Time-Dependent Black–Litterman

Abstract

The Black–Litterman method is widely used in the investment management industry to incorporate views in investment portfolios. The method applies when views are expressed as expected returns over the horizon for which allocation decisions are made, i.e., the investment horizon. In practice, however, the investor’s views are typically formulated for the near future while the investor’s investment horizon is much longer. To incorporate such views, we introduce the time-dependent Black–Litterman method and show that, in a time-dependent setting, a distinction should be made between unconditional and conditional views. Furthermore, we demonstrate its use for buy and hold investors. In an example, we show that the investor’s views have a plausible impact on resulting allocation decisions.

Appendices

Appendix 1: Derivation of the Bayesian Approach with Conditional Views

Here, we derive updating formula (23) for the posterior mean and covariance matrix in the Bayesian approach to the time-dependent Black–Litterman method with conditional views. Using the investor’s views as given by (11), the posterior mean, defined as the predicted mean conditional on the investor’s view, can be written as

$$\begin{aligned} \mu _{t|t} {:=}\left[ \left. \mu _{t|t-1} \right| v_t= P_t\mu _{t|t-1}+\xi _t \text { and } \mu _{t-1|t-1}=\pi _{t-1|t-1}\right] , \end{aligned}$$

(30)

where $\pi _{t-1|t-1}={\text {E}}\mu _{t-1|t-1}$. Using (16), it follows that

$$\begin{aligned} \mu _{t|t} = \left[ \left. \mu _{t|t-1} \right| v_t= P_t \pi _{t|t-1}+ P_t \theta _t+\xi _t \right] , \end{aligned}$$

(31)

where $\pi _{t|t-1}={\text {E}}\mu _{t|t-1}$. Again, we use (16) to determine the joint distribution of the predicted mean $\mu _{t|t-1}$ and the random variable $P_t \pi _{t|t-1}+ P_t \theta _t+\xi _t$:

$$\begin{aligned} \begin{bmatrix} \mu _{t|t-1}\\ P_t \pi _{t|t-1}+ P_t \theta _t+\xi _t \end{bmatrix}\sim N\left( \begin{bmatrix} \pi _{t|t-1}\\ P_t\pi _{t|t-1} \end{bmatrix}, \begin{bmatrix} \Pi _{t|t-1}&\tau \Phi _tP_t^T\\ P_t\tau \Phi _t&\Omega _t+P_t\tau \Phi _tP_t^T \end{bmatrix} \right). \end{aligned}$$

(32)

Using theory of conditional normal distributions leads to (23).

Appendix 2: Derivation of the Sampling Approach with Conditional Views

Here, we derive updating formula (24) for the sampling approach to the time-dependent Black–Litterman method with conditional views. In the sampling approach, the predicted mean $\mu _{t|t-1}$ is treated as if it were estimated with maximum likelihood from m future return observations. Based on the construction of conditional views, these observations are seen as observations of the return distribution at time t given that returns at time $t-1$ are equal to their expected value, i.e., they are observations from the conditional random variable

$$\begin{aligned} \left. r_{t|t-1}\right| \left( r_{t-1|t-1} = {\text {E}} r_{t-1|t-1}\right). \end{aligned}$$

(33)

Using (20), we conclude that $\Phi _t$ is the covariance matrix of this random variable. Thus, $\mu _{t|t-1}$ is treated as if it were estimated by maximum likelihood from m future return observations drawn from a normal distribution with covariance matrix $\Phi _t$ and an unknown true mean $\pi _{t|t-1}$. Similarly, the investor’s view $v_t$ is treated as if it were an estimate for $P_t\pi _{t|t-1}$ estimated with maximum likelihood from n observations drawn independently (also from the return observations) from a normal distribution with mean $P_t\pi _{t|t-1}$ and covariance matrix $\Omega _t$, the covariance matrix of the uncertainty in the view. As with the derivation of (6), we follow Mankert (2006, 2010) and Mankert and Seiler (2011) and conclude that the improved estimate for $\pi _{t|t-1}$ based on all observations is given by (24) with $\tau = m/n$.

Appendix 3: Dimension Reduction

Sampling approach applied to factor models

Here, we show that when the sampling approach is applied to factor model (25) with the investor’s view involving only a selection of variables called the view variables, it suffices to consider the system consisting of the factors and the view variables and let the remaining variables in the system be driven by the updated factors. For this, we split up the vector $X_t$ as in equation (26). Now, for unconditional views, the investor’s view involving only the view variables is, analogous to equation (9), written as

$$\begin{aligned} v_t=P_t \mu _{t}^V+\xi _t, \end{aligned}$$

(34)

where the superscript indicates the projection on the corresponding component. By writing the system consisting of (25a) and (26) as a $\mathrm {VAR}(1)$ model and applying the time-dependent Black–Litterman method, it follows that the gain $K_t$ breaks down into several parts. For the sampling approach to the time-dependent Black–Litterman method with unconditional views, the gain can be written as

$$\begin{aligned} K_t= \begin{bmatrix}K^F_t\\ K^V_t\\ K^W_t\end{bmatrix}= \begin{bmatrix}\Sigma _t^F B_V^T \\ \Sigma _t^V\\ B_W\Sigma _t^F B_V^T \end{bmatrix} \tau P^T_t\left( \Omega _t+P_t\tau \Sigma _t^V P_t^T\right) ^{-1}, \end{aligned}$$

(35)

where $\Sigma _t^F$ and $\Sigma _t^F$ are the forecasted covariance matrices using (25) and (26). With the gain as in (35), the mean is updated and predicted as in (21) and (22a). Now, since the last component $K^W_t$ can be written as $K^W_t = B_W K^F_t$, we conclude that the update of the mean $\mu ^W_{t|t}$ is solely driven by the updated factors:

$$\begin{aligned} \mu _{t|t} = \begin{bmatrix}\mu _{t|t}^F\\ \mu _{t|t}^V\\ \mu _{t|t}^W\end{bmatrix}= \begin{bmatrix}\mu _{t|t-1}^F + K_t^F \left( v_t-P_t\mu _{t|t-1}^V\right) \\ \mu _{t|t-1}^V + K_t^V \left( v_t-P_t\mu _{t|t-1}^V\right) \\ B_W\mu _{t|t}^F\end{bmatrix}. \end{aligned}$$

(36)

A similar result holds for the sampling approach with conditional views. The gain can now be written as

$$\begin{aligned} K_t= \begin{bmatrix}K^F_t\\ K^V_t\\ K^W_t\end{bmatrix}= \begin{bmatrix}\Phi ^F B_V^T \\ \Phi ^V + B_V\Phi ^F B_V^T\\ B_W\Phi ^F B_V^T \end{bmatrix} \tau P^T_t\left( \Omega _t+P_t\tau \left( \Phi ^V + B_V\Phi ^F B_V^T\right) P_t^T\right) ^{-1}. \end{aligned}$$

(37)

With the gain as in (37), the updated mean is given by (36). Thus, also in the sampling approach with conditional views, the mean $\mu ^W_{t|t}$ is solely driven by the updated factors.

From equations (36) and (37), we conclude that the updated remaining variables $W_{t|t}$ can be written as in equation (27).

Bayesian approach applied to factor models

Here, we show that when the Bayesian approach is applied to factor model (25) with the investor’s view involving only a selection of variables called the view variables, it suffices to consider the system consisting of the factors and the view variables and let the remaining variables in the system be driven by the updated factors. For this, we split up the vector $X_t$ as in equation (26). As discussed, the evolution equation has to be rewritten in the Bayesian approach as in (28) and (29). After rewriting, we are in the setting of the Bayesian approach to the time-dependent Back-Litterman method. With unconditional views, the gain can be written as

$$\begin{aligned} K_t= \begin{bmatrix}K^F_t\\ K^V_t\\ K^W_t\end{bmatrix}= \begin{bmatrix}\Pi _{t|t-1}^F B_V^T \\ \Pi _{t|t-1}^V\\ B_W\Pi _{t|t-1}^F B_V^T \end{bmatrix} P^T_t\left( \Omega _t+P_t\Pi _{t|t-1}^V P_t^T\right) ^{-1}, \end{aligned}$$

(38)

where $\mu ^F_{t|t-1}\sim N(\pi ^F_{t|t-1},\Pi ^F_{t|t-1})$, $\mu ^V_{t|t-1}\sim N(\pi ^F_{t|t-1},\Pi ^F_{t|t-1})$ and $\mu ^X_{t|t-1}\sim N(\pi ^F_{t|t-1},\Pi ^F_{t|t-1})$. Also here, the last component $K^W_t$ can be written as $K^W_t = B_W K^F_t$. Again this implies that the parts of the updated mean and covariance matrix involving W are solely driven by the updated factors:

$$\begin{aligned} \pi _{t|t}&= \begin{bmatrix}\pi _{t|t}^F\\ \pi _{t|t}^V\\ \pi _{t|t}^W\end{bmatrix}= \begin{bmatrix}\pi _{t|t-1}^F + K_t^F \left( v_t-P_t\pi _{t|t-1}^V\right) \\ \pi _{t|t-1}^V + K_t^V \left( v_t-P_t\pi _{t|t-1}^V\right) \\ B_W\pi _{t|t}^F\end{bmatrix}, \end{aligned}$$

(39a)

$$\begin{aligned} \begin{bmatrix} \Pi ^F_{t|t}&\Pi ^{FV}_{t|t} \\ \Pi ^{VF}_{t|t}&\Pi ^V_{t|t} \\ \end{bmatrix}&= \left( I-\begin{bmatrix}K_t^F\\K^V_T\end{bmatrix} \begin{bmatrix}0&P_t\end{bmatrix}\right) \begin{bmatrix} \Pi ^F_{t|t-1}&\Pi ^{FV}_{t|t-1} \\ \Pi ^{VF}_{t|t-1}&\Pi ^V_{t|t-1} \\ \end{bmatrix}, \end{aligned}$$

(39b)

$$\begin{aligned} \Pi _{t|t}&= \begin{bmatrix} \Pi ^F_{t|t}&\Pi ^{FV}_{t|t}&\Pi ^{FW}_{t|t}\\ \Pi ^{VF}_{t|t}&\Pi ^V_{t|t}&\Pi ^{VW}_{t|t}\\ \Pi ^{WF}_{t|t}&\Pi ^{WV}_{t|t}&\Pi ^W_{t|t} \end{bmatrix} = \begin{bmatrix} \Pi ^F_{t|t}&\Pi ^{FV}_{t|t}&\Pi ^{F}_{t|t}B_W^T\\ \Pi ^{VF}_{t|t}&\Pi ^V_{t|t}&\Pi ^{VF}_{t|t}B_W^T\\ B_W\Pi ^{F}_{t|t}&B_W\Pi ^{FV}_{t|t}&B_W\Pi ^F_{t|t}B_W^T+\Phi _W \end{bmatrix}. \end{aligned}$$

(39c)

With conditional views, we obtain a similar result: with the gain $K_t$ as in (37), the updated mean is as in (39a), the updated covariance matrix for the factors and the variables involved with view is given by

$$\begin{aligned} \begin{bmatrix} \Pi ^F_{t|t}&\Pi ^{FV}_{t|t} \\ \Pi ^{VF}_{t|t}&\Pi ^V_{t|t} \\ \end{bmatrix}= \left( I-\begin{bmatrix}K_t^F\\K^V_T\end{bmatrix} \begin{bmatrix}0&P_t\end{bmatrix}\right) \tau \begin{bmatrix} \Phi ^F&\Phi ^F B_V^T \\ B_V\Phi ^F&\Phi ^V + B_V\Phi ^F B_V^T \\ \end{bmatrix}, \end{aligned}$$

(40)

and the expression for $\Pi _{t|t}$ is as in (39c).

From equations (36) and (37), we conclude that the updated remaining variables $W_{t|t}$ can be written as in equation (27).

Appendix 4: Distribution of Cumulative Returns

To construct the optimal portfolios in Figure 5, the density forecast $X_t$ for quarterly log returns is converted to a density forecast of cumulative returns for the investment horizon in two steps. First, we use the methods described in Campbell and Viceira (2005a) to determine the mean $\mu ^Y_t$ and covariance matrix $\Sigma ^Y_t$ of the cumulative log returns $Y_t=\sum _{s=1}^t X_{s}$. Second, we convert the distribution of the cumulative log returns to a distribution of cumulative returns. Since cumulative log returns are normally distributed, the cumulative returns $Z=\exp (Y)-1$ have a shifted log-normal distribution with mean

$$\begin{aligned} \left( \mu ^Z_t\right) _i=\exp \left( \left( \mu ^Y_t\right) _i+\frac{1}{2}\left( \Sigma ^Y_t\right) _{ii}\right) \end{aligned}$$

and covariance matrix

$$\begin{aligned} \left( \Sigma ^Z_t\right) _{ij}= \left( \exp \left( \left( \Sigma ^Y_t\right) _{ij}\right) - 1\right) \exp \left( (\mu ^Y_t)_i+(\mu ^Y_t)_j + \frac{\left( \Sigma ^Y_t\right) _{ii}+\left( \Sigma ^Y_t\right) _{jj}}{2} \right) , \end{aligned}$$

where i and j refer to the components of the corresponding vectors or matrices. A similar analysis is performed in Van der Schans and Steehouwer (2015) which, for the interested reader, includes a discussion of the resulting descriptive statistics of the unadjusted density forecast.