performance maximization of managed funds
TRANSCRIPT
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
Monthly Observations
MKT 0.11 0.11 0.01 2084MKT,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.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Varying Observation Length
Factors Benchmark Attainable YearsSharpe Sharpe
Monthly Observations
MKT 0.11 0.11 2084MKT,SMB,HML,MOM 0.37 0.38 30
Quarterly Observations
MKT 0.19 0.2 694MKT,SMB,HML,MOM 0.63 0.71 9
Semi Annual Observations
MKT 0.27 0.28 346MKT,SMB,HML,MOM 0.88 1.12 4
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Liquid Index Options
Factors Benchmark Attainable YearsSharpe Sharpe
SPX 0.12 0.12 1803SPX,NDX 0.13 0.13 1148
SPX,NDX,RUT 0.13 0.13 1052
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% 2.21990.01-1994.12 4.5% 6.6% 4.1% 2.61995.01-1999.12 21.4% 14.3% 2.4% 0.92000.01-2005.12 -2.7% 0.8% 2.5% 1.2
Nonlinearity does not generate significant alpha in theBlack-Scholes model.
But call writing (BXM) or put writing (Lo, 2001) havesignificant alpha and high Sharpe ratio.
These strategies use actual option prices.
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&P500 averaged 16%, versus 20% of at-the-money volatilitymeasured by the VIX index.
Does this feature explain observed alpha?
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Alpha with Implied Volatility
Single benchmark:
St = S0e(µ−σ2
2)t+σ
√tψ (8)
Options still priced by the Black-Scholes formula, but withanother value for volatility σ̂ = λσ.
Nonspecification of a continuous-time dynamics.
Setting consistent with discrete-time model.
Market not complete.Option trading not equivalent to dynamic trading.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Alpha with Implied Volatility
Period Historical Implied Ratio MaxVol Vol Appraisal
1990.01-2005.12 16% 19% 1.21 5.77
1990.01-1994.12 12% 17% 1.39 14.011995.01-1999.12 16% 20% 1.27 7.962000.01-2005.12 19% 21% 1.11 1.48
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
The Discount Factor
Black-Scholes formula holds with implied volatility σ̂ = λσ, soψ is normal also under the risk-neutral measure Q.
The conditions:
EQ [St ] =ert (9)
VarQ(log St) =λ2σ2t (10)
imply that ψ ∼ N(δ√
t, λ2), where δ = −µ−rσ + σ2
2 (1− λ2).
The discount factor ma is:
ma = e−rt dQ
dP=
e−rt+ψ2
2− (ψ−δ
√t)2
2λ2
λ(11)
mb is the same as before, since it ignores option prices.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
The t statistic
The variance of ma is:
Var(ma) = e−2rt
eδ2t
2−λ2
λ√
2− λ2− 1
(12)
provided that λ ≤√
2, otherwise it is infinite.
A Taylor expansion shows that:
limn→∞
tmaxn√
n=√
Var(ma)− Var(mb) ≈
√1
λ√
2− λ2− 1+O(t)
(13)
Dominant term now of order zero.
Alpha does not disappear for small t.
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?
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.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Model Setting
One benchmark.IID Returns (xi )i≥1 with mean µ and variance σ2.
One uncorrelated payoff.IID Returns (zi )i≥1 IID with mean a and variance s2.
Managed portfolio holds a fixed unit of the payoff z , but atime-varying benchmark exposure.
Portfolio return is yi = βixi + zi .
βi arbitrary, but only depends on the pastβ1, x1, z1, . . . , βi−1, xi−1, zi−1.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Sample Quantities
After n periods, the evaluator estimates alpha and itssignificance with the usual estimators:
β̂n =1n
∑ni=1 xiyi − ( 1
n
∑ni=1 xi )( 1
n
∑ni=1 yi )
1n
∑ni=1 x2
i − ( 1n
∑ni=1 xi )2
α̂n =1
n
n∑i=1
yi − β̂n1
n
n∑i=1
xi
Make β̂n negatively correlated with benchmark return.
This makes α̂n positively biased.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Path Dependence Useless in Large Sample
Theorem
If E [x4i ] <∞, and the portfolio (βi )i≥1 satisfies:
limn→∞
1
n
n∑i=1
βi = b limn→∞
1
n
n∑i=1
β2i = b2 + v
then the following hold:
limn→∞
α̂n = a limn→∞
β̂n = b limn→∞
t̂n√n
=a√
s2 + v (µ2+σ2)2
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.
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 = βtdXt
Set leverage bounds: βt ∈ [βmin, βmax ].
Maximize expected alpha.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Theorem
Maximum alpha is:
E [α̂T ] ≤ σ√T
(βmax − βmin)1
3
√2
π
Optimal bang-bang strategy:
βoptt =
{βmin if Bt ≥ 0
βmax if Bt < 0
Keep low beta when return to date positive, and high betawhen negative.
σ = 15%, βmin = 0.5 and βmax = 1.5 deliver maximumexpected alphas of 1.78% for T = 5 years and 1.26% forT = 10.
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.
The Model Nonlinear Alpha Alpha and Volatility Small Sample Alpha
Thank You!