Appendix - Normally Distributed Admissible Choices are Optimal

Transcription

1 Appendx - Normally Dstrbuted Admssble Choces are Optmal James N. Bodurtha, Jr. McDonough School of Busness Georgetown Unversty and Q Shen Stafford Partners Aprl 994 latest revson September 00 Abstract Partcular utlty functons that are optmzers for a partcular admssble set element are dentfed. The optmalty of the normally dstrbuted admssble portfolos s shown n the second-order stochastc domnance ntegral context. Please address correspondence to the frst author at School of Busness, Georgetown Unversty, Old North 33, 37 th & O Streets, NW, Washngton, DC 0057, (0) , fax: (0) , and e-mal: bodurthj@gunet.georgetown.edu

2 For Normal Returns, Mean-Varance Admssble Choces are Optmal Abstract Partcular utlty functons that are optmzers for a partcular admssble set element are dentfed. The optmalty of the normally dstrbuted admssble portfolos s shown n the second-order stochastc domnance ntegral context. A-. Optmal Choces among Mutually Exclusve Alternatves We use a geometrc argument to llustrate constructon of the utlty functons that wll choose each element of the SSD admssble set. Proposton A-: For every member of the SSD admssble set of choces, there exsts a utlty functon n U that s the optmzer for that utlty functon. Φ, we can order the Proof: Gven an admssble set of normal dstrbutons = { F,F,,F } dstrbutons n the admssble set Φ n such a way that n < < < n, and µ < µ < < µ n. Furthermore, these strct relatons order the SD admssble set over the real lne. Consder any three adjacent dstrbutons, F, F, and F +, for nteger ndex, < <n. Construct two dstrbutons satsfyng the followng: ε = = ε ε N µ = µ + ε = + F ε ~ N µ ε µ, ε F + ε ~ + ε, + ε and (A-) We can always construct a smooth convex curve (maybe of second-degree) whch passes below the mean-varance pars assocated wth dstrbutons F ε, F, and F +ε, and above the mean-varance pars assocated wth dstrbutons F and F +. We use ths curve as the upper-most equ-utlty curve for some utlty functon. (Other equ-utlty curves n the same famly should be below ths curve.) Snce all dstrbutons other than F are below or on the rght sde of ths curve, F s the optmzer of ths utlty functon. Q.E.D. See Barron (977) for a thorough treatment of mean-varance utlty functons. --

3 For the mutually exclusve nvestment choce problem, we conclude that the SSD admssble set s the optmal set. Alternatve Proof of Proposton Φ, f Φ s a Proposton : Gven a set of normal dstrbutons = { F,F,,F,F } n n+ U admssble set, then t s also the CSSD admssble set and optmal. Proof: Φ s an admssble set; therefore, dstrbutons are mutually undomnated. Snce n the normal dstrbuton case, SSD s equvalent to the mean-varance decson rule, we can order the dstrbutons n Φ n such a way that < < < n, and µ < µ < < µ n. The mean and standard devaton of dstrbuton F n + may be anywhere n the sequence of F, F,. F n Case : max { j} µ = µ < µ + A dstrbuton may only be domnated by an alternatve wth n n j n hgher mean. Therefore, the dstrbuton wth the hghest mean cannot be domnated by a mxture of dstrbuton wth lower means. Case : µ n+ µ n max { µ j} < =. We dvde the set Φ n two parts: = { F,,F } j n { F,,F } Φ = k+ n, such that µ k< µ n+ < µ k+ and < < requre the followng: n r k r r ( ) ( ) ( ) j j j j n j= k+ j = k n+ k+ Φ, and k. For CSSD, we λ F t dt+ λ F t dt F + t dt, r, and< some r (A-) In the context of prevous Generalzed Locaton-Scale dstrbuton famly-related footnotes, mean-scale admssble t dstrbutons wth the same degree of freedom, stable dstrbutons wth the same characterstc exponent and skewness and log-normal dstrbutons are optmal. --

4 Dvdng through by the rght-hand sde of equaton (A-), r r k F n j( t ) dt Fj( t) dt λ j r + λ j r, r, and< some r j = F ( ) j k n t dt = + + Fn+ ( t) (A-3) dt Consder the frst term on the left hand sde of equaton A-3) whch s the weghted sum of SSD ntegrals over dstrbutons to k. By the dstrbuton domnance Lemma, all terms n ths sum go to zero as r goes to mnus nfnty. Analogously, consder the second term on the left hand sde of equaton A-3) whch s the weghted sum of SSD ntegrals over dstrbutons k+ to n. By the dstrbuton domnance Lemma, all terms n ths sum go to nfnty as r goes to mnus nfnty. Furthermore, at least one of the convex combnaton weghts assocated wth dstrbutons k+ to n must be greater than zero. Otherwse, the expected value of the domnatng mxture wll not exceed the expected value of the domnance canddate dstrbuton, whch s a necessary domnance condton. Therefore, the value of the left hand sde s far greater than one for suffcently small r, and approaches nfnty as r goes to mnus nfnty. Admssble normally dstrbuted choce n+ cannot be domnated under CSSD. Q.E.D. A-. Optmal Portfolo Choces In an n-securty portfolo problem, returns are represented by a random n-vector x, and a choce s a vector of portfolo nvestment weghts. We assume that the set of all feasble portfolo choces, P, s a closed bounded set: T n = { n } P = π= π, π,, π, : π =, max π < M, -3-

5 Investors have utlty functons n the U class, and the return vector follows a jont normal dstrbuton. In ths case, SSD domnance s equvalent to a mean-varance orderng. As s well known, the effcent set s the upper half of the mean-varance fronter. Snce nothng can domnate the unque portfolo that maxmzes the expected value of a U functon, the SSD admssble set s effcent. Furthermore, we can show that the SSD admssble set (the upper half of the mean-varance fronter) s optmal. Followng Huang and Ltzenberger (988), pp : 3 = ( ), V s the varance-covarance matrx of x, (,,, ) e E x A T V e, B e T V e, C T V, D BC A > 0 (A-) T The equaton for the mean-varance effcent fronter s the followng: C A ( µ ) C D C C A Therefore, = + µ C D C A T =, < µ < µ M, µ π e, and π T V π C (A-3) Proposton A-: For every member of the SSD admssble portfolo choce set, there exsts a utlty functon n U that s the optmzer for that utlty functon. Proof: To show that a gven member of the SSD admssble set s optmal, we construct the assocated U -class utlty functon. In a one-parameter utlty famly { e ax a> } a gven pont on the effcent fronter ( µ, ), we set : 0 and for 3 See Merton (97) and Roll (976) for full treatments of ths effcent portfolo set. Our effcent set s the open nterval above the mnmum varance portfolo and below some arbtrary upper bound. The upper bound s necessary to have a well-posed optmzaton problem. -4-

6 a = D > 0. Cµ A E e ax a = µ The assocated expected utlty s [ ] Substtutng (3) nto (4), [ ] E e ax = a C A µ C + µ D C (A-4) Ths expected utlty functon wll be maxmzed at µ max = D + A = µ. Q.E.D. ac C For the portfolo choce problem, we conclude that the mean-varance effcent fronter s both the SSD admssble set and optmal. A-3 CSSD Portfolo Choces For portfolo choces, Baron (977) has shown that a choce vector, π, domnates the assocated mxed strategy, λ π, for all strctly concave von Neumann-Morgenstern utlty functons. Ytzhak-Mayshar (997) have proven ths result n the context of Margnal Stochastc Domnance for general dstrbutons. We present a corollary to these results as Proposton A-3. We frst develop two addtonal lemmas: Lemma A-: The SSD ntegral () s convex. Proof: The SSD ntegral s a twce contnuously dfferentable real-valued functon on an open nterval. Furthermore, ts second dervatve s the normal densty and hence, non-negatve throughout ts doman. From Rockafellar (970), convexty follows by Theorem 4.4, and essentally strct convexty follows by Theorem 6.3 (the SSD ntegral gradent s the normal dstrbuton and s postve over the real lne.) Lemma A-: A portfolo of normally dstrbuted choces SSD domnates the assocated mxture of normally dstrbuted choces. -5-

7 Proof: Gven Lemma A- [convexty of the SSD ntegral ()], a convex combnaton (mxture) of these ntegrals s no less than the SSD ntegral defned over the lnear combnaton (portfolo) of the assocated random varables. Wth ntegraton by parts, we have the followng: x x µ x µ x ( ) µ F t dt = Φ +φ φ x µ, and x µ Φ are the standard normal dstrbuton and densty, respectvely. For a portfolo to CSSD domnate a mxture requres ( ) ( ) and (A-5) x µ p x µ p x µ p x µ x µ x µ p Φ p +φ p α Φ +φ p x µ x µ x µ + ( α ) Φ ( ) +φ ( ) x, and 0 <α<., ( ) Defnng the portfolo weghts to equal the mxture weghts, we have ( ) ( ) p (A-6) x =α x + α x (A-7) µ =αµ + αµ p ( ) ( ) ( ) =α + α αρ+ α α + α p However, settng the correlaton equal to one mples that the portfolo standard devaton s a convex combnaton of the other two standard devatons, and that ths standard devaton s an upper bound on the actual portfolo standard devaton: ( ) =α + α (A-8) Therefore, p p ρ= x µ p x µ p x µ p x µ p x µ p x µ p p Φ +φ p ρ= Φ p p p pρ= +φ pρ= pρ= x µ x µ x µ x µ x µ x µ α Φ ( ) +φ ( ) + ( α ) Φ ( ) +φ ( ), x (, ) and 0<α<. (A-9) -6-

8 The frst panel of Fgure depcts two SSD ntegrals. The second panel of ths fgure plots an example for the equally-weghted mxture and portfolo of the two normally dstrbuted choces. Our CSSD effcent portfolo proposton follows: Proposton A-3: The mean-varance effcent fronter choces are CSSD admssble. Proof: Gven Lemma A-, any mxture of alternatves s domnated by an assocated portfolo. Any portfolo not assocated wth the mean-varance effcent fronter s domnated by some element of the set of portfolos on the effcent fronter. Therefore, mean-varance effcent portfolo choces domnate mxtures of portfolo dstrbutons, and all such portfolos are CSSD admssble. Lke mutually exclusve choce CSSD Proposton, Proposton A-3 shows that the entre mean-varance effcent portfolo fronter s optmal. 4 4 Analogous results wll hold for t dstrbutons wth the same degree of freedom, Cauchy dstrbutons, and stable dstrbutons wth the same characterstc exponent and skewness parameter. Snce portfolos of log-normal returns are not log-normal, the mutually exclusve choce result for ths dstrbuton does not extend smlarly. -7-

9 Fgure Normal Dstrbuton-Based Second-Order Stochastc Domnance Integrals 5 Panel A: Two Alternatve Choce Elements S 0.30 S Panel B: Mxture and Portfolo of the Alternatves Mxture Portfolo x x µ x µ = µ Φ +φ. 5 Wth ntegraton by parts, F ( t) dt ( x ) φ x µ are the standard normal dstrbuton and densty, respectvely. Φ x µ and -8-

10 Addtonal References Barron, Davd P, "On the Utlty Theoretc Foundatons of Mean-Varance Analyss," Journal of Fnance, 3(5). December 977, Huang, Ch-fu and Robert H. Ltzenberger, Foundatons of Fnancal Economcs, New York, North-Holland, 988. Merton, Robert, "An Analytc Dervaton of the Effcent Portfolo Fronter," Journal of Fnancal and Quanttatve Analyss, 7(4), September 97, Rockafellar, R. Tyrell, Convex Analyss, Prnceton, NJ, Prnceton Unversty Press, 970. Roll, Rchard, A Crtque of the Asset Prcng Theory's Tests: Part I: On Past and Potental Testablty of the Theory, Journal-of-Fnancal-Economcs; 4(), March 977, pages