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1 ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth process never falls below a xed fraction of its maximum-to-date, and one strives to maximize the long-term growth rate of its expected utility. This problem was introduced and solved explicitly by Grossman and Zhou; we present an approach which simplies and extends their results. Introduction and Summary In a very interesting recent article, Grossman & Zhou (993) consider the classical portfolio optimization problem of Merton (97) under the \drawdown constraint" that the wealth process X () satisfy: X >max X (s); 8 t< (:) st almost surely. In other words, one admits only those portfolios () for which the corresponding wealth process X () never falls below % of its maximum-to-date, for some given constant IMA, University of Minnesota, Minneapolis, MN 55455, on leave from Columbia University, cvitanic@ima.umn.edu y Department of Statistics, Columbia University, New York, NY 7, ik@stat.columbia.edu

2 (; ). The objective is then to maximize the long-term growth rate R():= lim T T log E(X (T )) (:) of expected utility, for some power (; ), over portfolio rules () that satisfy (.). Using a mixture of analytical and probabilistic arguments, Grossman and Zhou provide an explicit solution to this problem, when investment is between a bond and one stock (modeled by geometric Brownian motion with constant coecients). They show that the optimal portfolio ^() always invests a constant proportion of the dierence X ^ max st X ^, t<, in the risky asset. Their insights are impressive, but the arguments are rather lengthy and detailed. We present in this paper an approach to the above problem, which simplies the results of Grossman & Zhou (993) { and extends them to the case of several stocks with general deterministic coecients. The model and the problem are introduced in sections and 3, respectively. The approach is based on an auxiliary nite-horizon stochastic control problem, formulated in section 4 (Problem 4., Remark 4.) in terms of the process N () in (4.). This problem admits an explicit optimal portfolio ^(), which is independent of the time-horizon T (; ) and can be found using \classical" martingale and duality arguments. It is then a relatively straightforward matter to show that this portfolio ^() is also optimal for the problem of maximizing (.); this is carried out in section 5. In section 6 we nd the optimal portfolio for the case of logarithmic utility function, and in section 7 we show that the same portfolio maximizes the long-term growth rate almost surely, not only in expectation. Moreover, this portfolio is optimal even if we allow random (adapted) market coecients. The Model Let us consider the following, by now standard, model of a nancial market M with one riskless asset (\bond", price P at time t) and a risky assets (\stocks"; prices P i at time t, id), modeled by the stochastic equations dp = P rdt; P () = (.) dp i = P i 4 bi dt + dx j= 3 ij dw j 5 ; Pi () = p i > ; (.) for i = ;:::;d: These equations are driven by the d-dimensional Brownian motion W = (W i ;:::;W d ), which generates all the randomness in the model { in the sense that the interest rate r(), the vector of appreciation rates b() =(b ();:::;b d ()) and the volatility matrix () =f ij ()g i;jd are bounded measurable processes, adapted to the augmentation F =

3 ffg t< of the ltration F W =(W(s);st), t<generated by W. We shall assume throughout that the matrix () isinvertible, and the \relative risk" process := [b ri],t<is bounded as well, where I is a d-dimensional vector with all entries equal to one. We shall also denote by := P = exp r(s)ds ; t< (:3) the \discount process" of this model. Consider now an economic agent who invests in this market according to a portfolio rule ()=( ();:::; d ()) in such away that his corresponding wealth process X ()isgoverned by the equation dx 3 dx = i X M dx 4 bi dt + ij dw j 5 " i= dx + i i= = rx dt + X X () = x; and satises the \drawdown constraint" Here (; ) is a given constant, and X M j= + M M [(b ri)dt + dw ] ; rdt (.4) P [X >M ; 8t<]=: (:5) M := max st ((s)x (s)): (:6) X M The interpretation is this: the agent does not tolerate the \drawdown of his discounted wealth, from its maximum-to-date", to be greater than or equal to the constant, at any time t ; thus, he imposes the (almost sure) constraint (.5). He invests a proportion i of the dierence X M > intheith stock, i =;:::;d, and invests P the remainder di= i X M + M of his wealth in the bond. With this interpretation in mind, we set up the formal model as follows.. Denition: For a given initial capital x>, let A (x) denote the class of measurable, F-adapted processes :[;)R d which satisfy k k dt < ; a.s. (:7) 3

4 for any given T (; ), and for which the stochastic functional/dierential equation (.4), (.6) has a unique F-adapted solution X () that obeys the constraint (.5). The elements of A (x) will be called \admissible portfolio processes". The class of Denition. is non-empty. In fact, it is shown in the Appendix that 8 >< >: for any measurable, F-adapted process :[;T]R d ; ^ =( ) is an admissible portfolio in A (x); for any x> 3 The Grossman-Zhou Problem 9 >= >; : (:8) Let U :(;)Rbe a utility function, i.e., a strictly increasing, strictly concave function of class C with U (+) = ;U () = and U(+). The convex dual of this function is given by ~U(y) := max[u(x) xy] =U(I(y)) yi(y);y > (3:) x> where I() is the inverse of U (). 3. Problem (Grossman & Zhou (993)): For some given <<, maximize the long-term rate of growth R():= lim T T log E(X (T )) (3:) of expected power-utility, over A (x). In particular, compute v() := and nd ^ A (x), for which the limit lim supremum in (3.3). T sup R() (3:3) A (x) log T E(X ^ (T )) = R(^) exists and achieves the Grossman and Zhou solved Problem 3. for d = and constant r;b ;, using rather lengthy analytical and probabilistic techniques. We present in section 5 a simple solution to this problem that allows general d and deterministic coecients r();b();() in (.), (.). 4 An Auxiliary Process and Problem For any portfolio process A (x) as in Denition (.), consider the auxiliary process N := X M (M ) ; t<: (4:) 4

5 Because the increasing process M () of (.6) is at o the set ft =X =M g, wehave from (.4), (.3), (4.): d(n ) = (N ) dw ; Consider also the processes Z :=exp (s)dw (s) W :=W+ (s)ds: (4:) k(s)k ds ; H:=Z: (4:3) From the product rule d(hn) = NdZ+Zd(N ) + dhz; Ni and (4.), (4.3) we obtain: d(hn) = HN( )dw. In other words, for any A (x) the process Z HN =( )x exp ( t )(s)dw (s) k k (s)ds (4:4) is a positive local martingale, hence supermartingale, which thus satises E [H(T )N (T )] ( )x ; 8T (; ): (4:5) We now pose an auxiliary stochastic control problem, involving the process N () of (4.). 4. An Auxiliary, Finite-Horizon, Control Problem: For a given T (; ) and utility function U :(;)R, denote by A (x; T ) the class of portfolios () that satisfy the requirements of Denition. on the nite horizon [;T], and nd ^() A (x; T ) that achieves V (; T;x):= sup EU(N(T)): (4:6) A (x;t ) There is a fairly straightforward solution to this problem along the lines of Karatzas, Lehoczky & Shreve (987), as follows: For any y>, A (x; T )wehave from (3.), (4.5): EU(N (T)) E ~ U(yH(T)) + ye(h(t)n (T)) E ~ U(yH(T)) + y( )x (4:7) The inequalities in (4.7) are equalities, if and only if y =^yand ()=^() are such that both N ^ (T ) = I(^yH(T)); (4.8) E[H(T )I(^yH(T))] = ( )x = (4.9) hold. Now ^yis uniquely determined from (4.9), and a portfolio ^ A (x) satisfying (4.8) can be found by introducing the positive martingale Q :=E[H(T)I(^yH(T))jF] = ( )x + Q(s)' (s)dw (s); t T: (4:) 5

6 The second equality follows from the representation theorem for Brownian martingales as stochastic integrals with respect to the Brownian motion W (e.g. Karatzas & Shreve (99), x3.4), and ' :[;T]R d is a measurable, F-adapted process with R T k'(s)k ds <, a.s. Comparing (4.) with (4.4) and recalling (.8), we see that In particular, (4.8) follows, and ^()= ( +' ) ()A (x; T ); H()N ^ () =Q(); a:s: (4:) V (; T;x)=E[(U I)(^yH(T))]: (4:) 4. Remark: Let us consider now Problem 4. with utility function Then, with := U(x) = x for := ( ); <<: (4:3), the formulae (4.9), (4.) become ^y ( )x = E[(H(T )) ] ; V (; T;x)= ( )x (E(H(T )) ) = : (4:4) If, in addition, the coecients r(), b(), () are deterministic, then " (H) = exp (s)dw (s) exp k(s)k ds r(s)+ + k(s)k ds and (4.), (4.), (4.4) give Q =( )x exp ( (s)dw (s) k(s)k ds ) ; ' = (4:5) ^ = ( + ) = ( ) ; independent oft; (4:6) V (; T;x)= ( )x exp ( r+ + kk dt) : (4:7) Clearly, the portfolio ^() of (4.6) is well-dened for all t<; it belongs to A (x) of Denition. for any x (; ), by (.8). 6

7 5 Solution of the Grossman-Zhou Problem We shall assume in this section that 8 >< >: the coecients r(), b(), () in the model of (.), (.) are deterministic, R and that r := lim T R T T r(s)ds; k k := lim T T T k(s)k ds exist and are nite. 9 >= >; (5:) 5. Theorem: Under the assumption (5.), the portfolio ^() of (4.6) is optimal for the Problem 3.. In fact, in the notation of (3.), (3.3), (4.7) and (5.) we have where lim T T log E(X ^ (T )) = R(^) =v()=v()+r ; (5:) V () := lim T T log V (; T;x)=r + ( + )k k = ( ) r + k k ( ) : ( ) (5:3) In order to establish this result, it will be helpful to consider the auxiliary problem v() := sup A (x) R (); " R () := lim T T log E(N (T )) ( ) : (5:4) From the fact that the portfolio ^() of (4.6) does not depend on the horizon T (; ), it is clear that lim T T log E(N (T ^ )) ( ) = R (^) =v()=v(): (5:5) It will also be helpful to note from (4.) that (N ) ( ) =() (X ) f M X ; (5:6) where the function f (x) := x ( x),x is strictly increasing on (;) and strictly decreasing on (; ). Proof of Theorem 5.: From (5.6) we obtain E(N (T )) ( ) ((T )) ( ) ( ) E(X (T )) ; (5:7) whence R () R() r v() r ; 8 A (x) (5:8) 7

8 and therefore V () v() r. In order to establish the reverse inequality, take (;) close enough to so that f () f (=), and observe from (5.6) that for an arbitrary A (x)(a (x)) we have E(N(T)) ( ) ((T )) (f (=)) E(X (T )) =((T)) ( =) E(X (T)) : (5:9) Consequently V () R () R() r ; 8 A (x); whence V () v() r ; letting " and invoking the continuity of the function V () in (5.3), we obtain V () v() r and thus the third equality of (5.): v() =V()+r = r + k k (( )) ( ) : (5:) To obtain the second equality in (5.), it suces to observe that (5.8), (5.5), (5.) imply v() R(^) R (^)+r = V ()+r = v(): Finally, the rst equality in (5.), i.e., the existence of the indicated limit, follows from the double inequality ( ) log( )+ T T T log( ( ) r(s)ds + T log E(N ^ (T )) ( )+ T ) T log E(X ^ (T )) r(s)ds + T log E(N ^ (T )) ( ) (5.) (a consequence of (5.7), (5.9)) in conjunction with (5.5), by passing to the limit as T and then letting " : 5. Remark: Formally setting = in (4.6), we recover the well-known optimal portfolio ^ = for the investment problem without the constraint (.5), with utility function U(x) = x from wealth and deterministic coecients. 6 Maximizing Long-term Rate of Expected Logarithmic Utility The methods of section 4-5 can also be used to show that the portfolio =( ) ; t< (6:) 8

9 is optimal for the problem of maximizing the long-term rate of expected logarithmic utility under the constraint (.5): lim T T E(log X (T )) lim T T E(log X (T )) = ( ) r + k k + r; 8 A (x): (6:) This problem was also considered by Grossman & Zhou (993) in their setting. It turns out that (6.) holds for general random, F-adapted coecients r(), b(), (), for which the conditions of section are satised and the limits r := lim T T Erdt; kk := lim T T exist and are nite. Indeed, consider u() = sup A(x) P(), P():= lim P () := lim T E(log T N Ekk dt (6:3) E(log T X (T )) and u() = sup A(x) P (); (T )) instead of the quantities in (3.), (3.) and (5.4), respectively. Solving the Problem 4. with U(x) =( ) log x leads to Q() ( )x =, '() in (4.), and thus the optimal portfolio of (4.) takes the form () in (6.), independent of the nite horizon T>; furthermore, (4.4) becomes V (; T;x) = log x + log( ) +( r(s)+ k(s)k )E R T ds, whence V (; T;x) u() = lim =( ) r+ k k T T Now one writes (5.6) with =, and uses exactly the same methodology as in section 5 to prove that lim T T E(log X (T )) = P( )=u()=u()+r,thus establishing (6.). 7 Maximization of Long-term Growth Rate from Investment More important than the optimality property (6.), however, is the fact that the portfolio () of (6.) maximizes the long-term growth rate from investment S() := lim T T log X (T ) lim T T log X (T )=( ) r + k k : +r ; a:s: (7:) over all A (x). Again, this comparison is valid for general random, F-adapted coecients as in section, under the proviso that the limits r := lim T T rdt; k k := lim T T 9 kk dt (7:)

10 exist and are nite, almost surely. In order to prove (7.), let us start by noticing that :=N=N, t< satises the stochastic equation d =( )dw ; () = from (4.) and It^o's rule, and is thus a positive supermartingale, for any A (x). It follows readily from this (e.g. Karatzas (989), p. 43) that lim t t log, or equivalently S ():= lim T T log(n (T )) lim T T log(n (T )) = (7.3) = ( ) r + k k =: s(); a.s. The existence of this last limit, and its value, follow from (A.6) and (7.). On the other hand, the inequality (5.6) with = leads to (as in (5.)) T log(n (T )) + r(s)ds log( ) T T T log X (T ) (7.4) Z T log(n (T )) + T r(s)ds T T log( ( =) ) almost surely, for any A (x)a (x) and any (;) suciently close to. In particular, (7.4) gives S ()+r s() := esssup S(); a.s. (7:5) A (x) in the notation of (7.)-(7.3), whence s()+r s(); similarly, S() r S () s(); whence s() r s() and in the limit as " : s() r s(), a.s. It develops that s() =s()+r = ( ) r + k k + r, and it remains to show the existence of the limit and the equality in (7.). But both of these follow by writing the double inequality (7.4) with, letting T to obtain in conjunction with (7.3) s() =s()+r lim T T log X (T ) lim T T log X (T ) s(); and then letting " to conclude lim log T T X (T )=s(), almost surely.

11 A Appendix We devote this section to the proof of the claim (.8). Clearly, it suces to show that for any x (; ) and () as in (.8), the stochastic equation d ^X =(^X ^M) dw ; ^M = max st admits a unique F-adapted solution that satises a.s. ^X(s); ^X() = x (A:) ^X >^M; 8t<: (A:) For then ^X()=() coincides with X ^ (), the wealth process corresponding to the portfolio ^ =( ) according to (.4), and (.5) is satised; and vice-versa. Suppose that ^X() isanf-adapted process that satises (A.), (A.). Following Grossman & Zhou (993), p. 69, observe that d log d Therefore, ^X ^M ^X(T ) ^M = = ^X ^M R := log( ) log dw d ^M ^M ; ^X ^M Clearly, the continuous increasing process K := log d ^M ^M ; whence := (s)dw (s) = + log ^M x ^M x k(s)k ds: : (A:3) is at away from the set n o t = ^X = ^M, i.e., away from the zero-set of the continuous nonnegative process R() of (A.3). From the theory of the Skorohod equation (e.g. Karatzas & Shreve (99), x3.6) we have then K = max (s), and from this and (A.3): st ^M M ~ :=xexp ( ) max (s) ; (A:4) st ) exp ^X X ~ :=xexp ( ) max (s) +( st Notice also (from (A.4), (A.5) and (4.)) that max (s) : (A:5) st N ^ =(^X ^M)( ^M) =( )x e : (A:6) It is straightforward to check that ~ X() satises (A.), (A.).

12 References [] S.J. Grossman and Z. Zhou, Optimal investment strategies for controlling drawdowns, Math. Finance 3 (3), (993), pp. 4{76. [] I. Karatzas, Optimization problems in the theory of continuous trading, SIAM J. Control & Optimization 7, (989) pp [3] I. Karatzas, J.P. Lehoczky and S.E. Shreve, Optimal portfolio and consumption decisions for a \small investor" on a nite horizon, SIAM J. Control & Optimization 5, (987), pp. 57{586. [4] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus ( nd edition), Springer-Verlag, New York (99). [5] R.C. Merton, Optimum consumption and portfolio rules in a continuous time model, J. Econom. Theory 3, (97), pp. 373{43, Erratum: Ibid 6 (973), pp. 3{4.

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