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# Are FX Swaps Mispriced?

### Juan Martínez

#### 28/07/2021

Given the exceptional liquidity of the FX swap market, with a daily turnover of 3 trillion USD [1], a consistent mispricing has always seemed rather unlikely. Precisely, this makes the possibility of such event particularly interesting, especially when the data during the last decade point in that direction.

In order to keep the scope of this post limited, we will draw from some of the concepts explained in our previous post, among others: Covered Interest Parity (CIP), the basis as a measure of CIP deviations and the risk composition of an FX swap. Feel free to come back to that text if anything in the following discussion doesn’t seem to make complete sense.

## Reconsidering Covered Interest Rate Parity

### LIBOR basis and credit risk differentials

Our interest in the topic of FX swap pricing began when we found out that the traditionally precise model of CIP (with LIBOR interest rates) started to fail after the Global Financial Crisis. The reason being that LIBOR rates started to incorporate (as they always should have had) a credit risk premium that is absent in FX swaps. Given the different risk composition of both instruments, one (LIBOR rates) cannot be used to accurately price the other (FX swaps).

As discussed in the previous post, given the spot $$S$$ the risk-free rate $$r$$, the liquidity risk premium $$lp$$ and the credit risk premium $$cp$$, the CIP-LIBOR model yields the following forward price $$F$$:

$$F = S \frac{1 + r_{\text{DOM}} + lp _{\text{DOM}} + cp _{\text{DOM}} }{1 + r_{\text{FOR}} + lp_{\text{FOR}} + cp_{\text{FOR}} }$$

Meanwhile, the actual correct price would be:

$$F = S \frac{1 + r_{\text{DOM}} + lp _{\text{DOM}} }{1 + r_{\text{FOR}} + lp_{\text{FOR}} }$$

What puzzled many researchers was the fact that the Libor basis persisted in the same magnitude after things calmed down, when credit risk premiums shouldn’t be that relevant. It’s important to note that this argument has mostly been made at a conceptual level and few have attempted to back it empirically. Below we will discuss one notable exception.

### GC repos

Since authors such as Whong and Zhang [2] still attribute the Libor basis to a credit risk differential, Du et al. [3] substitute LIBOR rates by general collateral (GC) repo rates in the CIP model showing that the basis remains. GC repo rates represent the cost of secured lending, so Du et al. consider this result proof that credit risk differentials aren’t the drivers of the LIBOR basis.

However, if we get into the details of what’s actually going on we can see this argument can be countered. Repos entail the temporary exchange of two assets (cash vs collateral) with different liquidity where the interest rate compensates the party that gives away a more liquid asset (cash).

$$\text{ir}^{\text{repo}} = \frac{1 + r + lp^{\text{GC}} }{1 + r + lp^{\text{cash}} } – 1$$

The key to this is that general collateral with different currency denomination will be traded in different markets with different liquidity conditions, thus having different liquidity premiums. Consequently, these terms ($$lp^{\text{GC}}_\text{DOM}$$ and $$lp^{\text{GC}}_\text{FOR}$$) won’t cancel out when we build the CIP model.

$$F = S \frac{1 + \text{ir}^{\text{repo}}_{\text{DOM}}}{1 + \text{ir}^{\text{repo}}_{\text{FOR}}} = S \frac{ \frac{1 + r + lp^{\text{GC}}_\text{DOM} }{1 + r + lp_{\text{DOM}} } }{ \frac{1 + r + lp^{\text{GC}}_\text{FOR} }{1 + r + lp_{\text{FOR}} } }$$

Thus although GC repo rates effectively eliminate the problem with credit risk differentials, they introduce another undesired factor: the liquidity premium of collateral in both currencies.

### Cross-currency repos

The limitation of GC repos was noted by Kohler & Müller [4] who turned instead to a more specific type of contracts: cross-currency repos. The key characteristic of these instruments is that they allow to provide collateral denominated in another currency. For instance, one can lend EUR receiving USD denominated collateral with a cross-currency repo, and then borrow USD providing that very same collateral with a standard repo.

Since the collateral used is the same in both transactions, its liquidity premium cancels out and we are left with a model that truly reflects the FX swap risk composition. It’s no surprise then that the basis using these rates is practically zero. However, this approach turns to be a dead end.

As the authors acknowledge, the cross-currency repo market is far less liquid than FX swaps, meaning that the former is priced according to the latter and not the other way around. In other words, the existence of a basis would most likely reflect a mispricing of cross-currency repos rather than FX swaps.

### Brief recap

As Heidorn & Mamadalizoda [5] point out, there’s no interest rate that can be directly used to price FX swaps accurately. Wong & Zhang [2] provide an interesting alternative but we will leave that for another time as it also has limitations. In summary, we have no model for pricing FX swaps, and yet we might still be able to answer the question that motivated this post.

## Arbitrage limits and departures from equilibrium

### Quarter-end spikes

The key resides in an interesting feature of FX swap prices reported by Borio et al. [6], namely quarter-end spikes. These have been linked to “the greater importance attached to quarter-end reporting and regulatory ratios following regulatory reforms”. In these periods, banks face tighter balance sheet constraints and have to partially step out of the market.

How does this relate to FX swap prices? Banks have the lowest funding costs and are in the best position to arbitrage FX swap mispricings. The fact that prices spikes when banks are temporarily more constrained indicates that they are keeping them under control the rest of the time through arbitrage. Nonetheless, even then they face certain limitations due to regulation put in place after the Global Financial Crisis leading to potential mispricings.

### Finding the equilibrium price

In order to correct these deviations from the equilibrium price we need to know both their direction and magnitude.

The first one is easy. Spikes are consistently in the same direction, pointing to a persistent asymmetric hedging demand that is driving FX swaps away from their fair price. We will not digress into the reasons behind this phenomenon but one can refer to Borio et al. [6] and Liao [7] for that matter. In any case, what this means is that the equilibrium price will be in the opposite direction (with respect to the normal market price) of the spikes.

The magnitude of the correction is harder to estimate since it depends on the constraints that regulation imposes on banks’ arbitraging activity. This regulation translates into additional funding costs that make arbitrage less (or not at all) profitable. An interesting workaround to approximate these shadow costs (named as such because of their difficult estimation) is presented below.

Du et al. [3] consider the most basic form of arbitrage for US banks, which consists on borrowing reserves from other institutions at the Fed Funds rate and putting them at the IOER deposit facility where the interest is higher. This operation is risk free and yet the spread between the two rates doesn’t narrow down because of the aforementioned shadow costs that banks face when increasing their balance sheet. Consequently, this spread serves as a lower bound for the cost of arbitrage under regulatory constraints.

Since different monetary systems have different regulations these costs will vary. Furthermore, not all types of leverage require the same amount of leverage and the financial instruments involved also have an impact. In summary, a lot more work can be done to make the correction more rigorous but we have presented the most basic approach.

## Conclusion

We can wrap up by briefly sumarising the two most important conclusions:

1. After the Global Financial Crisis it’s not straightforward to price FX swaps.
2. FX swaps are most likely mispriced due to limits on arbitrage. The direction and magnitude of these deviations can be indirectly inferred from market data.

Finally, the fact that mispricings can’t be arbitraged away doesn’t mean one can’t obtain profits by considering them into the asset allocation decision, which is what the Reserve Bank of Australia has been doing.

## References

[1] Bank for International Settlements. “BIS Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity in April 2019”.

[2] Wong, Alfred, and Jiayue Zhang. “Breakdown of covered interest parity: mystery or myth?”. Vol. 96. Bank for International Settlements, 2018.

[3] Du, Wenxin et al. “Deviations from Covered Interest Rate Parity”.
Working Paper 23170. National Bureau of Economic Research, 2017.

[4] Kohler, Daniel, and Müller, Benjamin. “Covered interest rate parity, relative funding liquidity, and cross-currency repos”. No. 2019-05. Swiss National Bank, 2019.

[5] Heidorn, Thomas, and Mamadalizoda, Nekruz. “Investigating the cross currency basis in EURUSD and EURGBP”. Frankfurt School – Working Paper Series, No. 227, Frankfurt School of Finance & Management, 2019

[6] Borio, Claudio, et al. “Covered interest parity lost: understanding the cross-currency basis.” BIS Quarterly Review September, 2016.

[7] Liao, Gordon Y. “Credit Migration and Covered Interest Rate Parity”. No. 1255, International Finance Discussion Papers. 2019.