Present Value with Stochastic Cash Flows
Overview
Stochastic cash flows are cash flows where the amount (and possibly timing) of the cash flow is not known ahead of
time. As a general rule, the discount rate can be taken to be the expected return of the cash flow. (see below)
Expected Return and the Discount Rate
Starting with todays price {% P_0 %}, we form the following equation.
{% \mathbb{E}(P_t) = P_0 e^{rt} %}
Note, at this point this equation is a tautology. That is, we have not yet defined the rate {% r %}
found in the equation. It just asserts that such a {% r %} exists. The following computes
what {% r %} has to be equal to.
{% e^{rt} = \frac{\mathbb{E}(P_t)}{P_0} %}
In order to quote the rate of return in annual terms, we assume that the time period is equal to 1 year,
or {% t=1 %}.
Then we get
{% r = log(\frac{\mathbb{E}(P_t)}{P_0}) \approx \frac{\mathbb{E}(P_t)}{P_0} %}
