You have a 5% UST maturing in 3 years selling at a price of 102, yielding 4.282% (BEY). A similar maturity UST with a coupon of 4.282% is selling at par. Show that premium is equal to the present value of the semi-annual coupon payment difference at the current market yield. Use a par value of $100.

Even if you can just guide me in how to get the answer, I would sincerely appreciate it.

Even before calculating anything, you know that the question's premise is correct. Both bonds return $100 at maturity (ignoring the final coupon payment), hence the PV of the maturity payoff is the same for both.

Therefore any price difference ($2 premium, in this case) must be attributable solely to the PV of the difference in their semi-annual coupon payments.

To actually verify this,

• Determine the six coupon payments of the premium bond

• Determine the six coupon payments of the par bond

• The difference between these two represents a cash flow stream which is an annuity (same amount each time) of six payments, one every six months.

Now calculate the PV of this annuity, using 4.282% / 2 as your discount rate. That present value will indeed be equal to the premium.

What am I doing wrong? I still can't figure this out...

Premium Bond

5/(1.04282) + 5/(1.04282)^2 +... + 105/(1.04282)^6

4.79 + 4.60 + 4.41 + 4.23 + 4.05 + 81.65 = $103.73

Par Bond

4.282/(1.04282) + 4.282/(1.04282)^2 +... + 104.282/(1.04282)^6

4.11 + 3.94 + 3.78 + 3.62 + 3.47 + 81.09 = $100.00

Did I do that part right or no? What is the next step? I should take the PV of the difference? So the PV of $3.73? Isn't my goal for the PV of the difference to equal $2.00?

Please help.

First thing is to note that the 5% rate and the 4.282% rate are both quoted annual rates. In this case you're dealing with semiannual cash flows, and so you convert this quoted annual rate to its 6-month equivalent (2.5% and 2.141%, respectively) in working with these semiannual cash flows.

You have two bonds, one of which is currently priced at 102 and the other at 100. The $2 price differential is due to the difference in the two bonds' cash flows; in fact, the difference is the present value of the cash flows' difference.

The premium-priced bond pays a cash flow stream of
2.50, 2.50, 2.50, 2.50, 2.50, 102.50.
This is on a six-month interval pattern.

The par-priced bond's counterpart CF stream is
2.14, 2.14, 2.14, 2.14, 2.14, 102.14.

Hence, the premium bond, in comparison with the par bond, pays an excess amount of
0.36, 0.36, 0.36, 0.36, 0.36, 0.36

Now calc the PV of that excess cash flow stream, using a discount rate of 2.141%. The answer won't be precise, of course, since we've used a bit 'o rounding at each step along the way.

Of course, I understand now! Thank you so much ArcSine, you have been extremely helpful! I really appreciate it.

Glad to help, amigo. Just as a side note, since the difference in the two bonds' cash flow streams is simply an annuity of 0.36 every 6 months for 6 periods, you could've also determined the PV of that difference using the PV-of-annuity formula...

0.36 \ \times \ \frac{1-1.02141^{-6}}{0.02141} \ \approx \ 2.00

But when there's only 6 cash flows involved the shortcut formula only buys you a few seconds advantage at best. And the approach you were using---determining the PV of each individual CF separately---provides more insight into what's really going on under the hood.

The point is to just be on the lookout for opportunities to use the shortcut annuity formulas in situations involving a large number of cash flows.