A US-based corporation has decided to make an investment in Sweden, for which it
will require a sum of 100 million Swedish kronor (SEK) in three-months’ time. The
company wishes to hedge changes in the US dollar (USD)-SEK exchange rate using
forward contracts on either the euro (EUR) or the Swiss franc (CHF) and has made the
following estimates:
• If EUR forwards are used: The standard deviation of quarterly changes in the
USD/SEK spot exchange rate is 0.007, the standard deviation of quarterly changes
in the USD/EUR forward rate is 0.018, and the correlation between the changes is
0.90.
• If CHF forwards are used: The standard deviation of quarterly changes in the
USD/SEK spot exchange rate is 0.007, the standard deviation of quarterly changes
in the USD/CHF forward rate is 0.023, and the correlation between the changes is
0.85.
Finally, the current USD/SEK spot rate is 0.104, the current three-month USD/EUR
forward rate is 0.471, and the current three-month USD/CHF forward rate is 0.602.
(a) Which currency should the company use for hedging purposes?
(b) What is the minimum-variance hedge position? Indicate if this is to be a long or
short position.
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(a) To determine which currency the company should use for hedging purposes, we need to compare the effectiveness of using EUR forwards and CHF forwards.
First, we calculate the covariance between the changes in the USD/SEK spot exchange rate and the changes in the USD/EUR forward rate:
Cov(USD/SEK, USD/EUR) = Corr(USD/SEK, USD/EUR) * Std Dev(USD/SEK) * Std Dev(USD/EUR)
= 0.90 * 0.007 * 0.018
= 0.0001134
Next, we calculate the covariance between the changes in the USD/SEK spot exchange rate and the changes in the USD/CHF forward rate:
Cov(USD/SEK, USD/CHF) = Corr(USD/SEK, USD/CHF) * Std Dev(USD/SEK) * Std Dev(USD/CHF)
= 0.85 * 0.007 * 0.023
= 0.0001311
Now, we can calculate the variance of the changes in the USD/SEK spot exchange rate:
Var(USD/SEK) = Std Dev(USD/SEK)^2
= 0.007^2
= 0.000049
We can use the formula for the minimum variance hedge ratio (h*) to determine the currency to use for hedging:
h* = Cov(USD/SEK, USD/CHF) / Var(USD/CHF)
= 0.0001311 / 0.000049
= 2.6796
The minimum variance hedge ratio for CHF forwards is 2.6796.
Now, we can calculate the variance of the portfolio using EUR forwards:
Var(portfolio) = Var(USD/SEK) + h^2 * Var(USD/EUR) + 2 * h * Cov(USD/SEK, USD/EUR)
Substituting the values:
Var(portfolio) = 0.000049 + 2.6796^2 * 0.018^2 + 2 * 2.6796 * 0.0001134
= 0.000729 + 0.02873697216 + 0.000060732768
= 0.029526732768
Similarly, we can calculate the variance of the portfolio using CHF forwards:
Var(portfolio) = Var(USD/SEK) + (1 - h)^2 * Var(USD/CHF) + 2 * (1 - h) * Cov(USD/SEK, USD/CHF)
Substituting the values:
Var(portfolio) = 0.000049 + (1 - 2.6796)^2 * 0.023^2 + 2 * (1 - 2.6796) * 0.0001311
= 0.000049 + (-1.6796)^2 * 0.023^2 + 2 * (-1.6796) * 0.0001311
= 0.000049 + 0.068601678176 + (-0.0000438212496)
= 0.068607856926
The variance of the portfolio using EUR forwards is 0.029526732768 and the variance of the portfolio using CHF forwards is 0.068607856926. Since the variance is lower for EUR forwards, the company should use EUR forwards for hedging purposes.
(b) The minimum-variance hedge position is given by h*, which is 2.6796. Since the company wants to hedge changes in the USD/SEK exchange rate, it should take a long position in the currency it is using for hedging, which in this case is the euro (EUR).
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