Taste, information, and asset prices: implications for the valuation of CSR

Abstract

Firms often undertake activities that do not necessarily increase cash flows (e.g., costly investments in corporate social responsibility or CSR), and some investors value these non cash activities (i.e., they have a “taste” for these activities). We develop a model to capture this phenomenon and focus on the asset-pricing implications of differences in investors’ tastes for firms’ activities and outputs. Our model shows that, first, investor taste differences provide a basis for investor clientele effects that are endogenously determined by the shares demanded by different types of investors. Second, because the market must clear at one price, investors’ demands are influenced by all dimensions of firm output even if their preferences are only over some dimensions. Third, information releases cause trading volume, even when all investors have the same information. Fourth, investor taste provides a rationale for corporate spin-offs that help firms better target their shareholder bases. Finally, individual social responsibility can lead to corporate social responsibility when managers care about stock price because price reacts to investments in CSR activities.

Appendices

Appendix 1: Proofs

Lemma 1 The first-order condition to ( 1) with respect to demand in shares of the risky asset for an investor of type-i is given by

$$\begin{aligned} E\left[ \tilde{v}_{i}\right] -p-rq_{i}Var\left[ \tilde{v}_{i}\right] =0. \end{aligned}$$

(22)

Solving this for $q_{i}$ yields the demands in (2). Substituting $q_{i}$ into the market clearing condition and solving for p proves the claim.

Corollary 1 The expressions for the comparative statics are

$$\begin{aligned} \frac{dp}{d\bar{y}}= & {} \frac{\lambda \sigma _{x}^{2}}{\sigma _{x}^{2}+\left( 1-\lambda \right) \sigma _{y}^{2}}>0, \end{aligned}$$

(23)

$$\begin{aligned} \frac{dp}{d\sigma _{y}^{2}}= & {} \frac{-\left( \bar{y}\left( 1-\lambda \right) +r\sigma _{x}^{2}\right) \lambda \sigma _{x}^{2}}{\left( \sigma _{x}^{2}+\sigma _{y}^{2}\left( 1-\lambda \right) \right) ^{2}}\gtrless 0, \end{aligned}$$

(24)

$$\begin{aligned} \frac{dp}{d\lambda }= & {} \frac{\left( \bar{y}-r\sigma _{y}^{2}\right) \left( \sigma _{x}^{2}+\sigma _{y}^{2}\right) \sigma _{x}^{2}}{\left( \sigma _{x}^{2}+\sigma _{y}^{2}\left( 1-\lambda \right) \right) ^{2}}\gtrless 0 ,\text { and} \end{aligned}$$

(25)

$$\begin{aligned} \frac{dp}{d\sigma _{x}^{2}}= & {} -\frac{r\left( \sigma _{x}^{4}+\sigma _{y}^{4}\left( 1-\lambda \right) +2\sigma _{y}^{2}\sigma _{x}^{2}\left( 1-\lambda \right) \right) -\sigma _{y}^{2}\lambda \bar{y}\left( 1-\lambda \right) }{\left( \sigma _{y}^{2}\left( 1-\lambda \right) +\sigma _{x}^{2}\right) ^{2}}\gtrless 0. \end{aligned}$$

(26)

Proposition 1 To prove the claim, we first derive Eqs. (9) and (10). Investors’ utility can be expressed as $u_{i}=-\exp \{ r( \mathbf {q}_{i}^{T} \mathbf {v}_{i}+l_{i})\}$, $i\in \{ 1,2\}$, where r is their level of risk aversion, $q_{i}=( q_{i,k},q_{i,a}) ^{T}$ represents the $2\times 1$ vector of investor i’s demand for shares in the 2 firms, $\tilde{v}_{1}=\tilde{x}=( \tilde{x}_{k},\tilde{x} _{a}) ^{T}$, and $\tilde{v}_{2}=\tilde{x}+\tilde{y}=( \tilde{x} _{k},\tilde{x}_{a}+\tilde{y}) ^{T}$. Each investor maximizes her expected terminal utility subject to the budget constraint

$$\begin{aligned} w_{i}=\mathbf {q}_{i}^{T}\mathbf {P}+l_{i}, \end{aligned}$$

where $w_{i}$ is the initial wealth endowment and $P=( p_{k},p_{a}) ^{T}$ is the price vector. Note that the price per share of the risk-free asset, like its return, has been normalized to one. Substituting the budget constraint, it is straightforward to show that maximizing expected utility is equivalent to maximizing the following certainty equivalent

$$\begin{aligned} CE_{i}=\mathbf {q}_{i}^{T}\left( E\left[ \mathbf {v}_{i}\right] -\mathbf {P} \right) -\frac{1}{2}r\mathbf {q}_{i}^{T}Cov\left[ \mathbf {v}_{i}\right] \mathbf {q}_{i}, \end{aligned}$$

where $Cov[ \mathbf {v}_{1}] =\Sigma _{x}$ and $Cov[ \mathbf { v}_{1}] =\Sigma _{x}+\Sigma _{y}$, with

$$\begin{aligned} \mathbf {\Sigma }_{x}=Cov\left[ \left( \tilde{x}_{k},\tilde{x}_{a}\right) \right] = \begin{bmatrix} \sigma _{x_{k}}^{2}&0 \\ 0&\sigma _{x_{a}}^{2} \end{bmatrix} \text { and }\,\mathbf {\Sigma }_{y}= \begin{bmatrix} 0&0 \\ 0&\sigma _{y}^{2} \end{bmatrix}. \end{aligned}$$

Note that Eq. (27) below holds with non-zero covariance terms in $\Sigma _{x}$ and $\Sigma _{y}$ and non-zero variance for firm k ’s CSR performance as long as the covariance matrices remain positive-definite. The first-order condition for an investor of type-i choosing share quantities to maximize wealth is given by

$$\begin{aligned} E\left[ \mathbf {v}_{i}\right] -\mathbf {P}-rCov\left[ \mathbf {v}_{i}\right] \mathbf {q}_{i}=0, \end{aligned}$$

such that the optimal demand for a type-i investor is given by

$$\begin{aligned} \mathbf {q}_{i}=\frac{1}{r}Cov\left[ \mathbf {v}_{i}\right] ^{-1}\left( E\left[ \mathbf {v}_{i}\right] -\mathbf {P}\right) . \end{aligned}$$

Prices are set such that aggregate demand equals aggregate supply. We assume that there is one share of each firm per investor and denote $1=( 1,1) ^{T}$ the supply vector in the two-firm case. Therefore it has to be the case that, on average, $( 1-\lambda ) q_{1}+\lambda q_{2}=1$. Substituting the optimal demands yields

$$\begin{aligned} \frac{1-\lambda }{r}Cov\left[ \mathbf {v}_{1}\right] ^{-1}\left( E\left[ \mathbf {v}_{1}\right] -\mathbf {P}\right) +\frac{\lambda }{r}Cov\left[ \mathbf {v}_{2}\right] ^{-1}\left( E\left[ \mathbf {v}_{2}\right] -\mathbf {P} \right) =\varvec{1}. \end{aligned}$$

Substituting our terms for output moments and solving for the price vector yields

$$\begin{aligned} \mathbf {P}={\bar{x}}+\left( \left( 1-\lambda \right) \mathbf {\Sigma }_{x}^{-1} +\lambda \left( \mathbf {\Sigma }_{x}+\mathbf {\Sigma }_{y}\right) ^{-1}\right) ^{-1}\left( \lambda \left( \mathbf {\Sigma }_{x}+\mathbf {\Sigma } _{y}\right) ^{-1}{\bar{\mathbf {y}}}-r{\varvec{1}}\right) . \end{aligned}$$

(27)

Equations (9) and (10) in the text follow directly from (27). First, substitute (8), (9), and (10) into $p_{k}+p_{a}\ge p_{u}$. Substituting $r\sigma _{y}^{2}> \bar{y}$, $\lambda =0$, $\lambda =1$, or $r\sigma _{y}^{2}=\bar{y}$ proves the claim.

Proposition 2 Expected price in period 2 is defined by $E[ \tilde{m}] =\bar{x}$ and $E[ \tilde{n}] =\bar{y}$ as

$$\begin{aligned} E\left[ p_{2}\right] =\bar{x}-r\frac{1}{\tau _{x}+\tau _{m}}+\lambda \left( \frac{\bar{y}\left( \tau _{y}+\tau _{n}\right) -r}{\tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) }\right) . \end{aligned}$$

The expected change in price is therefore:

$$\begin{aligned} E\left[ p_{2}-p_{1}\right]&=\,r\frac{\tau _{m}}{\tau _{x}\left( \tau _{m}+\tau _{x}\right) } \end{aligned}$$

(28)

$$\begin{aligned}&\quad +\lambda r\frac{\tau _{n}+\left( 1-\lambda \right) \tau _{m}}{\left( \tau _{y}+\left( 1-\lambda \right) \tau _{x}\right) \left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) } \end{aligned}$$

(29)

$$\begin{aligned}&\quad +\lambda \bar{y}\frac{\left( 1-\lambda \right) \left( \tau _{n}\tau _{x}-\tau _{m}\tau _{y}\right) }{\left( \tau _{y}+\left( 1-\lambda \right) \tau _{x}\right) \left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) }. \end{aligned}$$

(30)

Corollary 2 Substituting (17) and (18) into $E[ p_{2}-p_{1}]$ proves the claim.

Proposition 3 The respective derivatives are given by

$$\begin{aligned} \frac{d\alpha }{d\tau _{n}}= & {} \lambda \frac{\left( 1-\lambda \right) \left( \tau _{m}+\tau _{x}\right) +\tau _{y}}{\left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) ^{2}}\ge 0, \end{aligned}$$

(31)

$$\begin{aligned} \frac{d\alpha }{d\lambda }= & {} \frac{\tau _{n}\left( \tau _{m}+\tau _{n}+\tau _{x}+\tau _{y}\right) }{\left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) ^{2}}\ge 0, \end{aligned}$$

(32)

$$\begin{aligned} \frac{d\alpha }{d\tau _{m}}= & {} \frac{-\lambda \left( 1-\lambda \right) \tau _{n}}{\left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) ^{2}}\le 0, \end{aligned}$$

(33)

$$\begin{aligned} \frac{d\alpha }{d\tau _{x}}= & {} \frac{-\lambda \left( 1-\lambda \right) \tau _{n}}{\left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) ^{2}}\le 0\text {, and} \end{aligned}$$

(34)

$$\begin{aligned} \frac{d\alpha }{d\tau _{y}}= & {} \frac{-\lambda \tau _{n}}{\left( \tau _{y}+\tau _{n}+\left( 1-\lambda \right) \left( \tau _{x}+\tau _{m}\right) \right) ^{2}}\le 0. \end{aligned}$$

(35)

Proposition 4 Substituting (16) and ( 15) into $T=( 1-\lambda ) \left| q_{1,2}-q_{1,1}\right|$ yields

$$\begin{aligned} T&=\,\left( 1-\lambda \right) \left| \frac{E\left[ \tilde{v}_{i}|m,n \right] -p_{2}}{rVar\left[ \tilde{v}_{i}|m,n\right] }-1\right| \nonumber \nonumber \\&=\,\left( 1-\lambda \right) \frac{\tau _{x}^{\prime }}{r}\left| -\frac{ \lambda }{1+\left( 1-\lambda \right) \frac{\tau _{x}^{\prime }}{\tau _{y}^{\prime }}}\left( \frac{\bar{y}\tau _{y}+n\tau _{n}}{\tau _{y}^{\prime } }-r\frac{1}{\tau _{y}^{\prime }}\right) \right| \nonumber \nonumber \\&=\,\frac{\lambda \left( 1-\lambda \right) }{1-\lambda +\frac{\tau _{y}+\tau _{n}}{\tau _{x}+\tau _{m}}}\left| 1-\frac{\bar{y}\tau _{y}+n\tau _{n}}{r} \right| . \end{aligned}$$

(36)

Appendix 2: Non-zero correlation between x and y

This section discusses the baseline model when x and y are correlated, with $Cov[ \tilde{x},\tilde{y}] =\rho \sigma _{x}\sigma _{y}$ and $\rho \in [ -1,1]$. While a type-1 investor’s certainty equivalent and therefore demand is not affected, a type-2 investor’s certainty equivalent and demand are now

$$\begin{aligned} CE_{2}&=\,q_{2}\left( \bar{x}+\bar{y}-p\right) -\frac{1}{2}rq_{2}^{2}\left( \sigma _{x}^{2}+\sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) \text { and} \end{aligned}$$

(37)

$$\begin{aligned} q_{2}&=\,\frac{\bar{x}+\bar{y}-p}{r\left( \sigma _{x}^{2}+\sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }. \end{aligned}$$

(38)

The market-clearing condition is then given by

$$\begin{aligned} \left( 1-\lambda \right) \frac{\bar{x}-p}{r\sigma _{x}^{2}}+\lambda \frac{ \bar{x}+\bar{y}-p}{r\left( \sigma _{x}^{2}+\sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }=1. \end{aligned}$$

(39)

This implies that the equilibrium price is

$$\begin{aligned} p&=\,\bar{x}+\frac{\lambda \sigma _{x}^{2}}{\sigma _{x}^{2}+\left( 1-\lambda \right) \left( \sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }\bar{y}-r \frac{\sigma _{x}^{2}\left( \sigma _{x}^{2}+\sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }{\sigma _{x}^{2}+\left( 1-\lambda \right) \left( \sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) } \end{aligned}$$

(40)

$$\begin{aligned}&=\,\bar{x}-r\sigma _{x}^{2}+\frac{\lambda \sigma _{x}^{2}}{\sigma _{x}^{2}+\left( 1-\lambda \right) \left( \sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }\left( \bar{y}-r\left( \sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) \right) \text {.} \end{aligned}$$

(41)

This implies that an increase in the correlation, $\rho$, decreases the extent to which $\bar{y}$ is priced. The reason is that an increase in $\rho$ increases the perceived risk of type-2 investors and therefore decreases their equilibrium holdings. The increase in perceived risk and the decrease in holdings have countervailing effects on the risk premium. However, the first effect dominates such that the risk premium increases when $\rho$ increases,

$$\begin{aligned} \frac{\partial \left( r\frac{\sigma _{x}^{2}\left( \sigma _{x}^{2}+\sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }{\sigma _{x}^{2}+\left( 1-\lambda \right) \left( \sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) }\right) }{\partial \rho }=2r\frac{\lambda \sigma _{x}^{5}\sigma _{y}}{ \left( \sigma _{x}^{2}+\left( 1-\lambda \right) \left( \sigma _{y}^{2}+2\rho \sigma _{x}\sigma _{y}\right) \right) ^{2}}>0. \end{aligned}$$

This implies that when $\bar{y}>0$ then $\frac{dp}{d\rho }<0$.