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# The Coupon Effect

#### Eustice_Langham

##### New Member
I'm hoping that someone can explain this to me as to me it sounds very counter intuitive..."To understand these results, note that the yield to maturity is a complicated function of the spot rates applicable to each payment date. As the coupon increases, the average time until the bond holder receives cash flows decreases, and the spot rates for the early payment dates become relatively more important in determining the yield.".....Specifically the comment that as the coupon increases, the average time until the bond holder receives cash flows decreases ....this doesnt make alot of sense to me, I would have thought that the period between cash flows is fixed and known in advance as are the coupons ie 3% every 6 months, so therefore how does this make the statement accurate?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Eustice_Langham I interpret "the average time until the bond holder receives cash flows decreases" as a reference to duration (strictly speaking, Macaulay duration). Of course you are correct about the future stream of cash flows under a fixed rate bond: if we have a 3.0% annual-pay coupon bond, the future stream is $3.0 coupons; if the price drops, the yield (YTM) increases but the future cash flows are unchanged. My least favorite definition of duration--only because it is hardest for me to intuitively understand, I'm sure others understand it better--is "breakeven" duration. In this view, if bond has a Mac duration of 2.8 years, then the bond investor can expect to recover her investment in 2.8 years but only as a payment-weighted point in time abstraction. I prefer the definition of (Mac) duration as the bond's weighted average maturity (weighted by the PV of cash flows). Higher coupons imply a lower weighted average maturity (a lower duration), ceteris paribus. Consider a 2-year zero-coupon bond while the spot rate term structure is {1% @ 1 year; 3% @ 2 years). Super simple scenario, annual compound frequency. This bond has yield of 3.0% to match the 2-year spot rate because there is only the one cash flow (return of$100 or $1000 par). It's Mac duration is 2.0 years because that is the weighted average maturity (and also, per my less favored definition, you'd wait three years to recover your investment). Now switch to a 5.0% coupon bond, so the future cash flows are$5 and $105 (instead of zero and$100). This bond's duration is about 1.95 years (i.e., 5% * 1 year + 95% * 2 years = 1.95, but I am just ballparking, exact weights are PV of cash flows). As the coupon increases, the duration must decrease because the weighted average maturity decreases. To interpret "the average time until the bond holder receives cash flows decreases," I would say it this way: in the case of the zero, we waited the full 2.0 years to receive (any) cash flow, but with the coupon, because we receive a coupon in 1.0 year, we are now receiving some cash sooner, the payment-weighted point in time is sooner, and our bond's weighted average maturity (aka, duration) is slightly less.

The "coupon effect" is slightly differnt. In this example, the coupon effect refers to the fact that the zero-coupon bond has a yield of 3.0% but the 5.0% coupon bond (given the exact same spot rate term structure) has a slightly lower yield. The coupon bond, into the "complex average" that is yield, now must include the lower 1.0% 1-year spot rate so the "average" must be dragged down. Given an upward sloping spot rate term structures, an increase in the coupon rate assigns greater weight to the lower spot rates and therefore drags down the yield (intuitively: your bond includes more value at lower spot rates, versus the zero coupon bond which enjoys only the single highest spot rate). I hope that's helpful!

#### Eustice_Langham

##### New Member
David, mea culpa....I should have provided you with more background information, thankyou as well for the very thorough explanation.

I have attached a screenshot of the source of my question, its not from Duration but from the preceding chapter with the highlighted section the comment in question.

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