Active Risk and Information Ratio of Active Management Edward Qian Ronald Hua Putnam Investments Boston, MA
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Introduction
Quantitative Equity Research
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Previous Research Ø Ø Ø Ø Ø Ø Ø
Grinold, R.C. 1989, “Fundamental Law of Active Management.” JPM Grinold, R.C. 1994, “Alpha Is Volatility Times IC Times Score.” JPM Grinold & Kuhn, 1999, Active Portfolio Management, McGraw-Hill, New York Grinold, R.C., and R.N. Kahn. 2000. “The Efficiency Gains of LongShort Investing.” FAJ Clarke, R., H. de Silva, and S. Thorley. 2002. “Portfolio Constraints and the Fundamental Law of Active Management.” FAJ Sorensen, E.H., E. Qian, R. Hua, and R. Schoen. 2004. “Multiple Alpha Sources and Active Management.” JPM Qian & Hua. 2004. “Active Risk and Information Ratio of Active Management.” submitted 5/5/2004
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Key Practical Questions Ø
How to select quantitative factors – Information coefficient (IC)
Ø
How to translate factor score into alpha input to an optimizer – “Alpha is volatility times IC times score”
Ø
How to combine multiple alpha factors – IC & factor correlation
Ø Ø
What is the information ratio (IR) of a quantitative equity strategy What is the impact of various practical constraints 5/5/2004
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Key Previous Results Ø Fundamental
(FLOAM) Ø Modified
Law of Active Management IR = ICt N
FLOAM with transfer coefficient IR = TC ⋅ ICt N
Ø Alpha
is volatility times IC times score α i = ICtσ i zi
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Active Risk
Should We Trust Risk Models?
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A “Missing” Question Ø
What is the active risk of quantitative equity strategy –
Ø
If that is “wrong”, then the problem might be with the risk model (Hartmann et al 2002) i. ii. iii. iv.
Ø Ø
The obvious answer: the tracking error target given by a risk model
Estimation error in covariances in a risk model Time varying nature of covariances Serial auto-correlations of excess returns Drift of portfolio weights over a given period
What if the risk model is “perfect”? Why ex post tracking errors seem to ALWAYS exceed ex ante tracking error by a risk model 5/5/2004
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The Tale of Two Strategies Ø
The “perfect” strategy – It produces a stream of constant excess return every period,
1%, 1%, …, 1%, … – By definition, it has no active risk against benchmark – But a risk model would always tell us the strategy has active risk Ø
The bad strategy – Its excess returns are volatile, 1%, -1%, 1%, …, – Its active risk could easily exceed the tracking error estimate by
a risk model 5/5/2004
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Introducing Strategy Risk Ø Ø Ø Ø Ø
In addition to the generic risk model risk, each strategy has its own strategy risk The strategy risk reflects the variability of model’s ability to generate excess return It can be measured by the standard deviation of IC The true active risk of a strategy is a combination of riskmodel risk and its own strategy risk The information ratio should take it into account – This should not be too surprising since we already
acknowledge the difference in average IC 5/5/2004
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New Results on Active Risk Ø
True active risk consists of
σ = std( IC ) N σ model
– Risk-model target tracking error – Strategy risk – Square root of the breadth Ø
The strategy risk is different for different strategy – Different factors – Different combination of factors
Ø
In most cases, true active risk is greater than the riskmodel tracking error 1 – They are the same only if std ( IC ) = N – This is true if strategy risk is purely sampling error
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New Results on IR Ø
Average excess return remains roughly the same α t = IC t N σ model
Ø
The information ratio is then average IC divided by standard deviation of IC IR =
Ø Ø
IC std( IC )
The results applies to individual factors, different combinations of factors They are different for different stock markets, style groups, sectors 5/5/2004
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Quantitative Equity Models
A Portfolio Approach
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A New Approach Ø New
approach to build quantitative equity models Ø The new approach is descriptive in nature Ø We derive the new results based on the approach Ø It differs from the “normative” approach as in FLOAM
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A Normative Approach APT Risk Model MV Optimization
Optimal Weights
Excess Return
Forecasts APT
Actual Returns
Alpha factors
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Normative Versus Descriptive Ø
Derivation of fundamental law depends on normative framework of APT – Fundamental risk models are often based on APT framework – Expected excess returns of individual securities is also
assumed to follow APT: “beta” times alpha factor Ø
Descriptive approach – Practitioners need risk model to do mean-variance
optimization – But it is not necessary to APT will be true for each security – Based on “real” long-short portfolios 5/5/2004
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The New Approach APT Risk Model MV Optimization
Optimal Weights
Excess Return
Alpha factors Risk Factor Neutral
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Actual Returns
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Details Ø Ø Ø Ø
Select a risk model – BARRA, Northfield,APT ... Select a alpha factor – valuation, momentum, profitability, ... Select a fixed trading period – monthly, quarterly,.. MV optimization chooses optimal long-short portfolio – With fixed target tracking error – With all risk factor neutralized – Exact solution exists for the optimization
Ø Ø
Compute single period alpha Compute multi-period statistics – average, standard deviation, IR, ... 5/5/2004
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Limitations and Extensions Ø
Focus on strategies exploiting only the stock-specific returns with no factor bets – Mathematical convenience l l
Exact solution for optimal weights No need of historical factor return covariance matrices
– Most quantitative strategies have minimal risks Ø
Analysis can be extended to strategies with factor bets – We can expect the same qualitative conclusion such as
different strategy risk – The magnitudes of bias are not yet known 5/5/2004
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Single-period Results Ø
Optimal long-short portfolio weights wi = λ−1
Ø
f i − l1 − l2 β 1i − L − l M +1 β Mi σ i2
Single-period excess return αt = λ
Ø
−1 t
N
∑R i =1
i, t
Fi ,t
Risk-adjust forecasts and risk-adjusted returns Fi =
f i − l1 − l2 β 1i − L − lM +1 β Mi r − k − k β − L − k M +1 β M 1i Ri = i 1 2 1i σi σi
Ø
Single-period excess return in terms of IC and dispersions α t = λt−1 ( N − 1)ICt dis (R t )dis (Ft )
Ø
Final results 5/5/2004
α t ≈ ICt N σ model dis(R t ) 2004 Northfield Annual Conference
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Implications Ø Ø
The analysis is based on realistic long-short portfolios IC is correlation coefficient between risk-adjusted forecasts and risk-adjust returns – Not raw forecasts and raw returns – Not regressed forecasts
Ø Ø Ø
We do not impose any relationship (linear) between individual returns and individual forecasts, i.e., no APT For a consistent risk model, cross sectional dispersion of risk-adjusted return should be close to unity Therefore αt ≈ ICt N σ model 5/5/2004
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Empirical Results
Higher Active Risk
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Ex Post Active Risk 10
frequency
8
6
4
2
>13.5%
13.0%
12.5%
12.0%
11.5%
11.0%
10.5%
10.0%
9.5%
9.0%
8.5%
8.0%
7.5%
7.0%
6.5%
6.0%
5.5%
5.0%
4.5%
4.0%
3.5%
<3.0%
0
Almost all alpha factors produce tracking error higher than the target of 5%. Some are significantly higher. 5/5/2004
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Persistence of Strategy Risk 12
18 16
10
14
frequency
12
6 4
10 8 6 4
2
2
0
Second half active risk without adjustment
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Second half active risk with adjustment for strategy risk determined from the first half
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>13.5%
13.0%
12.5%
12.0%
11.5%
11.0%
10.5%
9.5%
10.0%
9.0%
8.5%
8.0%
7.5%
7.0%
6.5%
6.0%
5.5%
5.0%
4.5%
4.0%
3.5%
<3.0%
13.0%
>13.5%
12.5%
12.0%
11.5%
11.0%
10.5%
9.5%
10.0%
9.0%
8.5%
8.0%
7.5%
7.0%
6.5%
6.0%
5.5%
5.0%
4.5%
4.0%
3.5%
0 <3.0%
frequency
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Persistency of Strategy Risk out-of-sample strategy risk
5.50%
y = 0.4941x + 0.0138 R 2 = 0.5169 4.50%
3.50%
2.50%
1.50% 1.50% 2.00%
2.50% 3.00%
3.50% 4.00% 4.50%
5.00% 5.50%
in-sample strategy risk
Strong positive correlation between two samples. Factors having higher (lower) strategy risk in the first half tends to higher (lower) strategy risk in the second half too 5/5/2004
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Significant Difference Ø
Two valuation factors – Gross profit to enterprise value (GP2EV) – Forward earnings yield based on IBES FY1 consensus forecast
(E2P) GP2EV E2P
Ø
Average Alpha 6.2% 3.3%
STD of Alpha 6.9% 8.7%
IR of Alpha 0.90 0.38
Average IC 2.4% 1.4%
STD of IC 2.7% 3.4%
IR of IC 0.91 0.41
Average dis(R) 1.01 1.00
Average N 2738 2487
F-test shows that the variances of IC are significantly different at 5% level
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A Factor Example Quarterly Alpha 12.5% alpha 10.0%
ex post risk
GP2EV
7.5% 5.0% 2.5% 0.0% -2.5%
Ø Ø 5/5/2004
Dec-02
Dec-01
Dec-00
Dec-99
Dec-98
Dec-97
Dec-96
Dec-95
Dec-94
Dec-93
Dec-92
Dec-91
Dec-90
Dec-89
Dec-88
Dec-87
Dec-86
-5.0%
A decent factor But the active risk is consistently above target 2004 Northfield Annual Conference
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Information Coefficient Raw IC and Risk-adjusted IC 0.30
IC.raw
0.25
IC.refine
0.20
GP2EV
0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15
Ø Ø 5/5/2004
Dec-02
Dec-01
Dec-00
Dec-99
Dec-98
Dec-97
Dec-96
Dec-95
Dec-94
Dec-93
Dec-92
Dec-91
Dec-90
Dec-89
Dec-88
Dec-87
Dec-86
-0.20
The risk-adjusted IC is much more stable, indicating a better information ratio The raw IC carries risk-factor bias 2004 Northfield Annual Conference
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Multi-factor Models
“Optimize” the IC Ratio
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The Portfolio Approach Ø
Multiple forecasts for each N stocks – Previous research is unclear – Many practitioners resort to ad hoc weighting schemes based
on intuition or judgment Ø
Our approach provides a quantitative framework for combing multiple factors IC IR =
– For a multi-factor model, it is still true that std ( IC ) – We maximize the IR to find the optimal linear combination – The problem is very similar to optimal allocation among
multiple active managers Ø
Use optimal weights with caution and common sense 5/5/2004
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Important Inputs Ø
Each factor (manager) – Average IC, standard deviation of IC, IR – Analogous to expected alpha, active risk, IR
Ø
Among factors (managers) – IC correlation: time series correlations between different IC’s – Analogous to correlations between excess returns of different
managers – The correlations between different factors are not of primary importance – Factor correlation is not the same as IC correlation 5/5/2004
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Two-factor Case Ø
Combined IC of a single period IC t =
Ø Ø
IC q ,t ⋅ ω q + IC f ,t ⋅ ω f 1 − 2ω qω f (1 − ρ t )
The factor correlation is only a scale parameter and when it is constant throughout time it has no effect on the IR Expected IR IR ≈ ICt = IC q ,t ⋅ ω q + IC f ,t ⋅ ω f std (ICt )
Ø Ø
σ q2ω q2 + 2σ f , qω f ω q +σ 2f ω 2f
The optimal weights can be found For multi-factor case, there is a matrix solution 5/5/2004
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Summary Ø Ø Ø
Ø Ø Ø
Portfolio approach: simplified backtest Strategy risk contributes additionally to active risk. It can be measured by standard deviation of IC Use the proper definition of IC - the correlation between risk-adjusted forecasts and risk-adjusted returns, not the raw IC IR is often lower than what FLOAM predicts, even if there is no portfolio constraint and no model errors Select alpha factors based on average IC as well as standard deviation of IC Combine alpha factors based on the above as well as IC correlations
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