Download A New Method for Portfolio Performance

A New Method for Portfolio Performance Measurement
Robert Heinkel
Neal M. Stoughton1
University of British Columbia
University of California,
Irvine
Current Version: August, 1997
1
We are very grateful to SEI Financial Services for the data in this study, and we thank Marc Rouillard,
Gerhart Pahl and Patrick Walsh, in particular. We also appreciate helpful discussions with Cliff Ball, Glen
Donaldson, Mark Grinblatt, Alan Kraus, Philippe Jorion, and at seminar presentations at Amsterdam,
INSEAD, Lausanne, Mannheim, McGill, Norwegian School of Management, Odense, Simon Fraser, UBC,
UC Irvine and Vienna. The paper was also presented at the 1996 American Finance Association meetings in
San Francisco. We appreciate the research assistance of Lisa Kramer. Finally, we are grateful for the helpful
comments of an anonymous referee.
Abstract
An important facet of money management is the practice of tactical asset allocation. This paper
develops two measures of asset allocation skill, also known as market timing ability, which incorporate information present in both the portfolio weights as well as their contemporaneous rates of
return. By utilizing the standard normally distributed signal methodology, we develop and test a
model that is capable of ranking managers with differential information acquisition ability. Using
a sample of 34 balanced pension fund managers and quarterly asset data over a ten year period,
we find statistically significant evidence of superior market timing ability using each of our two
measures. We also compare and contrast our measures to the Henrikkson-Merton and GrinblattTitman approaches. We find that our measures are consistent with others in the literature, but
may have some advantages due to the simplicity of implementation.
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Introduction
A longstanding interest of both financial researchers and practitioners has been the issue of portfolio
performance measurement. As is well-known, performance is a key component in the decision by
clients to renew or terminate management contracts. The client/manager contracting model of
Heinkel and Stoughton (1994) established that interim performance monitoring and recontracting
may play an important role in resolving existing agency conflicts.
In recent years the amount of capital devoted to institutional management in the form of pension
and mutual funds has increased greatly. Index-tracking, in which money managers attempt to mimic
the returns on individual indexes, has gained an impressive market share. This is sometimes referred
to as “passive” management. The popularity of indexing and its growth, worldwide, continues in
spite of new evidence challenging beliefs in market efficiency and the capital asset pricing model.1
Whenever indexing is employed, the asset allocation decision becomes critical.2 The importance
of this function is exemplified by pension fund mandates for the utilization of specialist services for
direct asset allocation decisions. This reflects a general tendency in which asset allocation is being
unbundled from the remaining money management activities.
This paper implements two new measures of asset allocation skill, also known as market timing
ability. These new measures utilize information present in both the portfolio weights as well as
their contemporaneous rates of return. As such, this paper belongs to a newer generation of
approaches as opposed to models that developed measures by relying on rate of return data alone.
While these measures are closely related to previous academic work, we emphasize their practical
implementation in this paper. By testing these measures on a simulated data set and a sample of
actual pension fund data, we hope to demonstrate the efficacy of these two measures and to relate
them to other approaches in the literature.
The chief benefit of our parametric estimation approach is that it is capable of ranking multiple
managers with differential ability. The general approach to identifying managers with superior
information (as in Glosten and Jagannathan (1994) and Chen and Knez (1996)) is distinguished
1
For a provocative book on this subject, see Haugen (1995).
Asset allocation refers to the choice of portfolio percentages to be invested in various asset categories such as
cash, stocks and bonds.
2
1
from our model, in that we define a specific information precision parameter that is to be estimated.3
We utilize the well-established normal signal methodology, originally employed by Jensen (1972).
This information acquisition model is combined with a model of portfolio choice relative to a
benchmark.4
We then define two performance measures based on knowledge of the portfolio weights. Both of
the measures are derived from joint information present in the portfolio choices and contemporaneous rates of returns. By utilizing portfolio weights, we are able to avoid the difficulties identified by
Dybvig and Ross (1985), in which portfolio returns are non-linearly related to underlying market
returns when managers have timing information.5 Moreover, the use of portfolio weights allows
very simple econometric methods to be employed. This creates the potential for real practical
applicability. The original paper using portfolio weight information was Cornell (1979). Further
relevant discussions appeared in Admati and Ross (1985) and Grinblatt and Titman (1989).
The two performance measures developed in this paper are contrasted with the measures suggested by Henriksson and Merton (1981) and Grinblatt and Titman (1993). Our discrete measure
is closely related to the non-parametric model in Henriksson and Merton (HM).6 The HM nonparametric approach is based on estimating the likelihood of making correct forecasts by observing
if the manager has a tendency to be invested in an asset when its return is higher than the alternative asset. The basic principle behind our discrete measure is similar, except that by utilizing the
normally distributed signal structure, the likelihood function is non-linear in asset returns, unlike
HM. We believe that, by weighting the outcomes differently depending on ex post rates of return,
a manager’s portfolio actions are more accurately reflected in the measure.7
Our second performance measure, the continuous measure, is related to that of Grinblatt and
3
Our approach has been applied recently in a conditional model in Becker, Ferson, et al (1996).
Brennan (1993) looks at the consequences for security pricing when some investors are mean-variance optimizers
relative to a benchmark.
5
When using return data alone, some of these difficulties can be eliminated through the use of non-linear regressions
as discussed in Admati and Ross (1985), Bhattacharya and Pfleiderer (1983) and Admati, Bhattacharya, Pfleiderer
and Ross (1986). The use of factor models is explored in Connor and Korajczyk (1986).
6
Based on Merton (1981), HM also develop a parametric model which is based on an OLS regression with an
option term as an explanatory variable. One difficulty with the parametric approach as explored by Jagannathan
and Korajczyk (1986), among others, is that non- linear securities can confound the measurement.
7
In simulation experiments, Beebower and Varikooty (1991) tested a probit model that appears to be similar to
our model.
4
2
Titman (1993), which uses the covariance between actual portfolio returns and lagged changes
in portfolio weightings.8 Our continuous measure refines the approach of Grinblatt and Titman
(GT). First, their measure is affected not only by market timing skill, but can also be a function of
other manager-specific and contractual characteristics, such as the aggressiveness of the manager.
As originally suggested by Admati and Ross (1985) it is possible to disentangle these two effects
through a linear regression of portfolio weights on returns. We implement this idea by using the
R-squared. Second, our continuous model raises the issue of whether it is indeed possible to avoid
the use of a benchmark portfolio entirely as GT have argued. We discuss an interpretation of the
lagged weights in GT as a time-varying benchmark, as opposed to a constant benchmark.
As a test of the practical application of our two measures, we investigate the performance of a
sample of 34 balanced fund managers over a ten year period. The use of balanced fund data allows
us to concentrate our tests on the type of fund management for which the approach is designed.
We also apply our measures in a Monte Carlo simulation.
We find that all four of the measures that we examine indicate the presence of statistically
significant market timing for a subset of the population of managers in our pension fund sample.
By contrast, a large number of other studies have failed to find significant market timing ability.
Examples include Kon (1983), Henrikkson (1984), Chang and Lewellen (1984), Lee and Rahman
(1990) Coggin, et al (1990), and Weigel and Ilkiw (1991).
The outline of the paper is as follows. Section 2 contains a derivation of the optimal portfolio
choice. Section 3 discusses the data. Section 4 applies our measures to the pension fund data. A
comparison between our measures and two other models in the literature is provided in section 5,
including a simulation analysis. Section 6 concludes the paper.
2
The Model of Information Acquisition and Portfolio Formation
In this section, we provide the theoretical foundation for our measures of portfolio performance
in timing the market. Our methods are focused on the choice between two asset classes, such as
8
The Grinblatt-Titman measure is related to the Heinkel and Kraus (1987) insider trading measure.
3
equities and fixed income.9 We begin by setting forth the structure of information observed by
the portfolio manager. Then we specify a compensation contract for the manager and derive her
optimal portfolio choice. This portfolio choice forms the basis for the definition of two performance
measures of market timing.
The discrete measure relies only on limited information on the portfolio weights, namely whether
the portfolio is more risky than a benchmark. This measure is also applicable to situations where the
manager only attempts to forecast directions in the market such as in a newsletter. The continuous
measure employs the actual portfolio weight in assessing performance.
2.1
Information Acquisition
We assume that at the beginning of a portfolio formation period, the manager observes signals
about the rates of return of the asset classes over the period. For the sake of exposition, we will
refer to one asset as risky and the second asset as riskless; this is without loss of generality by
choosing one asset to be the numeraire with which excess returns are measured. The manager has
a prior expectation about the expected excess return, R̃, given by µ0 = E(R̃). The manager’s prior
2 = E[(R̃ − µ )2 ].
belief about the variance of the excess return is σR
0
The information structure is represented in a standard way (see e.g., Jensen (1972)). In each
period, the manager observes private information represented by a normally distributed signal, S̃,
with the property that
S̃ = kr̃ + η̃
(1)
where r̃ = R̃/σR , the normalized unit variance excess return, and where without loss of generality, η̃ is a zero mean unit variance disturbance term. Furthermore, we assume that r̃ and η̃ are
uncorrelated normal random variables. The coefficient, k, indicates the information content in the
signal. Clearly, as k becomes large, the signal is relatively more precise, i.e., less noisy. The ratio
of the variances in the two terms on the right hand side of equation (1) is defined by h = k2 , as
described in Heinkel-Stoughton (1994). It is precisely this parameter that we wish to estimate using
9
We can apply our methods to a portfolio with multiple asset classes by aggregating the asset classes into any two
categories, such as equity versus bonds or foreign versus domestic investment, etc.
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our measurement techniques. We shall therefore interpret h as the information processing ability
of the portfolio manager.
After observation of the signal, the manager forms her posterior distribution on the excess rate
of return. It is well-known that the posterior distribution is also normal. Denote µ as the posterior
mean, E[R̃|S̃]. From Bayesian statistics it is easy to compute the posterior mean as
µ = σR
h1/2
1
S̃ +
µ0 .
h+1
h+1
(2)
That is, the coefficient, k = h1/2 , indicates the amount of weight given to the signal. The posterior
variance, σ 2 , is deterministic and is equal to:
σ2 =
1
σ2 .
h+1 R
(3)
It is also of interest to compute the ratio of posterior mean to variance, which is similar to what
has been termed the Sharpe ratio in the literature.10 From equations (2) and (3) we obtain:
µ
µ0
h1/2
S̃ + 2 .
=
2
σ
σR
σR
(4)
That is, the posterior mean-variance ratio is equal to the prior mean-variance ratio plus an adjustment whose magnitude is jointly determined by the signal and the information processing ability
of the manager.
Now that we have described the information structure facing the manager, we employ this
structure in deriving the manager’s optimal portfolio choice and the resulting performance measures.
2.2
Optimal Portfolio Choice
We assume that the manager’s compensation is a linear function of the portfolio’s performance
in excess of a benchmark. Heinkel and Stoughton (1994) demonstrate that, in a repeated setting
with the possibility of termination, a linear contract can overcome agency problems associated with
10
The Sharpe ratio is usually taken as the expected return over the standard deviation of the return rather than
the variance.
5
moral hazard and adverse selection.
Specifically,
C̃ = φ0 + φ1 [W̃ − (x̄W0 R̃ + W0 (1 + RF ))],
or
C̃ = φ0 + φ1 (x − x̄)W0 R̃,
(5)
where C̃ is the manager’s cash compensation, φ0 and φ1 are the parameters of the compensation
contract, W0 is the initial portfolio value, W̃ is the ending portfolio value, and RF is the riskfree
rate. The benchmark portfolio is defined by the portfolio weight x̄, which is the proportion of the
risky asset in the benchmark.
Our model portrays portfolio choice in the standard exponential-normal framework. Admati
and Ross (1985) and Admati, Bhattacharya, Pfleiderer and Ross (1986) have utilized similar models.
Assuming the manager has exponential utility with a risk tolerance parameter τ , and given the
normal distribution of R̃, the manager’s optimal portfolio choice can be represented in the following
linear-quadratic form:
max φ1 (x − x̄)W0 µ −
x
1 2
φ (x − x̄)2 W02 σ 2 .
2τ 1
(6)
The resulting first order condition is
x − x̄ =
2.3
τ
µ
.
φ1 W0 σ 2
(7)
The Discrete Performance Measure
In the discrete performance measure, we ignore most of the quantitative information on the portfolio
weights. We simply record in each period whether the manager’s current risky portfolio weighting,
x, is greater than the benchmark (x > x̄).11 The choice of whether to adopt a more risky portfolio,
x > x̄, depends on the sign of the conditional mean, as seen in equation (7). Using equation (2),
11
This method is clearly applicable to situations where exact portfolio weights are unavailable, such as when looking
at analyst recommendations.
6
we see that this choice is equivalent to:
σR
h1/2
1
S̃ +
µ0 > 0.
h+1
h+1
Now, substituting from equation (1) and rearranging, we obtain the following condition for the
adoption of a portfolio more risky than the benchmark:
µ0
h
σR h1/2
+
R̃ +
η̃ > 0.
h+1 h+1
h+1
In order to put this expression into the standard form of a probit regression (Judge, et al (1985)),
we multiply by h + 1 and divide by σR h1/2 to get:
pr(x > x̄) = pr(
µ0
h1/2
R̃ + η̃ > 0).
+
σR
σR h1/2
(8)
Equation (8) is in the form of a probit expression:
pr(x > x̄) = pr(a + bR̃ + η̃ > 0),
where the coefficients are identified as
a=
µ0
σR h1/2
(9)
h1/2
.
σR
(10)
and
b=
This illustrates that in the discrete performance measure the estimate of the coefficient, b, will
identify the value of h. Therefore, if the coefficient on R̃ is significantly different from zero, the
manager will have been shown to have market timing ability.
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2.4
The Continuous Performance Measure
The previous measure utilizes only information concerning whether the portfolio is more risky
than the benchmark. It is therefore useful to have another measure that utilizes the quantitative
magnitude.
Using equation (7), the manager’s portfolio choice is equal to
x − x̄ = T
µ
,
σ2
(11)
where T = τ /(φ1 W0 ) represents the manager’s risk tolerance relative to the proportion of portfolio
value that is relevant to her.
Substituting from equation (4), we find that
"
#
h1/2
µ0
x − x̄ = T
S̃ + 2 .
σR
σR
Using the definition of S̃ from equation (1) we get the following linear regression equation:
x = x̄ + T
µ0
T h1/2
Th
R̃
+
+
η̃.
2
2
σR
σR
σR
(12)
Therefore, the continuous portfolio measure leads to a linear regression of the portfolio weights on
the excess return of the risky asset:
x = a + bR̃ +
T h1/2
η̃.
σR
The constant term in this regression is equal to the benchmark portfolio weight plus the unconditional optimal portfolio weight based upon prior information,
a = x̄ + T
8
µ0
2 ,
σR
while the slope coefficient is equal to
Th
2 .
σR
b=
In similar fashion with the probit expression, equation (8), we notice that the slope coefficient
is linearly related to the parameter, h, although the risk tolerance coefficient also enters here. Since
the risk aversion coefficient is not directly observable, this creates a slightly different estimation
problem than in the discrete choice case. The problem here is that a significant slope coefficient
could either indicate that the manager has superior informational abilities (high h) or simply that
she is very risk tolerant. Clearly one would like to disentangle these two effects.
Fortunately, it is possible to use the fact that the risk tolerance coefficient also impacts the error
term in equation (12) to remove the influence of T . As discussed in Admati and Ross (1985), both
T and h are identifiable. Consider the R2 from the above regression. The R2 may be computed as
follows:
R2 =
T 2 h2
2
σR
T 2 h2
2
σR
+
T 2h
2
σR
=
h
.
h+1
Hence,
h=
R2
,
1 − R2
which is monotonic in R2 . This illustrates that a test for a significant information processing ability
is simply a test of a significant regression relationship. Having completed our theoretical analysis,
we now turn to the data and estimation of the two models.
3
Data
Our data set, provided by SEI Financial Services, a Canadian performance measurement and
management consultant service, includes 34 Canadian pension fund accounts, each managed by a
different money management firm. The 34 money managers had approximately (Can)$120 billion
under management. These managed accounts are all categorized as balanced funds, since their
primary mission is to optimally allocate funds among various asset classes. Thus, this data set is
ideally suited as an application of our market timing measurement techniques.
9
The data consist of start-of-quarter commitments (i.e., portfolio weights) and quarterly rates of
return in seven asset allocation categories: short term (cash), bonds, private placements, mortgages,
convertibles, U.S. equities and Canadian equities. For most managers, the majority of assets are
in the form of Canadian equity and bonds. The next most predominant categories were U.S.
equity and cash. Mortgages, private placements and convertibles comprised a considerably smaller
fraction, usually less than five percent of the portfolio value. Indeed, the weights on these latter
asset categories are often equal to zero for many periods. The data are drawn from the time period
of 1982 (quarter 1) to 1991 (quarter 4), for a total of ten years, or 40 quarters. It is noteworthy that
this sample of 34 funds was randomly drawn from a much larger set of funds in the SEI Canadian
performance universe. In this respect, half of the funds were drawn from funds ranked above the
median according to SEI’s performance rankings and the other half were ranked below the median.
The data exhibit considerable variation in portfolio turnover. Some managers tended to keep
their asset class commitments fairly steady while others tended to show great variation in their
holdings. For example, some managers varied their portfolio weights by as much as 20 percent over
the course of three months.
To apply this data base to our theoretical model, we must define two asset classes in which the
managers invest. Thus, we must aggregate the seven asset classes in the data set in order to exhibit
a choice between two asset categories. This can be done in a number of ways, depending on the
nature of market timing being investigated. In our paper we have chosen to aggregate together
convertibles, U.S. equities and Canadian equities as the risky category and measure timing with
respect to a second asset category consisting of the sum of cash, bonds, mortgages and private
placements.12 Obviously other forms of aggregation can also be considered. Depending on the type
of aggregation assumed, the method provides a measure of timing relative to the alternative asset.
3.1
Asset Class Commitments
Given our asset class definition, the average risky asset commitment over all 34 managers was 41.99
percent, with a standard deviation of 6.66 percent. The average of the smallest risky commitment
12
We also have applied our methods to a second classification in which we focus only on Canadian equities and
cash. For brevity, we do not report these results here; they are available on request from the authors.
10
during the sample period was 26.61 percent; the average of the largest risky commitment during the
sample period was 54.51 percent. During the sample period, the highest risky exposure attained
by any one manager was 70.23 percent, while the smallest risky asset exposure was 9.21 percent.
We performed an augmented Dickey-Fuller test for unit roots in the portfolio weight data.
Assuming no time trend, we found that for 22 out of 34 managers we could not reject the hypothesis
of a unit root. However, Ball and Roma (1992) note that the standard unit root analysis may not
apply when the underlying process has reflecting barriers. Because of practical constraints on short
selling and other institutional restrictions, the risky asset commitment never leaves the [0,1] interval
in our data set. Applying the Dickey-Fuller test to the first differences in the portfolio weights shows
that the hypothesis of a unit root is rejected in 31 out of 34 managers for the differenced data series.
3.2
Asset Class Returns
We define the excess return on each manager’s two-asset-class portfolio as the value-weighted return
on the risky asset class (i.e., convertibles, U.S. equities and Canadian equities) less the valueweighted return on the “riskless” asset class (i.e., cash, bonds, mortgages and private placements).
Because the composition of both asset classes differs considerably by manager, our study investigates
asset class timing within the context of the manager’s individual portfolio. Thus, unlike a number
of other studies, we are not defining timing skill relative to the same market index.
We performed several diagnostic tests on the excess returns time series for each of the 34
managers in our sample. First, we regressed the excess return on its lagged value. The average
t-statistic on the slope coefficient across the 34 managers was 0.376, and only two t-ratios had
absolute values exceeding 1.0. The average regression R2 was 0.0085 and no R2 exceeded 0.05. The
Durbin h-statistic, which indicates autocorrelation in the residuals, was only greater than 2.0 in 6
of the 34 cases. We believe these simple tests indicate that serial correlation in the excess returns
is small.
Finally, for each manager we calculated the standard deviation of the excess return, σR , for the
full sample of 40 quarters, and also over the two subsample periods of the first and second halves
of the data. The averages, over managers, for the two subperiods are 7.0% and 8.3%. The sample
11
standard deviation was greater in the later subperiod than in the first subperiod for all except two
managers. However, a simple F -test of the ratio of the variances found that only two of the 34
managers had p-values below 5%, and none had p-values of 1%. This indicates that the standard
deviation of the return data appears to be stable over time.
4
Application of the Performance Measures
4.1
The Discrete Measure
In order to apply our discrete measure to this data sample, we must deal with the critical issue of
how the benchmark weights are defined. In practice, one would like to use the weights specified in
the contract between the client and the manager.13
In the absence of specific benchmark information, we define the benchmark weight, x̄, as the
manager’s average portfolio weight on the risky asset over the entire sample of commitments. The
use of the average commitment as a benchmark is well-documented in the practitioner literature.
See, for example, Brinson, Hood and Beebower (1986). In fact the term policy weights is often used
to describe these average portfolio weights. An important characteristic of the use of the average
weight over the sample period is that the benchmark weight is assumed to be fixed over time.
4.1.1
Sample Statistics
Table I contains some sample statistics and the results from the probit estimation for all 34 managers. First, it is noteworthy that the average commitment to equities, x̄, was less than 50 percent
for all but one manager in our sample. This is consistent with the fact that 22 out of 34 managers
had negative average excess returns, µ0 , over the sample period. That is, these managers’ fixed
income portfolios outperformed their equity portfolios. Manager 3 had the highest average excess
return, about 2.6 percent on an annual basis, while manager 16 had the lowest excess return, about
-4.8 percent annually. The standard deviations are relatively similar across managers and are in
the neighborhood of 14 percent on an annual basis. Thus, our sample does not contain evidence
13
For example, Fidelity states that their Asset Manager mutual fund has a benchmark of 40% stocks, 40% bonds
and 20% short term/money market instruments.
12
of extremely high risk in holdings as aggregated in this way. As usual, none of the average excess
returns are statistically different from zero.
4.1.2
Probit Estimates
The three columns on the right hand side of Table 1 contain the results of the probit estimation
procedure. The constant term, â, exhibits considerable variation across managers, with 21 being
positive and 13 being negative. All but two of these coefficients are statistically insignificant from
zero.
Our main focus is on the significance and magnitude of the slope coefficient, b̂. These coefficients
and the associated estimate of the information ability of the manager, ĥ, appear at the right of
Table I. These estimates of h pertain to formula (10). In the appendix, we derive the estimator, ĥ,
from the slope coefficient as an unbiased estimate of h. We show there that
2 2
2 2
ĥ = σR
b̂ − σR
σ ,
where σ is the standard error of the estimate.
Out of 34 managers, 8 have coefficients significantly different from zero at the .05 level of
significance. Manager 12 has superior market timing ability at the .01 level. This manager also has
the highest estimate of timing ability at ĥ = .477. Managers 13 and 25, on the other hand, have
estimates, ĥ, that are closest to zero. Given that ĥ is an unbiased estimator it is not surprising that
16 of the managers have estimates below zero. However, none of these are statistically significant,
indicating that this is merely due to estimation error.
Interestingly, Manager 12, with the greatest timing ability according to our discrete measure,
also has a fairly low excess return on her portfolio (µ0 = −1.086). This negative average excess
return results from the manager’s superior performance in her fixed income asset class relative
to her equity asset class performance. Nevertheless, the high timing measure indicates an ability
to switch between these two actively managed asset categories at the right times. Because our
measure does not employ passive asset class benchmarks, it is not designed to detect value added
13
from security selection, i.e., selectivity.
4.1.3
Cross-sectional Implications of the Discrete Model
An important issue in performance measurement is whether the method being investigated is capable of differentiating managers from one another. We now provide some evidence of differential
ability among managers using our discrete measure. We want to test the null hypothesis that the
results in Table I are due to random estimation error, rather than indicating that eight managers
have truly superior performance. That is, we want to know the probability that the series of
estimates could come from a situation where all managers have equal abilities.
If the error terms, η̃, across managers are independent, then we can utilize a fairly standard
chi-square test.14 Let us define b̂i as the slope coefficient estimate of manager i and σi as the
associated standard error. Also define b∗ to be a coefficient, which according to the null hypothesis
is the true common value of all manager’s slope coefficients. Then define κ as
∗
κ(b ) =
34
X
(b̂i − b∗ )2
I=1
σi2
(13)
Since the coefficients b̂i are normally distributed, then under the null hypothesis of equal slope
coefficients, κ should follow a chi-squared distribution with 34 degrees of freedom (Judge et al
(1985; 943)).
For the situation where b∗ = 0, we find that κ(0) = 55.00, indicating that the probability that
all managers have zero true timing ability is p = 0.013.
The same procedure can be used to test the hypothesis of all equal non-zero abilities. As an
extreme example, we find the value of b∗ that minimizes the sum of squared deviations over all
the managers. In this case, we find that b∗ = 0.025 and κ(b∗ ) = 25.93. The probability of this
event being true is p = 0.84, indicating that we cannot reject the hypothesis that all managers have
equal abilities consistent with this value of b∗ . This result also implies that we could not reject the
14
Since the probit estimation procedure involves non-linear estimation, we are unable to avoid the independence
assumption in the case of the discrete measure. However, we do relax this assumption in the continuous measure
later in this section.
14
hypothesis of equal abilities for certain positive values of b. This latter result could be due to the
independence assumption, however.
Therefore the discrete measure shows evidence that there is superior market timing performance
by a subset of the managers in our sample.
4.2
The Continuous Measure
Next, we turn to the application of the continuous measure of portfolio choice, utilizing linear regression procedures. We shall perform tests similar to those of the discrete measure, with two major
exceptions. First, because of the additional risk tolerance coefficient, the value of the estimates of
managerial ability will be based on the R2 from the regression, rather than the slope coefficient.
Second, because linear regression is so computationally efficient, we are able to perform a joint
estimation over all managers. This allows us to address the issue of dependence of the error terms.
4.2.1
Ordinary Least Squares Estimates
The form of the OLS equation (12) is:
x = a + bR̃ + ˜,
where ˜ = ((T h1/2 )/σR )η̃. Table II presents the estimates for this case. Consider first the estimates
of the constant term, â. These are interpreted as the benchmark portfolio weights plus a portfolio
commitment given by the average excess return, µ0 . One important difference between the probit
estimation and the OLS estimation is that we do not pre-specify the benchmark portfolio, but use
the estimation itself to identify the benchmark.
We are, of course, assuming that the benchmark is constant over time. An alternative assumption, which we will show in the next section to be consistent with the Grinblatt-Titman performance
measure, uses the previous period’s portfolio weight as the benchmark. The implication of this timevarying benchmark is to replace the portfolio weight with the difference in portfolio weights on the
left hand side of the OLS regression. As noted previously, this differencing resolves the existence of
15
unit roots in the levels. We have replicated the OLS regressions under the time-varying benchmark
assumption and obtained similar results.15
In a practical context, we view the setting of the benchmark as part of the contracting and
performance measurement process. Our measures allow the use of arbitrary benchmarks.
In computing the t-statistics on the slope coefficients, we utilized the procedure of White (1980)
to account for the possibility of heteroscedasticity.16 As mentioned in section 2, these are estimates
2 , and include both the effect of risk attitude as well as timing ability. In the OLS case,
of T h/σR
we find that 13 of the 34 managers have significant slope coefficients, as measured by the t-ratio
applied to the .05 confidence level. Also, 7 of the managers have significant slope coefficients at the
.01 confidence level.
Because of the confounding effect of risk aversion, we developed in section 2 an alternative
method of estimating the timing ability parameter, h, from the R2 of the regression equation.
These estimates, ĥ = R2 /(1 − R2 ) appear in the right hand column of Table II. We assess the
significance of the R2 using an F -test. From standard regression theory (Judge et al (1985; 30)), if
the errors are normally distributed,
"
R2
1 − R2
#
37
= 37ĥ,
1
should be distributed as an F random variable with 1 and 37 degrees of freedom. We therefore
computed the p values associated with these F values for each manager. It was found that 10 of
the managers had significant estimates of timing ability at the .05 confidence level, with 5 being
significant at the .01 level.
The number of managers with significant OLS slope coefficients exceeds both the number of
15
These results are available from the authors.
Three tests of model specification were performed on the continuous measure. First, a Goldfeld-Quandt test for
heteroskedasticity ordered the data by excess return and calculated residual variances for the first half and second
half subperiods. Using an F -test with (18,18) degrees of freedom revealed that none of the excess return residuals
exhibited heteroskedasticity at the 5% level. Second, using recursive residuals (see Harvey (1990), Ch. 5) the Harvey
heteroskedasticity test revealed that only one manager had a p-value below 5%. Lastly, the recursive residuals can be
used to test for departures from normality. While 6 managers had p-values below 5%, only 2 managers had p-values
below 1%. These results give us some assurance that the tests of the continuous measure are appropriate. We continue
to use the heteroskedastic-consistent estimation of the standard errors, which are very close to the standard errors of
the OLS estimation.
16
16
managers with significant OLS R2 and the number of managers with significant probit slope coefficients. The reason for this is most likely due to the fact that the OLS slope coefficient combines
the measurement of risk tolerance with timing ability. If our measures are correct, it would appear
from the data that a number of managers may simply be very risk tolerant, i.e., they tend to
move their portfolio weights considerably in response to information, rather than possessing really
superior information.
A comparison between the continuous and discrete measures appears in section 5.
4.2.2
Cross-sectional Implications of the Continuous Measure
As with the discrete measure, we would like to identify the probability that our results on the
significance of managerial timing ability arise from true differences rather than random estimation
error. We therefore provide the same chi-squared test results as with the discrete measure. In
addition, we perform a joint estimation in order to avoid the assumption of independence of errors.
Consider the slope coefficients of the OLS regression. If the errors are independent across
managers, then the chi-squared statistic, equation (13) may be applied. Evaluating this statistic
at b∗ = 0, we find that κ(0) = 116, with a p value of essentially zero. Thus, the hypothesis that all
managers have zero slope coefficients is soundly rejected. However, the hypothesis of equal slope
coefficients for any value of b cannot be rejected in the continuous model. For example, we find
that the value of b∗ that minimizes κ is b∗ = .153, and κ(.153) = 37.1, with associated p = .32.
This evidence indicates, as with the discrete model, there is a possibility that all managers have
equal and positive timing ability.
4.2.3
SUR Regression
In order to check the robustness of our results with the independence of errors across managers
assumption, we also performed a seemingly unrelated regression (SUR) analysis of the joint system
of regression equations:
xi = ai + bi R̃i + ˜i ,
17
i = 1, . . . , 34.
In this estimation procedure, the cross-managerial dependence on the error terms will be taken into
account. This may affect the parameter estimates. However, the main reason for investigating this
approach is that it is possible to construct a likelihood ratio test of the unconstrained system with
the constrained system where bi = b∗ for all i. This then can be used to test the hypothesis that
all managers have equal coefficients.
The parameter estimates in the SUR estimation are similar to those in the OLS regression of
Table II and are omitted for the sake of brevity.17 First of all, the Breusch-Pagan test described
in Judge, et al (1985; 476) shows that the hypothesis of a diagonal variance-covariance matrix for
the error terms is rejected. This indicates the presence of correlation in errors. Since these errors
form part of the signal received by each manager, and managers draw from information sets that
have some degree of conjunction, it is not unreasonable to expect this outcome.
Since the main purpose of the SUR regression is to see whether the covariance structure of the
errors affects the test of equality of the slope coefficients, we also estimated a restricted version of
the model in which bi = b∗ . In this case, the estimate of this value is b̂∗ = .1689. We construct the
likelihood ratio test in the following way. Let L(b), b = b1 , b2 , . . . , b34 , represent the log-likelihood
of the SUR estimation procedure in the unrestricted case. Let L(b̂∗ ) represent the log- likelihood
in the restricted case. Then, from asymptotic theory, the statistic given by
χ(b, b̂∗ ) = 2[L(b) − L(b̂∗ )]
has a chi-square distribution with 33 degrees of freedom. In this case we find that χ(b, .168) =
101.34, which essentially has a p-value equal to zero. Interestingly, exploiting the covariance structure of the errors shows that managers have significantly different slope coefficients, whereas this
was unable to be ascertained under the assumption of independence.
Even though there appears to be significant cross-correlations in the error terms, we find that
the estimates of managerial timing ability, ĥ, are very similar to the estimates from the OLS
regressions. Thus, we conclude that the ranking of managers is not affected by the more elaborate
17
The full set of estimations is available from the authors.
18
estimation procedure.
5
Comparison of Performance Measures
In this section we compare our two evaluation procedures with the two methods in the literature
which are most comparable to our two approaches. These approaches are the Henrikkson-Merton
(1981) nonparametric method and the Grinblatt-Titman (1993) portfolio change measure. Our
first comparison is on a theoretical basis, followed by comparisons using simulated data and actual
pension fund data.
5.1
5.1.1
Theoretical Comparisons
Henrikkson-Merton
The Henrikkson-Merton (HM) model specifies that the likelihoods of observing choices of the more
and less risky portfolio conditional on the excess return of the risky asset are, respectively:
p1 = pr(x > x̄|R̃ > 0)
p2 = pr(x ≤ x̄|R̃ ≤ 0).
HM obtain estimates of p1 + p2 by looking at the relative frequency of choosing the risky asset and
riskless assets, conditional on a positive or negative excess return on the risky asset. Then, the
significance of p1 + p2 > 1 is assessed.
By contrast, in our model, the likelihood of observing the risky portfolio is (by equation (8))
pr(x > x̄|R̃) = N (
µ0
h1/2
R̃),
+
σR
σR h1/2
where N is the standard normal distribution function. Notice that this is a monotonically increasing
function in R̃ with the information processing parameter, k = h1/2 , acting to strengthen the effect.
Forming estimates of p1 and p2 as they do, HM are assuming that these values are independent
19
of the level of R̃, whereas our measure has these probabilities dependent upon R̃ in a non-linear way.
The differences in approach are best illustrated by a situation in which the manager “correctly”
holds equities over a period in which they outperform cash by a large magnitude. The HM procedure
gives the same weight to this event as it does to an event in which equities barely beat cash. Our
probit estimation effectively gives greater weight to the latter event. The intuition is that marginal
outcomes tell more because there is a greater likelihood that good and bad managers will make
different decisions when the asset class returns are quite similar.
5.1.2
Grinblatt-Titman
The Grinblatt-Titman (GT) portfolio change measure derives from the intuitively appealing idea
that successful managers would exhibit positive covariances between their portfolio weights and
security returns, as discussed also in Heinkel-Kraus (1987). The GT definition of the portfolio
change measure features an assumption that expected portfolio weights are equal to past weights
lagged by some time period. If quarterly data are used with a single-period lag for expectations,
then their measure may be defined as follows:
M = E[(xt − xt−1 )R̃t ],
(14)
where the subscripts indicate the time period.
To see how this measure is related to our continuous measure, consider the optimal portfolio
choice of equation (12). Rewriting this equation for any period, t, under the assumption that the
benchmark portfolio is equal to the previous period’s portfolio weight:
xt − xt−1 = T
µ0
T h1/2
+
b
R̃
+
η̃t .
t
2
σR
σR
(15)
From the definition of the slope coefficient in an OLS regression,
b=
cov(xt − xt−1 , R̃t )
.
2
σR
20
(16)
Under the assumption that the manager’s signals are serially uncorrelated, E(xt ) = E(xt−1 ), we
know that
cov(xt − xt−1 , R̃t ) = E[(xt − xt−1 )R̃t ].
Using equation (16), this implies that the GT measure is related to the slope coefficient of the
2 . Rewriting this using the definition of b in the continuous measure
continuous model by M = bσR
shows that
M = T h.
(17)
Thus, we have shown that the GT measure can be interpreted as proportional to the slope
coefficient in the OLS regression of portfolio weights on excess return, taking the benchmark weight
to be the previous period’s portfolio weight. One important difference between our continuous
measure and the GT measure is that we utilize a constant benchmark, while GT use a time-varying
benchmark. We regard the manager as unconstrained in her portfolio choice in the sense that
she is able to respond optimally to new information in each period in order to select the optimal
deviation from a fixed benchmark. GT, on the other hand, essentially assume that the deviation
is measured relative to what was adopted in some previous period. Their view is consistent with
strong constraints imposed by transaction costs, for example, that limit the degree to which the
portfolio can be varied over time. Our approach is consistent with a situation in which the manager
is measured using a strong standard that would be true when the client does not want to provide
an allowance for transaction costs.
A second important distinction between the continuous and the GT measures is the presence of
an aggressiveness coefficient, T , in the GT measure, as indicated by equation (17). The GT measure
generates rankings that are sensitive to the combination of timing ability as well as risk tolerance.
By using the R2 from the OLS regression rather than the slope coefficient, our continuous measure
purges the managerial-specific risk characteristics from the performance rankings. Of course, this
interpretation of the comparison between the two methods is particular to the joint impact of
Grinblatt and Titman’s measure and our modeling of how information translates into asset choices.
Nevertheless, we feel it is important to understand how their measure fits into the context of our
21
model.
5.2
Simulation Results
In order to gain a better understanding of the statistical properties of our two measures we perform
a series of Monte Carlo simulations. In addition, we compare our measures to the HM and GT
measures using simulated data.
We employ the following simulation procedure.
For a given managerial ability, h:
• For the first trial:
– Generate 40 values of R̃ from a normal distribution with µ0 = 0.05 and σR = 0.14.
– Generate 40 values of η̃ from a standard unit normal distribution.
– Compute the optimal portfolio holdings according to
x = x̄ + T
µ0
T h1/2
Th
+ 2 R̃ +
η̃,
2
σR
σR
σR
with T = 5.0 and x̄ = 0.50.
– Compute estimated h in the probit regression by observing if x > x̄ or not.
– Compute the estimated h in the OLS regression from the regression R2 .
– Compute the HM and GT measures from the same 40 observations.
• Repeat for another simulation trial
Repeat for different managerial ability.
Table III provides the summary statistics for our two performance measures. We show the
mean, standard deviation and tenth and ninetieth percentiles for each true h corresponding to
0.001, 0.01, 0.1, 0.2 and 0.3. The mean estimate is monotonically increasing in the true h for
both measures. In addition, the majority of the mean estimates are within one standard deviation
of the true values. The continuous measure mean estimates always exceed the discrete measure
22
estimates. One possible explanation is that the probit estimate of h is unbiased whereas the OLS
estimate using R2 may be biased. The standard deviation of the estimates is also monotonically
increasing, indicating greater estimation error for higher values of the true h. The results of these
simple simulations provide sufficient reassurance that our measures have validity.
Table IV provides the results of comparing the four performance measures using simulated data.
For two managers, denoted by Type 1 (equal or lower ability) and Type 2 (equal or higher ability),
we have ranked them by their performance measures, and computed the proportion of simulations
in which Type 2’s measure ranked ahead of Type 1’s measure. For example, if Type 2 had h = 0.3
and Type 1 had h = 0.001, the simulations indicated that, under the discrete measure, the Type 2
manager ranked above the Type 1 manager in 73% of the simulation trials.
We find that for all measures the likelihood of identifying the better manager increases with
the magnitude of the difference between the true abilities. In addition, when abilities are equal, all
measures are close to equal (50%) proportions favoring one manager over the other. When there
are substantial differences in true ability, the GT measure appears to perform most accurately.
However, unlike the other measures, GT is affected by the value of T . In fact, as T is varied in the
simulations, all the measures except GT are unaffected. Using the simulated data, we were unable
to identify any measures as dominating the others.
5.3
Empirical Results
As a second exercise in comparing the performance measures, we employ the data described in
Section 3. Because the HM and GT methods do not provide direct estimates of h, we instead
compare the four measures by the rank order of managers. The four methods are compared in a
pairwise fashion by first finding the number of managers ranked in common as having significant
performance measures. Then, the top ten managers in each method are compared to see how
many appear in common in each pair of measures. This latter comparison allows us to derive a
significance test of the uniqueness of the managerial ranking.
Table V contains the results of the first comparison. Significance levels for HM and GT were
23
determined by applying the procedures documented in the respective papers.18 Notice that the
discrete measure using probit estimation identifies the fewest number of significant managers at the
1% level.
The bottom panel of Table V lists the number of managers at the 5% significance level that
match for each pair of managers. For example, the discrete measure identified 8 managers with
significant ability and the continuous measure identified 10 managers. Of these 18 managers, 5
were significant in both measures, while 18 − 5 = 13 managers were not common. We express this
overlap as the ratio 5/13 = 38%. The extent of commonality for these two measures is equal to
the comparative results of the continuous and HM measures. On the other hand, the HM and GT
measures yield rather disjoint sets of significant managers, since they agree on only one out of 19,
for a ratio of 1/18 = 6%.
Since we would also like to determine how significantly different each of these measures is from
each other, we also provide a second comparison. For each method of measuring performance, all
managers were ranked and the top ten were compared.19 The number of matches for each pair
of measures is listed in Table VI. According to the top ten rankings, the discrete and continuous
measures provide the greatest degree of agreement (with 7 out of 10 matches), while the GT model
seems to agree less well with the others.
In order to differentiate the measures from one another, we are interested in finding the probability that the rankings provided by these measures are uncorrelated and that therefore any agreement
in the top ten list is due to random chance. Consider the null hypothesis that ranking methods
are random. Given this hypothesis, in a population of N managers, the probability, p, that two
rankings would agree on n or more managers out of a sample of m can be computed as:
p=
m N −m
I
m−I
.
N
I=n
m
m
X
18
The significance levels in HM were established by determining whether the estimates of p1 + p2 were significantly
greater than one using a one-tailed hypergeometric test. The significance levels in GT were established by using
t-statistics which rely on the cross-sectional estimate of the standard error.
19
Note that the top ten managers are not necessarily the ones whose measures are most significantly different from
zero, because of differences in the methods by which the estimation errors are computed. However there is close
agreement.
24
Of these measures, the null hypothesis that the agreement between the discrete and continuous
measures is due to random chance is soundly rejected at the 1% level. The only other pair of
measures that rejects the null hypothesis of random agreement is between the continuous method
and the HM procedure. In this case the common degree of managerial rankings in the top ten
is significant at better than a 5% level. Some of the comparisons, namely between the HM and
discrete measures and the HM and GT measures indicate that they are so different it is likely that
any agreement is purely due to chance.
It is apparent from Tables V and VI that the GT measure provides more of a unique set of
rankings than do the others. Our model can address this question. Recall that the analysis earlier
in this section showed that the GT measure should be proportional to the slope coefficient from
the continuous model. We therefore computed the top ten rankings using the slope coefficient
(rather than the R2 ) from our continuous model and compared it to GT. We found that there was
agreement on 7 out of 10 managers, thus lending empirical support for our assertion that the GT
measure includes both risk attitude as well as timing ability.
We also computed the rank correlations between the various measures. These appear in Table
VII and indicate that while there is strong correlation between the rankings for the discrete and continuous measures, the highest correlation coefficient is for the comparison between the continuous
and HM methods. As before, the GT model is most dissimilar.
In summary, we find that our empirical results comparing four methods of performance measurement indicate that while our discrete and continuous methods are not identical, there is a
strong degree of correlation among them. Moreover, our two measures provide different predictions
regarding market timing skill from others in the literature, as evidenced here by comparisons with
Henrikkson-Merton and Grinblatt-Titman.
6
Conclusions
This paper has developed two new measures of tactical asset allocation ability, both derived from
a common model of the manager’s information structure. The difference in the measures lies
25
in the degree of quantitative information utilized with respect to the portfolio weights. Similar
to Henrikkson and Merton, the discrete measure categorizes the manager’s portfolio decision as
more or less risky than the benchmark. In contrast to HM, we allow the likelihood of correct
categorization to depend upon the rate of return realization.
The second measure we propose involves a continuous choice of weights among assets. The
resulting measure takes advantage of the linear relationship between weights and returns. Thus,
ordinary least squares regression can be used to determine an estimate of timing ability. Moreover,
basing the estimate on the R2 from such a regression allows the timing measure to be free from
contaminating influences such as selectivity issues or managerial tastes involving risk.
One purpose of this paper has been to develop both a common theoretical structure within
which to view timing measures such as ours as well as others in the literature. In this regard, we
believe that the discrete measure more fully utilizes the quantitative information on relative security
returns in the likelihood function. We believe the continuous measure features the capability to
disentangle market timing from other factors.
Regardless of the approach followed, we have illustrated the importance of specifying a benchmark portfolio. One implication of our analysis is that benchmark selection may be as significant
in the case of market timing as it has always been viewed in the case of selectivity. We believe
clients almost always have an alternative investment strategy in mind when turning their funds
over to professional managers. And, we believe these strategies are best represented by a stable
benchmark portfolio, as employed in our continuous measure.
A second purpose of this paper has been to illustrate the efficacy of the approach by estimating
the measures both on a simulated data set and on a new sample of balanced pension fund managers.
One conclusion from the analysis of the simulated data is that the discrete measure tends to
identify fewer managers as having market timing skills, while the GT measure appears to be the
most accurate. However, in the pension fund data the GT measure identifies more, but disparate,
managers as having timing skills. One possible explanation for this is that the manager’s specific
characteristics embodied in the portfolio choice are confounding the measure.
Results from the simulations and the empirical estimation indicate that the continuous measure
26
is most closely related to the discrete and HM measures. Given that these measures perform
similarly, the criterion of ease of application favors the continuous measure. Therefore, these
new measures appear to be capable of providing clients of pension funds, mutual funds and other
institutions with new and valuable techniques that can be used in selecting managers with the
highest asset allocation skill.
27
A
A.1
Appendix
The Unbiased Estimator in the Discrete Measure
We seek to derive an unbiased estimator, ĥ, from the probit estimation equation (10):
b=
h1/2
.
σR
The slope coefficient estimate, b̂, is measured with error:
b̂ = b + b .
Denote the standard error of the slope coefficient as σ so that σ 2 = E(2b ). Consider
2 2
2 2
σR
b̂ = σR
(b + 2bb + 2b )
2
= σR
(
h
2h1/2
+
b + 2b ).
2
σR
σR
Taking expectations,
2 2
2
E(σR
b̂ ) = σR
(
h
h1/2
+
2
E(b ) + σ 2 )
2
σR
σR
2 2
= h + σR
σ .
Thus an unbiased estimator for h, ĥ is
2 2
2 2
ĥ = σR
b̂ − σR
σ .
28
(18)
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30
Table I
Estimation of the Discrete Portfolio Model
Equity Composition: Convertibles, U.S. Equities, Can. Equities
The following Probit expression was estimated for each manager:
pr(x > x̄) pr(a bR̃ ˜ > 0)
where x>x̄ indicates choice of a portfolio more heavily weighted toward convertibles, U.S. equities or
Canadian equities than the average portfolio, R̃ is the excess return on the equity portfolio, and ˜ is the
normalized unit variance error term. The estimation was carried out on a time-series basis for a period of
40 quarters. In the table below, x̄ represents the average equity composition in percent over the sample
period, µ 0 is the average excess rate of return in percent, )R is the standard deviation of the excess
2
2
return in percent, and hˆ )R( bˆ )2) is the estimate of the manager's information processing ability,
where ) is the standard error of the slope estimate. t -statistics for the estimated parameters appear in
parenthesis. The values of the estimates different from zero with a 0.05 confidence interval are marked
with an asterisk (*) and those different from zero with a .01 confidence interval are marked with a double
asterisk (**).
Manager
µ
â
bˆ
hˆ
)
x̄
0
R
1
44.9
-.539
7.70
.0174
(.0856)
.0514*
(1.72)
0.1035
2
38.3
-.457
7.78
-.0573
(-.288)
.0119
(.465)
-0.0313
3
44.2
.665
8.29
.180
(.899)
.0138
(.568)
-0.0277
4
43.7
.539
6.47
.199
(.990)
-.0173
(-.565)
-0.0267
5
43.7
.131
9.35
.0617
(.310)
.0075
(.349)
-0.0353
6
42.7
-.428
7.49
.277
(1.35)
.0479
(1.58)
0.0772
7
43.6
-.265
7.30
-.125
(-.623)
.0374
(1.22)
0.0244
8
40.8
.257
7.36
-.138
(-.684)
.0382
(1.39)
0.0383
31
Table I (cont.)
Manager
x̄
µ0
)R
â
bˆ
hˆ
9
48.1
.554
8.07
.232
(1.11)
.0585*
(2.00)
0.1671
10
52.2
-.177
8.27
.194
(.971)
.0173
(.704)
-0.0208
11
44.0
.336
7.49
-.0734
(-.366)
.0316
(1.18)
0.0156
12
45.3
-1.086
8.04
-.0727
(-.339)
.0933**
(2.56)
0.4768
13
45.4
.253
8.20
-.134
(-.667)
.025
(1.01)
0.0016
14
35.3
-.308
6.43
.0707
(.354)
.0225
(.729)
-0.0185
15
32.8
-.228
8.10
.126
(.636)
.0030
(.121)
-0.0403
16
31.9
-1.194
7.11
-.0417
(-.204)
-.0329
(-1.17)
0.0148
17
45.6
-.244
6.71
-.253
(-1.26)
.0026
(.086)
-0.0413
18
48.7
-.163
7.36
.0031
(.015)
.0174
(.640)
-0.0235
19
39.9
-.552
7.76
.0815
(.404)
.0409
(1.41)
0.0500
20
46.7
-.666
7.58
.220
(1.08)
.0351
(1.31)
0.0297
21
42.5
.214
6.89
-.257
(-1.28)
.0120
(.409)
-0.0339
22
33.8
-.155
7.24
.125
(.626)
-.0092
(-.327)
-0.0368
23
40.1
.226
8.35
-.0678
(-.340)
.0200
(.784)
-0.0176
24
41.3
-.118
8.43
.283
(1.34)
.0661*
(2.15)
0.2431
25
44.1
-.313
8.22
.0725
(.362)
.0240
(.988)
-0.0010
32
Table I (cont.)
Manager
x̄
µ0
)R
â
bˆ
hˆ
26
43.6
-.046
7.75
.0581
(.285)
.0527*
(1.74)
0.1119
27
42.0
-.465
7.28
-.0492
(-.243)
.0499
(1.55)
0.0774
28
39.4
.167
7.68
.385*
(1.89)
.0031
(.114)
-0.0419
29
39.1
.193
7.86
.0587
(.295)
.0156
(.605)
-0.0261
30
33.4
.021
8.07
.0553
(.268)
.0623*
(2.09)
0.1948
31
42.2
-.522
6.43
-.329
(-1.55)
.0818*
(1.95)
0.2040
32
45.8
-.578
7.95
.0752
(.376)
.0198
(.784)
-0.0155
33
39.4
-1.147
6.113
-.313
(-1.49)
-.0431
(-1.27)
0.0262
34
43.1
-.522
7.21
.426*
(2.02)
.0538*
(1.69)
0.0984
33
Table II
Estimation of the Continuous Portfolio Measure
Equity Composition: Convertibles, U.S. Equities, Can. Equities
The following linear regression was estimated for each manager:
x a bR̃ ˜
using ordinary least squares, where x is the portfolio weight on convertibles, U.S. equities or Canadian
equities, R̃ is the excess return on the equity portfolio, and ˜ [(Th 1/2)/)R]˜ is the normally distributed
disturbance term . The estimation was carried out on a time-series basis for a period of 40 quarters. In
the table below, R 2 is the sample R-squared from the estimated equation and the estimate of the
manager's timing ability is determined as hˆ R 2/(1 R 2) . t -statistics for the estimated parameters
appear in parenthesis. The values of the estimates different from zero with a 0.05 confidence interval are
marked with an asterisk (*) and those different from zero with a .01 confidence interval are marked with
a double asterisk (**). The significance of the estimate of the manager's ability is determined by
evaluation of the F -distribution of 37hˆ at 1 and 37 degrees of freedom as detailed in the text.
Manager
â
bˆ
R2
hˆ
1
45.04**
(39.72)
0.2354
(1.48)
0.0588
0.0625
2
38.46**
(30.67)
0.2961
(1.52)
0.0777
0.0842
3
44.11**
(78.51)
0.0856
(1.32)
0.0367
0.0381
4
43.65**
(90.54)
0.0569
(0.56)
0.0149
0.0151
5
43.72**
(29.36)
0.1952
(0.95)
0.0352
0.0365
6
42.75**
(72.59)
0.1743**
(2.45)
0.1087
0.1220*
7
43.65**
(67.47)
0.1419*
(1.73)
0.0596
0.0634
8
40.79**
(101.13)
0.1205*
(1.76)
0.1080
0.1211*
34
Table II (cont.)
Manager
â
bˆ
R2
hˆ
9
47.87**
(35.62)
0.4685**
(2.80)
0.1649
0.1975**
10
52.26**
(130.85)
0.0863
(1.54)
0.0721
0.0777
11
43.92**
(61.37)
0.2128
(1.64)
0.1137
0.1283*
12
45.54**
(73.65)
0.2100**
(3.69)
0.1555
0.1841*
13
45.35**
(49.98)
0.2405*
(1.76)
0.1029
0.1147
14
35.37**
(47.35)
0.1198
(0.91)
0.0255
0.0262
15
32.75**
(21.98)
-0.0758
(-0.47)
0.0041
0.0041
16
31.86**
(23.21)
-0.0254
(-0.16)
0.0004
0.0004
17
45.61**
(27.24)
-0.0193
(-0.08)
0.0001
0.0001
18
48.74**
(39.14)
0.3350*
(1.69)
0.0860
0.0941
19
40.03**
(31.87)
0.1775
(1.38)
0.0293
0.0302
20
47.00**
(43.79)
0.4344*
(2.41)
0.1763
0.2140**
21
42.52**
(84.08)
0.1191*
(1.75)
0.0601
0.0639
22
33.83**
(29.39)
0.0624
(0.31)
0.0037
0.0037
23
40.08**
(38.04)
0.0054
(0.08)
0.0000
0.0000
24
41.40**
(30.93)
0.4473**
(2.59)
0.1621
0.1935*
25
44.16**
(23.74)
0.3037
(0.96)
0.0415
0.0433
35
Table II (cont.)
Manager
â
bˆ
R2
hˆ
26
43.57**
(49.13)
0.1355
(1.21)
0.0331
0.0342
27
42.11**
(56.72)
0.3069**
(3.35)
0.1828
0.2237**
28
39.35**
(43.72)
0.1795
(1.05)
0.0549
0.0581
29
39.14**
(31.51)
-0.0864
(-0.58)
0.0072
0.0073
30
33.43**
(80.69)
0.1808**
(3.29)
0.2323
0.3026**
31
42.36**
(72.19)
0.3579**
(3.85)
0.2857
0.4000**
32
45.92**
(62.06)
0.1721
(1.62)
0.0766
0.0830
33
39.14**
(26.88)
-0.2426
(-0.75)
0.0240
0.0246
34
43.18**
(48.78)
0.2217
(1.52)
0.0725
0.0782
36
Table III
Summary Statistics of the Simulated Distributions
This table presents the summary statistics of the probability distribution of estimates of h using a Monte
Carlo simulation of 100 trials. Parameters chosen for the simulation were µ 0
0.05 , )R
0.14 , x̄
0.5 ,
and T
5.0 . For these parameter values, five values of h were chosen ranging from 0.001 to 0.30. The
columns in the table correspond to the 10th percentile, mean, standard deviation and 90 th percentile values
of the distribution.
.10 percentile
Mean
Std. Dev.
.90 percentile
Discrete Measure
h = .001
-0.047
0.003
0.064
0.089
h=.01
-0.044
0.009
0.079
0.106
h=.10
-0.044
0.016
0.097
0.110
h=.20
-0.047
0.042
0.099
0.162
h=.30
-0.041
0.098
0.184
0.336
Continuous Measure
h=.001
0.000
0.028
0.040
0.081
h=.01
0.001
0.030
0.039
0.079
h=.10
0.001
0.044
0.056
0.124
h=.20
0.004
0.066
0.069
0.163
h=.30
0.006
0.113
0.113
0.268
37
Table IV
Comparison of the Measures Using Simulated Data
This table presents the difference in rankings based upon the estimated values of h for a Monte Carlo
simulation consisting of 1000 trials. The simulation involves two managerial ability types. Type 1 h
represents the true value of h for the first manager and type 2 h represents the true value of h for the
second manager. The data in the four rightmost columns refers to the proportion of trials in which the
type 2 manager ranked ahead of the type 1 manager.
Type 1 h
Type 2 h
Discrete
Continuous
HM
GT
0.001
0.3
.73
.83
.80
.90
0.001
0.2
.63
.72
.73
.80
0.001
0.1
.54
.57
.61
.66
0.001
0.01
.52
.52
.52
.52
0.3
0.3
.48
.49
.51
.51
0.2
0.2
.50
.50
.49
.48
0.1
0.1
.52
.51
.50
.52
0.01
0.01
.49
.49
.50
.47
38
Table V
Comparison of Performance Measures by Significance Levels
This table compares the methods by the number of managers out of the sample of 34 that have significant
timing measures. The top panel lists the number of managers whose performance is significant at the 1%
and 5% levels. The bottom panel compares the methods in a pairwise fashion. For each pair of measures,
the number, n , of managers in the 5% significance list that matches is listed. The percentage listed is
equal to n/(n m) , where m represents the unmatched number of managers. The four approaches are
discrete (probit), linear (OLS), HM (Henrikkson-Merton) and GT (Grinblatt-Titman).
Discrete
Continuous
HM
GT
Number Significant at
1%
1
5
4
5
Number Significant at
5%
8
10
8
11
Discrete Matches
8
100%
Continuous Matches
5
38%
10
100%
HM Matches
3
23%
5
38%
8
100%
GT Matches
3
19%
4
24%
1
6%
39
11
100%
Table VI
Comparison of Performance Measures by Top Ten Ranking
For each method, the top ten managers out of a population of 34 were selected. The table lists the number
from each top ten list that match. The p -values are listed in parenthesis. These correspond to the null
hypothesis that the number of matches for each pair of measures is due to chance (see the text for details).
Significant rejection of the null hypothesis is indicated by double-asterisk (**) at the 1% level and by
asterisk (*) at the 5% level.
Discrete
Continuous
HM
Discrete Matches
10**
(0.00)
Continuous Matches
7**
(.0019)
10**
(0.00)
HM Matches
3
(.6329)
6*
(.0190)
10**
(0.00)
GT Matches
4
(.3162)
5
(.1007)
3
(.6329)
40
GT
10**
(0.00)
Table VII
Rank Correlation Matrix
Discrete
Continuous
HM
Discrete
1.000
Continuous
.597
1.000
HM
.394
.649
1.000
GT
.323
.440
.221
41
GT
1.000