Department of Management Science & Engineering

Number 6/Spring 2000

Investment Science Newsletter

By Professor David G. Luenberger

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This is the sixth newsletter describing events in the Investment Science program at Stanford. There is a great deal of activity in the area and lots of new and exciting ideas are emerging. Our focus is primarily to pursue four areas: (1) Determine proper methods to evaluate complex cash flows, (2) Illuminate the unique features of commonly encountered options and decisions, (3) Develop effective computational methods for evaluation and strategy determination, and (4) Promote the use of advanced concepts in real firms by relating the new methods to standard accounting procedures and intuitive concepts. We have made progress in each of these areas, and the continued interaction with specific firms has been very helpful.

This issue of the Newsletter focuses on something in area (2) mentioned above. In the technical discussion below (The Option to Wait) we look at the one of the most basic of operational options. It has long been understood that the option to wait, or to delay, can have significant value because new information may become available or circumstances may change. However, as shown here, the option to wait may have substantial value even when all information is known and circumstances change only in a predictable and steady manner. One should always consider the option to wait.

The Website

As announced in Newsletter 4, the Investment Science web site is up. It is located at There has been a good deal of activity at the site, and we have plans to add more to it. For one thing, you will find copies of all past issues of this Newsletter at the site and some very useful software (see below). Your comments on the site are welcome.


An Excel add-in for drawing and evaluating triangular lattices was developed by José Carlos García Franco. We have used a version of this in the short course and we use it often to carry out simple analyses of all sorts of investment science situations, especially those related to real options, and to work with examples in the text, Investment Science. The newest version of the software is called LatticeMaker and it is located on the investment science web site available for (free) downloading.

Short Course News

Investment Science for Industry was given at a private firm in August, and at Stanford in September. Because the September session was held in the summer, we were able to use a larger room and accommodate a much larger group of attendees than in the past. It is satisfying to see the course grow, for it gives us the opportunity to interact with many other individuals who are interested in learning modern methods and using them in real situations. The next course will be April 27-28. Please refer colleagues to it (through the web site or to the web site mentioned above).

The Option to Wait

One of the most interesting results of real options theory concerns the option to wait. Perhaps surprisingly, it is sometimes best to delay a project even though it currently has positive present value.

This phenomenon occurs in situations of capital investment, switching of operations, and other long-term projects.

A nice example was pointed out to me on a recent trip to New York. There are a number of parking lots in Manhattan on which large income-producing structures (office buildings for example) could be built. The net present value of conversion from a parking lot to a building is positive, and there is no legal restriction against the process; but the lots are not being converted. It is better to wait until next year. In fact, next year it will be better to wait another year. This can go on year after year. This phenomenon has been studied in the real options literature under certain assumptions regarding the growth rates of economic variables and the uncertainty around those rates. However, the advantage of waiting does not necessarily depend on the presence of uncertainty. Sometimes it pays to wait even when all future prices are known; and this can be illustrated with a simple analysis.

To focus our discussion let us think in terms of the parking lot situation. Suppose that the annual profit of the parking lot is currently A and that this grows at a rate each year. Suppose also that a structure on the site would produce a rental profit of B and this profit increases at a rate of each year. For simplicity let us assume that conversion can be carried out immediately at a cost of S but this cost increases at a rate of for every year the project is delayed. Specific values we may wish to try are A = $1 million, B = $10 million, S= $195 million, and the rates

where is the risk-free rate at which to discount future cash flows.

Note first that if the parking lot is never converted, it will generate the revenue stream

The present value of this kind of stream (where the terms are discounted by

can be evaluated by a standard formula (the Gordon formula) and its value is


For the specific values we gave, this is =$22 million. This is the present value if the parking lot is never converted.

If a structure is immediately erected on the lot, the revenue stream will be

and so forth. The value of this stream is evaluated the same way as the stream from the parking lot and is

For our specific values this is =$25 million. The value gained by immediate conversion is

which for our example is $3 million.

Conversion has a higher present value than non-conversion. In fact, we see that

so the present value of conversion is about 13% greater than that of the status quo. Conventional analysis would dictate that the building should be converted.

The key mathematics of this example is simplest if we consider the benefit of waiting one year to convert. Suppose, for example, that we wait one year instead of converting at time 0. The difference in present value by this delay is given by the simple formula

This formula is easy to understand. A delay will give us A instead of B in the initial period, so the net change is A - B. However, we also delay the conversion cost. In discounted terms the cost would have been S but with the delay it is

The difference is

which we enter as a negative since it is a cost.

In our example, this produces = $8.727 million-a huge difference considering that the value of converting earlier is only $3 million.

Pursuing this idea we can find the relatively simple general formula for the difference between switching at k and waiting until k + 1. The same logic gives

This formula could be used at each time to determine if it is beneficial to convert or wait. We will come back to this later, since it helps explain the numerical results we shall obtain. It is possible to compute the various defined as the present value for conversion at time k. They are given by the (somewhat messy but still manageable) formula

A graph of these values is shown below.

The maximum possible present value is $73.883 million and is achieved by converting the parking lot to an income-producing building in year 14.

At year 14, the present value of going forward with the parking lot (using year 14 as the starting point) is $43.56 million. The corresponding value for going forward with the conversion is $240.58 million. Hence, conversion is carried out when the present value of that project is 5.52 as great as the status quo. At time 0 we found that conversion was worth 13% more than the status quo; that was not nearly enough; we need a ratio of 5.52 to justify immediate conversion.

The reason that one should wait to convert is that waiting postpones cost, and since in our example = 0, this postponement is valuable because of discounting. This savings initially outweighs the increased cash flow due to the structure. However, as time k increases, the rent increases and the cost advantage to delay becomes small compared to the difference in profit. This is clearly shown in the equation for which shows that the term involving B will for large k cause to become negative.

It is fun to consider the case where . Then if it is advantageous to wait at time 0 it is advantageous to wait every year---forever.

The conclusions of this example are of course dependent of the values of the parameters. And the advantage of waiting rapidly disappears if the total number of years considered is not infinity. Hence, projects in fast-moving industries may have short lives and should not be delayed. Competition is another factor that may tend to favor immediate action rather than delay.

The whole issue is more complex if there is uncertainty with respect to, say, the profits. In this case there may be an advantage to waiting, simply to see what these profits will be. This, however, is another story. The main point of the example presented here is that it may be valuable to wait even if the present value of current action is positive.