1) The document discusses performance evaluation of actively managed funds and strategies to maximize alpha and the t-statistic of alpha. 2) It presents a model where a manager chooses payoffs from available assets to maximize alpha based on observed benchmark returns over time. 3) Maximizing alpha relies on exploiting differences between the discount factors of available and benchmark payoffs, with the optimal strategy being the zero-beta portfolio of the two discount factors. 4) Implied volatility in option prices can generate alpha even in a Black-Scholes model by creating a difference between historical and risk-neutral discount factors.

1. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Performance Maximization
of Actively Managed Funds
Paolo Guasoni1 Gur Huberman2 Zhenyu Wang3
1 Boston University
2 Columbia Business School
3 Federal Reserve Bank of New York
European Summer Symposium in Financial Markets
July 21, 2008

2. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Portfolio Manager vs. Evaluator
Evaluator observes excess returns.
Over a fixed-interval grid
For a long time
Evaluator does NOT know positions.
Evaluator compares returns against benchmarks.
Manager aware of evaluation process.
Tries to manipulate performance.

3. 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 error
Sharpe ratio of hedged portfolio.

4. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Alpha without Ability
Return on index
8%
Return on index calls
Return on the fund
Regression line
Excess Fund Return
0%
Nonzero alpha!
-8%
-8% 0% 8%
Excess Market Return

5. 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.

6. 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 derivatives
or delta trading deliver a significant t-stat?
If options are priced by Black-Scholes, it will take many years.
Why does BXM out-perform?

7. 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 + bR m ).
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.

8. 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.

9. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Maximization of Alpha
The alpha of a strategy x ∈ Xa converges to:
1
α(x) = R f E [x(mb − ma )] (1)
The maximal t-statistic of alpha satisfies:
2
max
tn
s max = lim √ =R f E [(mb − ma )2 ] (2)
n
n→∞
=R f Var(ma ) − Var(mb ) (3)
Achieved by the payoffs:
3
x = ξ + l(mb − ma ) (4)
for arbitrary ξ ∈ Xb and l > 0.

10. 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)
R 2 of any payoff maximizing the Sharpe-ratio:
Var(mb )
R2 = (6)
Var(ma )
To generate highly significant alpha, the manager trades the
zero-beta portfolio mb − ma .
t statistic of alpha grows with gap in discount factor variance.
Increase in Sharpe ratio grows with t statistic.

11. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Geometric Brownian Model
A risk-free rate r and several benchmarks Sti .
d
dSti
σij dWtj 1≤i ≤d
=µi dt + (7)
Sti j=1
(Wti )1≤i≤d is a d-dimensional Brownian Motion,
t
µ = (µi )1≤i≤d is the vector of expected returns, and the
volatility matrix σ = (σij )1≤i,j≤d is nonsingular.
Market is complete.

12. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Discount Factors
Returns joint lognormal:
R f =e rt
√
Σii
R i =e (µi − )t+ tψi
1≤i ≤d
2
where Σ = σ σ, and ψ ∼ N(0, Σ).
Stochastic discount factors:
√
(µ−r ¯ Σ−1 (µ−r ¯
1) 1)
t+ t(µ−r ¯ Σ−1 ψ
− r+ 1)
2
ma =e
1 1
− f (E [R] − R f ) S −1 (R − E [R])
mb = f
R R
where S is the covariance matrix of simple returns.

13. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
t statistic of Black Scholes alpha
For one benchmark, a Taylor expansion shows that:
2
max µ−r
tn t
s max = lim √ ≈ √ + O(t 2 )
(µ − r ) +
σ
n
n→∞ 2
Dominant term of order t.
Alpha arises from the mismatch between trading and
monitoring frequencies.
Disappears in the continuous-time limit.
How big in practice?
Optimal payoff?

14. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Optimal Alpha Payoff
B. The Hedged Strategy
15%
10%
5%
0%
-5%
-10%
-15%
-20% -15% -10% -5% 0% 5% 10% 15% 20%
Rate of Return on the Benchmark

15. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Years to Significance
Factors Benchmark Attainable t stat Years
Sharpe Sharpe
Monthly Observations
MKT 0.11 0.11 0.01 2084
MKT,SMB,HML 0.27 0.27 0.06 103
MKT,SMB,HML,MOM 0.37 0.38 0.10 30
Factors estimated from 1:1963-12:2006.

16. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Varying Observation Length
Factors Benchmark Attainable Years
Sharpe Sharpe
Monthly Observations
MKT 0.11 0.11 2084
MKT,SMB,HML,MOM 0.37 0.38 30
Quarterly Observations
MKT 0.19 0.2 694
MKT,SMB,HML,MOM 0.63 0.71 9
Semi Annual Observations
MKT 0.27 0.28 346
MKT,SMB,HML,MOM 0.88 1.12 4

17. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Liquid Index Options
Factors Benchmark Attainable Years
Sharpe Sharpe
SPX 0.12 0.12 1803
SPX,NDX 0.13 0.13 1148
SPX,NDX,RUT 0.13 0.13 1052

18. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
BXM Performance: a contradiction?
Period S&P 500 BXM Alpha t-stat
1990.01-2005.12 7.1% 6.8% 2.7% 2.2
1990.01-1994.12 4.5% 6.6% 4.1% 2.6
1995.01-1999.12 21.4% 14.3% 2.4% 0.9
2000.01-2005.12 -2.7% 0.8% 2.5% 1.2
Nonlinearity does not generate significant alpha in the
Black-Scholes model.
But call writing (BXM) or put writing (Lo, 2001) have
significant alpha and high Sharpe ratio.
These strategies use actual option prices.

19. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Implied Volatility?
Implied volatility is consistently higher than realized volatility.
Over the period 1990-2004, historical volatility of the S&P
500 averaged 16%, versus 20% of at-the-money volatility
measured by the VIX index.
Does this feature explain observed alpha?

20. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Alpha with Implied Volatility
Single benchmark:
√
σ2
St = S0 e (µ− )t+σ tψ
(8)
2
Options still priced by the Black-Scholes formula, but with
another value for volatility σ = λσ.
ˆ
Nonspecification of a continuous-time dynamics.
Setting consistent with discrete-time model.
Market not complete.
Option trading not equivalent to dynamic trading.

21. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Alpha with Implied Volatility
Period Historical Implied Ratio Max
Vol Vol Appraisal
1990.01-2005.12 16% 19% 1.21 5.77
1990.01-1994.12 12% 17% 1.39 14.01
1995.01-1999.12 16% 20% 1.27 7.96
2000.01-2005.12 19% 21% 1.11 1.48

22. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
The Discount Factor
Black-Scholes formula holds with implied volatility σ = λσ, so
ˆ
ψ is normal also under the risk-neutral measure Q.
The conditions:
EQ [St ] =e rt (9)
22
VarQ (log St ) =λ σ t (10)
√ σ2
imply that ψ ∼ N(δ t, λ2 ), where δ = − µ−r + − λ2 ).
2 (1
σ
The discount factor ma is:
√
(ψ−δ t)2
ψ2
e −rt+ 2 −
dQ 2λ2
ma = e −rt = (11)
dP λ
mb is the same as before, since it ignores option prices.

23. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
The t statistic
The variance of ma is:
δ2 t
 
e 2−λ2
Var(ma ) = e −2rt  √ − 1 (12)
λ 2 − λ2
√
provided that λ ≤ 2, otherwise it is infinite.
A Taylor expansion shows that:
max
tn 1
lim √ = √
Var(ma ) − Var(mb ) ≈ − 1+O(t)
n λ 2 − λ2
n→∞
(13)
Dominant term now of order zero.
Alpha does not disappear for small t.

24. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Path-dependent Strategies
Two restrictive assumptions.
Large Samples.
Sample moments replaced by population values.
Constant strategies.
Manager chooses same payoff at each period.
Can a path-dependent strategy do better in the large sample?
And in a small sample?

25. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
The Limits of Path-dependent Alpha
Path-dependent strategies...
...are useless in large samples;
...have small alphas in small samples.

26. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Model Setting
One benchmark.
IID Returns (xi )i≥1 with mean µ and variance σ 2 .
One uncorrelated payoff.
IID Returns (zi )i≥1 IID with mean a and variance s 2 .
Managed portfolio holds a fixed unit of the payoff z, but a
time-varying benchmark exposure.
Portfolio return is yi = βi xi + zi .
βi arbitrary, but only depends on the past
β1 , x1 , z1 , . . . , βi−1 , xi−1 , zi−1 .

27. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Sample Quantities
After n periods, the evaluator estimates alpha and its
significance with the usual estimators:
n n n
1 1 1
i=1 xi yi − ( n i=1 xi )( n i=1 yi )
ˆ
βn = n n n
1 2 − (1 2
i=1 xi i=1 xi )
n n
n n
1 ˆ1
yi − βn
αn =
ˆ xi
n n
i=1 i=1
ˆ
Make βn negatively correlated with benchmark return.
This makes αn positively biased.
ˆ

28. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Path Dependence Useless in Large Sample
Theorem
If E [xi4 ] < ∞, and the portfolio (βi )i≥1 satisfies:
n n
1 1
βi2 = b 2 + v
lim βi = b lim
n→∞ n n→∞ n
i=1 i=1
then the following hold:
ˆn
t a
ˆ lim √ =
lim αn = a
ˆ lim βn = b
n
n→∞ n→∞ n→∞ 2 +σ 2 )2
s 2 + v (µ σ2
Alpha only comes from the uncorrelated payoff z.
Fluctuations in beta only add tracking error, as captured by v .
Better use βi = b, a constant strategy with v = 0.

29. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Bounding Small Sample Alpha
Take a continuous time approximation.
The benchmark return dXt = dSt /St follows the diffusion:
dXt = µdt + σdBt
where Bt is a Brownian Motion.
The portfolio return dYt is:
dYt = βt dXt
Set leverage bounds: βt ∈ [β min , β max ].
Maximize expected alpha.

30. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Theorem
Maximum alpha is:
σ 1 2
E [ˆ T ] ≤ √ (β max − β min )
α
3 π
T
Optimal bang-bang strategy:
β min if Bt ≥ 0
opt
βt =
β max if Bt < 0
Keep low beta when return to date positive, and high beta
when negative.
σ = 15%, β min = 0.5 and β max = 1.5 deliver maximum
expected alphas of 1.78% for T = 5 years and 1.26% for
T = 10.

31. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Conclusion
Alpha as the gap between evaluator and market pricing.
A zero-beta portfolio maximizes significance of alpha.
Nonlinearity alone does not explain observed alpha.
Nor do small sample effects.
Misspecifications are central.

32. The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Thank You!