Chapter 4. Growth Optimal Portfolio Selection with Short Selling and Leverage

Transcription

1 Chapter 4 Growth Optimal Portfolio Selection with Short Selling Leverage Márk Horváth András Urbán Department of Computer Science Information Theory Budapest University of Technology Economics. H-7 Magyar tudósok körútja 2. Budapest Hungary mhorvath@math.bme.hu urbi@shannon.szit.bme.hu The growth optimal strategy on non-leveraged long only memoryless markets is the best constantly rebalanced portfolio (BCRP) also called log-optimal strategy. Optimality conditions are derived to frameworks on leverage short selling generalizing BCRP by establishing noruin conditions. Moreover the strategy its asymptotic growth rate are investigated under memoryless assumption both from theoretical empirical points of view. The empirical performance of the methods was tested for NYSE data demonstrating spectacular gains for leveraged portfolios showing unimportance of short selling in the growth-rate sense both in case of BCRP dynamic portfolios. 4.. Introduction Earlier results in the non-parametric statistics information theory economics literature (such as [Kelly (956)] [Latané (959)] [Breiman (96)] [Markowitz (952)] [Markowitz (976)] [Finkelstein Whitley (98)]) established optimality criterion for long-only non-leveraged investment. These results have shown that the market is inefficient i.e. substantial gain is achievable by rebalancing predicting market returns based on market s history. Our aim is to show that using leverage through margin buying (the act of borrowing money increasing market exposure) yields substantially higher growth rate in the case of memoryless (independent identically distributed i.i.d.) assumption on returns. Besides a framework for leveraged investment we also establish mathematical basis for short selling i.e. creating negative exposure to asset prices. Short sell- 5

2 52 M. Horváth A. Urbán ing means the process of borrowing assets selling them immediately with the obligation to rebuy them later. It can be shown that the optimal asymptotic growth rate on a memoryless market coincides with that of the best constantly rebalanced portfolio (BCRP). The idea is that on a frictionless market the investor can rebalance his portfolio for free at each trading period. Hence asymptotic optimization on a memoryless market means that the growth optimal strategy will pick the same portfolio vector at each trading period. Strategies based on this observation are called constantly rebalanced portfolios (CRP) while the one with the highest asymptotic average growth rate is referred to as BCRP. Our results include the generalization of BCRP for margin buying short selling frameworks. To allow short leverage our formulation weakens the constraints on feasible set of possible portfolio vectors thus they are expected to improve performance. Leverage is anticipated to have substantial merit in terms of growth rate while short selling is not expected to yield much better results. We do not expect increased profits on short CRP strategy since companies worth to short in our test period should have already defaulted by now. Nonetheless short selling might yield increased profits in case of markets with memory since earlier results have shown that the market was inefficient (cf. [Györfi et al. (2006)]). In case of i.i.d. returns with known distribution [Cover (984)] has introduced a gradient based method for optimization of long-only log-optimal portfolios gave necessary sufficient conditions on growth optimal investment in [Bell Cover (980)]. We extend these results to short selling leverage. Contrary to non-leveraged long only investment in earlier literature in case of margin buying short selling it is easy to default on total initial investment. In this case asymptotic growth rate is minus infinity. By bounding possible market returns we establish circumstances such that default is impossible. We do this in such a way that debt positions are limited the investor is always able to satisfy his liabilities selling assets. Restriction of market exposure amount of debt is in line with the practice of brokerages regulators. Our notation for asset prices returns are as follows. Consider a market consisting of d assets. Evolution of prices is represented by a sequence of price vectors s s 2... R d + where s n = (s () n...s (d) n ). (4.) s (j) n denotes the price of the j-th asset at the end of the n-th trading period.

3 Growth Optimal Portfolio Selection with Short Selling Leverage 53 In order to apply the usual techniques for time series analysis we transform the sequence of price vectors {s n } into return vectors: where x n = (x () n...x (d) n ) x (j) n = s(j) n+ s (j) n Here the j-th component x (j) n of the return vector x n denotes the amount obtained by investing unit capital in the j-th asset during the n-th trading period Non-leveraged long only investment A representative example of the dynamic portfolio selection in the long only case is the constantly rebalanced portfolio (CRP) introduced studied by [Kelly (956)] [Latané (959)] [Breiman (96)] [Markowitz (976)] [Finkelstein Whitley (98)] [Móri (982)] [Móri Székely (982)] [Cover (984)]. For a comprehensive survey see also Chapters 6 5 in [Cover Thomas (99)] Chapter 5 in [Luenberger (998)]. CRP is a self-financing portfolio strategy rebalancing to the same proportional portfolio in each investment period. This means that the investor neither consumes from nor deposits new cash into his account but reinvests his capital in each trading period. Using this strategy the investor chooses a proportional portfolio vector b = (b ()...b (d) ) rebalances his portfolio after each period to correct the price shifts in the market. This way the proportion of his wealth invested in each asset at the beginning of trading periods is constant. The j-th component b (j) of b denotes the proportion of the investor s capital invested in asset j. Thus the portfolio vector has nonnegative components that sum up to. The set of portfolio vectors is denoted by d = b = (b()...b (d) ); b (j) 0 b (j) =. (4.2) Let S 0 denote the investor s initial capital. At the beginning of the first trading period S 0 b (j) is invested into asset j it results in position size S 0 b (j) x (j) after changes in market prices. Therefore at the end of the first

4 54 M. Horváth A. Urbán trading period the investor s wealth becomes S = S 0 b (j) x (j) = S 0 b x where denotes inner product. For the second trading period S is the new initial capital hence By induction for trading period n S 2 = S b x 2 = S 0 b x b x 2. n S n = S n b x n = S 0 b x i. (4.3) Including cash account into the framework is straight forward by assuming x (j) n = for some j for all n. The asymptotic average growth rate of this portfolio selection is i= W(b) = lim ln n S n = lim n n n lns n ( ) = lim n n lns 0 + n ln b x i n i= n = lim ln b x i n n i= if the limit exists. This also means that without loss of generality we can assume that the initial capital S 0 =. If the market process {X i } is memoryless i.e. is a sequence of independent identically distributed (i.i.d.) rom return vectors then the asymptotic rate of growth exists almost surely (a.s.) where with rom vector X being distributed as X i W(b) = lim n n n ln b X i = Eln b X a.s. (4.4) i= given that Eln b X is finite due to strong law of large numbers. We can ensurethispropertybyassumingfinitenessofelnx (j) i.e. E lnx (j) < for each j {...d}.

5 Growth Optimal Portfolio Selection with Short Selling Leverage 55 In fact because of b (i) > 0 for some i we have Eln b X Eln (b (i) X (j)) = lnb (i) +ElnX (i) > because of b (j) for all j we have ( ) Eln b X Eln dmaxx (j) j = lnd+emaxlnx (j) j lnd+emaxln X (j) j lnd+ j Eln X (j) <. From (4.4) it follows that rebalancing according to the best log-optimal strategy b arg max b d Eln b X is also an asymptotically optimal trading strategy i.e. a strategy with a.s. optimum asymptotic growth W(b ) W(b) for any b d. The strategy of rebalancing according to b at the beginning of each trading period is called best constantly rebalanced portfolio (BCRP). In the following we repeat calculations of [Bell Cover (980)]. Our aim is to maximize asymptotic average rate of growth. W(b) being concave we minimize the convex objective function f X (b) = W(b) = Eln bx. (4.5) To use Kuhn-Tucker theorem we establish linear inequality type constraints over the search space d in (4.2): for i =...d i.e. b (i) 0 b a i 0 (4.6) where a i R d denotes the i-th unit vector having at position i.

6 56 M. Horváth A. Urbán i.e. Our only equality type constraint is b (j) = 0 b e = 0 (4.7) where e R d e = (...). The partial derivatives of the objective function are f X (b) b (i) = E X(i) bx for i =...d. According to Kuhn-Tucker theorem ([Kuhn Tucker (95)]) the portfolio vector b is optimal if only if there are constants µ i 0 (i =...d) ϑ R such that for j =...d. This means that f X(b )+ µ i a i +ϑe = 0 i= µ j b a j = 0 E X(j) b X µ j +ϑ = 0 (4.8) µ j b (j) = 0 for j =...d. Summing up (4.8) weighted by b (j) we obtain: hence E b X b X µ j b (j) + ϑ =. ϑb (j) = 0 We can state the following necessary condition for optimality of b. If b arg max b d W(b)

7 Growth Optimal Portfolio Selection with Short Selling Leverage 57 then b (j) > 0 = µ j = 0 = E X(j) b = (4.9) X b (j) = 0 = µ j 0 = E X(j) b. (4.0) X Because of convexity of f X (b) the former conditions are sufficient too. Assume b d. If for any fixed j =...d either or E X(j) b X = b (j) > 0 E X(j) b X b (j) = 0 thenb isoptimal. Thelattertwoconditionsposeanecessarysufficient condition on optimality of b. Remark 4.. In case of an independent asset i.e. for some j...d X (j) being independent from the rest of the assets b (j) = 0 = E X(j) b X implies by b (j) = 0 that X (j) is independent of b X. This means that therefore b (j) = 0 = EX (j) E b X b (j) = 0 = EX (j) E. b X According to Kuhn-Tucker theorem for any fixed j =...d either or E X(j) b X = b (j) > 0 EX (j) if only if b is optimal i.e. E b X b (j) = 0 b arg max b d W(b). (4.)

8 58 M. Horváth A. Urbán Remark 4.2. Assume optimal portfolio b for d assets X = (X () X (2)...X (d) ) is already established. Given a new asset being independent of our previous d assets we can formulate a condition on its inclusion in the new optimal portfolio b. If then EX (d+) < E b X b (d+) = 0. This means that for a new independent asset like the cash we do not have to do the optimization for each asset in the portfolio we can reach substantial reduction in dimension of the search for an optimal portfolio. Remark 4.3. The same trick can be applied in case of dependent returns as well. If E X(d+) b X < then b (d+) = 0 which is much simpler to verify then performing optimization of asymptotic average growth. This condition can be formulated as ( ) EX (d+) E b X +Cov X (d+) b X which poses a condition on covariance expected value of the new asset. Remark 4.4. [Roll (973)] [Pulley (994)] [Vajda (2006)] suggested an approximation of b using lnz h(z) = z 2 (z )2 whichisthesecondordertaylorapproximationofthefunctionlnz atz =. Then the semi-log-optimal portfolio selection is b arg max b d E{h b X }.

9 Growth Optimal Portfolio Selection with Short Selling Leverage 59 Our new objective function is convex: f X (b) = E{h bx } = E{ bx 2 ( bx )2 } = E{ bx ++ 2 ( bx )2 } = E{ bx ++ 2 bx 2 bx + 2 } = E{ 2 bx 2 2 bx } = 2 be(xxt )b b2ex ( 2 ) } 2 = E{ bx 2 2 where X is column vector XX T denotes outer product. This is equivalent to minimizing f X (b) = E( bx 2) 2. Thus b can be simply calculated to minimize the squared distance from 2. In case of data driven algorithms the solution is using linear regression under the constraint b d. f X (b) = Var bx +(2 E bx ) 2 = Var bx +(2 bex ) 2. This means we minimize variance of returns while maximizing expected return. This is in close resemblance with Markowitz type portfolio selection. For a discussion of the relationship between Markowitz type portfolio selection the semi-log-optimal strategy(see[ottucsák Vajda(2007)] Chapter 2 of this volume). The problem can be formulated as a quadratic optimization problem as well where f X (b) = brb +4 4 bm R =E(XX T ) m =E(X).

10 60 M. Horváth A. Urbán Note that R is symmetric positive semidefinite since for any z R d z T Rz = z T E(XX T )z =E(z T X) 2 0. This means we face a convex programming problem again Short selling No-ruin constraints Short selling an asset is usually done by borrowing the asset under consideration selling it. As collateral the investor has to provide securities of the same value to the lender of the shorted asset. This ensures that if anything goes wrong the lender still has high recovery rate. While the investor has to provide collateral after selling the assets having been borrowed he obtains the price of the shorted asset again. This means that short selling is virtually for free. S = S C +P where S is wealth after opening the short position S is wealth before C is collateral for borrowing P is price income of selling the asset being shorted. For simplicity we assume hence C = P S = S short selling is free. In practice the act of short selling is more complicated. For institutional investors the size of collateral depends on supply dem on the short market the receiver of the more liquid asset usually pays interest. For simplicity we ignore these problems. Let us elaborate this process on a real life example. Assume the investor wants to short sell 0 shares of IBM at $00 he has $000 in cash. First he has to find a lender the short provider who is willing to lend the shares. After exchanging the shares the $000 collateral the investor sells the borrowed shares. After selling the investor has $000 in cash again the obligation to cover the shorted assets later. In contrast with our modelling approach where short selling is free it is also modelled in literature such that selling an asset short yields immediate cash this is called naked short transaction. This is the case in the Chapter

11 Growth Optimal Portfolio Selection with Short Selling Leverage 6 on Mean-Variance Portfolio Theory of [Luenberger (998)] in [Cover Ordentlich (998)]. Assume our only investment is in asset j our initial wealth is S 0. We invest a proportion of b ( ) of our wealth. If the position is long (b > 0) it results in wealth S 0 ( b)+s 0 bx (j) = S 0 +S 0 b(x (j) ) while if the position is short (b < 0) we win as much money as price drop of the asset: S 0 +S 0 b ( x (j) ) = S 0 +S 0 b(x (j) ). In line with the previous example assume that our investor has shorted 0 shares of IBM at $00. If the price drops $0 he has to cover the short position at $90 thus he gains 0 x $0. If the price rises $0 he has to cover at $0 loosing 0 x $0. Let b = (b (0) b ()...b (d) ) be the portfolio vector such that the 0-th component corresponds to cash. At the end of the first trading period the investor s wealth becomes S = S 0 b (0) + [ ] + b (j)+ x (j) +b (j) (x (j) ) (4.2) where (.) denotes the negative part operation. In case of the investor s net wealth falling to zero or below he defaults. Negative wealth is not allowed in our framework thus the outer positive part operation. Since only long positions cost money in this setup we will constrain to portfolios such that d j=0 b(j)+ =. Considering this it is also true that S = S 0 b (j)+ + j=0 = S 0 + [ ] + b (j)+ (x (j) )+b (j) (x (j) ) (4.3) [ ] + b (j) (x (j) ). (4.4) This shows that we gain as much as long positions raise short positions fall. We can see that short selling is a risky investment because it is possible to default on total initial wealth without the default of any of the assets in

12 62 M. Horváth A. Urbán the portfolio. The possibility of this would lead to a growth rate of minus infinity thus we restrict our market according to B +δ < x (j) n < +B δj =...d. (4.5) Besides aiming at no-ruin the role of δ > 0 is ensuring that rate of growth is finite for any portfolio vector (i.e. > ). For the usual stock market daily data there exist 0 < a < < a 2 < such that a x (j) n a 2 for all j =...d for example a = 0.7 with a 2 =.2 (cf. [Fernholz (2000)]). Thus we can choose B = 0.3. Given (4.4) (4.5) it is easy to see that maximal loss that we could suffer is B d b(j). This value has to be constrained to ensure no-ruin. We denote the set of possible portfolio vectors by ( B) d = b = (b(0) b ()...b (d) ); b (0) 0 b (j)+ = B j=0 b (j). (4.6) d j=0 b(j)+ = means that we invest all of our initial wealth into some assets buying long or cash. By B d b(j) maximal exposure is limited such that ruin is not possible rate of growth it is finite. b (0) is not included in the latter inequality since possessing cash does not pose risk. Notice that if B then d+ ( B) d so the achievable growth rate with short selling can not be smaller than in long only case. According to (4.4) (4.5) with B we show that ruin is impossible: [ ] + b (j) (x (j) ) > + = (B δ) δ b (j). [ ] b (j)+ ( B +δ )+b (j) (+B δ ) b (j)

13 Growth Optimal Portfolio Selection with Short Selling Leverage 63 If d b(j) = 0 then b (0) = hence no-ruin. In any other case δ d b(j) > 0 hence we have not only ensured no-ruin but also [ ] + Eln + b (j) (X (j) ) > Optimality condition for short selling with cash account A problem with ( B) d with is its non-convexity. To see this consider b = (0) ( ) b 2 = ( /2) ( ) b +b 2 2 = (/2/4) / ( ). This means we can not simply apply Kuhn-Tucker theorem on ( B) d. Given cash balance we can transform our non-convex ( B) d to a convex region ( B) d where application of our tools established in long only investment becomes feasible. The new set of possible portfolio vectors is a convex region: ( B) d = { b = +) ( b(0 b (+) b ( )... b (d+) b (d ) ) R + 2d+ 0 ; } b(j +) = B ( b (j+) + b (j ) ). j=0 Mapping from ( B) d (4.2) implies that to ( B) d happens by b = (b (0) (b () ) + (b () )...(b (d) ) + (b (d) ) ). (4.7) S = S b(0 +) 0 + [ b(j ] +) x (j) + b (j ) ( x (j) ) thus in line with the portfolio vector being transformed we transform the market vector too X = (X () X ()...X (d) X (d) ) +

14 64 M. Horváth A. Urbán so that S = S 0 b X. To use the Kuhn-Tucker theorem we enumerate linear inequality type constraints over the search space B ( b (j+) + b (j ) ) b(0 +) 0 b (i+) 0 b (i ) 0 for i =...d. Our only equality type constraint is b(j +) =. j=0 The partial derivatives of the convex objective function f X ( b) = Eln b X are f X ( b) = E b b X (0+) f X ( b) b (i+) = E X(i) b X f X ( b) b (i ) = E X(i) b X for i =...d. According to Kuhn-Tucker theorem (KT) the portfolio vector b is optimal if only if there are KT multipliers assigned to each of the former 2d+3 constraints µ 0+ 0µ i+ 0µ i 0ν B 0 (i =...d) ϑ R such that E +ϑ µ 0+ = 0 (4.8) b X E X(i) b X +ϑ µ i+ +ν B B = 0 E X(i) b X µ i +ν B B = 0

15 Growth Optimal Portfolio Selection with Short Selling Leverage 65 for i =...d while ν B [B µ 0+ b (0 +) = 0 µ i+ b (i +) = 0 µ i b (i ) = 0 ( b (j+) + b (j ) ) ] = 0 (4.9) ν B B ( b (j+) + b (j ) ) = ν B. Summing up equations in (4.8) weighted by b (0) b (i+) b (i ) we obtain: b X E b X +ϑ b (j +) +ν B B ( b (j+) + b (j ) ) = 0 j=0 +ϑ+ν B = 0 ϑ = ν B (4.20) ϑ. In case of B d ( b (j+) + b (j ) ) < because of (4.9) (4.20) we have that This implies ν B = 0 hence ϑ =. E b X + µ 0+ = 0 E X(i) b X + µ i+ = 0 E X(i) b X µ i = 0.

16 66 M. Horváth A. Urbán These equations result in the following additional properties b (0 +) > 0 = µ 0+ = 0 = E b X = b (0 +) = 0 = µ 0+ 0 = E b X b (i +) > 0 = µ i+ = 0 = E X(i) b X = b (i +) = 0 = µ i+ 0 = E X(i) b X b (i ) > 0 = µ i = 0 = E X(i) b X = 0 b (i ) = 0 = µ i 0 = E X(i) b X 0 for i =...d. We transform the vector b to the vector b such that (i =...d) while This way b (0) = b (0) + b (i) = b (i+) b (i ) min{ b (i+) b (i ) }. i= b (j) = we have the same market exposure with b as with b.

17 Growth Optimal Portfolio Selection with Short Selling Leverage 67 Due to the simple mapping (4.7) with regard to the original portfolio vector b this means b (0) > 0 = E b X = (4.2) b (0) = 0 = E b X. Also b (i) > 0 = E b X(i) X = E b X(i) X 0 which is equivalent to E b X E b X(i) X = b (i) = 0 = E b X(i) X E b X(i) X 0 which is equivalent to E b X E b X(i) X b (i) < 0 = E b X(i) X E b X(i) X = 0 which is equivalent to E b X E b X(i) X for i =...d.

18 68 M. Horváth A. Urbán 4.4. Long only leveraged investment No-ruin condition In the leveraged frameworks we assume (4.5) thus market exposure can be increased over one without the possibility of ruin. Again we denote the portfolio vector by b = (b (0) b ()...b (d) ) where b (0) 0 sts for the cash balance since no short selling b (i) 0i =...d. Assume the investor can borrow money invest it on the same rate r. Assume also that the maximal investable amount of cash L Br (relative to initial wealth S 0 ) is always available for the investor. In the sequel we refer to L Br as buying power. L Br is chosen to be the maximal amount investing of which ruin is not possible given 4.5. Because our investor decides over the distribution of his buying power b (j) = L Br. j=0 Unspent cash earns the same interest r as the rate of lending. The market vector is defined as X r = (X (0) X ()...X (d) ) = (+rx ()...X (d) ) so X (0) = +r. The feasible set of portfolio vectors is r +B d = b = (b(0) b ()...b (d) ) R + d+ 0 b (j) = L Br where b (0) denotes unspent buying power. Market evolves according to j=0 S = S 0 ( bx r (L Br )(+r)) + where S 0 r(l Br ) is interest on borrowing L Br times initial wealth S 0. To ensure no-ruin finiteness of growth rate choose L Br = +r B +r. (4.22)

19 Growth Optimal Portfolio Selection with Short Selling Leverage 69 This ensures that ruin is not possible: bx r (L Br )(+r) = b (j) X (j) (L Br )(+r) j=0 = b (0) (+r)+ > b (0) (+r)+ b (j) X (j) (L Br )(+r) b (j) ( B +δ) (L Br )(+r) = b (0) (+r)+(l Br b (0) )( B +δ) (L Br )(+r) = b (0) (r +B δ) L Br (B δ +r)++r +r (B δ +r)++r B +r = δ +r B +r Kuhn-Tucker characterization Our convex objective function the negative of asymptotic rate of growth is f +B X r (b) = Eln( bx r (L Br )(+r)). The linear inequality type constraints are as follows: b (i) 0 for i = 0...d while our only equality type constraint is b (j) L Br = 0. j=0 The partial derivatives of the optimized function are f +B X r (b) b (i) X (i) = E bx r (L Br )(+r). According to the Kuhn-Tucker necessary sufficient theorem a portfolio vector b is optimal if only if there are KT multipliers µ j 0 (j = 0...d) ϑ R such that X (j) E b X r (L Br )(+r) µ j +ϑ = 0 (4.23)

20 70 M. Horváth A. Urbán µ j b (j) = 0 for j = 0...d. Summing up (4.23) weighted by b (j) we obtain: b X r E b X r (L Br )(+r) µ j b (j) + j=0 b (j) ϑ = 0 (L Br )(+r) +E b X r (L Br )(+r) = L Brϑ + (L Br )(+r) E L Br b = ϑ. (4.24) X r (L Br )(+r) This means that L Br X (j) b (j) > 0 = µ j = 0 = E b = ϑ (4.25) X r (L Br )(+r) X (j) j=0 b (j) = 0 = E b X r (L Br )(+r) ϑ. For the cash account this means b (0) +r > 0 = µ j = 0 = E b = ϑ (4.26) X r (L Br )(+r) b (0) +r = 0 = E b X r (L Br )(+r) ϑ Short selling leverage For this case we need to use both tricks of the previous sections. The market evolves according to S = S 0 (b (0) (+r) + [ ] + b (j)+ x (j) +b (j) (x (j) r) (L Br )(+r)) over the non-convex region r ±B d = b = (b(0) b () b (2)...b (d) ); b (j) = L Br j=0

21 Growth Optimal Portfolio Selection with Short Selling Leverage 7 where L Br is the buying power defined in (4.22) b (0) denotes unspent buying power. Again one can check that the choice of L Br ensures no-ruin finiteness of growth rate. With the help of our technique developed in the short selling framework we convert to the following convex region: such that r ±B d = { b = +) ( b(0 b (+) b ( )... b (d+) b (d ) ) R + 2d+ 0 ; } b(0 +) + ( b (j+) + b (j ) ) = L Br b = ( b (0) b (+) b ( )... b (d+) b (d ) ) = (b (0) b ()+ b ()...b (d)+ b (d) ). Similarly to the short selling case we introduce the transformed return vector. Given we introduce X = (X ()...X (d) ) X ±r = (+rx () 2 X () +r...x (d) 2 X (d) +r). We introduce r in 2 X (i) +r terms since short selling is free hence buying powerspentonshortpositionsstillearnsinterest. Weuse2 X (i) +r instead of X (i) +r since while short selling is actually free it still limits our buying power which is the basis of the convex formulation. Because of r ±B d = r +B 2d we can easily apply (4.25) (4.26) hence b (0) +r > 0 = E = ϑ b X ±r (L Br )(+r) b (0) +r = 0 = E ϑ b X ±r (L Br )(+r) where ϑ is defined by (4.24) with X r = X ±r in place E X (i) b (i) > 0 = b X ±r (L Br )(+r) = ϑ 2 X (i) +r E b X ±r (L Br )(+r) ϑ

22 72 M. Horváth A. Urbán E X (i) b (i) = 0 = b X ±r (L Br )(+r) ϑ 2 X (i) +r E b X ±r (L Br )(+r) ϑ E X (i) b (i) < 0 = b X ±r (L Br )(+r) ϑ 2 X (i) +r E b X ±r (L Br )(+r) = ϑ. Note that in the special case of L Br = we have ϑ = because of (4.24) Experiments Our empirical investigation consider three setups each of which is considered in long only short leveraged leveraged short cases. We examine the BCRP strategy which chooses the best constant portfolio vector with hindsight its empirical causal counterpart the causal i.i.d. strategy. The latter strategy uses the best portfolio based on past data it is asymptotically optimal for i.i.d. returns. The third algorithm is asymptotically optimal in case of Markovian time series. Using nearest-neighborbased portfolio selection (cf. Chapter 2 of this volume) with 00 neighbors we investigate whether shorting yields extra growth in case of dependent market returns. The New York Stock Exchange (NYSE) data set [Gelencsér Ottucsák (2006)] includes daily closing prices of 9 assets along a 44-year period ending in The same data is used in Chapter 2 of this volume which facilitates comparison of algorithms. Interest rate is constant in our experiments. We calculated effective daily yield over the 44 years based on Federal Reserve Fund Rate from the FRED database. The annual rate in this period is 6.3% which is equivalent to r = daily interest.

23 Growth Optimal Portfolio Selection with Short Selling Leverage 73 Table 4.. Average Annual Yields optimal portfolios on NYSE data. Asset AAY b b B b +B b ±B Cash/Debt AHP 3% ALCOA 9% AMERB 4% COKE 4% DOW 2% DUPONT 9% FORD 9% GE 3% GM 7% HP 5% IBM 0% INGER % JNJ 6% KIMBC 3% MERCK 5% MMM % MORRIS 20% PANDG 3% SCHLUM 5% AAY 20% 20% 34% 34% Table 4.2. Average Annual Yields. Strategy BCRP IID Nearest Neighbor Annual Average Yield Long only 20% Short 20% Leverage 34% Short & Leverage 34% Long only 3% Short % Leverage 6% Short & Leverage 4% Long only 32% Short 30% Leverage 66% Short & Leverage 83%

24 74 M. Horváth A. Urbán Regarding (4.5) we chose the conservative bound B = 0.4 as the largest one day change in asset value over the 44 years has been This bound implies that in the case of r = 0 the maximal leverage is L Br = 2.5 fold while in case of r = L Br = Performance of BCRP algorithms improve further by decreasing B until B = 0.2 but this limit would not guarantee no-ruin. This property also implies that optimal leverage factor on our dataset is less than 5. Given our convex formalism for the space of portfolio vector convexity of log utility we use Lagrange multipliers active-set algorithms for the optimization. Table 4. shows the results of the BCRP experiments. Shorting does not have any effect in this case while leverage results in significant gain. Behavior of shorting strategies is in line with intuition since taking permanently short position of an asset is not beneficial. The leveraged strategies use maximal leverage they do not only increase market exposure but invest into more assets in order to reduce variation of the portfolio. This is in contrast with behavior of leveraged mean-variance optimal portfolios. 0 2 Leverage Short Leverage Long Only Short Wealth Time Fig. 4.. Cumulative wealth of the nearest neighbor strategy starting from 962.

25 Growth Optimal Portfolio Selection with Short Selling Leverage 75 Table 4.2 presents growth rates of the three setups we consider. BCRP being an optimistically anticipating estimate of possible growth our i.i.d. strategies do significantly underperform while the Average Annual Yields (AAYs) of the nearest neighbor strategies including leverage are spectacular. Figure 4. presents evolution of wealth in the latter case. While allowing short positions results in large drawdowns in the beginning these algorithms catch up later. Figure 4.2 shows the result of the algorithms starting from 980; it reveals that short selling does not offer any significant plus gain in this period benchmarking against the long-only approaches. 0 6 Leverage Short Leverage Long Only Short Wealth Time Fig Cumulative wealth of the nearest neighbor strategy starting from 980. References Bell R. M. Cover T. M. (980). Competitive optimality of logarithmic investment Mathematics of Operations Research 5 pp Breiman L. (96). Optimal gambling systems for favorable games in Proc. of the Fourth Berkeley Symposium on Mathematical Statistics Probability (University of California Press Berkeley) pp

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