Quantitative Analysis
Feb 11, 2016
Quantitative Analysis
Finding Alpha Fundamental Analysis
Forecast Dividends and find PV Look for stocks for which price PV
Ratio Analysis Compare accounting ratios Buy neglected stocks, short glamorous stocks
Quantitative Analysis Estimate regressions to identify equity styles that earn alpha
Quantitative Analysis Find anomalies to the CAPM Use regressions to forecast when the
anomalies will be particularly strong.
Tilt towards styles with positive alpha Tilt away from styles with negative alpha
Misvaluation View Misvaluation View:
Some stocks are just neglected by analysts Over-reaction by investors
Foundation of most hedge fund equity strategies
“Once an anomaly is discovered in academics, it takes about 5 to 10 years for traders to actually make it go away” (Bob Turley, Global Alpha Fund)
Value StrategiesAverage returns of Book-to-Market Portfolios
1927-2009
Growth Value
Value Strategies
Alphas of Book-to-Market Portfolios1927-2009
Growth Value
Momentum Portfolios Portfolio Formation at the end of month t:
Calculate the return for every stock over the period from month t-12 to month t-2. Rank all stocks by their past return Divide stocks into 10 value-weighted portfolios
Worst 10% in one portfolio Next 10% in next portfolio (etc.)
Momentum PortfoliosAverage Momentum Portfolio Returns
1927-2009
Momentum PortfoliosMomentum Portfolio Alphas
1927-2009
Quantitative Analysis Pros:
Straight forward to implement We know how to maximize Sharpe Ratios We don’t have to forecast dividends
Cons What has happened historically may not continue
in the future
Value and Momentum: Why do value and momentum strategies earn
positive alpha to begin with?
Misvaluation view Neglect and Over-reaction
Broken CAPM View: The CAPM doesn’t correctly model investor behavior. The CAPM was not built to tell us how people should
behave, but to model how they actually behave. Perhaps the model isn’t a realistic reflection of reality.
CAPM Assumptions1) Everyone has same forecast of expected
returns, standard deviations, and correlations. Probably not true When this assumption is relaxed, we still get the a
CAPM-like model, but in equilibrium, the VWP is no longer MSRP. Further, it is not obvious what portfolio is MSRP.
Roll’s Critique: Non-zero alpha can be the result of1) Prices out of equilibrium (markets inefficient)2) You are using the wrong portfolio in your test (Value-weighted
portfolio is not MSRP in equlibrium). It’s impossible to tell which is the true cause
CAPM Assumptions2) The R-Investor only cares about maximizing Sharpe
ratio Perhaps the R-investor has other objectives.
Arbitrage Pricing Theory The CAPM should hold
APT: Ross (1976)
Makes minimal assumptions: In equilibrium, no arbitrage opportunities Conclusion: alphas should be zero
Arbitrage Pricing Theory Notation:
Alpha is measured as the intercept in the following regression:
fme
m
fie
i
rrr
rrr
ie
me
i ebrr
Arbitrage Pricing Theory Suppose we measure the alpha of a “well
diversified” portfolio. Most of the variation is systematic Variance of ei is small Examples:
value portfolio top 30% (HW #17): 82% of the total variance is systematic
For Ford Stock (HW #16) 28% of the total variance is systematic
To an approximation, we can ignore ei in our regression for well diversified portfolios.
ExamplesY variables: returns on different stocksX variable: return on market
Asset 1:Lot’s of variation in e11% of variation is systematicR-squared=0.1189% of variation is unsystematic
Asset 2:Not a lot of variation in e82% of variation is systematicR-squared=0.8218% of variation is unsystematic
Extreme Example
Asset 3:All error terms are 0100% of variation is systematicR-squared=10 of variation is unsystematic
Examples For the “extreme example” there is no error
term:
For example 2 (well diversified portfolio): Error terms are small Most variation is systematic
em
e rar 33
emi
ei rr
Arbitrage APT:
Assume: In equilibrium there are no arbitrage opportunities (chance to make free money)
Claim: for well diversified portfolios, alpha must be approximately zero, or in other words, CAPM view of equilibrium is approximately correct.
Arbitrage Assume there is a well diversified portfolio
(A) whose alpha is 1% Assume beta=1.5 (arbitrary)
Assume there is some other well diversified portfolio (B) with alpha of -0.5% Assume beta=3 (arbitrary)
eM
eB
eM
eA
rr
rr
3005.
5.101.
Arbitrage The beta of a portfolio with weight w in A and
(1-w) in B has a portfolio beta of:
and a portfolio alpha of
BAp ww )1(
BAp ww )1(
Arbitrage Consider a portfolio (z) with a weight of 2 in
asset A and a weight of -1 in asset B
But since z is well diversified:
%5.2)005.()1(01.203)1(5.12
z
z
%5.2 zmzzez rr
Arbitrage But z by definition, is E[rz]-rf But portfolio z is risk-less (approximately)
Almost all systematic risk is diversified away, because it is composed of two well-diversified portfolios.
Systematic risk has been perfectly hedged Hence, E[rz]=rz
Because portfolio is approximately riskless, what you expect is what you get.
But since alpha=0.01 rz =rf +0.01 Borrow at risk-free rate, invest in this portfolio. Free money.
Arbitrage As many investors do this, what happens to
the cost of portfolio z? Goes up.
What happens to the realized return? Goes down.
In equilibrium rz=rf
Example 2 Suppose there are two well diversified portfolios:
How do you attack the arbitrage opportunity?
alpha betaA -0.06% 1.5B 0.50% 0.75
B)in (weight 2)1(A)in (weight 1
75.75.075).1(5.1
ww
www
Example 2 Portfolio z:
Weight in A = -1 Weight in B = 2 Approximately riskless Excess Return=2*.50+(-1*-.06)=1.06(%) Borrow all you can at risk-free rate Invest in portfolio z
Long-Short Hedge Funds Common hedge fund strategies:
Long value, short growth Long long/short momentum strategies
AQR Capital Management Barclay’s Global Investors Global Ascent Goldman Sachs Global Alpha Fund
Anomaly Even though many funds are attacking these
apparent arbitrage opportunities, opportunities apparently have not gone away.
Why? Industry view: pervasive mispricing
Anomaly Academic view:
These portfolios are not really close to risk-free
Why doesn’t the risk go away?
If the “e” in portfolio A is not independent to the “e” in portfolio B, then you can actually magnify the risk of the “arbitrage” portfolio by combining A and B. Related to stat rule number 5 - the variance of a sum of
random variables depends on the covariance.
the value-weighted portfolio is not the right portfolio to measure alpha against.
Arbitrage Pricing Theory Pros of APT:
Only relies on assumption of no-arbitrage Tells us the general form of any model of expected returns.
Expected returns must be a linear function of slope coefficients (betas).
Cons of APT What is the right portfolio or portfolios to use in
measuring alpha? APT tells us nothing about this.