# Black-Litterman: a soft introduction

The Black-Litterman (BL) model considers explicitly that investors do express their own views about risky asset returns by collectively holding the market portfolio according to their market capitalizations. An active investment manager aiming at beating the market often implements her own views tilting the portfolio weights towards over or under performing assets according to rules that require a certain amount of confidence about the very same views. Mean-variance optimization pitfalls have been thoroughly studied and referenced: error-maximization, portfolio concentration and high sensitivity to input (returns) changes. Thus, an investment manager is often confused when asked to translate her views in terms of portfolio tilts especially when confronting with counter-intuitive and contrasting outputs highly dependent on her own inputs. The BL model, starting from a “reverse” optimization provides investment managers with a simple, though effective, approach at factoring in their outlook about risky assets blending it with the required expected returns implied in the market allocations. Given the market capitalization ($w_m$) of risky assets, assuming investors collectively own the optimal portfolio and that they take on the desired level of risk ($V$), it is possible to “reverse” the returns required by them ($\Pi$) to maximize their utility according to their risk aversion degree ($\lambda$) using the following equation:

$\Pi = \lambda V \overline{w}_m$     (1)

where in (1) $\Pi$ is the vector of implied market returns, $\lambda$ is the risk aversion coecient, $V$ is the covariance matrix of asset returns, and $w_m$ is the vector of market capitalizations. The outright BL formula can be overwhelming to the qualitative manager who doesn’t have the inclination to delve into it and grasp its major quality in solving the portfolio allocation tilts puzzle.

$E\left(R\right) = \left[ \left(\tau V \right)^{-1} + P'\Omega^{-1}P\right]^{-1}\left[\left(\tau V \right)^{-1} \Pi+ P'\Omega^{-1}Q\right]$     (2)

Posted in Economics, Whitepapers
###### One comment on “Black-Litterman: a soft introduction”
1. R.Vanzini says:

Actually, in response to a comment I received in person, Black-Litterman isn’t opposed to Treynor-Black (core/swing portfolio model) and they’re not mutually exclusive. They’re just two different asset allocation models that can be be coupled to enhance excess-return.