A Dynamic Model for Investment Strategy

Transcription

1 A Dynamic Model for Investment Strategy Richard Grinold Stanford Conference on Quantitative Finance August

2 Preview Strategic view of risk, return and cost Not intended as a portfolio management tool Goal: analytical results What are the tradeoffs? 1

3 Information Ratio; IR Risk Aversion; λ Static Model Alpha; α Risk; ω 2

4 Half-life; HL Information Ratio; IR Risk Aversion; λ Transactions Costs Dynamic Model Alpha; α Risk; ω Cost; c 3

5 Applications Impact of additional assets under management on performance Estimate the value of diversifying investment themes Find the best mix of themes Product design: risk level, turnover, fast or slow ideas Improve myopic operational schemes of portfolio management 4

6 Some Principles Transactions costs imply time linkage Two costs The transactions costs you pay Opportunities lost (intimidation cost) From any initial conditions the system will go to an equilibrium 5

7 The Objective Alpha a Risk aversion λ p p W p κ TC z, p 2 Transactions cost amortization factor; TCAF Transactions costs ( ) Active position Active variance Initial active position 6

8 The No-Cost Ideal a= λ W q Ideal active position ignoring costs 7

9 Complete the Square U ( p) a p- λ p W p 2 The backlog U p = U q - λ q p W q p 2 ( ) ( ) { } { } Loss of risk adjusted return due to the backlog. 8

10 Cost Minimization λ p q W p q + κ TC z, p 2 { } { } ( ) Apples Oranges 9

11 Opportunity Loss 1.40% 1.20% 1.00% 0.80% 0.60% 0.40% Opportunity Loss t-cost frontier Opportunity Loss & t-cost frontier backlog risk = 1.20% backlog risk = 0.80% 0.20% efficient tradeoff backlog risk = 0.40% 0.00% 0.01% 0.11% 0.21% 0.31% 0.41% Transactions Cost 10

12 The Dynamic Framework Solves the apples and oranges problem Solves the equilibrium problem. We understand the significance of initial conditions. But there is no free lunch 11

13 B.S. Assumptions Half-life; HL Information Ratio; IR Risk Aversion; λ Transactions Costs Dynamic Model Alpha; α Risk; ω Cost; c 12

14 Key Assumption μ TC z, p = p z Ω p z 2 ( ) { } { } Spread costs are negligible Market impact is proportional to the variance. This, in turn, is proportional to the cost of hedging the position. Rudimentary empirical inquiry indicates it is not an absurd guess. Robert Solow, paraphrased: All of our results rest on assumptions that not quite true 13

15 Calibrating the t-cost parameter Forecast t-cost of trade portfolio Forecast t-cost versus risk of random trade lists Std dev of trade portfolio 14

16 Calibrating the t-cost parameter Forecast t-cost of trade portfolio Average forecast t-cost versus risk of random trade lists Std dev of trade portfolio 15

17 The Information Process a% t+ Δ t = γ a t + s% Δt ( ) ( ) γ 1 2 Δt HL New information, Uncorrelated with prior alpha γ is the auto-correlation of the alphas 16

18 Optimal Linear Decision Rule Policy Parameters: δ and ψ Decision Rule p= δ z+ 1 δ ψ q A scaled back ideal position 1 δ ψ = 1 γ δ ( ) { } m μ δ =, where m = m m λ Δ t 17

19 Myopic (single stage) Equivalent ψ scales back the alpha λ ψ a p p W p κ TC z, p 2 ( ) 1 δ κ = Δ t is the 'correct' amortization factor 18

20 Properties of the Optimal Policy: The Transfer Coefficient Information Ratio p q IR α α τ Corr( pq, ) IR ω = ω = p p q p q Transfer coefficient τ p 2 1 δ = < γ δ 1 19

21 Properties of the Optimal Policy: Risk Level Aggressiveness χ p ω ω p q ( 1 δ) ( 1+ γ δ) χ p = ψ < τ 1+ δ 1 γ δ ( ) ( ) P 20

22 Properties of the Optimal Policy: Alpha, Risk and Cost Annual Alpha { } 2 α = χ τ α = ψ α p p p q q Active Risk ω p = χp ωq Annual cost { 2 } c = U χ τ χ p q p p p 21

23 no cost low cost mid cost 9.00% 10.00% 8.00% 7.00% Predict After Cost Performance 16.0% 14.0% 12.0% 10.0% 8.0% 6.0% 4.0% 2.0% 0.0% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 22 Active Risk Alpha less Cost high cost

24 Impact of increased Assets under Management (AUM) 6% 5% risk 4% alpha 3% 2% 1% cost 0% AUM

25 Signal Diversification 14% 12% aggregate alpha 10% 8% aggregate objective 6% 4% 2% 0% 24 aggregate cost aggregate risk penalty Number of Signals

26 Generalization : Multiple Sources a% t t a t % t j J ( +Δ ) = γ ( ) + s Δ ; = 1, j j j j IR E α% Ω α% 2 1 j j j IR = λ ρ σ ; k = 1, J k k, j j j= 1, J 25

27 Generalization : Multiple Sources σ q Ω α q% Ω q% j 1 2 j j; σ j = E j j IR j ψ j 1 δ ; 0< ψ j < 1 1 γ δ j Decision Rule ( ) p= δ z+ δ ψ q 1 j j j= 1, J 26

28 ψ Down-weight 100% 80% Fast Signal Low Cost 60% 40% 20% High Cost Slow Signal 0% Half-Life (yrs.)

29 Summary The transactions cost opportunity loss trade-off The simplest model that captures the dynamic effects. Time linkage and equilibrium Explicit connection of outputs with inputs The nature of the optimal policy and its capture in our current practice Strategic studies: decrease cost by 10% or increase IR by 10%? Can be extended to general quadratic transactions costs A strategic grasp on the risk, return and cost problem; a new perspective. 28