ABSTRACT
Our analysis in Chapter 9 shows that the market value of a firm’s equity is comprised of two components. The first is called the recursion value of equity and is the present value of the future cash flows that the firm expects to earn, or equivalently the present value of the dividends that the firm expects to pay over its lifetime (as in §5-2 of Chapter 5), given that it is indefinitely constrained to operate within its existing investment opportunity set. There is, however, a second component of equity value, namely, the adaptation (or real option) value of equity. This is the option value that arises out of a firm’s ability to change its existing investment opportunity set by (for example) fundamentally changing the nature of its operating activities. Now suppose one follows the analysis of Chapter 9 in assuming that the recursion value of equity η(t) evolves in terms of a continuous-time branching process, namely,
dη(t)
dt = rη(t)+√η(t)dq(t)
dt
where r is the cost of equity capital and dq(t)/dt is a white noise process with variance parameter ζ 2. It will be recalled (as in §9-6) that under this process the recursion value of equity will grow at a rate equal to the cost of equity capital r, but that there will also be stochastic perturbations in the growth rate arising from the white noise process dq(t)/dt. Moreover, the variance associated with the stochastic component, Vart[
√ η(t)dq(t)] = ηζ 2dt,
will grow in line with the recursion value itself. This reflects the commonly held belief that the variance associated with increments in the recursion value of equity will become larger as the recursion value grows in magnitude.
