A Sunk Costs and the Value of Waiting

Suppose a project yields a profit of π₁ in year 1. In the second period it yields either no profit, with a probability of 1 - p, or a profit of π_H with probability p. [19] The rate of interest is r. Suppose there is a fixed cost of investing, in the first period of operation, of S. This cannot be recovered if, depending on the outcome in period 2, it is decided not to continue in production. But importantly the sunk cost may actually be higher than S because investing in the first period involves giving up the 'option value' of waiting to obtain more information. This can be seen as follows. The expected net present value (NPV) of the project is:

Hence the conventional NPV criterion suggests that investment is worthwhile if:

First, compare the fixed cost with an annual fee. The fee must be compared, not with the fixed cost, but with the equivalent annual fee, C, defined by . Hence:

Suppose there is an annual fee of F. The expected NPV in this case, in view of the fact that if the lower (zero) profit outcome arises, no production takes place and hence no fee is paid, is:

Starting the project in the first period is worthwhile so long as:

The maximum fixed cost, S_max, and its associated equivalent annual amount, C_max, under which investment in the project in the first period is not worthwhile, is obtained by replacing the inequality in (A.2) with an equality. This may be compared with the maximum fee, F_max, given by replacing the inequality in (A.5) with an equality, such that the project with a fee becomes not worthwhile. It can be seen that F_max > C_max if:

This clearly holds, since p < 1. [20] Thus the standard NPV expression in (A.1) does not provide an appropriate evaluation criterion. It is not only the fixed cost, S, that cannot be recovered after being incurred in period 1. Starting the project in the first period foregoes the option value of waiting. Hence it can be said that the sunk cost – the amount which cannot be recovered at any stage after starting the project – is greater than the fixed cost by an amount that reflects the option value.

Instead of having the choice of starting the project in the first period, or not at all, suppose instead that the investor waits until the second period before making a decision. If circumstances which give rise to zero profit arise, no investment takes place and no fixed cost needs to be lost. The expected NPV (evaluated in period 1) in this ‘waiting case' is thus simply:

Suppose that, from (A.2), the fixed cost is , so that it is not worth investing in the first period. Substituting this value in (A.7) gives:

Hence if π_H > π₁/, it is worth waiting and then investing in the project in the second period. In this case the value of waiting is E (NPV)_W since, by assumption, the value of acting in the first period, E (NPV), is zero. But in general the value of the option to wait, or the option value, V , is:

This option value need not, of course, always be positive. Similarly, the true sunk cost, S^*, exceeds the fixed cost, S, since it must allow for the foregone option value. Hence it is:

          S* = S + V           (A.10)