A Dynamic Model for Investment Strategy
A Dynamic Model for Investment A Dynamic Model for Investment StrategyStrategy
Richard Grinold
Stanford Conference on Quantitative Finance
August 18-19 2006
1
Preview
Strategic view of risk, return and cost
Not intended as a portfolio management tool
Goal: analytical results
What are the tradeoffs?
2
Static
Model
Information Ratio; IR
Risk Aversion; λ
Alpha; α
Risk; ω
3
Information Ratio; IR
Risk Aversion; λ
Transactions Costs
Half-life; HL Alpha; α
Risk; ω
Cost; c
Dynamic
Model
4
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
5
Some Principles
Transactions costs imply time linkage
Two costsThe transactions costs you payOpportunities lost (intimidation cost)
From any initial conditions the ‘system’ will go to an equilibrium
6
The Objective
( ),2
TCp p p z pλ κ′ ′⋅ − ⋅ ⋅ ⋅ − ⋅a W
Alpha
Active position
Risk aversion
Active variance
Transactions costs
Initial active position
Transactions cost amortization factor; TCAF
7
The No-Cost Ideal
qλ ⋅ ⋅a = W
Ideal active position ignoring costs
8
Complete the Square
Loss of risk adjusted return
due to the backlog.
( )2
U p p p pλ′ ′≡ ⋅ ⋅ ⋅ ⋅a - W
( ) ( ) 2
U Up q q p q pλ ′= ⋅ − ⋅ ⋅ −- W
The backlog
9
Cost Minimization
( ),2
TCp q p q z pλ κ′⋅ − ⋅ ⋅ − + ⋅W
Apples Oranges
10
Opportunity Loss & t-cost frontier
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
0.01%
0.11%
0.21%
0.31%
0.41%
Transactions Cost
Opp
ortu
nity
Los
s
efficient tradeoff backlog risk = 0.40%
backlog risk = 0.80%
backlog risk = 1.20%
Opportunity Loss & t-cost frontier
11
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
12
Information Ratio; IR
Risk Aversion; λ
Transactions Costs
Half-life; HL Alpha; α
Risk; ω
Cost; c
Dynamic
Model
B.S. Assumptions
13
Key Assumption
( ) ,2
TC z p p z p zμ ′= ⋅ − ⋅Ω ⋅ −
• 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’
14
Calibrating the t-cost parameter
Forecast t-cost versus risk of random trade lists
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Std dev of trade portfolio
Fore
cast
t-co
st o
f tra
de p
ortfo
lio
15
Calibrating the t-cost parameter
Average forecast t-cost versus risk of random trade lists
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Std dev of trade portfolio
Fore
cast
t-co
st o
f tra
de p
ortfo
lio
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
16
The Information Process
( ) ( )t t t tγ+ Δ = ⋅ + ⋅ Δsa a %%
• γ is the auto-correlation of the alphas
12
tHL
γΔ
⎛ ⎞≡ ⎜ ⎟⎝ ⎠
New information, Uncorrelated with
prior alpha
17
Optimal Linear Decision RulePolicy Parameters: δ and ψ
( ) Decision Rule
1p z qδ δ ψ= ⋅ + − ⋅ ⋅
, where 0.5 1 4
m mtm m
μδλ
= =⋅Δ+ + + ⋅
A scaled back ideal position1
1δψ
γ δ−
=− ⋅
18
Myopic (single stage) Equivalent
( ),2
TCp p p z pλψ κ′ ′⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅a W
1 is the 'correct' amortization factortδκ −
=Δ
scales back the alphaψ
19
Properties of the Optimal Policy:The Transfer Coefficient
2
2 2
Transfer coefficient
1 11p
δτγ δ−
= <− ⋅
( )
Information Ratio
,p qp p q
p q
IR Corr IRp qα α
τω ω
≡ = ⋅ = ⋅
20
Properties of the Optimal Policy:Risk Level
Aggressiveness
pp
q
ωχ
ω≡
( ) ( )( ) ( )1 11 1p P
δ γ δχ ψ τ
δ γ δ− ⋅ + ⋅
= ⋅ <+ ⋅ − ⋅
21
Properties of the Optimal Policy:Alpha, Risk and Cost
2
Annual Alpha
p p p q qα χ τ α ψ α= ⋅ ⋅ = ⋅
2
Annual cost
p q p p pc U χ τ χ= ⋅ ⋅ −
Active Risk
p p qω χ ω= ⋅
22
Predict After Cost Performance
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
10.00
%
Active Risk
Alp
ha le
ss C
ost low cost
mid cost
high cost
no cost
23
Impact of increased Assets under Management (AUM)
0%
1%
2%
3%
4%
5%
6%
0.01
0.11
0.21
0.31
0.41
0.51
0.61
0.71
0.81
0.91
1.01
1.11
1.21
1.31
1.41
1.51
1.61
1.71
1.81
1.91
alpha
risk
cost
AUM
24
Signal Diversification
0%
2%
4%
6%
8%
10%
12%
14%
1 2 3 4 5 6 7 8 9 10
aggregate objective
aggregate cost
aggregate risk penalty
aggregate alpha
Number of Signals
25
Generalization : Multiple Sources
( ) ( ) ; 1,j j j jt t t t j Jsγ+ Δ = ⋅ + ⋅ Δ =a a %%
2 1j j jIR E α Ω α−′≡ ⋅ ⋅% %
,1,
; 1,k k j jj J
IR k Jλ ρ σ=
= ⋅ ⋅ =∑
26
Generalization : Multiple Sources
( )1,
Decision Rule1 j j
j Jp z qδ δ ψ
=
= ⋅ + − ⋅ ⋅∑
1 ; 0 11j j
j
δψ ψγ δ−
≡ < <− ⋅
1 2;jj j j j j
j
EIR
q Ω α q Ω qσ
σ− ′≡ ⋅ ⋅ ⇒ = ⋅ ⋅% %
27
ψ Down-weight
0%
20%
40%
60%
80%
100%
0.02
0.13
0.25
0.37
0.48
0.60
0.71
0.83
0.94
Half-Life (yrs.)
Fast Signal
Slow Signal
Low Cost
High Cost
28
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.
