1. Risk Analysis “Risk” generally refers to outcomes that reduce return on an investment

2. Meaning of Uncertainty No basis for estimating probability of possible outcomes –No experiential data –Most often associated with innovations

3. Meaning of Risk Potential for revenue to be lower and expenditures to be higher than “expected” when investment was made. Measured by variation in these factors Causes –Physical risk – physical loss of growing stock due to “acts of God,” or uncontrollable acts of man –Market risk – changes in markets that cause variation in revenues and expenses –Financial risk – changes in interest rates and associated opportunity cost

4. How to Reduce Overall Risk Risk and rate of return are positively correlated Investors should hold a PORTFOLIO of investment options –Blend investments to maximize the rate of return for the level of risk an investor is willing to tolerate This is DIVERSIFICATION

5. Probability Distribution Relationship between possible outcomes and the percentage of the time that a given outcome will be realized if the process generating the outcomes is repeated 100’s of times.

6. Mean = $2,000 Probability of 25% Mean = $6,000 Probability of 50% Mean = $10,000 Probability of 25%

7. Expected Revenue E(R) = ∑ P m x R m n m = 1 Where, m = index of possible outcomes n = total number of possible outcomes P = probability of m th outcome R = possible revenue for m th outcome

8. Expected revenue of example E(R) = 0.25 x $2,000 + 0.5 x $6,000 + 0.25 x $10,000 = $6,000 Call this investment “risky”

9. Risk aversion Assume an investment with $6,000 future revenue that is guaranteed by US Government –E(R) = $6,000 x 1.0 = $6,000 –Call this investment “guaranteed” If an investor prefers the $6,000 guaranteed in the example above, to the $6,000 risky investment in the previous example they are “risk averse” –Have no tolerance for risk

10. Risk aversion If an investor is indifferent between the guaranteed $6,000 and the risky $6,000 then they are “risk neutral” If an investor prefers the risky $6,000 to the guaranteed $6,000 then they are “risk seekers” –They are willing to take a chance that they will get a return greater than $6,000

11. Risk-Return Relationship Because all investors have some risk aversion investment market must reward investors for taking higher risk by offering a higher rate of return in proportion to the risk associated with an investment

12. Variation Sum of squared deviations from expected revenue weighted by probability of outcome Variance = σ 2 = ∑ [R m – E(R)] 2 x P m Standard deviation = (σ 2 ) 1/2 n m=1

13. Example DeviationDeviation 2 x Probability $2,000 - $6,000 = -$4,000$16,000,000x.25 = $4,000,000 $6,000 - $6,000 = $0$0 x.50 = $0 $10,000 - $6,000 = $4,000 $16,000,000x.25 = $4,000,000 Variance =$8,000,000 Standard deviation =$2,828

14. Comparing standard deviations Risk is higher if standard deviation is higher, but If expected values vary can’t compare their variation Need measure of relative risk, –Coefficient of variation = Standard deviation / E(R) For example: $2,828/$6,000 = 0.47 –Standard deviation is 47% of expected value

15. Risk-free rate of return Risk-free rate assumption – r f = 3% used to be the standard, but today it is - 0.5% in real terms Correct PV is (risk-free revenue)/(1+ r f ) n Example $6,000/(1.03) 5 = $5,176 Buy U.S. Treasury bond for $5,176, get $6,000 at maturity in 5 years

16. Real Risk-Free Interest Rate 10-Yr. Treas. Sec., 3-Yr. Moving Average

17. Wall Street Journal, April 9, 2016, p. 1 This is r f

18. Risk Averse Investors Will only pay less than $5,176 for $6,000 5-year bond, i.e. –Discount $6,000 bond at rate of >3% –(risky E(R))/(1+RADR) n < (risk-free E(R)/(1+r f ) n Where r f is real risk-free interest rate How do we find risk-adjusted discount rate (RADR)? –Get investor’s certainty-equivalent (CE) –Example, what risk-free return is analogous to $6,000

19. Certainty Equivalent (CE) The dollar amount that, if received with certainty, would give the investor the same satisfaction or “utility” as the risky revenue with expected revenue E(R)

20. “Back Into” RDAR Correct present value = CE/(1+r f ) n = PV CE = (E(R))/(1+RADR) n (1+RADR) n = E(R)/PV CE RADR = (E(R)/PV CE ) 1/n -1 Example, CE = $4,000 Correct PV = $4,000/(1.03) 5 = $3,450 RADR = ($6,000/$3,450) 1/5 – 1 = 11.7%

21. Risk Premium (k) k = RADR – r f =11.7% - 3% = 8.7% No “general rule” about what risk premium is, or should be

22. Relative Measure of Risk Certainty-equivalent ratio, c r c r = CE / E(R) Example, c r = $4,000/$6,000 = 0.67 k = (1+r f )/(c r 1/n ) – (1+r f ) = 1.03/0.67 0.20 – 1.03 = 8.6% See Table 10-2 –Higher risk equates to smaller c r

23. Relative Measure of Risk See Table 10-2 –Higher risk equates to smaller c r –Risk premiums decrease with longer payoff periods If know an investors CE don’t need RADR

24. Rule of Thumb Estimate probability histograms for all cash flows Then ask the investor to state certainty- equivalents of these. Discount certainty-equivalents with a risk- free discount rate to arrive at a correct NPV for this particular investor.