Methods of incorporating risk factors into the design...

Methods for integrating risk factors into the design of discounting the expected benefits

Three variants of including risk in the formula of discounting of benefits can be recognized as correct. The two most popular methods are shown in Fig. 12.4.

Two variants of risk reflection in the formula for discounting future benefits

Fig. 12.4. Two options for reflecting risk in the discount formula for future benefits

In academic literature and in practice, advantage is given to method 1 - through adjusting the discount rate. Traditionally, systematic risks of the asset are revealed and the risk premium (risk premium) is justified.

The second risk-reflecting method is the risk-free equivalent method (or the risk-neutral cash flow equivalent to the guaranteed cash flow -

The method of risk-free equivalent assumes the existence of such cash flow streams, which will be, on the one hand, risk-free (that is, they are received by the investor with a 100% probability), and on the other - equivalent to the risky flows predicted for the investment object .

The third method is combined, when systematic risks are included in the discount rate in the form of a premium for systematic risks, and the remaining specific risks (unique, associated with this investment option) are built into cash flows, correcting them in a smaller direction.

Traditionally, the cash flow adjustment is realized using the indifference curve of the investor's utility:

where - the expected cash flow in the year t; - the value of the cash flow that the investor agrees to receive guaranteed in each year t instead of non-guaranteed cash flows from the investment asset; α - coefficient of equivalence, reflecting the attitude to the risk of the entity making the investment decision.

The more the investor does not accept the risk, the closer to zero the alpha value. Practical application of the method can be implemented for a series of projects of the company, among which are already adopted and implemented. If the cost of capital and the risk-free yield k f are known, then the analyst can estimate the equivalence coefficient.

Example 1

Consider a one-year medium-risk project with the expected net cash flow X = 5 million rubles. The required profitability of shareholders, taking into account the average risk is, for example, 15%, and the risk-free yield is 10%. The company uses only its own capital. The current estimate of the future cash flow is calculated using the formula

With this calculation, non-guaranteed cash flow appears. We are interested in what value of the equivalence coefficient this current estimate will be equal to the guaranteed value:

With we get a = 0.957. From the point of view of the owner of the capital, there is no difference in obtaining a non-guaranteed net cash flow for the project in the amount of 5 million rubles. or guaranteed in the amount of 4.785 million (5 ∙ 0.957).

Example 2

Consider the correspondence between the adjustments for the risk of the structure in cash flows and in the discount rate (Table 12.1)

Table 12.1

Comparison of the two risk reflection methods


Projected cash flow for the project

Estimated Guaranteed Equivalent Flow ( CECF)






4 ∙ (1.07/1.13) = 3.79



5 • (1.07/1.13) 2 = 4.48



5 • (1.07/1.13) 3 = 4.23

NPV (13%) = 0.92

PV (7%) = 0.92 (including rounding)

The risk-free return is 1%, the risk-adjusted rate of return is 13%. Methods give equivalent project estimates

Another option for implementing the risk-free equivalent method is using the CAPM model (the CAPM model is described in more detail in Section IV of the textbook). The method of risk-free equivalent with CAPM assumes a portfolio view of the asset's risk (through the introduction of flux correlation estimates but an investment project with market yield). The basic formulas of the method for the simplest situation of perpetual benefits of investment, obtained as an annuity, are given below. The return on investment (as the ratio of the annual cash flow to the capital according to the market valuation) is equated to the formula CAPM (in more detail the formula CAPM and the assumptions of the model are considered in Section IV of the textbook)

model CAPM where i is the company.

Hence , i.e. the expected return on investment depends on the covariance of the asset's cash flows and market return; - annual flows for the investment decision; - invested capital.

In the risk-free equivalent method, you should distinguish between the beta-ratio of return () and the beta-flow coefficient (beta cash flow), which shows the sensitivity of the capital received from investing cash flows to market changes (shifts in market returns):

- beta coefficient of cash flow.

Since the covariance between cash flow and market yield is measured in monetary units, the beta coefficient shows how the return of a money amount b to the market portfolio will compensate for the systematic risk - this is the investment fee in the market ().

There is a parity of two methods of applying CAPM in the cash flow and in the discount rate:

For an investment project with an estimated value of the beta yield taking into account investment costs in the amount of , which form the invested capital, the risk premium will be

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