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I have some notes that state that the higher the value of $N$ in an $N^\text{th}$ to default basket credit default swap, the lower the credit risk exposure (to the party selling the protection) because it is less likely that there will be a payoff.

Does this mean that the payoff of an $N^\text{th}$ to default basket credit default swap is only based on the loss amount of the $N^\text{th}$ default?

Presumably, though the payoff would be less likely, the payoff would otherwise be greater if it was based on the loss amounts of all reference entity defaults. And, for a sufficiently high default correlation, this would make the CDS more expensive as the value of $N$ increased?

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$n$th to default is OTC. Two counterparties can do any bespoke thing they like.

In all the NTD's that I've ever seen, and I've seen on the order of a few hundred, once $n-1$ credit events have already happened. (If $n=1$, this is just first to default.) Their recoveries don't affect anything. Also the running spread does not change.

The $n$th default works just like single-name credit default swap - the protection buyer receives notional minus the recovery of the $n$th default.

However, quoting John Hull and Alan White. "Valuation of a CDO and an $n$th to Default CDS Without Monte Carlo Simulation" http://www-2.rotman.utoronto.ca/~hull/downloadablepublications/HullWhiteCDOPaper.pdf :

An $n$th to default credit default swap (CDS) is similar to a regular CDS. The buyer of protection pays a specified rate (known as the CDS spread) on a specified notional principal until the $n$th default occurs among a specified set of reference entities or until the end of the contract’s life. The payments are usually made quarterly. If the $n$th default occurs before the contract maturity, the buyer of protection can present bonds issued by the defaulting entity to the seller of protection in exchange for the face value of the bonds. Alternatively, the contract may call for a cash payment equal to the difference between the post-default bond value and the face value.

(Footnote: This is how we will define an $n$th to default swap for the purposes of this paper. However $n$th to default swaps are sometimes defined so that there is a payoff for the first n defaults rather than just for the $n$th default. Also, sometimes the rate of payment reduces as defaults occur.)

I've never seen such NTDs, but you can write anything you like in a bespoke contract.

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