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- 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!

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