The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha Performance Maximization of Actively Managed Funds Paolo Guasoni 1 Gur Huberman 2 Zhenyu Wang 3 1 Boston University 2 Columbia Business School 3 Federal Reserve Bank of New York European Summer Symposium in Financial Markets July 21, 2008
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The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Performance Maximizationof Actively Managed Funds
Paolo Guasoni1 Gur Huberman2 Zhenyu Wang3
1Boston University
2Columbia Business School
3Federal Reserve Bank of New York
European Summer Symposium in Financial MarketsJuly 21, 2008
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Portfolio Manager vs. Evaluator
Evaluator observes excess returns.
Over a fixed-interval gridFor a long time
Evaluator does NOT know positions.
Evaluator compares returns against benchmarks.
Manager aware of evaluation process.Tries to manipulate performance.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Performance Evaluation
Evaluator observes the fund and benchmarks’ returns.Performs a linear regression.
Intercept alpha: excess preformance.
Sharpe ratio: average excess return / standard deviation
Appraisal ratio: alpha / tracking errorSharpe ratio of hedged portfolio.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Alpha without Ability
-8%
0%
8%
-8% 0% 8%
Excess Market Return
Ex
cess
Fu
nd
Ret
urn
Return on index
Return on index calls
Return on the fund
Regression line
Nonzero alpha!
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Superior Performance
Private information which predicts benchmarks payoffs.
Access to additional assets.
Access to derivatives on benchmarks.
Trades more frequent than observations.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
This Paper
An explicit strategy which maximizes the Sharpe ratio,delivers the highest asymptotic t-stat of alpha.
If benchmark prices follow Brownian motion, can derivativesor delta trading deliver a significant t-stat?
If options are priced by Black-Scholes, it will take many years.
Why does BXM out-perform?
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Model
Xb: payoffs spanned by benchmarks.(under CAPM, payoff of the form x = aR f + bRm).
Risk-free rate exists. 1 ∈ Xb.
Xa: payoffs available to the manager.
Xb ⊂ Xa.
mb ∈ Xb and ma ∈ Xa minimum norm SDFs.Attain Hansen-Jagannathan bounds.
No borrowing/short-selling constraints.Xb and Xa closed linear spaces.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Large Sample Alpha
Manager chooses the same payoff x from Xa at all periods.
Per-period returns are IID. Within period, not necessarily.
Evaluator observes IID realizations x1, . . . xn of x .
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Maximization of Alpha
1 The alpha of a strategy x ∈ Xa converges to:
α(x) = R f E [x(mb −ma)] (1)
2 The maximal t-statistic of alpha satisfies:
smax = limn→∞
tmaxn√
n=R f
√E [(mb −ma)2] (2)
=R f√
Var(ma)− Var(mb) (3)
3 Achieved by the payoffs:
x = ξ + l(mb −ma) (4)
for arbitrary ξ ∈ Xb and l > 0.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Sharpe Ratios and t statistic
The increase in squared Sharpe ratios is:
(R f )2(Var(ma)− Var(mb)) (5)
R2 of any payoff maximizing the Sharpe-ratio:
R2 =Var(mb)
Var(ma)(6)
To generate highly significant alpha, the manager trades thezero-beta portfolio mb −ma.
t statistic of alpha grows with gap in discount factor variance.
Increase in Sharpe ratio grows with t statistic.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Geometric Brownian Model
A risk-free rate r and several benchmarks S it .
dS it
S it
=µidt +d∑
j=1
σijdW jt 1 ≤ i ≤ d (7)
(W it )1≤i≤d
t is a d-dimensional Brownian Motion,µ = (µi )1≤i≤d is the vector of expected returns, and thevolatility matrix σ = (σij)1≤i ,j≤d is nonsingular.
Market is complete.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Discount Factors
Returns joint lognormal:
R f =ert
R i =e(µi−Σii2
)t+√
tψi 1 ≤ i ≤ d
where Σ = σ′σ, and ψ ∼ N(0,Σ).
Stochastic discount factors:
ma =e−(
r+ (µ−r 1̄)′Σ−1(µ−r 1̄)2
)t+√
t(µ−r 1̄)′Σ−1ψ
mb =1
R f− 1
R f(E [R]− R f )′S−1(R − E [R])
where S is the covariance matrix of simple returns.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
t statistic of Black Scholes alpha
For one benchmark, a Taylor expansion shows that:
smax = limn→∞
tmaxn√
n≈
((µ− r) +
(µ− r
σ
)2)
t√2
+ O(t2)
Dominant term of order t.Alpha arises from the mismatch between trading andmonitoring frequencies.Disappears in the continuous-time limit.
How big in practice?
Optimal payoff?
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Optimal Alpha Payoff
-15%
-10%
-5%
0%
5%
10%
15%
-20% -15% -10% -5% 0% 5% 10% 15% 20%
Exce
ss R
eturn
on t
he
Str
ateg
y
Rate of Return on the Benchmark
B. The Hedged Strategy
Figure: The payoff has zero-price and zero-beta, for µ = 11%, r = 5%,σ = 15%. The observation period is monthly, and the benchmark price atthe beginning of period is 100.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Years to Significance
Factors Benchmark Attainable t stat YearsSharpe Sharpe