Ruminations on Investment Performance Measurement

A Keynote Address to the 20th Annual Pacific Basin Finance,

Economics, Accounting and Management Meeting

By:Wayne Ferson,

September, 2012

Forthcoming in: European Financial Management, 2012

And Compiled From Various Sources:

1. Ferson, 2012, Investment Performance: A Review and Synthesis, The Handbook of the Economics of Finance (forthcoming).

2. Ferson and Lin, 2012, Alpha and Performannce Measurement: The Effects of Investor Heterogeneity (in review).

3. Ferson and Mo, 2012, Performance Measurement with Market and Volatility Timing and Selectivity, working paper, USC.

4. Ferson, 2010, Investment Performance Evaluation, Annual Reviews of Financial Economics 2, 207-234

5. Aragon and Ferson, 2008, Portfolio Performance Evaluation, Foundations and Trends in Finance 2, 1-111.

Main Observations / Claims:

1. "Traditional" Alphas are not to be trusted, but Stochastic Discount Factor (SDF) Alphas are Better.

2. Traditional = SDF alphas ONLY IF you use an "Appropriate Benchmark."

3. Mean Variance Efficient Portfolios are (almost never) Appropriate Benchmarks!

4. Sharpe Ratios Can be Justified!

5. Current Holdings-Based Approaches are Flawed, but I have some suggestions.

1. Traditional Alphas are NOT to be Trusted:

• We have easy familiarity with “Alpha:”

- CAPM Alpha- Three-factor Alpha- Four-factor Alpha- DGTW Alpha

Basic Question: Does a positive Traditional Alpha => Buy?

MEAN VARIANCE CASE: Sometimes, at the Margin (Dybvig and Ross 1985), but not necessarily (e.g. Gibbons et al. 1989).

DIFFERENTIAL INFORMATION: You would use the fund (But, maybe short a positive alpha fund!) (Chen and Knez, 1996)

And, you can have (Positive or Negative) alpha with Neutral Performance! egs: Jagannathan and Korajczyk (86), Roll (78), Green (86), Leland (99), Dybvig and Ross (85), Hansen and Richard (87), Ferson and Schadt (96), Brown, Goetzmann et al (2007), ….

And, you might buy even with negative alpha (Glode, 2011).

Literature Says Traditional Alphas are NOT to be Trusted!

The "Stochastic Discount Factor Approach" Produces Trustworthy Alphas:

Price = E m * Payoff |Z,

m = Stochastic Discount Factor,[e.g. βu'(c)/u'(c0)],

Z = Client's Information.

The "Right" Alpha: αp = E(mRp|Z) – 1

Ferson and Lin (2012) Show:

With the “Right” (i.e. client-specific SDF) Alpha, Positive Alpha Does Mean Buy!.

- Pretty general: [discrete response not just marginal, multiperiod not static, does not restrict timing information, consumption, constant risk aversion or require normality.]

- Special Case: Manipulation-proof (Brown, Goetzmann et al., 2007)

So, When Can "Traditional Alphas" be Trusted?

9

2. Alpha is Valid If you use an "Appropriate Benchmark" (Aragon and Ferson, 2008):

Research and real world practice, a benchmark return, RB, is used: Traditional returns-based Alpha:

αp = E(Rp – RB|Z) for some RB benchmark and Public Information, Z.

When is this equivalent to the Right, SDF alpha?

αp = E(mRp|Z) – 1, if αB=0, =>

= E(m[Rp – RB]|Z)

= E(m|Z) E(Rp – RB|Z) + Cov(m, Rp – RB |Z)

=> When Cov(m, Rp |Z) = Cov(m, RB |Z): You have an “Appropriate" bechmark, RB.

3. Mean variance benchmarks: (Almost) Never Appropriate

RB minimum variance efficient in R <=> RB has maximum (squared) correlation with m (Hansen Richard, EM 87)

<=> m = a + b RB + u, with E(uR|Z)=0.

=> Fund's Rp has Cov(Rp,m|Z) = bCov(Rp,RB|Z) + Cov(Rp,u|Z).

Appropriate Benchmark and MV Efficient => Cov(Rp,u|Z)=0.

Either: (i) RB minimum variance efficient in R, Rp => αp = 0. (Not Useful; e.g. Roll, 78) or

(ii) m exactly linear in RB (u=0) => Quadratic utility in RB:

• Without Quadratic Utility in RB, a mean variance efficient portfolio is NOT an Appropriate Benchmark!

4. Sharpe Ratios Can be Justified! The Sharpe ratio (SR):

SRp = E(rp)/σ(Rp), rp Rp - Rf

Traditionally: • Only for investor's total portfolio?• Inappropriate when returns are nonnormal? (Leland (1999), Brown, Goetzmann et al (2007), Lo (2008)).• Still, widely used in practice!

Claim: Can Justify the Sharpe ratio compared to an Appropriate benchmark: RB !

E(Rp - Rf)/σ(Rp) > E(RB - Rf) /σ(RB). If ρ>0,

=> E(Rp - Rf) > [ρσ(Rp)/σ(RB)] E(RB - Rf).

=> Rpt - Rft = αp + βp (RBt - Rft) + upt has αp > 0.

(Recall Point 2: Traditional Alphas on an Appropriate Benchmark are Justified.)Thus, SR fund > SR Appropriate benchmark <=> αp > 0 <=> SDF alpha >0

Holdings-Based Measures

Grinblatt and Titman (89,93):Returns conditionally joint normal with manager information,

Ω, nonincreasing absolute risk aversion =>

GT = Cov x(Ω)'r > 0,

(where cov(x'y) = Σi cov(xi,yi))

Implement as Covx(Ω)’r = E[x(Ω)-xB]'r

Ferson and Khang (2002): Explicit Public Information, Z:

CWM = E x(Z,Ω)’[r - E(r|Z)]

A (few) Holdings-based Performance Studies :)

Cornell (1979), Copeland and Mayers (1982), Brinson, Hood and Bebower (1986), Grinblatt and Titman (1989), Grinblatt and Titman (1993), Grinblatt, Titman, and Wermers (1995), Zheng (1999), Daniel, Grinblatt, Titman, and Wermers (1997), Wermers (2000), Ferson and Khang (2002), Kacperczyk, Sialm and Zheng (2005), Kacperczyk, Sialm and Zheng (2008), Kacperczyk, Veldkamp and Van Nieuwerburgh (2012), Kosowski, Timmerman, Wermers and White (2006), Kacperczyk and Seru (2007), Jiang, Yao and Yu (2007), Taliaferro (2009), Shumway, Szefler and Yuan (2009), Moneta (2009), Ferson and Mo (2012), Griffin and Xu (2009), Aragon and Martin (2009), Cohen, Coval and Pastor (2005), Blocher (2011), Wermers (2001), Chen, Hansen, Hong and Stein (2008), Gaspare, Massa and Matos (2006), Reuter (2006), Christophersen, Keim and Musto (2007), Qian (2009), Gaspare, Massa and Matos (2005), Ferreira and Matos (2008), Matos Starks et al (2011), Cici and Gibson (2012), Cohen, Frazzini and Malloy (2008), Huang and Kale (2009), Busse and Tong (2012).

Other Holdings-based Measures:

“Return gap” Kacperczyk, Sialm and Zheng (2008): Rp - x(Ω)'R

"Active Share," Cremers and Petajisto (2009):

|| x(Ω) - xB ||

Daniel, Grinblatt, Titman and Wermers (1997):

DGTW = Σi xit (Ri,t+1 - Rt+1bi) +

Σi (xit Rt+1bi - xi,t-kRt+1

bi(t-k)) + Σi xi,t-k Rt+1bi(t-k)

= CS + CT + AS

Each stock gets its own benchmark, Rt+1bi

Other Holdings-based Measures:

“Return gap” Kacperczyk, Sialm and Zheng (2008): Rp - x(Ω)'R

"Active Share," Cremers and Petajisto (2009):

|| x(Ω) - xB ||

Daniel, Grinblatt, Titman and Wermers (1997):

DGTW = Σi xit (Ri,t+1 - Rt+1bi) +

Σi (xit Rt+1bi - xi,t-kRt+1

bi(t-k)) + Σi xi,t-k Rt+1bi(t-k)

= CS + CT + AS

= Grinblatt Titman (89, 93)

Understanding Holdings-based Measures of Performance

How Does a Portfolio Manager Generate Alpha?

Managed Fund Gross Return: Rp = x(Ω)’R

Alphap = Em R’x(Ω)|Z – 1

= E(m R)’E(x(Ω)) + CovmR’ x(Ω)|Z)-1

= 1’E(x(Ω)) + CovmR’ x(Ω)|Z)-1

= CovmR’ x(Ω)|Z)

5. Why current Holdings-based Measures are Flawed:

What Should be Done:

αp = E(mRp|Z) – 1,

= Cov(mR' x(Ω)|Z)

What has been Done: Cov(R' x(Ω)), Estimated via:

E((R – RB)'x(Ω)) or E(R' (x(Ω)-xB)),

or "Conditional" versions of these

When are Current Holdings-based Measures Justified?

If αB=0, αp = E(m [Rp - RB]|Z)

αp = E(m|Z) E(Rp - RB|Z) + Cov(m; [Rp - RB]|Z)

(Appropriate Benchmark, => Cov(m; [Rp -RB]|Z)=0).

αp = E(m|Z) E(x(Ω)-xB|Z)’ E(R|Z) + Cov([x(Ω)-xB]’R|Z), | ---- current versions --|

=> E(x(Ω)|Z) = xB

=> "Appropriate benchmark weights xB:" 3 conditions:

(i) E(x(Ω)|Z) = xB

(ii) Cov(m,Rp)=Cov(m,RB)

(iii) αB=0.

Doing Holdings-based Performance

Measures Right: (Ferson and Mo (2012) SDF: m = a - b’rB, rB a vector of K portfolio excess returns.

A factor model regression: r= a + β rB + u, β = N x K matrix of "bottom up" betas. “Abnormal,” or idiosyncratic returns: v = a + u.

A fund’s portfolio weights, x, and return: rp = x’r = (x’β)rB + x’v. w’=x’β = asset allocation weights.

Definition of SDF alpha = Cov(mR’x) =>

αp = a Cov(w’rB) – b’ E [rBrB’ – E(rBrB’)] w+ E(a-b’rB) x’v GT measure Volatility Timing Selectivity (asset allocation level)

NOTE: If all three are going on, measures that leave one out are misspecified!

Summary:

1. The “Right” Alpha = SDF Alpha, in general investor specific. => Clientele-specific performance measures?

2. Traditional = SDF alphas ONLY IF you use an "Appropriate Benchmark."

3. Mean Variance Efficient Portfolios are (almost never) Appropriate

4. Sharpe Ratios Can be Justified! (Compare to an Appropriate Benchmark).

5. Current Holdings-Based Approaches are Flawed as they have been implemented, but they can be fixed!