Performance Maximization of Managed Funds

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


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.


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.


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!


Superior Performance

Private information which predicts benchmarks payoffs.

Access to additional assets.

Access to derivatives on benchmarks.

Trades more frequent than observations.


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?


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.


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 .


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.


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.


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.


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.


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?


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.


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.


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


Semi Annual Observations



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


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.


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?


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.


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 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 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.


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 Limits of Path-dependent Alpha

Path-dependent strategies...

...are useless in large samples;

...have small alphas in small samples.


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.


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.


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.


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.


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.


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.


Thank You!

Economy & Finance

Performance Maximization of Managed Funds