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I am doing the Interest Rate Models course on Coursera. In the third lecture of the second week, the lecturer provides this lemma:

Lemma 1

A coupon bond has par value at $T_0$ if and only if its coupon rates equal the corresponding swap rate:

$$1 = \sum_{i=1}^n P(T_0, T_i)\delta R_{\text{swap}}(T_0)+P(T_0, T_n)\text{.}$$

Proof

Exercise.

My question is what does "par value" mean in this Lemma? I did a google search of par value, and I got this definition from Investopedia:

Par value, also known as nominal value, is the face value of a bond or the stock value stated in the corporate charter.

The definition above makes it sound like all coupon bonds would have a par value, although that par value might be $0$. I don't know what that par value would have to do with the coupon rate. It seems to me that it would only make sense to say something like

A coupon bond has par value $X$ at $T_0\ldots$

so I am not sure what the claim is in the provided Lemma. What does "par value" mean here?

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    $\begingroup$ It simply means that the bond's current clean price is par (i.e., $P = 100$ assuming a notional amount of 100). $\endgroup$
    – Helin
    May 19 at 16:06
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    $\begingroup$ Bad choice of words by the instructor. Instead of "has par value" he/she should have said "is at par" or "has value equal to par" etc. $\endgroup$
    – noob2
    May 19 at 17:27
  • $\begingroup$ @Helin if you submit that as an answer, I will mark "accept" it. $\endgroup$
    – user54908
    May 19 at 17:59
  • $\begingroup$ The definition you found seems also confusing to me by badly choosing words. I agree with the explanation of @noob2 $\endgroup$
    – simzoor
    May 20 at 6:53
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Sloppy English + no editor.

The lemma really says that if you calculate the fair value of an instrument (FRN, or a floating leg of an interest rate swap..) that pays LIBOR (no spread added to it) by projecting the floating coupon cash flows using swap curve and discounting all the cash flows (coupons and principal) using the same curve, then this fair value is equal to the undiscounted face value of the remaining principal repayments.

The present value of the projected coupons is exactly the difference between the face value of the remaining principal repayments minus the present value of the principal discounted using the swap curve.

When swap rates go up (down), then the present value of your principal repayments will go down (up), but the coupon amount changes exactly to offset the change in the present value of the principal.

However if the coupon is being set in advance, as usually done with LIBOR, then all this is not quite true in the middle of a coupon period - only at the beginning of coupon period. Once current coupon effectively becomes fixed, the instrument's price can deviate a little from par as the short-term rates moves. Related question: Why does the valuation of the floating leg of a swap only use the next payment?

Note that the instrument can be amortizing. There's no need to assume that all the principal is repaid only at maturity.

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Par value is the principal payment made at maturity, versus present value which is essentially the price.

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  • $\begingroup$ This definition seems to be the the same as the one I provided. I'm not seeing how your definition makes the Lemma make sense. I would think that all bonds "have par value," and that this value might be $0$. I'm not sure what the par value would have to do with the coupon rate. Taking the definition "principle amount payed at maturity" doesn't line up with the the Lemma "A coupon bond has par value at $T_0$..." because $T_0$ is not the time of maturity. $\endgroup$
    – user54908
    May 19 at 14:27
  • $\begingroup$ I misquoted: "principle payment made at maturity" not "principle amount payed at maturity." $\endgroup$
    – user54908
    May 19 at 14:34
  • $\begingroup$ i don't see a difference in the language above, those statements mean the same thing to me. An amortizing bond can have a par value of say 100 and a principal payment or near 0 at maturity. In the case of a bullet bond the final principal payment can also include a coupon payment. $\endgroup$ May 19 at 14:40

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