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VNM utility and Risk Aversion

VNM utility and Risk Aversion

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VNM utility and Risk Aversion

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  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 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

  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) > 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) > 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

  5. 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) > 0.5U(W+h) +0.5U(W-h) U(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

  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) > 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) > 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?

  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) • Let’s 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 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

  16. 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)

  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  > 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

  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  > 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

  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, 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

  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 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)

  22. 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

  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 > 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 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

  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. 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

  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 INVESTOR’S WILLINGNESS TO HOLD A FINANCIAL ASSET SUCH AS STOCKS MAY BE SIGNIFICANTLY AFFECTED IF LOSSES HAVE BEEN EXPERIENCED IN PRIOR PERIODS