Risk Analysis

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 100s 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 s2 ? Rm E(R)2 Pm
• Standard deviation (s2 )1/2

N
m1
12
Example
Deviation Deviation2 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
13
Comparing standard deviations

• Risk is higher if standard deviation is higher,
  but
• If expected values vary cant 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 gt3
• (risky E(R))/(1RADR)n lt (risk-free E(R)/(1rf)n
• How do we find risk-adjusted discount rate
  (RDAR)?
• Get investors certainty-equivalent (CE)
• Example, what risk-free return is analogous to
  6,000

17
Back Into RDAR

• Correct present value
• CE/(1rf)n PVCE (E(R))/(1RADR)n
• (1RADR)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 (1rf)/(cr1/n) (1rf)
• 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 dont need RADR