# How can a deep discount bond with a longer time to maturity have a LOWER duration than an otherwise identical bond with a shorter time to maturity?

Here is a brief excerpt on the fixed income chapter from the 2020-2021 level 1 CFA curriculum:

1. Generally, for the same coupon rate, a longer-term bond has a greater percentage price change than a shorter-term bond when their market discount rates change by the same amount (the maturity effect).

[...]

There are exceptions to the maturity effect. [But they] are rare in practice. They occur only for low-coupon (but not zero-coupon), long-term bonds trading at a discount. The maturity effect always holds on zero-coupon bonds, as it does for bonds priced at par value or at a premium above par value.

I've tried wrapping my head around this for the better part of an hour. How on earth can this be?

(Macaulay) duration is the weighted average time until you get your money back. How then can more time to maturity result in you getting your money back sooner? Consider two bonds:

1. \$100 par value, 10% coupon paid annually, market discount rate (YTM) of 20%, 20 years to maturity. Macaulay duration: 6.20 years. Modified duration: 5.1695. 2. \$100 par value, 10% coupon paid annually, market discount rate (YTM) of 20%, 30 years to maturity. Macaulay duration: 6.08 years. Modified duration: 5.0629.

Why is this the case? Check it out in Excel: Probably easier to see with the \$Dur, which can be expressed as follows (assuming principal=1):

$${\rm Dur}=\frac{c}{y^2}\left(1-{\frac { yT+y+1}{ \left( 1+y \right)^{T+1} }}\right)+\frac{T} {\left( 1+y \right) ^{T+1}}={\rm Cpn \,Contrib+Princp\, Contrib}$$

The numerator in the principal component is linear (T), so as T grows, the denominator will start to outgrow the numerator, and hence this term will go down. In the coupon term, the numerator in the term after the minus sign is also linear in T so its denominator will start to outgrow, and hence this component will start to increase. If c is too small, then the increase in the coupon component may not fully offset the decrease in the principal component, and hence Duration will go down. You can do the same comparison for the price to get the answer in relative term.

It's a very good question. This is also mentioned in "Bond Math: the theory behind the formulas" - but the author doesn't get into a lot of details, he just mentions it as some kind of a mathematical oddity, if I remember correctly.

If there is a rigorous proof with a closed-form formula, the maths is beyond me.

Consider a bond that pays 5% annually. If you plot duration vs maturity for various values of the yield, you see something like this:

[UPDATE: the chart says 'modified duration' but, in fact, it's the Macaulay one - sorry for the typo] The coupon stays the same. As you increase the yield (the Y in the legend), i.e. as the price of the bond goes down, the curve goes down more and more.

Up to a certain threshold, as the maturity increase, the duration increased, but always less (positive 1st derivative, negative 2nd derivative).

Beyond that threshold, the chart first goes up then down. You see this very clearly in the intentionally extreme case of a 90% yield (the lowest curve above).

Why is this? You are correct that the Macaulay duration is a weighted average time, but, specifically, it is one in which you weigh the PV of each payment by time/price. In most cases, increasing the maturity increases this weighted average, but, when the bond is deeply discounted, the opposite happens. The best intuitive way I can think of to explain it is that the weights are such that the weighted average becomes lower (which is a bit circular as an explanation, I appreciate).

I have broken down some details below. 5% coupon, 90% yield. If the maturity is 4 years, the Macauly duration is 3.11

If it goes up to 5 years, the duration goes up to 3.24.

But if it goes up to 6 years, the duration goes back down to 3.11.

I hope this is useful. If someone knows of a more rigorous explanation, I'd be interested, too. Assuming continuously compounded interest rates, the terminal payoff of the bond is depended on : t*exp(-rt). If you plot this as a function of time you’ll see that it starts decreasing after t = 1/r. Assuming rates of 1%, it would mean after 100 years the terminal maturity payoff will start decreasing.

Now, a bond with very low coupons will have a large portion of its payoff on maturity and hence the duration will be largely effected by the terminal payoff. If the time scale is large enough (> 1/r), the f = discounted terminal payoff*maturity time of a bond with less maturity will be more than that of a larger maturity.

And since f plays a large role in duration of a low coupon bond, duration of low maturity bond will be more than large maturity bond.

This is more of a mathematical intuition, would love to see some financial/practical intuition for this.