VNM utility and Risk Aversion

1
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

2
What does the term risk aversion mean about an
agents 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

3
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) gt 1/2U(W h) 1/2U(W h) expected
  utility, where
• 1/2U(W h) 1/2U(W h) VNM utility

4
Risk aversion and utility

• U(W) gt 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) gt 0
• d2(U(W))/d(W)2 ? 0
• this is similar to our utility properties
  discussion

5
Measuring Risk Aversion
The utility of the linear combination is greater
than the linear combination
U(W)
U0.5(Wh) 0.5(W-h) gt 0.5U(Wh) 0.5U(W-h)
U(W h)
U0.5(Wh) 0.5(W-h)
0.5U(Wh) 0.5U(W-h)
U(W h)
W-h W Wh
W
6
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) gt 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

7
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) gt 1/2U(W
  h) 1/2U(W h)
• That is U(W) gt 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?

8

• 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

9

• 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

10
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

11
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

12

• 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

13

• This expression advances the parameter, v, as the
  natural measure of the degree of risk aversion
  appropriate to this set of preferences (utility
  function)
• Lets try another set of preferences such as the
  logarithmic utility function given by Ln(W)

14

• 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

15
Risk that is a proportion of the investors wealth

• In this case, h ?W, where ? 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,?) 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

16

• By a derivation similar to the pr(W,h) case
  (using advanced mathematics in finance)
• Pr(W,?) ½ 1/4?RR(W)
• Or the odds are a function of the degree of risk
  of wealth, ?, and the measure of relative risk
  aversion (not absolute risk aversion as in the
  previous case)

17
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

18

• In the general case for ? gt 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,?) ½ 1/4?? 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

19

• 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 ?
  gt 1
• Furthermore, if there are two investors, A and B,
  and ?A gt ?B, the investor with ? ?A will always
  demand a higher probability of success than will
  investor B with ? ?B, for the same fraction, ?,
  of wealth at risk

20

• 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, ?) , becomes pr(W, ?) ½ 1/4?
• In which case the requested odds of winning a
  payoff are not a function of initial wealth, W

21

• 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 ?), 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)

22

• But in the case where ? gt1 ----- then pr(W, ?)
  ½ 1/4??, where ? gt1, the investors demand
  higher probability of success than in the case
  where ? 1

23
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

24
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 gt 0

25

• 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

26

• This class of investors are indifferent to risk
• They are only concerned with an assets 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

27
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

28
UTILITY FUNCTION FOR PROSPECT THEORY
UTILITY
50
0 --
- 150 - 200
1000 W?
WEALTH W
29
INVESTORS UTILITY FUNCTION

• U(W) (W - W?)(1 - ?1)/(1-?1), IF W gt W?
• AND,
• U(W) -?(W-W?)(1-?2)/(1-?2), IF Wlt W?
• W? DENOTES THE BENCHMARK PAYOFF
• ? gt 1 CAPTURES THE EXTENT OF THE INVESTORS
  AVERSION TO LOSSES RELATIVE TO BENCHMARK
• ?1 AND ?2 NEED NOT COINCIDE

30

• 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

31

• NOT ALL TRANSACTIONS ARE AFFECTED BY LOSS
  AVERSION SINCE, IN NORMAL CIRCUMSTANCES, ONE DOES
  NOT SUFFER A LOSS IN TRADING A GOOD
• BUT AN INVESTORS WILLINGNESS TO HOLD A FINANCIAL
  ASSET SUCH AS STOCKS MAY BE SIGNIFICANTLY
  AFFECTED IF LOSSES HAVE BEEN EXPERIENCED IN PRIOR
  PERIODS