Forecasting

“Forecasting is very difficult, especially if it’s about the future.”

Niels Bohr

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?

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

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

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

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

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

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.

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.)

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

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

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.

Information Coefficient = 8.7%

Modeling Alphas

• Binary Model

• Elements are ±1 variables

• Example:– Monthly active returns

– Monthly forecast alphas

64

1i

i

15

11

1 jj

g

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

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%

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%

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

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

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

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

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%

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.

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

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

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.

Two Cases

• Compare StdTS{g} across stocks

• Case 1: Same for each stock

• Case 2: Proportional to stock volatility

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

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

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.

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.