# Why is the effective duration of a floating rate bond the time to the next payment?

I am wondering why the effective duration of a floating rate note is the time to the next payment. Effective duration is is defined as

$$\frac{V_{-\Delta y}-V_{+\Delta y}}{2V_0\Delta y},$$ for a small $$\Delta y$$. Why will this value be the time to the next payment for a floating rate note?

• Do you agree that a purely floating set of future payments would have zero duration? The payments would be automatically adjusted to maintain a constant V as i.r. change and therefore the numerator would be zero. Real life floating bonds are not quite like this: the first payment is announced ahead of time and will not be changed. Therefore that term (and that term alone) has non-zero duration. Because it is fixed and not floating. Commented Aug 26, 2020 at 19:49
• @noob2 How do you show this "a purely floating set of future payments would have zero duration? The payments would be automatically adjusted to maintain a constant V as i.r. change and therefore the numerator would be zero." I have heard this before, but I don't see how it holds. Can it be proven mathematically? Commented Aug 26, 2020 at 20:05
• @noob2 The problem is this: Assume that the interest rate changes, then the future cashflow changes, but we can also assume that the discountfactor changes, if we assume that the spread dont change, then the yield to maturity changes as much as the interest rate?, but these two might not cancel each other out? Commented Aug 28, 2020 at 8:58
• It can be shown mathematically that "When the coupon rate equals the yield to maturity the bond is priced at par" In other words if you want the ytm of the bond to be y (the current market level of yields on comparable investments), you can reset the coupon to c=y and magically the bond will be at par and will have the desired yield. Proof here quant.stackexchange.com/questions/42816/… Commented Aug 28, 2020 at 13:41
• @noob2 But the problem is that right after the coupon resets we don't know if the coupons YTM is the coupon rate as is the case in your link. It might be that the YTM is higher because the investors think the bond is riskier. And then you don't get that it resets to par right after coupon payment? And if we don't know that we don't know that the discounted future payments are independent of a rate increase or decrease? Commented Aug 28, 2020 at 14:00