# Interest rate risk of a bond as a function of the coupon

This SEC document claims that increasing the ocupon on a bond decreases the interest rate risk (bottom of page 3): And the Finra SIE exam states the same also.

I cannot understand the logic behind this statement, it just seems wrong to me. If we consider a simple example where we have a flat interest rate of $$r$$, and a bond that pays semiannually, then the value of the bond can be written as:

$$B = \frac{1}{(1+r)^{t_n}} + c \sum_{i=0 \ldots n} \frac{1}{(1+r)^{t_i}}$$

Where if we're just comparing two bonds to each other then, for the sake of comparison, the principal repayment can be ignored, and we can then look at the interest rate risk of the coupons. Here, we can happily say that the rate risk is linear in the coupons, and if we have larger coupons then we must have more risk.

So how is it that the SEC can say that a lower coupon bond has more interest rate risk? What am i missing?

• Although this does not add much to what @AIRacoon already wrote, it is generally called the coupon effect. Apr 24, 2022 at 23:03
• Ah okay, so is the actual statement that lower coupon bonds (but with the same value because they have higher credit spreads) have more interest rate risk?
– will
Apr 25, 2022 at 6:12
• @will Lower coupons (Risk free + credit spd) will have more interest rate risk. A rise in the risk free rate or the credit spd will cause the price of a bond to drop similarly. Some investors choose to generate both a Duration number for rates and a separate Credit Duration number to look at these risks separately Apr 25, 2022 at 21:19

Yes, the point made in the question is true; more fixed coupons all else equal leads to more interest rate risk. More precisely: more fixed coupons trivially (but well spotted) gives you more losses in USD per increase in the quoted market interest rate (in bps for example).

But what SEC refers to is the interest rate risk per invested USD, equal to the relative (in percentage point) loss you make per increase in the rate. And that increases with the maturity of the cashflows and is insensitive to the amount invested.

So the more you dilute the long bullet with the shorter coupons, the less average maturity you have and the less relative interest rate risk.

Still absolute interest risk increases the more you invest.

• My problem is with the statement "all else equal". This is not mentioned in the question, but is obviously what you would assume. The thing is, it's not all else equal. If you increase the coupons while changing nothing else, then your rate risk increases. You need to increase the coupons while simultaneously increasing the credit spread such that the bond's value remains par, at which point the statement is true. This is exactly not "all else equal".
– will
Apr 26, 2022 at 5:47
• You could increse the coupons while keeping market rates and spreads constant by increasing the price to face value and increasing your investment. That would increase absoulute interest rate risk while decreasing relative interest rate risk. Apr 26, 2022 at 9:39
• agreed absolute rate risk increases, and agreed rate risk per investment on the actual bond decreases, so rate risk per investment decreases - thanks. I guess this is what they were talking about in the documents and questions, thanks.
– will
Apr 26, 2022 at 20:37

Since duration is the primary risk of a bond, higher coupons tend to decrease the duration, and the risk of the bond.

• Yes they increase the duration, but if you increase the coupons while changing nothing else, then your rate risk must increase.
– will
Apr 26, 2022 at 5:49