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Active Portfolio Management: The Power of the Treynor-Black Model∗
Alex Kane Graduate School of International Relations and Pacific Studies (IR/PS)
University of California, San Diego 9500 Gilman Drive, La Jolla, CA 92093-0519 (Phone) 858-534-5969 (Fax) 858-534-3939
Tae-Hwan Kim School of Economics, University of Nottingham
University Park, Nottingham NG7 2RD, UK (Phone) 44-115-951-5466 (Fax) 44-115-951-4159
Halbert White Department of Economics, University of California, San Diego
9500 Gilman Drive, La Jolla, CA 92093-0508 (Phone) 858-534-3502 (Fax) 858-534-7040
December, 2003
Abstract: The performance of active portfolio methods critically depends on the forecasting ability of the security analyst. The Treynor-Black model provides an efficient way of implementing active investment strategy. Despite its potential benefits, the Treynor-Black model appears to have had little impact on the financial community, mainly because it has been believed that the precision threshold of alpha forecasts used as inputs to the model is too high. We seek to lower the threshold of forecast precision needed to beat a passive-portfolio strategy by improving the econometric methods used in constructing portfolios from security analyst forecasts. We apply shrinkage estimation to beta coefficients and to discount functions for forecasts of stock abnormal returns. OLS estimates, Least Absolute Deviations (LAD) estimates and shrinkage LAD estimates are compared by contribution to portfolio performance. Despite correlations between forecasts and realizations of abnormal returns as low as 0.04, the shrinkage LAD methodology yields superior performance in out-of-sample experiments. Key Words: Treynor-Black Model, Abnormal Returns, Sharpe Ratio, M2-measure, Least Absolute Deviations Estimator, Shrinkage LAD estimator.
∗ We would like to thank Clive Granger, Patrick Fitzsimmons, James Hamilton, Bruce Lehmann, Robert Trippi and Allan Timmermann for their helpful comments.
1
0. Introduction
The presumption of market efficiency is inconsistent with the existence of a vast industry engaged in
active portfolio management. Grossman and Stiglitz (1980) derive an information-inefficient capital
market equilibrium based on the cost of information and the fact that portfolio managers cannot
observe the asset allocations of competitors. Treynor and Black (1973) propose a model to construct an
optimal portfolio under such conditions, when security analysts forecast abnormal returns on a limited
number of securities. The optimal portfolio is achieved by mixing a benchmark portfolio with an active
portfolio constructed from the securities covered by the analysts. The original model assumes that
residuals from the market model are uncorrelated across stocks (the diagonal version), but it can easily
be extended to account for non-zero covariance across residuals (the covariance version).
The efficiency of the Treynor Black (TB) model depends critically on the ability to predict
abnormal returns. Its implementation requires that security analyst forecasts be subjected to statistical
analysis and that the properties of the forecasts be explicitly used when new forecasts are input to the
optimization process. It follows that security analysts must submit quantifiable forecasts and that they
will be exposed to continuous, rigorous tests of their individual performance. The entire portfolio is
also continuously subjected to performance evaluation that may engender greater exposure of
managers to outside pressures.
The TB model appears to have had little impact despite encouraging reports; e.g., Hodges and
Brealey (1973), Ambachtsheer (1974, 1977), Ferguson (1975), Ambachtsheer and Farrell (1979),
Dunn and Theisen (1983), Ippolito (1989), Goetzman and Ibbotson (1991), Kane et al (1999), to
mention a few of the listed references. Although theoretically compelling, the model has not been
widely adopted by investment managers. We suspect that portfolio managers and security analysts are
reluctant to subject their analysis to rigorous tests. This attitude may owe in no small measure to the
belief of many renowned scholars that the forecasting ability of most analysts is below the threshold
needed to make the model useful. This paper aims to identify this threshold, and to lower it by using
effective statistical methods.
Optimal portfolios are constructed with actual forecasts of abnormal returns and beta coefficients
obtained from a financial institution. We apply various ways of shrinking a robust estimator toward a
data-dependent point to identify and utilize predictive power. Since distributions of abnormal returns
are fat-tailed, we choose the Least Absolute Deviations (LAD) estimator as a benchmark. The quality
of estimates of stock betas affects the accuracy of the estimates of realized abnormal returns, needed to
2
measure the bias and precision of the forecasts. We also use Dimson’s (1979) aggregate coefficients
method to account for infrequent trading,
The analyst forecasts in our database show correlations between forecasts and realizations of
abnormal returns on the order of 0.04, and predictive ability generally declines over the sample period,
perhaps due to changing market conditions. The forecasts are biased and forecast errors are
asymmetric and correlated across stocks. Nevertheless, the application of shrinkage LAD estimation to
the TB model results in superior performance in both the diagonal and covariance versions. Portfolios
based on OLS estimates are dominated by the those that use LAD and shrinkage LAD estimators.
The paper is organized as follows. Section 1 presents the TB framework and Section 2 describes the
forecast data and sampling procedure. Section 3 elaborates on the estimation of beta coefficients and
abnormal returns from realized stock returns, and Section 4 treats the calibration of forecasts from the
history of forecasting records. Section 5 lays out the out-of-sample test procedures and Section 6
reports on portfolio performance. Section 7 provides a summary and conclusions.
1. The Treynor-Black Framework
To fix ideas and introduce notation, we briefly describe the model.1 Treynor and Black (1973) deal
with a scenario in which the mean-variance criterion (the Sharpe ratio) is used by investors; a specified
market index is taken as the default efficient (passive) strategy, and the security analysts of a portfolio
management firm cover a limited number of securities. Under these conditions, securities that are not
analyzed are assumed to be efficiently priced, and a portfolio of only the covered securities cannot be
efficient. The optimal portfolio must be a mix of the covered securities and the index portfolio. TB
identify the portfolio of only the covered securities (the efficient Active Portfolio, A) that can be
mixed with the index (Passive Portfolio, M) to obtain the optimal risky portfolio.2
With a risk-free asset (or a zero beta portfolio) whose rate of return is denoted rf, the weights of A
and M in the optimal risky portfolio, P, which maximizes the Sharpe ratio ( PPP RS σ= ) are given by
)R,)Cov(RRR(RR
)R,Cov(RRRw
MAMA2AM
2MA
MAM2MA
A +−+−
=σσ
σ ; wM = 1 - wA, (1)
where R is the expected excess return, E(r) – rf. Assuming the diagonal version of the market model
we have,
1 A more elaborate description can be found in Bodie et al (2001). 2 The existence of this portfolio can be inferred from Merton (1972) or Roll (1977).
3
Ri = αi+ βiRM + ei ; E(ei) = Cov (Ri, Rj) = 0, ∀ i ≠ j ; Var(ei) = σi2, (2)
where αi is the abnormal return expected by the analyst who covers the ith security and, except for σM,
σi denotes residual standard deviation of the ith security (or portfolio). We assume that the security
analysts cover n securities (i = 1, 2, …, n).
Substituting (2) into (1) yields
0A
0A )w(11
ww
β−+= ; 2
MM
2AA
0 Rw
σσα
= , (3)
where αA = ∑=
n
ii
1iw α , βA = ∑
=
n
ii
1iw β and the weight wi is given below in (5). Note that w0 is the
optimal weight in the active portfolio when its beta is average, βA = 1. The intuition of (3) is that the
larger the systematic risk of the active portfolio, the less effective is diversification with the index, and
hence, the larger the weight in A.
With wA from (3), the Sharpe ratio of the risky portfolio, P, is given by
SP2 =
[wA(αA + βARM ) + (1 − wA )R M ]2
wA2 (βA
2 σM2 + σA
2 ) + (1 − wA)2σ M2 + 2wA(1 − wA )βAσM
2 = SM2 +
αA2
σA2 , (4)
which reveals that the appraisal ratio (αA σ A ) of the active portfolio determines its marginal
contribution to the Sharpe ratio of the passive strategy. This appraisal ratio, in turn, is maximized by
choosing the weight, wi, on the ith covered security (out of n), to be
∑=
=n
1j2j
j
2ii
iw
σασα
. (5)
The reason βi is absent in (5) is that a correction is made for βA in wA of (3). Applying this solution to
(4) shows that the marginal contribution of an individual security to the risky portfolio’s squared
Sharpe measure is equal to its own squared appraisal ratio
∑=
+=+=n
i 12i
2i2
M2A
2A2
M2P SSS
σα
σα . (4a)
Thus, if forecast quality exceeds some threshold, there are economies to scale in the coverage of
securities that help explain large portfolios in the industry. Another important organizational
implication is that portfolio management can be organized into three decentralized activities; macro
forecasting to obtain RM, micro forecasting to obtain αi, and statistical analysis to obtain σM, βi and σi.3
3 Obviously, there is room for improvement by exchanging ideas among the staffs of the various activities.
4
The assumption of the diagonal model is obviously suspect. Yet it’s not a priori clear that even if
residuals are correlated across securities, the use of the generalized model will be profitable, since we
face a trade off between a somewhat flawed model (with assumed non-correlation) and a correct model
with estimation errors in the covariance matrix. We will put this tradeoff to the test; hence we need the
optimal portfolio for the generalized model. The first stage of the optimization with a non-diagonal,
residual covariance matrix is unchanged. We only need to redo the maximization of the appraisal ratio
of the active portfolio (see the analysis in Merton (1972), and Roll (1977)). Using matrix notation and
denoting the covariance matrix of residuals by Ω, the weight, wi , in the active portfolio is given by the
ith element of the following 1n × vector:
wc = [α’Ω -1 ι ]-1Ω -1 α, (6)
where α is the vector of expected abnormal returns and ι is a vector of ones. When the covariance
matrix is diagonal, wc reduces to wi in (5).
2. The Forecast Database and Sampling Procedures
The forecast data set used in our study has been provided by an investment firm active in the U.S. in
the early-mid 1990’s. We shall refer to this firm as “XYZ corp” hereafter. In that period, the firm
began extensively using artificial neural network-based statistical analysis to predict abnormal returns.
They used the S&P500 as a performance benchmark and, consequently, they mostly held large
company stocks that traded in relatively large volumes. The firm (XYZ) graciously provided us with
monthly4 forecasts of abnormal returns and beta coefficients for all stocks in its database for the period
December, 1992-December, 1995, 37 sets of monthly observations in all. These are the forecasts that
XYZ used in constructing their portfolios. Nevertheless, XYZ did not reveal how it went about
portfolio management.
In December, 1992, XYZ had 711 stocks in its database and additional stocks were added regularly
over the sample period, ending with 771 stocks in December, 1995. To simplify the test procedure, we
eliminated any stock for which one or more forecasts were missing, leaving 646 stocks for the test
databank. As the table below shows, the sample-period years, 1993-1995, were quite representative for
the US stock market with one each of average, bad and good years.
4 Forecasts were submitted by the last Friday of the month prior to the target month.
5
Annual Returns (%) 1993 1994 1995 1926-1999 Average SD (%)
Large Stocks 9.87 1.29 37.71 12.50 20.39
Small Stocks 20.30 -3.34 33.21 19.02 40.44
XYZ’s monthly forecasts of alpha were constrained to integer-values of percent per month, between
%12− and 14% so that any more extreme forecast would have been set to the appropriate limit. Figure
1 and Figure 2 show histograms of the alpha and beta forecasts for the 646 stocks in the sample over
the period, and Table 1 presents summary statistics for their location and dispersion.5 Alpha forecasts
were right-skewed and negative on average. Beta forecasts were distributed around one, typical for
large stocks.
[Figures 1, 2 and Table 1 here]
We confine our study to 105 of the 646 stocks in the database to reduce the computation time needed
for LAD estimation.6 Accordingly, we chose to work with a subset of 105 randomly selected stocks.7
Among many others, Rosenberg et al (1985) and Fama and French (1992) provide empirical
evidence that the equity book-to-market ratio (BE/ME) and market capitalization (SIZE) help explain
average stock returns.8 To account for this effect, we seek to preserve the distribution of BE/ME and
SIZE in our sub-sample. Annual data for these variables were obtained from Standard and Poor’s
COMPUSTAT tape. Using 7 categories of BE/ME and SIZE, we allocate the 646 stocks into 49
groups of similar BE/ME and SIZE, with about 13 stocks in each group. We randomly draw as close as
possible to an equal fraction of (2 or 3) stocks from each group, arriving at a random sample of 105
stocks that reflects the databank (population) distribution of the category variables. We use this sub-
sample in the subsequent sections. Figure 3 and Figure 4 show the population and sample histograms
for SIZE (Market Value) and BE/ME (Book/Market Value), respectively.
5 This is quite unusual. Most financial firms produce forecasts in the form of a ranking variable which must be converted to a scale variable, using the Information Correlation Adjustment proposed by Ambachtsheer (1977). 6 While LAD estimators can be obtained by simplex linear programming proposed by Charnes and Lemke (1954), this method is not efficient as the parameter space grows along with the number of observations and requires a long search time. Barrodale and Roberts (1974) proposed a modified version of the simplex algorithm (BR-L1) which is more efficient and greatly reduces computation time. Nevertheless, computation time for 646 stocks with BR-L1 is excessive. 7 As shown in section 1, reducing the number of covered stocks significantly lowers the contribution of the active portfolio and hence does not diminish the force of our positive test results. 8 Fama and French argue that these variables are proxies for the part of risk premiums not captured by market beta.
6
[Figures 3 and 4 here]
3. Estimation of Beta Coefficients and Realized Abnormal Returns
Forecasting accuracy derived from records of past forecasts is a critical input when using security
analysis to optimize portfolios. To determine the accuracy of abnormal return forecasts we need a time
series of realized abnormal returns.9 We obtained from DATASTREAM daily returns for the 105
stocks in the sample, the S&P500 index, and 3-month yields for T-bills for the period January 1, 1990
through March 31, 1996 with 1629 observations.
3.1 Beta Coefficients
We use XYZ’s beta forecasts to compute realized abnormal returns. Nevertheless, if we confine
ourselves to XYZ’s beta forecasts, we run the risk that tests of the quality of forecasts of abnormal
returns will depend too greatly on the quality of these beta forecasts. To avoid this risk we also use
standard estimates of betas from realized daily stock and market-index excess returns. For the in-
sample estimate of realized abnormal returns in any given month, we estimate beta from daily returns
over three years where the tested month is in the middle of the period. We estimate betas with the
following three alternative procedures.
(i) To account for infrequent trading we use Dimson’s (1979) aggregate coefficient method (AC). Beta
estimates are the sum of the contemporaneous plus K lead and lag coefficients from the regression
ti,ktm,
K
Kkki,iti, eRbaR ++= +
−=∑ ; ∑
−=
=K
Kkki,i bb ; t = 1, , ..., T. (7)
There is no obvious rule for selecting the number of lags and leads (K). An appropriate value for K can
be inferred from by regressing the market-model residuals (ex-post abnormal return minus alpha
forecast from XYZ) on a constant and the market excess returns, and testing for a zero slope
coefficient. If the correct number of lags and leads are used, then the slope coefficient from the
regression should be zero. We pool the residuals for all stocks over all the months in the sample
(approximately 10537 × = 3,885 observations). Table 2 shows the estimation results for 0 to 2 lags and
9 The term ‘realized’ for ex-post abnormal returns is somewhat misleading, since they are unobservable. We estimate them from realized returns, the market-model equation and estimated beta coefficients.
7
leads. We reject the hypothesis that the slope coefficient is equal to zero only for 0 and 1 lags/leads10
and it becomes significantly different from zero for K ≥ 2. Among the two candidates 0 and 1, we
choose K = 1 because our belief is that when K = 0, the infrequent trading problem may still be
present.
[Table 2 here]
(ii) Vasicek (1973) proposes the Bayesian estimate (V):
bV = wb + (1-w)b* ; w = 22
2
/1/1/1
* bb
b
vvv+
, (8)
where b = estimated market beta,
vb2 = estimate of variance of b,
b* = mean of prior distribution of market beta,
vb*2 = variance of prior distribution of market beta.
Vasicek suggests the cross-sectional mean and variance for b* and vb*2 respectively.
(iii) An alternative approach is a James-Stein shrinkage estimator (JS):
bJS = wb + (1-w)b* ; w = ,)()()(
1 *1*
+
− ⎥⎦
⎤⎢⎣
⎡−−
−bbbVarbb
h (9)
where h is a choice parameter and [a]+ = max(a,0). Note that (b-b*)Var(b)-1(b-b*) is an F-statistic with
(n-1,1) degrees of freedom, approximately a χ2-statistic with one degree of freedom for the implicit
hypothesis: H0: b = b*, and the weight w is now given by 1 −hF
⎡ ⎣
⎤ ⎦
+
. When F is large, (that is, we are
likely to reject the implicit null hypothesis), w is large (close to one), and we do not shrink. Instead of
specifying the variance of the prior distribution as in Vasicek’s method, the shrinkage factor, h, needs
to be specified. Noting that Prob[w ≥ 0] = Prob[F ≥ h], it is easily seen that as h → 0, Prob[w ≥ 0]
→ 1 while Prob[w ≥ 0] → 0 as h → ∞. We choose h = 0.45 such that Prob[w ≥ 0] = 0.5.
We apply both shrinkage methods to the AC estimates as two alternative estimates. To summarize,
we use four alternatives to XYZ’s beta forecasts: OLS, AC, V and JS, where V and JS are Vasicek’s
method and the JS method applied to the AC estimates. Figure 5 shows the distribution of beta
estimates from the four methods. It is evident that AC shifts the OLS distribution upwards, correcting 10 This can be explained by XYZ’s concentration in large-company stocks.
8
for the downward bias due to infrequent trading. V shrinks the tails toward the AC mean, while JS
leaves the tails almost unchanged; instead, it shifts the central mass of the distribution toward the
mean.
[Figures 5 and 6 here]
3.2 Realized Abnormal Returns
The five methods of estimating betas yield 5 sets of estimates of realized abnormal returns. For each of
the five methods, we have a 37105× matrix,b(⋅) , of monthly betas for the 105 stocks. To each element
in the matrix we obtain a corresponding element of the matrix of realized abnormal returns from
tMtititi RbR ,)(
,,)*(
,⋅⋅ −=α ; i = 1, ..., 105 ; t = 1, ..., 37 ; )(⋅ = (XYZ), (OLS), (AC), (V), (JS) (10)
A first glimpse at the quality of abnormal return forecasts is shown in Figure 6. The scatter of forecasts
and realization with the V specification (Dimson’s estimates with Vasiceck’s correction) show that the
constraint on the range of alpha forecasts [-12,14] may have been costly, particularly for positive
values.
[Figure 6 here]
4. Calibration of Alpha Forecasts
TB point out that we must explicitly account for the quality of forecasts when optimizing the portfolio.
This issue is taken up by Admati (1985), Dybvig and Ross (1985) and Kane and Marks (1990).
Assume, for example, a simple case where the realization, *α , and a forecast, α , are related by *α =
)ˆ(αf + ε where ε is white noise. Assuming that the function f is linear, we obtain the unbiased
forecast, αUB = E( *α | α ) = ρ2α , by discounting the raw forecast using the correlation, ρ, between *α
and α . We can obtain the appropriate discount function for α by estimating a well specified
regression of the forecasting record, e.g. *α = a + bα + η.
To assess overall forecast accuracy with various specifications and beta estimates, we pool all
10537 × = 3885 pairs of forecasts and realizations. We use three alternative specifications:
Linear LLL*L ˆba ηαα ++= (11a)
Parabolic P2
2P1PP*P ˆbˆba ηααα +++= (11b)
9
Kinked KKK*K ,0]ˆMax[ba ηαα ++= (11c)
The kinked-linear specification is a ‘no short sales’ alternative that may be required for many
institutions.
Pooling the sample across stocks and time to estimate the forecast discount function would affect
the test results in two important ways. First, applying a uniform discount function to all stocks is likely
unrealistic and would significantly reduce the potential contribution of the active portfolio. Hence,
pooling across stocks will, if anything, increase the power of the performance tests. However, pooling
observations across time entails the use of data that was unknown at the time the forecast was made.
For this reason we use pooling across time only to discuss issues of overall forecast quality. When we
test performance month-by-month we use only data from past months to estimate the discount
function, maintaining the pooling across stocks.
Figure 7 shows the fitted lines from the regressions of realizations on abnormal return forecasts
with the three alternative specifications. The linear specification reveals a severe correction for quality.
The graph of the parabolic specification shows that the correction would be more severe at the low end
of the forecast range. The extent of this differential overwhelms this specification. At the lowest end,
the discount function would convert negative alpha forecasts into positive signals, calling for long
positions in these stocks. The kinked line specification is a milder form of such correction, it converts
all negative forecasts to zero, effectively taking these stocks out of the active portfolio.
[Figure 7 here]
Table 3 presents three panels for the regression results of the three specifications, with five beta
estimators in each. The estimation results are almost identical for the various beta estimation methods.
With zero P-value for all cases, White’s heteroscedasticity test rejects the homoscedasticity
assumption. The standard errors in parenthesis in Table 3 are computed from the Heteroscedasticity-
Consistent Covariance Matrix Estimator11 (HCCME) proposed by White (1980). The adjusted R2 (RA2 )
is highest for the kinked-linear specification suggesting that the parabolic specification is not flexible
enough to handle the downward bias at the low end of the forecast range. The slope coefficients show
the superior power of the positive alpha forecasts. The inadequacy of the parabolic specification is
11 There are several ways of estimating the HCCME. We estimate: HCCME =(X’X)-1X’Ω*X(X’X)-1, where Ω*= Diag[et
2/(1-ht)], et = regression residual and ht = tth element of the hat matrix (X(X’X)-1X).
10
apparent from the insignificance of the slope coefficients on the squared forecast alpha. To complete
this picture, we draw similar conclusions from the intercepts. These are significant in the linear
specification and are smallest and insignificant in the kinked-linear specification. Finally, the RA2 is
quite small, never exceeding 0.00155.
[Table 3 here]
It is interesting to track the time-consistency of the quality of the forecasts. To do so, we regress
(11) on the 105 stocks, one month at a time -- 37 regressions for each specification and beta estimate.
Figure 8 shows the plots of the resultant RA2 for the V beta estimate only, as the other four plots are
very similar. The three plots reveal that, with the exception of the first two months, the quality of the
forecasts consistently deteriorated over time. This observation is somewhat surprising since the best
year of the sample period was the last, and Table 3 shows that positive forecasts fared best, indicating
that with overall low predictive power we must be careful in making generalization.
Armed with the various specifications for discounting individual forecasts we now turn to the
construction of optimal portfolios and performance evaluations.
[Figure 8 here]
5. Out-of-sample Test Procedures
The steps comprising our performance test are as follows:
(i) Estimate the discount function for forecasts of month t from paired observations of forecast and
realization of abnormal returns in month t-1. Abnormal returns of month t-1 are computed from the
market-model equation using beta coefficients estimated from past realized daily returns.
(ii) Obtain unbiased forecasts for month t by applying the discount function from month t-1 to the
forecasts of abnormal returns for month t.
(iii) Obtain macro forecasts of the mean and variance of the index portfolio.
(iv) Estimate the covariance matrix of residuals from past daily returns to use in the diagonal and
covariance versions of TB
11
(v) Construct the active portfolio using the unbiased forecasts and estimates of the residual variances
(covariance matrix) in the diagonal (covariance) version of TB. Construct the optimal risky portfolio
from the active and index portfolios.
(vi) Compute the realized return of the optimal portfolio in month t, using stock and market
realizations of excess returns in month t.
(vii) Use the realized monthly excess returns of the optimal risky TB portfolio and the market index (t
= 2, ..., 37) to evaluate performance.
5.1 Discount Functions for the Monthly Forecasts of Abnormal Returns
We use forecasts and realizations available each month. For each test month; t, t = 2, ..., 37, we first
estimate the four sets of 105 beta coefficients from realized daily returns over three years preceding
month t. XYZ’s 105 beta forecasts plus the four sets of estimates up to month t-1, bi,t −1(⋅) ; i = 1, ..., 105; t
= 2, ..., 37; (⋅) = OLS,AIT,AC,V,JS, are used in the market model equation (1) to compute the
realized abnormal returns, )*(1ti,
⋅−α ; i = 1, ..., 105; t = 2, ..., 37; JSV,AC,XYZ,OLS,)( =⋅ . Each of the
53× = 15 sets of 105 abnormal returns are then paired with the 105 forecasts, 1ti,ˆ −α ; i = 1, ..., 105; t =
2, ..., 37. For each set of the 15 combinations of KP,L,)( =o and JSV,AC,XYZ,OLS,)( =⋅ , we run
the following pooled regression (across i) for each t:
)(1ti,
)()()*(1t)i,( 1-t),(1-t),(1-t),(
+ˆba ⋅−
⋅⋅⋅− +=
oooo ηαα ; t = 2, ..., 37; KP,L,)( =o ; JSV,AC,XYZ,OLS,)( =⋅ (12)
where L,P,K stand for the linear, parabolic and kinked specifications. Hence, in the above regression,
we have 105 observations for t = 2, 210 observations for t = 3, and so on. Once the coefficients )()(
1-t),(1-t),(b,a ⋅⋅
oo are estimated for each test month t, the discounted alpha forecast for the ith security is
given by ti,)()( ˆba
1-t),(1-t),(α⋅⋅ +
oo. The critical role of the discount function motivates an elaborate estimation
scheme. We estimate (12) in 5 different ways: (i) OLS, (ii) LAD, (iii) Non-Random Combination LAD
(NRLAD), (iv) James-Stein LAD (JSLAD), and (v) Optimal Weighting Scheme LAD (OWLAD).12
12 Let b be the LAD estimator and g be the OLS estimator. Then, the JSLAD, NRLAD and OWLAD estimators are defined as follows:
gg)-)(b||g-b||/c-(1JSLAD 21 +=
gg)-)(bc-(1NRLAD 2 +=
gg)-)(b||g-b||/--(1OWLAD 221 += λλ ,
12
The NRLAD, JSLAD and OWLAD estimators (Kim and White (2001)) are obtained by optimally
mixing the OLS and LAD estimators. These estimators have smaller mean squared error than the OLS
and LAD estimators and are expected to improve out-of-sample performance of the forecasts. We
evaluate performance with these five estimation methods.
5.2 Residual Variances and Covariances
The optimal weights (5) of the diagonal version of TB require forecasts of residual variances, and
those of the covariance version (6) require a forecast of the full covariance matrix of residuals. Ideally,
these would be extrapolated from past forecasting errors. However, since we do not have a sufficient
number of observations, we estimate the covariance matrix from daily returns over three years ending
in the last trading day of month t-1. The five estimates of beta for each stock, obtained from the daily
returns of the prior 3-year period, yield five sets of deviations from the market model,
dM,)(mi,di,
)(di, RbR=e ⋅⋅ − ; d = day in the 3-year period; m = month in the 3-year period (the subscript for
the month for which the estimates are prepared is dropped for clarity). These daily residuals are used to
estimate the covariance matrix of daily residuals for the 3-year period prior to month t. The matrix is
multiplied by nt, the number of days in month t, to obtain the forecast of the residual covariance matrix
in month t.
5.3 Macro Forecasts
It appears from (4a) that macro forecasts of the mean and variance of the index-portfolio are not more
important than those of a single security. This appearance led Ferguson (1975) to argue that effort
spent on macro forecasting should not exceed that spent on any individual security. But this argument
is false. The Sharpe ratio in (4a) is conditional on optimal weights of all securities in the risky
portfolio. To the extent that a forecast for a security is of low quality and not properly discounted, (4a)
does not apply. The shortfall from the maximum Sharpe ratio will depend on the weight of the security
in the portfolio. Since it is likely that the weight on M will be substantially higher than that of any
individual security, the quality of macro forecasts will be substantially more important.
where g)-Q(b)g-(b||g-b|| 2 ′= and Q is a weighting matrix and the combination parameters ),,c,(c 2121 λλ are chosen to minimize the asymptotic risk of the corresponding estimator. See Kim and White (2001) for a detailed discussion.
13
The literature on estimating the market mean is sparse (see Merton (1980)), while the literature on
market volatility is quite rich. We use the AR(0)-GARCH(1,1) specification as in Engle et al (1993) to
forecast the market excess return and variance as follows:
RMt = E(RM) + εt (13)
21tM,
21-t
2Mt baw −++= σεσ ; t = 1, ..., T.
Daily returns and 3 year rolling estimation window are used in (13) so that T is the final day of the 3-
year period window. Once the estimation step for one iteration is completed, then the k-step ahead
volatility prediction ( )2kTT,M, +σ standing at time T is generated by
σM, T,T+12 = w + aσM, T
2 (14)
σM, T,T+ k2 = w + (a + b)σM, T,T+ k-1
2 ; k = 2, ..., nm
and the target month variance estimate isσM, m2 = σM, T+ k−1
2
k=1
nm
∑ where nm is the number of days in the
target month. Figure 9 shows the forecasts and realizations of the market index excess returns. The
RMSE of the excess-return forecasts is 2.20% per month.
[Figure 9 here]
5.4 Restrictions on Portfolio Weights
Despite the discount of the forecasts, the weights on the active portfolio derived from (3) and (6) turn
out to be excessive and volatile. It appears that the size and volatility of the active portfolio weights
result from the dynamics of the discounted abnormal-return forecasts. While many portfolio mangers
may take short positions in individual stocks (shorting against the box, in Wall Street terms), fewer can
use index futures to emulate short positions in the index. We therefore restrict the active portfolio
weight to the range [0,1] in order to rule out any unrealistic positions in the active portfolio that
generate superior performance.
5.5 Performance Measures
The TB model maximizes the Sharpe measure. However, most users have little intuition about the
value of an incremental improvement in the Sharpe measure. Modigliani and Modigliani (1997)
propose a transformation of the measure to a rate of return equivalent. This measure, which has come
to be known as M2, is computed from the Sharpe measure of portfolio P by
14
M P2 = SPσM – R M (15)
The first term SPσM gives the excess return on a mix of P with the risk free asset that would yield the
same risk (SD) as the market index portfolio M. By subtracting the average excess return on M we get
the risk adjusted return premium of P over M. M2 provides better intuition than S and has become
popular with practitioners; hence we include the measure in the performance evaluation reports.
We track the monthly optimal weights of the individual stocks and the index in the risky portfolios
derived from the various estimators and specifications. We use the daily returns on the stocks, index
and bills to compute realized daily excess returns on the risky portfolios over the 36 months of the
forecast period (January, 1992-December, 1995).13 The daily excess returns of each risky portfolio are
paired with the index excess returns to compute S and M2 for each portfolio over the entire period.
6. Portfolio Performance Evaluation
6.1 The Diagonal Version of the TB Model
Table 4 reports performance of portfolios constructed from the diagonal version of TB. It shows values
for the Sharpe and M2 measures for selected estimators and specifications. As could be predicted by
the poor results of the parabolic specification reported in Table 3 and Figure 7, portfolios derived from
the parabolic specification performed poorly and were eliminated from Table 4. We also eliminated the
inferior OLS and AC beta estimators. The major implications from Table 4 are as follows:
(i) All portfolios, except those derived from OLS estimation of the forecast-discount functions,
outperformed the passive strategy. The superior performance is of economic significance, providing
strong testimony to the value of even miniscule predictive ability of security analysts, and the power of
the TB model.
(ii) Portfolios derived from LAD estimators uniformly outperform portfolios from the OLS
specification of the discount functions. This indicates the existence of fat tails in the return
distributions and the value of better estimation methods.
(iii) The portfolios derived from the kinked specification of the discount functions uniformly
outperform the linear specification. This result may be unique to the unusual nature of the XYZ
forecasts that were better in the positive range. Still, the adjusted R-square of this specification (see
Table 3) shows that even positive forecasts were of low quality, and yet, with no short sales, portfolios
derived from these forecasts were significantly superior to the passive strategy.
15
[Table 4 here]
Table 5 presents the risk-return data of the optimal portfolio. The right-hand panel shows that with the
[0,1] restriction on the active portfolio weights, the managed portfolios risk is only slightly larger than
that of the passive strategy. Moreover, the risk of the kinked-specification portfolios is actually slightly
lower than that of the index portfolio. This means that superior performance is achieved by the
improvement in average returns from the identification of non-zero alpha stocks in the linear
specification, and positive-alpha stocks in the ‘no short sales’ (kinked) specification.
[Table 5 here]
6.2 The Covariance Version of the TB Model
In using the covariance version we face a trade off between a theoretically-advised improvement and
(low) estimation precision of the residual covariance matrix. Table 6 presents the performance
measures of the portfolios derived from the covariance version. Most portfolios (20 out of 30), and all
‘no short sales’ (kinked) specification portfolios, show improved performance over the portfolios from
the diagonal version. Here, too, the LAD-estimator portfolios perform best. This is another indication
that improved estimation will further increase the effectiveness of the model and the contribution of
security analysts. Table 7 shows that risk of the managed portfolios was increased relative to portfolios
from the diagonal version in most cases. This should be expected since the full covariance model
provides better utilization of forecasts and hence, larger positions in the active portfolio at the expense
of diversification.
Our sub-sample of 105 stocks out of the 646 stocks in XYZ’s databank was chosen randomly. This
suggests that the forecasting quality of the stocks that were left out is similar to the 105 stocks we
used. In that case, we can easily assess the incremental performance that would be obtained with the
diagonal model by expanding the universe of covered securities from 105 to 646. Table 4 shows the
value of the Sharpe measure of the JSLAD portfolio as 1.145, compared with .909 for the S&P500
index, resulting in M2 = 2.114%. Using (3) we obtain the contribution of the active portfolio to the
squared Sharpe measure by 1.1452 - 0.9092 = 0.485. This contribution would be expected to grow to
105646485.0 ÷× = 2.982, resulting in a Sharpe measure of 2.982 + .9092 =1.951, and M2 = 13 The first month is lost since we have no estimate of the discount function for this month.
16
166.8980.8951.1 −× = 9.354%. Obviously, the complexity of managing an organization 6 times as
large with the same production quality, and the difficulty in various statistical analyses can
substantially cut down the potential gain. At the same time, there can be a distinct diversification-like
advantage in estimation procedures with a larger universe of stocks.
[Tables 7,8 here]
7. Summary and Conclusions
The objective of this paper is to identify and reduce the threshold of profitable forecasting ability for
portfolio management. We suspect that the low precision of the forecasts of security analysts
contributes to the dearth of portfolio managers that efficiently use the security analysis afforded by the
Treynor Black model. We find that the threshold of profitable forecasts of abnormal returns is
extremely low, that is, security analysis that results with a correlation between forecasts and
realizations of abnormal returns as low as 0.04 can still be profitable in an organization that covers
more than 100 stocks. Nevertheless, this requires that econometric methods are utilized.
We experiment with a database of monthly forecasts of abnormal returns for 105 stocks over 37
months, provided by an investment firm that actually used these forecasts to construct its portfolios.
Using a database of realized returns on these stocks and the market index, we estimate forecast
discount functions and apply them to the forecasts prior to the construction of the risky portfolios. In
the process, we use the Dimson method to account for infrequent trading, and shrinkage Bayesian
estimators for beta coefficients. The discount functions are estimated with LAD estimators to account
for fat tails. Various specifications of the discount functions are used to account for the quality of the
forecasts. These methods significantly improve the performance of the risky portfolio.
We show that the key to profitability of low-precision forecasts is the use of sophisticated
econometric methods. Using OLS to estimate the market model and the forecast-discount functions
will not do. Shrinkage estimators for beta and LAD estimates of the discount functions, while
imposing organization-driven restrictions on the weights of the active portfolio, can endow a portfolio
derived from low-quality forecasts with profitability.
Our experiment was performed under adverse conditions. The forecast records were short, requiring
that we assign equal quality to all forecasts. Macro forecasting was not utilized and the substitute
extrapolation techniques were not as powerful as they could be. Finally, the estimates of residual
17
variances in the diagonal version and the covariance matrix in the covariance version of the TB model
can be improved. All this indicates that the threshold to profitability of forecast precision can be
further lowered.
These findings lend a real meaning to the concept of nearly-efficient capital markets. If XYZ was
representative of competitive investments firms, this suggests that competition leads to a degree of
information efficiency that reduces the forecast precision of super-marginal firms to a level as low as
what we observe in this experiment.
18
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20
Table 1. Summary Statistics for Alpha & Beta Forecasts
Beta Alpha Mean 0.982 -1.356 Std 0.274 4.089 Min 0.056 -12
25% 0.841 -5 50% 0.985 -1 75% 1.143 0
Max 2.087 14 Skewness -0.003 0.491 Kurtosis 3.268 3.32 JB 67.878 1005.369 P-value 0 0
Table 2. Lag Selection for the Aggregate Coefficient Method
K Intercept Slope R2 R2a Prob(F) Mean of
estimated betas
0 0.699 0.052 0.00024 -0.000012 0.334 0.8810
(0.123) (0.054)
1 0.700 -0.068 0.00041 0.000153 0.207 0.9995
(0.123) (0.054)
2 0.701 -0.113 0.00113 0.000876 0.036 1.0447
(0.123) (0.054)
Note: Standard errors are in parenthesis.
21
Table 3. Regression Results for the Calibration of Alpha Forecasts
Beta Intercept Alpha R2 R2a White’s
Heteroscedasticity Test Panel A: Linear Specification
OLS 0.3970 (0.1333) 0.1909 (0.0944) 0.001341 0.001084 92.6821
XYZ 0.3371 (0.1330) 0.1940 (0.0942) 0.001391 0.001134 92.7363
AC 0.3122 (0.1336) 0.1872 (0.0950) 0.001286 0.001028 94.9599
V 0.3210 (0.1334) 0.1911 (0.0947) 0.001342 0.001085 93.4382
JS 0.3154 (0.1336) 0.1892 (0.0950) 0.001314 0.001057 94.9233
Beta Intercept Alpha Alpha2 R2 R2a White’s
Panel B: Parabolic Specification Heteroscedasticity Test OLS 0.2808 (0.1523) 0.1914 (0.0946) 0.0554 (0.0455) 0.001841 0.001327 98.1026
XYZ 0.2257 (0.1521) 0.1945 (0.0943) 0.0531 (0.0453) 0.001851 0.001337 98.4132
AC 0.1936 (0.1526) 0.1877 (0.0952) 0.0565 (0.0462) 0.001804 0.001290 99.5293
V 0.2037 (0.1525) 0.1916 (0.0948) 0.0559 (0.0458) 0.001851 0.001337 98.4014
JS 0.1975 (0.1526) 0.1898 (0.0952) 0.0562 (0.0462) 0.001827 0.001313 99.6631
Beta Constant Alpha*(Alpha>0) R2 R2a White’s
Panel C: Kinked Linear Specification Heteroscedasticity Test
OLS 0.1823 (0.1198) 0.3939 (0.1960) 0.001793 0.001536 92.2511
XYZ 0.1225 (0.1194) 0.3896 (0.1951) 0.001762 0.001505 92.3440
AC 0.1000 (0.1200) 0.3910 (0.1979) 0.001762 0.001505 94.4778
V 0.1057 (0.1198) 0.3952 (0.1967) 0.001804 0.001547 92.9745
JS 0.1016 (0.1199) 0.3930 (0.1978) 0.001781 0.001524 94.4409
Note: Heteroscedasticity-consistent standard errors are in parenthesis.
Table 4. Sharpe Ratio and M2-measure: Diagonal Model (S&P500 Sharpe Ratio = 0.909)
Sharpe Ratio M2-measure OLS NRLAD JSLAD OWLAD LAD OLS NRLAD JSLAD OWLAD LAD Beta Line 0.895 0.922 1.033 0.921 1.025 -0.131 0.112 1.033 0.107 1.036 * Kinked 1.076 1.095 1.235 1.095 1.235 1.501 1.667 2.920 1.668 2.921 * Vasicek Line 0.905 0.957 1.058 0.955 1.058 -0.037 0.427 1.337 * 0.413 1.337 * Kinked 1.094 1.119 1.146 1.118 1.146 1.654 1.886 2.124 * 1,878 2.123 JS Line 0.904 0.965 0.941 0.963 0.940 -0.050 0.497 * 0.287 0.481 0.279 Kinked 1.095 1.120 1.145 1.120 1.145 1.665 1.894 2.114 * 1.890 2.112
Note: * indicates the best estimator in the row.
22
Table 5. TBP Return and Risk: Diagonal Model (S&P500 Return = 8.166: S&P500 Risk = 8.980)
TBP Return (Mean) TBP Risk (Std. Dev.) OLS NRLAD JSLAD OWLAD LAD OLS NRLAD JSLAD OWLAD LAD Beta Line 8.799 9.286 10.706 9.262 10.710 9.833 10.072 10.451 10.054 10.452 Kinked 9.539 9.581 10.794 9.585 10.796 8.861 8.750 8.744 8.752 8.744 Vasicek Line 8.872 9.294 11.637 9.277 11.641 9.800 9.713 10.996 9.712 11.000 Kinked 9.725 9.623 9.403 9.618 9.400 8.893 8.597 8.206 8.600 8.205 JS Line 8.899 9.323 11.577 9.307 11.583 9.846 9.664 12.299 9.666 12.317 Kinked 9.758 9.601 9.323 9.601 9.320 8.914 8.570 8.144 8.574 8.143
Table 6. Sharpe Ratio and M2-measure: Covariance Model (S&P500 Sharpe Ratio = 0.909)
Sharpe Ratio M2-measure OLS NRLAD JSLAD OWLAD LAD OLS NRLAD JSLAD OWLAD LAD Beta Line 0.765 0.744 1.019 0.747 1.019 -1.293 -1.482 0.980 -1.460 0.984 Kinked 1.205 1.232 1.435 1.232 1.435 2.659 * 2.897 * 4.717 * 2.898 * 4.718 * Vasicek Line 0.792 0.867 1.077 0.865 1.077 -1.054 -0.378 1.504 * -0.399 * 1.507 * Kinked 1.210 1.244 1.261 1.242 1.261 2.703 * 3.003 * 3.158 * 2.985 * 3.157 * JS Line 0.789 0.866 1.015 0.864 1.015 -1.082 -0.388 0.948 * -0.407 0.951 * Kinked 1.214 1.248 1.267 1.247 1.267 2.732 * 3.042 * 3.216 * 3.030 * 3.214 *
Note: * indicates the M2-measure is greater than its counterpart in the diagonal model in Table 4.
Table 7. TBP Return and Risk: Covariance Model (S&P500 Return = 8.166: S&P500 Risk = 8.980)
TBP Return (Mean) TBP Risk (Std. Dev.) OLS NRLAD JSLAD OWLAD LAD OLS NRLAD JSLAD OWLAD LAD Beta Line 7.958 8.170 11.972 8.162 11.983 10.398 10.976 11.754 10.930 11.760 Kinked 11.532 11.632 13.827 11.638 13.829 9.567 9.442 9.638 9.445 9.639 Vasicek Line 8.163 8.818 12.456 8.794 12.464 10.307 10.168 11.568 10.167 11.571 Kinked 11.526 11.390 10.978 11.378 10.976 9.523 9.158 8.706 9.162 8.705 JS Line 8.137 8.811 11.054 8.788 11.057 10.315 10.173 10.891 10.171 10.892 Kinked 11.550 11.365 10.927 11.358 10.925 9.517 9.106 8.621 9.110 8.621
23
Figure 1. Histogram of Alpha Forecasts
Figure 2. Histogram of Beta Forecasts
-15 -10 -5 0 5 10 150
1000
2000
3000
4000
5000
6000
7000
0 0.5 1 1.5 2 2.50
200
400
600
800
1000
1200
1400
1600
1800
2000
24
Figure 3. Market Value: Population vs. Sample
- 5 0 5 1 0
x 1 04
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0M a r k e t V a lu e : P o p u la t io n
1 9 9 4 ( u n it = m il l io n s o f $ )- 5 0 5 1 0
x 1 04
0
5
1 0
1 5
2 0
2 5
3 0
3 5
4 0
4 5M a r k e t V a lu e : s a m p le
1 9 9 4 ( u n it = m il l io n s o f $ ) Figure 4. Book/Market Value: Population vs. Sample
- 1 0 - 5 0 50
2 0
4 0
6 0
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0B o o k /M a r k e t V a lu e : P o p u la t io n
1 9 9 4 ( u n it = $ / $ )- 1 0 - 5 0 50
5
1 0
1 5
2 0
2 5
3 0B o o k / M a r k e t V a lu e : S a m p le
1 9 9 4 ( u n it = $ / $ )
25
Figure 5. Distribution of Beta Estimates Based on 4 Methods.
Figure 6. Scatter Diagram of Ex-Post Abnormal Returns vs. Alpha Forecasts
-1 -0.5 0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
OLS(dash) AC(solid) Vasicek(dashdot) JS(dotted)
Grand Mean: OLS=0.880 AC=0.999 Vasicek=0.989 JS=0.996
26
Figure 7. Fitted Lines Based on 3 Specifications
Figure 8. Predictive Power (Adjusted R2)
0 5 10 15 20 25 30 35 400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Predictive Power (Adjusted R2) over time (based on Vasicek + AC Method with n=1)
Line
(-)
Par
abol
a(+)
Kin
ked
Line
(*)
Date Forecasts Made (The First Adjusted R2 = 0.08)
27
Figure 9. Forecasting Monthly S&P 500 Index
0 5 10 15 20 25 30 35 40-5
-4
-3
-2
-1
0
1
2
3
4
Forecasting Monthly S&P500 Index (2/93-1/96) (AR(0)-GARCH(1,1) using 3 Year Daily Return Rolling Window)
Act
ual(*
) R
etur
n F
orec
ast(
+) V
olat
ility
For
ecas
t(o)
Prediction Root MSE = 2.2032
