# How to find an initial equilibrium benchmark portfolio that includes currencies for Black-Litterman model

We're working on a term project to adjust B-L model to yield robust results. From our readings, we resulted in that initial benchmark portfolio is constructed by using market capitalization weight. However, we want to include different currencies to our portfolio (U.S. dollar, Euro etc.) and we are clueless about how to determine market caps of currencies. Using total money supply for currencies didn't seem intuitive to us since, in that case, it would outweigh currencies unreasonably. Is there a correct way to determine this?

• Currencies are not really an asset like stocks, bonds, real estate, t-bills. They don't earn a return over the long run (some go up, some go down) so they probably should not be included in an equilibrium market portfolio. (An exception may be if you want to hedge foreign equities, bonds, etc. In this case you would go short the relevant currency to insulate yourself from currency fluctuations that would affect your foreign stocks, bonds, etc.). – Alex C Feb 28 '18 at 8:03

We first need to define the risk-free asset. One may assume that for a US investor, the riskfree asset is the USD, whereas, for a Japanese investor, it is JPY.

This is important because the perceived volatility of USD to a Japanese will cause him to require a higher return on a USD asset than on a JPY asset, thereby causing some home bias.

While the market weights for a US investor are a linear combination of USD cash and an efficient market portfolio including some JPY cash seen as a foreign asset. The converse is true for a JPY cash investor.

Cash is always subject to inflation, arguably, an eternal investor such as an endowment might use the world stock market cap as its riskfree asset, in which case any currency holding is seen as a risky asset.

The point is that the weights depend on the risk-free reference, so that market weights reflecting expected returns and covariance should not be based on narrow monetary mass such as M0.

On the topic of equilibrium weights and fx cash holdings, BL (Black Litterman) refer in their article to an ICAPM (international capital asset pricing model) and Universal Hedging methodology to derive the weights they use.

Ironically the BL (Black Litterman) model advocated against home bias at the worst point of history for US investors: it was published and marketed in the late 80s/early 90s just as the Japanese market had overtaken the US market. It would help embolden US investors and justify that they jump ship from the home-biased investment to buy Japanese stocks.

The investors who followed this strategy must have fared very badly: they would enter the Japanese market at the height of its overvaluation, at the beginning of a 30-year bear market and missed 2 decades of bull market in the US.

I think you have to use money supply in some regard, but I also agree with you that it is not a good proxy for total market value of that money.

Why not multiply money supply by exchange rate. You could also consider using purchasing power parity as a proxy for the value of money. The result of this operation is highly analogous to market capitalization (i.e., shares $\times$ price $\to$ M0 $\times$ FxRate).

For example, a country which has printed a lot of worthless money won't exert nearly as much effect on the initial equilibrium model. Likewise, a country with small, but valuable monetary reserves won't fall off the allocation scale.