Dumb Question on hedging International Portfolio

[Dave]

We can estimate the hedge ratios for hedging international portfolios by selecting the most independent major currencies with futures contracts available and calculate the hedge ratios jointly by running a multiple linear regression (MLR) of the domestic currency returns

R (computed in say USD) on the futures returns (Rf1, Rf2, Rf3) :



R = a + h1 Rf1 + h2 Rf2 + h3 Rf3 + e



Dumb question: say we find the hedge ratios to be

h1=x, h2=y, h3=z

So, we long/short x,y,z currency futures for each ______________ of the portfolio ?



If anyone here is familiar with this topic, then references will be appreciated.



Thanks

[marcos]

h1, h2, h3 can be interpretetd as the sensitivities of R to the foreign currencies. Thus, if h1=-.1, h2=.2, h3=-,3 you go long h1 with a weight of .1/(.1+.2+.3), long h3 with a weight of .3/(.1+.2+.3), and short h2 with a weight of .2/(.1+.2+.3) if you want to have the portfolio hedged.

[Dave]

Thanks marcos.



I guess one forgets this stuff if not used on a daily basis. Looking at this from the stat interpretation, sure, I see the sensitivity point of view since any coeff is the first partial derivative of R holding all others constant.



But how do we get the actual NUMBER of currency futures to long/ short? Would the following be correct: say we have 1b euros invested in Europe, 1b pounds in UK.



Number of Euro currency futures to long

= [.1/(.1+.2+.3) * 1b euros]/[euro currency futures price ]

[Dave]

Surely someone here must be a FX guy ?

[Tradenator]

Many places hedge international equities using benchmark (eg MSCI) weights, with either a 50% or a 100% static hedge that rolls on forever.  Forwards/swaps are good for this because you need to get away from the majors to hedge anything with size.  For a smaller fund, you would want to cull your basket of small positions, and then futures become viable.  If you are hedging a fund of funds, you need to account for what the individual managers are doing.  Running a static hedge like this also makes it easier to wrap the fund with an active currency program if you have one available.



I use futures to translate one fund from USD into AUD and NZD, but need forwards for the cross to put AUD back into NZD for this particular investor.  This is really just translating back to base currency, though, rather than hedging the exposure of the international equities by country.



EDIT: see [url=/Show%20Post.aspx?PostIDKey=52983]this thread[/url] for some reference materials and book recommendations

[aaron]

I think you're asking a simple question.



To remove the effect of currency f1 on your portfolio, you have to go short h1*NAV USD value of currency f1.



For example, h1 = -0.10, meaning if currency f1 strengthens by 1%, you expect your portfolio return to be 0.1% less than you would otherwise (note: this assumes you express the currency return as the return to a USD investor of holding the currency rather than the inverse, be very careful here). Your portfolio NAV is USD 1 billion. You want to go long f1 USD 100 million (that is, use USD 100 million to buy f1).



Now if f1 strengthens 1%, you make USD 1 million, which just offsets your expected USD 1 million portfolio loss.



The coefficients (h's) are not partial derivatives. They do not give the effect of currency returns assuming everything else is constant, they "assume" movements in the other currencies in the regression. This makes them unstable, especially if you have many currencies and closely-correlated currencies. I don't recommend multiple regression hedges as a general technique for this reason. You're better off doing univariate regressions first to determine the relative hedge amounts, then do a single univariate regression using the basket of currencies. You might do a bivariate regression instead, using currencies with low correlation, but once you get beyond that the statistical problems mount quickly.

[marcos]

What aaron wants to underline is the problem of multicollinearity in a regression analysis.



Google delivers a neat explanation on this issue:



"When two X variables are highly correlated, they both convey essentially the same information. In this case, neither may contribute significantly to the model after the other one is included. But together they contribute a lot. If you removed both variables from the model, the fit would be much worse. So the overall model fits the data well, but neither X variable makes a significant contribution when it is added to your model last. When this happens, the X variables are collinear and the results show multicollinearity."



Hence, the lower the correl between your indie currencies the better.

[sfca]

All this confuses the heck out of me.  Dave did not say what the underlying is.  In my deep dark past a hedged international portfolio was a non-USD bond portfolio that had its notional hedged with forwards.  The only portion not hedged then would be the monthly gain or loss since that was unpredictable.  Unless there are some currencies that don't have a good derivative representation, I'm not quite sure why the regression approach unless you're trying to do something with correlating the domestic returns and the currency moves and/or have derivatives with notionals not well defined here for hedging purposes.  If that first regression has portfolio returns as the dependent and the FX only as the independent variables, then the results would not be appropriate for hedging due to the omitted domestic security returns.  And before I would try to rely on statistical fits using regressions instead of notionals, I would first look at cointegration since that return specification is not going to pick up things like long term trends.  But as I said I get confused easily.

[Dave]

Aaron: The coefficients (h's) are not partial derivatives. They do not give the effect of currency returns assuming everything else is constant, they "assume" movements in the other currencies in the regression.



Thanks Aaron. Obviously I am not on top of my stats here, but aren't the regression coefficients supposed to be the partial derivatives in any MLR ?



I understand the Multicollinearlity aspect. From what I remember,  the X'X matrix does not invert (or is too large) when MC exists creating problems for computing the regression coeff matrix b, where b=(X'X)^-1 XY



sfca: point taken, but I do not have the book where I read this on me, but I will look it up when I have a chance.