Journal of Financial Economics 1 (1974) 67-94. e North-Holland Publishing Company

FALLACY OF THE LOG-NORMAL APPROXIMATION

TO OPTIhlAL PORTFOLIO DECISION-3lAKING OVER MAhi

PERIODS*

Robert C. MERTON and Paul A. SAMUELSON

Sloan School of Mu~magmrnf and Lkpartment of Economics.

M.I.T., Cambridge, Alms. 02139. U.S.A.

Received August 1973, revised version received October 1973

The fallacy that a many-period expected-utility maximizer should maximize (a) the expected logarithm of portfolio outcomes or (b) the expected average compound return of his portfolio is now understood to rest upon a fallacious use of the LawofLargc Nurtrbcrs. This paper exposes a more subtle fallacy based upon a fallacious use of the Cenfral-Limit Throrcnr. While the properly normalized product of indcpcndent random variables does asymptotically approach a log-normal distribution under proper assumptions, it involves a fallacious manipulation of double limits to infer from this that a maximizer of cxpcctcd utility after many periods will get a useful approximation to his optimal policy by calculating an cffrcicncy frontier based upon (a) the cxpcctcd log of wealth outcomes and its variance or(b) the cxpcctcd average compound rclurn and its variance. Lxpeclcd utilities cnlcuktted from the surrogate log-normal function dilTcr systcmaticnlly from the corrccl cxpcctcd utilities calculated from the true probability distribufion. A new concept of ‘initial wealth equivalent’ provides a transitive ordering of portfolios that illuminates commonly held confusions. A non-fallacious application of the log-normal limit and its associated mc:m-variance cllicicncy frontier is established for a limit whcrc nny/i.rcd/ horizon period is subdivided into cvcr more indcpcndcnt sub-intervals. Strong mutual-fund Separation Theorems are then shown to be asymptotically valid.

1. Introduction

Thanks to the revival by von Neumann and Morgenstern, maximization of the expected value of a concave utility function of outcomes has for the last third of a century generally been accepted as the ‘correct’ criterion for optimal portfolio selection. Opcrationnl theorems for the general case were delayed in becoming recognized, and it was appropriate that the seminal break- throughs of the 1950’s be largely preoccupied with the special case of mean- variance analysis.’ Not only could the fruitful Sharpe-Lintner-Mossin capital

*WC thank M.D. Goldman for scientitic assistance. Aid from the National Science Founda- tion is gratefully acknowlcdgcd.

‘Notable excepfions that-deal with the general case can be found in the works of Arrow (1965). Rothschild and Stiglitz (1970). and Samuelson (1967a). Alonn with mean-variance analysis, the theory of portfolio select-ion when the distributions are Pareto-Levy has been developed by Fama (1965) and Samuelson (1967b).

68 R.C. Merron. P.A. Samuelson, Log-twrmaf approximation to optimal portfolio choice

asset pricing model be based on it, but in addition, it gave rise to simple linear rules of portfolio optimizing. In the mean-variance model, the well-known Separation or Mutual-Fund Theorem holds: and with suitable additional assumptions, the model can be used to define a complete micro-economic framework for the capital market, and a number of empirically testable hypo- theses can be derived. As a result, an overwhelming majority of the literature on portfolio theory have been based on this criterion.’

Unfortunately, as has been pointed out repeatedly, the mean-variance criterion is rigorously consistent with the general expected-utility approach only in the rather special cases of a quadratic utility function or of gaussian distributions on security prices - both involving dubious implications. Further, recent empirical work has shown that the simple form of the model does not seem to fit the data as well as had been previously believed,3 and recent dynamic simulntionsJ have shown that the behavior over time of some efficient mean- variance portfolios can be quite unreasonable.

Aside from its algebraic tractability, the mean-variance model has interest because of its separation property. Therefore, great interest inhered in the Cass-Stiglitz (1970) elucidation of the broader conditions under which a more limited form of a ‘separation theorem’ must hold regardless of the probability distribution of returns. The special families of utility functions with constant- relative-risk aversions or constant-absolute-risk aversions further gained in interest.’ But it was realized that real-lift utilities riced not bc of so simple a form.

The dcsirc for simplicity of analysis Icd naturally to a search for approxi- mation thcorcms, particularly of the asymptotic type. Thus, even if mean- vnrinncc analysis were not exact, would the error in using it become small:’ A dcfcnsc of it W;IS the demonstration that mean-variance is asymptotically correct if the risks are ‘small’ (i.e., for ‘compact’ probabilities); closely related was the demonstration of the same asymptotic equivalcncc when the trading interval becomes small (i.e., continuous trading). More recently, it has been shown that as the number of assets becomes large, under certain conditions, the mean-variance solution is asymptotically optimnL6 And the present authors in another paper (1974) have developed a generalized mean-variance approxi- mation procedure that is somewhat described by that paper’s title. This enables

‘The literature is so extensive that we rcfcr the reader to the bibliography. Additional references can bc found in the survey articles by Fnma (1970) and Jcnscn (1972). and the book by Sharpe (1970).

‘See Black, Jensen and Scholes (1972) and Friend and Blumc (1970). ‘SW tlnkansson (1971~). ‘Cass and Stiglitz (1970) showed that these utility functions were among the few which

satisfied the separation property with respect to initial wealth for arbitrary distributions. Other authors [ttakansson (1970). Leland (1968). Mcrton (1969, 1971) and Samuclson (1969)] have previously made extensive use of thcsc functions.

*For the first, see Samuelson (1970); for the second, see Mcrton (1969, 1971); for the last, see Ross (19723).

R.C. ,tferron. P.A. Samuelson, L.og-normal approximation to optimal porffolio choice 69

all those with utility functions that are near to a particular utility function to

develop close approximations to optimal portfolios along a mean-variance

trade-off frontier generated by, a series expansion around that base function; this not only generalizes the standard Markowitz-Tobin mean-variance analysis of money outcomes, but can sometimes also validate the Hakansson efficiency

frontier of expected average compound return and of its variance. A particularly tempting hunting ground for asymptotic theories was thought

to be provided by the case in which investors maximize the expected utility of

terminal wealth when the terminal date (planning horizon) is very far in the

future. Recourse to the Law of Large Numbers, as applied to repeated multi- plicative variates (cumulative sums of logarithms of portfolio value relatives),

has independently tempted various writers, holding out to them the hope that one can replace an arbitrary utility function of terminal wealth with all its

intractability, by the function U( WT) = log( IV,): maximizing the geometric

mean or the expected log of outcomes, it was hoped, would provide an asympto- tically exact criterion for rational action, implying as a bonus the efficiency of a diversification-of-portfolio strategy constant through time (i.e., a ‘myopic’ rule, the same for every period, even when probabilities of direrent periods were interdependent!). So powerful did the max-expected-log criterion appear to be, that it seemed even to supersede the general expected utility criterion

in cases where the latter was shown to be inconsistent with the max-cxpcctcd-log

criterion. For some writers. it was a case of simple errors in reasoning: they mistakenly thought that the sure-thing principle sanctified the new criterion.’

For others, an indefinitely large probability ofdoing bcttcr by M&hod A than by Method B was taken as conclusive cvidcncc for the superiority of A.” Still other converts to the new faith ncvcr rcnlizcd that it could conllict with the plausible

postulates of von Neumann maximizing; or, still others, more sophisticatedly. hnvc tried to save the approximation by appealing to bounded utility functions.“

Except to prepare the ground for a more subtlc frlllncy of the same asymptotic

genus, the present pnpcr need not more than review the simple max-cxpcctcd-log fallacy. It can concentrate instcatl on the asymptotic fallacy that involves, not primarily the Law of Large Numbers, but rather the Central-Limit Theorem.

It is well known that portfolio stratcgics that are uniformly the same in every period give rise to a cumulative sum of logarithms of returns that do approach,

when normalized under specified conditions cusily met a gaussian distribution - sunecsting heuristically :I Log-Normal Surrogate, whose two parameters : Zc each period’s expected log of return, and variance of log of return - will bccomc ‘asymptotically sufficient parameters’ for efficient portfolio managing, truly

an enormous simplification in that all optimal portfolios will lie on a new efficiency frontier in which the first paramctcr is maximized for each different value of

‘See. for example. Auump (1971). ‘Set Urciman (1960) and La~clnc (1959). %x Markowirz (1959,1972).

70 R.C. Murton. P.A. Samuelson. Log-normal approximation to optimal portfolio choice

the second. This frontier can be generated solely, by the family U(IVr,) = (l+‘,)‘/y, and for y # 0, this leads away from the simple max-expected-log

portfolio. So this new method (if only it were valid!) would avoid the crude fallacy that men with little tolerance for risk are to have the same long-run

portfolio as men with much risk tolerance. Furthermore, since the mean and

variance of average-return-per-period are asymptotic surrogates for the log

normal’s first two moments (in a sense that will be described), Hakansson

(197lb) average-expected-return - which to many [such as us in paper (1974)) has no interest as a criterion for its own sake - seems to be given a new legiti-

macy by the Central-Limit Theorem. Furthermore, suppose the Log-Normal

Surrogate were valid, so that the true portfolio distribution could then be validly replaced by a log-normal with its E{log It’,} and var{log IV,]. In that case, the criterion of espected average compound return (which has no interest for

any thoughtful person whose utility function is far away from log bV,) could when supplemented by the variance of average compound return, be given for the first time a true legitimacy for all persons with U = IVYhi. Furthermore,

this would hold whether 1 -y is a small positive number or whether y is a very

large negative number. The reason for this is that mean and variance of average

compound return can be shown to be ‘asymptotic surrogates’ for the Log- Normal Surrogate’s two suficicnt parameters, TE(log W,} and Tvar{log W,},

CY, being any single period’s portfolio outcome. But. as WC shall show, it is a fallacy of double limits to try to use the (unnormalizcd) Log-Normal Surrogate.

And for pcoplc with U = P/y and y far from zero, the result can bc dis- nstcrously bad. I” What about the argument that cxpcctcd average compound

return dchcrvcs analysis bccausc such analysis may bc rclcvant to those decision

makers who do not have a max I:‘{U( )} criterion and who just happen to bc intcrcstcd in avcragc-compound-return ? After some rcflcction, WC think an

appropriate reaction would go as follows: It is a free country. Anybody can set up whatcvcr criterion hc wishes. Howcvcr, the analyst who understands the

implications of various criteria has the useful duty to help people clarify the goals they will, on reflection, really want. If, after the analyst has done his duty

of explaining the arbitrariness of the usual arguments in favor of average

compound return, any dccisionmakcr still persists in being interested in average compound return, that is his privilege. In our experience, once understanding

of the issues is realized, few decisionmakers retain their interest in average

compound return. One purpose of this paper is to show, by counter-examples and examination

of illcgitimatc interchange of limits in double limits, the fallacies involved in the above-described asymptotic log-normal approximation. What holds for

‘Oln ;! spin-ofT paper [Snmuclson and Merton (197J)] to this one. WC show that for U = W/e and c wry small, a perturbation technique involving E:log rt’, } and var {log W, ) can -quite indrpcndcntiy of any log-normality - give ;1 gcncralizcd mean-variance frontier for efficient local approximation. The two asymptotic moments of average compound return can, properly normalized. serve the same local-approximation purpose for l/Tsmall enough.

R.C. Mcrton. P.A. Samuelson, Log-normal approximafion to optimal portfolio choice 7 I

normalized variables is shown to be generally not applicable to octuul terminal

wealths. Then, constructively, we show that, as any fixed horizon planning

interval is subdivided into a number of sub-interval periods that goes to infinity (causing the underlying probabilities to belong to a gaussian infinitely-divisible continuous-time probability distribution), the mean-log and variance-log

parameters are indeed asymptotically sufficient parameters for the decision; so that one can prepare a (il. a) efficiency frontier that is quite distinct from the Markowitz mean-variance frontier of actual returns and not their logarithms;

this logarithmic (ii, a) frontier now provides many of the same two-dimensional simplifications (such as separation properties). The limit process involving

breakdown of a finite T into ever-more subperiods is, of course, quite different

from a limit process in which the number of fixed-length periods T itself goes

to infinity. There is still another kind of asymptotic approximation that attempts, as

T -, 00. to develop in Leland’s (1972) happy phrase a ‘turnpike theorem’

in which the optimal portfolio proportions are well approximated by a uniform strategy that is appropriate to one of the special family of utility functions, Cr( KiT) = ( W,)‘/y, where y- I is the limiting value of the elasticity of marginal

utility with respect to wealth as If’,. 4 co. Letting y then run the gamut from

I to --oo generates a new kind of efficiency frontier, distinct from that of ordinary mean-variance or of [E(log W’}, var{log IV}], but which obviously gcncralizcs the single-point criterion of the would-bc cxpcctcd-log maximizers.

It gcncralizcs that criterion in that wc now rationally trade-off mean return against risk, dcpcnding on our own subjcctivc risk tolerance paramctcr y.

Thcrc is some question ’ ’ as to the robustness of the Lcland theorem for utility functions ‘much diffcrcnt’ from mcmbcrs of the iso-elastic family; but the main

purpose of our paper is to uncover the booby traps involved in log-normal and other asymptotic approximations, and WC do not examine this rlucstion in any depth.

2. Exact solution

In any period, investors face II securities, 1, . . ., u. One dollar invested in the

jth security results at the end of one period in a value that is Z,(l), a positive random variable. The joint distribution of thcsc variables is specified as

Prob{Z,(I) 5 z,, . . ., Z”(I) 5 2”) = F[L!,, . . ., Z”] = F[z], (1)

where F has finite moments. Any portfolio decision in the first period is defined by the vector [n!,(l), . . ., w,(l)], L;wj(l) = I; if the investor begins with initial wealth of W,, his wealth at the end of one period is given by the random variable

W, = W,[w,(l)Z,(l)+ . . . +w”(l)z”(l)]. (2)

“Ross (1971b) provides a rather simple example where the Leland result does not obtain.

72 R.C. Merron. P.A. Samuelson. Log-normal approximation to optimal portfolio choice

By the usual Stieltjes integration over F[c,, . . ., zn], the probability distribution

of W, can be defined, namely

Prob{log(Ct’,/W,) 5 x} = P,[x; w,(l), . . ., w,,(l)]

= P,[x; w(l)].

(3)

An investment program re-invested for T periods has terminal wealth, W,.,

defined by iterating (2) to get the random variable

= W,[w(l), . . .( w(T)].

It is assumed that the vector of random variables [Z(t)] is distributed in- dependently of Z(f+k), but subject to the same distribution as Z(l) in eq. (I). Hence, the joint probability distribution of all securities over time is given by the product

Prob(Z(1) 5 z(l), . . ., Z(T) 6 z(T)}

= f+,(l), . . .* z.(l)]. . . F[z,(T), . . ., Z”(T)]

= F[z(l)] . . . F[z(T)].

(3

Since Wr/W, consists of a product of indcpendcnt variates, lo_c(W,./W,,)

will consist of a sum of indcpcndcnt variates. Thcrcforc. its probability distri- bution is, for each T, dcfinnblo rccursivcly by the following convolutions,

Prob(log(CVr/W,) 5 X} = P&; w(l), . . ., w(T)] (6)

Pz[x; w(l). w(2)] = f”_P ,[x-s;w(2)]P,[ds; w(l)]

P&V; w(l), w(2), w(3)] = &,P,[x-s; w(3)]P,[ds; w(l), w(2)]

I’,[.r; uil), . . ., w.(T)]

= ~‘?,/‘,[x-s; tv(T)]P,_ ,[ds; w(I), . . ., w(T- l)].

Here, as a matter of notation for Stieltjcs integration, ]“,f(s)dg(s) =

I”,f(Mds). The investor is postulated to act in order to maximize the expected value of

(concave) terminal utility of wealth

mnx E{ U,( W,[w(l), . . ., w(T)])} (W(O)

= max J?,_U?.( W,e’)P,[dx; w(l), . . ., w(T)] (WC’))

(7)

G O,[w**(l), . . ., w**(T); IV,].

Here, U,( ) is a concave function that can be arbitrarily specified, and [w,(f)], for each I, is understood to be constrained by Xyw,(f) = 1.

R.C. Merton, P.A. Samuelson, Log-normal approximarion IO optimal portfolio choice 73

For a general UT, the optimal solution [. . ., w**(t), . . .] will not involve portfolio decisions constant through time, but rather optimally varying in

accordance with the recursive relations of Bellman dynamic programming, as discussed by numerous authors, as for example in Samuelson (1969). But, here, we shall for the most part confine our attention to uniform strategies

[w,(t), . * *, W”Wl = [H’l, . . *, WA. (8) I

For each such uniform strategy, log (IV,/ IVe) will consist ofa sum ofindependent and identically distributed variates. We write the optimal uniform strategy as the vector w(r) s wf, and abbreviate

P,[s; w.] 3 P,[x; w, . . ., w], T = 2, 3, . . ., (9)

W,[w] 3 W,[w, , . .) M’],

O&v*; Wo] 55 &Jw*, . . .( w*; Ii’“].

Actually, for the special utility functions,

U,(IV) = kY’/y,y < 1,y # 0

= log rv, y = 0,

(10)

it is well known that w**(f) s WV*, indcpcndcnt of T, is a ncccssnry result for

full optimality.

3. Max-cxpcctcd-log (gcomctric-mean) fallacy

SupposcZ, rcprcscnts a’safc security’ with certain return

Z,, = e’ EE R 2 I. (I I)

If the other risky sccuritics arc optimally held in positive amounts, together their uncertain return must have an expcctcd value that exceeds R. Cons&r now the parameters

EUog( CvIIty,)]

var{log(LV,/W,)}

= E{log(~wjzj)~ = /(‘VI, . , ., IV”) = p(w), (12)

= Ei[logl~,*~jzj)-/l(“‘)]~} = fJ+,, . . ., WJ

= a’(w).

For w = (w,, . . ., IV,) = (0. 0, . . ., 0, I), /1(w) = r 2 0. As w, declines and the sum of all other H’j become positive, /d(w) must bc positive. However, j’(w) will reach a maximum; call it I, and recognize that wtt is the max- expected-log strategy.

71 R.C. dlrrfon, PA. Sumuelson. Log-normul approximation to optimal purrfolio chaice

As mentioned in sect. I, many authors fallaciously believe that w+? is a good

approximation to W* for T large, merely because

lim Prob{IV,[~~~+‘] > W,[w])= 1, w # w++. (13) T-I1

A by now familiar counter-example occurs for any member of the iso-elastic family. eq. (IO). with 7 # 0. For a given 7. w*([; = IV*, the same strategy,

independent of T; since each 7 is easily seen to call for a difTerent w.*, it must

be that W* # wtt for all T and p # 0, in as much as )I’* = tvtt only when y = 0 and Lr7(kI’) = log IV. Hence, the vague and tacit conjecture that w*

converges to IV+’ asymptotically as T + co, is false.

Others, e.g. hlarkowitz (1972. p. 3). who are aware of the simple fallacy

have conjectured that the max-expected-log policy will be ‘approximately’

optimal for large T when U,( ) is bounded (or bounded from above). I.e.. if U,( ) is bounded (or bounded from above), then the expected utility maximizer, it is argued, will be ‘approximately indifferent’ between the {NJ*} and {w++}

programs as T becomes large. The exact meaning of ‘approximate indiffcrencc’ is open to interpretation.

A trivial meaning would bc

lim EU,( W,Z., [IV*]) = M = ;,iJIln EU, ( CVoZT(“‘++]), (14) T-n

whcrc i\I is the upper bound of U,,( ). This dclinition mcrcly rcllccts the fact, that cvcn :I sub-optimal strategy (unless it is too absurd) will Icad as T -_) co

to the upper bound of utility. I:or example, just holdin, cr the risklcss asset with positive return per period R = cr > I, ~iil, for large enough 7’. get one arbitra-

rily &SC to the bliss Icvcl of utility. f ICSlCC, even if eq. (14)‘s dclinition of indi[l~rcncc ~crc‘ 10 m;lkc the corl,;ccturc trus, its implicalion:, h:ivc practically

II0 COl~lCI~l. I\ meaningful intcrprstation would be indillkcncc in terms of an ‘initial

\ve;~lth cquic;il~nt’. That is, ict ni,ri’,, bc the initial wealth ccluivaknt of a

pr~>,cra*n {IV’; rclatikc to prugram :~j; cklincd such thaw, for each ‘T,

f~Ur(II,,II’,,%, [Wi]) z EU,( lI’,%[[W’]). (15)

(rI,,- I,lV,, is the amount of additional initi:ll uc:llth the investor would

rcquirc to be indilkrcnt to giving up the {w’] program for the {w’] program. Thus, WC‘ could IJW fl, :(T; it’,) as a mcaurc’ ’ of how ‘close’ in optimality

terms the (N,’ j -.- (11 ’ ,+*; program is to the (No’) --- {bc*i program. The con-

jecture th3t max-cxpccted-lo, 11 is asymptotkllly optimal in this modified sense

would be true only if it could be shotkn that n, JT; Wo) is a dccrcasing function

’ *n,,t ; ) could be used to rnnk portfolios bccausc it provides a complrfc and transitive or&ring. Note: if ll,,( ; ) > 1 then fl,,( ; ) < 1. Howcvcr n,,( ; J i I/n,t( ; ).

R.C. Merron. P.A. Samuelson. Log-normd upproximurion M optimal portfolio choice 75

ofTand

lim II I r(T; IV’,) = 1, T-w

or even if II, JT; Wo) were simply a bounded function of T.

Consider the case when cI,( ) = ( )‘/y, y < 1. Then,

ECI,( bYJ,[W*]) = W,rE[(Z, [W*])‘]/y (16)

= w,~{E[(Z,[,~*l)‘l)TIY,

by the independence and identical distribution of the portfolio return in each period. Similarly, we have that

,?u,(n, 2 Ct’,Z,[w++]) = (n, 2 cv”)‘{E[(Z,[w++])‘))r/y. (17)

Now, {IV*} ma.uimizes the expected utility of wealth over one period (i.e., for the iso-elastic family IV** = w*), and since \I’* # w++ for y # 0, we have that

E[(Z,[rv*])‘) 5 E[(Z,[W++]i’] as y gj 0. (18)

From eqs. (I 5). (I 6), and (I 7). we have that

l-l, JT; IV,,) = [i.(y)]“‘, (19) where

l(Y) = E[(Z, [~v*l)yJIEIG, [w”l,‘l.

From eq. (IS), I.(y) > 1 and T/y > 0. for y > 0; 2.b) < I and T/y < 0. for y < 0; and, since A is indcpcndcnt of ‘I’and IV,, WC’ have from (19) that, for y # 0 and cvcry IV, > 0,

G7l-l ,z(‘T; w,):c!-r > 0, (‘01

lim ff , z(T; It’,,) = ~3. 7 -.,n

llcncc, even for l/.l( ) with ;III upper bound (as when y < 0). ;III investor

would require as T 3 a, ~111 ckcr-larger initial payment to give up his (II’*}

program. Similar results obtain for U,( ) functions \\ hich arc bourdcd front

above ad b&w. I3 Thcrcforc, the {w”} program is dclinitcly not ‘approsi- matcly’ optimal for large T.

Further, the sub optimal (~~r”j policy will, for every finite T however large,

be in a clear sense ‘behind’ the best strategy, {IV*}. Indeed, lr‘t 11s apply the tcbt used in the Eisenhower Administration to compare U.S. and U.S.S.R. grow, 111. How many years after the U.S. reached each real GNP level did it tahc the U.S.S.R. to reach that kvcl? This defines a function AT =/(GNP,), and some Kremlinologists of that day took satisfaction that AT WIS not declining in time.

In a new calculation similar to the initial wealth equivalent analysis above. Ict

us for each level of E[UT( IV,)] define AT = 7 *++-T* as the di!Tercncc in time

L3Scc Goldman (1972) for sonic cxamplcs.

76 R.C. Merton. P.A. Samuelron, Log-normal approximation to optimal portfoLio choice

periods needed to surpass that level of expected utility, calculated for the optimal strategy {w*} and for the max-expected-log strategy {wtt); then it is not hard to show that AT 4 co as T---c co. Again the geometric-mean strategy proves to be fallacious.

Finally, we can use the initial wealth equivalent to demonstrate that for sufficiently risk-averse investors, the max-expected-log strategy is a ‘bad’ program: bad, in that the {w’+} program will not lead to ‘approximate’ opti- mnlity even in the trivial sense of (14) and hence, will be dominated by the program of holding nothing but the riskless asset.

Define {IV’} s [0, 0.0, . . ., I] to be the program which holds nothing but the riskless asset with return per period, R 2 1. Let (w’] be the max-expected- log program {w’+} as before. Then, for the iso-elastic family, the initial wealth equivalent for the {w’} program relative to the {w’} program, II,, W,, is defined by

(21)

or l-l , J = [dcrll- T’y, (22)

where C/I(~) = E[(Z, [w~‘])~]/R~.

To cxaminc the propcrtics of the (b(r) function, we note that since {w++} G [IV+‘, IV++, . . ., wtt] dots maximize E log(Z[~; ;I]) E E lo@,Z~-‘wj[Zj(I)-R]

+ R), it must satisfy

f~‘([~,(l)-x]:Z[~~~‘+; I]]. = 0, j = I. 2,. . .,jI. (23)

Multiplying (23) by IV,++ and summing over j = I, 2. . . ., )I, WC have that

E{(Z[W++; I]-R)/%[w+‘; I]] = 0 (24) or that

E{(Z[rc++; I])- ‘} = R-‘. (25)

From eqs. (22) and (25), we have that

cl,(y) > 0; (b(O) = (/I(- 1) = I.

Further, by dilfercntiation,

(26)

c//(y) = E{(Z[w+‘; 1])‘log2(Z[w”; 11)) > 0, (27)

and so, 4 is a strictly convex function with a unique irltcrior minimum at

Ymin. From (Xi), - I < y”Ii” < 0, and thcrcfore, 4’(y) < 0 for y < Y,,,~“.

Hencc,sincc&(-1) = l,+(y) > 1 fory < -1. But, from (22). (f,(y) > 1 for 7 < - 1 implies that for any R, Cc/, > 0, and

y< -1, lim n,,(T)= Co, (28) r--o0

R.C. Merton, Pd. Samuelson. Log-normal approximation to optima/portfolio choice 77

and therefore, such risk-averse investors would require an indefinitely large initial payment to give up the riskless program for the max-expected-log one.

Further, in the case where R = 1 and the riskless asset is non-interest bearing cash, we have that for y c - 1, E[(Z,[w++; I])‘] > 1 which implies that

lim EUr( W&[w++; r]) = lim IV,{E[(Z[w++; 1 ])y]}r/y r-.m T-a2

= -al, (29)

so that such a risk-averse person’s being forced into the allegedly desirable max-expected-log strategy is just as bad for infinitely large T as having all his initial wealth taken away! Few people will opt to ruin themselves voluntarily once

they understand what they are doing. I4

4. A false log-normal approximation

A second more subtle, fallacy has grown out of the more recent literature on optimal portfolio selection for maximization of (distant time) expected terminal

utility of wealth. After giving arguments based on maximizing expected average-compound-

return that imply myopic and uniform strategies, w(l) 3 wr, Hakansson (1971 b, pp. 868-869) proceeds to use the Central-Limit Theorem to argue that for large T, the distribution of terminal wealth will be approximately log- normal. Then, he approximates the mean and variance of avcrage-compound- return by the first two moments of the associated Log-Normal Surrogate. To obtain an approximation to the cfhcient set for the mean and variance ofaveragc- compound-return, he (p. 871) substitutes the Log-Normal Surrogate for the true random variable portfolio return in an iso-elastic utility function and computes its cxpcctcd value prior to any maximization. This done, he is able to use the property that maximization of WY/y under the log-normal distribution reduces to a simple linear trade-off relationship between expected log of return and the variance of log return with y being a measure of the investor’s risk-return trade off. More generally, if a portfolio is known to have a log-normal distribution

with parameters [p. cr’] = [Elog( W,/ W,J, var log( WI/ W,,)], then for all utilities, it will be optimal to have a maximum of the first parameter for any fixed value of the second. While it is not true that for a fixed value of p, all concave utility maximizers would necessarily prefer the minimum a2, it is true that for a fixed value of a = log[E( W,/ W,)] = jd ++a2, all concave utility maximizers would optimally choose the minimum 0 . ’ I5 Hence, the Hakansson derivation can

‘%oldman (1972) derives a similar result for a bounded utility function. Thus. Samuelson (1971. p. 2495) conceded too much in his criticism of the geometric-mean policy when he stated that such a policy would asymptotically outperform any other uniform policy for utility functions bounded from above ‘and below.

“See Merton (1973. appendix 2) for a proof.

78 R.C. Merron. P.A. Samuelson. Log-normal approximation IO optimal portfolio choice

suggest that there exists an asymptotic ‘efficient frontier’ in either of the two related parameter spaces b, 0’1 or [z, CT’]. In the special case of the WY/y

family, the y determines the point on that frontier where a given investor’s optimal portfolio lies. One of us, independently, fell into this same trap.16

Unfortunately, substitution of the associated log-normal for the true distri- bution leads to incorrect results, as will be demonstrated by counter-example. The error in the analysis leading to this false conjecture results from an improper interchange of limits, as we now demonstrate.

For each uniform portfolio strategy {w}, define as in eq. (2), the one-period portfolio return in period t by

Z[t; W] E $WiZi(t) = W,[W]/ Wf_ 1 e (30)

Given the distributional assumption about asset returns in sect. 2, the Z[t; ~71 will be independently and identically distributed through time with

Prob(log(Z[t; w]) I; x} E P,(x; w). (31) I

The T-period return on the portfolio is defined, for any T 1 1, by

ZT[W] z fi z[f; w] = w&4/ w0 9 131

(32)

with

Prob(log(ZJw]) 6 x} EE P,(x; w) (33)

as in eq. (6) with the w’s indcpcndcnt of time. Dclinc Z$[w] to be a log-normally distributed random variable with para-

mctcrs PTand a”Tchoscn such that

117. z E(tog(z$.[w]} = Jw%(Z,[wl)~ = ~m%$z,[wl)~ (34)

(r2T s var{log(Z$[w])} = var{log(Z,[w])} = Tvar{tog(Z,[w])}.

We call Z$.[w] the ‘surrogate’ log-normal to the random variable ZT[w]; it is the log-normal approximation to Z,[w] fitted by equating the first two moments in the classical maximum-likelihood Pearsonian curve-fitting proce-

dure. Note that by definition

p = /J(W) = J’?‘,&‘,(dx; w), (35)

u2 = a2(w) = J”,,,fx-/~)2PI(dx; w).

lbThe neatness of the heuristic arguments and the attractiveness of the results has led B number of authors to assert and/or conjecture their truth. See Samuelson (1971, p. 2496) where P(w) and a*(w) are misleadingly said to be ‘asymptotically suHicient parameters’, an assertion not correct for T- 00. but rather for increasing subdivisions of time periods leading to diflusion-type stochastic processes.

R.C. Merton, P.A. Sumuelson, Log-normal approximation to optimal portfolio choice 79

Since each log(Z(t; w]) in 1: log(Z[r; w]) is an identically distributed, indepen-

dent variate with finite variance, the Central-Limit Theorem applies to give us the valid asymptotic relation.”

Central-Limit Theorem. As T gets large, the normalized variable, Y, E [Iog(ZT[w] - ~(w)Tj/a(wWT, approaches the normal distribution. I.e.,

lim Prob{Yr s JJ} = lim P&~(w)l/Ty+~(w)T; w] T-CO T-m

(36)

= N(y) G (2x)- ‘/‘p pe-f” dr,

where N( ) is the cumulative distribution function for a standard normal variate.

Since Iog(Zr[w]) = log(Wr[w]/W,), this is the valid formulation of the log- normal asymptotic approximation for the properly standardized distribution of terminal wealth.

In a more trivial sense, both Pr(x; w) for Z,[w] and also its surrogate, q(x; pT, dT) E N[(s-p(w)T)/o(wr)dT] forZ:[w], approach the same common limit, namely

where

lim P,(x; w) = L = lirn q(s; /lT, aZT), (37) T-m T-m

L = 1, /l(W) < 0, (38)

= 0, /1(W) > 0,

= 4, /i(W) = 0.

But it is not the case that

lim {[L-P&K; w)]/[L--(x; pT, a2T)]} = 1. i-*m

(39)

From the fcrrajirma of the valid Central-Limit thcorcm, a false corollary tempts the unwary.

False Corollary. If the returns on assets satisfy the distributional assumptions of sect. 2, then, as ‘f --* 00, the optimal solution to

mm E[U,,(W,[w])] = max IY)mUTIWO e”]P,(d.r; H’), (w) Iv)

over all’* unijorm strategies {w}, will bc the same as the optimal solution to

ya)x E[ U,( W,Z,[w])[ = max jE’_UT[ W, eQ’(x; ,uT, a2T) dx, W (W)

“To apply the Central-Limit Theorem correctly, the choice for (w) must be such that Z[r; w] is a positive random variate with its logarithm well defined. Our lntcr discussion of optimal politics is unaffected by this restriction since the class of utility functions considered will rule out Z[r; w] with a positive probability of ruin. Cf. Clakansson (1971b, p. 868).

“With reference to footnote 17. the corollary is false even if we restrict the set of uniform strategies considered to those such that Prob (Z,(w] = 0) = 0 for all finite T.

80 R.C. Merton. P.A. Samuelson, Log-normal approximation to optimal portfolio choice

where n’(x; pT, a*T) = (Zrra*T)-I’* exp[-(x-pT)*/2cr2T] = N’[(x-p(w)T)/ o(wWal.la(wWT. I.e., one can allegedly find the optimal portfolio policy for large T, by ‘replacing’ Z,[w] by its surrogate log-normal variate, Zi[w].

Let us sketch the usual heuristic arguments that purport to deduce the false corollary. From eqs. (36) and (37). one is tempted to reason heuristically, that since Yr is approximately distributed standard normal, then log(Z,[w]) = adTY,+pT E X, is approximately normally distributed’ 9 with mean FT

and variance a*T, for large T. Hence, if Xr has a density function P,‘(x;w) then, for large T, one can write the approximation

P,‘(x; w) x rj(x; /tT, o*T), (40)

and substitute the right-hand expression for the left-hand expression whenever T is large. As an example, from eq. (40), one is led to the strong (and false!) conclusion that, for large T,

jZ m U( W, e’)P;(x; w) dx z j?mU(Wo e%‘(x; pT, a*T)dx, (41)

in the sense that

lim [JZDo U( W, e*)P;(x; w) dx- jZmCI( CY, e”)q’(x; pT, a*T) dxJ T-m

= j?_C.J(W, c’) Jim_ [P;(x; w)-q’(x; FT. a*T)] dx = 0. (42)

But, actually cq. (42) is quite false as careful analysis of the Central-Limit Thcorcm will show.

A correct analysis immcdiatcly shows that the heuristic argument leading to (41) - (42) involves an incorrect limit intcrchangc. From cq. (36). for each T,

the random variable Y, will be seen to have a probability function F=(y; w) =

Pr(o(w)d?y++~(rv)T; w). By dclinition, WC have that

0,. = I”,U( IV, e”)P.,.(dx; w)

= jZ’,UIWO exp(&Ty+llT)]t;;(dy; w), (43)

and, for the surrogate-function calculation,

j?_LI( IV, e”)q’(x; PT. o*T) dx (44)

=j?‘_,CJ[W,, exp(o%@y+lT)]N’(y) dy.

Further, from the Central-Limit Theorem, we also have that, in the case where

PA ; w) and Fr( ; W) have densities, aP,/dx = P;(x; w) and aF,/ay =

F;(Y; 4, lim F+(y; w) = N’(y). (45)

T-m

‘The argument is that a constant times a normal variate plus a constant, is a normal variate.

R.C. Merron. PA. Samuelson. Log-normal approximation to optimal portfolio choice 8 I

However, to derive eq. (42) from (43) - (45), the following limit interchange would have to be valid for each y,

lim {UIWO exp(dTy+@)]F;Ot; w)} T-m

(46)

In general, as is seen from easy counter-examples, such an interchange of limits will be illegitimate, and hence, the False Corollary is invalid. In those cases where the limit interchange in eq. (46) is valid (e.g., lJ is a bounded function), the False Corollary holds only in the trivial sense of (14) in sect. 3. I.e., as already noted, in the limit as T + co, there will exist an infinite number of

portfolio programs (including holding one hundred percent of the portfolio in the positive-yielding, riskless asset) which will give expected utility levels equal to the upperbound of CJ. As we now show by counter-example. it is not true that portfolio proportions, w*, chosen to maximize expected utility over the PT( ; w) distribution will be equal to the proportions, wt, chosen to maximize expected utility over the surrogate q( ; pT, a*T),even in the limit.

To demonstrate our counter-example to the False Corollary, first, note that for the iso-elastic family, the expected utility level for the surrogate log- normal can bc written as

E( W,Z$[w])‘/y = W,‘exp[yrf(w)T+jyZaZ(w)T]/y. (47)

As Hakansson (1971 b) has shown, maximization of eq. (47) is cquivalcnt to the maximization of

[P(W) + tYcr*(W)l. (48)

Hence, from eq. (48). the maximizing wt for cq. (47) depend only on the mean and variance of the logarithm of one-period returns.

Second, note that bccausc the portfolio returns for each period arc indcpcn- dently and identically distributed, we can write the expected utility level for the true distribution as

L.-(W,Z,[w])7/y = W,l{E(Z,[w]‘)}‘/y.

Hcncc, maximization ofeq. (49) is equivalent to the maximization of

(49)

E(Z 1 [w.P)/Y 9 (50)

which also depends only on the one-period returns.” Consider the simple two-asset case where Z[r; w] G w(_JJ~- R)+ R and

where the y, are indcpcndent, Bernoulli-distributed random variables with

“‘This is an important point because it implies that if the conjecture that {IV+) = (IV*) could hold for the iso-elastic family for large r. then it would hold for Tsmall or r = I ! I.e., p(w) and u(w) would be suffkient parameters for the portfolio decision for any time horizon.

82 R.C. Merton, P.A. Samuelson. Lag-normal approximation to opfimal portfolio choice

Prob{y, = A} = Prob{y, = 6) = l/2 and I > R > 6 > 0. Substituting into eq. (50), we have that the optima! portfolio rule, w*, will solve

max ([wfd-R)+R]‘+[s(S_R)+R]‘}/27, (51) (WI

which by the usual calculus first-order condition implies that w* will satisfy

0 = (&R)[w*(l-R)+RJ’-‘+(&-R)[w*(S-R)+R)y-’. (52)

Rearranging terms in eq. (52), we have that

[(R - @/(A - R)] = [(w*(A - R) + R)/(w*(b - R) + R)]‘- ’ (53) or

where w* = w*(y) = (A- l)R/[(I-R)+A(R-a)], (54)

A E [(R-S)/(k-R)]“Y-’ .

From eq. (35), the surrogate log-normal for Zr[w] will have parameters

and /.I = /l(w) E 1/2(log[w(A-R)+ R]+log[w(S-R)+R]}, (55)

02 = r?(W) z l/2{log2[w(A-R)+R]+logZ[~(6-R)+R]}-jr*.

(56) The optimal portfolio rule relative to cq. (48), wt, will bc the solution to

0 = l/2 R-R S-R

wt(i. - R) + R + w’(d - R) + R

I -; log[(w’(lt- R)+ R)(w+(S- R)+ R)] >

+; log[~+(A- R)+ R] ( wt(f$+R)

+,~~[w’(s-R)+Rl(~+~~~~+~~},

which can be rewritten as

0 = (~)[l+;log(o)]-B[l-flog(B)],

(57)

(58)

where B 3 [w’(I.- R)-t- R]/[w+(G- X1+ R]. Note that since both w’* and W’ are independent of T, if the False Corollary had been valid, then IV* = wt.

Suppose W* = wt. From the definitions of A and B and from eq. (53), we have that B = A and (A- R)/(R-5) = A *-l. Substitute A for Bin eq. (58) to get

I/(y) = A~-‘[~+~log(n)]-n[l-qlog(A)]. (59)

R.C. Mrrton, PA. Same!son, Log-normal approximation to optimal portfolio choice 83

For arbitrary 1, b, and R, H(g) = 0 only if y = 0 (i.e., if the original utility function is logarithmic). Hence, w* # wt for y # 0, and the False Corollary is disproved. The reason that w* = wt for y = 0 has nothing to do with the log-normal approximation or size of T since in that case, eqs. (48) and (50) are identities.

This effectively dispenses with the false conjecture that for large T, the log- normal approximation provides a suitable surrogate for the true distribution of terminal wealth probabilities; and with any hope that the mean and variance of expected-average-compound-return can serve as asymptotically sufficient decision parameters.

5. A chamber of horrors of improper limits

It is worth exposing at some length the fallacy involved in replacing the true probability distribution, Pr(x; w), by its surrogate log-normal or normal approximation, ~(x; pT, o*TI = N[(x-pT)/adTJ. First as a salutary warning against the illegitimate handling of limits, note that by the deEnition of a pro- bability density ofP,(x; w), dP,(x; w)/Sx, where such a density exists

j”,P;(x; w) dx E 1. W

(60)

Also, as is well known when summing independent, identically-distributed, non-normalized variates, the probabilities spread out and

lim P;(x; w) dx z 0. T-m L

(61)

Combining cqs. (60) and (61), we see the illegitimacy of interchanging limits in the following fashion:

1 = lim jZ_P;(x; w) dx = I?‘_ lim P;(x; w) dx

= J;,mg.dx = 0.

r-o3

(62)

Similarly, as was already indicated for a non-density discrete-probability example in the previous section, from the following true relation,

lim [P;(x; w)-~‘(x; ptr, a*r)] = 0, (63) T-UJ

it is false to conclude that, for 0 c y < 1,

lim j’Z,e”[P;(x; w)--)7’(x; pr, a’r)] dx T-m

_eyX lim [P>(x; w)-~‘(x; pT, a*T)] dx

(64)

84 RX, Merton, PA Sanudon, Lq-normal approximation to optimal portfolio choice

Hence, trying to calculate the correct Or[w], in eq. (7) even for very large T, by relying on its surrogate

j’?=eT’q’(x; pT, 02T) dx = exp(Irc44 + 1/2u*c~*(w)lT} (65)

leads to the wrong portfolio rules, namely to wt # w* 3 w**(r). Thus, we hope that the fallacy of the surrogate log-normal approximation

with respect to optimal portfolio selection has been laid to rest. In concluding this debunking of improper log-normal approximations,

we should mention that this same fallacy pops up with monotonous regularity in all branches of stochastic investment analysis. Thus, one of us, Samuelson (19651, had to warn of its incorrect use in rational warrant pricing.

Suppose a common stock’s future price compared to its present price,

V(r+T)/V(t) E Z[T], is distributed like the product of Tindependent, uniform probabilities P[Z]. For T large, there is a log-normal surrogate for

Prob{log(Z[T]) d x} E H,(x). (66)

Let the ‘rational price of the warrant’ be given, as in the cited 1965 paper, by

FJV] = e-“TJ?‘,max[O, Vex-C]fI;(x) dx, (67)

where C is the warrant’s exercise price and e” = E{ V(r+ 1)/V(t)}. Although the density of the normalized variate y G (log(Z[Tj)-pT)/adT approaches N’(y), it is false to think that, even for large T,

err + j?‘,c^q’(_r; pT, o*T) dx = exp{[p+ 1/2a*]T}. (68)

Nor, for finite T, will we in other than a trivial sense, observe the identity

WV] = FWI (69)

= exp{(-jc- 1/2cr*]T}j?,max[O, Vex-C]$(x; pT, a*T)dx.

It is trivially true that lim,,, {F.r[V]-F{.[V]} = Y- Y = 0, but untrue

that lim,,,{(V-F,[V])/(V-Fi[V]} = 1. Of course, if H,(x) is gaussian

to begin with, as in inlinitcly-divisible continuous-time probabilities, there is no need for an asymptotic approximation and no surrogate concept is involved to betray one into error.

Similar remarks could be made about treacherous log-normal surrogates misapplied to the alternative 1969 warrant pricing theory of Samuelson and Mcrton. * ’

To bring this chamber of horrors concerning false limits to an end, we must stress that limiting inequalities can be as treacherous as limiting equalities. Thus, consider two alternative strategies that produce alternative random

2*Samuclsonand Mcrton (1969).Since Black-Scholes (1973)warrant pricing is basedsquarely on exact log-normal definitions, it is inexact for non-log-normal surrogate reasoning. See Mcrton (1973) for further discussion.

R.C. Merton. PA. SamueLson, Log-normal approximation to optimalportfolio choice 85

variables of utility (or of money), written as U, and U, I respectively and satis- fying probability distributions P,(U; T) and P, ,(U; T). Too often the false inference is made that

lim Prob{U, > U,,} = 1 T-W

(70)

implies (or is implied by) the condition

lim {E[U,]-E[U,,]} > 0 (71) T-m

Indeed, as was shown in our refuting the fallacy of max-expected-log, yielding U, rather than the correct optimal U,, that comes from use of w* # wtt, it is thecase that eq. (70) is satisfied and yet it is also true that

E[U,- U,,] < 0 for all Thowever large. (72)

Or consider another property of the tvtt strategy: Namely,

Prob{Z,[w”] 5 x} 5 Prob{Z,[w] 5 x} for allx < M(r; w), (73)

where M(T; w) is an increasing function of T with lim,,,M(T; w) = co. Yet eq. (73) does not imply eq. (71) nor does it imply asymptotic First Order Stochastic Dominance. I.e., it does not follow from eq. (73) that, as T -+ 00,

Prob{Z,[w”] 5 x} s Prob{Z,[w] s x} for al/x independently of T. (74)

The moral is this: Never confuse exact limits involving normalized variables with their naive formal extrapolations. Although a=b and b = c implicsa = c; still u = b and b z c c:innot reliably imply (I z c without careful restrictions put on the interpretation of’ NN ‘.

6. Continuous-time portfolio selection

The enormous interest in the optimal portfolio selection problem for in- vestors with distant time horizons was generated by the hope that an inter. temporal portfolio theory could be developed which, at least asymptotically, would have the simplicity and richness of the static mean-variance theory without the well known objections to that model. While the analysis of previous sections demonstrates that such a theory does not validly obtain, we now derive an asymp- totic theory involving limits of a different type which will produce the conject- ured results.

We denote by T the time horizon of the investor. Now, we denote by N the number of portfolio revisions over that horizon, so that h = T/N will be the length of time between portfolio revisions. In the previous sections, we examined the asymptotic portfolio behavior as T and N tended to infinity, for a iixed h(= I). In this section, we consider the asymptotic portfolio behavior for a

86 R.C. Merton, P.A. Samuelson, Log-normal approximation to optimal portfolio choice

fixed T, as N tends to infinity and h tends to zero. The limiting behavior is interpreted as that of an investor who can continuously revise his portfolio. We leave for later the discussion of how realistic is this assumption and other assumptions as a description of actual asset market conditions, and for the moment, proceed formally to see what results obtain.

While the assumptions of previous sections about the distributions of asset returns are kept, we make explicit their dependence on the trading interval length, h. I.e., the per-dollar return on the ith asset between times t--5 and t when the trading interval is of length h, is Z,(t, T; h), where T is integral in h.

Suppose, as is natural once time is continuous, that the individual Zj( ) are distributed log-normally with constant parameters. I.e.,

where

and

Prob{log[Zj(l, r; h)] g X} = N(x; PIT, ~fr),

E{log[Zj(l, r; h)l] = Cljr

var{log[Zj(f, r; A)]} = Q:T,

(75)

(76)

with 11, and CJ: independent of h. Further, define

a,r E (rl, + l/2+

= log[E{Z,(t. r; IJ))],

(77)

and o,iT = COV{lOg[z,(t, 7; jr)], lOg[z,(f, T; h)]}. (78)

As in eq. (I I), the nth asset is risklcss with a, = r and ~J;T = 0. WC dcnotc by [w*(l; /I)] the vector of optimal portfolio proportions at time I

when the trading interval is of length h, and its limit as h + 0 by ,v*(t; 0). Then the following separation (or ‘mutual fund’) thcorcm obtains:

Theorem. Given n assets whose returns are log-normally distributed and given continuous-trading opportunities (i.e., h = 0), then the {w*(r; 0)) are such that: (1) There exists a unique (up to a non-singular transformation) pair of ‘mutual funds’ constructed from linear combinations of these assets such that indepen- dent of preferences, wealth distribution, or time horizon, investors will be indif- ferent between choosing from a linear combination of these two funds or a linear combination of the original n assets. (2) If Z, is the return on either fund, then Z, is log-normally distributed, (3) The fractional proportions of respective assets contained in either fund are solely a function of the a, and ai,. (i,j = 1, 2, . . .( n) an ‘efficiency’ condition.

Proof of the theorem can be found in Merton (1971, pp. 384-386). It obtains whether one of the assets is riskless or not. From the theorem, we can always work here with just two assets: the riskless asset and a (composite) risky asset

R.C. Merton, PA. SamueLon. Log-normal approximation to optimal portfolio choice 87

which is log-normally distributed with parameters (a, c?).” Hence, we can reduce the vector [w*(t; O)] to a scalar w*(r) equal to the fraction invested in the risky asset. If we denote by a*( E w*(f)(x-r)+r) and u:( E WHACK), the (in- stantaneous) mean gain and variance of a given investor’s optimal portfolio, an efficiency frontier in terms of (s*, a,) or (p*, a,) can be traced out as shown in either fig. la or 1 b. The (z*, a,)-frontier is exactly akin to the classical Markowitz

(a+, cI) 1 - EFFIC!ENCY FRC)UTIER

an)

a*=r+ X0,

Fig. la

(pa. (ra 1 - EFFICIENCY FRONTIER

Fig. I b

single-period frontier where a ‘period’ is an instant. Although the frontier is the same in every period, a given investor will in general choose a difTerent point on the frontier each period depending on his current wealth and T-f.

In the special case of iso-elastic utility,23 UT = W/y, y < 1, w*(r) = w*, a constant, and the entire po:tfolio selection problem can be presented graphi-

“See Met-ton (1971. p. 388) for an explicit expression for a and u* as a function of the 01 and CT,]. f ./ = 1. 2. . ., n.

“See Merton (1969. p. 251 and 1971, pp. 388-394).

88 R.C. Merton. P.A. Samuelson. Log-normal approximation lo optimal portfolio choice

tally as in fig. 2. Further, for this special class of utility functions, the distribu- tion of wealth under the optimal policy will be log-normal for all t.24

Hence, from the assumptions of log-normality and continuous trading, we have, even for T finite and not at all large, a complete asymptotic theory with all the simplicity of classical mean-variance, but without its objectionable assumptions. Further, these results still obtain even if one allows intermediate consumption evaluated at some concave utility function, V(C).~’

Having derived the theory, we now turn to the question of the reasonableness of the assumptions. Since trading continuously is not a reality, the answer will depend on ‘how close’ w*(r; 0) is to w*(r; 11). I.e., for every S > 0, does there

PORTFOLIO SELECTION FOR ISO-ELASTIC UTILITI FUNCTION5

exist an /I > 0, such that I(w*(;; 0)- w*(r; /~)li -C 5 for some norm [I 1, and

what is the nature of the S = S(l)) function ? Further, since log-normality as

the distribution for returns is not a ‘known fact’, what arc the conditions such that one can validly use the log-normal as a surrogate?

Since the answer to both questions will turn on the distributional assumptions for the returns, WC now drop the assumption of log-normality for the Zj( ; h),

but retain the assumptions (maintained throughout the paper) that, for a

‘%ee Mcrton (1971, p. 392). ‘“Further the log-normality assumption can bc weakened and still some separation and

cfflcicncy conditions will obtain. See Mcrton (forthcoming).

R.C. Merton. PA. Samuekon. Log-normal approximation IO optimal portfolio choice 89

giuen h, 26 the one-period returns, Zj( , h; h) have joint distributions identical

through time, and the vector [Z(r, h; h)] E [Z,(r, h; hl, Z,(r, h; h), . . ., Z,(t,h; h)] is distributed independently of [Z(t + s, h; h)] for .F > h.

By definition, the return on the ith security over a time period of length Twill be

Z,(t, T; h) s fi Z,(kh, h; It). (79) k=l

Let X,(k, h) E log[Zj(kh, h; h)]. Then, for a given h, the {Xi@, h)} are inde- pendently and identically distributed with non-central moments

mj(i; h) z E{[X,(k, h)]‘}, i = 0, 1, 2,. . ., k

and moment-generating function

Jlj(A; h) 7 E{exo[JX,(k, A)])

= E(fZ,(k/r, h; h)]“}.

(81)

Define the non-central moments of the rule of return per period, Z,(kh, h; II) - I,

by Mj( 1 ; It) = aj(lr)h 3 EfZj(kir. /I; A)- I), (82)

k

M,(2; /I) = 0:(/1)/r + (a,(h)h) * 7 E{[Z,(kl/, It; /I)- l]Z},

M,(i; /I) 5 E([Z,(klf, h; A)- I]‘}, i = 3,4,. , .,

k

whcrc a,(It) is the cxpcctcd rate of return per unit time and o;(A) is the variance of the rate of return per unit time.

Samuelson (1970) has dcmonstratcd that if the moments M,( ; /I) satisfy

M,( 1 ; h) = O(h), (83)

M,(2; II) = O(A),

M,(k; II) = o(h), k > 2)

then I(w*(l; 0)- w*(r; A)\\ = 0(/r) where’0’ and’o’aredefined by

g@) = 0(/J) I if (g/h) is bounded for all h 5 0 (84)

‘61t is not required that the distribution of one-period returns with trading interval oflcngth hl will be in the same family of distributions as the one-period returns with trading interval of length /I*(# h,). Le.. the distributions riced not bc infinitely divisible in time. However, we do require sufficient regularity that the distribution of X,(P: h) is in the domain of attraction of the normal distribution, and hence. the limit distribution as /I + 0 will be infinitely divisible in time.

90 R.C. hlerton. P.A. Samuelson. Log-normal approximation to optimal portfolio choice

and g(h) = o(h), if lim(glh) = 0.

h-0 (85)

Thus, if the distributions of returns satisfy eq. (83), then, for every 6 > 0, there exists an h > 0 such that 11 w*(t; 0) - w*(r; h! /I < S, and the continuous- time solution will be a valid asymptotic solution to the discrete-interval case. Note that if eq. (83) is satisfied, then cXj(h) = O(1) and 4(F) = O(l), and ccj E iim,,,a,(h) wit1 be finite as will u2 E lim,,,@).

Given that Mj( ; h) satisfies eq. (83). we can derive a similar relationship for mj( ; /I). Namely, by Taylor series,”

mj(1 ; h) = E{lOg(l + [Zj- I])} (86)

= E{ Jr (- l)“‘[Zj- l]‘/i}

= Mj( I ; /I) - l/2Mj(2; A)+ o(h). from eq. (83)

= 0(/l),

and mj(2; h) = E{log2(l + [Z,- 11)) (87)

= E{ ,f, ,T, (- l)‘+p[z,- l]“‘/i/J} = X

= M,(2; /1)+0(/l)

= O(h),

and, in a similar fashion,

mj(k ; /I) = o(h), for k > 2. (88)

Thus, the mj( ; hj satisfy eq. (83) as well. If we detinc ~~(11) E m,(l; 11)//r as the meart logarithmic return per unit time and o!(h) z~ [m,(2; h)-&/r)h2]/h as the variance of the log return per unit time, then from eqs. (82) (86) and

(87), we have that

and

a, = limcc,(h) = limbcj(/l) + l/24(/1)] = ~j + l/24 h-0 h*O

0: E limos = limaj(h) = crj. h-0 h-0

(89)

(90)

Eq. (89) dcmonstrrites that the true distribution will satisfy asymptotically the exact relationship satisfied for all k by the log-normal surrogate; from eq. (90), the variance of the arithmetic return will equal the variance of the

37The series expansion is only valid for 0 < 2 5 2. Hence, a rigorous analysis would dc- vclop a second expansion for 2 > 2. However. as is shown later in eq. (95). the probability that 2 > 2 can bc made arbitrarily small by picking /I small enough. Thus, the contribution of the 2 > 2 part to the moments o(h).

R.C. Merton, P.A. Sam&son, Log-normal approximation to optimal portfolio choice 91

logarithmic return in the limit. Hence, these important moment relationships match up exactly between the true distribution and its log-normal surrogate.

If we define Y,(T; h) = log[Zj(T, T; h)], then from eq. (79), we have that

Yj(T; h) = i X,(k, h), where N = T/h, (91) k=l

and from the independence and identical distribution of the Xi< ; h), the moment-generating function of Y, will satisfy

4j(A; h, T) E E(exp[A Yj(T; h)]) (92)

= [l(lj(A; h)lr’lh.

Taking logs of both sides of eq. (92) and using Taylor series, we have that

log[$j(J; hv r)l = (Tlk)loE[$j(~; h)l (93)

= (T//z)log[ ,Fo +$k’ (0; /z)Ak/k!] z

= (T/h)log[ &nj(k; h)l’/k !] ?A4

= (T/h)log[ I +nrj( I ; h)A+ 1/2nr,(2; h)A2 +0(/t)]

= T[J(mj( 1; h)/h) + (A*/~)(vI,(~; h)/ll) + O(h)] e

Substituting ~c,(h)h for m,(l ; II) and a~(lr)h+p~(h)l~~ for nrj(2; II) and taking the limit ash + u in eq. (93) WC have that

log[g,(A; 0, T)] = lit-n log[&A; h, T)] (94) h-l

and thercforo, 4,(1; 0, ‘f) is the moment-generating function for a normally- distributed random variable with mean p,Tand variance ojT.

Thus, in what is essentially a valid application of the Central-Limit Theorem, we have shown that the limit distribution for Y,(T, T; II) as h tends to zero, is under the posited assumptions gaussian, and hence. the limit distribution for Z,(T, T; 11) will be log-normal for dljnife T. Further from eqs. (89)-(90), the surrogate log-normal, fitted in the Pearsonian fashion of earlier sections, will, for smaller and smaller /I, be in the limit, the true limit distribution forZ,(T, T; 0).

It is straightforward to show that if the distribution for each of theZj( , h; h),

j = 1,2,. . .( n, satisfy eq. (83), then for bounded Wj(l; A), the ‘single-period’ portfolio returns, Z[r; H’(I). h], will for each f satisfy eq. (83). However, unless the portfolio weights are constant through time [i.e., w,(f) = w,], the resulting limit distribution for the portfolio over finite time, will not be log-normal.

How reasonable is it to assume that eq. (83) will be satisfied? Essentially

92 R.C. Merton. PA. SamueLson, Log-normal approximation IO optimal portfolio choice

eq. (83) is a set of sufficient conditions for the limiting continuous-time sto- chastic process to have a continuous sample path (with probability one). It is closely related to the ‘local Markov property’ of discrete-time stochastic processes which allows only movements to neighboring states in one period

(e.g., the simple random walk). A somewhat weaker sufficient condition [implied by eq. (83); see also Feller

(1966, p. 321)] is that for every 6 > 0,

Prob{-b 5 Xj( ;h) S 6) = l-o(h), (95)

which clearly rules out ‘jump-type’ processes such as the Poisson. It is easy to show for the Poisson that eq. (83) is not satisfied because M,(k; h) = O(h) for all k and similarly, eq. (95) is not satisfied since Prob(-b 5 Xj( ; h) 4 6) = 1 -O(h).

In the general case when eq. (95) is satisfied but the distribution of returns are not completely independent nor identically distributed, the limit distribution will not be log-normal. but will be generated by a diffusion process. Although certain quadratic simplifications still occur, the strong theorems of the earlier part of this section will no longer obtain.

The accuracy of the continuous solution will depend on whether, for reason- able trading intervals, compact distributions are an accurate representation for asset returns and whcthcr, for these intervals, the distributions can be taken to be independent. Examination of time series for common stock returns shows that skcwncss and higher-order moments tend to bc negligible relative to the first two moments for daily or weekly observations, which is consistent with eq. (83)‘s assumptions. Howcvcr, daily data tend also to show some negative serial correlation, significant for about two weeks. While this finding is inconsistent with indcpcndcncc, the size of the correlation coctlicicnt is not large and the short-duration of the correlation suggests a ‘high-speed of adjustment’ in the auto-correlation function. Hence, while we could modify the continuous analysis to include an Ornstcin-Uhlenbcck type process tocapturc theseeffects, the results may not differ much from the standard model when empirical esti- mates of the correlation are plugged in.

‘Vf. Merton (1971. p. 401).

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