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If we calculate the par rate for n periods, why does the nth swap rate equal the par rate? A mathematical formulation would be helpful apart from an intuitive answer.

Edit: Example:- A 2 year bond pays semiannual coupons and has a par value of $100.

Swap Rates- 0.65% (0.5year), 0.8% (1year), 1.02% (1.5 years), 1.16% (2years).

Computing discount rates from this gives- 0.9968, 0.9920, 0.9848, 0.977 respectively.

Now the par rate would be -

$Par/2 * [(d(0.5)+d(1.0)+d(1.5)+d(1.5)+d(2.0)] + 100 * d(2.0) = 100$

Gives par = 1.16%

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  • $\begingroup$ It is not clear what you are asking. Where do you get the conclusion that the nth swap rate equal the par rate? You may see it somewhere, but it is usually related to certain background explained in detail. $\endgroup$
    – Gordon
    Oct 21, 2015 at 15:40
  • $\begingroup$ @Gordon I've added the missing information. Should this be enough? $\endgroup$
    – Piyush Shandilya
    Oct 21, 2015 at 17:34

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Since your 2-year bond is at par, the fixed coupon payments over the 2 years match the payments in the fixed leg of the 2-year swap exactly. Hence the par rate of the bond is the same as the par swap rate.

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    $\begingroup$ Thanks. I was confused about the swap rate itself. Now that it is clear, it seems silly. Your explanation was clear and easy to understand. $\endgroup$
    – Piyush Shandilya
    Oct 21, 2015 at 17:53

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