Download Risk Analysis - Purdue Agriculture

Risk Analysis
“Risk” generally refers to
outcomes that reduce return on an
investment
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
Meaning of Uncertainty
• No basis for estimating probability of
possible outcomes
– No experiential data
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.
Probability Distribution
Mean = $6,000
Probability of 50%
0.14
0.12
0.08
Mean = $2,000
Probability of 25%
Mean = $10,000
Probability of 25%
0.06
0.04
0.02
Future Revenue
$1
0,0
00
$1
1,0
00
$1
2,0
00
$9
,00
0
$8
,00
0
$7
,00
0
$6
,00
0
$5
,00
0
$4
,00
0
$3
,00
0
$2
,00
0
0
$1
,00
0
Probability
0.1
Expected Revenue
N
•
EVR = E(R) =
∑P R
m
m
Where, m = index of possible outcomes
N = total number of possible outcomes
P = probability of mth outcome
R = possible revenues
m
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”
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 previous example, to the $6,000 guaranteed,
the example above, then they are “risk averse”
– Have no tolerance for risk
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
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
Variation
• Sum of squared deviations from expected
revenue weighted by probability of
outcome
N
• Variance = σ2 =
∑ [R
2P
–
E(R)]
m
m
m=1
• Standard deviation = (σ2 )1/2
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
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
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
Real Risk-Free Interest Rate
10-Yr. Treas. Sec.,
3-Yr. Moving Average
12.00
10.00
8.00
6.00
4.00
2.00
-4.00
-6.00
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
1966
-2.00
1964
0.00
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
“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%
Risk Premium
• k = RADR –rf
=11.7% - 3% = 8.7%
• No “general rule” about what risk premium
is or should be
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
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