Portfolios with Trading Constraints and Payout Restrictions John R. Bi - PDF document

1Portfolios with Trading Constraints an
1Portfolios with Trading Constraints and Payout Restrictions John R. BirgeNorthwestern University(joint work with Chris Donohue, Xiaodong Xu, and Gongyun Zhao)•(Very) long-term investor (example: university endowment)•Payout from portfolio over time (want to keep

•Invest in various asset categories•Dec
•Invest in various asset categories•Decisions:•How much to payout (consume)?•How to invest in asset categories?•Complication: restrictions on asset trades2•Basic formulation•General infinite horizon solution method•Simplified problem and continuous time •Results

for restricted-trading portfolio•Future
for restricted-trading portfolio•Future issues•Notation:x –current state (x u (oru) –current action given x (u (oru–single period discount factorx,u–probability measure on next period state y c(x,u) –objective value for V(x) –value function of optimal expected fu

ture rewards •Problem: Find V such that
ture rewards •Problem: Find V such that V(x) = maxU(x)[V(y)] }for all x 4•V*•Contraction–V*||•Unique Fixed Pointif TV, then V•Value Iteration•Distributed Value IterationX infinitely often, (Here, random choice of x, use concavity.)•Deepest Cut to maximize Vof do

main of V*)6Feasibility: Ax + Bu Trans
main of V*)6Feasibility: Ax + Bu Transition:u for some realization i with probability pIteration k Problem:) = maxs.t. A xb, -Eu) -eFrom duality:max,u) -u) + emax,u) -)) for optimal for xAx +Cuts:equal to the constant terms.•Determine asset allocation and con

sumption policy to maximize the exp•Stat
sumption policy to maximize the exp•State and Action x=(cons, risky, wealth) u=(cons_new,risky_new)•Two asset classes •Risky asset, with lognormal return distribution•Riskfree asset, with given return •Power utility function•Consumption rate constraicons_new cons1

__1newconsnewconsc7•Dybvig ’95*•Contin
__1newconsnewconsc7•Dybvig ’95*•Continuous-time approach•Solution Analysis•Consumption rate remains constant until wealth reaches a new maximum•The risky asset allocation is proportional to w-c/r, which is the excess of wealth over the perpetuity value of current

consumptiondecreases as wealth decrease
consumptiondecreases as wealth decreases, approaching 0 as wealth approaches c/r(which is in absence of risky investment sufficient to maintain consumption indefinitely). •Dybvig’01•Considered similar problem in which consumption rate d (soft constrained problem)*

“Duesenberry'sRatcheting of Consumption
“Duesenberry'sRatcheting of Consumption: Optimal Dynamic Consumption and Investment Given Intolerance for any Decline in Standard of Living” Review of Economic Studies62, 1995, 287-313.•Replicate Dybvigcontinuous time results •Evaluate the effect of trading restr

ictions for •Consider additional problem
ictions for •Consider additional problem features•Transaction Costs•Multiple risky assets10•Can formulate infinite-horizon investment problem in stochastic programming •Solution with cutting plane method•Convergence with some conditions•Results for trade-restrict

ed assets significantly different from m
ed assets significantly different from market assets •Application of typical stochastic programming approach complicated by •Initialization.•Define a valid constraint on Q(x)xTbAxtsxQexcpxQiiiiiiiit..max00exExQiRequires problem extremely high rate of consumpti