Measuring the Timing Ability of Fixed Income Mutual Funds

by Yong Chen Wayne Ferson Helen Peters first draft: November 25, 2003 this revision: March 9, 2006 * Ferson and Peters are Professors of Finance at Boston College, 140 Commonwealth Ave, Chestnut Hill, MA. 02467, and Ferson is a Research Associate of the NBER. He may be reached at (617) 552-6431, fax 552-0431, wayne.ferson@bc.edu, www2.bc.edu/~fersonwa. Peters may be reached at (617) 552-3945, helen.peters.1@bc.edu. Chen is a Ph.D student at Boston College, and may be reached at yong.chen@bc.edu. We are grateful to participants in workshops at Baylor University, Brigham Young, Boston College, DePaul, the University of Florida, Georgetown, the University of Iowa, the University of Massachusetts at Amherst, Michigan State University, the University of Michigan and Virginia Tech. and the Office of Economic Analysis, Securities and Exchange Commission for feedback, as well as to participants at the following conferences: The 2005 Global Finance Conference at Trinity College, Dublin, the 2005 Southern Finance and the 2005 Western Finance Association Meetings. We also appreciate helpful comments from Nadia Voztioblennaia.

ABSTRACT

Measuring the Timing Ability of Fixed Income Mutual Funds

This paper evaluates the ability of US fixed income mutual funds to time common factors related to bond markets. A methodological challenge is to control for potential non-timing-related sources of nonlinearity in the relation between fund returns and common factors. Nonlinearities may arise from dynamic trading strategies or derivatives, funds' responses to public information, they may appear in the underlying assets held by the fund, or they may be related to systematic patterns in stale pricing. Controlling for these effects we find that funds' returns are more typically nonlinear in nine common factors than are fixed-weight benchmark returns. This is an incomplete draft

1. Introduction

The relative share of academic work on bond funds is dwarfed by the share and

importance of these funds and assets in the economy. Elton, Gruber and Blake (1993,

1995) and Ferson, Henry and Kisgen (2006) study US bond mutual fund performance,

concentrating on the funds' risk-adjusted returns, or alphas. These studies find that the

typical average performance after costs is negative and on the same order of magnitude as

the funds' expenses. However, the total performance may be decomposed into

components, such as timing and selectivity ability. If investors place value on timing

ability, for example a fund that can mitigate losses in down markets, they may accept a

total performance cost for this service. To better understand performance we need to

examine its components. We argue that a logical place to start, in the case of fixed income

funds, is to study timing ability.1 That is the goal of this paper. This is one of the first

papers to comprehensively study the ability of US bond funds to time their markets.2

Timing ability on the part of a fund manager is the ability to use information about

the future realizations of common factors affecting security returns. Selectivity refers to

the use of security-specific information. If common market factors explain a relatively

large part of the variance of a typical bond return, it follows that a relatively large fraction

of the potential performance of bond funds is likely attributed to timing. Thus, a logical

step in looking at performance is to concentrate on timing ability. However, measuring

1 We do not explicitly study market timing in the sense recently taken to mean trading by investors in a fund to exploit stale prices reflected in the fund's net asset values, or late trading -- potentially fraudulent trading after the close of the market. But we will see that these issues can affect measures of a fund manager's ability.

2 Brown and Marshall (2001) develop an active style model and an attribution model for fixed income funds, isolating managers' bets on interest rates and spreads. Comer, Boney and Kelly (2005) study timing ability in a sample of 84 high quality corporate bond funds, 1994-2003, using variations on Sharpe's (1992) style model. Comer (2006) measures bond fund sector timing using portfolio weights. Aragon (2005) studies the timing ability of balanced funds for bond and stock indexes.

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the timing ability of fixed income funds is a subtle problem.

Traditional models of market timing ability rely on convexity in the relation

between the fund's returns and the common factors.3 In bond funds, perhaps even more

clearly than in equity funds, convexity or concavity can arise for various reasons unrelated

to timing ability. Our empirical analysis controls for other sources of nonlinearity. These

include convexity or concavity that may be inherent in the benchmark assets held by a

fund, the use of dynamic trading strategies or derivatives, portfolios that respond to

publicly available information and stale prices in the measured returns.

Most versions of the models with the controls for non-timing-related nonlinearities

find that funds' returns are more typically nonlinearly related to nine common factors

than are fixed-weight benchmark returns. We find evidence of positive timing ability for

some factors, including credit spreads and the relative value of the dollar. There is also

evidence of "negative" timing ability for some factors, including interest rates, mortgage

spreads and bond market liquidity. Negative timing is similar to what previous studies

have found for equity funds relative to the stock market. However, unlike the case of

equity funds, conditional models (e.g. Ferson and Schadt, 1996) do not explain the

nonlinearity.

The rest of the paper is organized as follows. Section 2 describes the models and

methods. Section 3 describes the data. We study timing relative to a set of seven bond

market factors and two equity market factors. Section 4 presents our empirical results and

Section 5 offers some concluding remarks.

3 The alternative approach is to examine managers' portfolio weights and trading decisions to see if they can predict returns and factors (e.g. Grinblatt and Titman, 1989). Comer (2006) is a first step in this direction for bond funds.

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2. Models and Methods

A traditional view of performance separates timing ability from security selection ability,

or selectivity. For equity funds, timing is closely related to asset allocation, where funds

rebalance the portfolio among asset classes and cash. Selectivity essentially means picking

good stocks within the asset classes. Like equity funds, fixed income funds engage in

activities that may be viewed as selectivity or timing.

Fixed income funds may attempt to predict institution-specific supply and demand

or changes in credit risks associated with particular bond issues. Funds can also attempt

to exploit liquidity differences across bonds, for example on-the-run versus off-the-run

issues. These trading activities are akin to security selection. In addition, managers may

tune the interest rate sensitivity (e.g., duration) of the portfolio to time changes in the level

of interest rates or the shape of the yield curve in anticipation of the influence of economic

developments. They may vary the allocation to asset classes (e.g. corporates versus

governments versus mortgages) that are likely to have different exposures to economic

factors. These activities may naturally be considered as attempts at "market timing," since

they relate to anticipating market-wide factors.

Classical models of market-timing ability consider an equity manager who chooses

a portfolio consisting of a market index and Treasury bills. The manager observes a signal

about the future relative performance of the market over bills, and adjusts the market

exposure of the portfolio. If the response is assumed to be a linear function of the signal as

modelled by Admati, Ross and Plfeiderer (1986), the portfolio return is a convex quadratic

function of the market return as in the model of Treynor and Mazuy (1966). If the

manager shifts the portfolio weights discretely, as in the model of Merton and Henriksson

(1981), the resulting convexity may be modelled with call options. Jiang (2003) develops a

timing measure based on the probability, but not the magnitude of a convex relation. In

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each case the convexity in the relation between the fund return and the market return

indicates the timing ability. Our models modify this classical set up to accommodate

nonlinearities unrelated to managers' timing ability.

2.1 Classical Market Timing Models

The classical market-timing regression of Treynor and Mazuy (1966) is:

rpt+1 = ap + bp ft+1 + Λp ft+12 + ut+1, (1)

where rpt+1 is the fund's portfolio return, measured in excess of a short-term Treasury bill.

With equity market timing, as considered by Treynor and Mazuy, ft+1 is the excess return

of the stock market index. Treynor and Mazuy (1966) argue that Λp>0 indicates market-

timing ability. The logic is that when the market is up, the successful market-timing fund

will be up by a disproportionate amount. When the market is down, the fund will be

down by a lesser amount. Therefore, the fund's return bears a convex relation to the

market factor.

In a fixed income context it seems natural to replace the market excess return with

systematic factors for bond returns. However, if a factor is not an excess return, the

appropriate sign for the coefficient might not be obvious. For example, bond returns

move in the opposite direction as interest rates, so a signal that interest rates are about to

rise means bond returns are likely to be low. We show below that market timing ability

still implies a positive coefficient on the squared factor.

Stylized market-timing models confine the fund to a single risky-asset portfolio and

cash. This makes sense theoretically, from the perspective of the Capital Asset Pricing

Model (CAPM, Sharpe, 1964). Under that model's assumptions there is two-fund

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separation and all investors hold the market portfolio and cash. But two-fund separation

is generally limited to single-factor term structure models, and there is no central role for a

"market portfolio" of bonds in most fixed income models. In practice, however, fixed

income funds often manage to a "benchmark" portfolio that defines the peer group or

investment style. We use style-specific benchmarks to replace the market portfolio.

Assume that the fund manager combines a benchmark portfolio with return RB and

a short-term Treasury security or "cash" with known return Rf, using the portfolio weight

x(s), where s is the private timing signal. The managed portfolio return is Rp = x(s)RB + [1-

x(s)]Rf. The signal is observed and the weight is set at time t, the returns are realized at

time t+1, and we suppress the time subscripts for simplicity. In the simplest example the

factor and the benchmark's excess return are related by a linear regression (we allow for

nonlinearities below):

rBt+1 = μB + bB ft+1 + uBt+1, (2)

where rB = RB - RF is the excess return, μB=E(rB), the factor is normalized to have a mean

of zero and uBt+1 is independent of ft+1. Assume that the signal s = f + v, where v is an

independent, mean zero noise term with variance, σv2. The manager is assumed to

maximize the expected value of an increasing, concave expected utility function,

E{U(rp)|s}. Finally, assume that the random variables (r,f,s) are jointly normal and let σf2 =

Var(f). With these assumptions the optimal portfolio weight of the market timer is:4

4 The first order condition for the maximization implies: E{U'(.)rB|s} = 0 = E{U'(.)|s} E{rB|s} + Cov(U'(.),rB|s}, where U'(.) is the derivative of the utility function. Using Stein's (1973) lemma, write the conditional covariance as: Cov(U'(.),rB|s} = E{U''(.)|s} x(s) Var(rB|s). Solving for x(s) gives the result, with λ = -E{U'(.)|s}/E{U''(.)|s} > 0.

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x(s) = λ E(rB|s)/σB2, (3)

where λ > 0 is the Rubinstein (1976) measure of risk tolerance, which is assumed to be a

fixed parameter, and σB2 = Var(rB|s), which is a fixed parameter under normality.

The implied regression for the managed portfolio's excess return follows from the

optimal timing weight x(s) and the regression (2). We have rp = x(s) rB, then substituting

from (2) and (3) and using E(rB|s) = μB + bB [σf2/(σf2 + σv2)] (f+v), we obtain equation (1),

where ap = λμB2/σB2, bp = (λμBbB/σB2) [1 + σf2/(σf2+σv2)] and Λp = (λ/σB2)bB2

[σf2/(σf2+σv2)]. The error term upt+1 in the regression is a linear function of uB, v, vuB, fuB

and vf. The assumptions of the model imply that the regression error is well specified,

with E(up f) = 0 = E(up) = E(up f2).

The simple model illustrates that timing ability implies convexity between the

fund's return and a systematic factor that is linearly related to the benchmark return. Note

that if λ>0 the coefficient Λp ≥ 0, independent of the sign of bB. Also, if the manager does

not receive an informative signal, then Λp=0, as E(rB|s) and x(s) are constants in that case.

However, the simple model does not consider nonlinearity in the fund's return that may

arise for reasons unrelated to timing ability. In order to obtain reliable measures of timing

ability, it is necessary to extend the model to control for these other potential sources of

nonlinearity.

2.2 Nonlinearity Unrelated to Timing

Convexity or concavity is likely to arise when the underlying assets held by a fund

bear a nonlinear relation to the common factors. While this may occur in equity funds,

such nonlinearities are likely in bond funds. Even simple bond returns are nonlinearly

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related to interest rate changes, and securities such as mortgage products are described in

terms of their "negative convexity" with respect to interest rate factors.

A second potential cause of nonlinearities is "interim trading," as described by

Goetzmann, Ingersoll and Ivkovic (2000), Ferson and Khang (2002) and Ferson, Henry and

Kisgen (2006). This refers to a situation where fund managers trade more frequently than

the fund's returns are measured. With monthly fund return data, interim trading

definitely occurs. Derivatives may often be replicated by dynamic trading, so the use of

derivatives is a closely related issue. For example, a fund that holds call options bears a

convex relation to the underlying asset, and that can be confused with market timing

ability (Jagannathan and Korajczyk, 1986).5

A third reason for nonlinearity, unrelated to timing ability, arises if there is public

information about future asset returns, and if managers' portfolios respond to this public

information. As shown by Ferson and Schadt (1996), even if the conditional relation is

linear, the response to public information can induce nonlinearities in the unconditional

relation between the fund and a benchmark return. In this case the portfolio weights at

the beginning of the period are correlated with returns over the subsequent period, due to

their common dependence on the public information, and the products of the weights and

the returns are nonlinear. Nonlinearity from public information trading can occur even if

there is no interim trading problem.

The fourth potential reason for nonlinearity unrelated to timing ability is stale

pricing of a fund's underlying assets. Thin or nonsynchronous trading in a portfolio has

5 In another example, Brown et al. (2004) explore arguments that incentives and behavioral biases can induce managers to engage in option-like trading within performance measurement periods, even if they have no superior information. The fund's investors are "short" any option-like compensation provided to fund managers, and the fund's measured return, which is net of manager compensation, could bear a nonlinear relation to the benchmark.

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long been known to bias beta estimates for the portfolio (e.g., Scholes and Williams, 1977).

We show that if the degree of nonsynchronous trading is related to a common factor, this

"systematic" stale pricing can create additional spurious concavity or convexity. The

following subsections describe our approaches to controlling for these issues.

2.3 Addressing Nonlinearity in Benchmark Assets

We generalize the classical market timing model to handle nonlinearity in the

relation between the benchmark asset returns and the common factors. To do this we

replace the linear regression (2) with a nonlinear regression:

rBt+1 = aB + bB(ft+1) + uBt+1, (4)

where bB(f) is a nonlinear function, specified below, and we assume that uB and bB(f) are

normal and independent. The manager's market-timing signal is now assumed to be s =

bB(f) + v, where v is normal independent noise with variance, σv2. This captures the idea

that the manager focusses on the return implications of information about the common

factor.

With these modifications the optimal weight function in (3) obtains with:

E(rB|s) = μB [σv2/(σf*2 + σv2)] + [σf*2/(σf*2 + σv2)][aB + bB(f) + v], where σf*2=Var(bB(f)).

Substituting as before we derive the nonlinear regression for the portfolio return:

rpt+1 = ap + bp [bB(ft+1)] + Λp [bB(ft+1)]2 + ut+1, (5)

with: ap = λ aB(μB σv2 + aB σf*2)/[σB2(σf*2+σv2)],

bp = λ(μB σv2 + 2aB σf*2)/[σB2(σf*2+σv2)],

and Λp = (λ/σB2)[σf*2/(σf*2+σv2)].

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The model restricts the coefficients in the nonlinear regression of the fund's return on the

factors. The intuition is that nonlinearity of the benchmark return partly determines the

nonlinearity of the fund's return. If there is no market-timing signal, then x(s) is a

constant, Λp=0 in (5) and the nonlinearity of the fund's return mirrors that of the

benchmark. A successful market timer has a convex relation, relative to the benchmark,

and thus Λp>0. For example, if the benchmark is convex in the factor, a market-timing

fund is more convex than the benchmark. We combine equations (4) and (5), and estimate

the model by the Generalized Method of Moments (Hansen, 1982).

One of the forms for bB(f) that we examine is a quadratic function. This has an

interesting interpretation in terms of systematic co-skewness. Asset-pricing models

featuring systematic coskewness with a market factor are studied, for example, by Kraus

and Litzenberger (1976). Equation (1) is, in fact, equivalent to the quadratic "characteristic

line" used by Kraus and Litzenberger, when the factor is the market portfolio excess

return. The coefficient on the squared market factor measures the systematic coskewness

risk of the portfolio, and this may have nothing to do with market timing. Thus, a fund's

return can bear a convex relation to the factor because it holds assets with positive

coskewness risk. Equation (5) allows the benchmark to have coskewness risk when bB(f)

is quadratic. When Λp=0 and the manager has no timing ability, the fund inherits its

coskewness from the benchmark. If the fund has timing ability, Λp>0 measures the

additional convexity generated by the timing ability.

2.4 Addressing Interim Trading

Interim trading means that fund managers trade more frequently than the fund's

returns are measured. This can lead to spurious inferences about market timing ability, as

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shown by Goetzmann, Ingersoll and Ivkovic (2000) and Jiang, Yao and Yu (2005), and can

also lead to incorrect inferences about total performance, as shown by Ferson and Khang

(2002). Ferson, Henry and Kisgen (2006) propose a solution, starting with a continuous-

time asset pricing model. The continuous-time model prices all portfolio strategies that

may trade within the period, provided that the portfolio weights are nonanticipating

functions of the state variables in the model. If the continuous-time model can price

derivatives by replication with dynamic strategies, the use of derivatives is also covered

by this approach.

Ferson, Henry and Kisgen derive the stochastic discount factor (SDF) for a set of

popular term structure models, that applies to discrete-period returns and allows for

interim trading during the period from time t to time t+16:

tmt+1 = exp(a - Art+1 + b'Axt+1 + c'[xt+1 - xt]), (6)

where x is the vector of state variables in the model. The terms Axt+1 = Σi=1,...1/Δ x(t+(i-1)Δ)Δ

approximate the integrated levels of the state variables. The measurement period

between t and t+1 is one month, matching observed fund returns, and the period is

divided into pieces of length Δ=one trading day. Art+1 is the similarly time-averaged level

of the short-term interest rate. The empirical "factors," ft+1, in the SDF thus include the

discrete monthly changes in the state variables, their time averages and the time-averaged

short term interest rate: ft+1 = {[xt+1-xt], Axt+1, Art+1}.

With the approximation ef ≈ 1+f, we can consider a model in which the SDF is

linear in the expanded set of empirical factors. Using a linear function for the SDF is

6 A stochastic discount factor is a random variable, tmt+1, that "prices" assets through the equation Et{tmt+1 Rt+1}=1.

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equivalent to using a beta pricing model for expected returns, written in terms of the

relevant factors (Dybvig and Ingersoll (1982), Ferson, 1995). This motivates including the

time-averaged variables in regressions like (5), as additional factors to control for interim

trading effects.7

2.5 Addressing Public Information

A nonlinear relation between a fund's return and common factors may arise if

managers respond to public information and there is time variation in factor risk

premiums. Such time variation may be even more likely for bond returns than for stock

returns. Conditional timing models have been designed to control for these effects by

allowing portfolio weights and funds' betas to vary over time with public information.

Using equity funds, Ferson and Schadt (1996) and Becker, et al. (1999) find that such

conditional timing models are better specified than models that do not control for public

information. In particular, Ferson and Schadt (1996) propose a conditional version of the

market timing model of Treynor and Mazuy (1966):8

rpt+1 = ap + bp rBt+1 + Cp'(Zt rBt+1) + Λp rBt+12 + ut+1. (7)

7 Equation (6) also motivates the use of an exponential function for bB(f) in Equation (4). Under an exponential function the squared term in a market timing model like (5) is perfectly correlated with the linear term, so we don't estimate this case separately. An exponential function for bB(f) is closely approximated by a linear function when the factor changes are small numbers.

8 Ferson and Schadt (1996) also derive a conditional version of the market timing model of Merton and Henriksson (1981), which views successful market timing as analogous to producing cheap call options. This model is considerably more complex than the conditional Treynor-Mazuy model, and they find that it produces similar results.

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In Equation (7), the new term Cp'(Zt rBt+1) controls for nonlinearity due to the

public information, Zt. The expected benchmark return and the manager's optimal

portfolio weights both depend on Zt. This induces common time-variation in the

benchmark return and the fund's "beta" on the benchmark. The term C'(ZtrBt+1) captures

this common variation, controlling for the "spurious" timing effect due to public

information.

In a regression like (7) we measure timing with respect to the benchmark return, rB.

However, we are also interested in timing with respect to interest rates and other

economic factors that may bear a nonlinear relation to the benchmark, as described above.

We study timing ability for these factors by replacing rB in Equation (7) with the nonlinear

function bB(f) and estimating the equations (4) and (7) together in a system.

2.6 Addressing Stale Prices

Thin or nonsynchronous trading in a portfolio biases estimates of the portfolio beta

(e.g., Scholes and Williams, 1977). A similar effect occurs when the measured value of a

fund reflects stale prices (e.g. Getmansky, Lo and Makarov, 2004). If the extent of stale

pricing is related to a common factor, we call it "systematic" stale pricing. Systematic stale

pricing can create spurious concavity or convexity and thus a spurious impression of

market timing ability.

To address the stale pricing issue we use a simple model in the spirit of Getmansky,

Lo and Makarov. Let pt be the natural log of the fund's "true" net asset value per share

with the last period's dividends reinvested, so that rt = pt - pt-1 is the true return. The true

return would be the observed return if no prices were stale. We assume rt is independent

over time with mean μ. The measured price, pt* is given by δt pt-1 + (1-δt) pt, where the

coefficient δt ε [0,1] measures the extent of stale pricing during the month t. The

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measured return on a fund, rt*, is then given by:

rt* = δt-1rt-1 + (1-δt)rt, (8)

Assume that the market or factor return rmt, with mean μm and variance σm2, is iid over

time and measured without error.9

To model stale pricing that may be systematic, consider a regression of the extent of

stale pricing at time t on the market factor:

δt = δ0 + δ1(rmt - μm) + εt, (9)

where we assume that εt is independent of the other variables in the model. We are

interested in moments of the true return, like Cov(rt,rmt2), which measures the timing

ability. Straightforward calculations relate the moments of the observable variables to the

moments of the unobserved variables, as follows.

E(r*) = μ (10a)

Cov(rt*,rmt) = Cov(r,rm)(1- δ0 + 2δ1μm) - δ1 {μ σm2 + Cov(r,rm2)} (10b)

Cov(rt*,rmt2)= Cov(r,rm2)(1- δ0 + δ1μm) - δ1{ μE(rm3) + Cov(r,rm3) - [Cov(r,rm)+μμm](μm2+σm2)} (10c)

Cov(rt*,rmt) + Cov(rt*,rmt-1) = Cov(r,rm) (10d)

Cov(rt*,rmt2) + Cov(rt*,rmt-12) = Cov(r,rm2) (10e)

Cov(rt*,rmt3) + Cov(rt*,rmt-13) = Cov(r,rm3) (10f)

9 We have considered models in which the benchmark return rm also has stale pricing, and find that the number of parameters proliferates to the point of intractability. Thus, our estimates of this model should in practice capture fund staleness relative to any staleness in the benchmark return.

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Equation (10a) shows that stale prices will not affect the measured average returns

in this model, even if stale pricing is systematic.10 Equation (10b) captures the bias in the

measured covariance with the market. The market beta, of course, is this covariance

divided by the market variance. If δ1=0 stale pricing is not systematic, and the effect on

the measured beta reduces to an attenuation bias as in the model of Scholes and Williams

(1977). If δ1 is not zero stale pricing is systematic, and the measured beta is affected by

both stale pricing and the true timing ability. Equation (10c) shows that the measured

timing coefficient is also biased. If the stale pricing is not systematic, however, the bias

reduces to a scale factor. If δ1=0 we can therefore test the null hypothesis of no timing

ability using the measured covariance, as the true timing coefficient is zero if and only if

the measured covariance with the squared market factor is zero (assuming that δ0≠1).

When stale pricing is systematic the bias of the timing coefficient is complex as it depends

on the true beta, the value of δ1, higher moments and other parameters. We need to

control for this complicated bias in order to measure the timing ability.

Fortunately, equation (10e) reveals a simple way to control for a biased market

timing coefficient due to stale prices. The sum of the covariances of the measured return

with the squared market factor and the lagged squared factor equals the true covariance

or timing coefficient. This is similar to the bias correction for betas in the models of

Scholes and Williams (1977) and Dimson (1979).

If we append moment conditions to the system (10) to identify μm, σm2 and E(rm3)

we have a system of nine equations that allow us to identify the nine parameters, {μm, σm2,

10 Qian (2005) studies the effects of stale prices on the measured average performance of equity style funds, and finds that if staleness is correlated with new money flows there can be effects on average returns. See Zietowitz (2003) and Edelin (1999) for estimates of the size of potential losses to buy-and-hold investors, attributed to other investors trading on stale net asset values.

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Cov(r,rm), Cov(r,rm2), E(rm3), Cov(r,rm3), δ0, δ1, and μ}. We estimate the system by the

generalized method of moments (GMM, Hansen, 1982). We measure the timing ability,

adjusted for the effects of stale pricing, via the parameter Cov(r,rm2).

2.7 Combining the Effects

We have models for four different sources of convexity or concavity unrelated to

timing ability, and we may wish to control for all of these effects. The models for interim

trading and public information suggest control variables that may be included in the

market timing regressions, while the nonlinear-benchmark models imply a system of

equations. The general form of the models is a system including Equation (4) and

Equation (11):

rpt = a + β'Xt + Λp [bB(ft)2] + upt, (11)

where Xt is a vector of control variables observed at t or before, and which includes bB(ft),

the benchmark's nonlinear projection on the factor. In the presence of stale pricing we

observe rpt* and not the true return, rpt. However, according to Equation (10e) we can still

estimate the timing coefficient easily if we do not require estimates of the deeper

structural parameters of the model. Simply replacing bB(ft)2 in (11) with the sum, bB(ft)2 +

bB(ft-1)2, delivers a coefficient that should be proportional to Λp.

3. The Data

We first describe our sample of bond funds. We then describe the interest rate and other

economic data that we use to construct the common factors relative to which we study

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timing ability. Finally, we describe the fund style-related benchmark returns.

3.1 Bond Funds

The fund data are from the Center for Research in Security Prices (CRSP) mutual

fund data base, and include returns for the period from January of 1962 through

December of 2002. We select open-end funds whose stated objectives indicate that they

are bond funds.11 We exclude money market funds and municipal securities funds.

There are a total of 23,281 fund-year records in our initial sample. In order to

address back-fill bias we remove the first year of returns for new funds, and any returns

prior to the year of fund organization, a total of 1345 records. Data may be reported prior

to the year of fund organization, for example, if a fund is incubated before it is made

publicly available (see Elton, Gruber and Blake (2001) and Evans, 2004). Extremely small

funds are more likely to be subject to back-fill bias. We delete cases where the reported

total net assets of the fund is less than $5 million. This removes 3205 records. We delete

all cases where the reported equity holdings at the end of the previous year exceeds 10%.

This removes another 644 records. After these screens we are left with 18,087 fund-years.

The number of funds with some monthly return data in a given year is five at the

beginning of 1962, rises to 17 at the end of 1973, to 617 by 1993 and to 1756 at the end of

11 Prior to 1990 we consider funds whose POLICY code is B&P, Bonds, Flex, GS or I-S or whose OBJ codes are I, I-S, I-G-S, I-S-G, S, S-G-I or S-I. We screen out funds during this period that have holdings in bonds plus cash less than 70% at the end of the previous year. In 1990 and 1991 only the three digit OBJ codes are available. We take funds whose OBJ is CBD, CHY, GOV, MTG or IFL. If the OBJ code is other than GOV, we delete those funds with holdings in bonds plus cash totalling less than 70%. After 1991 we select funds whose OBJ is CBD, CHY, GOV, MTG, or IFL or whose ICDI_OBJ is BQ, BY, GM or GS, or whose SI_OBJ is BGG, BGN, BGS, CGN, CHQ, CHY, CIM, CMQ, CPR, CSI, CSM, GBS, GGN, GIM, GMA, GMB, GSM or IMX. From this group we delete 116 fund years for which the POLICY code is IG or CS.

17

the sample in December of 2002.

We group the funds into equally-weighted portfolios according to fund style. We

define seven styles on the basis of the various objective codes on the CRSP database. The

styles are Global, Short-term, Government, Mortgage, Corporate, High Yield and Other.12

Summary statistics for the style-grouped funds are reported in Panel A of Table 1. The

mean returns are all between 0.31 and 0.56% per month. The standard deviations of

return range between 0.47% per month, for Short-term Bond funds, to 1.66% per month

for High-yield funds. The first-order autocorrelations of the returns range from 8%, for

Global Bond funds, to 30% for High Yield funds. The minimum return across all of the

style groups in any month is -6.7%, suffered in October of 1979 by the corporate bond

funds. The maximum return is 10.3%, also earned by the Corporate bond funds in

November of 1981.

Table 1 also reports the second order autocorrelations of the fund returns. The

stylized thin trading model assumes that thin trading is resolved (all the assets trade)

within two months. This implies that the measured returns have an MA(1) time-series

structure, and the second order autocorrelations should be zero. The largest second order

autocorrelation in the panel is 7.1%, with an approximate standard error of 1/√T = 1/√144

= 8.3%. For the portfolio of all funds, where the number of observations is the greatest,

the second order autocorrelation is -2.2%, with an approximate standard error of 1/√492 =

4.5%. Thus, none of the second order autocorrelations is significantly different from zero,

12 Global funds are coded SI_OBJ=BGG or BGN. Short-term funds are coded SI_OBJ=CSM, CPR, BGS, GMA, GMS or GBS. Government funds are coded OBJ=GS POLICY=GOV, ICDI_OBJ=GS or GB, or SI_OBJ=GIM or GGN. Mortgage funds are coded ICDI_OBJ=GM, OBJ=MTG or SI_OBJ=GMB. Corporate funds are coded as OBJ=CBD, ICDI_OBJ=BQ, POLICY=B&P or SI_OBJ=CHQ, CIM, CGN or CMQ. High Yield funds are coded as ICDI_OBJ=BY, SI_OBJ=CHY or OBJ=CHY or OBJ=I-G and Policy=Bonds. Other funds are defined as funds that we classify as bond funds (see the previous footnote), but which meet none of the above criteria.

18

consistent with the assumptions of the model.

We also group the funds into equally-weighted portfolios according to various

fund characteristics, measured at the end of the previous year. Since the characteristics

are likely to be associated with fund style, we form groups within each of the style

classifications. The characteristics include fund age, total net assets, percentage cash

holdings, percentage of holdings in options, reported income yield, turnover, load

charges, expense ratios, the average maturity of the funds' holdings, and the lagged return

for the previous year. The Appendix provides the details.

3.2 Common Factor Data

We use daily and weekly data to construct monthly empirical factors, including the

discrete changes measured from the ends of the months and the time-averaged values

used as controls for interim trading bias. Most of the data are from the Federal Reserve

(FRED) and the Center for Research in Security Prices (CRSP) databases. The daily

interest rates are from the H.15 release. The factors reflect the term structure of interest

rates, credit and liquidity spreads, exchange rates, a mortgage spread and two equity

market factors.

Three factors represent the term structure of Treasury yields: A short-term interest

rate, a measure of the term slope and a measure of the curvature of the yield curve.

Studies such as Litterman, Sheinkman and Weiss (1991) find that three factors explain

most of the variance of Treasury bond returns across the maturity spectrum, and that

level, slope and curvature provide a convenient way to represent three factors.

The short-term interest rate is the three-month Treasury rate, converted from the

bank discount quote to a continuously compounded yield. Chapman, Pearson and Long

(2001) argue that the three month Treasury rate is a good empirical proxy for the

19

theoretical short-term interest rate. The slope of the term structure is the ten-year yield

less the one-year yield. The curvature measure is the concavity: y3 - (y7 + 2y1)/3, where yj

is the j-year fixed-maturity yield.

Since some of our funds hold corporate bonds subject to default risk and mortgage

backed securities subject to prepayment risks, we construct associated factors. Our credit

spread series is the yield of Baa corporate bonds minus Aaa bonds, from the FRED. This

series is measured as the weekly averages of daily yields. We use the averages of the

weeks in the month for our time-averaged version of the spread. For the discrete changes

in the spread we use the first differences of the last weekly values for the adjacent months.

The first difference series may not be as clean as with daily data, but we are limited by the

data available to us. Our mortgage spread is the difference between the average contract

rate on new conventional mortgages, also available weekly from the FRED, and the yield

on a three-year, fixed maturity Treasury bond. Here we use daily data on the Treasury

bond and weekly data on the mortgage yield to construct the time averages and discrete

changes.

Liquidity is likely to be a relevant factor for bond funds. Liquidity varies both

across issues and over time. For example, on-the-run Treasury issues are more liquid than

off-the-run issues. Also, credit markets have periodic episodes where liquidity gets tight

or loose. For market timing we are interested in market-wide fluctuations in liquidity.

Our measure follows Gatev and Strahan (2005), who advocate a spread of commercial

paper over Treasury yields as a measure of short term liquidity in the corporate credit

markets. We use the yield difference between three-month nonfinancial corporate

commercial paper rates and the three month Treasury yield. The commercial paper rates

are measured weekly, as the averages over business days, and we convert them from the

bank discount basis to continuously compounded yields.

20

Some of the funds in our sample are global bond funds, so we include a factor for

currency risks. Our measure is the value of the US dollar, relative to a trade-weighted

average of major trading partners, from the FRED. This index is measured weekly, as the

averages of daily figures, and we treat it the same way we treat the credit spread data.

Corporate bond funds, and high-yield funds in particular, may be exposed to

equity-related factors. We therefore include two equity market factors in our analysis. We

measure equity volatility with the VIX-OEX index implied volatility. This series is

available starting in January of 1986. We also include an equity market valuation factor,

measured as the log dividend ratio for the CRSP value-weighted index. The dividends are

the sum of the dividends over the past twelve months, and the value is the cum-dividend

value of the index. The level of this ratio is a state variable for valuation levels in the

equity market, and its monthly first difference is used as a factor. The first difference of

this series approximates the (negative of the) continuously-compounded capital gains

component of the index return.

Table 2 presents summary statistics for the monthly common factor series starting

in February of 1962 or later, depending on data availability, and ending in December 2002.

Missing values are excluded and the units are percent per year (except for the dollar

index). Panel A presents the levels of the variables, panel B the monthly first differences

and Panel C presents the time-averaged values.

The average term structure slope was positive, at just under 80 basis points during

the sample period. The average credit spread was about one percent, but varied between

32 basis points to more than 2.8%. The average mortgage spread over Treasuries was just

over 2% for the period starting in 1971, and varied between about -1% to more than 7%.

The liquidity spread averaged 4.7%, with a range between 1.3% and 6.8% over the 1997-

2002 period.

21

In their levels the factors are highly persistent time series, as indicated by the first

order autocorrelation coefficients. Six of the nine are in excess of 90%. The time averaged

factors are also highly persistent, with autocorrelations greater than 84%. Moving to first

differences, the series look more like innovations.13

3.3 Style Index Returns

We construct style-specific benchmarks for the funds from a set of asset-class

returns, using a methodology similar to Sharpe (1992). The problem is to combine the

asset class returns, Ri, using a set of portfolio weights, {wi}, so as to minimize the "tracking

error" between the return of the fund (or fund style group), Rp, and the style-matched

benchmark portfolio, ΣiwiRi. The portfolio weights are required to sum to 1.0 and must be

non-negative, which rules out short positions:

Min{wi} Var[Rp - Σi wiRi], (13)

subject to: Σi wi = 1, wi ≥ 0 for all i,

where Var[.] denotes the variance. We solve the problem numerically, deriving a set of

weights for each fund style group. The asset class returns include US Treasury bonds of

three maturity ranges from CRSP (less than 12 months, less than 48 months and greater

than 120 months), the Lehman Composite Global bond index, the Lehman US Aggregate

13 Given the relatively high persistence and the fact that some of the factors have been studied before exposes us to the risk of spurious regression compounded with data mining, as studied by Ferson, Sarkissian and Simin (2003). However, Ferson, Sarkissian and Simin (2005) find that biases from these effects are largely confined to the coefficients on the persistent regressors, while the coefficients on variables with low persistence are well behaved. Thus, the slopes on our persistent control variables and thus the intercepts may be biased, but the timing coefficients should be well behaved.

22

Mortgage Backed Securities index, the Merrill Lynch High Yield US Master index and the

Lehman Aaa Corporate bond index.14

Panel B of Table 1 presents the style-index weights for each of the style portfolios.

The weights reported in the table do not sum exactly to 1.0 due to rounding errors, but we

carry many more digits of precision in the return calculations. The weights present

sensible patterns in most cases, suggesting that both the style classification of the funds

and Sharpe's procedure are reasonably valid. The global funds load most heavily on

corporate bonds and global bonds, then on low quality bonds. Short term funds have

most of their weight in bonds with less than 48 months to maturity. Mortgage funds have

their greatest weights on mortgage backed securities and shorter term bonds. Corporate

funds have more than 50% of their weight in high or low grade corporate bonds. High

yield funds have 70% of their weight in low grade corporate bonds. Government funds

load highly on mortgage-backed securities (33%). This is consistent with the portfolio

weights described by Comer (2006), who finds that government style bond funds allocate

about 1/3 of their portfolios to mortgage related securities.

4. Empirical Results

We first examine the empirical relations between the various factors, the fund groups and

passive investment strategies proxied by the Sharpe style benchmarks. We estimate the

model of stale pricing and evaluate the effects of the other controls for nonlinearities

unrelated to timing. Finally, we apply the adjusted timing measures to individual funds.

14 We splice the Blume, Keim and Patel (1991) low grade bond index prior to 1991, with the Lehman High Yield index after that date. We splice the Ibbotson Associates 20 year government bond return series for 1962-1971, with the CRSP greater than 120 month government bond return after 1971.

23

4.1 Factor Models

We begin the empirical analysis with regressions of the Sharpe style benchmark

returns on changes and squared changes in the common factors, looking for convexity or

concavity in these relations. Any nonlinearity would appear in a naive application of the

timing regression (1) for funds, if funds simply held the benchmarks. These regressions

are shown in Panel A of Table 3. Panel B of Table 3 shows regressions for the mutual

funds' returns, grouped by the same style categories. Comparing the funds with the style

benchmarks provides a feel for the affects of active management. The table shows the

coefficients on the squared terms for the regressions, run one factor at a time.

The coefficients for the style benchmarks in Panel A of Table 3 present some

interesting patterns. Out of 63 cases (seven styles x 9 factors) there are only five

heteroskedasticity-consistent t-ratios on the squared factors with absolute values larger

than two. However, 21 of the absolute t-ratios are between 1.6 and 2.0, which provides

some evidence of nonlinearity. Most of the coefficients on the squared factors are positive

-- only eight of the 63 are less than zero. This means that we would expect to measure

weak convexity or timing ability, based on regression (1), if funds simply held the

benchmarks.

The coefficients for the mutual funds in Panel B of Table 3 tell a different story. We

find 13 absolute t-ratios larger than 2.0. This is more than expected by chance; thus, the

mutual fund returns are significantly nonlinearity related to the factors.15 Furthermore,

the coefficients on the squared factors are negative in about half the cases; whereas, most

15 Using a simple binomial model assuming independence, the t-ratio associated with finding 13 "rejections," when the probability of observing a rejection is 5%, is (13/63 - .05)/(.05(.95)/63)0.5 = 5.69.

24

of the coefficients were positive for the benchmarks. Thus, overall the fund returns

appear more concave in relation to the factors than the benchmark returns. This suggests

poor market timing ability on the part of the mutual funds. Alternatively, the concavity

could reflect derivatives or dynamic trading, public information or stale prices.

The right-hand column of Table 3 presents the R-squares of the factor model

regressions, using all of the factors simultaneously. The fractions of the fund return

variances explained by the factors are interesting because large fractions suggest that the

ability to time common factors could represent a large portion of funds' potential

performance. In the first row of R-squares the changes in the factors are the independent

variables. In the second row the squared factor changes are also included in the

regression.

The R-squares for the funds in Panel B are between 33% and 62%, using all the

factors, and exceed 45% for five of the seven styles. The squared factors contribute more

to the explanatory power of the regressions for the funds than they do for the benchmark

returns (Panel A). The adjusted R-squares rise by almost 20% for global bond funds, 8.6%

for short term funds and 3.5% for government funds when the squared factor changes are

included in the regressions. This makes sense if trading by fund managers leads to

nonlinearity in the funds' returns, relative to the benchmarks.

Panel A of Table 3 reports the adjusted R-squares for the benchmark returns, which

range from 28% to 72%. Including the squared terms typically reduces the adjusted R-

squares a little. The R-squares are somewhat higher if we include the lagged values of the

state variables in the regressions to capture expected return variation over time.

4.2 Stale pricing

This section summarizes the results of estimating the stale pricing model, system

25

(10), for the excess returns of the funds grouped by style. The Sharpe benchmark excess

return for the fund style group serves as rm. We also estimate the model for the excess

returns of the characteristics-based fund groups for each style. Table 4 shows the

estimated coefficients in Equation (9), the model for systematic stale pricing, along with

the estimate of the timing coefficient adjusted for stale pricing effects. We scale the timing

coefficients similar to a regression coefficient, reporting the estimates of Cov(r,rm2)/E{rm3}.

The right-hand column of the table reports a Wald test of the hypothesis that there is no

stale pricing effect: δ0=δ1=0.

The δ0 coefficients, which measure the average level of stale pricing, are positive

and between 0.137 and 0.271. One interpretation of the model is that, on average, the

reported prices for 13-27% of the securities held by the funds are stale at the end of the

month. The government style funds are the least stale by this measure and the corporate

bond funds are the most stale. However, the standard errors of these estimates are large,

and none are significantly different from zero.

The δ1 coefficients measure the sensitivity of stale pricing to the return of the

benchmark index. The δ1 coefficients are not statistically significant. The hypothesis that

δ1=0 says that stale pricing is not systematic, and implies that we can test for timing ability

using the measured covariance with the squared benchmark return. The Wald test does

reject the hypothesis that there are no stale pricing effects (δ0=δ1=0) for five of the seven

styles. Examining 147 cases, including funds grouped by characteristics within each style,

we find p-values of the Wald test less than 5% in 79 of the cases. Thus, the results leave

open the possibility that stale pricing might effect the estimates of timing ability.

The timing coefficients should measure timing ability purged of the stale pricing

effects. The standard errors of these coefficients are also large, and none are significantly

different from zero. However, all of the values in the table are positive, which suggests

26

positive timing ability after adjusting for stale prices. Estimating the model on the

characteristics-grouped fund portfolios, we find that 111 of the 147 timing coefficients are

positive. Thus, controlling for stale pricing alone shifts the evidence in favor of positive

timing ability.

4.3 Controls for Nonlinearity

Table 5 summarizes the impact of the various treatments for non-timing-related

nonlinearity in funds' returns. We evaluate the marginal impacts by comparing a full

model with all of the treatments with the model that removes one of the treatments. We

evaluate each type of control as a separate treatment. Our objective is to assess the

empirical importance of the various treatments.

The model combines equations (4) and (11) in a system. To control for interim

trading we include the time-averaged variables Ar and Ax in the regressions. To control

for public information we include the lagged level of the factor as Zt, multiplied by the

change in the factor as suggested by Equation (7). For thin trading we use the sum of the

squared factor changes and the lagged squared changes in place of the squared factor

changes. To control for nonlinearity in the benchmarks we consider three specifications

for the function bB(f): linear, quadratic and exponential. As described earlier, a quadratic

function can be motivated by co-skewness, and an exponential function can be motivated

by a model with interim trading or derivatives.

There are many ways to evaluate the importance of a treatment in regression

models. Ultimately we are interested in the magnitude of the timing coefficients. We

choose a simple measure of the impact of a treatment on the estimated coefficient. This

allows us to compare treatments in a consistent way, even where the models are not

nested and/or the degrees-of-freedom does not change across treatments. We form a "t-

27

ratio" to compare the timing coefficient in the full model with all the treatments, or Λf, to

the model without the treatment in question, or Λ0:

t = (Λf - Λ0)/|σ(Λf) - σ(Λ0)|, (14)

where σ(Λ) is the consistent standard error of the estimator. The denominator of the t-

ratio uses an approximation, assuming that the two estimates are extremely highly

correlated (ρ=1). The estimators will be very highly correlated in most of the cases, and

assuming the correlation is 1.0 is conservative in the sense of leading us to include more

treatments in the final models.16

The first result relates to nonlinearity in the benchmark returns. As described

above, when bB(f) is exponential the squared term, bB(f)2, that captures the timing ability is

essentially collinear with bB(f). Furthermore, ef≈1+f when f is numerically small so the

linear function well approximates the exponential. Thus, there is no empirically

measurable impact of an exponential function, compared with a linear function, in the

reduced-form model.

The second result relates to the impact of nonlinearity in the benchmark returns on

the timing coefficients, as captured by a quadratic function bB(f), compared with a linear

function. The squared factor now changes appear in the full model, even without the

market timing term. The timing coefficient is identified by a combination of the third and

fourth power of the factor changes in the quadratic bB(f) model. These terms are never

significant in the regression and they are very imprecisely estimated. As a result, all of the

timing coefficients have huge standard errors and the t-ratios estimating the impact on the

16 With a lower correlation the standard error of the difference between the estimates is larger, the t-ratio is smaller and we would include fewer treatments.

28

timing coefficients are small. It appears to be unreliable to use higher order products than

the square of the factor changes to identify the model parameters. We therefore leave out

the terms of higher order than squared in the subsequent analysis. The rest of the analysis

uses the linear bB(f) model as the "full" model and the importance of the treatments for

nonlinearity in the fund returns is evaluated relative to a linear bB(f) model for the

benchmark returns.

Table 5 summarizes the marginal importance of the treatments for nonlinearity in

the fund returns, as measured by the t-ratios. Panel A sorts the results by fund style and

Panel B sorts by the common factors, which enter the models one at a time. The table

reports the fractions of the cases where the squared t-ratios exceed the 5% critical value for

a chi-squared variable with one degree of freedom. In Panel A there are 180 possible cases

for each style (9 factors times 20 characteristics-sorted portfolios of funds), and in Panel B

there are 140 possible cases for each factor. Cases where all the required variables share

fewer than 36 monthly observations are excluded.

In Panel A of Table 5 the fraction of the cases where the treatments for stale pricing,

derivatives/dynamic trading, or pubic information generate significant changes varies

from less than one percent to 17% across the fund style groups. Many of the entries are

close to the 5% expected, under the null that the timing coefficient does not change. Using

a binomial approximation, the standard errors of the fractions in Table 5 are about 2%.

We find nine examples where the fractions exceed 9%, roughly statistically significant, and

each treatment produces at least two examples with fractions greater than 9%. Thus, even

though the model does not admit an important role for nonlinearity in the benchmark

returns, the treatments for nonlinearity in fund returns all seem to impact the timing

coefficients. This impression is consistent with the story of the factor model regressions,

where there was stronger nonlinearity indicated in the funds than in the benchmark

29

returns.

Panel B of Table 5, sorting by factor, shows a much greater range of outcomes. This

says that the impact of the treatments on the measured timing ability -- which does not

vary much across the style groups according to Panel A -- varies more across the factors.

Since the total variation in the estimated timing coefficients in the full model is fixed

across the panels, this says that there is relatively small variation in the coefficients due to

style effects for a given factor, and large factor effects for a given style. This makes sense,

given that the characteristics-grouped portfolios within a given style show relatively

limited heterogeneity in their returns (see the Appendix), while the effects of the different

factors on the coefficients is likely to be very different.

None of the treatments has much affect on the timing coefficients for the mortgage,

dividend-price ratio or liquidity factors, but the coefficients for the other factors are

affected by at least one of the treatments. The fractions of the cases where a treatment

significantly changes the timing coefficient varies from zero to 73% across the factors, and

eleven of 27 entries in the panel are greater than 9%. Every treatment produces fractions

larger than 9% for at least three factors. Overall, the evidence suggests that the treatments

for non-timing-related nonlinearity in fund returns has a potentially significant impact on

the results.

4.4 Timing Coefficients

The point estimates of the timing coefficients are imprecisely estimated in most of

the models, and less than 5% of the cases yield absolute t-ratios for a zero coefficient larger

than 2.0. We examine the fractions of the timing coefficient estimates that are positive or

negative. In the full model with all of the treatments, most of the fractions are near 50%

when sorting by fund style. Overall, just under 50% of the timing coefficients are positive.

30

Thus, the overall patterns in the point estimates are consistent with the presence of

unbiased estimators and the null hypothesis of no timing ability. However, in some slices

of the data high concentrations of positive and negative timing coefficients are found.

The short term and high yield fund portfolios produce more negative coefficients

(74% and 60%, respectively, of the characteristics and factor combinations) than the other

styles. The catch-all "other" bond fund group delivers the largest number of positive

timing coefficients. Both patterns are found in the full model and in most of the models

which remove one of the treatments.17 The frequency of positive and negative timing

coefficients differ from 50% by highly significant fractions, based on an approximate

binomial standard error of about 4%, once we group the cases by fund style or by factor.

So, the evidence suggests there may be pockets of timing ability, and there may be cross-

sectional variation in ability that the model can measure.

Sorting by factor, the fractions of positive timing coefficients range from zero to

85% across the factors. Large proportions of positive coefficients are found for the

convexity, credit and dollar factors (each producing 76-85% positive), suggesting

widespread timing ability for these factors. Frequently-negative timing coefficients are

found for the interest rate level, mortgage and dividend yield factors, suggesting "negative

timing," with respect to these factors. In summary, we find evidence of positive timing

ability for some factors, including credit spreads and the relative value of the dollar.

There is also evidence of "negative" timing ability for some factors, including interest rates,

mortgage spreads and bond market liquidity. Negative timing is similar to what previous

studies have found for equity funds relative to the stock market. However, unlike the case

of equity funds, conditional models (e.g. Ferson and Schadt, 1996) do not explain the

17 The single exception is the stale price treatment for high yield funds, where removing the treatment results in 40% negative coefficients.

31

nonlinearity.

4.5 Cross-sectional Analysis

We study fund level timing coefficients and performance to understand the relation

of the timing coefficients to fund characteristics, and to see if the timing measures predict

future performance. If a fund with a larger timing coefficient earns higher excess returns

it suggests that funds use timing ability to enhance returns and pass some of the spoils on

to investors. If the timing coefficients can predict subsequent performance, they may be

useful in investor decision making. If we find a negative relation it suggests that funds

pay for convexity at the cost of lower average returns. This is consistent with a model in

which the coskewness captured by the timing coefficient is priced in the market.

For the analysis at the fund level we identify and combine cases where funds report

multiple share classes. Multiple classes are identified when two ICDI codes for the same

year have a common fund name and a different share class code on the CRSP database.

There are combined under the first ICDI code, value-weighting the share classes using

their total net assets to measure the values. We combine both the monthly returns and the

annual characteristics of multiple share classes in the same way.

We first study the cross-sectional relations between the timing coefficients and the

funds' characteristics. Stay tuned.

5. Concluding Remarks

If common factors explain a large part of the variance of a typical bond return, a

large fraction of the potential performance of fixed income funds should be attributable to

timing the common factors. Few studies have explored the market timing ability of fixed

income funds, and we find that the problem of measuring this ability is a subtle one.

32

Traditional models of market timing measure convexity in the relation between the fund's

return and the common factors. However, convexity or concavity is likely to arise for

reasons unrelated to timing ability, especially so in fixed income funds. We adapt classical

market timing models to fixed income funds by controlling for other sources of

nonlinearity, such as the use of dynamic trading strategies or derivatives, portfolio

strategies that respond to publicly available information, nonlinearity in the benchmark

assets and systematically stale prices.

We find evidence of positive timing ability for some factors, including credit

spreads and the relative value of the dollar. There is also evidence of "negative" timing

ability for some factors, including interest rates, mortgage spreads and bond market

liquidity. Negative timing is similar to what previous studies have found for equity funds

relative to the stock market. However, unlike the case of equity funds, conditional models

(e.g. Ferson and Schadt, 1996) do not explain the nonlinearity. A future draft will study

the cross sectional relations at the fund level.

Appendix: Funds Grouped by Characteristics

The fund characteristics include age, total net assets, percentage cash holdings, percentage

of holdings in options, reported income yield, turnover, load charges, expense ratios, the

average maturity of the funds' holdings, and the lagged return for the previous year. Each

year we sort the funds of a given style with nonmissing characteristic data from high to

low on the basis of the previous year's value of a characteristic and break them into thirds.

We form equally weighted portfolio returns from the funds in the high group and the low

group for each month of the next year.

We plot in Figure 1 the end-of-year time-series of the cutoff values that define the

upper and lower thirds of the distributions for selected characteristics. For the purposes

33

of these figures only, we define the cutoffs without regard to fund style, pooling all of the

funds. (The results for the styles are similar except as noted.)

Figure 1 shows that bond funds have experienced some trends that are different

from equity funds. The graph of the fund age breakpoints shows that the age distribution

was the youngest during 1992-1994, then grew older. Equity style funds present a

different picture, growing younger with the large number of new equity funds starting in

the mid-1980s (e.g. Ferson and Qian, 2004).

The turnover distribution for our funds has remained relatively stable, with the

lower and upper-third cutoffs rising from 61% and 133%, respectively, in 1992 to 75% and

176%, respectively, in 2001. (Turnover among Global Bond Funds has trended downward

since 1993.) Equity fund turnover at the end of the sample period is at similar levels, but

has increased dramatically since the 1960's, according to Ferson and Qian.

Figure 1 shows that bond funds have similar patterns with respect to their fees and

expenses as are found for equity funds. Total load fees have declined, with the upper

cutoff starting the period at more than 8% and ending at 3.75%, while the lower-third

cutoff went from 4.15% to zero. Expense ratios show an upward trend over much of the

sample, with the upper-third cutoff finishing the period at 1% per year, and the lower-

third cutoff rising from 0.3% to 0.8% over the sample.18 No funds report option holdings

greater than zero before 1994, and the upper cutoff value stays below 5.5% to the end of

the sample. The High-yield funds tend to report more options holdings than the other

styles. The cutoffs for average maturity and cash holdings remain relatively stable over

the sample period. Exceptions include the Short-term Bond funds, and Corporate Bond

18 The exceptions here are Global Bond funds, where expenses are tightly clustered at just under 1% per year since 1993; High-yield Bond funds, where expenses and turnover have been stable since 1992; and Short-term bond funds, where expense ratios have slightly declined since 1992.

34

funds where cash holdings have trended downwards since 1992. The average maturities

of the Short-term Bond funds and Global Bond funds peaked in 1998 and ended the

period slightly higher than they started in 1992.

Income yield breakpoints for the funds start in the 4% to 5% range in 1962 and rise

to a peak during 1980-1984, where the upper-third cutoffs are in the 11% to 14% range. By

the end of the sample the yield cutoffs are back down to the 5% to 6% range. Finally, the

lagged annual returns show more random variation over time than the other

characteristics, highlighting the good years for fixed income in 1967, 1975-76, 1985, 1991

and 1995 and low-return years in 1969 and 1994.

We examine summary statistics for the returns of the characteristics-grouped

portfolio returns, separately for each of the style classifications (these are available by

request). The average return differences between the high and low-characteristics groups

are never more than two standard deviations of the mean apart, the maximum difference

being less than 15 basis points per month. However, the low-expense group offers higher

average returns than the high expense group for six of the seven styles. This should not be

surprising, as the returns are measured net of funds' expenses and trading costs. Elton,

Gruber and Blake (1993) and Blake, Elton and Gruber (1995) emphasize the importance of

expenses for fixed income funds' average risk-adjusted returns, and Ferson, Henry and

Kisgen (2006) find that expenses are related to funds' conditional alphas.

References Admati, Anat, Sudipto Bhattacharya, Paul Pfleiderer and Stephen A. Ross, 1986, On timing and selectivity, Journal of Finance 61, 715-732. Aragon, George, 2005, Timing multiple markets: Theory and evidence from balanced mutual funds, working paper, Arizona State.

35

Becker, C., W. Ferson, D. Myers and M. Schill, 1999, Conditional Market timing with Benchmark investors, Journal of Financial Economics 52, 119-148. Blake, Christopher R., Edwin J. Elton and Martin J. Gruber, 1993, The performance of bond mutual funds, Journal of Business 66, 371-403. Blume, Marshall, D. Keim and J. Patel, 1991, Returns and volatility of low-grade bonds: 1977:1989, Journal of Finance 46, 49-74. Brown, Stephen, David R. Gallagher, Ono W. Steenbeck and Peter L. Swan, 2004, Double or nothing: Patterns of equity fund holdings and transactions, working paper, New York University. Brown, David T. and William J. Marshall, 2001, Assessing fixed income manager style and performance from historical returns, Journal of Fixed Income 10 (March). Chapman, David A., John B. Long and Neil D. Pearson, 2001, Using proxies for the short rate: When are three months like an instant? Journal of Finance Comer, George, 2006, Evaluating Bond Fund Sector Timing Skill, working paper, Georgetown University. Comer, George, Vaneesha Boney and Lynne Kelley, 2005, High quality bond funds: Market timing ability and performance, working paper, Georgetown University. Cornell, Bradford and Kevin Green, 1991, The investment performance of low-grade bond funds, Journal of Finance 46, 29-48. Dimson, Elroy, 1979, Dybvig, Phillip H. and Jonathan Ingersoll, 1982, Mean variance theory in complete markets, Journal of Business 55, 233-251. Edelen, Roger, 1999, Investor flows and the assessed performance of open-ended mutual funds, Journal of Financial Economics 53, 439-466. Elton, Edwin, Martin J. Gruber and Christopher R. Blake, 1995, Fundamental Economic Variables, Expected returns and bond fund performance, Journal of Finance 50, 1229-1256. Elton, Edwin, Martin J. Gruber and Christopher R. Blake, 2001, A first look at the Accuracy of the CRSP and Morningstar Mutual fund databases, Journal of Finance 56, 2415-2430. Evans, Richard, 2005, Mutual fund incubation: The market for fund returns, working paper, Boston College.

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Farnsworth, Heber K., Wayne Ferson, David Jackson and Steven Todd, Performance Evaluation with Stochastic Discount Factors, 2002, Journal of Business 75, 473-504. Ferson, Wayne E., 1995, Theory and Empirical Testing of Asset Pricing Models, Chapter 5 in Finance, Handbooks in Operations Research and Management Science, by Jarrow, Maksimovic and Ziemba (editors), Elsevier, 145-200. Ferson, W., Tyler Henry and Darren Kisgen, 2006, Evaluating Goverment Bond Funds using Stochastic Discount Factors, Review of Financial Studies (forthcoming). Ferson, W. and Kenneth Khang, 2002, Conditional performance measurement using portfolio weights: Evidence for pension funds, Journal of Financial Economics 65, 249-282. Ferson, Wayne E. and Meijun Qian, 2004, Conditional Performance Evaluation Revisited, Research Foundation Monograph of the CFA Institute ISBN 0-943205-69-7, 84 pages. Ferson, W. and Rudi Schadt, 1996, Measuring fund strategy and performance in changing economic conditions, Journal of Finance 51, 425-462. Ferson, Sarkissian and Simin, 2003, Spurious regressions in Financial Economics? Journal of Finance 58, 1393-1414. Ferson, Sarkissian and Simin, 2005, Is stock return predictability spurious? Journal of Investment Management vol. 1, no. 3, 10-19. Ferson, W. and Vincent A. Warther, 1996, Evaluating Fund Performance in a Dynamic Market," Financial Analysts Journal 52, no. 6, pp.20-28. Gatev, Evan and Phillip Strahan, 2004, Banks' advantage in supplying liquidity: Theory and evidence from the commercial paper market, Journal of Finance (forthcoming). Getmansky, Mila, Andrew W. Lo and Igor Makarov, 2004, An econometric model of serial correlation and illiquidity in hedge fund returns, Journal of Financial Economics 74, 529-610. Goetzmann, W., Ingersoll J., Ivkovic Z., 2000, Monthly measurement of daily timers, Journal of Financial and Quantitative Analysis 35, 257-290. Grinblatt, M., Titman, S., 1989. Mutual fund performance: an analysis of quarterly portfolio holdings. Journal of Business 62, 393-416.

37

Hansen, Lars P., 1982, Large sample properties of the generalized method of moments estimators, Econometrica 50, 1029-1054. Jagannathan, R., Korajczyk, R., 1986. Assessing the market timing performance of managed portfolios. Journal of Business 59, 217-236. Jensen, M. C., 1968, "The Performance of Mutual Funds in the period 1945-1964," Journal of Finance, 23, 389-416. Jiang, George T., Tong Yao and Tong Yu, 2005, Do mutual funds time the market? Evidence from portfolio holdings, working paper, University of Arizona. Jiang, Wei, 2003, A nonparametric test of market timing, Journal of Empirical Finance 10, 399-425. Krau, Alan and Robert Litzenberger, 1976, Skewness preference and the valuation of risky assets, Journal of Finance 31, 1085-1100. Litterman, R. and J. Sheinkman, 1991, Common factors affecting bond returns, Journal of Fixed Income 1, Malone, Matthew, 2002, Risk manipulation in bond mutual fund tournaments, unpublished Ph.D dissertation, Boston University. Merton, Robert C. and Roy D. Henriksson. 1981. "On market timing and investment performance II: Statistical procedures for evaluating forecasting skills." Journal of Business, 54: 513-534. Qian, Meijun, 2005, Stale prices and the performance evaluation of mutual funds, working paper, Boston College. Scholes, Myron and Joseph Williams, 1977, Estimating betas from nonsynchronous data, Journal of Finance 5, 309-327. Sharpe, William. F., 1992, Asset Allocation: Management style and performance measurement, Journal of Portfolio Management 18, 7-19. Treynor, Jack, and Kay Mazuy, 1966, "Can Mutual funds Outguess the Market?" Harvard Business Review, 44, 131-136. Zitzewitz, Eric, 2003, Who cares about shareholders? Arbitrage-proofing mutual funds, Journal of Law, Economics and Organizations 19, 225-266.

38

Table 1

Mutual Fund Monthly Returns: Summary Statistics. The sample periods for the fund returns are January of the year indicated under Begin through December of 2002. Nobs is the number of time series observations, Begno is the number of funds at the start of the sample period and Endno is the number in December of 2002. The returns are percent per month. Mean is the sample mean, std is the sample standard deviation, ρ1 is the first order sample autocorrelation and ρ2 is the second order autocorrelation. Panel B presents the portfolio weights of Sharpe style benchmarks associated with each group of funds. The benchmarks are formed from the returns to US Treasury bonds with less that or equal to 12 (le12) 48 (le48) months to maturity, greater than 120 months to maturity (gt120), a high-grade corporate bond index (cb), a low-grade corporate bond index (junk), a global bond index (global) and a mortgage-backed securities index (mort). The weights are estimated using all available months from 1976 through 2002. Panel A: Equally-weighted Portfolios of Mutual Funds Style Begin nobs Begno Endno Mean Min Max Std ρ1 ρ2 All 1962 492 5 1720 0.530 -5.019 9.357 1.338 0.245 -0.022 Global 1994 108 357 224 0.318 -4.988 3.531 1.144 0.080 0.063 Short Term 1991 120 434 442 0.386 -1.387 1.650 0.469 0.200 0.009 Government 1986 204 2 812 0.462 -3.000 3.000 0.914 0.152 -0.067 Mortgage 1991 144 13 954 0.426 -1.168 2.010 0.612 0.153 0.071 Corporate 1962 492 4 822 0.540 -6.667 10.33 1.525 0.198 -0.026 High Yield 1991 144 47 489 0.462 -5.064 6.593 1.659 0.304 0.068 Other 1964 408 7 591 0.558 -4.765 9.240 1.586 0.193 -0.066 Panel B: Sharpe Style Weights Funds Weights assigned to benchmarks: le12 le48 gt120 cb junk global mort All 0.0 0.14 0.02 0.32 0.31 0.03 0.18 Global 0.00 0.00 0.00 0.48 0.17 0.35 0.00 Short Term 0.27 0.49 0.03 0.00 0.09 0.03 0.09 Government 0.00 0.16 0.09 0.25 0.07 0.09 0.33 Mortgage 0.00 0.37 0.03 0.19 0.02 0.03 0.37 Corporate 0.00 0.20 0.06 0.39 0.24 0.00 0.10 High Yield 0.08 0.15 0.00 0.00 0.70 0.00 0.07 Other 0.06 0.00 0.00 0.40 0.29 0.01 0.24

39

Table 2

Summary Statistics for the common factor data. The sample periods begin as indicated under Starts (yyyymm), and all series end in December of 2002. Nobs is the number of time series observations, excluding missing values. The units are percent per year. Mean is the sample mean, std is the sample standard deviation and ρ1 is the first order sample autocorrelation of the series. Panel A: Levels of the Factors Factor Starts Nobs mean min max std ρ1 short rate 196202 491 6.145 1.220 16.27 2.726 0.974 term slope 196202 490 0.783 -3.160 3.310 1.086 0.948 curvature 196907 402 0.190 -1.097 0.773 0.294 0.900 credit spread 196202 491 1.001 0.320 2.820 0.439 0.961 mortgage spread 197105 380 2.163 -1.049 7.106 0.980 0.780 liquidity spread 199701 72 4.711 1.332 6.810 1.699 0.961 US dollar 197101 384 1.101 0.845 1.677 0.157 0.985 Equity Values 196312 469 3.474 2.815 7.265 0.408 0.871 Equity Volatility 198601 204 21.55 9.820 61.41 7.495 0.784 Panel B: First differences of the Factors short rate 196203 490 -0.0032 -4.114 2.580 0.559 0.130 term slope 196203 488 0.0040 -1.460 2.800 0.344 0.132 curvature 196908 401 -0.0001 -0.447 0.770 0.131 -0.083 credit spread 196203 490 0.0012 -0.550 0.680 0.121 0.066 mortgage spread 197106 379 -0.0034 -2.970 2.672 0.356 0.266 liquidity spread 199702 71 -0.0580 -0.970 0.594 0.237 0.353 US dollar 197102 383 -7.5E-5 -0.081 0.087 0.024 0.155 Equity values 196401 468 -0.0072 -1.909 0.164 0.109 0.180 Equity volatility 198602 203 0.0613 -15.38 39.03 4.882 -0.172 Panel C: Time-averaged Factors short rate 196202 491 6.160 1.211 17.12 2.761 0.977 term slope 196202 491 0.778 -3.032 3.273 1.076 0.959 curvature 196907 402 -0.186 -0.778 0.824 0.288 0.932 credit spread 196202 491 0.999 0.315 2.648 0.434 0.969 mortgage spread 197104 381 2.140 0.061 5.428 0.786 0.857 liquidity spread 199701 72 4.730 1.334 6.780 1.682 0.962 US dollar 197101 384 1.101 0.860 1.692 0.158 0.987 Equity Values 196312 469 3.478 2.764 7.416 0.423 0.881 Equity Volatility 198601 204 21.50 9.790 58.22 7.438 0.841

40

Table 3

Regressions of fixed income funds and benchmark index returns on changes in factors and their squares. The sample starts in February of 1962 or later, depending on the factor, and ends in December of 2002. The coefficients on the squared factor changes are shown on the first line and their consistent standard errors are on the second line. The factors are measured in monthly percentage units. R2 is the adjusted coefficient of determination. The R2 in the first row refers to a regression that includes the changes in all nine factors but no squared changes. In the second row the regression includes the squared factor changes.

Style short slope curve credit mort. liquid exchange eq.value eq. vol. R2(%) group Panel A: Style Benchmarks Global 0.124 0.796 12.6 10.9 0.044 -1.60 1.22E-05 0.017 0.00231 57.4 0.212 0.453 8.00 6.25 0.124 1.19 9.91E-05 1.14 0.00112 53.5 Short Term 0.099 0.446 9.67 7.81 0.029 -0.350 3.40E-05 0.160 0.00120 65.9 0.0986 0.238 4.38 3.58 0.076 0.581 5.41E-05 0.551 0.00059 62.9 Government 0.137 0.842 13.5 11.0 0.037 -0.994 5.36E-05 -0.021 0.00200 67.5 0.188 0.449 7.88 6.01 0.122 0.834 9.12E-05 1.01 0.00101 63.7 Mortgage 0.125 0.711 13.3 10.5 0.030 -0.398 5.57E-05 0.014 0.00182 71.8 0.162 0.400 7.10 5.45 0.112 0.682 8.24E-05 0.872 0.00083 70.0 Corporate 0.119 0.753 12.3 10.6 0.038 -1.54 4.69E-05 0.225 0.00113 47.2 0.191 0.425 7.42 5.79 0.115 1.56 9.76E-05 0.982 0.00133 39.0 High Yield 0.150 0.608 9.68 9.89 0.129 -2.68 7.93E-06 0.832 -0.00030 27.6 0.207 0.400 7.02 6.11 0.092 4.15 0.000136 1.01 0.00211 21.5 Other 0.156 0.803 11.8 11.1 0.078 -1.51 4.43E-05 0.311 0.00093 36.8 0.210 0.433 7.71 6.10 0.108 1.87 0.000103 1.04 0.00142 28.6 Panel B: Funds by Style Global -2.64 -0.12 -13.9 -4.69 -2.23 -1.50 3.28E-05 -30.7 -0.0118 33.7 1.39 1.88 7.39 4.77 1.35 1.37 0.000497 47.1 0.0025 53.6 Short Term -0.460 0.16 -7.03 -2.95 -0.888 -0.185 8.81E-05 -7.98 -0.0012 45.5 0.294 1.11 4.99 1.78 0.299 0.403 0.000110 23.5 0.00065 54.1 Government -0.403 1.19 -7.91 -5.14 -0.718 -1.13 0.000147 -6.11 0.00144 58.9 0.520 1.29 4.67 2.85 0.348 0.791 6.54E-05 14.4 0.00100 62.4 Mortgage -0.828 0.057 -6.40 -3.34 -1.28 -0.260 0.000101 43.3 -0.00120 61.8 0.362 1.51 2.85 2.31 0.357 0.454 5.04E-05 18.8 0.00130 59.9 Corporate 0.102 0.589 8.18 4.80 0.041 -0.880 -4.07E-05 -0.341 0.00016 59.9 0.166 0.290 5.72 4.96 0.099 0.652 7.12E-05 0.796 0.00097 58.1 High Yield -3.57 -1.43 -3.27 28.4 -1.09 -1.81 0.000374 -95.7 -0.0158 35.1 4.07 4.32 14.1 7.39 1.54 4.34 0.000167 90.2 0.0062 34.4 Other 0.100 0.511 7.91 8.00 0.048 -1.61 2.27E-05 0.345 9.9E-05 54.5 0.132 0.266 4.63 4.31 0.076 2.14 7.70E-05 0.795 0.00160 54.6

41 Table 4 Estimates of Market Timing Coefficients and Systematic Stale Pricing

The sample starts in February of 1962 or later, depending on the fund group, and ends in December of 2002. The coefficients on the squared benchmark excess returns, Cov(r,rm2), are estimated by the Generalized Method of Moments applied to the system (10). The parameters δ0 and δ1 are the means and slope coefficients, respectively, of the regression model for the extent of stale pricing, regressed on the excess return of the style benchmark index, rm. Consistent standard errors are on the second line. The Wald test examines the hypothesis that δ0=δ1=0. The right-tail p-value of the Wald test is shown in the far right column. Fund group δ0 δ1 Cov(r,rm2) Wald Test Nobs E(rm3) (p-value) Global 0.2004 2.998 0.8028 108.0 6.680 6.789 3.607 0.0000 Short Term 0.1845 3.522 0.2751 120.0 13.40 13.43 1.687 0.0007 Government 0.1370 4.134 0.3835 204.0 21.45 21.29 3.146 0.0000 Mortgage 0.1458 -3.062 0.04682 144.0 113.5 113.0 2.384 0.0026 Corporate 0.2711 -0.3547 0.6717 490.0 15.15 15.47 11.93 0.9952 High Yield 0.2287 1.891 0.04053 144.0 41.28 41.20 64.37 0.0346 Other 0.2539 -0.0533 0.5643 406.0 8.483 8.880 5.893 0.9703

42 Table 5

The effects of controls for fund return nonlinearity on timing coefficient estimates. The table records the percentage of cases in which the treatment has a statistically significant effect at the 5% level. In panel A there are between 111 and 180 cases for each fund style and in Panel B there are between 6 and 136 of 160 possible cases for each factor. A case requires at least 36 monthly observations on each variable in the full model and the model without the indicated treatment. The models are estimated on monthly data for 1962-2002, with 490 or fewer observations depending on the fund group and factor combination. Treatment stale interim public prices trading information Panel A: By fund style groups Global 0.833 14.17 10.00 Short Term 6.667 2.500 12.50 Government 6.667 7.500 7.500 Mortgage 10.83 3.333 4.167 Corporate 8.904 2.740 10.96 High Yield 10.83 8.333 10.83 Other 1.802 17.12 13.51 Panel B: By Factor Short rate 15.67 5.970 5.224 Term slope 2.206 16.91 5.147 Curvature 0.000 40.00 30.00 Credit Spread 0.000 15.00 40.00 Mortgage Spread 0.735 5.882 25.00 Liquidity Spread 0.000 0.000 1.504 US Dollar 0.000 10.29 5.882 Equity Values 0.000 0.000 0.000 Equity Volatility 24.26 1.471 9.559

43

44

45

46

47

48

49

50