Transcript VNM and Risk Aversion

VNM utility and Risk Aversion
 The desire of investors to avoid risk, that
is variations in the value of their portfolio
of holdings or to smooth their
consumption across states of nature is a
primary motive for financial contracting
 Now we use the VNM framework and
place some restrictions on it to capture
some elements of risk
What does the term risk
aversion mean about an agent’s
utility function?
 Consider a financial contract where the
potential investor either receives an
amount h with probability pr = ½ or
must pay an amount h with probability
pr = ½
We would not accept this offer
 The most basic sense of risk aversion
implies that for any level of wealth, W, a
risk-averse investor would not wish to
own such a security
 In utility terms, this proposition means
 U(W) > 1/2U(W + h) + 1/2U(W – h) =
expected utility, where
 1/2U(W + h) + 1/2U(W – h) = VNM
utility
Risk aversion and utility
 U(W) > 1/2U(W + h) + 1/2U(W – h) says
 that the slope of the utility function decreases
as the agent becomes wealthier
 The marginal utility, d(U(W))/d(W), decreases
with increasing W
 d(U(W))/d(W) > 0
 d2(U(W))/d(W)2  0
 this is similar to our utility properties
discussion
Measuring Risk Aversion
The utility of the linear combination is greater
than the linear combination
U(W)
U[0.5(W+h) +0.5(W-h) >
U(W + h)
0.5U(W+h) +0.5U(W-h)
U[0.5(W+h) +0.5(W-h)
0.5U(W+h) +0.5U(W-h)
U(W – h)
W-h
W
W+h
W
The Arrow-Pratt Measures of
Risk Aversion
 Absolute risk aversion
 - U΄΄(W)/U΄(W) = RA(W)
 Relative risk aversion
 -WU΄΄(W)/U΄(W) = RR(W)
 Risk aversion means U΄(W) > 0 and U΄΄(W)  0
with U΄ = first derivative (slope) and U΄΄ =
second derivative or change in slope
 The inverse of these measures gives a measure
of risk tolerance
The risk averse concept
 We learned earlier, that a risk averse
investor will not accept the proposition
 1/2U(W + h) + 1/2U(W – h), since U(W) >
1/2U(W + h) + 1/2U(W – h)
 That is U(W) > prU(W + h) + (1-pr)U(W
– h) for h = some payoff or payout
 So what odds of the combination of
payoff or payout will they accept?
 But note that any investor will accept
such a bet if pr is high enough,
particularly if pr = 1
 And reject the offer if pr is small, and
surely reject if pr = 0
 The willingness to accept this opportunity
presumably is related to the level of
current wealth
 Let pr = pr(W, h) be the probability at which
an agent is indifferent between accepting or
rejecting the investment
 It can be shown (using mathematics of more
advanced finance) that
 pr(W, h) ≈ ½ + 1/4hRA(W)
 The higher the measure of absolute risk
aversion, RA(W), the more favorable odds the
agent will demand to take up the offer
Comparing agents
 If we have two investors, say A and B,
and
 If RA(W)A ≥ RA(W)B , then investor A
will always demand more favorable odds
than investor B
 In this sense, investor A is more risk
averse
An Example:
 Consider the family of VNM utility-in-
money functions of the form
 U(W) = -(1/v)e(-vW) { the exponential
utility function} for v = a parameter
 For this case, pr(W,h) ≈ ½ + 1/4hv
 Since RA(W) = -U΄΄/U΄ = -ve(-vW)/[(-v/v)e(-vW) = v by just forming the ratio of
the appropriate second and first
derivatives of this utility function
 So the odds requested by an agent with
this type of preference (utility) are
independent of the initial level of wealth,
W
 On the other hand, the more wealth at
risk (h), the greater the odds of a
favorable outcome demanded
 This expression advances the parameter,
v, as the natural measure of the degree of
risk aversion appropriate to this set of
preferences (utility function)
 Let’s try another set of preferences such
as the logarithmic utility function given
by Ln(W)
 Again, RA(W) = -U΄΄/U΄, but this gives us
 RA(W) = 1/W, if we take the appropriate
second and first derivatives of Ln(W)
 Why? -U΄΄/U΄ = -(-1/W2 )/(1/W) =1/W
 So pr(W,h) ≈ ½ + 1/4hRA(W) =
 ½ + 1/4h(1/W), or ½ + (¼)h/W
 So in this case, the odds that the agent must
have are related to h relative to initial wealth,
W
Risk that is a proportion of the
investor’s wealth
 In this case, h = өW, where ө is some
constant of proportionality, like 0.3 or
0.5, in which the payoff or the payment
would be 30% or 50% of wealth
 Now, pr(W,ө) represents the odds that an
investor would have to have to take up an
offer such as we have been representing
as 1/2U(W + h) + 1/2U(W – h), if the
investor is risk averse
 By a derivation similar to the pr(W,h)
case (using advanced mathematics in
finance)
 Pr(W,ө) ≈ ½ + 1/4өRR(W)
 Or the odds are a function of the degree
of risk of wealth, ө, and the measure of
relative risk aversion (not absolute risk
aversion as in the previous case)
An example
 Now let the utility function be given by a
somewhat more complicated utility
function as
 U(W) = [W(1-)/(1-)], for  being a
parameter that is greater than 1
 Just a note here--- if  = 1, then U(W) =
Ln(W), like the last example
 This general function is also a VNM
utility function
 In the general case for  > 1, we find RR(W) = -
WU΄΄/U΄ = -[W(-W(--1))/W-] = -(-W/W) = ,
by taking the appropriate second and first
derivatives of the utility function
 So pr(W,ө) ≈ ½ + 1/4ө are the odds that an
investor has to have in order take up the
proposition of an investment that gives a payoff
and also can require a payment -- h
 In this case, the investor demands a probability
of success that is related to the proportion of
wealth at risk and the utility parameter , and 
>1
 Furthermore, if there are two investors, A and
B, and A > B, the investor with  = A will
always demand a higher probability of success
than will investor B with  = B, for the same
fraction, ө, of wealth at risk
 In this sense, a higher  denotes a greater
degree of risk aversion for this investor
class
 Now, with the case of  = 1, the
probability expression pr(W, ө) , becomes
pr(W, ө) ≈ ½ + 1/4ө
 In which case the requested odds of
winning a payoff are not a function of
initial wealth, W
 The odds in this case depend on the
proportion of wealth that is at risk
 The lower is the fraction of wealth that is
at risk (the lower is ө), the more investors
are willing to consider entering into a fair
bet ( a risky opportunity where the
probabilities of success or failure are
both ½) as in the investment 1/2U(W + h)
+ 1/2U(W – h)
 But in the case where  >1 ----- then
pr(W, ө) ≈ ½ + 1/4ө, where  >1, the
investors demand higher probability of
success than in the case where  = 1
The odds have to be greater than
even to accept, under risk aversion
 Under the assumption of risk aversion,
then what we have been developing is the
fact that a risk averse investor has to
have greater than even odds to accept a
proposition of 1/2U(W + h) + 1/2U(W –
h), which is even odds of a payoff versus
a payment
Risk neutral investors
 One class of investors demands special
mention --- these are the risk neutral
investors (like banks in some cases)
 This class of investors has considerable
influence on the financial equilibria in
which they participate
 This class of investor is identified with
utility functions of linear form U(W) =
cW + d, for c, d = constants and c > 0
 Both of our measures of risk aversion give the
same results for this class of investor
 RA(W) = 0 = RR(W)
 Whether measured as a proportion of wealth or
as an absolute amount of money at risk, these
investors do not demand better than even odds
when considering risky investments of the type
we have been considering
 This class of investors are indifferent to
risk
 They are only concerned with an asset’s
expected payoff
 Depending on the portfolio under
consideration, it is generally considered
that banks belong to this class --- they
certainly do have weight in the conditions
of financial equilibrium
Prospect Theory
 UNDER VNM EXPECTED UTILITY,
THE UTILITY FUNCTION IS
DEFINED OVER ACTUAL PAYOFF
OUTCOMES
 UNDER PROSPECT THEORY,
PREFERENCES ARE DEFINED, NOT
OVER ACTUAL PAYOFFS, BUT
RATHER OVER GAINS AND LOSSES
RELATIVE TO SOME BENCHMARK
UTILITY FUNCTION FOR
PROSPECT THEORY
UTILITY
50
0 --
- 150
- 200
1000
= W
WEALTH = W
INVESTOR’S UTILITY
FUNCTION
 U(W) = (|W - W|)(1 - 1)/(1-1), IF W > W
 AND,
 U(W) = -λ(|W-W)(1-2)/(1-2), IF W<= W
 W DENOTES THE BENCHMARK PAYOFF
 λ > 1 CAPTURES THE EXTENT OF THE
INVESTOR’S AVERSION TO LOSSES
RELATIVE TO BENCHMARK
 1 AND 2 NEED NOT COINCIDE
 SO THE CURVATURE MAY DIFFER
FOR DEVIATIONS ABOVE AND
BELOW THE BENCHMARK
 SO THE PARAMETERS COULD
HAVE A LARGE IMPACT ON THE
RELATIVE RANKING OF
UNCERTAIN INVESTMENT PAYOFF
 NOT ALL TRANSACTIONS ARE
AFFECTED BY LOSS AVERSION SINCE, IN
NORMAL CIRCUMSTANCES, ONE DOES
NOT SUFFER A LOSS IN TRADING A
GOOD
 BUT AN INVESTOR’S WILLINGNESS TO
HOLD A FINANCIAL ASSET SUCH AS
STOCKS MAY BE SIGNIFICANTLY
AFFECTED IF LOSSES HAVE BEEN
EXPERIENCED IN PRIOR PERIODS