constructing a minimum variance portfolio

Question

Assume a US-based company has sold something to a Norwegian company. It will receive 1M Norwegian Kroner in two months, and would like to hedge this future cash flow against currency exchange risk. It's not possible to hedge Kroner and Dollars directly, but the company can hedge with Euros, since Euros behave similarly to Kroner. Following data is given:

Exchange rates:

0.164\$/Kroner $=: K$;

0.625\$/€ $=: M$;

0.262€/Kroner.

correlation coefficient between $K,M: \sigma_{K,M}=0.8$

standard deviations:

$\sigma_{M}= 3\%$, $\sigma_{K}=2.5\%$ per month.

I am confused since we actually have not two assets, where we have to find the optimal weights in the portfolio, but we have only one asset, which is the futures between Euros and Dollars. Can someone show me how I work with this data, and explain the solution?

Answer 1

Strictly speaking, this is a proxy hedging problem. You have to hedge one currency with another. The one period covariance matrix is assumed to be $$\Sigma_{1}=\left[\begin{array}{cc} 0.03 & 0\\ 0 & 0.025 \end{array}\right]\left[\begin{array}{cc} 1 & 0.8\\ 0.8 & 1 \end{array}\right]\left[\begin{array}{cc} 0.03 & 0\\ 0 & 0.025 \end{array}\right] $$

So you can calculate the two period covariance matrix as $$\Sigma_{2}=2\Sigma_{1}$$

By your assumptions, you have a portfolio that is effectively $-100\%$ short the Krona. You could set this up an optimization problem to find $$w\equiv argmin\left\{ \left(w-w_{b}\right)'\Sigma_{2}\left(w-w_{b}\right)'\right\} $$ where $w_{b}\equiv\left[\begin{array}{c} 0\\ -1 \end{array}\right]$ and constrain to weights so that $w$ makes no investment in the Krona.

However, in this case, you can do it more simply. You can write out the portfolio variance (using values from $\Sigma_{2}$) as $$\sigma^{2}=w_{1}^{2}\sigma_{1}^{2}+w_{2}^{2}\sigma_{2}^{2}+2w_{1}w_{2}\sigma_{1,2}$$ and since $w_{2}$ is already known to equal $-100\%$ you can minimize that directly to find $$w_{1}=-w_{2}\frac{\sigma_{1,2}}{\sigma_{1}^{2}} $$ which works out to $w_{1}=\frac{2}{3}$. Then you would just convert that weight into an amount of euros to buy.

Answer 2

The reason for hedging the exchange risk is that the deal was made given the current exchange rate, and the company wants to make sure that they receive the same amount of dollars (or more) in 2 months that they agreed to now.

Assuming that the exchange rates stay the same, when they get paid:

K 1M = $ 164,000 or € 262,000

The variance that you want to minimize is the $/K variance, which is the variance of our future cash flow to be paid in 2 months. We are saying that we can't just buy the NOK/USD futures, and since the Euro is highly correlated with Kroners, we can use Euro.

Remember you also have dollars, so you actually have 2 assets. If you buy all Euros then you are exposed to the risk of the Dollar gaining on the Euro, if you just keep all dollars, you are exposed to the Euro (and Kroner) gaining on the dollar. So I think the minimum variance of your future cash flow will be attained by buying some combination of dollars and euros.