Markowitz Principles for Multi-Period Portfolio Selection Problems - PowerPoint PPT Presentation

Markowitz Principles for Multi-Period Portfolio Selection Problems with Moments of any Order and Constraints Thamayanthi Chellathurai Enterprise Risk & Portfolio Management Bank of Montreal, Toronto Disclaimer: The opinion expressed in this talk is that of the author only, and not that of the Bank of Montreal

Outline 1. Single-Period Portfolio Selection 2. Limitations 3. Multi-Period Portfolio Selection 4. Multi-Period Portfolio Selection with Non- Negative Wealth Constraints 5. Numerical Results 6. Conclusion 2

Markowitz Single-Period Model = t t 0 t = T th → = S Price of one unit of i asset at t 0 i , 0 th → = S Price of one unit of i asset at t T i , T → r Interest rate → T Planning horizon 3

Markowitz Single-Period Model Risk-free Asset rT S S e 0 , 0 0 , 0 Risky Assets ( N ) S i , 0 S , i T 4

U U 0 , 0 0 , 0 S S 0 , 0 0 , T U U 1 , 0 1 , 0 W ( 0 ) S S , W ( T ) 1 , 0 1 T t = t = T t 0 U U N , 0 N , 0 S S N , T N , 0 = + + + ( 0 ) ... W U S U S U S 0 , 0 0 , 0 1 , 0 1 , 0 N , 0 N , 0 = + + + W ( T ) U S U S ... U S 0 , 0 0 , T 1 , 0 1 , T N , 0 N , T th → U Number of Shares of i asset i , 0 = transacted at t 0 5

    N S S W ( T ) W ( 0 ) ∑ i , T i , 0   = + − U , 0   i S S S S   = i 1 0 , T 0 , 0 0 , T 0 , 0 = Problem : Given W ( 0 ), U ? i , 0 6

  S S 1 , T 1 , 0 −   S S     U 0 , T 0 , 0 1 , 0         S S     U 2 , T 2 , 0 − 2 , 0     S S = = A , Q 0 , T 0 , 0     . .     .     . .     .  U    S S , 0 N N , T N , 0   − S S   0 , T 0 , 0 7

W ( T ) W ( 0 ) T = + Q A S S 0 , 0 , 0 T     ( ) ( 0 ) W T W T   = + ( ) E E Q A    S  S 0 , T 0 , 0     W ( T )   T = Var A C A   S   0 , T { } [ ] [ ] T = − − C E Q E ( Q ) Q E ( Q ) → Covariance matrix 8

Markowitz Problem Find A that minimizes     1 W ( T ) 1   T = Var A C A   2 S 2   0 , T subject to     G W ( T ) W ( 0 ) W ( T )   T = + = E E ( Q ) A    S  S S 0 , T 0 , 0 0 , T 9

  G W ( T ) W ( 0 )   − S S   0 , T 0 , 0 − 1 = A C E ( Q ) opt T − 1 E ( Q ) C E ( Q )         G W ( T ) W ( 0 ) W ( T ) W ( 0 )     − = − E     S S S S     0 , T 0 , 0 0 , T 0 , 0         G W ( T ) W ( 0 ) W ( T ) W ( 0 ) 1     − = − Std      S S   S S  T − 1 E ( Q ) C E ( Q ) 0 , T 0 , 0 0 , T 0 , 0 Efficient Frontier         W ( T ) W ( 0 ) W ( T ) W ( 0 ) −     T 1 − = − ( ) ( ) E Std E Q C E Q     S S S S     0 , T 0 , 0 0 , T 0 , 0 10

● Short-sale restrictions ● Bounds on asset holdings ● Transaction costs ● Sensitivity to E(Q) and C ( Perold 1984, Best & Grauer 1991, Best & Hlouskova 2008, Roman & Mitra 2009, Best 2010 ) 11

Issues 1. Single Period • Now or Never Volatility is Risk NOT an • Opportunity (Fernholz & Shay 1982, Luenberger 1998) 2. Moments • First and Second only 12

Multi-Period Markowitz Principles ( Proc. Roy. Soc., 2002 ) t …… 0 1 2 M M+1 th → = Price of one unit of i asset at t t S i , j j th → U Number of shares of of i asset held i , j = at t t after tran saction j 13

Multi-Period Markowitz Principles U U , U U U U , − 0 , M 1 0 M 0 , 0 0 , 0 0 , 1 0 M S S S , S 0 , 0 + 0 , 1 0 M 0 , M 1 U U , U U , − 1 , M 1 1 M U U 1 , 1 1 M S , S 1 , 0 1 , 0 S S 1 M 1 , 0 1 , 1 1 , + 1 M U , U U , N M U U U − N , M 1 N M N , 0 N , 0 N , 1 S S S S , , + 1 N M , 0 , 1 N N N M 14

U 0 , 0 S 0 , 0 U 1 , 0 W ( 0 ) S 1 , 0 t = t 0 U N , 0 S N , 0 = + + + W ( 0 ) U S U S ... U N S 0 , 0 0 , 0 1 , 0 1 , 0 , 0 N , 0 15

U U 0 , 0 0 , 1 S S 0 , 1 0 , 1 U U 1 , 0 1 , 1 S S 1 , 1 1 , 1 t = t 1 U U N , 0 N , 1 S S N , 1 N , 1 N N ∑ ∑ ● Instant Transaction = U S U S i , 0 i , 1 i , 1 i , 1 ● Frictionless = = i 0 i 0 16

U 0 , M S + 0 , M 1 U , 1 M W ( T ) S + 1 , M 1 = t t M + 1 = T U N , M S + N , M 1 Wealth Dynamics N ∑ = ( 0 ) W U S i , 0 i , 0 i = 0 N N ∑ ∑ = = U S U S , j 1 , 2 , ... , M − i , j 1 i , j i , j i , j = = i 0 i 0 N ∑ = W ( T ) U S i , M i , M + 1 17 = i 0

t = t j U U 0 , j − 0 , j 1 α = − U U S − 0 , j 0 , j 0 , j 0 , j 1 U U , − 1 , j 1 1 j α = − U U S 1 , j − 1 , j 1 , j 1 , j 1 U U α = − U U , − N j 1 N , j − N , j N , j N , j 1 S N , j 18

    N S S W ( t ) W ( t ) ∑ i , 1 i , 0   1 0 = + − U   i , 0 S S S S   = i 1 0 , 1 0 , 0 0 , 1 0 , 0 = + k 2, 3, ... , M 1     N S S W ( t ) W ( t ) ∑ i , k i , 0   k 0 = + − U   i , 0 S S S S   = i 1 0 , k 0 , 0 0 , k 0 , 0     − S k 1 N S ∑ ∑ i , j i , k   + − α i , j   S S   = = j 1 i 1 0 , k 0 , j ( Proc. Roy. Soc., 2002 ) = → U , i 1, 2, ... , N Determinis tic Controls i , 0 α = = → , i 1, 2, ... , N, j 1, 2, ... , M Stochastic Controls i , j 19

Multi-Period Markowitz Principles     N S S W ( t ) W ( t ) ∑ + i , M 1 i , 0   M + 1 0 = + − U   i , 0 S S  S S  = i 1 + + 0 , M 1 0 , 0 0 , M 1 0 , 0     S M N S ∑ ∑ + i , M 1 i , j   + − α ,   i j S S   = = j 1 i 1 + 0 , M 1 0 , j = → U , i 1, 2, ... , N Determinis tic Controls i , 0 α = = → , i 1, 2, ... , N, j 1, 2, ... , M Stochastic Controls i , j → Non - Anticipati ve Processes 20

Parametrization of Stochastic Controls (Proc. Roy. Soc., 2008)     N S S ∑ − k k , j k , j 1   α = + − Linear : V B i , j i , j i , j   S S   k = 1 − 0 , j 0 , j 1 Quadratic :     N S S ∑ − k k , j k , j 1   α = + − V B i , j i , j i , j   S S   k = 1 − 0 , j 0 , j 1         N l S S S S ∑∑ − − l , k l , j l , j 1 k , j k , j 1     + − − D     i , j S S S S     = = l 1 k 1 − − 0 , j 0 , j 1 0 , j 0 , j 1 k l , k V , B , D Determinis tic Parameters i , j i , j i , j Un knowns to be determined 21

Multi-Period Markowitz Principles W ( t ) W ( t ) T + M 1 0 = + Q A S S + 0 , M 1 0 , 0 → Q Stochastic vector of Asset Prices, linear & quadratic terms k → A Determinis tic vector of Unknowns, U , V , B i,0 i, j i, j     W ( t ) W ( t )   T + M 1 0 = + E E ( Q ) A    S  S 0 , + 1 0 , 0 M     W ( t ) T  +  M 1 = Var A C A   S   + 0 , M 1 { } [ ][ ] T = − − C E Q E ( Q ) Q E ( Q ) → E ( Q ) Means as well as higher order moments → C Second order and higher order moments 22