**Mad_Mac (Feb 16th, 2014):** What makes this question particularly confusing is that the implied interest rate curve has such a bizarre shape.

We're told in the question that we're dealing with treasury notes maturing exactly one year from now, and that there is one (annual) coupon remaining. At first glance it would appear that this is a straight-forward question, with the coupon being paid together with the principal, and that there is only one zero-coupon interest rate that is of relevance.

However, when you back out what this zero-coupon rate is, you find different values for the two bonds. We can therefore infer that there are several cash flows, and that the bonds are not simply the present value of a combined principal & coupon payment in one year from now.

So in other words, it has to be the case that were dealing with bonds that will pay a coupon at some point before maturity. We're told that the coupons are paid annually, so we can therefore conclude that there are only two cash flows remaining: the payment of the coupon, and the payment of the principal.

We're not given the exact payment date for when the coupon is paid. All we know is that it is not paid together with the principal.

Let:

DFCP = DiscountFactorCouponPayment ; and

DFPP = DiscountFactorPrincipalPayment

2.875 x DFCP + 100 x DFPP = 98.4

6.25 x DFCP + 100 x DFPP = 101.3

This solves for DFCP = 0.859259 and DFPP = 0.959296

Now we can find the price of the 4 1/2 Treasury Note: 4.5 x 0.859259 + 100 x 0.959296 = 99.796

Other Notes:

DFPP is 0.959296 , which translates into an annual interest rate of 4.2%. Nothing strange with that value.

However, lets now turn to DFCP. With a factor of 0.859259, the rate, before annualising, is 16.4%.

Again, we don't know when the coupon will be paid, but no matter what assumptions you make, the shape of the interest rate curve will be very strange looking.

Say for example that the 16.4% rate is for a cash payment 11 months from now. The annualised rate would then be almost 18% vs the 4.2% 12-month rate.

If the 16.4% rate is for a cash payment 6 months from now, then the annualised rate would be almost 33%.

I can appreciate that the point of the question is to illustrate how the law of one price, and interpolation work. The point of this reply is rather to show that the traditional approach that you'd take when solving fixed income questions still works. However, linear interpolation is clearly a quicker way to solve the problem.

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