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27 Aug 2008
Aug
28
Foreign Exchange: Volatility-minimizing hedge (2007 FRM)
by David Harper, CFA, FRM, CIPM
Learning Outcome
- LO 19.9: Explain the implications of perfect positive correlation, zero correlation, and perfect negative correlation between price risk and quantity risk for the optimal hedge ratio and the risk of the hedged versus the un-hedged cash flows.
Hedging foreign currency quantity risk with futures
In the previous post, I used German auto makers referenced in the Wall Street Journal to illustrate the quantity risk of foreign exchange exposure. Specifically, due to price elasticity of demand, German auto makers cannot sufficiently raise U.S. prices on their exported cars in order to compensate for a stronger Euro (their reported solution is to build cars in the U.S., a natural hedge).
Where the previous learning outcome (19.8) illustrated the problem of how quantity risk compounds the price risk we normally associate with foreign currency, the above learning outcome (19.9) illustrates a potential solution: the use of futures contracts to hedge the foreign exchange risk. The key idea is that the number of futures contracts ought to vary depending on the correlation between the exchange rate (or foreign currency futures prices) and the so-called quantity risk. I think the term 'quantity risk' is a little awkward. To remind, quantity risk refers to the quantity of foreign currency demanded, which fluctuates with the exchange rate. Due to price elasticity, changes in the exchange rate (price) may impact the volume of currency demanded (quantity). The source for all of this is the tortuous prose in Rene Stulz's Risk Management & Derivatives (Chapter 8).
Optimal Hedge Ratio
The volatility-minimizing hedge for a random cash flow, where C = cash flows and where G = the futures price (in our example below, the exchange rate) at maturity of the hedge, is given by:
This is an important, all-purpose formula. Please note it is the same minimum variance hedge ratio for a futures contract provided by John Hull. Replace the covariance in the numerator with the product of: correlation [cash flow, futures price], the standard deviation of cash flows, and the standard deviation of the exchange rate, and the hedge ratio becomes:
Which is the same formula, but more familiar to us as the hedge ratio for futures: correlation multiplied by the ratio of volatilities.
Example: U.S. Auto Maker Exports to Germany
Let's take the example of a U.S. car maker exporting to Germany (I reversed the countries, I think better from my home country!).
- We simplify by assuming the spot exchange rate = the future exchange rate, given a short period like three months. Assume the current exchange rate is $1.5 dollars per 1 Euro.
- We happen to know that next period, the exchange rate will either go to $1.00 dollar per Euro (strong Dollar/weak Euro) or $2.00 dollars per Euro (weak Dollar/strong Euro).
- Then assume three correlation scenarios. That is, correlations between the exchange rate (price risk) and units of foreign currency collected (quantity risk): perfect, negative and no correlation. Perfect correlation implies that if the exchange rate goes to $1 dollar per Euro, the "quantity" of Euros collected also goes to 1. Under negative correlation, a $1 exchange rate corresponds to a greater "quantity" of two (2) Euros collected
To restate what this means, the first column below ("the dollar price of the Euro") is the exchange rate and the realized futures price under two scenarios(here is the price risk). The next three columns ("Cash flow in Euros based on correlation between [FX, Euros]") represent quantity risk. Perfect positive correlation implies that a shift in the exchange rate to $1 dollar per Euro also impacts the number of Euros our company can collect. The product of the first column and the second set of columns is the translated US dollar cash flow. For example, under positive correlation and a $2 exchange rate, 2 Euros are collected which earns $4 in U.S. dollars (2 Euros x $2 exchange rate). In short, the final set of three columns represents price multiplied by quantity.
If that is what we know, the formula for the volatility-minimizing hedge (h) tells us that we should short foreign currency futures contracts. Specifically,
- Under perfect positive correlation, short 3 FX futures (covariance between cash flows and futures price of 0.75 divided by variance of futures price of 0.25)
- Under no correlation, short 1.5 FX futures (covariance between cash flows and futures price of 0.375 divided by variance of futures price of 0.25)
- Under perfect negative correlation, no futures are needed.
To test this under perfect positive correlation, note the U.S. dollars collected (D) are offset by gains/losses on the futures contracts. For example, if the exchange rate goes to $1, the cash flow is $1 (1 Euro x $1) but the short futures gain is $1.50 (3 contracts x $0.50 gain per). But if the exchange rate goes to $2, the cash flow is $4 (2 x $2) and the futures loss is $1.50 (3 contracts x $0.50 loss per). Because of the futures contracts, the gain is $2.50.
The optimal number of contracts under no correlation is to short 1.5 futures contracts, but the hedge cannot be perfect: note the dispersion in the net profit outcomes. However, by shorting 1.5 futures, the worst outcome ($1.75) is better than the worst unhedged outcome ($1.00).
Finally, the negative correlation is naturally hedged:
Summary
Price risk is the risk due to fluctuating foreign currency; in our example, we don't know if the exchange rate will go to $1 or $2. But there is an additional dimension. The currency change, due to the price elasticity of demand, is likely to impact the quantity of German Euros demanded. So, our total exposure is price (exchange rate) and quantity (foreign currency units demanded). We will collect price multiplied by quantity; the volatility of [price x quantity] depends on their correlation. Their correlation informs the optimal hedge. We can short futures to deploy the hedge, but as the no correlation scenario illustrates, a perfect hedge is unlikely. In other words, our hedged cash flows will still be volatile but less volatile than our un-hedged cash flows.
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