Why is the hedging cost using forwards $\frac{F-S}{S}$ and how is this related to Carry?

I'm a bit confused about hedging a foreign exposure and how this is related to carry.

I've read that the hedging cost $$h$$ is $$h=\frac{F-S}{S}$$

where $$F$$ is the forward and $$S$$ is the spot price. On the other hand Pedersen defines the following as the carry $$c$$:

$$c = \frac{S-F}{S}$$

Lets for the moment assume we are interested in currency forward as this is the case for currency hedging. Further if we are a European investor who wants to a purchase a dollar denominated bond. So $$S$$ is the spot USD/EUR rate and $$F$$ the corresponding forward. Let $$r_u$$ and $$r_e$$ the available US and European interest rate, respectively. Using the covered interest rate parity I can derive the forward price as

$$F = S\frac{1+r_e}{1+r_u}$$

Now several questions how market participants use the vocabulary in reality:

1. Why do market participants use $$h$$ as the cost and not simply the forward price $$F$$?
2. In simple terms carry is the return from an asset of just holding it. Why is this exactly $$-h$$? What is the intuition behind it?
3. Why do market participants often just take the difference of the interest rate as hedging cost (which is not exact as seen above). I.e. if you hold a USD denominated bond as a European investor they calculate it as $$r_e-r_u$$, which is different from $$h$$.
• @noob2 many thanks makes sense. If you want you can add this as an answer. So far your comments fit closest my questions.
– math
Feb 11 '20 at 18:55

Re-reading the famous Carry article by Koijen, Moskowitz, Pedersen, Vugt that you linked (at least the online version) I find that they define Carry of foreign currency as (equation 7 on Page 9):

$$C_t = \frac{S_t-F_t}{F_t}$$

However, I have also seen sources where the carry is defined (like you did) as

$$C_t = \frac{S_t-F_t}{S_t}$$

In both cases Carry is intended to describe the return on forward long position of the foreign currency if "all things remain the same", i.e. in this case if the spot rate remains the same (i.e. $$S_T=S_t$$, where t is now and T is the maturity date of the forward). The difference between these two definitions is in the denominator, i.e in the "amount invested". With a derivative like a forward, the "amount invested" is unclear to begin with; Pedersen assumes that you set aside the full value of the forward contract, while the other definition assumes you set aside the amount S, the spot value of the foreign currency. Either convention will do, and S or F are of the same order of magnitude in any case.

Now we come to hedging foreign bonds with forwards. Since you are long bonds, you are long the foreign currency, to hedge your FX risk you must short the forward, i.e. sell the foreign currency in the forward market. The amount of foreign currency you sell should be your forecast of what the bonds will be worth at time T. Usually this will be a bit higher than the current value of the bonds, since the bonds will accrue some interest, which we can assume will be kept in the foreign currency and/or re-invested in foreign bonds. If you are wrong in your estimate you will have a "hedging error" (an undesired impact from foreign exchange fluctuations) but this will be quite small in most cases.

People refer to $$h_t$$ as the "hedging cost" but this is confusing terminology. For one thing $$h_t$$ can be positive or negative, unlike the everyday meaning of "cost" which is always a detraction from your wealth. The hedge leg will earn a return over time (the "hedge return"), which depends on the movements of the spot rate. $$h_t$$ is the hedge return when the spot rate remains unchanged ($$S_T=S_t$$). With this definition we see that this is exactly the same as the carry, but of the opposite sign, because the hedge is short the foreign currency, while the carry is calculated for a person long the currency. So $$h_t=-C_t$$.

Why do market participants "often take the interest rate differential as the hedging cost"? It is an approximation. We have $$h=(S−F)/S=1−F/S=1−\frac{1+r_e}{1+r_u}=\frac{r_u−r_e}{1+r_u}\approx r_u−r_e$$ because the denominator $$1+r_u$$ is close to 1. It is easy to subtract two numbers, and although wrong, it is usually not too far off.

Q1. What exactly do you mean with "$$h_t$$ is the hedge return when the spot rate remains unchanged" ?

When you insure an automobile for 1 year, the "cost of insurance" refers to the fact that you have to pay H to the insurance company even if your car is not damaged. If you had no insurance and the car was not damaged, you would have had to pay nothing for insurance and nothing for repairs. So the "cost of insurance" is a kind of "regret" you feel when you insure fearing something bad, but nothing happens. Of course in other cases the insurance may pay off, for example if your car is heavily damaged in an accident.

By (rough) analogy, when you hedge your bonds you enter into a currency forward which will earn a return (make money or lose money). This return depends on the evolution of the spot rate and is equal to $$-\frac{S_T-F_t}{S_t}$$ (the minus sign is because of a short position). Suppose the spot rate remains unchanged $$S_T=S_t$$ then (1) You regret hedging, it was completely unnecessary (2) You still "pay a cost" (which could be a negative, i.e. a profit to you). This amount $$h_t = \frac{F_t-S_t}{S_t}$$ is called by some people the "cost of hedging" by analogy with the car insurance case I mentioned earlier.

Q2. they say: "For hedging you will receive domestic interest rate and essential pay foreign interest rate." Why is this the case?

Normally a company enters into a forward with a Bank. But they could also "make their own (short) forward" by: (1) borrowing the foreign currency from a foreign bank, (2) converting the currency immediately into the domestic currency at the rate $$S_t$$, (3) Investing the domestic currency at the domestic rate until time T. In fact the proof of the CIP Theorem makes use of the fact that the Bank forward and the "do it yourself" forward are equivalent and must be priced the same. Because the foreign and domestic interest rates are not equal this will usually result in some gain or loss (except in the case where the future spot rate turns out to be exactly equal to the forward rate F i.e. $$S_T=F_t$$, when it will be zero).

[To help remember this, you might imagine a very troubled emerging market country with very high inflation and very high interest rates. In these cases people say "Oh my god, it is so expensive to hedge country X". What they mean is that $$r_{foreign} >> r_{domestic}$$ and $$F< so hedging amounts to borrowing expensively and investing cheaply, or selling the currency forward at a poor value (i.e. below the current value). And it is true that if (by some miracle) the exchange rate does not devalue you will pay a big cost of hedging. But of course the market is not stupid and in these cases there is a high chance of devaluation, it is practically a certainty sooner or later.]

• 1. thanks for the detailed answer. I have two small questions afterwhich I'm happy to accept yours. What exactly do you mean with "$h_t$ is the hedge return when the spot rate remains unchanged" ?
– math
Feb 15 '20 at 10:31
• 2. in this paper, page 2 under "point 1 Interest rate differential" they say: For hedging you will receive domestic interest rate and essential pay foreign interest rate." Why is this the case?
– math
Feb 15 '20 at 10:33
• I answered the questions. Feb 15 '20 at 14:51
• thanks this was very helpful
– math
Feb 15 '20 at 15:21

Starting from the top of your Q and moving down, $$h$$ (or really $$h - 1$$) in your first example represents forward yield. Depending on which side you're on, this is either a cost (of hedging) or a profit.

I haven't read it, but in all likelihood, Pedersen represents it in inverse because he's considering a 'short' position (eg, in your $$EUR/USD$$ example, Pedersen would be speaking to holding $$USD/EUR$$).

Why do market participants use h as the cost and not simply the forward price F?

Because as part of a forward position, you also tacitly hold (or are short) the underlying. Hence, the cost is the difference (the ratio) between the forward and spot.

n simple terms carry is the return from an asset of just holding it. Why is this exactly −h? What is the intuition behind it?

As above, $$F$$ is the theorized value of the underlying at some future date. If we hold $$S$$ to expiry, it will be worth $$F$$ (though, in reality probably not exactly), and our return will be $$\frac{F}{S} - 1 = h$$. This will be $$-h$$ for a short (eg, $$USD/EUR$$) position.

Why do market participants often just take the difference of the interest rate as hedging cost (which is not exact as seen above). I.e. if you hold a USD denominated bond as a European investor they calculate it as re−ru, which is different from h.

Would need additional context, namely where you're seeing this, to answer explicitly but practically, this is likely because while you hold a USD-demoninated bond as a european investor, assuming you haven't hedged, you're exposed to changes in the exchange rate between your home . Using the difference between the abiding rates (eg, $$r_usd$$ - $$r_eur$$) is related to interest rate parity and would require a much longer explanation. The Wiki article for interest rate parity does a decent job laying the framework.

Provided you haven't hedged, your currency return is the difference between the two, either a cost or profit depending on your position and movement of the exchange rate. Hedging to remove that exposure would, because of IRP, cost you the difference in the two interest rates (hence, cost to hedge).

Suppose that your local currency is USD and you go long a 1 USD bond in EUR market so the nominal of your EUR bond is $$\frac{1}{S}$$ EUR where $$S=EUR/USD$$.

1. You pay $$\frac{1}{S} EUR=1 USD$$ as nominal for the bond.
2. You buy a $$EUR/USD$$ forward with $$\frac{1}{S}$$ EUR as nominal with the same maturity as your bond. Thus, at the maturity date you will get back the nominal which is $$\frac{1}{S} EUR$$ then you convert it to your local currency USD using the forward. So, you will get back $$\frac{F}{S}$$ USD.

The price of the hedge is then $$\frac{F}{S}-1$$ USD as you payed $$1 USD$$ at the begining and get back $$\frac{F}{S} USD$$ at maturity.

Of course, if you sell a bond the cost of the hedge will be $$\frac{S-F}{S}$$

• So you are assuming $S=EUR/USD$ same for F right?
– math
Feb 11 '20 at 21:03
• Why do I get $F$ USD at maturity? I want to short USD and be long EUR in the forward market, no?
– math
Feb 11 '20 at 21:15
• Yes i'm assuming $S=EUR/USD$ and you should be short USD as you get back your bond nominal which is S USD. I edited my answer for more details. Feb 13 '20 at 17:31