When does 'All Eggs in One Risky Basket' Make Sense?

Paper prepared for the Irish Economics Association’s Annual Conference 2005 When does ‘All Eggs in One Risky Basket’ Make Sense? GERRY BOYLE and DENIS CONNIFFE Department of Economics, NUI, Maynooth Abstract: In an important paper comparing expected utility and mean-variance analysis, Feldstein (1969) examined a simple portfolio problem involving just two assets, one riskless and one risky. He concluded there could easily be ‘plunging’, that is, investment in the risky asset alone. His background assumptions were that the risky asset’s yield was log normally distributed and that the investor’s attitude to risk was expressible by a logarithmic utility. We look at how conclusions are affected by choice of distribution and utility function. While conclusions can depend on choice of distribution, they are remarkably robust to choice within the range of plausible positive distributions. In contrast, conclusions are sensitive to choice of utility function and we find the key determinant to be how much the investor’s relative risk aversion differs from unity and in what direction. Based on historical stock market returns, our analysis implies that the prevalence of diversification that is observed is consistent with a relative risk aversion coefficient of about 2.5. I INTRODUCTION Feldstein (1969) published an important and much cited paper criticising the use of meanvariance (or   ) analysis rather than expected utility in the theory of economic behaviour under uncertainty. He based his analysis on a simple portfolio problem involving just two assets, one riskless and one risky. But since the risky asset could itself be considered to be the optimal, or market, portfolio of risky assets, the separation theorem of mean-variance portfolio analysis suggests the importance of the problem. There were three components to Feldstein’s paper. The first outlined the possibility that the    indifference curves of a risk averter might not be convex downwards. The second argued that an investor choosing a combination of a riskless and a risky asset might be much more likely to opt or ‘plunge’ for a risk-only portfolio than previous analysis had suggested, so that, contrary to Tobin’s (1958) analysis, diversification between money and bonds need not generally follow from risk aversion. The third criticised the use of    analysis for multi-asset portfolios. Subsequent 2 discussion of Feldstein’s paper has concentrated on the first and third components. The second, which appears to have received relatively little attention, provides the motivation for this paper. Feldstein assumed a log normal distribution1 of the return on the risky asset and a logarithmic utility function. He demonstrated that a utility-maximising strategy would lead to plunging for the risky asset given values of the parameters of the log normal that he considered far from unlikely. We will review his results in section II along with some discussion of subsequent comments. We continue to examine how much plunging depends on the specification of the probability distribution and utility function. We show in section III that if the logarithmic utility function is retained, Feldstein’s results are remarkably robust across a range of distributions. However, in section IV, we show that the situation is very different with regard to choice of utility function and we relate the variations in the likelihood of plunging to the degree of risk aversion embedded in the utility functions. Finally, in section V we consider some implications of these findings. II FELDSTEIN’S ANALYSIS Feldstein, following Tobin (1958), considered an investor with initial wealth A, allocating a proportion p to the risky asset, so that, after one period, wealth becomes y = (1-p)A + pAx, where the riskless asset is assumed non-interest bearing and x is random with a mean presumed greater than unity2. Feldstein took x as log normal with parameters *  and *  , that is, a density of 2* 2 2 *)(log 1 * 1 2 1 )(        x e x xf ,    x0. (1) Then the mean of x 2** 2 1    e and the variance of x   1 22** 22     ee . For investment in the risky asset to have any attraction, Feldstein further assumed that the investor is risk averse with a logarithmic utility function. So the investor wants to choose p so as to 1 Since, as is well known, expected utility and    analysis are compatible given normality, Feldstein needed to assume some non-normal distribution. However, log normality, or other positive distributions, seem more plausible than the normal and have more empirical support. 2 Obviously, the mathematical nature of the problem is unchanged by attaching a non-stochastic yield to the riskless asset and adjusting the yield of the risky asset accordingly. 3 maximise }])1[log{()}({ pAxApEyuE    )}]1(1[log{log    xpEA . (2) The derivative            )1(1 1)( xp x E p uE is obviously positive at p = 0, so that there will certainly be some investment in the risky asset. At p = 1 it is dxxf xx E)( 1 1 1 1         (3) which is 2** 1 2 1   e=         2 2 1 1 1    . So the derivative is still positive, implying plunging, that is, all funds invested in the risky asset, if         2 2 1    . (4) Feldstein maintained this would often be the case, giving the example of 05.1   , when plunging occurs unless  is well more than four times the expected bond yield of 5%. He contrasted this situation with the ubiquity of diversification implied by Tobin’s (1958) mean variance analysis of the same two asset model and linked it to non convexity of indifference curves, so as to emphasise that mean variance analysis is not always an adequate substitute for expected utility maximisation. While many subsequent papers, whether defending or further criticising mean-variance analysis, have discussed Feldstein’s paper (including, among others, Tobin (1969), Tsiang (1972), Bierwag (1974), Borch (1974), Levy (1974), Mayshar (1978) Felstein (1978) and Meyer (1987)), the plunging phenomenon itself has not featured prominently. Mayshar accepted the correctness of Feldstein’s analysis of plunging, but maintained it was really unrelated to issues of the convexity of indifference curves, or validity of mean-variance analysis. One of his arguments was that Feldstein had analysed indifference curves generated by when x is assumed log normal, but in the )}({ xuE portfolio problem considered a variable )}1(1(    xpAy , which being a translation of a log 4 normal has a somewhat different distribution. Another was the more general point that if the investor restricted to choosing values of p and will obtain the same value of x irrespective of choice of p, then whatever the distribution of x, the distributions of the variables is )( )()( p ppy y y    are obviously identical, where )}1(1{)(      pAp y and   App y  )( . Then whatever the istribution and whatever the (concave) utility function, the optimal p, assuming it < 1, could be btained from mean variance analysis in terms of d yy   and o. While more roundabout than direct etermination by equating the derivative of with respect to p to zero, it is certainly ompatible with it3. But whether p is < 1, or whether plunging occurs is ignored in this argument nd it is not at all clear that it should be. There would seem no sense in using mean variance analysis hen p=1 and, if plunging is possible, interest should focus on how distribution and utility function ffect its occurrence. Certainly, condition (4) was derived assuming log normality and log utility. ote too that if p = 1, y is log normally distributed if x is. In contrast to Mayshar, Meyer (1987) isputed the validity of Feldstein’s result on plunging, claiming it resulted from an implicit borrowing striction in the model. But if the investor could borrow without cost, this would just increase A with lunging still occurring. o no detailed examination seems to have been conducted into how the occurrence of lunging might change with the distribution or how it might depend on the precise utility function hosen. III VARYING THE PROBABILITY DISTRIBUTION etaining, for the present, the logarithmic utility, formulae (2) and (3) remain unchanged. lunging will occur if )}({ yuEd c a w a N d re p S p c R P 0 1 1       x E, (5) hile a mixture of assets will be optimal if it is negative. If x could take negative values, the xpectation of the reciprocal of x may not exist, which can usually be interpreted as , plying that plunging cannot occur. So, for example, assuming a normal distribution for x would w E e im 3 Other authors have made this point including (implicitly) Bierwag (1974) and Meyer (1987). 5 exclude plunging. But if we assume that the most that can be lost by investment in a risky asset is the mount invested, a non-negative distribution is appropriate for x. aking x as positively distributed , the well known delta method for approximation of xpectations proceeds a T e                                         ...1 1 1 111 3 3 2 2 1          EEE x E ... 1 4 3 3 2      , (6) here 3  w denotes the third moment about the mean. Now if we ignore terms with denominators volving powers of  in greater than 3, this gives               2 2 1 11    x E, which implies (4) again as the determinant of plunging. Admittedly, this approximation assumes that e coefficient of variation is not large, but note that if the distribution is positively skew, implying 3  th positive, this reduces (6) making it more likely that (5) is positive and plunging occurs. owever, we need not rely on approximation arguments to show that plunging can occur for ther than the log normal. Suppose x follows a Pearson Type 3 distribution (often called the two arameter gamma) H o p     x exxf    1 )( 1 )(    , x0 here the mean and variance can be shown to be    w and . Then 22       2 1 )1( 11           x E. his will be less than unity and plunging will occur if . (7) o plunging is quite plausible unless variation is very large, although not quite as likely as with a log since T  /1 2  S normal 6                           2 2 4 4 2 2 1 2 2 21 1 ...1 1 1 11               . However, taking 05.1  as in Feldstein’s example, (7) shows plunging occurs unless 22.  , again more than least four times the expected bond yield of 5%. Again, suppose x follows a Weibull distribution, which, like the log normal and Gamma distributions, has been found in various empirical studies to provide a good fit to wealth distributions. The density is                         x e x xf 1 )( ,    x0. It is evident that f(x)/x tends to infinity as x goes to zero for 2   , implying that the expectation goes to infinity also and that plunging cannot occur. For 2  exact integration gives              1 1 11 x E. (8) The mean of a Weibull is         1 1 (9) and so for large  , since , the right hand side of (8) becomes the reciprocal of1)1(   . This must be less than unity, as  was presumed greater than unity, so plunging must then occur. Practically plausible values of  are best judged from the coefficient of variation. The variance of a Weibull is                    1 1 2 1222 and, obviously, large  implies small  . The square of the coefficient of variation is 1 1 1 2 1 2 2 2                  . (10) So for any  and  ,  can be deduced from (10) and then  from (9), so that (8) can show if plunging occurs. Taking the case of 05.1   and  = .2, or four times the expected yield, we find (8) takes the value .996 so that plunging occurs even with that much variation, which is essentially the 7 same as Feldstein found for the log normal distribution. Similar findings follow for other plausible positive distributions for which algebraic exact solutions are obtainable. The multiple of the expected yield that variation must exceed before plunging is ruled out can vary somewhat, but the overall situation is clear. Plunging ought not to be an infrequent prediction with distributions that are usually considered plausible candidates for wealth, at least when we believe an investor’s risk aversion can be represented by logarithmic utility. IV VARYING THE UTILITY FUNCTION To point up the difference that choice of utility function can make, we first contrast logarithmic utility with the extremely popular negative exponential utility , with y eyu   1)( 0  , where, as before, y = (1-p)A + pAx. This is often taken in conjunction with a normal distribution for x and then it is well known that then the optimal p is 2 1    A p  which must be less than unity if A is large enough and so the amount invested in the risky asset is Ap, constant irrespective of the amount available for investment. However, this absence of plunging also occurs with the distributions considered in the previous sections. Taking the two-parameter gamma, for example, the expected utility is dxexeyuE x xpA         1)}1(1{ )( 1 1)}({ . dxxee ex Apx pA 1 ) 1 ( )1( )( 1            . This can be integrated exactly to give     1 1)1(    Ap epA . Differentiating with respect to p gives           1 1 1 )1(       Ap Ap Ae pA 8 which, if A is large, will obviously be negative at p = 1. The optimum investment Ap is 2 )1(1         Ap , again a constant, irrespective of A. The contrast between results for the logarithmic utility and the negative exponential utility must be due to the different degrees of risk aversion they embody. The coefficients of absolute risk aversion and relative risk aversion are y R1  and , for 1 r Ryu log  and  R and yRr   , for . y eu   1 Furthermore is unbounded and yu log  as y  , while the negative exponential utility 1 as y . There is so much less to be gained from large increments in wealth with the  negative exponential utility. For a finer exploration of the influence of risk aversion on plunging, we take the utility function , with     yu 11    . For  negative this is unbounded as y    and for  positive it is bounded as y . 1  In fact, 0  corresponds to and, obviously, yu log1    gives risk neutrality. The coefficients of risk aversion are y R1   and 1    r R, showing decreasing absolute risk aversion and constant relative risk aversion. So this utility function is not at all as risk averse as the negative exponential utility, but it can be either less risk averse than the logarithmic utility (if  is negative) or more risk averse (if  is positive). So it should be a sensitive criterion for examining plunging. For any distribution density f(x), the expected utility is .  dxxfxpAyuE )()}1(1{1)}({     Differentiating with respect to p and setting p = 1 gives . (11) dxxfxxA )()1( 12     If this is positive plunging occurs. It can be evaluated by integration for any of the positive distributions mentioned earlier. For (1), the log normal density, (11) becomes 9       2**2* 2 *) 2 1 ( 2 21      eeA . So plunging occurs if 2** ) 2 1 (   , or expressed in of  and  1 2 2 1              , (12) which reduces to Feldstein’s criterion (4) if  = 0. For  negative, plunging is even more likely than with logarithmic utility, in the sense that it will occur in spite of ever greater variability. At  = -1 plunging is certain. Conversely, for  positive and increasing, plunging becomes less likely and will not occur if  is large enough. So, since decreasing absolute risk aversion and constant relative risk aversion holds for this utility function whatever the value of  , it is not these characteristics in themselves that determine plunging. Rather, since 1    r R, it is the difference from unity (and direction) of the relative risk aversion, which of course is the (negative) elasticity of the slope of the utility function. High implies absence of plunging. r R These findings are not dependent on the log normal alone. Returning to (11), but assuming f(x) follows the two parameter gamma, gives dxexxx Ax           11 2)1( )( . Integration gives  )1()( )( 1 2         A.  1)1( )( )1( 1 2          A So plunging occurs if 1)1(     , or in terms of  and   11 2      . (13)