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: $\frac{\partial J}{\partial t}=J_{t},$ $\frac{\partial J}{\partial W}=J_{W},$ $\frac{\partial^{2} J}{\partial W^{2}}=J_{WW}.$

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

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

3. 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: $J(t,W_{t})=h(t)^{\alpha}\frac{W_{t}^{1-\alpha}}{1-\alpha}.$

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

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

3. 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 $$h(T)=0$$.

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

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