non-linear asset pricing

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The two estimated parameters of the CL are the alpha and the beta; the former ... documented in the financial literature that small stocks have higher risk and ...... 1.10. MTW. 2515. 0.05. 0.60. 0.60. 0.32. 0.05. 1.21. 0.31. 0.05. 1.25. MUR. 2515.
NON-LINEAR ASSET PRICING Péter ERDŐS Department of Finance Budapest University of Technology and Economics, Hungary Műegyetem rkp. 9., Budapest, 1111, HUNGARY [email protected] Mihály ORMOS Department of Finance Budapest University of Technology and Economics Műegyetem rkp. 9., Budapest, 1111, HUNGARY [email protected] Dávid ZIBRICZKY Department of Computer Science and Information Theory Budapest University of Technology and Economics Műegyetem rkp. 9., Budapest, 1111, HUNGARY [email protected]

Abstract We test the nullhypothesis of linearity of the CAPM. We aim to investigate whether or not there is a linear relationship between risk and return (security market line) and between the return of a given stock and the return of the market portfolio (characteristic line). On the other hand, we derive new semi-parametric risk measures for the case of non-linearity. We apply our analysis on a random sample of 150 stocks of the Standard and Poor’s large-, mid- and small cap components. We show that if linearity of the characteristic line does not hold, the CAPM betas are significantly downward biased, thus the standard market risk measure cannot be used. We argue that portfolio managers can beat the market only when extreme market movements occur.

Keywords: asset pricing, kernel regression, risk measures, semi-parametric models JEL codes: C14, C51, G12, G32

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Introduction The Capital Asset Pricing Model (CAPM, Sharpe, 1964; Lintner, 1965; Mossin, 1966) is one of the most broadly applied equilibrium models in the financial literature. The CAPM and the standard asset pricing models such as the APT (Arbitrage Pricing Theory) (Ross, 1976), the Fama-French three-factor-model (Fama and French, 1996) and the Carhart (1997) four-factormodel assume linear relationship between risk and return. Stapleton and Subrahmanyam (1983) verify the linear relationship between risk and return in the case of the CAPM. In our study we investigate linearity of two relationships; first the one between the return of a given security and the return of the market portfolio which relation is known as the characteristic line (CL); and the other one between the asset return and its market risk (measured by the slope of the CL); that is the security market line (SML). The standard asset pricing tests decide on the validity of the model based on linear regressions which is a correct approach if the relationship between risk and return is indeed linear. If this assumption does not hold, then the estimated parameters by the Ordinary Least Squares (OLS) or by any other linear estimators are biased and inconsistent. The two estimated parameters of the CL are the alpha and the beta; the former measures the abnormal return (see Jensen, 1968), the latter measures the relevant risk of a given asset. If the capital market is in equilibrium, securities gain returns according to their market risk, that is, the expected value of the abnormal return is equal to zero. We approach the sketched questions from two different directions; first, we investigate the nullhypothesis of linearity in the case of the two relationships which can be derived from the CAPM; second, we consider deriving new semiparametric risk measures (”alpha” and ”beta”) if we reject the nullity. We employ our analysis on a random sample from the S&P (Standard & Poor’s) large-, mid- and small cap components (S&P 500, S&P MidCap 400, S&P SmallCap 600). We obtain the daily returns from the Center for Research in Security Prices (CRSP) database. We pick 50-50-50 randomly chosen stocks from each size index. The linear nullity of the CL as a general rule for the whole market can be rejected at any usual significance level. We estimate four SMLs; one for the chosen components of each index and another one for all companies. Linearity of the SMLs cannot be rejected. The slope of the SML for the S&P small companies is negative, that is our results confirm the small company effect (see, e.g., Banz, 1981; Basu, 1983; Fama and French, 1995). It is welldocumented in the financial literature that small stocks have higher risk and because of it they provide higher expected return, thus firm size is a risk factor besides market risk. It is striking that linearity of the CL of large cap stocks can be rejected at any usual significance level, while those of small and mid companies cannot be rejected at 95% significance level which means the small company effect does not explain the fallacy of linearity. The semi-parametric performance measurement shows that portfolio managers can beat the market only when extreme market movements occur.

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Applied methodology The difference between linear and non-linear regression Assuming two random variables, regression is a conditional expected value, in the most general form E (Y | X ) = m( X )

(1)

where Y is the dependent and X is the explanatory variable. Assuming linear relationship, simple linear estimators can be applied (e.g., OLS Ordinary Least Squares, ML Maximum Likelihood, GMM – Generalized Method of Moments); however, if linearity does not hold the linear estimators induce biased and inconsistent parameter estimations. We need a distribution free, robust estimator for the tests sketched in the introduction section which induces precise estimations even in a non-linear environment. If we assume that Eq. (1) is linear, the CAPM is

Yi , j = αˆ j + βˆ j X i , j + εˆ i , j i = 1,2,3,...,n;

j = 1,2,3,...,N

(2)

where αˆ j and βˆ j are the intercept and slope coefficients of the jth security respectively and εˆ j ,t is the residual series of the regression. Yi,j and Xi,j are the risk premium of the jth security and the market in period i, respectively. We do not assume linear relationship between the variables, that is linear regression is not a suitable method for estimating Eq. (1). Nadaraya (1964) and Watson (1964) derive a kernel based regression estimator, which can estimate Eq. (1) without assuming any specific form of the relationship between the variables. The Nadaraya-Watson estimator is

ˆ h( x ) = m

1 n ∑Whi ( x ) Yi n i =1

(3)

where Whi ( x ) is the so-called Nadaraya-Watson weighting function, that is Whi ( x ) =

Kh ( x − X i ) 1 n ∑ Kh ( x − X j ) n j =1

,

(4)

where K h (u ) is the kernel function, h is a properly selected bandwidth1. The estimation at each x is the average of Yi observations in window h. The Nadaraya-Watson estimator determines the weighting vector based on distance, that is the weight of Yi is proportional to the distance x-Xi. If an X i observation is far from the observation being estimated, the weight of Yi will be smaller and vice versa. Choice of the kernel function Härdle et al. (2004) show that choice of the kernel function is only of secondary importance, the focus is rather on the right choice of bandwidth. Most of the frequently used kernel functions (uniform, triangular, Epanechnikov, triweight, cosine, Gaussian) include an indicator I, which is equal to one if the condition embedded in the function meets and is equal to 1

1

⎛ x − Xi ⎞ ⎟. ⎝ h ⎠

For brevity we use that K h ( x − X i ) = K ⎜ h

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zero otherwise. Härdle et al. (2004) argue that the Epanechnikov kernel has the fastest convergence. Estimating our results by different kernels, we find no significant difference. 2 If linearity does not meet, semi-parametric risk measures (“alpha” and ”beta”) should be used which are derived by derivative estimation, hence we need a kernel function which is differentiable in each point and satisfies the following conditions: (1.) K ( u ) continuous in the interval of [ −1,1] , (2.)

K ( u ) = 0 , K ′ ( u ) = 0 and K ′′ ( u ) = 0

at u = −1;1 points.

The above conditions are valid only for the triweight and the Gaussian kernels. We use the indicator free Gaussian kernel, which is in the form

K (u ) =

1 ⎛ 1 ⎞ exp ⎜ − u 2 ⎟ . 2π ⎝ 2 ⎠

(5)

The bandwidth selection As we argue earlier, the bandwidth selection has superior importance over the kernel choice, so we focus on this issue more closely. Smoothing depends on bandwidth h, the kernel function becomes flatter as its value is growing, thus the impact of closer values declines while the impact of more distant values grows at each point of estimation. If we select a too large bandwidth, the curve is oversmoothed and the bias increases while the standard deviation of the estimation decreases because the estimated curve becomes flatter. Choosing a too small bandwidth (undersmoothing), the impact of the nearby points strengthen and the estimation approaches Yi as h approaches zero. In this case the variance of the estimation increases and the bias can decline or rises; however, departing from the optimal bandwidth, the mean squared error increases. We present the cases of under/oversmoothing in Figure 1; at the tails of the curve where data points are sparse, the undersmooth curve links the observations, while the oversmoothed curve is too far from the data points because the nearest points are underweight.

2

These results are not presented in this study but they are available upon request.

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Smoothing problems

Oversmoothed curve Undersmoothed curve Optimally smoothed curve

Figure 1 Over/undersmoothed kernel curve The characteristic line of the Lowe’s Companies Inc. (randomly chosen) has been estimated by the kernel regression; the market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. The model, the non-parametric CAPM is ˆ h ( rm − r f ) + ηˆ LOW where rLOW-rf and rM-rf are the risk premium of the Lowe’s Companies Inc. and the rLOW − r f = m market respectively, h is an optimally selected bandwidth by Cross Validation (CV). We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression. The undersmoothed (oversmoothed) curve has a bandwidth five times (one fifth of) the CV optimal.

We aim to select an optimal bandwidth which minimizes the estimation error which is measured by the Average Square Error (ASE) in the form

ASE ( h ) = ASE {mˆ h } =

2 1 n mˆ h ( X i ) − m ( X i )} w ( X i ) , { ∑ n i =1

(6)

ˆ h ( X i ) is the estimated form of Eq. (1) and m ( X i ) is the true value of the same. The where m

w ( X i ) is a trimming function which can lower the impact of the outliers and we do not use this throughout the paper since we do not want to discount the weight of extremes, thus we take w( X i ) = 1 for each i. Since ASE (h ) is a random variable, we can take its conditional expectation. Härdle et al. (2004) show that this is the sum of squared bias and variance, that is

MASE ( h ) = E ( ASE ( h ) | X 1 = x1 ,..., X n = xn ) = b 2 ( h ) + v ( h ) ,

(7)

where the bias is

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⎫⎪ 1 n ⎧⎪ 1 n K h ( X i − X j ) − b (h) = m X m X ( ) ( ) ⎨ ∑ ∑ ˆf ( X ) j i ⎬ w( Xi ) , n i =1 ⎪ n j =1 h i ⎪⎭ ⎩ 2

(8)

and the variance is v (h) =

1 n ⎡1 ∑⎢ n i =1 ⎢ n 2 ⎣

⎤ ⎧⎪ K h ( X i − X j ) ⎫⎪ 2 ⎥w ( X i ) . σ X ⎨ ∑ ˆ (X ) ⎬ ( j) ⎥ f j =1 ⎪ h i ⎩ ⎭⎪ ⎦ n

(9)

The m(•) function is unknown, thus the bias cannot be calculated; a method is required which can approximate the MASE (Mean Average Squared Error) and can be calculated by the available data. MASE takes its minimum value where ASE does. Substituting the suitable values of Y into Eq. (6), we obtain p( h ) =

2 1 n ˆ h ( X i )} w ( X i ) . Yi − m { ∑ n i =1

(10)

Eq. (10) approximates MASE but it has a shortcoming; Yi estimates itself by mˆ h ( X i ) , thus if h → 0 , p (h ) can be minimized arbitrarily, so mˆ h (•) would approach Yis. Härdle et al. (2004) eliminate this problem by using a penalizing function in the form G( h ) =

2 1 n 1 ˆ h ( X i )} Ξ ⎛⎜ Whi ( X i ) ⎞⎟ w ( X i ) , Yi − m { ∑ n i =1 ⎝n ⎠

(11)

where Ξ(h ) grows as h declines, that is it adjusts the error emerging from the naive approximation of Yi ~ mh ( x ) . Let us assume the Generalized Cross-Validation penalizing function in the form

Ξ GCV ( u ) = ( 1 − u ) . −2

(12)

If we substitute this into Eq. (11), we obtain −2

2⎛ 1 n 1 ⎞ CV ( h ) = ∑ {Yi − mˆ h ( X i )} ⎜1 − Whi ( X i ) ⎟ , n i =1 ⎝ n ⎠

(13)

which is known as the Cross-Validation (CV) function. Härdle et al. (2004) show that the ASE is minimal when CV (h ) is minimal, thus the bandwidth of the kernel regression is optimal. We use the simplex search method for the minimizing problem (see Lagarias et al., 1998). The iteration process can be accelerated if we choose an initial value close to the optimum, for example, based on the Silverman’s rule of thumb3. An adjusted version of the Silverman’s (1986) rule of thumb is in the form

It is necessary if time series are long since the computational time of algorithm minimizing the CV (h ) function is proportional to the fourth power of the number of observations. 3

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⎧⎪ 1 n hˆ rot = 1.06 min ⎨ ∑ Xi − X ⎩⎪ n − 1 i =1

(

)

2

R ⎫⎪ − 51 , ⎬n , 1.34 ⎭⎪

(14)

where R is the difference between the third and the first quartile of Xis. The closer the distribution to the normal, the more accurate the rule is. Since our time series are not normally distributed, the Silverman selected bandwidth is not optimal; however, it is suitable for serving as an initial value for the minimizing algorithm which reduces the computational complexity efficiently.4 Figure 2 shows the squared error in the function of the number of iterations; the closer the initial value to the optimal, the faster the convergence is. Figure 2 shows that the algorithm finds the optimal value the fastest way if the initial value is equal to the bandwidth selected by the Silverman rule. It can be seen that the squared error does not decline iteration by iteration, since the algorithm excludes the possibility to converge to a local minimum. Figure 3 shows that during the optimization process the bandwidth oscillates around the optimal value, while the amplitude vanishes the fastest in the case of the Silverman choice. Squared error in the function of number of iterations with different initial values Optimal value hSilvarman 2hSilvarman πhSilvarman 5hSilvarman

Squared error

7.5hSilvarman

Number of iterations

Figure 2 Minimizing the mean average squared error in the function of number of iterations with different bandwidth initial values We test the optimization process on the characteristic line of the Lowe’s Companies Inc. (randomly chosen). The line has been estimated by the kernel regression; the market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. The model, the nonˆ h ( rm − r f ) + ηˆ LOW where rLOW-rf and rM-rf are the risk premium of the Lowe’s parametric CAPM is rLOW − r f = m Companies Inc. and the market respectively, h is an optimally selected bandwidth by Cross Validation (CV). We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression. The figure shows the process of optimal bandwidth selection, that is the mean average squared error in the function of the number of iterations. The initial values for the bandwidth optimization are the optimal; the bandwidth selected

4

We reject the normality of all of the time series based on the Jarque-Bera test. This result is available upon request.

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according to the Silverman (1986) rule of thumb and 2, π, 5 and 7.5 times the Silverman selected. The minimization is solved by the simplex search method (see Lagarias et al., 1998) in 14, 15, 16, 21 and 22 steps.

Bandwidth in the function of number of iterations with different initial bandwidth Optimal value hSilvarman 2hSilvarman πhSilvarman 5hSilvarman

Bandwidth (h)

7.5hSilvarman

Number of iterations

Figure 3 The bandwidth selection process in the function of the number of iterations with different initial values We test the optimization process on the characteristic line of the Lowe’s Companies Inc. (randomly chosen). The line has been estimated by the kernel regression; the market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. The model, the nonˆ h ( rm − r f ) + ηˆ LOW where rLOW-rf and rM-rf are the risk premium of the Lowe’s parametric CAPM is rLOW − r f = m Companies Inc. and the market respectively, h is an optimally selected bandwidth by Cross Validation (CV). We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression. The figure shows the process of optimal bandwidth selection, that is the bandwidth in the function of the number of iterations. The initial values for the bandwidth optimization are the optimal; the bandwidth selected according to the Silverman (1986) rule of thumb and 2, π, 5 and 7.5 times the Silverman selected.

Goodness of fitting To sustain comparability with the linear regression, we use R2 as the measure of the goodness of fitting. By definition, R2 is

R2 ≡ 1 − n

SSE , SST n

(15)

ˆ h ( X i ) ) and SST = ∑ (Y i −Y ) . where SSE = ∑ (Y i −m i =1

2

2

i =1

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This definition is equivalent to the R2 used for linear regression; the difference is only that ˆ h ( X i ) instead of the parametric estimation, αˆ + βˆ X i in SSE. there is the kernel estimation, m Since we calculate both measures in the same way, they are comparable5. Confidence band The kernel regression only estimates m(x) , thus we cannot state anything on the real process with certainty, the value of the estimation is a random variable at each x point. To be able to limit the location of the real process at a given confidence level we estimate a confidence band in the form

⎡ ⎢m ˆ h ( x) − z α 1− ⎢ 2 ⎣

K 2 σˆ 2 ( x ) ˆ h ( x) + z α ,m 1− nhfˆ ( x )

2

h

K 2 σˆ 2 ( x ) ⎤ ⎥, nhfˆh ( x ) ⎥ ⎦

(16)

where the variance at x point is n

ˆ h ( x )}2 , σˆ 2 ( x ) = ∑ Whi ( x ){Yi − m

(17)

i =1

K gauss

2

is the Gaussian norm-square, that is ∞

K gauss

2

=

∫

1

K dx = 2

2 π

−∞

,

(18)

fˆh ( x) is the estimated density, and α is the confidence level (see Härdle et al., 2004). If the number of observations decline in the neighborhood of x, the confidence band flares. As the distribution of market returns are obviously not uniform; diverging from zero, the probability of occurrence declines, so the confidence band flares at the tails. (see later in Figure 7).

Hypothesis testing and test statistic We test whether or not the CL and the SML are linear by applying the nullhypothesis of linear regression against the kernel regression alternative. Let us assume the parametric model in the form

E (Y | X = x ) = mθ ( • ) ,

(19)

where θ is a vector of parameters, thus the nullhypothesis is H 0 : m( x ) ≡ mθ ( x ) , which is tested against the H : m( x ) ≠ m ( x ) alternative. The θˆ vector is the estimation of θ which can be 1

θ

estimated by standard parametric regressions. m( x ) is unknown, thus we use mˆ h ( x ) to approximate it. If we cannot reject Ho, it means that the kernel regression does not differ significantly from the parametric one. The difference between the two estimations can be measured by ˆ ( X i ) − mθˆ ( X i )} w ( X i ) . h ∑ {m n

2

(20)

i =1

5 In the case of linear regression it would be more adequate to use adjusted R2 since we lose several degrees of freedom because of parameter estimation (in our case we lose two). We have to note that it has no significant impact because we use a relatively large sample.

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While mθˆ (•) is asymptotically unbiased and the speed of convergence of the parameters is

n,

the non-parametric estimation is biased because of smoothing and the speed of convergence is only nh . Härdle and Mammen (1993) introduce artificial bias into the parametric estimation to solve this problem. They use kernel weighted regression in the form

∑K (X n

ˆ θˆ ( X i ) = m

j =1

h

i

− X j ) mθˆ ( X j )

∑K (X n

j =1

h

i

,

−Xj)

(21)

instead of mθˆ (x ) and based on Eq. (21) they derive

ˆ ( Xi ) − m ˆ θˆ ( X i )} w ( X i ) T = h ∑ {m n

2

(22)

i =1

test statistic. The distribution of T is unknown; however, it can be determined by the wild bootstrap approach (see Härdle and Mammen, 1993). The wild bootstrap method generates new Yi* samples based on the residuals of the parametric regression. The approach assumes that the first three moments of the residuals of the original and the new regressions are equal. The process of the hypothesis testing (1.) Compute the value of the T statistic defined in Eq. (22). (2.) Generate the residual series of the parametric regression. 1− 5 (3.) Determine ε i*, j based on the golden ratio, that is let ε i*, j = εˆ i , j with probability 2 1+ 5 q and ε i*, j = εˆ i , j with probability 1-q where εˆ i , j is the residual of the regression 2 defined in Eq. (2).6 (4.) Generate for each j stock new Yi*, j , X i , j samples based on

{(

)}

i =1,...,n

Y = mθˆ ( X i ) + ε . * i, j

(5.)

j

* i, j

n

{

Compute the T* test statistic, that is T * = h ∑ mˆ ( X i ) − Yi*

}. 2

i =1

(6.)

Generate nboot different samples repeating steps (3.)-(5.).

(7.)

H 0 cannot be rejected if T < P(1−α )*100 T * , that is T is smaller than a suitable

( )

quantile of T * . Supposing linear regression, the Eq. (19) is in the form

mθˆ ( x ) = αˆ + βˆ x . 7

6

The sample generated this way fulfil all the three necessary conditions, that is E

( )

(23)

(ε ) = 0 , E (ε ) = εˆ * i

*2 i

2 i ,

E ε i*3 = εˆi3 . 7

For simplicity we do not use indexes in these equations.

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Substituting Eq. (21) and (23) into Eq. (22), T statistic is n ⎧ n K X X Y K h ( X i − X k ) αˆ + βˆ X k − ( ) ∑ k i n ⎪∑ ⎪ k =1 h i T = h∑ ⎨ n − k =1 n i =1 ⎪ K X X Kh ( X i − X k ) − ( ) ∑ h i k ∑ ⎪⎩ k =1 k =1

(

)

2

⎫ ⎪⎪ ⎬ w( Xi ) , ⎪ ⎪⎭

(24)

where w( X i ) = 1 for each i. The testing procedure is according to the previously presented sevenstep hypothesis testing. If the nullhypothesis is rejected, the linear estimators cannot be used for parameter estimation since parameters would be biased and inconsistent.

Risk and performance measurement ”Beta” and “alpha” estimation The relevant, non-diversifiable risk can be measured linearly by βˆ , that is the slope coefficient of the CL. In the case of non-linearity, this beta estimation cannot be used, so we approximate the market risk by a semi-parametric method. Härdle et al. (2004) show that T βˆ ∗ ( x ) = βˆ ( x ) , βˆ ( x ) ,..., βˆ ( x ) can be estimated by minimizing

(

0

1

)

p

∑ {Y − βˆ n

min ˆ ˆ

βˆ 0 ,β1 ,...,β p ,

i =1

i

∗ 0

p − βˆ 1∗ ( X i − x ) − ... − βˆ ∗p ( X i − x )

} K (x − X ). 2

h

i

(25)

βˆ ∗ ( x ) can be estimated by the Weighted Least Squares (WLS) in the form

βˆ ∗ ( x ) = ( X T WX ) X T WX , −1

⎛1 X − x 1 ⎜ ⎜1 X2 − x where the weights defined in Eq. (4), X = ⎜ M ⎜M ⎜⎜ ⎝1 Xn − x

p is the power of the regression,

⎛ Y1 ⎞ ⎜ ⎟ Y Y =⎜ 2⎟ ⎜M ⎟ ⎜⎜ ⎟⎟ ⎝ Yn ⎠

(26)

( X1 − x) 2 ( X 2 − x) 2

M

( X n − x) ⎛ Kh ( x − X 1 )

and

⎜ W =⎜ ⎜ ⎜ ⎜ ⎝

2

L L O L

( X 1 − x ) ⎞⎟ p ( X 2 − x ) ⎟⎟ , p

M

( Xn − x) 0

0

Kh ( x − X 2 )

M

M

0

0

p

⎟ ⎟⎟ ⎠

⎞ L 0 ⎟ L 0 ⎟. ⎟ O M ⎟ L K h ( x − X n ) ⎟⎠

The estimation defined in Eq. (26) is called local polynomial regression (see Härdle et al., 2004). The βˆ ∗ ( x ) vector has as many elements as the power of the estimated equation, thus, for example, βˆ ∗ ( x ) is the local constant estimation of the mˆ ( x) regression function, which is itself h

0

the Nadaraya-Watson kernel regression. βˆ 1 ( x ) approximates the derivative of m ( x ) on which the average slope can be determined. The CAPM is a linear model, thus it assumes the power of the regression is one, so we take the power of the polynomial regression to be one and estimate the ”beta” this way. Blundell (1991) shows that ”beta” is simply the expected value of the derivative estimation, that is ∗

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(

)

ˆ h′ ( x ) ≈ βˆ ∗ = E m

1 n ˆ ∑ β1 ( X i ) . n i =1

(27)

Eq. (27) is adequate for estimating market risk even if linearity does not hold. The advantage of this procedure is that it also considers cases if linearity is not valid only over certain intervals; it is not necessary that the risk of a given asset is constant under any circumstances and it makes possible to estimate extreme risk under extreme circumstances, thus this risk measure is more realistic than the simple CAPM beta. Jensen (1968) measures the difference between the normal return and the realized return by the intercept of the CL. If linearity does not hold, this approach cannot be used since the estimated alpha would be biased and inconsistent. Similarly to the derivative estimation, we derive a semi-parametric measure. The average performance of an asset can be determined by the surplus over its risk-adjusted return (calculated by the CAPM substituting the estimated semiparametric beta) and this is called ”alpha” or semi-parametric alpha, that is

αˆ ∗ = E (αˆ ∗ ( x ) ) ≈

(

)

1 n Yi − βˆ ∗ X i , ∑ n i =1

(28)

where βˆ ∗ is defined in Eq. (27). The Jensen alpha is an adequate performance measure only if the characteristic ”curve” is over/under the theoretical CL (zero intercept) exactly by alpha, which is true only if linearity holds. On the other hand Eq. (28) estimates the abnormal return at each point which can vary point to point, and the average performance is the mean of the point estimations.

Data For the analysis we use 50-50-50 randomly selected stocks from the S&P 500, the S&P MidCap 400 and the S&P SmallCap 600 index components. These indexes represent the return of the large-, the mid- and the small capitalization stocks. The market return is the one available in the CRSP database which is capitalization weighted and adjusted with dividend (VWRETD). This index tracks the return of the New York Stock Exchange- (NYSE), the American Stock Exchange- (AMEX) and the NASDAQ stocks. The risk-free rate is the return of the one-month Treasury bill from the CRSP. We use daily returns for a ten-year period of 1999-2008. Our data are not free of survivorship bias (see, e.g., Elton, 1996), that is only those companies are eligible for insertion into the database which are still on the market at the end of the investigated period. This can introduce some bias into the estimated parameters since those companies which go bankrupt, bear larger risk and might underperform the market significantly. However, we have to note that the survivorship bias can be smaller than the average since we include mid- and small cap firms besides large cap ones. For presenting our results we choose one stock randomly, the Lowe’s Companies Inc. (LOW), which is an S&P 500 component and another one which exhibits non-linearity in the CL, the National Oilwell Varco Inc (S&P 500 component).

Results

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In this section we present our results applying the estimations and testing presented in previous sections. Figure 4 Panel A shows the CL of the Lowe’s Inc. The bold curve is the kernel regression defined in Eq. (3), the dotted-dashed line is the linear regression defined in Eq. (2), the dotted curves are the confidence band of the kernel regression at 95% level. The R2 of the kernel regression is almost 4% higher (0.369 vs. 0.356), the alphas are significantly not different (they are the same up to four decimals); however, the linear beta is significantly downward biased (1.15 vs. 1.09). We cannot reject the linearity of the CL of the Lowe’s Inc., which result is also maintained by the 95% confidence band.8 On the other hand, linearity of the National Oilwell Varco Inc. in Figure 4 Panel B can be rejected at 95% level. Characteristic „curve” Panel A

Linear regression Kernel regression 95% Confidence band

8

We generate 250 different samples for the T test applying the wild bootstrapping method.

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Characteristic „curve” Panel B

Linear regression Kernel regression 95% Confidence band

Figure 4 The characteristic curve of the Lowe’s Companies Inc. (Panel A) and the National Oilwell Varco Inc. (Panel B) We estimate the characteristic curve of both companies by two different methods: 1, by the kernel regression, the ˆ h ( rm − rf ) + ηˆ j where ηˆ are the residuals, rj-rf and non-parametric CAPM is estimated (bold curve), that is rj − rf = m j rM-rf are the risk premium of the stock and the market respectively, h is an optimally selected bandwidth by Cross Validation (CV) (We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression); 2, by linear regression (dashed line), that is rj − rf = αˆ j + βˆ j ( rm − rf ) + εˆ j is estimated where εˆ j is the residual series. The market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. The dotted lines represent the 95% confidence bands.

We apply the above calculations for all the 150 randomly selected stocks in our database. The results are divided into three parts based on market capitalization: large-, mid- and small cap. The average R2 is higher for the kernel regression in all three segments. Considering the large cap stocks, linearity of the CLs can be rejected in 9 cases out of 50 at 95% significance level. This result is significant on which we can reject the linear relation between the returns of S&P 500 components and the market return. In the case of mid- and small capitalization stocks we can reject linearity of the CL in two cases each which means we cannot reject the nullhypothesis. However, we have to note that at 94% confidence level linearity can be rejected also for the small and mid cap equities. Altogether in 13 cases (8.7%) out of 150 we can reject the nullhypothesis, that is at 95% confidence level linearity of the CLs of US stocks can be rejected (see Table 1). Table 1 The results of kernel and linear regressions of characteristic curves In Panel A,B,C the estimated parameters and statistics of characteristic curves of 50-50-50 randomly chosen companies from the S&P 500, S&P MidCap 400, S&P SmallCap 600 universe can be seen respectively. We estimate the characteristic curve of all the companies by two different methods: 1, by the kernel regression, the nonˆ h ( rm − r f ) + ηˆ j where ηˆ are the residuals, rj-rf and rM-rf are the risk parametric CAPM is estimated, that is r j − r f = m j premium of the stock and the market respectively, h is an optimally selected bandwidth by Cross Validation (CV)

577

(We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression); 2, by linear regression, that is rj − rf = αˆ j + βˆ j ( rm − rf ) + εˆ j is estimated where εˆ j is the residual series. The market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. In the first column the tickers of each stock in the database, in column 2 the number of observations, in column 3 the mean return for the estimation period, in column 4 the p-value of the linearity test, in column 5 the optimal bandwidth of the kernel regression and then the R2s, alphas and betas of the kernel and the linear regressions can be seen. The number of asterisks means rejection of the linear nullhypothesis at 90%, 95%, 99% confidence level.

578

Panel A - S&P 500 Ticker ACAS AES APH BA BAX BJS BMC BRL CPWR D DD DHI DOV FDX FO GM HCBK HCP HPC HSP KFT LMT LOW LSI LXK MDT MI MTW MUR MWW NOV NUE ORCL PAYX PG QCOM S S SLE SPG SRE SYMC TDC TE TGT TLAB WFR WLP WLP ZMH Average

No. of obs. 2515 2515 2515 2515 2515 2515 2515 2509 2515 2515 2515 2515 2515 2515 2515 2515 2383 2515 2483 1175 1898 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 1565 2515 2515 2515 2515 316 2515 2515 2515 2515 1805 1486 1860 2362

E(r)

p-value

0.01 0.26 0.05 0.52 0.11 0.50 0.04 0.25 0.05 *0.01 0.09 0.28 0.04 **0.02 0.10 0.31 0.01 **0.05 0.05 0.56 0.00 0.36 0.07 0.42 0.02 0.36 0.04 **0.04 0.04 0.14 -0.05 0.47 0.11 0.64 0.08 *0.06 0.02 0.27 0.02 0.16 0.01 0.26 0.05 0.48 0.05 0.20 0.05 *0.06 0.02 ***0.00 0.01 0.72 0.02 0.19 0.05 0.60 0.09 *0.07 0.08 0.12 0.12 **0.04 0.11 0.30 0.09 **0.02 0.04 ***0.00 0.03 0.21 0.16 0.24 -0.06 *0.06 0.06 0.60 -0.01 0.14 0.07 0.28 0.05 0.31 0.12 **0.02 -0.15 0.32 0.01 0.25 0.04 0.39 -0.01 ***0.00 0.18 0.45 0.06 0.21 0.10 0.72 0.03 0.22 0.05 -

h 0.39 0.58 0.42 0.43 0.49 0.84 0.40 0.44 0.40 0.70 0.41 0.36 0.38 0.39 0.39 0.77 0.79 0.66 0.37 0.37 0.49 0.61 0.41 0.39 0.41 0.43 0.55 0.60 0.71 0.39 0.38 0.39 0.40 0.39 0.56 0.39 0.53 0.28 0.55 0.70 0.51 0.40 0.63 0.54 0.39 0.39 0.36 0.36 0.65 0.38 0.48

R 2KR 0.34 0.15 0.34 0.29 0.15 0.19 0.25 0.10 0.23 0.19 0.36 0.30 0.44 0.31 0.28 0.29 0.26 0.32 0.23 0.24 0.22 0.13 0.37 0.33 0.18 0.21 0.40 0.32 0.23 0.30 0.26 0.36 0.36 0.32 0.15 0.31 0.29 0.17 0.21 0.37 0.24 0.22 0.46 0.19 0.36 0.30 0.17 0.24 0.06 0.22 0.26

αˆ KR 0.00 0.04 0.10 0.03 0.04 0.09 0.04 0.09 0.01 0.04 -0.01 0.06 0.01 0.03 0.03 -0.06 0.10 0.07 0.02 0.01 0.00 0.04 0.04 0.05 0.02 0.00 0.01 0.05 0.08 0.07 0.11 0.10 0.09 0.03 0.02 0.16 -0.07 0.04 -0.02 0.06 0.04 0.12 -0.04 0.00 0.03 -0.02 0.17 0.05 0.08 0.03 0.04

βˆ KR 0.92 1.13 1.19 0.82 0.55 0.85 1.49 0.65 1.50 0.43 0.84 1.53 1.01 0.93 0.66 1.13 0.55 0.64 0.95 0.67 0.39 0.44 1.15 1.91 1.09 0.61 0.93 1.21 0.60 1.98 1.04 1.25 1.59 1.11 0.43 1.72 1.01 0.90 0.50 0.68 0.55 1.27 0.81 0.58 1.09 1.76 1.77 0.66 0.45 0.69 0.97

R2LR 0.29 0.14 0.33 0.27 0.13 0.17 0.24 0.09 0.22 0.19 0.35 0.29 0.44 0.30 0.26 0.28 0.26 0.27 0.20 0.21 0.20 0.12 0.36 0.31 0.17 0.20 0.35 0.31 0.20 0.29 0.22 0.34 0.35 0.30 0.13 0.30 0.25 0.16 0.18 0.30 0.21 0.21 0.42 0.16 0.34 0.28 0.16 0.19 0.05 0.20 0.24

αˆ LR 0.00 0.04 0.10 0.03 0.04 0.09 0.04 0.09 0.01 0.04 -0.01 0.06 0.01 0.03 0.03 -0.06 0.10 0.07 0.02 0.01 0.00 0.04 0.04 0.05 0.02 0.00 0.01 0.05 0.08 0.07 0.11 0.10 0.09 0.03 0.02 0.16 -0.06 0.04 -0.02 0.06 0.04 0.12 -0.04 0.00 0.03 -0.02 0.17 0.05 0.08 0.03 0.04

βˆ LR 1.09 1.17 1.18 0.83 0.52 0.98 1.30 0.63 1.41 0.49 0.86 1.34 0.97 0.87 0.66 1.29 0.61 0.82 0.93 0.62 0.49 0.51 1.09 1.76 0.94 0.64 1.10 1.25 0.77 1.77 1.21 1.27 1.49 1.01 0.44 1.51 1.25 0.90 0.56 0.90 0.62 1.19 0.85 0.61 1.06 1.50 1.75 0.68 0.42 0.65 0.98

579

Panel B - S&P MidCap 400 Ticker AAI AMG ARG AVCT BRO CLF CMG CPT CR CWTR CXW DCI ELY ENR FMER FNFG FTO HBI HE HMA HNI HRC HRL IDXX IEX IRF JBHT JBLU KMT LRCX MEG MRX MTX MVL NHP OII ORLY OSK PSD RS RYL SKS SRCL TECD THG UTHR UTR VARI WBS WOR Average

No. of obs. 2515 2484 2515 2118 2515 2515 737 2515 2515 2515 2515 2515 2515 2168 2515 2363 2515 581 2515 2515 2515 2515 2515 2515 2515 2515 2515 1673 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2515 2347 2515 2406 2515 2515 2400

E(r)

p-value

0.11 0.60 0.07 0.34 0.10 0.49 0.03 *0.06 0.08 0.21 0.13 *0.06 0.10 ***0.00 0.05 0.27 0.01 0.29 0.12 0.16 0.05 0.51 0.07 0.68 0.04 0.41 0.07 0.12 0.03 0.38 0.09 0.30 0.15 *0.10 -0.03 0.32 0.03 0.89 -0.02 *0.08 0.02 0.52 0.00 0.35 0.05 0.25 0.07 0.63 0.06 0.68 0.08 *0.07 0.10 0.32 0.02 0.67 0.06 0.10 0.14 **0.02 -0.06 0.26 0.02 0.31 0.02 0.58 0.14 0.41 0.07 0.14 0.10 0.15 0.07 *0.08 0.07 0.65 0.03 0.43 0.08 *0.08 0.09 0.53 -0.01 0.32 0.14 0.44 0.01 0.27 0.02 0.40 0.15 0.26 0.01 0.28 0.10 0.32 0.01 0.14 0.05 0.18 0.06 -

h 0.63 0.57 0.48 0.53 0.47 0.40 0.38 0.83 0.31 0.36 0.62 0.43 0.46 0.54 0.33 0.53 0.37 0.69 0.62 0.82 0.74 0.45 0.36 0.46 0.40 0.35 0.43 0.52 0.41 0.38 1.05 0.42 0.44 0.40 0.54 0.53 0.38 0.49 0.63 0.44 0.45 0.39 0.39 0.40 0.37 0.34 0.50 0.33 0.44 0.41 0.48

R 2KR 0.17 0.45 0.24 0.26 0.21 0.32 0.29 0.32 0.39 0.15 0.07 0.31 0.22 0.18 0.40 0.24 0.22 0.32 0.18 0.11 0.25 0.14 0.11 0.16 0.35 0.31 0.23 0.21 0.38 0.35 0.16 0.17 0.32 0.09 0.31 0.18 0.22 0.20 0.13 0.37 0.29 0.22 0.11 0.22 0.23 0.09 0.36 0.25 0.39 0.32 0.24

αˆ KR 0.10 0.06 0.09 0.04 0.07 0.13 0.14 0.04 0.00 0.12 0.04 0.06 0.04 0.07 0.02 0.08 0.15 0.01 0.02 -0.03 0.01 -0.01 0.04 0.06 0.05 0.08 0.10 0.00 0.06 0.13 -0.07 0.01 0.01 0.13 0.06 0.09 0.06 0.06 0.02 0.07 0.08 -0.01 0.13 0.01 0.02 0.16 0.00 0.11 0.00 0.04 0.06

βˆ KR 1.41 1.32 0.99 1.52 0.65 1.13 1.40 0.60 0.94 1.31 0.71 0.82 1.02 0.71 0.96 0.74 0.92 0.91 0.41 0.51 0.84 0.47 0.46 0.77 0.90 1.71 1.06 1.22 1.04 2.04 0.82 0.77 0.82 0.76 0.77 0.86 0.94 0.94 0.41 1.12 1.40 1.09 0.67 1.19 0.93 0.97 0.82 1.27 0.93 1.12 0.96

R 2LR 0.16 0.43 0.23 0.25 0.20 0.26 0.24 0.27 0.33 0.13 0.06 0.31 0.21 0.17 0.35 0.22 0.18 0.28 0.18 0.06 0.25 0.13 0.09 0.16 0.34 0.29 0.22 0.21 0.37 0.32 0.15 0.16 0.32 0.09 0.29 0.16 0.20 0.18 0.13 0.34 0.28 0.19 0.10 0.20 0.22 0.08 0.32 0.23 0.37 0.30 0.22

αˆ LR 0.10 0.06 0.09 0.04 0.07 0.13 0.13 0.04 0.00 0.12 0.04 0.06 0.04 0.07 0.02 0.08 0.15 0.01 0.02 -0.03 0.01 -0.01 0.04 0.06 0.05 0.08 0.10 0.00 0.06 0.13 -0.07 0.01 0.01 0.13 0.06 0.10 0.06 0.06 0.02 0.07 0.08 -0.01 0.13 0.01 0.02 0.16 0.00 0.11 0.00 0.04 0.06

βˆ LR 1.24 1.38 1.03 1.44 0.64 1.35 1.00 0.76 0.92 1.25 0.69 0.84 0.98 0.71 1.01 0.77 1.09 1.00 0.39 0.61 0.93 0.50 0.38 0.75 0.90 1.51 1.01 1.14 1.09 1.77 1.07 0.80 0.84 0.79 0.83 0.94 0.89 0.94 0.38 1.29 1.27 1.05 0.64 1.02 0.93 0.83 0.92 1.16 1.00 1.15 0.96

580

Panel C - S&P SmallCap 600 Ticker ABM ACLS CASY CCRN DNEX EE GFF GTIV HMSY HOMB HTLD INT ISYS KNOT MANT MFB MNT MNT MOH MTH NILE NOVN NPK NSIT NTRI NWK ONB PBY PEI PENX PRFT PRGS PVTB RADS RBN SAFM SKT SLXP SNS SNS SSS SUP SYMM TSFG TXRH UEIC UNS VECO VSEA ZEP Average

No. of obs. 2454 2077 2426 1766 2473 2373 2396 2142 2313 627 2440 2412 2405 1292 1716 850 2461 398 1366 2417 1152 2440 2431 2478 1327 2404 2387 2428 2397 2396 2228 2461 2255 2439 2424 2402 2426 1997 2397 106 2434 2464 2432 2446 1046 2426 2381 2482 2429 287 2052

E(r)

p-value

0.06 0.38 0.02 0.63 0.04 *0.06 0.04 0.28 0.06 0.39 0.03 0.42 -0.01 0.38 -0.01 0.31 0.05 0.37 0.09 0.36 0.09 0.11 0.04 0.30 0.06 0.96 0.05 0.84 0.02 0.24 0.13 0.65 0.09 0.28 -0.01 0.30 0.05 0.22 0.10 0.27 0.14 0.25 0.12 0.52 0.07 0.24 0.05 0.65 0.13 0.25 0.10 0.35 0.10 0.64 0.00 0.31 0.04 **0.03 0.06 0.35 0.05 0.62 0.09 0.52 0.14 0.67 0.10 0.49 0.22 *0.08 0.16 *0.08 -0.06 0.14 0.10 0.28 0.00 0.61 0.11 0.44 0.04 0.36 0.04 0.54 -0.05 0.32 0.05 0.13 0.00 0.14 -0.02 0.48 0.10 0.19 0.02 0.21 0.20 **0.03 0.30 0.54 0.07 -

h 0.40 0.54 0.44 0.41 0.50 0.54 0.51 0.40 0.90 0.75 0.40 0.77 0.56 0.91 0.48 0.57 3.31 1.01 0.90 0.70 0.57 0.50 0.67 0.43 0.56 0.54 0.67 0.35 0.86 0.64 0.65 0.40 0.54 0.40 0.41 0.35 0.56 0.58 0.64 1.21 0.55 0.49 0.47 0.38 0.32 0.40 0.52 0.46 0.32 1.22 0.63

R 2KR 0.25 0.27 0.23 0.19 0.20 0.21 0.20 0.15 0.03 0.30 0.21 0.23 0.11 0.04 0.11 0.31 0.01 0.01 0.10 0.25 0.16 0.09 0.28 0.19 0.05 0.11 0.26 0.21 0.34 0.08 0.04 0.20 0.12 0.17 0.21 0.11 0.29 0.09 0.15 0.05 0.34 0.26 0.17 0.31 0.30 0.16 0.21 0.28 0.30 0.39 0.19

αˆ KR 0.03 -0.03 0.06 0.00 0.04 0.05 0.03 0.16 0.13 0.13 0.08 0.13 0.07 -0.03 0.11 -0.01 0.09 0.02 0.03 0.14 0.05 0.12 0.06 0.03 0.05 0.04 0.02 0.01 0.01 0.04 0.22 0.04 0.10 0.09 0.05 0.10 0.09 0.09 -0.01 0.27 0.06 -0.01 0.08 -0.02 0.00 0.09 0.05 -0.01 0.14 0.40 0.07

βˆ KR 0.85 2.20 1.05 1.25 0.95 0.74 0.92 0.82 0.46 1.06 1.02 0.79 0.87 0.68 0.94 1.25 0.68 0.23 0.61 1.56 1.32 1.04 0.57 1.44 1.03 1.32 0.77 1.30 0.75 0.65 0.94 1.08 0.74 1.39 0.90 0.79 0.68 1.15 0.96 0.51 0.68 0.87 1.53 1.06 1.43 1.13 0.54 1.88 1.90 1.21 1.01

R2LR 0.23 0.26 0.21 0.18 0.20 0.21 0.20 0.13 0.03 0.28 0.20 0.19 0.11 0.04 0.10 0.30 0.04 0.01 0.10 0.21 0.14 0.08 0.26 0.18 0.04 0.11 0.25 0.19 0.28 0.08 0.03 0.20 0.11 0.15 0.19 0.09 0.26 0.08 0.15 0.06 0.31 0.25 0.16 0.24 0.20 0.16 0.20 0.24 0.27 0.38 0.17

αˆ LR 0.03 -0.04 0.06 0.00 0.04 0.05 0.03 0.16 0.13 0.13 0.09 0.12 0.07 -0.03 0.11 -0.01 0.09 0.02 0.03 0.14 0.05 0.12 0.06 0.03 0.05 0.04 0.02 0.01 0.01 0.04 0.22 0.04 0.10 0.09 0.05 0.10 0.09 0.09 -0.01 0.27 0.06 -0.01 0.08 -0.02 0.00 0.09 0.05 -0.01 0.14 0.38 0.07

βˆ LR 0.80 1.84 0.86 0.97 0.86 0.66 0.90 0.70 0.51 0.81 0.85 1.02 0.86 0.85 0.71 1.19 0.48 0.19 0.70 1.48 0.99 0.94 0.65 1.31 0.78 1.09 0.75 1.16 1.01 0.72 1.01 0.99 0.65 1.41 0.89 0.67 0.72 0.90 0.82 0.57 0.78 0.82 1.33 1.20 0.87 1.02 0.61 1.56 1.67 1.07 0.92

We reject the linear nullhypothesis at 18% of the S&P 500 companies and at 8.7% of all the companies, in which cases the parameters estimated by linear regression are biased and inconsistent, thus the market risk (”beta”) and the abnormal return (”alpha”) should be estimated by non-parametric methods. The parametric and semi-parametric alphas are not significantly 581

different; however, the average difference between the linear and the kernel betas is 11% which is a significant result.9 The linear CL of mid- and small cap firms cannot be rejected, thus in these cases linear regression can be used. The abnormal returns of large cap stocks estimated by both methods do not differ significantly; the average alphas of those stocks which exhibit non-linearity are 0.05 vs. 0.05 and which exhibit linearity are 0.04 vs. 0.04. However, the average non-linear kernel betas are significantly higher than the OLS estimated (1.13 vs. 1.21), while if linearity holds there is not such a significant difference (0.90 vs. 0.92). There is no significant difference between the alphas of the mid cap stocks, if linearity can be rejected (0.14 vs. 0.13) and if cannot be rejected, they are the same (0.05). Betas are different, if linearity does not hold, they are 1.72 vs. 1.39; however, if we are not able to reject the linearity, there is no significant difference (0.93 vs. 0.94). In the case of small firms, the average alphas do not differ if linearity holds (0.07); however, betas are slightly different, 1.00 vs. 0.91. If we reject linearity, alphas are still the same on average (0.08) and betas do not differ significantly (1.33 vs. 1.34). Summarizing the results of parameters estimation, we can argue that the kernel and the OLS alphas are almost the same; it does not matter whether or not linearity holds. On the other hand, the OLS parameter estimation is significantly downward biased when linearity of the CL can be rejected. The average beta is inversely related to firm size confirming the small firm effect (see, e.g., Banz, 1981; Basu, 1983). Those stocks which feature non-linear characteristic curve have higher “betas” which is a sign that extremes cause invalidity of linearity since outliers raise the risk; on the other hand, the extremes are hard to be explained by the market linearly. It is a striking result that if linearity can be rejected, the difference in betas estimated by both methods is not the largest among small firms but among mid caps; however we have to note that there are only two mid cap stocks in the database whose characteristic curve exhibit non-linearity. If linearity can be rejected the difference is also significant at large cap stocks which along with the previous result indicate that linearity is not connected with firm size. Figure 5 shows the derivative estimation of the characteristic curve of Lowe’s Inc. in the function of the market risk premium. It can be seen that the estimated relevant risk is not constant, it behaves very volatile at the tails, in addition, on the positive tail risk rises along with the market risk premium. Figure 5 Panel B shows the derivative estimation of the National Oilwell Varco Inc.. The estimated market risk is similar to the one in Panel A; under normal circumstances, beta is constant; however, under extreme circumstances, estimation is very volatile. ”Beta” is constant at the central part of the distribution maintaining the CAPM; however, at the tails it behaves differently. The non-constant risk estimation has several reasons. First, it can be imagined that linearity does not hold at the tails causing non-constant derivative estimation. Second, the number of observations is low at the tails which introduces noise in the estimation.

9

The average difference is calculated by

(

1 150 ∑ abs 1 − βˆ LR / βˆ KR 150 j =1

). 582

Semiparametric derivative estimation

Slope

Panel A

Semiparametric derivative estimation

Slope

Panel B

Figure 5 Semi-parametric derivative estimation. The characteristic curves of the Lowe’s Companies Inc. (Panel A) and the National Oilwell Varco Inc. (Panel B) (randomly chosen stocks) are estimated by the kernel regressions. The market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. ˆ h ( rm − rf ) + ηˆ j where ηˆ are the residuals, rj-rf and rM-rf The model, the non-parametric CAPM is in the form rj − rf = m j

are the risk premium of the stock and the market respectively, h is an optimally selected bandwidth by Cross Validation (CV). We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression. The figure shows the estimated derivative in the function of the risk premium. The semiparametric derivative estimation is obtained by the weighted least squares estimator using the Nadaraya-Watson kernel weights.

583

Figure 6 Panel A shows the performance estimation of Lowe’s Inc. in the function of the market risk premium. Under normal circumstances, “alpha” is stable and close to zero; however, security reacts differently to the extreme negative and positive movements of the market. The stock price can overreact the market movements on the extreme negative side inducing significant negative ”alpha”; however, on the positive tail we measure significant positive performance, that is the bigger the extreme market movement, the larger the abnormal return is. We have to note that the regression estimate is less precise at the tails, since the probability of extreme movements is relatively low and the probability of occurrence declines departing from the normal level of market movements. Figure 6 Panel B shows the performance estimation of the National Oilwell Varco Inc. Results are similar in the case of the Lowe’s Inc., under normal circumstances, the ”alpha” is relatively stable and significantly not different from zero; while under extreme circumstances, the performance is significantly negative at the negative tail and significantly positive on the positive tail. This result has an important notice for the mutual fund industry; managers can beat the market only when extreme market movements occur. Semiparametric alpha estimation

Abnormal performance (alpha)

Panel A

584

Semiparametric alpha estimation

Abnormal performance (alpha)

Panel B

Figure 6 Semi-parametric alpha estimation The characteristic curves of the Lowe’s Companies Inc. (Panel A) and the National Oilwell Varco Inc. (Panel B) (randomly selected stocks) are estimated by the kernel regressions. The market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. ˆ h ( rm − rf ) + ηˆ j where ηˆ are the residuals, rj-rf and rM-rf The model, the non-parametric CAPM is in the form rj − rf = m j

are the risk premium of the stock and the market respectively, h is an optimally selected bandwidth by Cross Validation (CV). We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression. The figure shows the estimated semi-parametric alpha (the abnormal return) in the function of the risk premium. The semi-parametric derivative estimation is obtained by the weighted least squares estimator using the Nadaraya-Watson kernel weights.

We use the semi-parametric betas in Table 1 as explanatory variable and the average daily returns as the dependent variable to estimate the SMLs. Table 2 shows the estimated parameters of the SMLs by market capitalization.

585

Table 2 Estimations of SMLs

We estimate SMLs for the S&P 500, S&P MidCap 400, S&P SmallCap 600 stocks and for all the stocks in our database. The rows 1,2,3,4 include the estimated parameters and statistics of the SMLs for the S&P 500, the S&P MidCap 400, the S&P SmallCap 600 and for all the stocks, respectively. We estimate the SMLs by two methods: 1,

( )

ˆ h βˆ ∗j + λˆ j , where r is the average return of stock j in the investigated by the kernel regressions in the form r j = m j period, βˆ ∗ is the semi-parametric beta of stock j, λˆ is the residual series and h is an optimally selected bandwidth j

j

by Cross Validation (CV) (We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression.); 2, by linear regression in the form rj = αˆ ′j + βˆ ′j βˆ ∗j + κˆ j , where κˆ j are the residuals. The market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. In the third column the mean return of the given index for the estimation period, in column 4 the p-value of the linearity test, in column 5 the optimal bandwidth of the kernel regression and then the R2s, alphas and betas of the kernel and the linear regressions can be seen. Security Market Lines Segment S&P 500 S&P MidCap 400 S&P SmallCap 600 All companies

No. of obs.

E(r)

p-value

50 50 50 150

0.04798 0.06083 0.06749 0.05877

0.144 0.728 0.760 0.688

h

R 2KR

αˆ ′KR

βˆ ′KR

R2LR

αˆ ′LR

βˆ ′LR

20.504 15.896 18.728 14.751

0.1169 0.1311 0.0885 0.0424

0.0401 0.0217 0.1022 0.0504

0.0081 0.0407 -0.0344 0.0085

0.0516 0.1043 0.0263 0.0100

0.0192 0.0141 0.0957 0.0439

0.0296 0.0486 -0.0280 0.0152

The mid cap SML has the highest slope coefficient (0.0407) as, the slope of the large cap line is very small (0.0081) and the curve is almost flat. The slope of the SML estimated for S&P SmallCap 600 stocks is negative (-0.0344), that is larger risk would induce smaller expected return. This result is connected to the small firm effect; small firms gain relatively large expected return (see, e.g., Banz, 1981; Basu, 1983; Fama and French, 1995; 1996). The SML estimated for all the 150 companies in our database has a slope coefficient of 0.0085 which is still very flat. Figure 7 shows the SMLs for the S&P size indexes and for all the stocks. We apply the wild bootstrap linearity test for the SMLs and we can conclude that linearity cannot be rejected at any usual significance level. Security market line – S&P 500

Average return

Panel A

Linear regression Kernel regression 95% Confidence band Risk (beta)

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Security market line – S&P MidCap 400

Average return

Panel B

Linear regression Kernel regression 95% Confidence band Risk (beta) Security market line – S&P SmallCap 600

Average return

Panel C

Linear regression Kernel regression 95% Confidence band Risk (beta)

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Security market line – All companies

Average return

Panel D

Linear regression Kernel regression 95% Confidence band Risk (beta)

Figure 7 Security market lines for the S&P 500, the S&P MidCap 400, the S&P SmallCap 600 and for all the stocks

We estimate security market lines for the S&P 500, S&P MidCap 400, S&P SmallCap 600 stocks and for all the stocks in our database in Panel A, B, C, D respectively. We estimate the security market lines by two methods: 1, by ˆ h βˆ ∗j + λˆ j , where r is the average return of stock j in the the kernel regressions (bold curves) in the form rj = m j

( )

investigated period, βˆ ∗j is the semi-parametric beta of stock j, λˆ j is the residual series and h is an optimally selected bandwidth by Cross Validation (CV) (We use the Gaussian kernel and the Nadaraya (1964) and Watson (1964) weighting function in the kernel regression.); 2, by linear regression (dashed lines) in the form rj = αˆ ′j + βˆ ′j βˆ ∗j + κˆ j , where κˆ j are the residuals. The market return is the Center for Research in Security Prices (CRSP) value weighted index return; the risk free rate is the return of the one-month Treasury bill. The dotted curves represent the 95% confidence bands.

Concluding remarks and future research indications Based on our results, linearity of the CL can be rejected in 8.7% of the US stocks, so the standard linear estimators cannot be used for parameter estimations of market risk and performance. We propose using semi-parametric approaches for estimating “alphas” and “betas” since our study shows that they do not differ significantly from the linear measures when linearity holds, while they provide good alternatives when linearity can be rejected. Both risk measures, that is alpha and beta are not constant when extreme market movements occur; however, standard measures would indicate constant estimations. The linear beta is significantly downward biased when linearity does not hold, which problem can be solved by our semi-parametric beta measure. Our semi-parametric alpha shows that managers can beat the market only under extreme market conditions.

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