# Optimizing a currency only portfolio with negative weights

I am testing various optimization methods for a currency-only portfolio. I have a vector of expected returns for the major developed currencies vs. the USD each week (based on a proprietary model). I then take the annualized vol of each currency and dump it all into an optimizer, looking to run a portfolio of currency bets that maximize my return for a given level of risk. I have multiple issues and would appreciate any insights:

Firstly, the forecast model will sometimes generate across-the-board short positions in all currencies in my model against the USD. Since currencies are a relative value game, how do I incorporate exposure to the USD in the optimizer? Should the expected returns and the vol on the USD simply be an unweighted inverse average of all the other currencies? Also, since I cannot be net long or short the currency market, do I impose the constraint that the weights must sum to zero? Am I even thinking about this problem correctly?

Secondly, when I do the above, the weights swing dramatically and often assign positive weights when the model of expected returns is indicating a short (and vice versa). It's typically happens for positions with a low expected return and one could argue these are hedging positions. Is the ideal solution here to impose a constraint that the sign of the weight needs to match the sign of the expected returns?

Alternatively should I try Black Litterman? What are the market equilibrium returns for currencies?

Or should I drop the vector of expected returns from the optimizer, use it to simply generate long-short positions, and then simply run the optimizer to minimize my risk?

Appreciate any suggestions

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folks, incredibly helpful and thughtful answers..much appreciated. the scales have come off my eyes... - i think i remove the USD from the optimsation process and remove the constraint that the weights must sum fo flat...the inverse of the net sum of the weights then effectively becomes the USD position. also, both a question and a statement for Freddy - i am optimizing FX pairs, all vs the USD....i am often generating weights that are the opposite sign to the expected return - if say EUR and CHF are very highly correlated (ie >80%) and i have a positive expected return for both, the optimizer – user3522 Dec 31 '12 at 12:36

For Black-Litterman, an equilibrium no arbitrage condition such as interest rate parity suggests investors would be indifferent between investing in either the foreign or domestic currency. Thus, you could use a constant zero return for all currencies in your opportunity set as the equilibrium model. What this ultimately would do is act like a shrinkage estimator for your active views on the currency markets. This could actually improve the performance of your forecasts measurably.

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• first of all, there is nothing wrong in a currency-only portfolio to be dollar long and short the cross currencies. If that is what your model predicts and if you have a high confidence in the predictions and standard error being low then why do you have issues being dollar long and short the other currencies?

• You can implement boundary conditions, such as a maximum exposure per currency, so I do not see anything wrong with optimizing expected risk-adjusted return.

• Of course you can be net long or short any of the currencies you trade in your portfolio, hence there is no logical reason why the weights have to sum to zero.

• I assume you make mistakes in your optimization. I do not see a way how your expected returns can be negative but the recommended weight is positive for a given currency. your expected returns are inputs to the optimization, the weight the output, if they are not of equal sign then you are doing something wrong. But beware, if the expected returns are on fx pairs and not on the currencies themselves then it could happen that you get an expected return of -1% for USD/JPY, 2% for USD/EUR, and 1% for GBP/JPY, given those are the only currencies you trade. Then your optimization will most likely assign a positive weight to USD/JPY (if you do the optimization right) because the net expected return in USD alone is positive and the net expected return in JPY alone is negative (this is all under a lot of simplifying assumptions...)