**Prof.Dr. Ömer ARIÖZ**

İnşaat Mühendisliği Bölüm Başkanı

Head of the Department of Civil Engineering

Hasan Kalyoncu Üniversitesi

Hasan Kalyoncu University

**Prof. Dr. Kemal Yıldırım**

Anadolu Üniversitesi

Anadolu University

**TIME-UNCERTAINTY RELATION IN DECISION- MAKING PROCESS**

Companies plan on many investment projects to increase their turnover and profits and do analysis as to for which price their products will be sold and how much profit these prices will yield. Likewise, when the companies plan their investments based on the income they will gain in the future, the concept of time becomes quite important together with uncertainty. Therefore, the concept of time should be combined with uncertainty and involved in the decision-making process. The concept of time is involved in the decision-making process by means of calculating “net present value” in certainty cases. As for uncertainty cases, the concept of time can be associated with decision-making through two methods in particular: “risk-adjusted discount approach” and “certainty-equivalent approach”. Expected utility and discounted utility models considerably contribute to the combination of time and uncertainty.

**1. Net Present Value**

The objective of the companies is to maximize the “net present value” (NPV) of the profit they expect in the future. Therefore, when a company wants to compare many investment projects with uncertain profit to the sales amount related to these, one of the most common methods is to calculate the expected “net present value” for each project. In order to take the risk and uncertainty into account, planners will assign probability or probability frequency. Making the value expected in the future “net present value” is possible through taking proper discount rates as well as time into account. Therefore, while calculating “net present value”, the future level of the profits, sales and costs that are certain must be discounted by taking a certain rate and time into consideration. In an uncertainty case, the value expected for different periods of time must be found through the use of the subjective probabilities of the events. In a certainty case, “net present value” of the profit expected to be gained in the future can be calculated for n year(s) as follows:

Assume that the expected profit amounts, which a cement producer gets for 4 years from any product, are 12, 15 and 18 million TL respectively and the discount rate is 10%. The “net present value” of the expected profit that the company will gain for 4 years can be calculated as follows:

As for an investment project that needs to be determined, the initial investment cost must be included as well unlike the formula above. “The net present value” of the expected returns of an investment project for each following year is calculated as follows:

Assume that a cement producer plans to make an investment through purchasing a plant in two different regions abroad, which are A and B and wants to get into the market, and investment costs are 120 and 150 million TL respectively in the regions A and B, and the discount rate is 10%. Assume that the investment to be made in the region A will return for five years 24, 28, 34, 38, 42 million TL, and in the region B will return for five years 38, 40, 42, 45, 48 million TL. In this case, the net present value of the two alternatives that the cement producer has regarding the investments to be made in the region A and B can be calculated as follows;

In this example, the cement producer gains 2,537 million TL abroad by purchasing a plant in region A and 9,698 million TL by making investment through purchase of a plant in region B as net present value (NPV). When an executive or investor with limited resources would like to make a choice between alternative projects, he or she should choose the alternative with the highest NPV, that is, the alternative of making investment in the region B.

**2. Risk-adjusted Discount Approach**

In the equation [2] given above, the “net present value” of an investment project was calculated by using the discount rate and time together. However, as discount rates in uncertainty cases are not certain, this equation will not accurately represent certainty case under uncertainty. In uncertainty cases, “net present value” can be calculated by means of “risk-adjusted discount rate”. However, since time and risk are variables different from each other, combining the effects of these two variables over risk-adjusted discount rate in [K] value requires a special assumption with regard to the updated relation between the effects of time and risk on present value. According to this, the equation [2] given above can be written this time as follows:

The symbols here mean as follows; NPV: net present value

K: risk-adjusted discount rate

Rt: expected return in t time

t : time

C0: initial investment cost.

Here, K is “risk-adjusted discount rate” and it can be calculated through the sum of “risk-free discount rate” and “discount risk premium”. Graph 1 shows the risk-return/ trade-off curves of the three risk-averse companies with different reactions towards risk. On these curves, the risk (σ) is shown on the horizontal axis, and the rate of return is shown on the vertical axis. As the risk-free discount rate is 5% here, the curves U1, U2 and U3 are drawn with an intersection at 5%. Since the Company-3 is the risk-averse at the highest level, it needs the highest rate of return at the given level of risk. (σ*). When the “discount risk premium” values are accepted as 10% for the Company-1, 15% for the Company-2 and 20% for the Company-3 at the risk level σ* in this example, the “risk-adjusted discount rate” values belonging to these three different companies are calculated as K1=%15, K2=%20 and K3=%25 respectively.

Assuming that these companies assess an investment project which has an initial cost of 120 million TL and is expected to return 100 million TL for 5 years, the “net present value”s of the companies for this investment project can be calculated as follows:

In this example, it is less likely that the Company-3, the risk-adverse at a higher level, will undertake this investment project compared to the Company-1, the risk adverse at a lower level because the “net present value” of the returns expected from the investment projects are calculated for Company-1, Company-2 and Company-3 as 215, 179 and 149 million TL respectively. Therefore, as the Company-1 is the risk-adverse at a lower level among the others, it will get a higher “net present value” and be more likely to undertake this project.

**3. Certainty-equivalent Approach**

In an uncertainty case, the “net present value” can be calculated through above-mentioned “riskadjusted discount approach” However, the discount rate in the denominator in [3] equation given in this approach, is comprised of the modification of the risk-adjusted discount rate (K). As for the certainty-equivalent approach, the numerator is modified as follows without the denominator changing in the equation at risk-free level.

Certainty-equivalent coefficient, (α) changes from 0 to 1 based on the investor’s consideration and higher values of the coefficient α mean that the investor considers the risk smaller. If the coefficient is zero, which is an extreme value, this means that the investor considers the project very risky, which shows that it is slightly possible that the expected return will realize. In the event that the certainty-equivalent coefficient is 1, the other extreme value, this shows that the investor considers the project risk-free.

Assume that the certainty-equivalent coefficient for the investment of the three different projects regarding the products which a cement producer wishes to produce are α1=0,1, α2=0,5 and α3=0,9 respectively, and the returns of these investments for 5 years will be 30, 15 and 5 million TL respectively and the initial investment cost is 10 million TL. In the event that the risk-free discount rate is 10%, the “net present value”s can be calculated for these projects as follows:

Given the fact that NPV2 > NPV3 > NPV1 , it seems more appropriate to make investment in Project-2 with the “net present value” of 18.43 million TL. Certaintyequivalent coefficient can also be shown as follows:

To explain this with an example, if an investor or executive, considers a certain return of 9 million TL equivalent to a risky return of 10 million TL in terms of utility, the certaintyequivalent coefficient can be calculated as α= =0,9. Both risk-adjusted discount and certainty-equivalent approach are formed following the subjective assessment of the risk by the investor. However, the certainty-equivalent approach is preferred more as it combines the investor’s attitude towards the risk with the net present value more clearly.