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# Should you have an FX hedge for a US-listed stock?

A cursory online search for “FX hedge and stocks” throws up reams of articles about currency exposure from foreign investments. There is, however, very little guidance offered to investors regarding domestic stocks. And yet, there are times when it makes sense to have an FX hedge even for a stock that’s quoted in the investor’s default currency. Over 40% of sales for S&P 500 companies come from markets outside the US and so currency movements will figure meaningfully into the overall earnings picture for many socks in the index.

How do you think about this problem, and determine the right amount of FX exposure to hedge? Let’s illustrate with an example: a hypothetical US-based, US-listed company that is an exporter and has meaningful revenues and costs in a foreign currency. Although we will lay out all of the algebra for those interested, we heavily draw on the intuition behind the formulas to make sure following detailed calculation isn’t essential.

To begin, we assume the company value is a function of all its future cashflows:

where V is the value of the firm, CF is current cashflow, **ρ** is the discount rate (prevailing cost of capital at the time), **π** is the company’s current profit and **τ** is the rate of applicable corporate tax. Since the company is a significant exporter its operations in part depend on revenue in a foreign currency. S denotes the exchange rate of the foreign currency in US Dollars (for example, price of 1 GBP in US Dollars: thus higher S means the value of the US dollar has gone down with respect to the British Pound.)

We will focus on the easiest case where none of the other variables (tax rate, discount rate etc.) change in response to fluctuations in S. This is a good assumption for most relevant cases but there are times – especially if S moves are large – when these assumptions need to be revisited. We’ll address this question at the end and discuss directional impact of relaxing such assumptions.

__CASE 1__

## Discount factor ρ and tax rate τ stay fixed

If you assume discount rates and tax rates don’t change with the exchange rate (an assumption that makes sense so long as these changes are small), the main impact on company value V is through the profit channel.

WARNING: TEDIOUS MATH AHEAD

Define the following

X, X* - sales of the domestic good at home and abroad

Z, Z* - sales of the foreign – produced good at home and abroad

C, C* - cost functions, denominated in US dollars and foreign currency, of producing X and X*

K, K* - cost functions, denominated in US dollars and foreign currency, of producing Z and Z*

Then the total profit for the company can be restated as

and after some manipulation you get

and further

where **δ** is now the sensitivity of the percent change in company’s current profits to a percent change in the exchange rate S. You can rewrite this

where h1 is foreign-derived revenue as percent of total revenue, h2 is foreign-derived cost as percent of total cost and r is the profit rate/margin (profit π as percent of total revenue.) This is **hugely simplified** for the case where the company produces all its goods domestically, and therefore h2=0. In that case, we get

Using this in the calculation we did before, we find that

And

We are nearly finished. The last step is to connect sensitivity of company’s value V to the change in exchange rate S. This is straightforward. As you recall from above,

Then

And finally

which, since

simplifies to

END OF TEDIOUS MATH

The sensitivity of the company’s value (in percentage terms) to changes in S (also in percentage terms) simply equals h1/r. In the extreme – when all of the company’s profits are derived abroad – the right amount of FX hedge is simply the total value of the company. That’s because V and S move in tandem. The company’s earnings are a direct function of the exchange rate. However, when only a portion of the earnings are derived from abroad, the multiplier is higher the higher the proportion of earnings is derived from abroad. In contrast, profit margin mitigates the impact currency has on the value of the company. **The intuition here is that the larger the profit margin, the larger the cushion that can absorb losses from the impact on the top line derived from abroad. **For example, if the company derives 10% of its earnings abroad – assume it earns 10 dollars out of every 100 from abroad – and has 100% profit margin (entire revenue is profit), fluctuations in the currency only impact 10% of the earnings. Therefore the multiplier is 0.1, and only 10% of the company’s market value needs to be hedged against FX exposure.

HOWEVER, why isn’t this multiplier always true empirically? If you test a company’s past returns on changes in the exchange rate, you’ll often find that the sensitivity is actually less than predicted above. One reason is the assumption we made initially: that the tax rate and the discount rate don’t change with the exchange rate. Assuming a fixed tax rate is probably fine but the discount rate is quite likely impacted by the exchange rate, especially for big moves. So a more realistic scenario is to assume that ρ also varies with the exchange rate: if GBP really appreciates against the USD it’s likely that the US domestic discount rate will RISE. Let’s take a look at this second case.

__CASE 2__

## Discount factor ρ CAN change with the exchange rate

TEDIOUS MATH

Now we have two terms when we take the derivative

and after some cleaning up you get

END OF TEDIOUS MATH

As may by now be apparent, the actual sensitivity of the company’s value will be LESS than in the original case 1. Why? When S rises precipitously, the value of earnings from abroad rises, driving up the company’s value. However, US domestic rate ρ will likely rise in response (because the value of the US Dollar is dropping relative to the foreign currency) as Federal Reserve aims to shore up confidence in the currency. That will drive the value of the stock down by discounting its future cash flows at a much higher rate.