Portfolos wth Tradng Constrants and Payout Restrctons John R. Brge Northwestern Unversty (ont wor wth Chrs Donohue Xaodong Xu and Gongyun Zhao) 1 General Problem (Very) long-term nvestor (eample: unversty endowment) Payout from portfolo over tme (want to eep payout from declnng) Invest n varous asset categores Decsons: How much to payout (consume)? How to nvest n asset categores? Complcaton: restrctons on asset trades 2 1
Outlne Basc formulaton General nfnte horzon soluton method Smplfed problem and contnuous tme soluton Results for restrcted-tradng portfolo Future ssues 3 Problem Formulaton Notaton: current state ( X) u (or u ) current acton gven (u (or u ) U()) δ sngle perod dscount factor P u probablty measure on net perod state y dependng on and u c(u) obectve value for current perod gven and u V() value functon of optmal epected future rewards gven current state Problem: Fnd V such that V() ma u U() {c(u) + δ E Pu [V(y)] } for all X. 4 2
Approach Defne an upper bound on the value functon V 0 () V() X Iteraton : upper bound V Solve for some TV ( ) ma u c( u) + δ E Pu [V (y)] Update to a better upper bound V +1 Update uses an outer lnear appromaton on U 5 Successve Outer Appromaton V 0 V 1 TV 0 0 V* 6 3
Propertes of Appromaton V* TV V +1 V Contracton TV V * δ V V* Unque Fed Pont TV*V* f TV V then V V*. 7 Convergence Value Iteraton T V 0 V* Dstrbuted Value Iteraton If you choose every X nfntely often then V V*. (Here random choce of use concavty.) Deepest Cut Pc to mamze V ()-TV () DC problem to solve Convergence agan wth contnuty (cauton on boundary of doman of V*) 8 4
Detals for Random Choce Consder any Choose and s.t. < ε Suppose V K V () V*() V ()-V ( )+V ( )- V*( )+V*( )-V*() 2 ε K + δ V -1 ( )-V*( ) 2 ε K + δ V 0 ( 0 ) V*( 0 ) 9 Cuttng Plane Algorthm Intalzaton: Construct V 0 ()ma u c 0 (u) + δe Pu [V 0 (y)] where c 0 c and c 0 concave. V 0 s assumed pecewse lnear and equvalent to V 0 ()ma {θ θ E 0 + e 0 }. 0. Iteraton: Sample X (n any way such that the probablty of A s postve for any A X of postve measure) and solve TV ( ) ma u c( u) + δ E Pu [V (y)] where V (y) ma{θ θ E l y + e l l0.} Fnd supportng hyperplanes defned by E +1 and e +1 such that E +1 + e +1 TV (). +1. Repeat. 10 5
Specfyng Algorthm Feasblty: A + Bu b Transton: yf u for some realzaton wth probablty p Iteraton Problem: TV ( ) ma uθ c( u) + δ p θ s.t. A + B u b - E l (F u) - e l + θ 0 l. From dualty: TV ( ) nf µλl ma uθ c( u) -µ(a +Bu-b) + δ (p θ + l λ l (E l (F u) + e l - θ )) ma uθ c( u) -µ (A +Bu-b) + δ (p θ + l λ l (E l (F u) +e l - θ )) for optmal µ λ l for c( u ) + c( u ) T (- -u ) -µ A +µ b + ( l λ l e l ) Cuts: E +1 c( u ) T -µ A e +1 equal to the constant terms. 11 Investment Problem Determne asset allocaton and consumpton polcy to mamze the epected dscounted utlty of spendng State and Acton (cons rsy wealth) u(cons_newrsy_new) Two asset classes Rsy asset wth lognormal return dstrbuton Rsfree asset wth gven return r f Power utlty functon c ( cons _ new) cons _ new 1 γ 1 γ Consumpton rate constraned to be non-decreasng cons_new cons 12 6
Estng Research Dybvg 95* Contnuous-tme approach Soluton Analyss Consumpton rate remans constant untl wealth reaches a new mamum The rsy asset allocaton α s proportonal to w-c/r f whch s the ecess of wealth over the perpetuty value of current consumpton α decreases as wealth decreases approachng 0 as wealth approaches c/r f (whch s n absence of rsy nvestment suffcent to mantan consumpton ndefntely). Dybvg 01 Consdered smlar problem n whch consumpton rate can decrease but s penalzed (soft constraned problem) * Duesenberry's Ratchetng of Consumpton: Optmal Dynamc Consumpton and Investment Gven Intolerance for any Declne n Standard of Lvng Revew of Economc Studes 62 1995 287-313. 13 Obectves Replcate Dybvg contnuous tme results usng dscrete tme approach Evaluate the effect of tradng restrctons for certan asset classes (e.g. prvate equty) Consder addtonal problem features Transacton Costs Multple rsy assets 14 7
100 Results Non-decreasng Consumpton 90 80 As number of tme perods per year ncreases soluton converges to contnuous tme soluton Allocaton to Stoc 70 60 50 40 30 Dybvg N 1 N 2 N3 N4 N 6 N12 20 10 0 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 Consumpton Rate 15 Results Non-Decreasng Consumpton wth Transacton Costs 60 50 40 No Transacton Cost Transacton Cost (ntal stoc allocaton 0%) Transacton Cost (ntal stoc allocaton 100%) Stoc Allocaton 30 20 10 0 3.5 3.7 3.9 4.1 4.3 4.5 4.7 4.9 Consumpton Rate 16 8
Observatons Effect of Tradng Restrctons Contnuously traded rsy asset: 70% of portfolo for 4.2% payout rate Quarterly traded rsy asset: 32% of portfolo for same payout rate Transacton Cost Effect Small dfferences n overall portfolo allocatons Optmal m depends on ntal condtons 17 Etensons Soft constrant on decreasng consumpton Allow some decreases wth some penalty Lag on sales Watng perod on sale of rsy assets (e.g. 60- day perod) Multple assets Allocaton bounds 18 9
Conclusons Can formulate nfnte-horzon nvestment problem n stochastc programmng framewor Soluton wth cuttng plane method Convergence wth some condtons Results for trade-restrcted assets sgnfcantly dfferent from maret assets wth same rs characterstcs 19 Approach Applcaton of typcal stochastc programmng approach complcated by nfnte horzon Q δt ( ) ma p ( c + e Q( ) s. t. A b T Intalzaton. Defne a vald constrant on Q() 0 0 ( ) E e Q + Requres problem nowledge. For optmal consumpton problem assume etremely hgh rate of consumpton forever 20 10
11 21 Approach (cont.) Iteraton ( ) ( ) ( ) where V U Fnd mn γ Epensve search over possble for the optmal consumpton problem because of small number of varables ( ) { } and e E V mn 0 + ( ) ( ) 1 0.. ma + Θ Θ + e E T b A s t e c p U t K δ ( ) + > e b p e T p E defne a new cut Else If ρ µ µ ε γ termnate.