Solving a nonlinear equation with random variable

Question

I would like to solve an equation that looks like this

UPDATE

$E[(R^{1-\gamma})(r_k+\theta-r_z)]=0$ , where $R=\phi r_z+(1-\phi)(r_k+\theta)$ and $\phi\in[0,1]$,

$\theta$, is a random variable normaly distributed with zero mean, and some variance $\sigma^2$. Thus, the $E[.]$ stands for the expectation operator over this random variable.

An alternative version of the problem, solves a similar equation, but now $\theta$ is lognormally distributed.

The computational task, is to find $\phi$, such as the condition/equation above holds, for some given numbers/values for $r_z,r_k$ and $\gamma$ all of which are positive real numbers.

Can someone help on how I can calculate this, ideally in MATLAB ?

Answer 1

Since $\phi$ is a scalar between $0$ and $1$, the easiest method for finding a root is bisection. If you cannot calculate the expectation of the nonlinear function $$f_\phi(\theta) = \left(\phi r_z +(1-\phi)(r_k+\theta)\right)^{1-\gamma}(r_k+\theta-r_z)$$ in terms of $\phi$ analytically, you can use quadrature to approximate. For the first variant, this amounts to $$E[f_\phi(\theta)] = \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-\frac{t^2}{2\sigma^2}} f_\phi(t)\,dt \approx \sum_{i=1}^N \frac{w_i}{\sqrt{\pi}} f_\phi(\sqrt2 \sigma t_i),$$ where $t_i$ and $w_i$ are the Gauss-Hermite quadrature nodes and weights of order $N$. If $\theta$ is log-normally distributed, then $\theta=e^\beta$, where $\beta$ is normally distributed, so you can just replace $t_i$ by $s_i:=e^{t_i}$ in the above to get a quadrature rule (with the same weights) for the log-normal variant.

This procedure is straightforward to implement in Matlab; you can download functions to calculate the quadrature weights and nodes here or here.