CAPM-anomalies: quantitative puzzles

Abstract

The stochastic discount factor is a crucial determinant of the equity premium as well as the cross-sectional distribution of stock returns. This gives rise to the idea that there is a tight link between the equity premium puzzle and cross-sectional asset pricing puzzles. This paper examines similarities between the low-beta premium, the value premium, the small-size premium and the equity premium in a special case of a static consumption-based asset pricing model with constant relative risk aversion and lognormal dividends that can be solved in closed form. The results show that cross-sectional asset pricing puzzles are quantitative puzzles just like the equity premium puzzle: The model generates a premium for stocks with a low beta, a high dividend-price ratio and a small market capitalization but the size of the premium is too small. Furthermore, the size of the premium rises together with the equity premium as the risk aversion coefficient or consumption risk is increased.

Appendix

1.1 Derivation of the Expectation

In the proofs, we often encounter expectations of the form $\mathbf {E}\left[ e^{a\Delta y+b\left( \Delta y\right) ^2}\right]$ with different coefficients a and b and $\Delta y \sim \mathcal {N} (\mu -y_0,\sigma ^2)$. Here, we show the common derivation of this expectation using the completion of the square method.

$$\begin{aligned} \mathbf {E}\left[ e^{a\Delta y+b(\Delta y )^2}\right]&=\frac{1}{\sigma \sqrt{2\pi }}\int _{-\infty }^{\infty }e^{a\Delta y+b(\Delta y )^2}e^{-\frac{\left( (\Delta y- (\mu -y_0)\right) ^2}{2\sigma ^2}}\mathrm{d}\Delta y\\&=\frac{1}{\sigma \sqrt{2\pi }}\int _{-\infty }^{\infty }e^{-\frac{1}{2}\left( \frac{1}{\sigma ^2}-2b\right) \left( (\Delta y)^2-2\frac{a\sigma ^2+(\mu -y_0)}{1-2b\sigma ^2}\Delta y+\frac{(\mu -y_0)^2}{1-2b\sigma ^2}\right) }\mathrm{d}\Delta y. \end{aligned}$$

We can rewrite this expression as

$$\begin{aligned} \frac{e^{-\frac{1}{2}\left( \frac{1}{\sigma ^2}-2b\right) \left( \frac{(\mu -y_0)^2}{1-2b\sigma ^2}-\left( \frac{a\sigma ^2 +(\mu -y_0)}{1-2b\sigma ^2}\right) ^2\right) }}{\sigma \sqrt{2\pi }}\int _{-\infty }^{\infty }e^{-\frac{1}{2}\left( \frac{1}{\sigma ^2}-2b\right) \left( \Delta y-\frac{a\sigma ^2+(\mu -y_0 )}{1-2b\sigma ^2}\right) ^2}\mathrm{d}\Delta y. \end{aligned}$$

so that the integrand becomes proportional to a normal probability density function with mean $\frac{a\sigma ^2+(\mu -y_0)}{1-2b\sigma ^2}$ and standard deviation $\sqrt{\frac{\sigma ^2}{1-2b\sigma ^2}}$. Therefore, the expectation is equal to

$$\begin{aligned} \mathbf {E}\left[ e^{a\Delta y+b(\Delta y )^2}\right]&=\frac{1}{\sqrt{1-2b\sigma ^2}}e^{-\frac{1}{2}\left( \frac{1}{\sigma ^2}-2b\right) \left( \frac{(\mu -y_0)^2}{1-2b\sigma ^2}-\left( \frac{a\sigma ^2 +(\mu -y_0)}{1-2b\sigma ^2}\right) ^2\right) }\nonumber \\&=\frac{1}{\sqrt{1-2b\sigma ^2}}e^{\frac{1}{2\sigma ^2}\frac{\left( a\sigma ^2 +(\mu -y_0)\right) ^2}{1-2b\sigma ^2}-\frac{1}{2\sigma ^2}(\mu -y_0)^2}. \end{aligned}$$

(1)

$\square$

Proof of Proposition 1

The price–dividend ratio of an asset of type $\rho$ is

$$\begin{aligned} \frac{q^{\rho }}{D_0^{\rho }}&=\mathbf {E}\left[ \delta \left( \frac{c_0}{c_1}\right) ^{\gamma }\frac{D_1^{\rho }}{D_0^{\rho }}\right] \\&=\delta \mathbf {E}\left[ e^{-\gamma (y_1-y_0) -\frac{\gamma }{2}\sigma _{\rho }^2 (y_1^2-y_0^2)}e^{\rho (y_1-y_0)}\right] . \end{aligned}$$

To shorten notation, let us define $\Delta y\equiv y_1-y_0$ and $\Delta y \sim \mathcal {N}\left( \mu -y_0,\sigma ^2\right)$. The price–dividend ratio then is

$$\begin{aligned} \frac{q^{\rho }}{D_0^{\rho }}&=\delta \mathbf {E}\left[ e^{-\gamma \Delta y -\frac{\gamma }{2}\sigma _{\rho }^2\Delta y (y_1+y_0)}e^{\rho \Delta y}\right] \\&=\delta \mathbf {E}\left[ e^{-\gamma \Delta y -\frac{\gamma }{2}\sigma _{\rho }^2 \left( (\Delta y )^2 +2y_0\Delta y \right) }e^{\rho \Delta y}\right] \\&=\delta \mathbf {E}\left[ e^{(\rho -\gamma -\gamma \sigma _{\rho }^2y_0)\Delta y -\frac{\gamma }{2}\sigma _{\rho }^2 (\Delta y )^2 }\right] . \end{aligned}$$

Using the formula for the expectation (1), we obtain

$$\begin{aligned} \frac{q^{\rho }}{D_0^{\rho }}&= \frac{\delta }{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}e^{\frac{1}{2\sigma ^2}\frac{\left( (\rho -\gamma -\gamma \sigma _{\rho }^2y_0)\sigma ^2 +(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2\sigma ^2}-\frac{1}{2\sigma ^2}(\mu -y_0)^2}. \end{aligned}$$

(2)

$\square$

Proof of Proposition 2

The expected return of an asset of type $\rho$ is

$$\begin{aligned} \mathbf {E}\left[ R^{\rho }\right]&=\mathbf {E}\left[ \frac{D_1^{\rho }}{q^{\rho }}\right] =\frac{\mathbf {E}\left[ e^{\rho \Delta y}\right] D_0^{\rho }}{q^{\rho }}\\&=\frac{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}{\delta }e^{\rho (\mu -y_0)+\frac{1}{2}\rho ^2 \sigma ^2-\frac{1}{2\sigma ^2}\frac{\left( (\rho -\gamma -\gamma \sigma _{\rho }^2y_0)\sigma ^2 +(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2\sigma ^2}+\frac{1}{2\sigma ^2}(\mu -y_0)^2}. \end{aligned}$$

The price–dividend ratio of the market portfolio is

$$\begin{aligned} \frac{q^{M }}{D_0^{M }}&=\mathbf {E}\left[ \delta \left( \frac{c_0}{c_1}\right) ^{\gamma }\frac{D_1^{M }}{D_0^{M }}\right] \\&=\delta \mathbf {E}\left[ e^{(1-\gamma )(y_1-y_0) +\frac{1-\gamma }{2}\sigma _{\rho }^2 (y_1^2-y_0^2)}\right] \\&=\delta \mathbf {E}\left[ e^{(1-\gamma )\Delta y +\frac{1-\gamma }{2}\sigma _{\rho }^2 \Delta y(y_1+y_0)}\right] \\&=\delta \mathbf {E}\left[ e^{(1-\gamma )\Delta y +\frac{1-\gamma }{2}\sigma _{\rho }^2 \left( (\Delta y )^2 +2y_0\Delta y \right) }\right] \\&=\delta \mathbf {E}\left[ e^{(1-\gamma )(1+\sigma _{\rho }^2y_0)\Delta y +\frac{1-\gamma }{2}\sigma _{\rho }^2 (\Delta y )^2 }\right] . \end{aligned}$$

Using the formula for the expectation (1), we obtain

$$\begin{aligned} \frac{q^{M }}{D_0^{M }}&= \frac{\delta }{\sqrt{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}e^{\frac{1}{2\sigma ^2}\frac{\left( (1 -\gamma )(1+ \sigma _{\rho }^2y_0)\sigma ^2 +(\mu -y_0)\right) ^2}{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}-\frac{1}{2\sigma ^2}(\mu -y_0)^2}. \end{aligned}$$

(3)

This step assumes that $\sigma ^2/\left( 1-\left( 1-\gamma \right) \sigma _{\rho }^2\sigma ^2\right)$ is the variance of a normally distributed random variable, which requires $\sigma _{\rho }^2\sigma ^2<1$ in case $\gamma =1$.

Therefore, the expected return on the market portfolio is

$$\begin{aligned} \mathbf {E}\left[ R^{M }\right]&=\mathbf {E}\left[ \frac{D_1^{M }}{q^{M }}\right] \\&=\frac{\mathbf {E}\left[ e^{\Delta y +\frac{1}{2}\sigma _{\rho }^2 (y_1^2-y_0^2)}\right] D_0^{M }}{q^{M }}\\&=\frac{\mathbf {E}\left[ e^{(1+\sigma _{\rho }^2y_0)\Delta y +\frac{1}{2}\sigma _{\rho }^2 (\Delta y )^2 }\right] D_0^{M }}{q^{M }}\\&=\frac{\sqrt{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}{\delta \sqrt{1-\sigma _{\rho }^2\sigma ^2}}e^{\frac{1}{2\sigma ^2}\left( \frac{\left( (1+\sigma _{\rho }^2 y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2\sigma ^2}-\frac{\left( (1-\gamma )(1+\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-(1-\gamma )\sigma _{\rho }^2 \sigma ^2}\right) }. \end{aligned}$$

Here, we use again the formula for the expectation given in (1) assuming that $\sigma ^2/(1-\sigma ^2 \sigma _{\rho }^2)$ is the variance of a normal distribution, which requires $\sigma _{\rho }^2\sigma ^2<1$.

The riskless rate is

$$\begin{aligned} R^f&=\frac{1}{\mathbf {E}\left[ \delta \left( \frac{c_0}{c_1}\right) ^{\gamma }\right] }\\&=\frac{1}{\delta \mathbf {E}\left[ e^{-\gamma (y_1-y_0) -\frac{\gamma }{2}\sigma _{\rho }^2 (y_1^2-y_0^2)}\right] }\\&=\frac{1}{\delta \mathbf {E}\left[ e^{-\gamma \Delta y -\frac{\gamma }{2}\sigma _{\rho }^2 \Delta y(y_1+y_0)}\right] }\\&=\frac{1}{\delta \mathbf {E}\left[ e^{-\gamma \Delta y -\frac{\gamma }{2}\sigma _{\rho }^2 \left( (\Delta y )^2 +2y_0\Delta y \right) }\right] }\\&=\frac{1}{\delta \mathbf {E}\left[ e^{-\gamma (1+\sigma _{\rho }^2y_0)\Delta y -\frac{\gamma }{2}\sigma _{\rho }^2 (\Delta y )^2 }\right] }. \end{aligned}$$

Using the formula for the expectation (1), we obtain

$$\begin{aligned} R^f&= \frac{\sqrt{1+\gamma \sigma _{\rho }^2 \sigma ^2}}{\delta }e^{-\frac{1}{2\sigma ^2}\frac{(-\gamma (1+\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0))^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}+\frac{1}{2\sigma ^2}(\mu -y_0)^2}. \end{aligned}$$

$\square$

Proof of Proposition 3

The CAPM beta $\beta ^{\rho }$ is

$$\begin{aligned} \beta ^{\rho }&=\frac{\text {Cov}\left( R^{\rho }, R^M\right) }{\text {Var}\left( R^M\right) }=\frac{q^M}{q^{\rho }}\frac{\text {Cov}\left( D_1^{\rho },D_1^M\right) }{\text {Var}\left( D_1^M\right) }\\&=\frac{q^M}{q^{\rho }}\frac{\mathbf {E}\left[ e^{(1+\rho )y_1+\frac{1}{2}\sigma _{\rho }^2y_1^2}\right] -\mathbf {E}\left[ e^{\rho y_1}\right] \mathbf {E}\left[ e^{y_1+\frac{1}{2}\sigma _{\rho }^2y_1^2}\right] }{\mathbf {E}\left[ e^{2y_1+\sigma _{\rho }^2y_1^2}\right] -\mathbf {E}\left[ e^{y_1+\frac{1}{2}\sigma _{\rho }^2y_1^2}\right] ^2}\\&=\frac{q^M}{q^{\rho }}e^{(\rho -1)y_0-\frac{1}{2}\sigma _{\rho }^2y_0^2}\frac{\mathbf {E}\left[ e^{(1+\rho +\sigma _{\rho }^2y_0)\Delta y+\frac{1}{2}\sigma _{\rho }^2(\Delta y)^2}\right] -\mathbf {E}\left[ e^{\rho \Delta y}\right] \mathbf {E}\left[ e^{(1+\sigma _{\rho }^2y_0)\Delta y+\frac{1}{2}\sigma _{\rho }^2(\Delta y)^2}\right] }{\mathbf {E}\left[ e^{2(1+\sigma _{\rho }^2y_0)\Delta y+\sigma _{\rho }^2(\Delta y)^2}\right] -\mathbf {E}\left[ e^{(1+\sigma _{\rho }^2y_0)\Delta y+\frac{1}{2}\sigma _{\rho }^2(\Delta y)^2}\right] ^2}. \end{aligned}$$

Using the formula for the expectations (1) and multiplying the numerator and denominator by $e^{\frac{1}{2\sigma ^2}(\mu -y_0)^2}$, we obtain

$$\begin{aligned} \beta ^{\rho }&=\frac{q^M}{q^{\rho }}e^{(\rho -1)y_0-\frac{1}{2}\sigma _{\rho }^2y_0^2}\frac{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (1+\rho +\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1-\sigma _{\rho }^2\sigma ^2}}-\frac{e^{\rho (\mu -y_0)+\frac{1}{2}\rho ^2 \sigma ^2 +\frac{1}{2\sigma ^2 }\frac{\left( (1+\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1-\sigma _{\rho }^2\sigma ^2}}}{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( 2(1+\sigma _{\rho }^2 y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-2\sigma _{\rho }^2\sigma ^2}}}{\sqrt{1-2\sigma _{\rho }^2\sigma ^2}}-\frac{e^{\frac{1}{\sigma ^2}\frac{\left( (1+\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2\sigma ^2}-\frac{1}{2\sigma ^2}(\mu -y_0)^2}}{1-\sigma _{\rho }^2\sigma ^2}}. \end{aligned}$$

The assumption $\sigma _{\rho }^2 \sigma ^2<1/2$ allows us to interpret $\sigma ^2/(1-\sigma _{\rho }^2\sigma ^2)$ as well as $\sigma ^2/(1-2\sigma _{\rho }^2 \sigma ^2)$ as variances of a normal distribution and to apply (1).

Using the formulas for the price–dividend ratio (2) and (3), we obtain

$$\begin{aligned} \beta ^{\rho }&=\frac{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (1-\gamma )(1+\sigma _{\rho }^2 y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}}{\sqrt{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}\left( \frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (1+\rho +\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1-\sigma _{\rho }^2\sigma ^2}}-\frac{e^{\rho (\mu -y_0)+\frac{1}{2}\rho ^2 \sigma ^2 +\frac{1}{2\sigma ^2 }\frac{\left( (1+\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1-\sigma _{\rho }^2\sigma ^2}}\right) }{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (\rho -\gamma -\gamma \sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}\left( \frac{e^{\frac{1}{2\sigma ^2}\frac{\left( 2(1+\sigma _{\rho }^2 y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-2\sigma _{\rho }^2\sigma ^2}}}{\sqrt{1-2\sigma _{\rho }^2\sigma ^2}}-\frac{e^{\frac{1}{\sigma ^2}\frac{\left( (1+\sigma _{\rho }^2y_0)\sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2\sigma ^2}-\frac{1}{2\sigma ^2}(\mu -y_0)^2}}{1-\sigma _{\rho }^2\sigma ^2}\right) }. \end{aligned}$$

$\square$

1.2 Positivity of the equity premium

In order to show that the equity premium is positive, we show that the ratio of the expected market return and the riskless rate is greater than one. Dividing the expression of the expected market return by the expression of the riskless rate as given in Proposition 2, we obtain

$$\begin{aligned} \frac{\mathbf {E}[R^M]}{R^f}=\frac{\sqrt{1+(\gamma -1)\sigma _\rho ^2\sigma ^2}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}\sqrt{1-\sigma _{\rho }^2\sigma ^2}}e^{-\frac{\left( \left( \gamma -1\right) \sigma ^2-\mu \right) ^2}{2\left( \left( \gamma -1\right) \sigma _{\rho }^2\sigma ^2+1\right) \sigma ^2}+\frac{(\gamma \sigma ^2-\mu )^2}{2(\gamma \sigma _{\rho }^2\sigma ^2+1)\sigma ^2}+\frac{(\sigma ^2+\mu )^2}{2(1-\sigma _{\rho }^2\sigma ^2)\sigma ^2}-\frac{\mu ^2}{2\sigma ^2}}. \end{aligned}$$

First, we note that $\sqrt{1+(\gamma -1)\sigma _\rho ^2\sigma ^2}\ge \sqrt{(1+\gamma \sigma _{\rho }^2\sigma ^2)(1-\sigma _{\rho }^2\sigma ^2)}$. Second, we calculate the derivative of the exponent with respect to $\sigma ^2$:

$$\begin{aligned} \frac{(\mu \sigma _{\rho }^2+1)^2}{2(1-\sigma _{\rho }^2\sigma ^2)^2}+\frac{(2\gamma (\gamma -1)\sigma _{\rho }^2 \sigma ^2+2\gamma -1)(\mu \sigma _{\rho }^2+1)^2}{2(1+\gamma \sigma _{\rho }^2\sigma ^2)^2\left( 1+(\gamma -1)\sigma _{\rho }^2\sigma ^2 \right) ^2}, \end{aligned}$$

which is positive because all squared terms are positive and $2\gamma (\gamma -1)\sigma _{\rho }^2\sigma ^2$ and $2\gamma -1$ are nonnegative for $\gamma \ge 1$. Thus, the exponent of the exponential function is increasing in $\sigma$ and minimal for $\sigma \rightarrow 0$, where it takes the value zero as can be calculated using L’Hopital’s rule. The exponential function therefore takes only values greater than one unless $\sigma =0$. $\square$

Proof of Proposition 4

The CCAPM beta $\beta _\mathrm{con} ^{\rho }$ is

$$\begin{aligned} \beta _\mathrm{con}^{\rho }&=\frac{\text {Cov}\left( R^{\rho },(R^M)^{-\gamma }\right) }{\text {Cov}\left( R^M,(R^M)^{-\gamma }\right) }=\frac{q^M}{q^{\rho }}\frac{\text {Cov}\left( D_1^{\rho },\left( D_1^M\right) ^{-\gamma }\right) }{\text {Cov}\left( D_1^M,\left( D_1^M\right) ^{-\gamma }\right) }\\&=\frac{q^M}{q^{\rho }}\frac{\mathbf {E}\left[ e^{(\rho -\gamma )y_1-\frac{\gamma }{2}\sigma _{\rho }^2y_1^2}\right] -\mathbf {E}\left[ e^{\rho y_1}\right] \mathbf {E}\left[ e^{-\gamma y_1- \frac{\gamma }{2}\sigma _{\rho }^2y_1^2}\right] }{\mathbf {E}\left[ e^{(1-\gamma )y_1+\frac{1-\gamma }{2}\sigma _{\rho }^2y_1^2}\right] -\mathbf {E}\left[ e^{y_1+\frac{1}{2}\sigma _{\rho }^2y_1^2}\right] \mathbf {E}\left[ e^{-\gamma y_1-\frac{\gamma }{2}\sigma _{\rho }^2y_1^2}\right] }\\&=\frac{q^M}{q^{\rho }}\frac{e^{(\rho -1)y_0-\frac{1}{2}\sigma _{\rho }^2y_0^2}\left( \mathbf {E}\left[ e^{(\rho -\gamma \eta )\Delta y-\frac{\gamma }{2}\sigma _{\rho }^2(\Delta y)^2}\right] -\mathbf {E}\left[ e^{\rho \Delta y}\right] \mathbf {E}\left[ e^{-\gamma \eta \Delta y-\frac{\gamma }{2}\sigma _{\rho }^2(\Delta y)^2}\right] \right) }{\mathbf {E}\left[ e^{(1-\gamma )\eta \Delta y+\frac{1-\gamma }{2}\sigma _{\rho }^2(\Delta y)^2}\right] -\mathbf {E}\left[ e^{\eta \Delta y+\frac{1}{2}\sigma _{\rho }^2(\Delta y)^2}\right] \mathbf {E}\left[ e^{-\gamma \eta \Delta y-\frac{\gamma }{2}\sigma _{\rho }^2(\Delta y)^2}\right] }, \end{aligned}$$

where $\eta$ is defined as $\eta \equiv 1+\sigma _{\rho }^2y_0$. Using the formula for the expectations (1) and multiplying the numerator and denominator by $e^{\frac{1}{2\sigma ^2}(\mu -y_0)^2}$, we obtain

$$\begin{aligned} \beta _\mathrm{con}^{\rho }&=\frac{q^M}{q^{\rho }}\frac{e^{(\rho -1)y_0-\frac{1}{2}\sigma _{\rho }^2y_0^2}\left( \frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (\rho -\gamma \eta )\sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}-\frac{e^{\rho (\mu -y_0)+\frac{1}{2}\rho ^2 \sigma ^2 +\frac{1}{2\sigma ^2 }\frac{\left( -\gamma \eta \sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}\right) }{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (1-\gamma )\eta \sigma ^2+(\mu -y_0)\right) ^2}{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}}{\sqrt{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}-\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( \eta \sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2\sigma ^2}+\frac{1}{2\sigma ^2}\frac{\left( -\gamma \eta \sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2\sigma ^2}-\frac{1}{2\sigma ^2}(\mu -y_0)^2}}{\sqrt{1-\sigma _{\rho }^2\sigma ^2}\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}}. \end{aligned}$$

Using the formulas for the price–dividend ratio (2) and (3), we obtain

$$\begin{aligned} \beta _\mathrm{con}^{\rho }&=\frac{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (1-\gamma )\eta \sigma ^2+(\mu -y_0)\right) ^2}{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}}{\sqrt{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}\left( \frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (\rho -\gamma \eta )\sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}+\frac{1}{2\sigma ^2}(\mu -y_0)^2}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}-\frac{e^{\frac{1}{2\sigma ^2 }\frac{\left( -\gamma \eta \sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}+\frac{1}{2\sigma ^2}\left( \rho \sigma ^2+(\mu -y_0)\right) ^2}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}\right) }{\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (\rho -\gamma \eta )\sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2 \sigma ^2}}}{\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}\left( \frac{e^{\frac{1}{2\sigma ^2}\frac{\left( (1-\gamma )\eta \sigma ^2+(\mu -y_0)\right) ^2}{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}+\frac{1}{2\sigma ^2}(\mu -y_0)^2}}{\sqrt{1-(1-\gamma )\sigma _{\rho }^2\sigma ^2}}-\frac{e^{\frac{1}{2\sigma ^2}\frac{\left( \eta \sigma ^2+(\mu -y_0)\right) ^2}{1-\sigma _{\rho }^2\sigma ^2}+\frac{1}{2\sigma ^2}\frac{\left( -\gamma \eta \sigma ^2+(\mu -y_0)\right) ^2}{1+\gamma \sigma _{\rho }^2\sigma ^2}}}{\sqrt{1-\sigma _{\rho }^2\sigma ^2}\sqrt{1+\gamma \sigma _{\rho }^2\sigma ^2}}\right) }. \end{aligned}$$

$\square$