# Can Black-Litterman-type expected return estimation be used for regional ETFs?

The Black-Litterman approach to return estimation overcomes the problems associated with estimating expected returns via historical averages by determining the equilibrium returns implied by the Capital Asset Pricing Model (CAPM).

The CAPM explains the return on an asset as a function of the risk premium offered by the market only. The market here is, in theory, an index of all investable securities but broad indices are often used.

Many of the canonical sources on Black Litterman use in their example not individual assets (like equities) but abstract "regions", implying the use of regionally distinct broad-based index funds.

My question is the following: Is the CAPM (to the extent that it is suitable for, e.g., equities) suitable for estimating the equilibrium expected returns of regional equity and bond index ETFs relative to a self-built global securities market index, comprising, for example the following:

• 65% MSCI All Country World Index,
• 15% Citigroup World Government Bond Ex-US Index,
• 15% Citigroup US Government Bond Index,
• 3% Merrill Lynch US High Yield Cash Pay Constrained Index,
• 2% JP Morgan (EMBI) Emerging Markets Bond Index Global?

Is a $\beta$ calculated for regional and asset class-specific ETFs relative to the performance of such an index meaningful and are there any known applications of such a method? What are some potential problems I should think of before constructing such an index?

• (1) the CAPM doesn't work, and (2) and there isn't a constant term in the CAPM. Under the CAPM $\mathbb{E}[R_i - R_f] = \beta_i \mathbb{E}[R_m - R_f]$. CAPM has long ago been killed in academia as an empirical failure. In data, high beta stocks have lower returns, not higher. In multi factor models, the market factor is mostly useful to capture the equity premium. – Matthew Gunn Jan 24 '18 at 16:31
• By constant I was referring to the risk-free rate in the $\mathbb{E}[R_i] = R_f + \beta_i \mathbb{E}[R_m - R_f]$ formulation of the CAPM. Note that my question is not whether the CAPM itself is valid or whether multifactor models perform better but whether the reasoning carries over to ETFs or whether there is anything that prevents this conceptually. One might argue that the reason the CAPM doesn't work in practice is that there is no observable market portfolio. An index constructed as described may approximate such a market portfolio better, but again that's not the focus of my question. – Constantin Jan 24 '18 at 16:42

Under the logic of the CAPM, the equation $\operatorname{E}[R_i - R_f] = \beta_i \operatorname{E}[R_m - R_f]$ would hold for any return, whether it's a stock return, bond return, portfolio return, call option return, etc....

If the CAPM holds for a set of assets, it's easy to see that CAPM would hold for any portfolio over those assets. Let $\mathbf{w}$ be a vector of security weights in a portfolio. Then.

\begin{align*} \operatorname{E}[R_p - R_f] &= \operatorname{E}\left[\sum_i w_i (R_i - R_f) \right] \\ &= \sum_i w_i \beta_i\operatorname{E}[R_m - R_f]\\ &= \beta_p \operatorname{E}[R_m - R_F] \end{align*}

And the CAPM holds for the portfolio return. Note that $\beta_p = \sum_i w_i \beta_i$ because of the linearity of covariance.

### Major problems with the CAPM

1. Other variables (eg. value, momentum, and investment) besides market beta give information on the cross-section of expected stock returns.
2. The security market line with respect to market betas doesn't go the proper direction! It's either flat or perhaps downward sloping.

It's an old 80s/90s argument that the CAPM isn't testable because the return on the overall market portfolio isn't observable. But then the CAPM isn't even a scientific theory in the Karl Popper sense of generating testable implications. Most damning to the CAPM is the downward sloping security market line which basically means the CAPM's central prediction goes the wrong direction.

As Box famously said, "all models are wrong but some are useful." The CAPM just isn't useful for describing the data. I'm not saying market betas are useless. In broader factor models, the market beta is basically there to capture that equities have higher average returns than bonds.

Anyway, these problems have been known a long time... see Fama and French "Cross-section of Expected Stock Returns," Cochrane's, "New Facts of Finance, Frazinni "Betting Against Beta," etc.... Empirical asset pricing has moved on towards multifactor models, statistical approaches using PCA etc.... Pure macro-finance economic theory has moved towards consumption based approaches.

Basically, I don't think you want to get stuck in hackneyed 90s debates on the CAPM.

Constructing a global market factor is fine. I endorse that. It's at least useful for risk management and may be useful as part of a broader factor model. But I doubt you'll find the past 20-30 years of empirical asset pricing was all wrong, and all we needed for the CAPM to work was $R_m = .65 R_{MSCI} + \ldots$.

There's another line of literature you may want to look at as to whether global financial markets are integrated, whether you want regional or global factors. Are the factors for Japan different than those of the US?