A Bayesian framework for combining valuation estimates

Abstract

Discounted cash flow, method of comparables, and fundamental analysis typically yield discrepant valuation estimates. Moreover, the valuation estimates typically disagree with market price. Can one form a superior valuation estimate by averaging over the individual estimates, including market price? This article suggests a Bayesian framework for combining two or more estimates into a superior valuation estimate. The framework justifies the common practice of averaging over several estimates to arrive at a final point estimate.

Appendix: Proof of Theorem 1

This section establishes that the specific choice of weights κ _I and κ _C in Theorem 1 maximizes the precision of $\mathop{V}\limits^{\frown}$ or, equivalently, minimizes var[$\mathop{V}\limits^{\frown}$]. To this end, consider an arbitrary linear weighing scheme $\mathop{V}\limits^{\frown}=\kappa _P P+\kappa _I V_I +\kappa _C V_C,$ where κ _P , κ _I , and κ _C are yet to be determined real numbers. Plugging in V = P + e, V _I  = V + e _I , and V _C  = V + e _C and requiring $E[\mathop{V}\limits^{\frown}]=V$ immediately yields κ _P  = 1−κ _I −κ _C , which implies $\mathop{V}\limits^{\frown}=V+(-1+\kappa _I +\kappa _C )e+\kappa _I e_I +\kappa _C e_C.$ Computing the variance of this expression yields var$[\mathop{V}\limits^{\frown}]=(1-\kappa _I -\kappa _C )^{2}\sigma ^{2}+\kappa _I^2 \sigma _I^2 +\kappa _C^2 \sigma _C^2-2(1-\kappa _I-\kappa _C )\kappa _C \rho \sigma \sigma _C.$ The first order conditions for minimizing var$[\mathop{V}\limits^{\frown}]$with respect to κ _I and κ _C may be written in matrix form as

$$\left( \begin{array}{ll} \sigma^2+\sigma_I^2 & (\sigma+\rho\sigma_c)\sigma \\ (\sigma+ \rho \sigma_c)\sigma & \sigma^2+\sigma_c^2 + 2\rho\sigma\sigma_c \end{array}\right) \left( \begin{array}{l}\kappa_I \\ \kappa_C \end{array}\right) = \left( \begin{array}{l} \sigma^2\\ (\sigma +\rho\sigma_c)\sigma\\ \end{array}\right).$$

Inverting the matrix recovers the expressions for κ _I and κ _C given in the Theorem. These weights minimize var$[\mathop{V}\limits^{\frown}],$ which means they maximize $(\hbox{var}[\mathop{V}\limits^{\frown}])^{-1}$, the precision of $\mathop{V}\limits^{\frown}.$ ■

1.1 Generalization of Theorem 1

Theorem 1 assumes that error e _l from the intrinsic valuation estimate is uncorrelated to market noise e. If an analyst believes that corr(e _I ,e) = ρ _I   =  ρ _I  ≠ 0, he or she should replace the weights in Theorem 1 with the following formulas:

$$\begin{aligned} \kappa_I=&\frac{\left[(1-\rho^2)\sigma\sigma_C+(\rho\sigma+\sigma_C) \rho_I\sigma_I\right]\sigma\sigma_C}{(1-\rho^2)\sigma^2\sigma^2_C +2(\rho\sigma+\sigma_C)\rho_I\sigma\sigma_C\sigma_I+\left[(1-\rho^2_I)\sigma^2+2\rho\sigma\sigma_C+\sigma^2_C\right]\sigma^2_I}\\ \kappa_C=&\frac{\left[(1-\rho^2_I)\sigma\sigma_I+(\rho_I\sigma+\sigma_I) \rho\sigma_C\right]\sigma\sigma_I}{(1-\rho^2)\sigma^2\sigma^2_C +2(\rho\sigma+\sigma_C)\rho_I\sigma\sigma_C\sigma_I+\left[ (1-\rho^2_I)\sigma^2+2\rho\sigma\sigma_C+\sigma^2_C\right]\sigma^2_I}\\ \end{aligned} $$

and

$$(1-\kappa_I-\kappa_C)=\frac{\left[(\sigma_C+\rho\sigma)\sigma_I+\rho_I\sigma\sigma_C\right]\sigma_I\sigma_C}{(1-\rho^2)\sigma^2\sigma^2_C+2(\rho\sigma+\sigma_C)\rho_I\sigma\sigma_C\sigma_I+\left[(1-\rho^2_I)\sigma^2+2\rho\sigma\sigma_C+\sigma^2_C\right]\sigma^2_I}$$

Proof

Same logic as the proof of Theorem 1. When corr(e _I ,e) = ρ _I , the new first order optimization condition becomes

$$\left( \begin{array}{ll} \sigma^2+\sigma_I^2 + 2\rho_I\sigma\sigma_I & (\sigma + \rho\sigma_c + \rho_I \sigma_I)\sigma \\ (\sigma + \rho\sigma_c + \rho_I\sigma_I)\sigma & \sigma^2 + \sigma_c^2 + 2\rho\sigma\sigma_c\end{array}\right)\left( \begin{array}{l}\kappa_I \\ \kappa_C \end{array}\right) =\left( \begin{array}{l} (\sigma+\rho_I\sigma_I)\sigma\\ (\sigma+\rho\sigma_c)\sigma\\ \end{array}\right).$$

Inverting the matrix recovers the expressions for κ _I and κ _C . These weights maximize the precision of the triangulation estimate $\mathop{V}\limits^{\frown}.$