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A question on the yield curve for coupon-paying vs non-coupon-paying bonds

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asked Oct 20, 2016 in BUS 2016H - Financial Mathematics by simon_rigby (4,220 points)
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I have a question from Tutorial 13, Question 5, b (iii):

The question reads:
Explain why the gross redemption yield curve for coupon paying bonds will slope down with a less steep gradient than the zero coupon bond yield curve  

The memo answer is:
The gross redemption yields from coupon paying bonds are, in effect, weighted averages of the earlier zero coupon bond yields. Later terms to redemption yields will be affected by earlier higher zero coupon yields and hence will not decrease as quickly as the zero coupon bond yield curve.  
If possible, could you please explain the bold section of the memo?

1 Answer

+2 votes
answered Oct 20, 2016 by simon_rigby (4,220 points)
selected Oct 24, 2016 by Richard van Gysen
Best answer
So the question is specific to the scenario described at the top of Q5(b), in which the expected short-term rates are decreasing with time. I include the information here:

Short-term, one-year annual effective interest rates are currently 10%; they are expected to be 9% in one year’s time, 8% in two years’ time, 7% in three years’ time and to remain at that level thereafter indefinitely.

The line in bold is probably best understood with an example, or at least by writing down the price equations.

The price of a 2-year ZCB, with these expected short-term rates, is \( (1.10)^{-1}(1.09)^{-1}\), while a 3-year ZCB is priced at \((1.10)^{-1}(1.09)^{-1}(1.08)^{-1}\).

The yields are \(9.5\%\) and \(9\%\) respectively, which we'd expect because it's just a geometric mean of the short-term rates during that time.

The prices of coupon-paying bonds with terms 2 and 3 are
$$P_2 = C(1.10)^{-1} + (1+C)(1.10)^{-1}(1.09)^{-1} $$
$$P_3 = C(1.10)^{-1} + C(1.10)^{-1}(1.09)^{-1}  + (1+C)(1.10)^{-1}(1.09)^{-1}(1.08)^{-1}$$
where \(C\) is the rate of the annual coupon.

So when we calculate the gross redemption yield (equivalent interest rate that makes the discounted cash flows equal to the price) for coupon-paying bonds, the earlier short-term rates are more "weighty" than the later ones, if \(C > 0\).

(Notice that \(1.10\) appears often in the equations for \(P_2\) and \(P_3\) but only once in the prices of the ZCBs).