Pricing Dual-Currency Bonds with FX Forwards and Credit Risk

# Pricing Dual Currency Bond with Forwards instead of Cross Currency Swap


# Pricing Dual Currency Bond with Forwards instead of Cross Currency Swap












i got the task to price a bunch of dual currency bonds (EUR/GBP/CHF/USD...) and i am a bit puzzled. As the notional of the bond is in EUR but the repayment is in USD, i assumed that for pricing purpose i could price the dual currency bond as a combination of a EUR-bond plus a cross currency swap. However, this does't seem to account for variations in the cross currency spread, thus i got the hint from our pricing provider to price each part of the cash flow with the associated FX forwards. My question is: is this correct and can i capture the term structure of the ccy spread with this? Is this the state of the art and whether there is further literature on these dual currency bonds (i also have some other bonds that classify as dual or multi currency bonds plus some optionalities such as callability etc.). Your feedback is highly appreciated!

## Answer by Dimitri Vulis (score 1, accepted)

https://quant.stackexchange.com/a/67802

A (credit-risk-free) cross-currency swap is just a portfolio of fx forwards. But bonds may have credit risk. There's a non-zero probability that the bond issuer will default, in which case all coupons disappear, and the bond holder is likely to be left with an accelerated claim on the remaining principal.

Let's assume for simplicity that 1 your accounting is in USD 2 your DCBs don't have any exotic embedded optionality, but are just a promise to pay some pre-determined cash flows of one of two kinds: either on this day, we pay some USD; or, on this day, we owe you some amount of foreign currency, so we'll look up the FX rate, and pay you the corresponding amount of USD.

You want to get the fair value of each cash flow.

But because of the credit risk, your USD cash flows are worth less than e.g. U.S. treasury debt promising to pay the same amount. Likewise you can't just price each foreign currency cash flow like an FX forward.

Instead, you have to further discount each cash flow by the probability of surival (i.e., of the bond issuer not defaulting) at the time of the cash flow.

You add these up, and you also add the recovery value (the fair value of what the bond holder would receive if the issuer does default - if it's not zero) to get the fair value of the entire bond.

Edit: for example, let us consider the BRL-denominated, external-law, USD-settled bonds issued by Brazil sovereign, e.g. US105756BL31 - 12.5% 2022s or US105756BT66 - 8.5% 2024s or US105756BJ84 - 12.5% 2028s. They pay semi-annually fixed coupon denominated in BRL. They don't amortize - the entire principal is repaid at maturity. The only unusual feature is that instead of paying the BRL to the bond holder (like NTN-F bonds), there is an embedded non delivery forward: 2 business days before each cash flow we observe the USD-BRL rate, using the same rules as other USD-BRL NDFs, and send the corresponding USD amount to the bond holder. These bonds trade frequently, although not on Trace, their market price is easy to observe. They have their own yield curve. The market sees relatively high probability that Brazil sovereign will default on these bonds (as well as on its USD-denominated bonds). On the other hand, the sovereign can continue paying on local-law BRL-denominated bonds (like NTN-Fs) by just printing more BRL.

So you would calculate the fair price of each cash flow by discounting it with 1 the risk-neutral survival probability from CDS on Brazil's external USD debt, for the cash flow date, and 2 the sum of onshore swap curve (CDI in Brazil, works similar to libor) and the BRL cross-currency spread - just like NDFs. (The xccy spread has term structure, of course.)

Pricing the cash flows of these bonds as if they were just NDF's, without considering the credit risk; or discounting with onshore NTN-F yield curve, would both give you a price further from where these bonds actually trade.