An alternative derivation of the Berk and Green model using the Kalman Filter

This note provides an alternative derivation of Proposition 1 in Berk and Green (2004) without consulting the mentioned reference for all those that do not have the referenced textbook at hand (like myself). All errors are my own.

Kalman Filter general formulas

The general idea of the Kalman filter and related approaches is that we want to infer an unobserved state variable (e.g. a manager's ability) from something that we can actually observe, albeit with some noise (e.g. a manager's fund return). The state transition equation and measurement equation in matrix notation are:

$\begin{align} \tag{1} x_{t+1} & = A_{t} x_{t} + B_{t} u_{t} + q_t \\ \tag{2} y_{t} & = C_t x_t + D_t u_t + r_t \end{align}$

with $q_t \sim N(0,Q_t)$ and $r_t \sim N(0,R_t)$. I am borrowing the notation from a post here. The heart of the Kalman filter is the updating of the state estimate $\hat{x}_t$ and how certain one is about the state estimate, as measured by the (co-)variance:

$\begin{align} \tag{3} \hat{x}_{t+1} & = \hat{x}_t + P_t C_t^T ( C_t P_t C_t^T + D_t U_t D_t^T + R_t)^{-1} (y_{t+1} - C_t \hat{x}_t - D_t \hat{u}_t) \\ \tag{4} P_{t+1} & = P_t - P_t C_t^T ( C_t P_t C_t^T + D_t U_t D_t^T + R_t)^{-1} C_t P_t \end{align}$

Note that $P_t$ is actually known, contrary to $x_t$, which may or may not be a reasonable assumption for particular applications of the basic Kalman filter. Given some initializations, we can use equations (3) and (4) to update our optimal estimate of the state(s) and the associated (co-)variance at every point in time.

Berk and Green application

The general formulas simplify a lot in the Berk and Green case. In the Berk and Green case the unobserved state, a manager's ability, is constant over time. That is, $A_t = 1$, $B_t = 0$ and $x_t=\alpha_t=\alpha \ \ \forall t$. The decreasing returns to scale specification for returns governs the measurement equation, and $C_t = 1$ and $D_t = -1$ with $u_t=K(AUM_t)$ where $K$ denotes the cost function as a function of fund size (assets under management). So state transition and measurement equations are (using their notation, except for the cost function):

$\begin{align} \tag{5} \alpha_{t+1} & = \alpha_{t} = \alpha \\ \tag{6} r_{t+1} & = \alpha - K(AUM_t) + \epsilon_{t+1} \end{align}$

Equation (3) then implies that, using the fact that $R_t = \sigma^2$ in the Berk and Green notation:

$\begin{align} \tag{7}\hat{\alpha}_{t+1} & = \hat{\alpha}_t + P_t (P_t + \sigma^2)^{-1} (r_{t+1} - \hat{\alpha}_t + K(AUM_t)) \\ \tag{8} \hat{\alpha}_{t+1} & = \hat{\alpha}_t + P_t (P_t + \sigma^2)^{-1} r_{t+1} \end{align}$

where the second line uses the equilibrium condition $\hat{\alpha}_t = K(AUM_t) \ \ \forall t$ (their equation (4)). Equation (8) is basically already the solution to Proposition 1 in the original paper (their equation (5)). It remains to solve the recursion for $P_t$ and express it as a function of the parameters. For that, use equation (4) with $P_1 = Q_{1} = \eta^2$. Then

$\begin{align} \tag{9} P_2 & = P_1 - P_1 (P_1 + \sigma^2)^{-1} P_1 = P_1 \sigma^2 (P_1 + \sigma^2)^{-1} \\ \tag{10} P_3 & = P_1 \sigma^2 (2 P_1 + \sigma^2)^{-1} \\ … \\ \tag{11} P_t & = P_1 \sigma^2 ((t-1)P_1 + \sigma^2)^{-1} \\ \tag{12} & = 1 / ((t-1) \omega + \gamma) \end{align} $

where the last line uses precisions (their notation) instead of variances. Plug back in into equation (8) and simplify to get:

$\begin{align} \tag{13} \hat{\alpha}_{t+1} & = \hat{\alpha}_t + \omega (\gamma + t\omega)^{-1} r_{t+1} \end{align}$

which is their equation (5) and provides the optimal updating rule as a function of the underlying parameters. It is easy to see that the optimal estimate for a manager’s ability consists of two terms: i) the previous perception of a manager’s ability, and ii) a term that accounts for new information embedded in this period’s return. The second component diminishes as more data is observed.

References

Mutual Fund Flows and Performance in Rational Markets, Jonathan B. Berk and Richard C. Green, Journal of Political Economy, 2004, vol. 112, no. 6