You have been asked to check the arbitrage opportunity in the Treasury bond market by comparing the cash flows of selected bonds with cash flows of combinations of other bonds. If a 1 year zero coupon bond is priced at USD 96.12 and 1 year bond paying a 10% coupon semi annually is priced at USD 106.20, what should be the price of 1 year treasury bond that pays 8% coupon semi annually ?

You can duplicate the cash flows with 80% of the coupon bond (so the coupon will be USD4 (= ½ × 10% × USD100 × 80%) and 20% of the zero (so the par value will be USD100 (= USD100 × 80% + USD100 × 20%); the price of this portfolio is:

I have a coupon bond paying 10% and I’m trying to match the coupon payments of a bond paying 8%; thus, I need 8% / 10% = 80% of the coupon bond. That gives me only 80% of the par value, and I need 100% of the par value; I’m 20% short, so I add 20% of a zero-coupon bond.

More generally, suppose that you had two bonds, the first paying a coupon rate of C1 (0% in our example) and the second paying a coupon rate of C2 (10% in our example). You have to solve the following system of equations:

w1×C1 + w2×C2 = 8%

w1×100 + w2×100 = 100

The first equation ensures that the coupon payments for your portfolio match those of the 8% bond, and the second ensures that the par value of your portfolio matches that of the 8% bond.

Once you solve for w1 and w2, use those to get the price (P1 is the price of the first bond, USD96.12 in our example; P2 is the price of the second bond, USD106.20 in our example):

Price = w1×P1 + w2×P2

If they’d given us, say, a 9% bond and a 5% bond, then we’d solve: