Duration and Convexity [Concepts Series]

Duration is a fundamental financial concept, but it’s more part of the fixed income investor toolkit than the equity investors’. For reasons I’ll get into, duration is usually a misleading concept for equity investors, but there are a few cases where that’s changing.

Duration in bond investing refers to two different measures, which are, conveniently, measured roughly the same way:

  1. The weighted average time in the future at which a bond pays its returns — i.e. the number of years from now when you’ll realize half of the net present value of the bond, and
  2. A bond price’s sensitivity to changes in interest rates, all else being equal.

It’s intuitive that these concepts would be related: the longer in the future a given stream of cash flows is, the more sensitive its value is to interest rates.

Here’s a quick table, looking at a bond with a 4% coupon paid every six months:

The important things to pay attention to are the discount rates (at the top) and the PV/Duration numbers at the bottom. The PV number is the present value of the cash flows at a given discount rate; the duration is the weighted average time at which those cash flows are received.

There are some obvious wrinkles to this, however. A somewhat uninteresting one is that bonds with different terms can have different duration profiles: if a borrower has the right to buy back a bond at a fixed price, the bond is insensitive to interest rate changes once it reaches that price (because the company could refinance it at a lower rate).

More interesting, and germane to equity investors: that “all else being equal” line hides some complications. All else is not equal: interest rates respond to external variables. Specifically, they tend to be low when the economy is slow, and high when it’s doing well. For a risk-free bond, the economy doesn’t matter, but for risky bonds, default risk is important.

To factor this in, investors look at effective duration: how much a bond’s price actually changes when interest rates rise or fall. Effective duration equals theoretical duration for zero-risk bonds, but effective duration is shorter as a bond’s credit quality gets lower: the circumstances that cause lower interest rates also cause higher defaults. Think of how a junk bond issued in mid-2000 might have performed over the next three years as the federal funds rate dropped from 6.5% to 1.0%.

All else being equal, that’s good news for the bond. But all else was not equal: high-yield bonds more sensitive to economic growth, and the spread between high-yield bonds and risk-free ones exploded from just over 5% in early 2000 to 11% by 2002.

As it turns out, not only do bonds’ effective durations decline as their credit rating gets lower, but the lowest-rated bonds have a negative effective duration. They’re more sensitive to changes in GDP growth than to the inversely-correlated changes in risk-free rates, so they do better in an environment where rates rise.

Any discussion of duration is incomplete without talking about convexity. Convexity is the second derivative: the change in the rate of price change as rates change. This describes the phenomenon the table above illustrates: for our hypothetical 4% coupon bond, a 1% change in rates increases its value by 7.4% if the change is from 10% to 9%, but increases its value by 8.9% if rates drop from 1% to 0%.

You can broaden the use of “convexity” to cover any case where an investor’s exposure to a trade changes as the trade plays out. If you buy out-of-the-money call options, as the underlying stock price rises your exposure rises. If you sell stocks short, your exposure to the trade declines as it works out (if a stock drops from $10 to $5, the size of your short position is halved). For short sellers who are betting on a stock falling to zero, part of the job is to continuously increase exposure — $10 to $0 and $5 to $0 each yield a 100% profit, so short sellers make their money when shareholders admit, one by one, that the stock they’re holding is indeed worthless.

But convexity extends even beyond that. Consider Salesforce (which joined the Dow last week, about two decades after it was founded). One of the valuable parts of their economics is that they’ve turned their product into a job requirement:

Every seat Salesforce sells increases the demand for Salesforce as a skill, and everyone who gets good at using Salesforce raises demand for the product. Convexity also applies to learning, since you pick up mental models, analogies, and useful tools. It’s easier to analyze a software business if you’ve built models and made investments in twenty other ones beforehand; it’s easier to learn options pricing theory if you picked up the math you need while learning physics; it’s easier to understand contemporary politics if your categorization of world leaders doesn’t amount to Just-Like-Hitler, Just-Like-FDR, and Other.

Managing convexity, in finance and in life, means controlling what percentage of your opportunities you convert into the outcomes you want. Since most people don’t pursue their best ideas with sufficient vigor, and don’t concentrate their investment portfolios in line with their confidence, you could argue that poor convexity management explains most of the gap between modest and massive success.

This post is part of an ongoing series explaining financial concepts that I cite on Medium and in my daily newsletter, The Diff. In this morning’s subscribers-only post, I use duration and convexity to explain the appeal of a new alternative asset class.

I write about technology (more logos than techne) and economics. Newsletter: https://diff.substack.com/

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