Department of Management Science & Engineering

Number 4/Spring 1999

Investment Science Newsletter

By Professor David G. Luenberger

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This is the fourth newsletter describing events in the Investment Science program at Stanford. Each issue reports on the nature of recent research and projects. There is also a short article of substance related to Investment Science concepts. The article last month on the two-rate method of discounting generated a number of favorable comments from readers. We appreciate such feedback, for it helps foster the primary objective of this newsletter which is to keep in touch after the two-day course. The article in this issue is "Risk-Time Discounting." This describes a most general method of discounting, capable of handling options and other decision-oriented structures in projects.

Short Course News

The short course continues to evolve. New examples are added each time it is offered and the (somewhat) complex mathematics that is standard in this field is being replaced by simple arguments that relate to familiar business concepts. As an example, the esoteric theory of valuation, which is usually taught in classrooms in terms of partial differential equations or continuous matching of securities, is now presented in the course in terms of discounting. It is also shown how proper valuation relates to ROE (return on equity) and to firm growth. There is also a stronger emphasis on the structure of projects-the various types of options and how they can be treated easily.

April and September Investment Science for Industry will be offered at Stanford April 22-23 and September 16-17. Detailed information can be obtained from the web site:

For detailed information about Investment Science for Industry Course

Past participants are encouraged to bring this course to the attention of their colleagues. We have had several such referrals in the past.

Projects and Research

In addition to the projects that we have discussed in previous newsletters, a new initiative has been generated this year. A special project course (for Stanford students) will be offered in the Spring quarter. This course is available to a select number of students who have completed two quarters of Investment Science in the department. The course is being taught primarily by Robert Luenberger. The course will use real examples from industry, so if your company would like to have a group of 5 students work on a short (3 month) investment project, please contact me at Stanford, or Rob at Onward ( Of course we are available for more comprehensive projects, but this special opportunity may be a good way to get Investment Science started in your company.

Our current research and outside applications include involve banking decisions, environmental compliance issues in electric power, electric power trading, contract design, general project evaluation, capital investments in cyclic industries (such as the petrochemical industry), and the strategic deployment of Information Technology. New applications are being discovered every month, and of course many of these ideas have evolved from participants' discussions at the short course.

Risk-Time Discounting

The last issue of this newsletter discussed the "two-rate" method of discounting, which is a simple way of modifying traditional discounted cash flow analysis to properly account for degrees of uncertainty. This method seems to appeal to many people.

Now, we shall discuss an even more powerful method of discounting: Risk-Time Discounting, (referred to as the Two Step Method in the last newsletter). This method is still simple to understand and simple to implement. In some respects it builds on the two-rate method, but it can handle complex cash flows that are subject to saturation, curtailment, expansions, options, and other nonlinearities.

The basic idea of the method is simple: discount all component cash flows first for risk and then for time; hence the name, "risk-time discounting".

To describe the method, let us review the two basic principles of discounting.

1. Market Uncertainty

Suppose a cash flow is a perfect market cash flow; that is, it is proportional to the value of a stock. For example, the revenue of a new product one year from now might be estimated to be equal to that of 1,000 shares of the company stock in one year--reflecting the fact that if the market is strong for the company in general it is likely to be strong for the new product as well. The present value of such a market cash flow is equal to current value of the market variable. In the example that means that the cash flow has a present value equal to the current value of 1,000 shares of stock. (The reason is that the stock and the project are perfect substitutes in terms of what cash they generate in a year. Therefore, the current values of this cash flow must be equal.)

Now translate this into discounting. If the company stock is expected to increase at some rate, say 14%, then we discount the expected value of the cash flow by this same 14%. For example if the current value of the stock is $100, we expect it will be $114 at the end of the year. If we expect the project cash flow to be equal to 1,000 shares, that will be $114,000. The present value of the cash flow is $100,000, obtained by discounting back by the 14% that it rose.

2. Private Uncertainty

Suppose now that the cash flow is a private cash flow, which means that it is independent of any variable in the market. For example, the cash flow might be the cost of repairs to machinery, which depends only on whether or not the machinery breaks down. The expected values of such cash flows should be discounted at the risk-free rate. Hence if the risk-free rate is 5%, and the expected value of a private cash flow in one year is $100, then the present value is $100/1.05 = $95.24 .

These two types of uncertainty and the related discounting principles are the basis for all discounting. All cash flows can be represented as combinations of these two types.

Now let us describe the risk-time discounting procedure. We first discount all cash flow processes for risk. For a market variable this means that we discount by its growth rate minus the risk-free rate. For the stock that increases by 14% per year when the risk-free rate is 5%, we discount by 14%- 5% = 9%. For a private variable we discount by the risk-free rate, which is 5% in our example--no risk-adjustment is required. Then, after necessary risk discounts are made, we find the aggregate resulting cash flow and discount that for time, using the risk-free rate.

It may seem like a belabored process to discount the market cash flows in two steps: 9% for risk followed by 5% for time. But the advantage is that then the result is valid for cash flows subject to options, saturation, expansions, and various other nonlinearities, which are the meat of modern project evaluation, representing a large part of a project's value.

The reason this risk-time technique works is that we discount the underlying market process for risk before it enters our spreadsheet as a cash flow. It is discounted before being subject to options and other distortions inherent in our project. It is discounted for market risk "out in the market" not in our project. It is a market uncertainty, not a private one, and should be discounted in the market, not in our private project. This explanation is of course merely suggestive; the real explanation is based on some fairly deep mathematics. But this simple explanation is easy to remember. Discount market uncertainty for risk out in the market, not in your private project.

Let us see how to do this in a simple example. Consider a call option on a stock. (An example that is similar to one worked several ways in the text Investment Science.)

Suppose the stock has a current price of $62.00, is expected to grow at 20%, and has a volatility of 20%. The risk-free rate is 5%. The option has a strike price of $60 and a time to maturity of 5 months. This is a nonlinear payoff that might be typical of projects as well. In this case the payoff is positive only if the stock ends up higher than $60, in which case we get the surplus over $60.

The value of the option can be found by risk-time discounting. In this case the only cash flow occurs at the end of 5 months and is a pure market uncertainty. We discount the market variable "in the market" before it gets to the option. The expected growth rate of the stock is 20% and the risk-free rate is 5%; hence we discount the stock by 20%-5% =15%. The remaining expected growth rate of the stock is then equal to the risk-free rate (of 5%). Now, we can use simulation or the lattice method to carry out the discounting. In simulation, we would run many simulations of the stock (after it is discounted for risk down to a 5% expected growth rate). We would find the value of the payoff (cash flow) in each simulation run and discount that for time (at 5%). The average of all these discounted values (averaged over the various simulation runs) is the value of the option.

Remember the principle: Discount for risk and then for time. But discount market risk outside-in the market-before it encounters non-market characteristics such as project options.