19.4. Merton’s portfolio problem - Example

Merton’s portfolio problem aims at finding the optimal value of the fraction the risky asset \(\pi_t\), in a 2-asset portfolio: the risky asset and a risk-free asset. The optimal solution is the fraction \(\pi^*_t\) that maximizes the value function

\[V(x,t) = \mathbb{E} \left[ \left. \int_{s=t}^{T} e^{- \rho (s-t)} u(c_s) \, ds + e^{-\rho(T-t)} B(T) u(X_T) \right| X_t = x \right] \ ,\]

i.e. a discounted return of a utility function \(u(c_s)\), depending on the consumption \(c_s\) at time \(s \in [t, T]\) with a weight of utility function evaluated for the final value of the wealth \(X_T\), modeling the bequest, weighted with \(B(T)\), and subject to the dynamics of wealth \(X_t\) goverend by the SDE

\[\begin{aligned} dX_t & = \underbrace{\left[ r_t + \left( \mu_t - r_t \right) \pi_t \right] X_t \, dt}_{\text{expected return}} - \underbrace{c_t \, dt}_{\text{consumption}} + \underbrace{\pi_t \sigma_t X_t \, dW_t}_{\text{volatility of the return}} \ . \end{aligned}\]

Under the assumption of CCRA, i.e. with the expression of the utility function

\[u(x) = \frac{x^{1-\gamma}}{1 - \gamma} \ ,\]

being \(\gamma\) the personal risk-adversion factor, the optimal solution provides:

• a value of optimal fraction invested in the risky asset \(\pi^*\) constant in time (did you say rebalancing?),

  \[\pi^*_t = \frac{\mu - r}{\gamma \sigma^2} \ ,\]

  with \(\mu\), \(\sigma^2\) the expected value and the variance of the return of the risky asset, and \(r\) the risk-free return

• the expression of the consumption \(c_t\), that’s proportional to the wealth \(X_t\),

  \[c_t = \frac{X_t}{f(t)} \ ,\]

  through a function of time \(f(t)\).

Function \(f(t)\) is

\[\begin{split}f(t) = \begin{cases} \dfrac{1}{\nu} + \left( \varepsilon - \dfrac{1}{\nu} \right) e^{-\nu (T-t)} & \text{,} \quad \text{ if $\nu \ne 0$} \\ T - t + \varepsilon & \text{,} \quad \text{ if $\nu = 0$} \\ \end{cases}\end{split}\]

with

\[\nu = \dfrac{1}{\gamma} \left\{ \rho - (1 - \gamma) \left( \dfrac{(\mu-r)^2}{2 \sigma^2 \gamma} + r \right) \right\} \ .\]