The European Journal of Finance Vol. 12, No. 6–7, 605–626, September–October 2006
Using Irregularly Spaced Returns to Estimate Multi-factor Models: Application to Brazilian Equity Data ÁLVARO VEIGA∗ & LEONARDO R. SOUZA∗∗ ∗ Department
of Electrical Engineering, Catholic University of Rio de Janeiro, Brazil, ∗∗ Department of Mathematics, Catholic University of Rio de Janeiro, Brazil
ABSTRACT Multi-factor models are useful tools to explain cross-sectional covariance in equities returns. In this paper a new estimation method is proposed that makes use of irregularly spaced returns and an empirical example is provided with the 389 most liquid equities in the Brazilian Market. The market index shows itself capable of explaining equity returns while the US$/Brazilian real exchange rate and the Brazilian short interest rate do not. The example shows the usefulness of the estimation method in further using the model to fill in missing values and to provide interval forecasts. KEY WORDS: Multi-factor model, missing data, irregularly spaced returns
1.
Introduction
Emerging markets frequently suffer from low liquidity and tend to concentrate most transactions on a few liquid assets.1 For instance, the Brazilian equity market comprises about 1190 different stocks but almost 40% have not been traded in the past year and 32% of the remainder have been traded less than once a month. As a consequence, for many stocks there are no prices for a large proportion of the trading days. Still worse, traditional missing value imputation methodologies do not take into account prices of transactions occurring on a given day but not in previous and subsequent days. As they work with daily returns, they discard information on prices of these ‘isolated’ days since the computation of daily returns requires the existence of transactions on two consecutive days. Financial institutions, however, need these prices and returns everyday in order to fulfil regulatory requirements and implement their methodologies of quantitative analysis of risk and return.2 Moreover, given some statistical properties of the data, widespread accurate pricing of non-traded equities diminishes arbitrage opportunities. A market model can provide the expected values of missing prices and returns. Sharpe (1964) proposed the Capital Asset Pricing Model (CAPM) to explain asset returns. However, a number of papers have provided empirical evidence against the CAPM (for example, Bhandari, 1988; and Chan, Hamao and Lakonishok, 1991). Furthermore
Correspondence Address: Álvaro Veiga, Department of Electrical Engineering, PUC Rio Rua Marquˆes de S˜ao Vicente, 225, Rio de Janeiro, Brazil. Email:
[email protected] 1351-847X Print/1466-4364 Online/06/06–70605–22 © 2006 Taylor & Francis DOI: 10.1080/13518470600763489
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Fama and French (1992) and Jegadeesh (1992) show that the market beta has little power in explaining cross-sectional asset returns, meaning that some additional common ‘factor’ could better explain the returns. Ross (1976) proposed the Arbitrage Pricing Theory (APT), which allowed more than one factor to explain the assets returns and consequently diversify risk premia. However, in all these studies the factors are non-observable. Reinganum (1981) and Mei (1993) use an autoregressive approach to explain the hidden factors. Chen, Roll and Ross (1986) introduced macroeconomic variables to explain monthly stock returns in a multi-factor linear regression. However, this approach is limited by the availability of macroeconomic data, in which many of the variables have an observation frequency no higher than once a month, while financial institutions need daily estimates and forecasts. Even so, it is common practice in these institutions to use a similar model to explain asset returns, where the most used risk factors are Market indexes, foreign exchange rates and those related to interest rates. Then a circular problem may arise: the Market model must be estimated before computing the expected value of missing prices and returns, but many numerical problems may arise and bias the estimation if the proportion of missing data is high. A biased estimation will lead to poor price filling. Also, since the market model uses returns as the dependent variable, instead of prices, the proportion of missing values will be greater than the proportion of missing prices. For instance, note that a stock that is negotiated every second day will generate no daily return at all. In this paper, to circumvent the problem mentioned above, we propose an estimation method for the multi-factor model, which makes use of irregularly spaced returns, enabling the use of every historical price available and thus increasing efficiency. Furthermore, we allow for the weights assigned to the data to decrease exponentially with age, so that the distant past does not significantly influence the estimates of the ever-changing market. Other authors have tried different approaches to estimate ‘betas’ for infrequently traded assets. Scholes and Williams (1977), Dimson (1979), and Fowler and Rorke (1983) propose lagged regressions to have their OLS estimates combined in some way. Brooks et al. (2002a, 2002b) correct the beta OLS estimates for bias and inconsistency using a sample selectivity model. None of them, however, seem to have used irregular returns. Marsh (1979) proposes an estimation method using irregular returns to help testing some market hypothesis. His approach is similar to ours, but he does not take into account the age of the observations. Dimson and Marsh (1983) use the same approach as Marsh (1979), but they study the stability of the estimates across non-overlapping periods. Consider a practical illustration: there are equities with as low as 5% of transaction days, for which the multi-factor model can be used to predict both the missing prices and the τ -steps-ahead interval forecast for returns, conditioned on the risk factors (that must be taken into account exogenously). We work with τ = 1, 3, 5, 10; having as motivation to use up to three steps ahead the fact that the S˜ao Paulo Stock Exchange Market (Bovespa) liquidity system (represented by its clearinghouse, CBLC) determines that parties involved in a transaction have three days to liquidate it; and five and ten days ahead to satisfy the needs of the financial institutions, corresponding roughly to one and two weeks. To evaluate the estimation method, we ran a back test with the daily closing prices of 389 stocks over the period ranging from 2 January 1995 to 28 February 2001. Two prime aspects of the estimated model were examined, both having practical implications for financial institutions. First, we examined its ability to produce good predictions of missing prices. This is done by comparing each observed price with the predicted value that would be produced by the model if this particular price was missing. Second, the τ -steps-ahead interval forecasts were evaluated by comparing the nominal and observed frequencies of values in the 5% tails, which is an important measure of risk (Jorion, 1997). For comparison purposes, the same experiment was run on other commonly used imputation methodologies, such as repeating
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the last price (naïf method) or mimicking the market index return, as well as using the multifactor model estimated with only the available daily returns. All these are nested under the same multi-factor class of models, possibly with time-varying coefficients. We conclude that the multifactor model estimated using irregular returns globally outperforms all the others, including the traditional regular returns estimation. Furthermore, the São Paulo Stock Exchange Market Index (IBV) shows itself to globally explain equity returns but the US Dollar/Brazilian Real exchange rate and the Brazilian inter-bank overnight interest rate (CDI) do not, influencing only a few specific equities. The next section presents the notation used throughout this paper and Section 3 describes the methodology proposed here. Section 4 shows an empirical example with Brazilian equity data. Section 5 offers some concluding remarks. 2.
Basic Framework
In this section we describe the notation and the methodology used to estimate the multi-factor j model making use of irregularly spaced returns. The price of equity j at time t is denoted by Pt and the (log-)return of the same equity at time t by j Pt j j j = ln(Pt ) − ln(Pt−1 ) Rt = ln (1) j Pt−1 where j = 1, . . . , J (J = 389 in the experiment) and t = 1, . . . , T . In our formulation, using logreturns is necessary to preserve the additivity of daily returns over periods of τ days (τ integer and greater than one). 2.1 Multi-factor Model The usual multi-factor model with p factors describes a linear relation between the return of equity j and some market indicators, taking the form j
j
j
Rt = (βt )T Xt + εt j
j
j
j
(2)
where βt ∈ B ⊂ p+1 , βt = [β0t , . . . , βpt ], is the vector of coefficients of equity j at time t, Xt = j [1, X1 , . . . , Xp ] is the vector of market indicators (possibly log-returns) at time t and εt errors j 2 supposedly iid, εt ∼ N (0, σj ). We allow the coefficients to vary in time to let the relationship between returns and indicators be dynamic. However, the study of this variation is beyond the scope of the paper. To deal with this variation in a simple fashion, we estimate all models within the exponentially weighted moving average (EWMA) framework (J.P. Morgan, 1995), which is briefly explained in Section 2.3. Given there is no missing value in the estimation window, the estimation is obtained via weighted least squares, with the weights exponentially decreasing with the age of the data. The missing value of equity j occurs when there is no trade of the equity j at time (day) t. Its imputation at time t based on this model uses the returns expected values given the current (daily) market indicators at time t. The VaR (value at risk, which ‘measures the worst expected loss over a given time interval under normal market conditions at a given confidence level’, Jorion, 1997, p. xiii) prediction τ days-ahead, however, requires forecasts (possibly density forecasts) of the market indexes.
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2.2 Irregularly Spaced Returns Emergent markets contain a number of illiquid equities which are not negotiated every day, unlike the market indicators (the risk factors), which are available every business day. For these equities, the estimation window is scattered with missing values. We assume in this paper that the data are missing at random (MAR), whether or not this randomness is endogenous or exogenously implied by information3 arrivals that are randomly distributed across time. The latter would incorporate microstructure effects as in Easley et al. (1996), for example, in the error term. There would be, thus, some fraction of information that is common to all stocks, which might be explained by the risk factors, and other fraction of information concerning only a single stock or a small group of stocks, which would be modelled primarily in the error term. Other studies (e.g. Engle and Russell, 1998) model the duration (time between subsequent trades) as an autoregressive process and find empirical evidence of transaction clustering. These results, however, refer to high-frequency data (therefore liquid equities), whereas our paper refers to daily data. Lo and MacKinlay (1990) treat the same problem as here. However, instead of considering the returns of nontraded equities as missing, as we do here, they consider them to be zero and accumulate all information in the return associated to the next observed price. This induces a spurious negative autocorrelation in their ‘observed’ data, also found in empirical applications by Lo and MacKinlay (1988) and Atchison et al. (1987). In our estimation procedure we use returns over the irregular periods between transactions that are explicitly observed. This means that the information is spread over the nontransaction period, avoiding, by construction, the spurious autocorrelation. It is straightfoward to show that if the observed irregular returns are used, the same calculations in Lo and MacKinlay (1990) will yield zero correlations. Let us consider the estimation window of a market model for a specific equity. Suppose the risk factors are observable every trading day, but there are days in which the equity was not traded and thus no price was observed. Figure 1 illustrates how missing data determines the use of irregular returns. Define the τ days return at time t for equity j as j
j
j
Rt,τ ≡ ln Pt − ln Pt−τ
(3)
Moreover, setting the right-hand sides of equations (1) and (2) equal and rearranging yields: j
j
j
j
ln Pt = ln Pt−1 + (βt )T Xt + εt
Figure 1. Estimation window with missing data and irregular returns
(4)
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Iterating (4) for τ periods and summing up gives: j
j
ln Pt = ln Pt−τ +
τ −1
j
(βt−i )T Xt−i +
i=0
τ −1
j
εt−i
(5)
i=0
j
2 For practical purposes, βt and σj,t are considered constant on the interval [t − τ , t]4 between two subsequent transactions. This does not imply any assumption on the time constancy of the coefficients. Otherwise, the EWMA estimation scheme we propose in the next section considers j a random walk dynamics for the βt . Using (3), (5) may be rewritten as j
j
Rt,τ = (βt )T
τ −1
j
Xt−i + wt,τ
(6)
i=0
where j
wt,τ =
τ −1
j
εt−i ;
j
wt,τ ∼ N (0, τ σj2 ).
i=0
τ −1
The sum i=0 Xt−i in (6) represents the cumulative value of the factors during the period with no transaction. In the case of the present paper, only log-returns are considered as factors so that the cumulative value represents the τ -days log-return. 2.3 Exponentially Weighted Moving Average (EWMA) The EWMA framework has been widely used in practice since the RiskMetrics (J.P. Morgan, 1995) methodology was proposed. Its motivation is that the process generating the data may change smoothly through time. So, the older the data, the less weight it must have attached to it. These weights decrease exponentially with time, according to the smoothing parameter λ, 0 < λ < 1. To day t ∗ is thus given weight λ times the weight given to the day t ∗ + 1. The parameter K, in turn, determines the number of effective days to be used in the estimation window. The mean of equity j returns is then estimated at time t by j µˆ t
=
K−1
j
λi (1 − λ)Rt−i
(7)
i=0
The covariance between returns of equities i and j , as well as the variance of equity i are estimated at time t by K−1 j j 2 i σ ˆ = C λm (1 − λ)(Rt−m Rt−m ) − µˆ it µˆ t ij,t m=0 (8) K−1 2 i λm (1 − λ)(Rt−m )2 − (µˆ it )2 σˆ i,t = C m=0
where C is such that the sum of weights Cλm (1 − λ) is the unity (C → 1 as K → ∞). We do not take into account the loss of a degree of freedom in calculating the mean, but this has little effect since we use K = 252 (corresponding to one year of data). It is usual to consider the mean of
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returns as being zero, further simplifying these equations. However, this assumption is not made here. An overview on EWMA is given in Alexander (1996). Muth (1960) was the first to show that the EWMA procedure gives the same estimates as the j Kalman filter when the mean µt follows a random walk.
3.
Methodology
3.1 Estimation with Irregular Returns Some equities have trading prices for all days in the estimation window. In these cases, (2) is considered and a weighted least squares procedure, with EWMA weights, yields the estimates of j βt for each equity j at the current time t. This is done through the equation: j βˆt = (X T WX)−1 X T WRj
(9)
where j
j
j
R j = RKx1 = [Rt−K+1
···
Rt−K+2
j
Rt−1
j
R t ]T
is the vector of returns of equity j ; 1 X1,t−K+1 (Xt−K+1 )T 1 X1,t−K+2 .. . = = . .. (Xt−1 )T .. . 1 X1,t (Xt )T
X = XKx(p+1)
· · · Xp,t−K+1 · · · Xp,t−K+2 .. . ···
Xp,t
is the design matrix (the same for all equities); and W = WKxK
0
0 .. .
λK−2
0
···
=
λK−1
··· 0 .. . .. . 0 0 1
is the weights matrix5 . In this case, p market indicators are considered. Note that the lines of R j , X and W correspond to one day return. However, if there is a missing value in the estimation window, a line will correspond to τ days return since (6), instead of (2), is considered. Note that t is the first day, after t − τ , where the asset considered is traded. The redefined matrices R j , X and W , where all lines corresponding to a same τ days return collapse
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into one single line, are as follows: j
X = XKj x(p+1)
τ1 τ2 = . . . τKj
j
τ 1 −1
X1,t−i1 −m
m=0 τ 2 −1
j
···
Xp,t−i1 −m τ −1 2 ··· Xp,t−i2 −m m=0 .. . τKj −1 ··· Xp,t−iKj −m
X1,t−i2 −m .. .
τKj −1
X1,t−iKj −m
m=0
W = WKj xKj
τ 1 −1 m=0
m=0
;
j R i−i2 ,τ2 = .. . j Rt−iK ,τK
R j = RKj x1
j
Rt−i1 ,τ1
m=0
i1
λ τ1 0 = .. . 0
0
···
λ i2 τ2
..
..
.
..
···
0
. .
0 .. . 0 λiKj
(10)
τK j
where there are Kj + 1 prices (Kj possibly irregularly spaced returns) available in the estimation window for equity j ; τk , k = 1, 2, . . . , Kj , is the time interval where the kth return, of equity j , is built; ik is the time between the kth return of equity j and the current time t (in other words, the age of the return, measured in days), so that ik = ik+1 + τk+1 , ∀k ≥ 1. So, in addition, to account for exponentially decaying weights, W assigns weights inversely proportional to the variances of j the errors wt,τ 6 . Li (2003) shows that this procedure yields the same estimates as the Kalman filter in the case where the β’s are modelled as a random walk. 3.2 Missing Values Imputation In the more general case, the imputation of missing values can occur at any point of the estimation window, although the results with real data shown here are only from imputation at j the end of it (current day). For the sake of simplicity, consider pt = ln Pt . Following (5), the j expected value of pt , conditional on βt , the matrix of market indicators X˜ t+τ2 = X˜ t+τ2 (p+1,t+τ2 ) = [X1 , X2 , . . . , Xt , . . . , Xt+τ2 ], and the next existing price (forward or backward respectively for
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equations 11a and 11b) is given by E(pt /pt−τ1 , βt , X˜ t+τ2 ) = pt−τ1 + (βt )T j
j
τ 1 −1
Xt−i
(11a)
Xt+τ2 −i
(11b)
i=0
and j j E(pt /pt−τ2 , βt , X˜ t+τ2 ) = pt+τ2 − (βt )T
τ 2 −1 i=0
with respective variances:
Var(pt /pt−τ1 , βt , X˜ t+τ2 ) = E j
τ 1 −1
2 εt−i = τ1 σj2
(12a)
i=0
and j Var(pt /pt+τ2 , βt , X˜ t+τ2 ) = τ2 σj2
(12b)
j j j Using (11a) and (11b), and replacing βt by its estimate βˆt (unbiased if βt is constant), we obtain the final expression for the conditional expected value of a missing price:
pˆ t/t−τ1 = pt−τ1 + (βˆt )T j
τ 1 −1
Xt−i
(13a)
Xt+τ2 −i
(13b)
i=0 j pˆ t/t+τ2 = pt+τ2 − (βˆt )T
τ 2 −1 i=0
If the missing price is placed before the first observed price, then (13b) is used. On the other hand, if the missing price is after the last observed price, (13a) is used. Otherwise, when the missing price has observed prices both before and after it, the minimum variance combination of (11a) and (11b) is given by pˆ t = γ pˆ t/t+τ2 + (1 − γ )pˆ t/t−τ1 =
τ1 τ2 pˆ t/t+τ2 + pˆ t/t−τ1 τ1 + τ 2 τ1 + τ 2
(14)
Note that we do not take into account the uncertainty on the estimated coefficients in (13a) and (13b) to minimize the variance in (14). On the contrary, the minimization is performed in (11a) and (11b), simplifying considerably the calculation. For details on the minimum variance estimator see Neter et al. (1996, p. 400). 3.3 Prediction of Interval Forecasts Factor models are not tailored to produce forecasts since they will depend on forecasts of the risk factors themselves. In this paper, we consider the interval forecast (IF) prediction as if the market factors were known at the forecast date, since we aim mainly to evaluate the estimation of the betas, rather than forecast the risk factor volatilities. Proceeding otherwise would make unclear whether our method or a poor volatility estimate was responsible for some possible inadequacy in the IF coverage. Furthermore, concerning missing value imputation, it corresponds to the situation occurring in practice in financial institutions.
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If one is to consider the problem from the view of the clearing house, which guarantees the transactions, two-sided IFs must be supplied. This is because any of the two parties may default, which means that the clearing house may have to buy or sell the equity in the market three days after the transaction is agreed. The IF is computed as the following. The expected value of equity j j τ days return at time t + τ , given βt and X˜ t+τ is given by τ τ j j j ˜ j T (βt+i ) Xt+i + wt,τ = (βt+i )T Xt+i (15) E(Rt+τ,τ |βt , Xt+τ ) = E i=1
i=1
j
j
As the relationship in (15) is linear on βt , and considering an almost constant βt from t − τ to t, it is straightforward to forecast the τ days return from time t by j j Rˆ t+τ,τ = (βˆt )T
τ
(16)
Xt+i
i=1
Now we estimate the variance τ days ahead. Since it was assumed that the coefficients β are (almost) constant and the errors are serially independent, the variance of equity j τ days return at time t + τ may be approximated by T τ τ ∼ (17) Xt+i j Xt+i σ 2j = τσ2 + R ,t+τ,τ
j,t
βt
i=1
i=1
where βtj is the variance–covariance matrix of the betas. This variance can be estimated at time t by T τ τ 2 ˆ j (18) + Xt+i Xt+i σˆ R2 j ,t+τ,τ ∼ = τ σˆ j,t βt i=1
i=1
2 ˆ j and σˆ j,t are obtained by the regression. In our exercise, the observed values where the estimates βt of the risk factors are considered as given, instead of being forecasted, in order to evaluate only the irregular returns methodology, instead of volatility estimation. If the risk factors variance– covariance matrix (VCV) were to be estimated, there would be a number of methods to do so. Alexander and Leigh (1997) study the accuracy of some of these methods considering the proportion of returns that fall below the estimated VaR. The IF must then be based on a predictive distribution, the exact form of which is non-trivial to obtain. It could be estimated by bootstrap or by using the historic method (considering the empirical distribution of residuals observed as the distribution of errors), as is usually done by financial institutions. With supposedly normal errors j j j as in (2), and considering that βˆt is an unbiased estimator of βt (which is true in the case βt is j j constant in time), the conditional distribution of the forecast errors Rt+τ,τ − Rˆ t+τ,τ is a Student’s t with Kj − p − 1 degrees of freedom multiplied by σˆ R2 j ,t+τ,τ . Nonetheless, this parametric form will be used instead of a nonparametric alternative (as bootstrap or the historic method). The lower and upper bounds for a (1 − α)100% IF are given then by j j L = exp[ln(Pt ) + Rˆ t+τ,τ + t −1 (α/2, Kj − p − 1)σˆ Rj ,t+τ,τ ]
and U = exp[ln(Pt ) + Rˆ t+τ,τ + t −1 (1 − α/2, Kj − p − 1)σˆ Rj ,t+τ,τ ] j
−1
j
(19)
where t (., v) is the inverse cumulative distribution function of a Student’s t random variable with v degrees of freedom, and Kj + 1 is the number of observed prices of equity j in the estimation
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window. The lower bound, if desired, can be calibrated using the semi-variance (see Gastineau 2 and Kristzman, 1996, p. 250, for example) to estimate σˆ j,t . The semi-variance is the mean of the squared negative deviations from the mean, and its inspiration comes from the asymmetry observed in financial returns (‘leverage effect’). However, we tried using the semi-variance in this irregular returns context with no satisfactory results. The VaR at 1 − α/2 confidence level is given by the lower bound of the 1 − α IF.
4.
Empirical Example with Brazilian Equity Data
The database consists of 389 stock closing prices (the most liquid in the Brazilian Market) over the period ranging from 2 January 1995 to 28 February 2001 (over six years of daily data, 1540 trading days). The period under study comprises sub-periods with high volatility and others with low volatility. Figure 2 displays the returns and the squared returns of the Market Index (IBV), where one notices periods of high and low volatility: the former, when high returns (in absolute value) are clustered together; whereas the latter, when there are calm periods of low returns. More than 250 thousands of (τ -days) returns are used in the comparison. That means, all these returns are erased and filled in, as well as having the coverage of their IF’s verified. This includes equities with different liquidities; from very liquid stocks, the prices of which are observed every day; to very illiquid ones, with average time between subsequent trades as long as roughly two weeks. Considering closing prices as daily prices (i.e. regularly spaced) is an approximation, since the last trade needs not to be near the end of the trading period. However, the time in the day at which each equity’s last transaction occurs is unavailable to us.
Figure 2. Raw and squared returns from the IBOVESPA Index (IBV), from 2 January 1995 to 28 February 2001
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Figure 3. Schematization of the experiment
The results are divided into two sections, the first dedicated to the imputation of missing values and the second to the IF prediction. To access the accuracy of the missing values imputation, each single price in the database observed τ days after another existing price (τ = 1, 3, 5, 10) is deleted and filled in as schematized in Figure 3. This enables the computation of error statistics on the imputation. In turn, the IF is predicted for every observation in the database occurring τ days after another existing observation (τ = 1, 3, 5, 10), and its coverage verified. The values τ = 1, 3 are motivated by the fact that the parties involved in a transaction have three days to liquidate it. Thus, a clearing house must predict IFs for the equity price up to three days ahead in order to assess its risk in the case of default. The values τ = 5, 10 are roughly related to weeks, corresponding to periods used for risk analysis by financial institutions. In order to illustrate the behaviour of the estimated betas, Figure 4 displays the time series of the estimates of the IBV coefficient for four different equities, each one representing a liquidity level. They were chosen based on the average interval between consecutive existing prices. The first one has an average interval of 9.2 days (roughly one price for each two weeks); the second, 4.8 days (roughly one price a week); the third displays no missing value (average interval equal to 1); and the fourth, 2.1 days (one closing price each two days). Clearly, the time series is smoother for more liquid equities, and the assumption of almost constant betas between two adjacent returns may seem at first sight unreasonable for very illiquid equities. However, taking a more comprehensive point of view, a noisier behaviour of estimates is expected from illiquid equities since they are based on less data (less observed prices in the last 252 days), which makes the estimation less accurate. Furthermore, it is the best approximation we can provide without a burdensome increase in model complexity. One must bear in mind that model complexity is undesirable if there is a great amount of missing data. 4.1 Missing Values Imputation The imputation of missing values must be performed at the end of each day, so that the values of market indexes in the irregular returns factor model are known up to t. The accuracy of the imputation given by the proposed method is compared with the following methods: the naïf,
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Figure 4. Time series behaviour of the estimated IBV betas, for four different equities with different average τ ’s (days between subsequent trades). Panels: (a) very illiquid equity; (b) illiquid equity; (c) liquid equity; (d) semi-liquid equity
where the last price available is repeated to fill in a missing value (predicted returns are zero); and the market proxy (the IBV), where the equity is supposed to follow the market return in the j absence of the real price. Note that both methods are nested in the multi-factor (naïf : βi,t = 0, j j i = 0, 1, . . . , p; proxy: βk,t = 1, where βk,t is the coefficient referring to the market index return, j and βi,t = 0, i = k). A further comparison includes the multi-factor estimated conventionally, that is, only with regular (daily) returns. In a companion paper, Souza and Veiga (2001) implemented an E-M algorithm (Dempster, Laird and Rubin, 1977) with principal component analysis to compete against the multi-factor with irregular returns on the same database. Their conclusion is that the multi-factor outperforms the E-M for the equities with less than 95% of data available. To find the best multi-factor configuration, we tested three values of the EWMA smoothing constant, λ = 0.98, 0.99 and 1, as well as an automatic λ selection based on minimizing in-sample squared errors, restricting it to values between 0.90 and 1. We tried to find some relationship of the automatically selected λ with some indicator, such as the supi∈{1,...,Kj} [τi ], but this attempt was unsuccessful. We did not use fixed values of λ smaller than 0.98, in order to enable estimation for very illiquid equities, as there is a minimal number of prices (returns) to estimate the regression, such that illiquid equities require longer estimation windows. For example, if λ = 0.97, the weight assigned to a 100 days-old return is less than 5% of that assigned to the current return; and about 1% if the age of the return is 150 days. If λ = 0.95, a return aged 90 days is assigned a weight less than 1% of the weight assigned to the current return. To sum up, these high fixed values of λ are justified by the size of the estimation window (252 business days, approximately one year of data), which in turn is justified by the inclusion of many low liquidity equities in the comparison.7 The factors used to explain the equities returns were the IBV, index based on the most liquid equities negotiated in the S˜ao Paulo Stock Market; the CDI short interest rate; and the US
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Dollar/Brazilian Real exchange rate (US$). These were chosen because they are the ones usually considered by Brazilian financial institutions. Different combinations of these three factors were compared against each other, as well as a stepwise selection of factors. The comparison is done by means of the RMSSE (root mean square of standardized errors); the DC (direction of change statistic); and the MAPE (mean absolute percentage error). The RMSSE is computed as follows: 2 ˆj j (Rt,τ − Rt,τ ) 1 (20) RMSSE(τ ) = j N t,j σˆ Rt,τ where N is the total number of cases and τ is the time horizon, τ = 1, 3, 5 and 10. In order to j is estimated by EWMA and is the same for all methods. Errors are keep the comparison fair, σˆ Rt,τ standardized by the volatility estimate because some equities are more volatile than others, and so their prediction errors can be compared without some equities dominating the statistic in spite of others. While the RMSSE measures the error in the return prediction, the MAPE measures the percentage error in imputed prices. In turn, the DC statistic is the difference between the number of times the predicted return has the same sign as the observed return, and the number of times the direction of change is predicted incorrectly, relative to the total number of cases. Table 1 displays the one-day-ahead RMSSE for the naïf, the proxy and the multi-factor with irregular returns using one factor (the IBV return with λ = 0.99, λ = 1 and with the automatically selected λ) and using automatically selected factors (with λ = 0.99; and factors selected by a stepwise procedure). The results for λ = 0.98, as well as for the other possible fixed configuration of factors, are not shown as they are, in general, worse than those displayed, and do not bring any relevant insight to the comparison. The results are grouped by the adjusted R 2 (of IBV, λ = 0.99) as this was the feature that best explained the difference between methods. The column on the Table 1. RMSSE one day ahead by adjusted R 2 Adj. R 2
Naïf
Proxy
IBV (λ = .99)
Autom. factors
IBV (λ = 1)
IBV (λ autom.)
No. of cases
% Signif. betas
<0% 0–5% 5–15% 15–30% 30–45% 45–55% 55–65% 65–75% 75–85% 85–90% 90–95% 95–100% All
1.16 1.09 1.04 1.03 0.98 0.98 0.95 0.95 0.93 0.90 0.99 0.98 1.04
1.43 1.17 1.08 1.01 0.89 0.78 0.72 0.60 0.48 0.37 0.31 0.24 1.05
1.16 1.07 1.00 0.93 0.82 0.75 0.69 0.59 0.48 0.35 0.28 0.23 0.96
1.48 1.08 1.02 0.96 0.88 0.76 0.72 0.64 0.63 0.49 0.27 0.21 1.01
1.16 1.07 1.00 0.93 0.82 0.74 0.69 0.58 0.48 0.35 0.28 0.23 0.96
1.22 1.12 1.03 0.96 0.84 0.77 0.71 0.60 0.49 0.35 0.29 0.24 1.00
12755 54406 64960 49187 25704 10043 6244 4164 2746 1280 966 377 232832
0% 48% 94% 98% 99% 99% 99% 99% 100% 100% 100% 100%
Comparison of the root mean square standardized (by the volatility) error – RMSSE – in filling in missing one-day returns (best in bold), grouped by the adjusted R 2 of using IBV, λ = 0.99. Naïf is predicting a zero return; proxy is predicting the same return as the market index; and the remaining are multi-factor models estimated using irregular returns, where the factors are IBV (the market index, using λ = 0.99, λ = 1 or automatically selected λ), CDI (interest rate) and US$ (US$/Brazilian Real exchange rate). ‘Autom. factors’ refers to stepwise selection of factors (λ = 0.99), and ‘λ autom.’ refer to automatic λ selection.
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right gives the percentage of significant (at 5% significance level) betas for the IBV factor with λ = 0.99. Note that this factor is significant in the vast majority of cases for adjusted R2 over 5% (94% of the times for adjusted R2 between 5% and 15%, and in nearly all the cases for values above 15%). It is still significant in approximately half the cases the adjusted R2 lies between 0% and 5%. The irregular returns multi-factor with this as the only factor (IBV, both with λ = 0.99 and λ = 1) dominates the other methods, achieving the lowest RMSSEs for all cases, except adjusted R 2 above 90%, where the automatic selection of factors outperforms the use of this single factor. However, note that these are the cases where more than 90% of the variance is explained solely by the IBV factor. Figure 5 illustrates the results of Table 1, but referring to three days ahead. It is clear from the magnitude of the RMSSE that the differences between methods may be subtle in some cases. Table 2 displays the MAPE for five-days-ahead imputation. The single IBV factor with λ = 0.99 and λ = 1 remains the best overall method. However, as the forecast horizon increases, the naïf method becomes best for adjusted R2 below 5%, and the proxy starts to be an attractive method for adjusted R2 above 90%, together with the automatically selected factors. Table 3 displays the same results as Table 2, but considering ten days ahead. The naïf method continues to be the best choice for adjusted R2 below 5%, and the single factor IBV remains a reliable method, but now the proxy yields very attractive results for adjusted R2 above 30%. From these results, we conclude that, as the forecast horizon increases, fixing the value of beta equal to unity may be the best policy if there is a significant correlation between the market index and the equity returns. The RMSSE and the MAPE show that, overall, the multi-factor with irregular returns using only the market index returns performs best. This means that, in general, the market index explains equity returns well, whereas the US$/Brazilian Real exchange rate and the Brazilian inter-bank overnight interest rate do not. However, the reader must keep in mind that 50 of these equities form the index, so that the experiment is biased in favour of this index (each of the two assets with most weight make up around 10% of the index, while the following five make up between 4% and 6% each). On the other hand, the majority of these equities have negligible weight in the index. In sum, the naïf outperforms the multi-factor by a slight margin when the adjusted R 2
Figure 5. RMSSE three days ahead by adjusted R 2
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Table 2. MAPE five days ahead by adjusted R 2 Adj. R 2
Naïf
<0% 0–5% 5–15% 15–30% 30–45% 45–55% 55–65% 65–75% 75–85% 85–90% 90–95% 95–100% All
7.7% 6.7% 6.0% 6.1% 6.2% 6.6% 6.5% 6.7% 6.9% 6.6% 5.9% 5.9% 6.4%
Proxy
IBV (λ = .99)
Autom. factors
IBV (λ = 1)
IBV (λ autom.)
No. of cases
% Signif. betas
8.6% 7.4% 6.2% 5.7% 5.2% 5.2% 4.7% 4.4% 3.8% 3.0% 2.0% 1.7% 6.2%
8.2% 7.0% 5.9% 5.5% 5.1% 5.1% 4.7% 4.5% 3.9% 2.9% 2.0% 1.9% 5.9%
8.7% 342% 7.5% 5.6% 5.2% 5.3% 4.9% 4.6% 4.9% 3.7% 2.0% 1.7% 83.0%
8.1% 6.9% 5.9% 5.5% 5.0% 5.1% 4.7% 4.4% 3.9% 2.9% 2.0% 1.9% 5.9%
9.5% 8.0% 6.5% 5.9% 5.3% 5.4% 5.0% 4.8% 4.1% 3.1% 2.1% 2.1% 6.5%
12043 51599 62903 48474 25449 9963 6178 4142 2734 1278 966 377 226106
0% 49% 94% 98% 99% 99% 99,6% 99,4% 100% 100% 100% 100%
Comparison of the mean absolute percentage error – MAPE – in filling in missing prices, where the last price available is five days before (best in bold), grouped by the adjusted R 2 of estimating IBV. Methods are as in the previous table.
is low, and so does the proxy when the R2 is high, for a forecast horizon over one day ahead. However, as the R 2 rises the naïf tends to be outperformed by all others. If the forecast horizon is one day ahead, the single factor IBV (both with λ = 0.99 and λ = 1) is the best method, except for adjusted R2 above 90%. Automatic selection of factors is useful for high values of the adjusted R2 , but automatic selection of λ is never superior to using λ = 1 (it is at most equal), using the IBV as the single factor. This can be explained by the nature of the irregular returns, which make the selection procedure rather unstable. Figure 6 shows the DC statistic for three days ahead. No method is particularly the best one, if we consider only high values of adjusted R2 . However, considering also low values of this Table 3. MAPE ten days ahead by adjusted R 2 Adj. R 2
Naïf
Proxy
IBV (λ = .99)
Autom. factors
IBV (λ = 1)
IBV (λ autom.)
No. of cases
% Signif. betas
<0% 0–5% 5–15% 15–30% 30–45% 45–55% 55–65% 65–75% 75–85% 85–90% 90–95% 95–100% All
10.8% 9.3% 8.6% 8.7% 9.0% 9.1% 8.6% 8.9% 9.2% 8.2% 8.6% 7.9% 9.0%
11.5% 9.9% 8.5% 7.8% 7.2% 7.1% 6.4% 5.9% 4.9% 3.9% 3.0% 2.3% 8.4%
11.7% 9.8% 8.4% 7.8% 7.3% 7.1% 6.5% 6.2% 5.3% 4.3% 2.9% 3.0% 8.4%
13.1% 100% 85.8% 8.4% 9.6% 7.7% 7.0% 6.4% 8.0% 7.0% 2.9% 2.4% >100%
11.5% 9.6% 8.3% 7.7% 7.2% 7.0% 6.4% 6.0% 5.2% 4.2% 2.9% 2.9% 8.3%
14.7% 11.6% 9.5% 8.6% 7.9% 7.9% 7.2% 6.8% 5.8% 4.7% 3.1% 3.6% 9.6%
11818 50799 62147 47972 25311 9897 6132 4120 2727 1276 966 377 223542
0% 49% 94% 98% 99% 100% 100% 100% 100% 100% 100% 100%
Comparison of the mean absolute percentage error – MAPE – in filling in missing prices, where the last price available is one day before (best in bold), grouped by the adjusted R 2 of estimating IBV. Methods are as in the previous table.
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Figure 6. DC three days ahead by adjusted R 2
statistic, the proxy (the market index return) can be considered the best indicator for the rise or fall of each equity price three days ahead. On the other hand, by definition, the naïf yields a zero value for this statistic, and is the worst method, together with using no factor but a constant. The results for other horizons are qualitatively similar. 4.2 Irregular Versus Regular Returns In this section, the irregular returns multi-factor model is compared with the conventional multifactor with daily returns. The simple fact that the conventional multi-factor needs two consecutive days of trade to provide a daily return compares favourably with the multi-factor with irregular returns. The regular returns multi-factor discards the information of single prices (with no negotiation of the equity in the previous or the next day) and part of the information is contributed by a price at the end of a block of prices, while the irregular returns version uses all prices. For this reason, there can be equities for which the coefficients β can be estimated by the irregular returns version but not conventionally. As long as there are degrees of freedom enough to reasonably estimate the coefficients β, the irregular returns multi-factor can be used, and in the present paper we considered 10 returns (less than 4% of data available, considering an estimation window of 252 days) as a lower bound to estimate the regression. Table 4 compares the number of cases in the database for which there were enough data so that each version could estimate the coefficients β, considering a time span from 25 August 1998 to 28 February 2001. Note that regular returns enable the estimation only one-third of the times that irregular returns do when there are at most 5% of data (prices) available, and approximately half the times when there are between 5% and 15% of data available. Above about 65% of data available, both enable the same number of estimations in the database. The comparison reported below takes into account only the cases where both versions were able to estimate the coefficients. Since we stipulated 10 returns as a minimum to estimate the regression, the case where between 0% and 5% of the prices are available is restricted to 11 or 12 prices existing in the 252 days estimation window. In view of
Regular Irregular
0–5%
5–15%
15–30%
30–45%
45–55%
55–65%
65–75%
75–85%
85–90%
90–95%
95–100%
100%
33 100
824 1446
3628 4006
6073 6278
5027 5097
5976 6003
7049 7049
10260 10260
7262 7262
9131 9131
24248 24248
30822 30822
Using Irregularly Spaced Returns to Estimate Multi-factor Models
Table 4. Frequency of 252 days estimation windows with more than 10 (regular or irregular) returns available, with an existing price in the end. The cases are grouped by percentage of existing prices within the window
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Figure 7. RMSSE of filled in data from regular and irregular returns multi-factor, three days ahead, grouped by percentage of existing prices in the estimation window
this, 33 cases where 10 daily returns could be computed out of 100 where 11 or 12 prices were available may seem too many, as one would expected existing prices scattered randomly over the estimation window. However, there are many cases in the database where a liquid equity started being traded during the period under study, having less than 13 prices in the past year because it was less than 13 days old then, justifying the high proportion. Figures 7 and 8 show, respectively, the RMSSE and the MAPE of imputed missing values, for three days ahead. Note that the results are now grouped by percentage of existing prices in the estimation window, unlike the comparison with the naïf and the proxy methods. The advantage of the irregular returns version over the conventional one is apparent. The irregular returns version almost always yields better results. A significant difference appears when there are between 0% and 75% of prices available for imputation three days ahead, with the greatest difference appearing for between 15% and 30% of existing prices. When all prices are available, both versions are equal by definition and so are their results. This also means that their results approximate better, the more data exist in the estimation window.
Figure 8. MAPE of filled in data from regular and irregular returns multi-factor, three days ahead, grouped by percentage of existing prices in the estimation window
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It is clear from these results that the multi-factor with irregular returns outperforms the conventional multi-factor, in addition to enabling the estimation in cases where the conventional one cannot be applied. 4.3 IF Prediction In Section 4.1, it is clearly shown that the irregular multi-factor outperforms the naïf in filling in missing values, especially for higher values of the adjusted R 2 . The proxy is slightly outperformed, especially for lower values of that statistic. From the previous results, we chose the best configuration to be the one that uses only the IBV as a factor. The 90% two-sided IF (obtained via equation (19)) from this configuration is tested in this section, together with a benchmark. The benchmark is the use of plain EWMA (λ = 0.99) to predict the IF of the equity price, under the assumption of normal errors. The percentage of cases the price fell below (above) the 90% two-sided IF is shown in Tables 5 and 6, grouped by adjusted R2 , for five and ten days ahead, respectively. Starred and double starred numbers are significantly different from the nominal percentage of 5% at respective confidence levels of α = 0.05 and α = 0.01. The significance was obtained using the likelihood ratio test of unconditional coverage proposed by Christoffersen (1998). However, it is important to bear in mind that the higher the number of observations, the higher is the test power to detect deviations from the nominal tail percentages, even if these deviations are so small that they are irrelevant in practice. The great amount of data explains the high number of rejections, including cases with coverage that is reasonable in practice (as, for instance, 5.4% and 4.7%). The main objective of this exercise is to get reasonable coverage, and this can be obtained using these two methods. In general, the parametric approach, using both EWMA and the irregular returns multi-factor, produced conservative lower bounds for the two-sided IFs. As for the upper bounds, the same is true, but the EWMA appears as an exception to the rule for adjusted R 2 below 30%. As with the Table 5. Percentage of cases below or above the two-sided 90% five days ahead IF grouped by adjusted R 2 Below 90% IF Adj.
R2
<0% 0–5% 5–15% 15–30% 30–45% 45–55% 55–65% 65–75% 75–85% 85–90% 90–95% 95–100%
Above 90% IF
EWMA
IBV
EWMA
IBV
No. of cases
2.2%∗∗ 2.2%∗∗ 2.9%∗∗ 3.9%∗∗ 4.6%∗∗ 4.0%∗∗ 3.8%∗∗ 4.5% 3.6%∗∗ 3.2%∗∗ 5.4% 3.2%
1.3%∗∗ 1.5%∗∗ 2.1%∗∗ 3.0%∗∗ 3.3%∗∗ 3.8%∗∗ 3.9%∗∗ 5.2% 5.4% 4.7% 4.7% 9.8%∗∗
5.5% 5.4%∗∗ 5.4%∗∗ 6.0%∗∗ 5.1% 5.4% 4.3%∗ 4.4% 3.2%∗∗ 3.8% 3.5% 3.2%
3.4%∗∗ 3.6%∗∗ 3.6%∗∗ 4.7%∗∗ 4.4%∗∗ 5.0% 4.7% 5.7% 4.8% 5.2% 3.8% 6.4%
12043 51515 62903 48474 25449 9963 6178 4142 2734 1278 966 377
Percentage of cases outside (above or below) the 90% five-days-ahead interval forecasts, grouped by the adjusted R 2 of estimating IBV. Cases below the 90% interval forecast are equivalent to cases below the 95% VaR. The multi-factor model with only the market index as a factor (the best in filling in missing values) is represented by ‘IBV’, whereas EWMA corresponds to the use of plain EWMA. Starred and double-starred values correspond to 5% and 1% rejections of the Christoffersen (1998) test of unconditional coverage, respectively.
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Table 6. Percentage of cases below or above the two-sided 90% ten days-ahead IF grouped by adjusted R 2 Below 90% IF Adj.
R2
<0% 0–5% 5–15% 15–30% 30–45% 45–55% 55–65% 65–75% 75–85% 85–90% 90–95% 95–100%
Above 90% IF
EWMA
IBV
EWMA
IBV
No. of cases
1.9%∗∗ 2.0%∗∗ 2.9%∗∗ 4.2%∗∗ 4.6%∗ 3.6%∗∗ 3.1%∗∗ 3.6%∗∗ 3.7%∗∗ 2.2%∗∗ 5.2% 3.7%
1.2%∗∗ 1.3%∗∗ 1.9%∗∗ 3.0%∗∗ 3.2%∗∗ 3.6%∗∗ 3.8%∗∗ 4.5% 4.7% 5.1% 5.5% 6.6%
5.9%∗∗ 5.5%∗∗ 5.5%∗∗ 5.8%∗∗ 4.8% 4.8% 3.8%∗∗ 3.7%∗∗ 2.2%∗∗ 3.4%∗ 3.7% 1.1%∗∗
3.4%∗∗ 3.5%∗∗ 3.4%∗∗ 4.2%∗∗ 4.1%∗∗ 4.6% 4.1%∗∗ 5.2% 3.3%∗∗ 4.3% 3.2%∗ 6.1%
11818 50722 62147 47972 25311 9897 6132 4120 2727 1276 966 377
Legend: the same as Table 5, but referring instead to ten days-ahead interval forecasts.
missing values imputation, the EWMA (which somehow corresponds with the naïf method) is better at producing IFs for low values of the adjusted R 2 (below 45% for the lower bound and below 15% for the upper bound). On the other hand, as the adjusted R 2 increases, the irregular returns multi-factor starts to yield better results (very reasonable ones, indeed), with a curious exception in the lower bound for adjusted R 2 above 95%. Some classes (of adjusted R 2 ) see small but statistically significant deviations from the nominal tail percentage, from where we conclude that, although the coverage is in general close to the nominal, it is different. In general, the methods tended to be more conservative at the IF lower bound than at the upper bound, which means that an asymmetric predictive distribution could fare better. However, using the semi-variance approach (results not shown here but available on request) did not yield any better result. The multi-factor model, estimated with irregular returns, has shown itself to be a useful tool to predict risk.
5.
Conclusions
In this paper we propose the use of returns which are computed from prices irregularly spaced in time to estimate the multi-factor model for equity returns. The multi-factor model is a simple but efficient tool to explain cross-sectional covariance in equities returns. The model showed itself useful for forecast missing data, as well as providing interval forecasts for future returns. Furthermore, the use of irregular returns enables the estimation in cases where using only regular (daily) returns would not. An empirical example with data from the 389 most liquid equities in the Brazilian Market confirmed the superiority of the multi-factor estimated with irregular returns over the traditional regular returns version, as well as two benchmark methods (the naïf and mimicking the return of the market index). Moreover, the market index showed itself capable of explaining equities returns whereas the US$/Brazilian real exchange rate and the Brazilian inter-bank overnight interest rate did not.
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Acknowledgements This work was carried out in 2001 when both authors were working as consultants to Algorithmics Brazil and through Algorithmics Brazil to the São Paulo Exchange Market (BOVESPA) and its clearing house (CBLC). The authors would like to thank two anonymous referees, as well as seminar participants at the Graduate School of Economics/Getulio Vargas Foundation, and conference attendants at the Computation in Economics and Finance 2002 and at the Forecasting Financial Markets 2003 for useful comments on previous versions of this work. The second author (Souza) acknowledges the financial support received afterwards by FAPERJ, which enabled him to write the academic version of the paper. Notes 1 2 3 4 5 6 7
Subramanian (2001, p. 77), for example, observes this effect in bond markets. For an example in the Brazilian Market one can cite the resolution number 2804, which regulates liquidity risk, decreed by the Brazilian Central Bank in December 21, 2000. Possibly private information. There is no data in between t − τ and t to estimate how β and σ 2 change. Furthermore, if τ is small the coefficients β and the variance σ 2 are believed to change little. In fact, if all the weights in W are multiplied by the same constant, the estimates do not change, as the change in the inversion of X T WX compensates the change in X T WRj . That is why the weights do not match with those in 2.3. In order to minimize the estimator variance (see, for example, Neter et al., 1996, p. 400). Some require one year to have, say, 6 or 10 days on which they are traded.
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