Financial Economics: Risk Aversion and Investment Decisions

Transcription

1 Financial Economics: Risk Aversion and Investment Decisions Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY March, / 50

2 Outline Risk Aversion and Portfolio Allocation Portfolios, Risk Aversion, and Wealth 2 / 50

3 The portfolio problem Setup Let s now put our framework of decision-making under uncertainty to use. Consider a risk-averse investor with vn-m expected utility who divides his or her initial wealth Y 0 into an amount a allocated to a risky asset say, the stock market and an amount Y 0 a allocated to a safe asset say, a bank account or a government bond. 3 / 50

4 The portfolio problem Setup Y 0 = initial wealth a = amount allocated to stocks r = random return on stocks r f = risk free return Ỹ 1 = final wealth Ỹ 1 = (1 + r f )(Y 0 a) + a(1 + r) = Y 0 (1 + r f ) + a( r r f ) 4 / 50

5 The portfolio problem Objective Function The investor chooses a to maximize expected utility: max a E[u(Ỹ1)] = max a Eu[Y 0 (1 + r f ) + a( r r f )] If the investor is risk-averse, u is concave and the first-order condition for this unconstrained optimization problem, found by differentiating the objective function by the choice variable and equating to zero, is both a necessary and sufficient condition for the value a of a that solves the problem. 5 / 50

6 The portfolio problem First-order Condition The investor s problem is max a Eu[Y 0 (1 + r f ) + a( r r f )] The first-order condition(f.o.c.) is E[u [Y 0 (1 + r f ) + a ( r r f )]( r r f )] = 0 Note: we are allowing the investor to sell stocks short (a < 0) or to buy stocks on margin (a > Y 0 ) if he or she desires. 6 / 50

7 The portfolio problem Risk Aversion and Portfolio Allocation Theorem Theorem 5.1 If the Bernoulli utility function u is increasing and concave, then a > 0 if and only if E( r) > r f a = 0 if and only if E( r) = r f a < 0 if and only if E( r) < r f Thus, a risk-averse investor will always allocate at least some funds to the stock market if the expected return on stocks exceeds the risk-free rate. 7 / 50

8 The portfolio problem Proof To prove the theorem, let W (a) = E[u [Y 0 (1 + r f ) + a( r r f )]( r r f )] so that the investor s first-order condition can be written more compactly as it follows that W (a ) = 0 W (a) = E[u [Y 0 (1 + r f ) + a( r r f )]( r r f ) 2 ] < 0 since u is concave. This means that W is a decreasing function of a. 8 / 50

9 The portfolio problem Proof Finally, with W (a) = E[u [Y 0 (1 + r f ) + a( r r f )]( r r f )] W (0) = E[u [Y 0 (1 + r f )]( r r f )] = u [Y 0 (1 + r f )]E( r r f ) = u [Y 0 (1 + r f )][E( r) r f ] Since u is increasing, this means that W (0) has the same sign as E( r) r f. 9 / 50

10 The portfolio problem Proof We now know that: W (a) is a decreasing function W (0) has the same sign as E( r) r f. W (a ) = 0 10 / 50

11 The portfolio problem Proof E( r) r f > 0 implies W (0) > 0, and since W is decreasing, W (a ) = 0 implies a > 0. Since W (0) has the same sign as E( r) r f, E( r) r f > 0 11 / 50

12 The portfolio problem Proof 12 / 50

13 The portfolio problem Proof 13 / 50

14 Example Example suppose u(y ) = ln(y ), then u (Y ) = 1/Y, assume that stock returns can either be good or bad: r = { r G r B with probability π with probability 1 π where r G > r f > r B defines the good and bad states and E( r) = πr G + (1 π)r B > r f so that E( r) > r f and the investor will choose a > 0 14 / 50

15 Example Example The first-order condition is specializes to E[u [Y 0 (1 + r f ) + a ( r r f )]( r r f )] = 0 π(r G r f ) Y 0 (1 + r f ) + a (r G r f ) + (1 π)(r B r f ) Y 0 (1 + r f ) + a (r B r f ) = 0 implies a = (1 + r f )[π(r G r f ) + (1 π)(r B r f )] Y 0 (r G r f )(r B r f ) which is positive since r G > r f > r B and E( r) r f = π(r G r f ) + (1 π)(r B r f ) > 0 15 / 50

16 Example Example In this case, a a = (1 + r f )(E( r) r f ) Y 0 (r G r f )(r B r f ) Rises proportionally with Y 0 Increases as E( r) r f rises Falls as r G and r B move father away from r f, holding E( r) constant; that is, in response to a mean preserving spread. 16 / 50

17 Example Example a = (1 + r f )(E( r) r f ) Y 0 (r G r f )(r B r f ) r f r G r B π E( r) a Y The fraction of initial wealth allocated to stocks rises when stocks become less risky or pay higher expected returns. 17 / 50

18 Example Risk Aversion and Portfolio Allocation Before moving on, return to the general problem max Eu[Y 0 (1 + r f ) + a( r r f )] a but assume now that the investor is risk-neutral, with u(y ) = αy + β and α > 0, so that more wealth is preferred to less. The risk-neutral investor solves max Eα[Y 0 (1 + r f ) + a( r r f )] + β a = max αy 0 (1 + r f ) + a[e( r) r f ] + β a So long as E( r) r f > 0, the risk-neutral investor will choose a to be as large as possible, borrowing as much as he or she is allowed to in order to buy more stocks on margin. 18 / 50

19 Portfolios, Risk Aversion, and Wealth The previous examples call out for a more detailed analysis of how optimal portfolio allocation decisions, summarized by the value of a that solves max a Eu[Y 0 (1 + r f ) + a( r r f )] are influenced by the investor s degree of risk aversion and his or her level of wealth. 19 / 50

20 Portfolios, Risk Aversion, and Wealth The following result was proven by Kenneth Arrow in The Theory of Risk Aversion, published in the 1971 volume Essays in the Theory of Risk-Bearing and reprinted in 1983 in volume 3 of the Collected Papers of Kenneth J. Arrow (Harvard University Press). Theorem Theorem 5.2 Consider two investors, i = 1 and i = 2, and suppose that for all wealth levels Y, R 1 A(Y ) > R 2 A(Y ) where RA i (Y ) is investor i s coefficient of absolute risk aversion. Then a 1(Y ) < a 2(Y ) where ai (Y ) is amount allocated by investor i to stocks when he or she has initial wealth Y. 20 / 50

21 Portfolios, Risk Aversion, and Wealth Recall that the coefficients of absolute and relative risk aversion are R A (Y ) = u (Y ) u (Y ) and R R (Y ) = Yu (Y ) u (Y ) Thus R 1 A(Y ) > R 2 A(Y ) i.e., u u 1 1(Y ) > 2(Y ) u (Y ) u 2 (Y ) implies Yu u 1 1(Y ) > 2(Y ) Yu (Y ) u 2 (Y ) or RR(Y 1 ) > RR(Y 2 ) 21 / 50

22 Portfolios, Risk Aversion, and Wealth Arrow s result applies equally well to relative risk aversion: Theorem Theorem 5.3 Consider two investors, i = 1 and i = 2, and suppose that for all wealth levels Y, R 1 R(Y ) > R 2 R(Y ) where RR i (Y ) is investor i s coefficient of relative risk aversion. Then a1(y ) < a2(y ) where ai (Y ) is amount allocated by investor i to stocks when he or she has initial wealth Y. 22 / 50

23 Example Example with CRRA utility FN Let s test Arrow s proposition out, by generalizing our previous example with logarithmic utility to the case where u(y ) = Y 1 γ 1 1 γ with γ > 0. For this Bernoulli utility function, the coefficient of relative risk aversion is constant and equal to γ. 23 / 50

24 Example Example with CRRA utility FN Hence u(y ) = Y 1 γ 1 1 γ, thus u (Y ) = Y γ and stock returns can either be good or bad r = { r G with probability π r B with probability 1 π where r G > r f > r B defines the good and bad states and E( r) = πr G + (1 π)r B > r f so that E( r) > r f and the investor will choose a > / 50

25 Example Example with CRRA utility FN With CRRA utility and two states for r, the F.O.C. specializes to E[u [Y 0 (1 + r f ) + a ( r r f )]( r r f )] = 0 π(r G r f ) [Y 0 (1 + r f ) + a (r G r f )] γ + (1 π)(r B r f ) [Y 0 (1 + r f ) + a (r B r f )] γ = 0 25 / 50

26 Example Example with CRRA utility FN a Y 0 = (1 + r f )([π(r G r f )] 1/γ [(1 π)(r f r B )] 1/γ ) (r G r f )[(1 π)(r B r f )] 1/γ (r B r f )[π(r G r f )] 1/γ γ r f r G r B π E( r) a Y Consistent with Arrow s theorem, higher coefficients of relative risk aversion are associated with smaller values of a. 26 / 50

27 Example Portfolios, Risk Aversion, and Wealth a Y 0 = (1 + r f )([π(r G r f )] 1/γ [(1 π)(r f r B )] 1/γ ) (r G r f )[(1 π)(r B r f )] 1/γ (r B r f )[π(r G r f )] 1/γ Note that with constant relative risk aversion, a rises proportionally with wealth. Two additional theorems, also proven by Arrow, tell us more about the relationship between a and wealth. 27 / 50

28 absolute risk aversion v.s. absolute wealth Portfolios, Risk Aversion, and Wealth Theorem Theorem 5.4 Let a (Y 0 ) be the solution to max a Eu[Y 0 (1 + r f ) + a( r r f )] If u(y ) is such that (a) R A (Y ) < 0 then da (Y 0 ) dy 0 > 0 (b) R A (Y ) = 0 then da (Y 0 ) dy 0 = 0 (c) R A (Y ) > 0 then da (Y 0 ) dy 0 < 0 28 / 50

29 absolute risk aversion v.s. absolute wealth Portfolios, Risk Aversion, and Wealth Part (a) R A(Y ) < 0 then da (Y 0 ) dy 0 > 0 describes the normal case where absolute risk aversion falls as wealth rises. In this case, wealthier individuals allocate more wealth to stocks. 29 / 50

30 absolute risk aversion v.s. absolute wealth Portfolios, Risk Aversion, and Wealth Part (b) R A(Y ) = 0 then da (Y 0 ) dy 0 = 0 means that investors with constant absolute risk aversion u(y ) = 1 ν e νy allocate a constant amount of wealth to stocks.... so a CARA investor finds a bet of the ideal size and sticks with it, even when income increases. 30 / 50

31 absolute risk aversion v.s. absolute wealth Portfolios, Risk Aversion, and Wealth Part (c) R A(Y ) > 0 then da (Y 0 ) dy 0 < 0 describes the case where absolute risk aversion rises as wealth rises. The implication that wealthier individuals allocate less wealth to stocks makes this case seem less plausible. 31 / 50

32 relative risk aversion v.s. relative wealth Portfolios, Risk Aversion, and Wealth Consistent with our results with CRRA utility, the next result relates changes in relative risk aversion to changes in the proportion of wealth allocated to stocks. Define the elasticity of the function a (Y 0 ) as η = d(lna (Y 0 )) d(lny 0 ) = Y 0 d(a (Y 0 )) a (Y 0 ) dy 0 The elasticity measures the percentage change in a (Y 0 ) brought about by a percentage-point change in Y / 50

33 relative risk aversion v.s. relative wealth Portfolios, Risk Aversion, and Wealth Theorem Theorem 5.5 Let a (Y 0 ) be the solution to max a Eu[Y 0 (1 + r f ) + a( r r f )] If u(y ) is such that (a) R R (Y ) < 0 then η > 1 (b) R R (Y ) = 0 then η = 1 (c) R R (Y ) > 0 then η < 1 The theorem confirms what we know about CRRA utility: it implies that a rises proportionally with Y 0. But the theorem extends the results to the cases of decreasing and increasing relative risk aversion. 33 / 50

34 relative risk aversion v.s. relative wealth Portfolios, Risk Aversion, and Wealth With CRRA where a Y 0 = K K = Hence (1 + r f )([π(r G r f )] 1/γ [(1 π)(r f r B )] 1/γ ) (r G r f )[(1 π)(r B r f )] 1/γ (r B r f )[π(r G r f )] 1/γ ln(a (Y 0 )) = ln(k) + ln(y 0 ) η = d(lna (Y 0 )) = 1 d(lny 0 ) 34 / 50

35 relative risk aversion v.s. relative wealth Summary absolute value a relative to wealth Y 0 excess return Theorem 5.1 absolute risk aversion Theorem 5.2 Theorem 5.4 relative risk aversion Theorem 5.3 Theorem / 50

36 Risk Aversion and Saving Behavior So far, we ve assumed that investors only receive utility from the terminal value of their wealth, and asked how they should split their initial wealth accumulated, presumably, through past saving across risky and riskless assets in order to maximized the expected utility from terminal wealth. Now, let s take the possibly random return on the investor s portfolio of assets as given, and ask how he or she should optimally determine savings under conditions of uncertainty. 36 / 50

37 Risk Aversion and Saving Behavior Suppose there are two periods, t = 0 and t = 1, and let Y 0 = initial wealth s = amount saved in period t=0 c 0 = Y 0 s = amount consumed in period t=0 R = 1 + r = random, gross return on savings c 1 = s R = amount consumed in period t=1 Suppose also that the investor has vn-m expected utility over consumption during periods t = 0 and t = 1 given by u(c 0 ) + βe[u( c 1 )] = u(y 0 s) + βe[u(s R)] where the discount factor β is a measure of patience 37 / 50

38 Risk Aversion and Saving Behavior The solution to the investor s saving problem max u(y 0 s) + βe[u(s R)] s is described by the first-order condition u (Y 0 s ) + βe[u (s R) R] = 0 or u (Y 0 s ) = βe[u (s R) R] 38 / 50

39 Risk Aversion and Saving Behavior u (Y 0 s ) = βe[u (s R) R] We can use this optimality condition to investigate how optimal saving s responds to an increase in risk, in the form of a mean preserving spread in the distribution of R. Intuitively, one might expect there to be two offsetting effects: The riskier return will make saving less attractive and thereby reduce s ; The riskier return might lead to precautionary saving in order to cushion period t = 1 consumption against the possibility of a bad output and thereby increase s. 39 / 50

40 Risk Aversion and Saving Behavior u (Y 0 s ) = βe[u (s R) R] To see which of these two effects dominates, define g( R) = u (s R) R so that the right-hand side becomes βe[g( R)] Jensen s inequality will imply that after a mean preserving spread the distribution of R in this expectation will fall if g is concave and rise if g is convex. 40 / 50

41 Risk Aversion and Saving Behavior When g is concave, a mean preserving spread in the distribution of R will decrease E[g( R)]. 41 / 50

42 Risk Aversion and Saving Behavior When g is convex, a mean preserving spread in the distribution of R will increase E[g( R)]. 42 / 50

43 Risk Aversion and Saving Behavior The definition g( R) = u (s R) R suggests that the concavity or convexity of g will depend on the sign of the third derivative of u. The product and chain rules for differentiation imply g ( R) = u (s R)s R + u (s R) g ( R) = u (s R)(s ) 2 R + u (s R)s + u (s R)s = u (s R)(s ) 2 R + 2u (s R)s implies that g ( R) has the same sign as u (s R)s R + 2u (s R) 43 / 50

44 Risk Aversion and Saving Behavior To understand precautionary saving behavior, the concept of prudence is defined by Miles Kimball, Precautionary Saving in the Small and in the Large, Econometrica Vol.58 (January 1990): pp Whereas risk aversion is summarized by the second derivative of the Bernoulli utility function u, prudence is summarized by the third derivative of u. 44 / 50

45 Risk Aversion and Saving Behavior Kimball defines the coefficient of absolute prudence as P A (Y ) = u (Y ) u (Y ) and the coefficient of relative prudence as P R (Y ) = Yu (Y ) u (Y ) thereby extending the analogous measures of absolute and relative risk aversion. 45 / 50

46 Risk Aversion and Saving Behavior g ( R) has the same sign as or u (s R)s R + 2u (s R) u (Y )Y +2u (Y ) = u (Y )[ Yu (Y ) u (Y ) +2] = u (Y )[2 P R (Y )] g ( R) is positive if 2 < P R (Y ); g ( R) is negative if 2 > P R (Y ) 46 / 50

47 Risk Aversion and Saving Behavior Hence, if 2 < P R (Y ), then g ( R) > 0. Since g is convex, a mean preserving spread in the distribution of R increases the right hand side of the optimality condition u (Y 0 s ) = βe[u (s R) R] and s must increase to maintain the equality. The precautionary saving effect dominates if the coefficient of relative prudence exceeds / 50

48 Risk Aversion and Saving Behavior Conversely, if 2 > P R (Y ), then g ( R) < 0. Since g is concave, a mean preserving spread in the distribution of R decreases the right hand side of the optimality condition u (Y 0 s ) = βe[u (s R) R] and s must decrease to maintain the equality. The precautionary saving effect dominates if the coefficient of relative prudence is less than / 50

49 Risk Aversion and Saving Behavior, Example To apply these results, let s calculate the coefficient of relative prudence implied by the CRRA utility function u(y ) = Y 1 γ 1 1 γ where γ > 0, since u (Y ) = Y γ imply P R (Y ) = Yu (Y ) u (Y ) u (Y ) = γy γ 1 u (Y ) = γ(γ + 1)Y γ 2 γ 2 Y γ(γ + 1)Y = = γ + 1 γy γ 1 49 / 50

50 Risk Aversion and Saving Behavior, Example Hence, the CRRA utility function u(y ) = Y 1 γ 1 1 γ implies both a constant coefficient of relative risk aversion equal to γ and a constant coefficient of relative prudence equal to γ + 1. If γ > 1, saving rises in response to a mean preserving spread in the distribution of R. When γ < 1, saving falls. In the special case γ = 1 of logarithmic utility, saving is unaffected. 50 / 50