Utility generators and the universal t

3. Utility generators and the universal t

For a fixed number P, consider the following utility function 3 defined on security returns y: U y s ymax 0, P y y ; y` - y - `. 4 Ž . Ž . Ž . P The graph of U, plotted in Fig. 1, corresponds to the end of period pay-off to the writer of a put option with strike price P, with the initial premium income X Ž . 4 disregarded. Apart from this, note that U y G 0 for all y and the function U is P P concave. Ž . Let F R denote the distribution of the benchmark return R. In what follows, we shall allow F to have jumps, e.g. to refer to empirical distribution functions, Ž . but of course, it remains continuous to the right, similarly, denoted by G r , the return distribution function for the subject fund or portfolio. Integrating by parts, P E U R s y P y R d F R Ž . Ž . Ž . H P y` P s y F R d R. Ž . H y` Similarly, P E U r s y G r d r . Ž . Ž . H P y` P Ž . Recall that r dominates R by SSD if and only if for any number P, H G r d r F y` P Ž . Ž Ž .. H F R d R see Rothschild and Stiglitz 1970 . Hence, r dominates R if and y` only if: E U r E U R ; all P . 5 Ž . Ž . Ž . r P R P As the expectational limiters indicate, the SSD condition is cast in terms of the marginal distributions of r and R, so that aspects of the joint distribution are not involved. The SSD property means that r will be preferred to R by investors with SSD Ž . Ž . any risk averse utility function, i.e. r R m E U r E U R , any concave r R U. In this sense, the utility functions U could be said to span the set of arbitrary P concave utility functions. We now show that a similar sort of property holds for the equivalent margin measure, although joint distributions are now involved in a more essential way. 3 The fact that strictly positive utility is not attained is immaterial, as U is equivalent in decision P theoretic terms to a q b U for a arbitrary and any b 0. P 4 X Ž . Strictly, U P does not exist but the point has measure zero and we simply define it as unity. P Ž . Fig. 1. The utility function U y . P Ž . 5 The equivalent margin t P associated with the utility function U is defined P by: X E e R y R U R Ž . Ž . Ž . P t P s , Ž . X E U R Ž . P where, U X R s 1 R F P Ž . P s 0 otherwise. X Ž . Ž . As EU R s F P , it follows that, P 1 P t P s e R y R d F R . 6 Ž . Ž . Ž . Ž . Ž . H F P Ž . y` Ž . In what follows, function 6 will be called the COMD between r and R. Note in particular that, t ` s lim t P s E e R y R Ž . Ž . Ž . R P ™` w x s E E r N R y R Ž . R r s m y m , r R Ž . Ž . where m s E r and m s E R are the marginal means. Theorem 1 shows that r R the equivalent margin t for an arbitrary concave twice differentiable utility U function U can be expressed as a weighted average of the generating t . P 5 Ž . Notationally, one ought to use t instead of t P to be consistent with general notation t for a U U P Ž . utility function U. However, t P will be required at a later point to indicate a function as well as a Ž . functional, so the notation t P is adopted at the outset, and will be used only in connection with the Ž . Ž . utility generator indexed by P. In such contexts, the notation t P or t P is later used to distinguish R r the utility generators associated with different returns R and r. X Ž . X Ž . Theorem 1. Suppose that lim U P s U ` exists and let U be a twice P ™` Y differentiable utility function of returns for which U y F 0,y ` - y - `. Then, the associated equiÕalent margin t is giÕen by: U X ` U ` Ž . t s t ` q w P t P dP , 7 Ž . Ž . Ž . Ž . H X U EU R Ž . y` where, U Y P Ž . w P s y F P . 8 Ž . Ž . Ž . X E U R Ž . ` The weights w P are such that w P G 0 and H w P dP F 1, with equality if y` X U ` s 0. Ž . Ž Ž . . Ž . Proof. From Eq. 6 above and changing P to R, e R y R d F R s w Ž . Ž .x d t R F R , whence, X X t EU R s E e R y R U R Ž . Ž . Ž . Ž . U ` X s U R d t R F R . Ž . Ž . Ž . H y` Ž . Integrating by parts permissible for Stieltjes integrals and dividing both sides by X Ž . E U R , we get: X ` U ` t ` 1 Ž . Ž . Y t s y F R U R t R d R. 9 Ž . Ž . Ž . Ž . H X X U E U R E U R Ž . Ž . y` Ž . Y Ž . X Ž . Ž . Let w R s yU R rEU R F R . Then, ` ` 1 ` X X w R d R s y F R U R y U R d F R Ž . Ž . Ž . Ž . Ž . H H X y` ½ 5 E U R Ž . y` y` 1 X X s y U ` y EU R 4 Ž . Ž . X E U R Ž . U X ` Ž . s 1 y . X E U R Ž . X X Ž . X Ž . As U is a decreasing function, 0 F U ` F EU R . Finally, changing the Ž . ` Ž . dummy variable of integration in Eq. 9 back to P, we find that H w P y` X Ž . d P F 1, with equality if U ` s 0. I Corollary. If t P G 0 for y` - P - `, then t G 0 for an otherwise arbitrary, U twice differentiable risk aÕerse utility function U. Several remarks may be useful at this point. Ž . X Ž . Ž . 1 If U ` s 0 meaning global strict concavity , then by the second mean value theorem of differential calculus, there must exist a value P such that Ž . t s t P . There is therefore an option style utility function that has exactly the U same equivalent margin as the given utility function, strengthening the interpreta- tion of the U as a set of Autility generatorsB for arbitrary concave utility P functions. Ž . Ž . 2 An alternative expression for the weight w P is, U Y P Ž . w P s z F P , Ž . Ž . Y E U R Ž . Y Ž . X Ž . Ž . where z s yEU R rEU R is the Rubinstein 1973 coefficient of absolute risk aversion. The pattern of weights as P varies will therefore depend on two factors. The first is the second derivative of the utility function at P relative to the Ž . average. Consider, for example, the constant relative risk aversion family U R s Ž . j 1 q R rj ; j - 0, with j ™0 indicating the logarithmic utility function log Ž . Ž . Ž . jy2 1 q R . For such functions, the weights w P diminish with 1r 1 q P , so that the log utility function has weights that diminish more slowly relative to other members of the family, which correspond to greater risk aversion. The second factor is the cumulative probability mass up to P. If U Y is diminishing while Ž . F P is increasing, a trade-off between the two factors is therefore implied, and an Ž . interior maximum of w P will usually exist. It is generally apparent that in Ž . certain zones, notably the smallest and highest value of P, t P - 0 over some small interval will not necessarily conflict with t G 0 for any realistic U. U

4. The COMD and stochastic dominance