forecasting. “forecasting is very difficult, especially if it’s about the future.” niels bohr
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Forecasting
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“Forecasting is very difficult, especially if it’s about the future.”
Niels Bohr
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Introduction
• How do we move from signals to forecast returns?
• How do we combine signals?
• What is our intuition about the magnitude of our forecasts?
• What “real world” issues must we take into account?
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From signals to forecasts
• Signals don’t even have the right units
• Basic forecasting formula
• Best linear unbiased estimator (BLUE)
1| ,E E Cov Var r g r r g g g E g
|
0T
E
E
Min
q r r g
q
q q
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Basic Forecasting of Active Returns
• Replace r with .
• Assume that unconditional (consensus) expected active returns are zero, and remember that alphas are conditional expected active returns:
1,Cov Var E α δ g g g g
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Forecasting Intuition
• One asset, one signal, linear estimate:
• Expectations: t a b g t t
|
E a b E g
a E b E g
E g a b g
E b g E g
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Intuition, cont.
• Use regression result for coefficient, b
• Assume that the unconditional expected active return is zero, and that alpha is the conditional expected active return:
1| ,E g E Cov g Var g g E g
1,Cov g Var g g E g
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Rule of Thumb
• Remember that:
• Hence:
, ,Cov g Corr g Std Std g
IC Std g
g E gIC IC z IC Volatility Score
Std g
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Forecast controls for:
• Skill– If IC=0, alpha=0
• Expectations– If g=E{g}, alpha=0
• Volatility– If two stocks have the same score, we expect
the more volatile stock to rise more.
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Portfolio Construction
*2
2
*
2
2 2
2
nn
n
n n n
n n
n n n
h
IC z zIC
ICh z
Assume uncorrelated active returns. (This isn’t required, it just facilitates intution.)
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IC Intuition
• Simple model: only care about being directionally correct on active return.– N Stocks– = ±1– g = ±1
, ,
2 1
correct incorrectN NCov g Corr g
NIC fr
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IC Intuition
• Roulette wheel– fr=52.6%, hence IC=0.052
• Active equity management– IC=0.05 is quite good– IC=0.10 is fantastic– IC=0.00 is average
• Many investors significantly over-estimate available levels of skill.
“A good forecaster is not smarter than everyone else, he merely has his ignorance better organized.” Anon
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IC Intuition, cont.
• What do these IC’s look like?
• An IC=0.1 implies a regression R2 of 0.01
• Let’s look at an example of a promising idea.
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Information Coefficient = 8.7%
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Modeling Alphas
• Binary Model
• Elements are ±1 variables
• Example:– Monthly active returns
– Monthly forecast alphas
64
1i
i
15
11
1 jj
g
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Building Alphas in the Binary Model
• We can see that monthly stock volatility is 8%, monthly signal volatility is 4%, and expected signal value is 1.
• What is the IC:
• What is the Alpha:
, 1
0.0332
Cov gIC
Std g
10.03 8 0.06 1
4
gIC Score g
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Rule of Thumb Examples
• Stock Tip
• Stock volatility is 20%
• Rule of thumb provides structure in classic unstructured situation
ScoreIC 1 20% 0% 0%5% 1% 2%10% 2% 4%
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Rule of Thumb Examples
• Broker BUY/SELL recommendations
• Assume overall IC=0.05
• Note the substantial scaling to account for skill.
Stock Recommendation Score AlphaExxon 21% BUY 1 1.05%
GM 28% BUY 1 1.40%GE 17% SELL -1 -0.85%
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How do we handle multiple signals?
• Go back to basic forecasting formula.
1 2 3, , , , , , ,T
Cov Cov g Cov g Cov g
g
IC S
1 1 2
1 2 2
,
,
Var g Cov g g
Var Cov g g Var g
g
S ρ S
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Multiple signals
• Substitute into basic forecasting formula
• We need to know the IC of each signal, as well as the correlation of the signals.
1
1
1
,
T
T
Cov Var E
E
g g g g
IC S S ρ S g g
IC ρ z
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Multiple Signals
• What happens with two signals?1
12 1212
12 1212
1 11
1 11
ρ
1
1 1 2 2
T
IC z IC z
IC ρ z
1 12 21 2
121
IC ICIC
2 12 12 2
121
IC ICIC
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Multiple Signals
• Check limiting cases.
• This becomes more interesting with 3 signals.
• What happens to the overall IC?
• Is this useful, or just lots of algebra?
1totalIC TIC ρ IC
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ExampleMMI Company Residual Risk Broker Rating Research Service AlphaAMERICAN EXPRESS CO 17.12% BUY 0.99%AT&T CORP 33.35% BUY 1.51%CHEVRONTEXACO CORP 19.27% BUY 0.48%COCA COLA CO 20.15% SELL -0.99%DISNEY WALT CO 25.52% BUY 0.33%DOW CHEM CO 19.61% SELL -0.30%DU PONT E I DE NEMOURS 17.90% BUY 1.12%EASTMAN KODAK CO 26.26% SELL -1.02%EXXON MOBIL CORP 21.03% BUY -2.05%GENERAL ELEC CO 16.63% BUY 0.07%GENERAL MTRS CORP 27.89% BUY -1.07%INTERNATIONAL BUSINESS 18.46% BUY -1.12%INTL PAPER CO 17.36% BUY -0.67%JOHNSON & JOHNSON 18.06% BUY 1.79%MCDONALDS CORP 19.09% SELL 0.47%MERCK & CO INC 21.90% BUY 1.47%3M CO 17.94% BUY 0.01%ALTRIA GROUP INC 24.07% BUY -0.82%PROCTER & GAMBLE CO 18.23% BUY -0.56%SEARS ROEBUCK & CO 31.14% SELL 0.11%
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Rule of Thumb confronts Real World: Cross-sectional Scores
• Our analysis so far implicitly focused on time-series analysis:– Active returns for 1 stock over time– Signal(s) for 1 stock over time– Score, z, is a time-series score; involving time-series
expectations and standard deviations
• But the typical active management problem involves multiple assets at one time, with managers looking to pick relative winners and losers.
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Cross-sectional Scores: Example
• Stocks in S&P 500
• Calculate FY1 E/P for each stock
• Calculate mean and standard deviation across stocks
• Cross-sectional score:
cs
cs
cs
E EMean
P Pz
EStd
P
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Dilemna
• Do we “volatility adjust” these scores?
• Is alpha proportional to the cross-sectional scores, or to the cross-sectional scores, times volatility?
• Or, to put it more generally:
• What is ?
~ csz
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Analysis
• The rule of thumb provides alpha as a function of time-series scores.
• We need to relate these two statistics.
• Ignore mean signal, either time-series or cross-sectional.
• Ignore signal correlations across stocks.
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Two Cases
• Compare StdTS{g} across stocks
• Case 1: Same for each stock
• Case 2: Proportional to stock volatility
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Case 1
TS n CS n
n n nTS
TS n CS n
TS CS
CS
Std g c Std g c
g g gz
Std g c Std g
z z
IC z
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Case 2
~
~
nTS n n CS
n
n n nTS
TS n n nCS n
n
CSTS
CS
gStd g c c Std
g g gz
Std g c gStd
zz
IC z
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Practical Experience
• Anecdotal evidence: Case 2 is more prevalent than Case 1.
• Case 1 mainly happens with 0/1 signals.
• Empirically we do see the connection between volatility scaling and modeling the time-series signal standard deviations.
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Summary
• Forecasting analysis controls raw signals for expectations, skill, and volatility.
• It also tells us how to combine signals.• Rule of thumb provides intuition, and
structure in unstructured situations.• Skill level is low in stock selection.• Don’t confuse cross-sectional with time-
series scores.