# How to quantify the coupon effect?

I'm reading Moorad Choudhry's book "Advanced Fixed Income Analysis"

The first chapter briefly touches on the coupon effect which I understand from other sources is the effect of pricing an annuity (the coupons) and a zero-coupon bond (the repayment of the notional), and so in an upward sloping curve the higher coupon bond will have a lower yield. But I'm not sure I understand the below (from section 1.2.3), and I can't find other sources for this type of analysis.

One method used to identify relative value is to quantify the coupon effect on the yields of bonds. The relationship between yield and coupon is given by (1.2):

$$rm = rm_P + c \cdotp max(C_{PD} - rm_P,0) + d \cdotp min(C_{PD} - rm_P,0)$$ (1.2)

where

$$rm$$ is the yield on the bond being analysed

$$rm_P$$ is the yield on the par bond of specified duration

$$C_{PD}$$ is the coupon on an arbitrary bond of similar duration to the part [sic] bond

and $$c$$ and $$d$$ are coefficients. The coefficient $$c$$ reflects the effect of a high coupon on the yield of a bond. If we consider a case where the coupon rate exceeds the yield on the similar-duration par bond ($$C_{PD} > rm_P$$), (1.2) reduces to (1.3):

$$rm = rm_P + c \cdotp (C_{PD} - rm_P)$$. (1.3)

Equation (1.3) specifies the spread between the yield on a high coupon bond and the yield on a par bond as linear function of the spread between the first bond's coupon and the yield and coupon of the part bond.

Is anyone better able to explain what I am looking at? Or provide a better source?

Thanks

• What he proposes to do is to estimate a linear regression, to find the coefficients c and d. Then to identify bonds that have a yield higher than implied by the regression line (i.e. bonds that are attractive even after considering the coupon effect). Are you familiar with regression as a statistical technique? – Alex C Apr 8 at 16:48
• Hi Alex C. Thanks, but that's not my issue. What I don't understand is the intuition involved behind (1.2), where the equation itself is derived from. And even when it's simplified to (1.3) is it not the wrong way around. Naively had I not seen this then to express as a linear function I would have written something like: Bond Yield - Par = (Bond Coupon - Par) * coefficient + constant. But even that seems a leap. – BG25 Apr 8 at 19:06
• I agree with you that this equation does not have a firm theoretical basis. It is an ad-hoc, empirical approach. A sounder method would be based on an estimated yield curve IMO. – Alex C Apr 8 at 20:38

## 1 Answer

Equations don't look intuitive. To quantify coupon effect for the same credit risk and maturity, discount the cash flows of a bond to get the price and then yield. Then discount the second set of cash flows to get the price and then yield. The difference between the yields is your coupon effect.

Use zero coupon rates to discount the cash flows.

• Thanks VanillaCall – BG25 Apr 15 at 8:56