gives rise to a one factor representation: $$\beta_{i}(\bar{\mu}-\rho)=\mu_{i}-\rho,$$ for all $i$, where $\beta_{i}$ is the beta of asset $i$ with respect to portfolio $\bar{\pi}$. Show that we have the following vectorial relationship: $$\Sigma \bar{\pi}=\gamma \mu + \delta e,$$ where $\gamma$ and $\delta$ are constants associated to the one factor representation.

8. Conclude that portfolio $\bar{\pi}$ (the portfolio which generates the given one factor representation) is a solution to the problem defined in the first question. As a reminder, in the context of convex objective function and linear constraints, the first order condition attached to the Lagrangian is both necessary and sufficient to define a solution.

Exercice 2

We consider the following continuous time investment problem. The investment universe is composed of two assets, cash with constant rate of return: $$\frac{dD_{t}}{D_{t}}=r dt,$$ and a risky asset that follows a geometric diffusion process: $$\frac{dP_{t}}{P_{t}}=\mu dt+\sigma dB_{t}=r dt+(\mu-r)dt+\sigma dB_{t},$$ where $B_{t}$ is a scalar Brownian motion. The price of risk is defined as $\lambda=(\mu-r)/\sigma$.

At each point in time, wealth $W_{t}$ is invested to finance a consumption flow $C_{t}dt$ over the time interval $[t,t+dt]$. The fraction of wealth invested on the risky asset at time $t$ is noted $x_{t}$.

1. Give the stochastic differential equation followed by wealth assuming consumption is zero. As a reminder, this is the infinitesimal version of the discrete time equation. Give $E_{t}[dW_{t}]$ (the drift of wealth) and $d[W]_{t}$ (the quadratic variation of wealth).

2. Same question without assuming $C^{t}=0$.

At each time $t$, the investor maximizes: $$E_{t}\left[\int_{t}^{T}e^{-\rho (u-t)}u(C_{u})du\right],$$ by making consumption - $C_{t}$ - and investment - $x_{t}$ - choices. The associated value function is noted $J(t,W_{t})$. It is admitted that the dynamic programming principle implies the following partial differential equation (HJB): $$0=\max_{(C_{t},x_{t})}\left[ u(C_{t})-\rho J+\frac{\partial J}{\partial t}+\frac{\partial J}{\partial W}E[dW_{t}]+\frac{1}{2}\frac{\partial^{2} J}{\partial W^{2}}d[W]_{t} \right].$$

We will use the following notations: $$\frac{\partial J}{\partial t}=J_{t},$$ $$\frac{\partial J}{\partial W}=J_{W},$$ $$\frac{\partial^{2} J}{\partial W^{2}}=J_{WW}.$$

3. Give the optimal consumption rate $C_{t}^{*}$ as a function of $J_{W}$ and the utility function.

4. Give the optimal risky asset weight $x_{t}^{*}$, outlining the relevant property of the value function underlying your reasoning.

5. Describe the structure of this solution.

We now assume the utility function is: $$u(C)=\frac{C^{1-\alpha}}{1-\alpha},$$ with $\alpha>1$ and admit that the value function has a similar structure: $$J(t,W_{t})=h(t)^{\alpha}\frac{W_{t}^{1-\alpha}}{1-\alpha}.$$

6. Show that: $$\frac{C^{*}_{t}}{W^{*}_{t}}=h(t)^{-1},$$ and: $$x_{t}^{*}=\frac{1}{\alpha}\frac{\lambda}{\sigma}.$$

7. Show that at each date $t$: $$u(C^{*})-J_{W}C^{*}=\frac{\alpha}{1-\alpha}J_{W}^{(\alpha-1)/\alpha},$$ $$J_{W}Wx^{*}(\mu-r)+\frac{1}{2}J_{WW}W^{2}\sigma^{2}x^{*2}=-\frac{1}{2}\frac{J_{W}^{2}}{J_{WW}}\lambda^{2}.$$

8. Injecting the above results into HJB, prove that $h(\cdot)$ solves the following differential equation: $$h'+\frac{1}{\alpha}\left[-\rho+(1-\alpha)r+\frac{1-\alpha}{2\alpha}\lambda^{2}\right]h+1=0,$$ with $h(T)=0$.

9. Find the solution $h(\cdot)$.

10. Give the stochastic differential equation followed by log wealth and log consumption.

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