Experience, information asymmetry, and rational forecast bias

Abstract

This study examines whether it is ever rational for analysts to post biased estimates and how information asymmetry and analyst experience factor into the decision. Using a construct where analysts wish to minimize their forecasting error, we model forecasted earnings when analysts combine private information with consensus estimates to determine the optimal forecast bias, i.e., the deviation from the consensus. We show that the analyst’s rational bias increases with information asymmetry, but is concavely related with experience. Novice analysts post estimates similar to the consensus but as they become more experienced and develop private information channels, their estimates become biased and deviated from the consensus. Highly seasoned analysts, who have superior analytical skills and valuable relationships, need not post biased forecasts.

Appendices

Appendix A: Proofs

1.1 A. The objective function:

Given that the analyst’s forecast is a weighted average of the analyst’s unconditional estimate and the consensus, F = w E + (1 − w) E _c , the objective function can be expressed as:

$$ \mathop {\min }\limits_{w|b} \left[ {w^{2} \left( {\tau_{0} + \tau (b)} \right)^{ - 2} + (1 - w)^{2} \tau_{c}^{ - 2} + 2\rho w(1 - w)\left( {\tau_{0} + \tau (b)} \right)^{ - 1} \tau_{c}^{ - 1} } \right] $$

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The first order condition then is:

$$ 2w\left( {\tau_{0} + \tau (b)} \right)^{ - 2} - 2(1 - w)\tau_{c}^{ - 2} + 2\rho (1 - w)\left( {\tau_{0} + \tau (b)} \right)^{ - 1} \tau_{c}^{ - 1} - 2\rho w\left( {\tau_{0} + \tau (b)} \right)^{ - 1} \tau_{c}^{ - 1} \equiv 0 $$

by collecting terms, we then have:

$$ w\left\{ {\left( {\tau_{0} + \tau (b)} \right)^{ - 2} + \tau_{c}^{ - 2} - 2\rho \left( {\tau_{0} + \tau (b)} \right)^{ - 1} \tau_{c}^{ - 1} } \right\} = \tau_{c}^{ - 2} - \rho \left( {\tau_{0} + \tau (b)} \right)^{ - 1} \tau_{c}^{ - 1} $$

this means that the optimal weight is:

$$ w = \frac{{\left( {\tau_{0} + \tau (b)} \right)^{2} - \rho \tau_{c} \left( {\tau_{0} + \tau (b)} \right)}}{{\left( {\tau_{0} + \tau (b)} \right)^{2} + \tau_{c}^{2} - 2\rho \tau_{c} \left( {\tau_{0} + \tau (b)} \right)}} $$

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1.2 B. Proof of proposition 1:

By taking the derivative of Eq. 7 with respect to τ ₀, we have:

$$ \frac{\partial w}{{\partial \tau_{0} }} = \frac{{2\tau_{c}^{2} \left( {\tau_{0} + \tau (b)} \right) - 2\rho \tau_{c} \left( {\tau_{0} + \tau (b)} \right)^{2} - \rho \tau_{c}^{3} }}{{\left[ {\left( {\tau_{0} + \tau (b)} \right)^{2} + \tau_{c}^{2} - 2\rho \tau_{c} \left( {\tau_{0} + \tau (b)} \right)} \right]^{2} }} $$

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Clearly, since the denominator of ∂w/∂τ ₀ is positive, then the sign is only a function of the numerator. This implies that the sign changes when the numerator, $2\tau_{c}^{{}} \left( {\tau_{0} + \tau (b)} \right) - 2\rho \left( {\tau_{0} + \tau (b)} \right)^{2} - \rho \tau_{c}^{2} ,$ is at maximum. To find the maximum, we solve for τ ₀ that satisfies the first order conditions of the numerator. The first order condition yields$\tau_{c} - 2\rho \left( {\tau_{0} + \tau (b)} \right) \equiv 0$. Thus, at optimal weight $\tau_{0} + \tau (b) = 0.5\rho^{ - 1} \tau_{c} .$

1.3 C. Proof of proposition 2:

By taking the derivative of Eq. 7 with respect to bias, we have:

$$ \frac{\partial w}{\partial b} = \frac{{\left[ {2\tau_{c}^{2} \left( {\tau_{0} + \tau (b)} \right) - 2\rho \tau_{c} \left( {\tau_{0} + \tau (b)} \right)^{2} - \rho \tau_{c}^{3} } \right]\frac{\partial \tau }{\partial b}}}{{\left[ {\left( {\tau_{0} + \tau (b)} \right)^{2} + \tau_{c}^{2} - 2\rho \tau_{c} \left( {\tau_{0} + \tau (b)} \right)} \right]^{2} }} $$

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Clearly, since the denominator of ∂w/∂b is positive, then the sign is only a function of the numerator. This implies (1) that since ∂τ/∂b is positive, then the optimal weight would be monotonically increasing with ∂τ/∂b or information asymmetry, and (2) that the optimal weight is nonlinearly, concavely related to private information precision. Since the first term in the numerator is a quadratic function of analyst’s own precision, the maximum in the function is the point at which the numerator changes sign. This point, however, is exactly the same point at which ∂w/∂τ ₀ maximizes. For biases at which $\tau_{0} + \tau (b)$ falls below $0.5\rho^{ - 1} \tau_{c} .$, then so long as bias increases so does the optimal weight.

Appendix B

See Table 6.

Table 6 Alternate samples

Appendix C

See Table 7.

Table 7 Alternate proxies for information asymmetry for Table 5