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Understanding Risk Analysis and Risk-Return Relationship

Dive into the complexities of risk analysis, the concepts of risk and uncertainty, and the risk-return relationship in investments. Learn about risk aversion, risk-free rates, and calculating risk premiums effectively.

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Understanding Risk Analysis and Risk-Return Relationship

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Presentation Transcript

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

  2. 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 costs • Financial risk – changes in interest rates and associated opportunity cost

  3. Meaning of Uncertainty • No basis for estimating probability of possible outcomes • No experiential data

  4. 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.

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

  6. Expected Revenue • EVR = E(R) = ∑ PmRm N m Where, m = index of possible outcomes N = total number of possible outcomes P = probability of mth outcome R = possible revenues

  7. 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”

  8. 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

  9. 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

  10. 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

  11. Variation • Sum of squared deviations from expected revenue weighted by probability of outcome • Variance = σ2 = ∑ [Rm – E(R)]2 Pm • Standard deviation = (σ2 )1/2 N m=1

  12. Example

  13. 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

  14. Risk-free rate of return • Risk-free rate assumption – rf = 3% is still a valid assumption • Correct PV is (risk-free revenue)/(1+ rf)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

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

  16. 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+rf)n • How do we find risk-adjusted discount rate (RDAR)? • Get investor’s certainty-equivalent (CE) • Example, what risk-free return is analogous to $6,000

  17. “Back Into” RDAR • Correct present value = CE/(1+rf)n = PVCE = (E(R))/(1+RADR)n (1+RADR)n = E(R)/PVCE RADR = (E(R)/PVCE )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%

  18. Risk Premium • k = RADR –rf =11.7% - 3% = 8.7% • No “general rule” about what risk premium is or should be

  19. Relative Measure of Risk • Certainty-equivalent ratio, cr cr =CE/E(R) Example, cr =$4,000/$6,000 = 0.67 k = (1+rf)/(cr1/n) – (1+rf) = 1.03/0.670.20 – 1.03 = 8.6% • See Table 10-2 • Higher risk equates to smaller cr

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

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