Dynamic Porfolio Opimisaion wih Trading Coss James Sefon and Sylvain Champonnois London Quan Conference Sepember 2012
Tracabiliy and Transparency Imporan Quans have needed o upgrade heir approach To rebalance more frequenly maybe coninuously To explicily accoun for he impac of rade coss To combine signals of differen horizons Inroduce faser signals (news, shor erm reversals) Difficul o rely on black box opimisaion (especially dynamic) Need o be able o jusify he posiions in our porfolios If a hedge agains risk If a hedge agains coss Or wheher o ake advanage of a shor or long erm opporuniy 1
Conribuions of work Our explici soluion o a sylised dynamic porfolio opimisaion 1. Sraegy is o rade (in proporion o coss) owards a arge porfolio Or opimally exi unwaned asses. 2. Targe porfolio has a clear srucure I is a weighed average of insananeous opimal smoohing porfolios Weighs are a funcion of rading coss and he discoun rae 3. Risk aversion inroduces (as well as wihin period diversificaion) Inerermporal hedging moives à la Meron Disrus cosly asses whose posiion is sensiive o volaile fuure opporuniies 4. Exend he idea of sochasic discoun facor (SDF) o problems wih rading coss 2
Dynamic Porfolio Opimisaion There is now a large lieraure on his problem wihou a clear seminal conribuion (perhaps Garleanu and Pedersen (2011)). A reason for he plehora of papers is here is no clear sylised model of 1. The rading cos funcion a) Linear Coss - Small Trades b) Quadraic Coss Large Trades 2. The Crierion o be opimised a) Mean variance wihin period (no ineremporal hedging) b) Mean variance on erminal wealh (dynamically inconsisen) c) A Uiliy Funcion on erminal wealh 3. The dynamic model underlying sock reurns 1. Time-varying and predicable risk premiums (asse allocaion) 2. Time-varying exposures o consan risk facors (syle-churning) 3
Our Sylised Model Insananeous reurns of he k asses are normally disribued ( µ C r ) d dε, dp = + s + p + s p where he economic saes evolve s, d As d s, and Cov ε s = s + ε = Σ ε p, Invesor holds n shares of each asse and choses her rading sraegy T τ Λτ dn = τd which coss d 2 o maximise he expeced value of her discouned uiliy consan risk aversion. 4
Or o rephrase T Define wealh w = n p hen our invesor choses rading sraegy τ and consumpion c given dw rw c C T T T = + n ss d + n εp, εs, ds = As s + dεs, where Cov ε = Σ p, dn = τd τ Λτ 2 and wealh says posiive, so as o maximise her expeced discouned uiliy δs max ( βc ) s J = E e e ds τ, c 0 5
.. and hen as a conrol problem s The problem saes are x = which evolve εs, n ε ε = p, As 0 As 0 0 dx = d + dε + τd 0 0 x 0 0 I A B A Wealh evolves 1 T T 1 T C 0 0 I Cs 0 s τ Λτ dw = rw c + x d 2 0 I I 0 0 I x 2 T C R C T T C 0 0 0 s I I + x dε 0 I I 0 0 0 C T R D The conrol variables are τ and c chosen o maximise uiliy. 6
The Soluion Procedure 1. The problem is now phrased as a sandard risk sensiive conrol problem. 2. We can wrie down he Hamilon-Jacobi-Bellman (HJB) equaion. 3. Assume ha he value funcion is quadraic funcion of he saes x ( 1 T δ γ x x ) 2 J = K exp w + Π and subsiue ino he HJB equaion 4. Show ha his solves he HJB if Π saisfies a Riccai equaion. 7
The Soluion Procedure Theorem: The opimal rading inensiy τ and consumpion c are τ = Λ 1 B x T 2 δ r c = r w + 2 Π + + Σ Π 2 γ ( 1 T ) 1 ( T x x race B ) 1 B1 Wealh + PV of fuure ne resources Where γ=rβ and Π saisfies T T T T T 1 1 2 2 ( ) ( ) 1 rπ = Π A + AΠ γ Π B + C RD Σ Π B + C RD + C RC + ΠB Λ B Π 8
The Opimal Porfolio 1. The arge porfolio Weighed average of an insananeous Meron hedging porfolio. n saic meron, The sraegy is o 1. Trade owards he arge porfolio in proporion o he inverse of he coss Move owards arge n arge, As coss ge lower n n saic meron, 9
The opimal rading rule Define he arge porfolio as he porfolio ha maximises he presen value of fuure resources i.e max n x T Πx hen he rading rule can be wrien ( n ) τ = Λ B ΠB n 1 T Targe 2 2 10
The Targe Porfolio Define he Meron Saic Porfolio as he porfolio ha maximises he insananeous mean variance problem ( ) ( ) T T T T T T max xc RCx γ x Π B1 + C RD Σ Π B1 + C RD x n Expeced reurn o Adjused Insananeous Variance Porfolio of Ne Resources hen he opimal porfolio is weighed average of fuure Meron Saic Porfolios Targe r Saic Meron r n = E e Hn d e Hd 0 0 where H ( 1 T B ) 2 B 2 = exp Λ Π 11
An example in discree ime Timing a Value Til
Model: Expeced Reurns o a Long-Shor Value Sraegy 1. r +1 - Reurns o op 3 rd minus boom 3 rd socks sored by book o price. 2. S +1 - Difference in logs of book o price of op 3 rd minus boom 3 rd. 3. M +1 - A weighed average of lagged reurns r +1 Sae Equaions ( S+ 1 S) ϕ( S S) Oupu Equaion = + S M = r + ϕm + 1 + 1 M ( ) r+ = µ + α S S + αm + ε 1 S M η 2 ε σε ρσσ ε η Cov η = 2 ρσσ ε η ση η < 0 implies lower fuure reurns. However as Cov(η, ε)< 0, hedge by holding more of value porfolio. ε < 0 implies lower fuure reurns due o momenum. However can hedge by holding less of value porfolio. Mean reversion leads o posiive, momenum o negaive, hedging demands 13
Esimae he model by maximum likelihood 1. r +1 - Reurns o op 3 rd minus boom 3 rd socks sored by book o price. 2. S +1 - Difference in logs of book o price of op 3 rd minus boom 3 rd. 3. M +1 - A weighed average of lagged reurns r +1 Sae Equaions ( S 1.45) 0.968 ( S 1.45) + 1 (0.017) Oupu Equaion = + M = r + 0.75M + 1 + 1 (0.13) ( ) r = 0.0018 + 0.028 S 1.45 + + 0.132( M 0.008) + ε 1 (0.0007) (0.011) (0.032) η ε 11.1% 0.43 Cov η = 23.5% Diagonal- Annualised Volailiy Off Diagonal - Correlaions Esimaed on monhly daa from Jan 1994 April 2012 The half-life of he momenum signal is jus over 3 monhs There is srong negaive correlaion beween he innovaions o reurns and innovaions o he opporuniy se. The R 2 of he oupu equaion is 0.12 14
The Invesmen Objecive The invesor chooses his il o value. He is benchmarked o he marke. Thus his fully invesed porfolio reurns are: r + wr Mk + 1 + 1 We assume a consan reurn o he marke of 4% per year, and covariance η ε = Cov Mk ε 23.5% 0.43 0.15 11.1% 0.33 20% Diagonal- Annualised Volailiy Off Diagonal - Correlaions He dynamically adjus his ils o maximise he CRA of his wealh afer T periods Coss are calibraed so ha a 1% increase in he value il in a monh coss 0.1bps, whereas a 10% increase coss 10bps. 15
Targe Porfolio Longer he horizon he lower he weigh on value, due o anicipaed rading coss. Trading momenum is more expensive, so allocaion fall faser wih horizon. 0.6 0.4 0.2 0-0.2-0.4 Weigh on Long-Shor Value Porfolio Valuaion spreads a 1 s.d (+0.26) Momenum sae a 1 s.d (+0.05) Base when saes a equilibrium Monhs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Momenum sae a -1 s.d (-0.05) Valuaion spreads a -1 s.d (-0.26) Trading Horizon 16
Impac of Ineremporal Hedging Moives on Targe Porfolio Weigh on value if he Valuaion Spread is +1 s.d (0.26) above mean. Over a long horizon, ineremporal hedging increases weigh on value. 0.6 0.5 0.4 0.3 0.2 0.1 Base case Correlaion = -0.43 Correlaion = 0.0 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 Weigh on value if he Momenum sae is +1 s.d (0.05) above mean. Over a long horizon, ineremporal hedging significanly reduces weigh on value. 0.6 0.5 0.4 0.3 0.2 0.1 Base case Correlaion = 1.0 Correlaion = 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 M o nhs 17
Impac of Coss on Targe Porfolio Weigh on value if he Valuaion Spread is +1 s.d (0.26) above mean. Over a long horizon, coss genly reduce he weigh on value 0.6 0.5 T = 0.01 0.4 0.3 0.2 T = 1 T = 0.1 Increasing Coss 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Weigh on value if he Momenum sae is +1 s.d (0.05) above mean. Over a long horizon, coss significanly reduce he weigh on momenum. 0.6 0.5 0.4 T = 0.01 0.3 0.2 0.1 T = 1 T = 0.1 Increasing Coss 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 M o nhs 18
Impac of Coss on Trading Speed Trading Speeds increase as horizon increases as he accruing benefi increases Trading Speeds reduce significanly as coss increases (perhaps no very surprising!) 0.6 T = 0.01 0.5 0.4 0.3 0.2 0.1 0 T = 0.1 T = 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Monhs Increasing Coss 19
Conclusions When comparing single period opimisaion o a dynamic opimisaion 1. No accouning for ineremporal hedging will 1. Seriously overweigh Momenum or rend following sraegies 2. Underweigh value sraegies 2. No accouning for rading coss in relaion o horizon of he sraegy 1. Overweighs momenum again far more heavily han a value sraegy 3. Suggess a separaion sraegy: 1. Esimae he opimal long run arge porfolio on he basis of coss and hedging demands 2. Esimae he rading speed a he second sage based on maximizing ne reurns. 20
Par 3: Appendix
Transacions Coss Linear in he size of rade Leland (1996, 1999) and Akinson (2004, 2010) analysed opimal rebalancing when coss are proporional o change in porfolio weighs, TC = k w Reasonable model if rades are small Cos is of rading a block of shares is 1 2 k ( Bid-Ask Spread) ( Number of Shares) = 1 2 ( Bid-Ask Spread )/( Price of Share) This model of coss implies no-rade zone. In zone no rade Ouside rade back o edge of he zone 22
Transacions Coss No rade-zone Inuiion is easy: Coss from a non opimal porfolio will be convex in deviaion from opimaliy Cos of rading are linear in he size of he rade w * (opimal Level) w Gain from Trade = L w Loss from Trade = -k w No Trade Zone TC = w Trade if L w> k w Loss Loss due deviaion from opimaliy Capial gains can be analysed similarly (effecively jus anoher cos) In muli-asse problem, numerically difficul o find borders of no rade-zone. 23
Transacion Coss Large Trades Trading coss can be divided ino: 1. Explici Coss: Order processing coss (compensaed hrough he bid-ask spread) 2. Implici Coss : Invenory and informaional coss. These cause prices o move o compensae he inermediary for risk of a large book and being he wrong side of an informed rade respecively. Madhaven (2000) saes ha while mos researchers recognize ha quoed spreads are small, implici rading coss can acually be economically significan because large rades move prices. 1 0.9 0.8 0.7 0.6 0.5 =0 Temporary Impac And similarly, Almgren (2005) 0.4 For large rades, he mos imporan componen of hese is he impac of he rader s own acions on he marke. These coss are nooriously difficul o measure. 0.3 0.2 0.1 0-0.1 Permanen Impac Time 24
The Assumpion of Quadraic Transacion Coss A block rade w is spli ino n smaller rades, wih each of hese rades execued a prices, p +i for i=0,1..n-1, where p pɶ = S sgn( w ) 1 i 2 pɶ = α S sgn( w ) + ε 1 i + 1 2 pɶ is mid price and S is he bid-ask he spread. The coefficien α is he informaion ransfer of he rade. Soll and Huang (97) and Soll (00) esimae α o be beween 5-45%. Aggregae over he n rades o find he cos of rading he block. ns n TC= p np = + + noise i= n 1 + i ɶ i= 0 α α ( 1 2) 2 2 So if n is large and α>0, cos is approximaely quadraic in he size of he block, n. 25
Wha he research says: Almgren (2005) is he only work on large daase wih deails of breakdown of orders o individual rades. He finds ( ) ( ) 2 1.6 1 2 Permanen Temporary Impac Impac TC= k w + k w where k 1 and k 2 are funcion of sock s volailiy, liquidiy, average daily volume and paricipaion rae. Kissel e al (2005, 2008), I-sar model: ( ) ( ) 1.36 2.05 1 2 Permanen Temporary Impac Impac TC= k w + k w where again k 1 and k 2 are funcion of sock s volailiy, liquidiy and average daily volume. Conclusion Quadraic reasonable assumpion for large rades - bu channel is hrough he price impac 26
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