Why FX Forwards and Cross-Currency Basis Swaps Have Different Risks
Summary
The document asks whether an FX forward and a float-to-float cross-currency swap can be treated as equivalent for hedge-effectiveness checks immediately after repricing. It contrasts a forward’s value, based on the difference between current and contracted forward rates, with a swap valuation built from discounted floating cash flows in two currencies. The question suggests that cross-currency basis might appear through discount factors and affect the relationship.
The answer says the instruments are not equivalent because cross-currency basis is a term-dependent risk factor, and it can change the notionals implied by the exchange. It explains that a simplified swap exchanging overnight risk-free rates differs from market structures using benchmark rates; conceptually, the latter can be decomposed into an overnight-rate basis swap and domestic and foreign money-market basis swaps. The response is qualitative and does not provide a numerical valuation or a full hedge-effectiveness test.
Key ideas
- An FX forward and a cross-currency basis swap should not be assumed to have equivalent values after repricing.
- Cross-currency basis is a term-dependent risk factor that can affect implied notionals.
- Benchmark-rate swaps differ from a simplified exchange of overnight risk-free rates.
- The swap can be understood as involving overnight-rate and domestic and foreign money-market basis components.
- The document gives a conceptual explanation but no numerical hedge test.
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Full text
# Equivalency of FX forwards and FX basis swaps for risk-management purposes
# Equivalency of FX forwards and FX basis swaps for risk-management purposes
Can one deem an FX float-to-float swap and a FX forward equivalent on dates immediately after repricing? The reason I am asking, I am hedging something that can be modeled via an FX forward, I was given an FX basis swap to hedge it. I want to safe myself some work and claim that they are equivalent on repricing days (or immediately after) when effectiveness of the hedge would be assessed. I believe, it comes down to how forward points are reflected in the the two instruments. An FX forward is valued as follows:
$$ (F_{t_i,t_M}-F_C) \times N \times \text{df}_{\text{FWD}} $$
I don't know how to do the classic "where" block, so in a sentence, the above means: (FX forward rate at time $i$ until time $M$ (maturity) $F_{t_i,t_M}$ minus FX contracted forward rate $F_C$) times notional $N$ times discount factor $\text{df}_{\text{FWD}}$
A swap's value would be:
$$\sum_{i=k}^{n}{N_{\text{DC}}\times \frac{r_{\text{DC}}}{\text{days}}\times \text{df}_{\text{i,SWP,DC}}} - S_{0} \left(\sum_{i=k}^{n}{N_{\text{FC}}\times \frac{r_{\text{FC}}}{\text{days}}\times \text{df}_{\text{i,SWP,FC}}}\right) $$
Where "DC" = domestic currency, "FC" = foreign ccy, $r$ the applicable interest forward rate and "df" = discount factor. You might have notices I ommitted the final exchange of principal which would add some more complexity and is not important for the question.
The point is that the formulas are obviously different, yet I am trying to claim that the values will be equivalent immediately after repricing. Here's why:
We know that the value of each variable leg will be $100$ in their respective currencies (as any floating bond would be, where forward and discount rates are interconnected. So one could claim that the FX swap's value will be primarily driven by the spot rate $S$, used to translate the foreign currency leg. At this point one would say that the forward will be driven by the difference between the actual forward rate $F$ and the contracted rate so they should NOT be the same. But. Here comes my twisted thinking (or lack thereof). I found out, that in the reality of markets, the forward rate difference in the swap would be hidden in the discount factors as the basis spread, and I am wondering if that, somehow, would make the value equivalent to the forward. Unfortunately don't have access to SWPM to test the "theory". Thoughts?
## Answer by user7056 (score 3, accepted)
https://quant.stackexchange.com/a/4053
The two are not equivalent, because of the cross-currency basis spread (CCBS), which became a risk factor in itself sice 2007, and does depend on term. This practically leeds to a difference in your constantly-assumed notionals (the notional is not constant anymore).
What it happens is that you assume having a constant notional cross-currency swap that exchanges a risk-free overnight rate in one curreny for a risk-free overnight rate in another currency. In reality, one exchanges benchmark rates (say LIBOR), so you should imagine having three swaps: a cross-currency basis spread overnight risk-free rates, a money market basis swap of domestic LIBOR versus domestic overnight risk-free rates and a money market basis swap of foreign LIBOR versus foreign overnight risk-free rates.
This change in notional can actually be seen when doing a mark-to-market cross currency swap instead of your standard one.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.