# 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

I agree with much of what Freddy has said, but I just want to flesh out some issues with regards to multi-currency portfolio optimization.

1. I find it helpful to think in terms of one currency, the currency that your investor is evaluating you based on. In your case, this would be U.S. dollars.
2. The general approach to portfolio optimization is to first forecast the distribution of your invariants/risk factors to the horizon and then to convert them into prices at the horizon. For your application, the invariants would be log changes in currencies AND the interest rates you would receive by borrowing or lending in those currencies (you may need to incorporate different borrowing or lending rates, but for the sake of simplicity we can assume them to be equal). You would then need to convert the currency/interest rate distributions at the horizon into positions that represent the return at the horizon in dollars of changing dollars to foreign currency today, earning the foreign interest rate, and then changing back to dollars at the horizon. Similarly, investing in U.S. dollars would avoid the currency component, but maintain the domestic interest rate. You would then consider these to be your universe, find the mean and covariance matrix, and optimize as appropriate (avoiding leverage would require weights to sum to 1, i.e. you either invest in the U.S. asset or convert dollars to foreign assets). This approach can also easily be adapted to forward currency contracts.
3. The above discussion primarily focused on simple currency allocation, but it is a useful framework for international equity and bond portfolio optimization as well. The investor would convert dollars to equities/bonds in foreign currency and then convert back to dollars at the horizon. One major difficulty is when the investor wishes to hedge their foreign currency exposure. The problem is that the foreign currency exposure is different in every time period (it grows or shrinks as the underlying asset grows or shrinks). So you have some options for how to handle this in practice. Perhaps the simplest is a two-step approach to find the optimal portfolio excluding the hedge and then put on the hedge in a second step given the optimal portfolio.
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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...)