From reading 31 in the CFAI text (bottom of page 449):
“Without showing the details, we shall simply state the result that a floating-rate bond’s duration is approximately the amount of time remaining until the next coupon payment. For a bond with quarterly payments, the maximum duration is 0.25 years and the minimum duration is zero. Consequently, the average duration is about 0.125 years. From the perspective of the issuer rather than the holder, the duration of the position is –0.125.”
Anyone know how these numbers were arrived at and whether there is a generic formula around the calculations?
Thanks cpk. I understood the average numbers… that’s not the issue. I was talking more about this: “For a bond with quarterly payments, the maximum duration is 0.25 years and the minimum duration is zero.” Where the heck did they come up with max of 0.25? Are they implying that the bond portfolio under consideration is composed of only zeros (which would explain the 1 year = duration of 1 logic)?
A floating rate bond will reset its coupon payments at each coupon. As such there is practically no duration. As the bond resets to a new rate reflecting the new current rate environment it is not really exposed to interest rate fluctuations and so no duration (or very little)
First, we’re talking _ effective _ duration here. (A floating-rate bond’s modified duration and Macaulay duration are longer, but very difficult to calculate.)
Second, this calculation assumes that the floating-rate bond’s coupon will reset to the market rate, so that its price will reset to par. If the market’s perception of the risk of the bond changes – so that the required spread over the market rate isn’t the same as the coupon spread over the market rate – then its price won’t reset to par and the effective duration will change. This could happen as well if the bond’s coupon rate has a cap or a floor.
Care to briefly expand on the first point? I understand why the modified/Macaulay duration will be longer (say, in the case of a callable bond where price changes are subdued in the effective duration calculation due to the call), but given nature of floating rate bonds (cash flows are always changing), aren’t modified/macualay measures not particularly meaningful due to the underlying linkage between rates and coupons?
Exactly. Modified duration and Macaulay duration assume that cash flows do not change; effective duration allows for changes in cash flows. Thus, effective duration is an appropriate measure of interest-rate sensitivity of price for bonds with variable cash flows (e.g., bonds with embedded options, floating-rate bonds), while modified duration is not. (Macaulay duration, of course, is not a measure of interest-rate sensitivity of price.)
^Not knowing much about duration (beyond what was taught in exam)–are modified, effective, and Macaulay all different? Do they have different uses? Wouldn’t you expect them to be about the same for a given bond or portfolio?
Macaulay duration is the weighted average time to receipt of (promised) cash flows, where each weight is the present value of the cash flow to be received divided by the price of the bond. It doesn’t measure interest-rate sensitivity of price (though it is very close).
Modified duration is an adjustment to Macaulay duration that does measure interest-rate sensitivity of price, assuming that cash flows are made as promised.
Effective duration is a measure of interest-rate sensitivity of price, allowing that cash flows may change.
For a bond with known, immutable cash flows (no embedded options, fixed rate), modified duration and effective duration are identical. For a bond with (potentially) variable (and unknown) cash flows (those with embedded options, those with floating rates), modified duration and effective duration will (likely) be different; when they are different, effective duration is the appropriate measure of interest-rate sensitivity of price, because it incorporates the possibility of changing cash flows.
Thanks for the explanations. However, how does someone know, in terms of magnitude, whether 0.25 is a small number or a huge number. What would be considered a “large” duration for a floating rate bond? Also, why 0.25? Why not 0.1 or 0.05 for that matter? If we take the 2009-2012 period, interest rate certainty (at least with regards to the Fed’s actions) was almost a given… so wouldn’t you choose a super-low number like zero for your duration?
I’d imagine that floating rate bonds reset their coupons either every coupon date, or every year (more likely the former, I’d guess), so 0.25 years is fairly typical for a floater. Of course, it’s quite small compared to most fixed-rate bonds.
That’s a guess: probably a simplification for the CFA curriculum. To compute it accurately you’d have to create a binomial interest rate tree, guess at the interest rate volatility, and so on. It’s complicated.
If you expect that interest rates aren’t going to change, then duration doesn’t matter: there will be no difference in the price change for bonds with 0.25 years duration and bonds with 25 years duration. You’d be more concerned with the coupon rate: if a long-duration bond will give you a few extra basis points of coupon and no greater price risk, you’d go for the coupon.