A Dynamic CAPM with Supply Effect Theory and Empirical Results

Abstract

Breeden (1979) and Grinols (1984) and Cox et al. (1985) have described the importance of supply side for the capital asset pricing. Black (1976) derives a dynamic, multiperiod CAPM, integrating endogenous demand and supply. However, Black’s theoretically elegant model has never been empirically tested for its implications in dynamic asset pricing. We first theoretically extend Black’s CAPM. Then, we use price, dividend per share, and earnings per share to test the existence of supply effect with US equity data. We find the supply effect is important in US domestic stock markets. This finding holds as we break the companies listed in the S&P 500 into ten portfolios by different level of payout ratio. It also holds consistently if we use individual stock data.

A simultaneous equation system is constructed through a standard structural form of a multiperiod equation to represent the dynamic relationship between supply and demand for capital assets. The equation system is exactly identified under our specification. Then, two hypotheses related to supply effect are tested regarding the parameters in the reduced form system. The equation system is estimated by the seemingly unrelated regression (SUR) method, since SUR allow one to estimate the presented system simultaneously while accounting for the correlated errors.

Appendices

Appendix 1: Modeling the Price Process

In Sect. 93.2.3, Eq. 93.16 is derived from Eq. 93.15 under the assumption that all countries’ index series follow a random walk process. Thus, before further discussion, we should test the order of integration of these price series. From Hamilton (1994), we know that two widely used unit root tests are the Dickey-Fuller (DF) and the augmented Dickey-Fuller (ADF) tests. The former can be represented as P _t = μ + γP _t−1 + ε _t , and the latter can be written as ΔP _t = μ + γ P _t−1 + δ ₁ΔP _t−1 + δ ₂ΔP _t−2 + … + δ _p ΔP _t−p + ε _t .

Similarly, in the US stock markets, the Phillips-Perron test is used to check whether the value-weighted price of market portfolio follows a random walk process. The results of the tests for each index are summarized in Table 93.7. It seems that one cannot reject the hypothesis that all indices follow a random walk process since, for example, the null hypothesis of unit root in level cannot be rejected for all indices but are all rejected if one assumes there is a unit root in the first-order difference of the price for each portfolio. This result is consistent with most studies that find that the financial price series follow a random walk process.

Table 93.7 Unit root tests for P_t

Appendix 2: Identification of the Simultaneous Equation System

Note that given G is nonsingular, Π = −G⁻¹ H in Eq. 93.19 can be written as

$$ \begin{array}{l}\mathrm{where}\ \mathrm{A}=\left[\begin{array}{cc} \mathrm{G} & \mathrm{H} \end{array}\right]=\left(\begin{array}{cccccccc} {\mathrm{g}}_{11} & {\mathrm{g}}_{12} & \dots \dots & \hfill {\mathrm{g}}_{1\mathrm{n}} & {\mathrm{h}}_{11} & {\mathrm{h}}_{12} & \dots \dots & {\mathrm{h}}_{1\mathrm{n}} \\ {} {\mathrm{g}}_{21} & {\mathrm{g}}_{22} & \dots \dots & {\mathrm{g}}_{1\mathrm{n}} & {\mathrm{h}}_{21} & {\mathrm{h}}_{22} & \dots \dots & {\mathrm{h}}_{2\mathrm{n}} \\ {}. & & & & & &. & \\ {}. & & & & & &. & \\ {} {\mathrm{g}}_{\mathrm{n}1} & {\mathrm{g}}_{\mathrm{n}2} & \dots \dots & {\mathrm{g}}_{\mathrm{n}\mathrm{n}} & {\mathrm{h}}_{\mathrm{n}1} & {\mathrm{h}}_{\mathrm{n}2} & \dots \dots & {\mathrm{h}}_{\mathrm{n}\mathrm{n}} \end{array}\right)\\ {}\kern1.68em \mathrm{W}=\left[\begin{array}{cc} \Pi & {\mathrm{I}}_{\mathrm{n}} \end{array}\right]\hbox{'}=\left(\begin{array}{cccccccc} {\pi}_{11} & {\pi}_{12} & \dots & {\pi}_{1\mathrm{n}} & 1 & 0 & \dots \dots & 0 \\ {} {\pi}_{21} & {\pi}_{22} & \dots & {\pi}_{1\mathrm{n}} & 0 & 1 & \dots \dots & 0 \\ {}. & & & & & &. & \\ {}. & & & & & &. & \\ {} {\pi}_{\mathrm{n}1} & {\pi}_{\mathrm{n}2} & \dots & {\pi}_{\mathrm{n}\mathrm{n}} & 0 & 0 & \dots \dots & 1 \end{array}\right)\end{array} $$

(93.24)

That is, A is the matrix of all structure coefficients in the model with dimension of (n × 2n), and W is a (2n × n) matrix. The first equation in Eq. (93.24) can be expressed as

$$ {\mathrm{A}}_1\mathrm{W}=0, $$

(93.25)

where A₁ is the first row of A, i.e., A₁ = [g₁₁ g₁₂ …. g_1n h₁₁ h₁₂ ….. h_1n].

Since the elements of Π can be consistently estimated, and I_n is the identity matrix, Eq. 93.25 contains 2n unknowns in terms of πs. Thus, there should be n restrictions on the parameters to solve Eq. 93.25 uniquely. First, one can try to impose normalization rule by setting g₁₁ equal to 1 to reduce one restriction. As a result, there are at least n−1 independent restrictions needed in order to solve Eq. 93.25.

It can be illustrated that the system represented by Eq. 93.17 is exactly identified with three endogenous and three exogenous variables. It is entirely similar to those cases of more variables. For example, if n = 3, Eq. 93.17 can be expressed in the form

$$ \begin{array}{l}-\kern0.36em \left(\begin{array}{ccc}\hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{11}+{\mathrm{a}}_1{\mathrm{b}}_1\hfill & \hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{12}\hfill & \hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{13}\hfill \\ {}\hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{21}\hfill & \hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{22}+{\mathrm{a}}_2{\mathrm{b}}_2\hfill & \hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{23}\hfill \\ {}\hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{31}\hfill & \hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{32}\hfill & \hfill \mathrm{r}*\mathrm{c}{\mathrm{s}}_{33}+{\mathrm{a}}_3{\mathrm{b}}_3\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\mathrm{p}}_{1\mathrm{t}}\hfill \\ {}\hfill {\mathrm{p}}_{2\mathrm{t}}\hfill \\ {}\hfill {\mathrm{p}}_{3\mathrm{t}}\hfill \end{array}\right)\\ {}\kern2.16em +\kern0.36em \left(\begin{array}{ccc}\hfill \mathrm{c}{\mathrm{s}}_{11}+{\mathrm{a}}_1\hfill & \hfill \mathrm{c}{\mathrm{s}}_{12}\hfill & \hfill \mathrm{c}{\mathrm{s}}_{13}\hfill \\ {}\hfill \mathrm{c}{\mathrm{s}}_{21}\hfill & \hfill \mathrm{c}{\mathrm{s}}_{22}+{\mathrm{a}}_2\hfill & \hfill \mathrm{c}{\mathrm{s}}_{23}\hfill \\ {}\hfill \mathrm{c}{\mathrm{s}}_{31}\hfill & \hfill \mathrm{c}{\mathrm{s}}_{32}\hfill & \hfill \mathrm{c}{\mathrm{s}}_{33}+{\mathrm{a}}_3\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\mathrm{d}}_{1\mathrm{t}}\hfill \\ {}\hfill {\mathrm{d}}_{2\mathrm{t}}\hfill \\ {}\hfill {\mathrm{d}}_{3\mathrm{t}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\mathrm{v}}_{1\mathrm{t}}\hfill \\ {}\hfill {\mathrm{v}}_{2\mathrm{t}}\hfill \\ {}\hfill {\mathrm{v}}_{3\mathrm{t}}\hfill \end{array}\right)\end{array} $$

(93.26)

where

• r* = scalar of risk-free rate

• s_ij = elements of variance-covariance matrix of return

• a_i = inverse of the supply adjustment cost of firm i

• b_i = overall cost of capital of firm i

For example, in the case of n = 3, Eq. 93.17 can be written as

$$ \left(\begin{array}{l}{\mathrm{g}}_{11}\kern0.5em {\mathrm{g}}_{12}\kern0.5em {\mathrm{g}}_{13}\\ {}{\mathrm{g}}_{21}\kern0.5em {\mathrm{g}}_{22}\kern0.5em {\mathrm{g}}_{23}\\ {}{\mathrm{g}}_{31}\kern0.5em {\mathrm{g}}_{32}\kern0.5em {\mathrm{g}}_{33}\end{array}\right)\left(\begin{array}{l}{\mathrm{p}}_{1\mathrm{t}}\\ {}{\mathrm{p}}_{2\mathrm{t}}\\ {}{\mathrm{p}}_{3\mathrm{t}}\end{array}\right)+\left(\begin{array}{l}{\mathrm{h}}_{11}\kern0.5em {\mathrm{h}}_{12}\kern0.5em {\mathrm{h}}_{13}\\ {}{\mathrm{h}}_{21}\kern0.5em {\mathrm{h}}_{22}\kern0.5em {\mathrm{h}}_{23}\\ {}{\mathrm{h}}_{31}\kern0.5em {\mathrm{h}}_{32}\kern0.5em {\mathrm{h}}_{33}\end{array}\right)\left(\begin{array}{l}{\mathrm{d}}_{1\mathrm{t}}\\ {}{\mathrm{d}}_{2\mathrm{t}}\\ {}{\mathrm{d}}_{3\mathrm{t}}\end{array}\right)=\left(\begin{array}{l}{\mathrm{v}}_{1\mathrm{t}}\\ {}{\mathrm{v}}_{2\mathrm{t}}\\ {}{\mathrm{v}}_{3\mathrm{t}}\end{array}\right). $$

(93.27)

Comparing Eq. 93.26 with Eq. 93.27, the prior restrictions on the first equation take the form g₁₂ = −r*h₁₂ and g₁₃ = −r*h₁₃ and so on.

Thus, the restriction matrix for the first equation is of the form

$$ \Phi =\left(\begin{array}{cccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathrm{r}}^{*}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathrm{r}}^{*}\hfill \end{array}\right) $$

(93.28)

Then, combining Eq. 93.25 and the parameters of the first equation gives

$$ \begin{array}{ccc}\hfill \left[\begin{array}{cccccc}\hfill {\mathrm{g}}_{11\ }\hfill & \hfill {\mathrm{g}}_{12}\hfill & \hfill {\mathrm{g}}_{13}\hfill & \hfill {\mathrm{h}}_{11\ }\hfill & \hfill {\mathrm{h}}_{12\ }\hfill & \hfill {\mathrm{h}}_{13}\hfill \end{array}\right]\hfill & \hfill \left(\begin{array}{ccccc}\hfill {\uppi}_{11}\hfill & \hfill {\uppi}_{12}\hfill & \hfill {\uppi}_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {\uppi}_{21}\hfill & \hfill {\uppi}_{22}\hfill & \hfill {\uppi}_{13}\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill {\uppi}_{31}\hfill & \hfill {\uppi}_{32}\hfill & \hfill {\uppi}_{33}\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \mathrm{r}*\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \mathrm{r}*\hfill \end{array}\right)\hfill & \hfill \left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\hfill \end{array}. $$

(93.29)

That is, extending Eq. 93.29, we have

$$ \begin{array}{l}{\mathrm{g}}_{11}{\pi}_{11}+{\mathrm{g}}_{12}{\pi}_{21}+{\mathrm{g}}_{13}{\pi}_{31}+{\mathrm{h}}_{11}=0,\\ {}{\mathrm{g}}_{11}{\pi}_{12}+{\mathrm{g}}_{12}{\pi}_{22}+{\mathrm{g}}_{13}{\pi}_{32}+{\mathrm{h}}_{12}=0,\\ {}{\mathrm{g}}_{11}{\pi}_{13}+{\mathrm{g}}_{12}{\pi}_{23}+{\mathrm{g}}_{13}{\pi}_{33}+{\mathrm{h}}_{13}=0,\\ {}{\mathrm{g}}_{12}+{\mathrm{r}}^{*}{\mathrm{h}}_{12}=0,\mathrm{and}\\ {}{\mathrm{g}}_{13}+{\mathrm{r}}^{*}{\mathrm{h}}_{13}=0.\end{array} $$

(93.30)

The last two (n−1 = 3−1 = 2) equations in Eq. 93.30 give the value h₁₂ and h₁₃, and the normalization condition, g₁₁ = 1, allows us to solve Eq. 93.25 in terms of πs uniquely. That is, in the case n = 3, the first equation represented by Eq. 93.25, A₁W = 0, can be finally rewritten as Eq. 93.30. Since there are three unknowns, g₁₂, g₁₃, and h₁₁, left for the first three equations in Eq. 93.30, the first equation A₁ is exactly identified. Similarly, it can be shown that the second and the third equations are also exactly identified.

Appendix 3: Derivation of the Formula Used to Estimate d_t

To derive the formula for estimating d_t, we first define the partial adjustment model as

$$ {D}_t={a}_1+{a}_2{D}_{t-1}+{a}_3{E}_t+{u}_t $$

(93.31)

where D _t = dividend per share in period t, D _t−1 = dividend per share in period t−1, E _t = earnings per share in period t are dividends and earnings, and u_t = error term in period t. Similarly,

$$ {D}_{t+1}={a}_1+{a}_2{D}_{\mathrm{t}}+{a}_3{E}_{t+1}+{u}_{t+1}. $$

(93.31′)

And thus,

$$ {\mathrm{E}}_{t-1}\left[{D}_t\right]={a}_1+{a}_2{D}_{t-1}+{a}_3{\mathrm{E}}_{t-1}\left[{E}_t\right], $$

(93.32)

$$ {\mathrm{E}}_t\left[{D}_{t+1}\right]={a}_1+{a}_2{D}_t+{a}_3{\mathrm{E}}_t\left[{E}_{t+1}\right], $$

(93.33)

$$ {\mathrm{E}}_{t-1}\left[{D}_{t+1}\right]={a}_1+{a}_2{\mathrm{E}}_{t-1}\left[{D}_t\right]+{a}_3{\mathrm{E}}_{t-1}\left[{E}_{t+1}\right]. $$

(93.34)

Substituting Eq. 93.32 to Eq. 93.34, we have

$$ {E}_{t-1}\left[{D}_{t+1}\right]={a}_1+{a}_1{a}_2+{a}_2^2{D}_{t-1}+{a}_2{a}_3{\mathrm{E}}_{t-1}\left[{E}_t\right]+{a}_3{\mathrm{E}}_{t-1}\left[{E}_{t+1}\right]. $$

(93.34′)

Subtracting Eq. 93.34′ from Eq. 93.33 on both hand sides, we have

$$ {\mathrm{E}}_t\left[{D}_{t+1}\right]-{\mathrm{E}}_{t-1}\left[{D}_{t+1}\right]=-{a}_1{a}_2+{a}_2{D}_t-{a}_2^2{D}_{t-1}-{a}_2{a}_3{\mathrm{E}}_{t-1}\left[{E}_t\right]+{a}_3{\mathrm{E}}_t\left[{E}_{t+1}\right]-{a}_3{\mathrm{E}}_{t-1}\left[{E}_{t+1}\right]. $$

(93.35)

Equation Eq. 93.35 can be investigated depending upon whether E_t is following a random walk.

Case 1

E_t follows an AR(p) process.

If the time series of E _t is assumed to be stationary and follows an AR(p) process, then after taking the seasonal differences, we obtain

$$ d{E}_t={\rho}_0+{\rho}_1d{E}_{t-1}+{\rho}_2d{E}_{t-2}+{\rho}_3d{E}_{t-3}+{\rho}_4d{E}_{t-4}+{\varepsilon}_t, $$

(93.36)

where dE _t = E _t − E _t−4.

The expectation adjustment in seasonally differenced earnings, or the revision in forecasting future seasonally differenced earnings, can be solved as

$$ {\mathrm{E}}_t\left[d{E}_{t+1}\right]-{\mathrm{E}}_{t-1}\left[d{E}_{t+1}\right]={\rho}_1\left(-{\rho}_0+d{E}_t-{\rho}_1d{E}_{t-1}-{\rho}_2d{E}_{t-2}-{\rho}_3d{E}_{t-3}-{\rho}_4d{E}_{t-4}\right). $$

(93.37)

Since _t [dE _t+1] − _t−1[dE _t+1] = _t [E _t+1] − _t−1[E _t+1], we have

$$ {}_t\left[{E}_{t+1}\right]-{}_{t-1}\left[{E}_{t+1}\right]={\rho}_1\left(-{\rho}_0+d{E}_t-{\rho}_1d{E}_{t-1}-{\rho}_2d{E}_{t-2}-{\rho}_3d{E}_{t-3}-{\rho}_4d{E}_{t-4}\right). $$

(93.38)

Furthermore, from Eq. 93.36, we have

$$ {}_{t-1}\left[d{E}_t\right]={\rho}_0+{\rho}_1d{E}_{t-1}+{\rho}_2d{E}_{t-2}+{\rho}_3d{E}_{t-3}+{\rho}_4d{E}_{t-4}. $$

(93.39)

Similarly, _t−1[dE _t ] = _t−1[E ₁ − E _t−4] = _t−1[E _t ] − E _t−4; thus, _t−1[E _t ] can be found by

$$ {}_{t-1}\left[{E}_t\right]={\rho}_0+{\rho}_1\left(d{E}_{t-1}\right)+{\rho}_2\left(d{E}_{t-2}\right)+{\rho}_3\left(d{E}_{t-3}\right)+{\rho}_4\left(d{E}_{t-4}\right)+{E}_{t-4}. $$

(93.40)

Finally, the expectation adjustment in dividends, d _t , can be found by plugging Eqs. 93.38 and 93.40 into Eq. 93.35:

$$ \begin{array}{c}{d}_t\equiv {}_t\left[{D}_{t+1}\right]-{}_{t-1}\left[{D}_{t+1}\right]=-{a}_1{a}_2+{a}_2{D}_t-{a}_2^2{D}_{t-1}\\ {}-{a}_2{a}_3\left({\rho}_0+{\rho}_1d{E}_{t-1}+{\rho}_2d{E}_{t-2}+{\rho}_3d{E}_{t-3}+{\rho}_4d{E}_{t-4}+{E}_{t-4}\right)\\ {}+{a}_3{\rho}_1\left(-{\rho}_0+d{E}_t-{\rho}_1d{E}_{t-1}-{\rho}_2d{E}_{t-2}-{\rho}_3d{E}_{t-3}-{\rho}_4d{E}_{t-4}\right)\end{array} $$

(93.41)

Or

$$ {d}_t={C}_0+{C}_1{D}_t+{C}_2{D}_{t-1}+{C}_3d{E}_t+{C}_4d{E}_{t-1}+{C}_5d{E}_{t-2}+{C}_6d{E}_{t-3}+{C}_7d{E}_{t-4}+{C}_8{E}_{t-4} $$

(93.42)

where C ₀ to C ₈ are functions of a ₁ to a ₁ and ρ ₀ to ρ ₄.

That is, the expectation adjustment in dividends, d _t , can be found by the coefficients estimated in Eqs. 93.31 and 93.36, i.e., a ₁ to a ₃ and ρ ₀ to ρ ₄, and the observable data from the time series of D _t and E _t .

Case 2

E _t follows a random walk process.

If the series of earnings, E _t , follows a random walk process, i.e., _t [E _t+1] = E _t , _t−1[E _t ] = E _t−1, and _t−1[E _t+1] = E _t−1, then Eq. 93.35 can be redefined:

$$ {d}_t\equiv {}_t\left[{D}_{t+1}\right]-{}_{t-1}\left[{D}_{t+1}\right]={C}_0+{C}_1{D}_t+{C}_2{D}_{t-1}+{C}_3{E}_t+{C}_4{E}_{t-1} $$

(93.43)

where C ₀ = −a ₁ a ₂ C ₁ = a ₂, C ₂ = − a ²₂ , C ₃ = a ₃, and C ₄ = − a ₃(1 + a ₂).

That is, the expectation adjustment in dividends, d _t , can be found by the observable data from the time series of D _t and E _t .

In this study, we assumed that E _t follows a random walk process. Therefore, we used Eq. 93.43 instead of Eq. 93.42 to estimate d _t in Eqs. 93.22 and 93.23 in the text.