Despite the thrilling mathematical and scientific components the financial world has developed in the last decades, the core of the craft remains mostly the same: do not lose money.

And the first step towards not losing money is definitely knowing how to count it.

Believe it or not, certain kinds of contracts and obligations are not so easy to boil down to a figure given in percentual terms. In today’s post, we will deal with a specific scenario: currency hedging with variable exposure.

For the sake of example, we will accompany Olivia, a European worker with businesses all around the globe, in her endeavor along with currency movements. We will start by getting to know her and her currency problems, to then present a numerical example at the end.

Let’s start out a walkthrough, should we? 😉

## When is currency risk created?

In the FX market, one can exchange one currency for another at a given rate: the spot price. It changes by the second, as world economies trade and market participants buy or sell. This is the number most people refer to when they talk about the “cable”, the “aussie-euro” or the famous EURUSD rate.

Olivia, whose base currency is EUR, among many others, makes business with Australian companies; from where she receives payments in AUD (the common jargon says *“Olivia is long AUD”*). If Olivia converted the payments at the prevailing rate on the day she receives them, she would be completely exposed to market fluctuations of the AUDEUR spot rate. She might have done really well in business terms, but if the rate swings against her, she could lose a lot of value: the amount of EUR she expected to receive the day the payment was announced is different from the amount of EUR she gets the day she actually converts it, despite both of them representing the same amount of AUD!

Because she is aware of such currency risks and why they are not worth taking, she decides to hedge them. And she does so with FX forward contracts.

## FX Forward Contracts

Through a currency manager, **Olivia locks in the exchange rate for a given date in the future.** When a FX forward position is opened, two parties agree to exchange currencies in a future date (a.k.a. the *maturity date*) at a given *forward price*, usually different from the existing spot price on the opening date of the contract. Why the forward price is different from the spot price is a long story by itself, about which we may talk about another day. **Just keep in mind that forward prices fluctuate in time just like spot prices do.** Additionally, money settlement only takes place at the maturity date.

Now, because Olivia’s business is thrilling, her *exposure*, that is, how many AUD she expects to receive in the payments, changes with time too. The changes are produced when the payments settle. As you can see, at the end of the year, all payments are settled.

Therefore, her currency manager will adjust the hedge ratio if it goes too far away from a target value, to make sure Olivia is hedged to the right level all the time (this is quite similar to investing money in the same position every week, somehow you are averaging the entry price).

The currency manager will be using *one month forward contracts*. This means that every month the currently open contracts will have to be closed near their maturity, and new ones opened due for next month’s maturity date: this is known as a **rollover**.

**When Olivia’s currency manager executes the rollover, a flux of money between both parties has to take place.** The flux is proportional to the exposure, and for Olivia’s interests, it is settled in EUR. If the spot price would have moved in her favor, she will have to pay; if she would have lost value with the currency swing, she will receive money. Because in the rollover Olivia did not end but rather renewed her obligations, she doesn’t need to deliver the complete amount of currency agreed to in the first place, but rather the difference between the average price of her position and the price at which it’s closed.

Summing up, over time, **Olivia’s bank account will see monthly deposits or withdrawals in EUR, according to the movement of the forward prices.** In a frictionless world, Olivia would get back or give the exact amount of money she would have lost or gained because of the spot movements.

But we are not in a frictionless world, aren’t we?

## Money Result vs. Dimensionless Result

By the end of the year, Olivia would like to know if hiring the currency manager has being beneficial or not.

If she adds up all the withdrawals and deposits *due to the rollovers*on her bank account, she will get the net value of money she received or delivered that year. But since her exposure is variable, how much she gets paid in AUD, how could she compare two years where her profits were outstandingly different?

The answer is simple: you need to cast your results in a dimensionless form, you need to **compute a percentage, *** a rate of return*.

### Time-Weighted Returns vs. Money-Weighted Returns

The usual rate of return that pops up in financial calculations is the Time-Weighted Rate of Return (TWRR). This number removes the effects of inflows and outflows, so it shows the evolution one unit of value would have had for a given period. It is a good metric to compare funds, portfolios, hedge funds, etc.

However, for an individual investor or user of a financial product, it could have a misleading drawback. Let’s say you invest in a fund, and along with the year deposit or withdraw money from it, without ever closing your position. You could end up with a negative result, even if the fund performs well that year. The reason is that you might have had a lot of money invested during a bad period, withdrawn out of fear or need, and then have little money invested in when the fund performs again. As a result, the TWRR for the period is positive, but you look at your numbers and you have lost money.

To go around this discrepancy, one can opt for the Money-Weighted Rate of Return (MWRR), which will give you the equivalent rate of return of a bank deposit with the same pattern of deposits, withdrawals, and final result [1].

### Internal Rate of Return

There are multiple approaches to compute a MWRR, and the one we find that suits best is the computation of the Internal Rate of Return (IRR) :

The internal rate of return is a discount rate that makes the net present value (NPV) of all cash flows equal to zero in a discounted cash flow analysis.

Source: Investopedia

This formulation can be problematic at times, for the nonlinear equation you solve can have multiple solutions. To work around this problem, I would encourage you to read [2]. An alternative formulation of the same equation is presented, in my opinion, easier to grasp, and several arguments are given in favor of the largest solution in case of multiplicity.

## Olivia’s Rate of Return

Finally, let’s get to see all this in numbers! To compute the MWRR, we need to translate all the above in a stream of fluxes.

You need to follow three simple steps:

- Obtain the PnL settlements for each rollover day.

- Obtain the impact the currency would have had in the exposure variations.

- Put them all together in the IRR equation and solve it.

As we can see, for this particular year, Olivia was smart to hedge her currency risk. And with this valuation methodology, she will be able to compare the results with next year’s business activity.

Thanks for reading, and see you next time!

## Bibliography

[1] *The Fixed Rate Equivalent (FREQ): Measuring the Performance of Financial Accounts in the Presence of Deposits and Withdrawals*, A GreaterThanZero White Paper, 2012.

[2] *Choosing the Right solution of IRR equation to measure investment success*, Journal of Performance Measurement, 2013.