In the world of investing, it seems most people’s energy is focused squarely on stocks. Massive amounts of research goes into stock investing every day and even casual investors have an incredible amount of information at their fingertips. Combine that wealth of data with a long term growth pattern in the stock market since 2009 that means anyone who has been in the market less than ten years has no recollection of a single meaningful bear market, and stocks are so ingrained on our collective investing consciousness that many people don’t give other assets a second thought.

As a result, while bonds were once staples of any self-respecting asset allocation lots of investors just aren’t into them like they used to be. Some of it is definitely recency bias, and I think another factor is that many people don’t truly understand how bonds work. But I’ll concede that the bond market is also different than it was more than a decade ago, and I can understand why people might think twice about relying too much on historical data. After all, bonds sound like awesome choices when they paid 10% interest, but consistent declining rates over time surely boosted any backtested numbers while depressing yields today to something a lot less desirable. So it’s no surprise that I see questions along this line all the time:

With record low interest rates today that are even negative in some situations, what’s the point of having bonds in a portfolio at all?

That’s a very good question. And the full answer is kinda complicated and includes some advanced finance mechanics that fly under the radar even for very experienced investors. But explaining complicated concepts is kinda my thing, so let’s talk about a little thing called bond convexity.

To do this topic justice requires covering a lot of ground, and I encourage you to read the whole thing. But to give you a sneak peek of where it’s all going, here’s what you can expect to learn:

How bonds make money The effect of convexity on bond returns How to apply bond convexity to your own portfolio

Sound interesting? Let’s get started.

How Bonds Make Money

Before we jump in the deep end, I think it helps to first talk about how a bond works. A bond is basically a loan that you provide for a fixed amount of time (called the “maturity”) in exchange for regular interest payments (called the “coupon” payment) along the way. Interest payments are something almost everyone is familiar with, and although the repayment of principal works differently in a bond than in a mortgage or student loan, paying and receiving interest on a loan is something we can all relate to. And the money we receive from interest payments on a bond is pretty simple.

Here you can see the 1-year return on a new bond held for one year. If the interest rate is 10%, the money received is 10% of the initial bond value. And if the interest rate is 0%, the money received is 0%. Things get a little more complicated if you talk about different definitions of overall profit (or “yield”) based on how long the bond is held, but for now let’s keep it simple and think about new issues.

Stare at that chart long enough and you’ll eventually notice a problem that bothers a lot of potential bond investors. Why on earth would someone lend money at very low interest when there are higher-returning options available? In fact, in some countries it’s not uncommon to see bonds offered with negative interest where you pay the government to hold your money! There are a few reasons that might make sense including the relative safety of other investments, but major institutional investors continue to buy bonds in huge volumes and they have way more options than your average investor. That seemingly irrational behavior gives us a clue that something else is going on.

That something else is the idea of capital appreciation. Think of a 30-year bond with a 10% interest rate as a contract where someone has agreed to pay you $1,000 a year on your $10,000 loan for the next 30 years. You can choose to collect those payments for a full 30 years, or you can choose to sell that contract to another investor. Because interest rates for new bonds change all the time based on the prevailing market conditions, the value of your fixed contract changes along with them. That change in underlying market value for your bond is called capital appreciation, and it’s the second way bonds make money. Your bond is worth more or less than you paid for it, and if you’re lucky you can sell it for a profit.

The thing that drives the capital appreciation of the bond is the difference between the interest rate of your bond and the prevailing interest rates of a similar bond on the open market. When market rates fall, your bond with a high coupon is more valuable. And when rates rise, your bond with a low coupon is less valuable. So to track the value of your bond, you simply have to track interest rates over time.

In order to illustrate how much a 1% move in interest rates affects the value of a bond, the financial industry reports a measure called duration. I won’t bore you with all the math that goes into it, but I’ll give an example of how to use it. Looking up the home page for the intermediate term treasury fund IEF, the “effective duration” (as of today) is about 7.4 years. That means that if the interest rate changes by 1% the value of the bond is expected to change by 7.4%. We’ll study that expectation in more depth in just a moment, but for now think about how that expected change in value affects how you evaluate the bond.

So to quickly recap, bonds make money in two ways: interest payments and capital appreciation. That explains how bonds can make money even with negative interest rates, but it’s completely understandable that many bond investors will look at the combined interest rate and duration of a bond and assume diminishing returns as rates continue to fall. At some point it just doesn’t sound worth it.

The Effect Of Convexity On Bond Returns

Still with me? I realize that’s a lot of new info for young investors to process, and I also understand that it may seem a bit elementary for experts already well-educated about bonds. All of that background is important to paint the picture for the next point that I wager may surprise both sets of investors.

The assumption of diminishing returns with falling rates is wrong

Like really wrong. What if I told you that a theoretical 30-year bond with a negative interest rate experiences double the profit of one with a 10% interest rate even if the change in rates is an identical -1%?

It’s absolutely true, and it’s due to something called bond convexity. Bond convexity basically means that the sensitivity of a bond to interest rate changes is not constant. It’s also not linearly related to rates. In fact, it accelerates as rates drop, and the amount of acceleration depends on how much time is left on the bond. The duration listed in the fund description is not wrong by any means, but like glancing at the speedometer while your car accelerates down the highway, it’s simply a snapshot of an ever-moving target that follows a much more complex set of mathematical rules than you realize.

To understand bond convexity, let’s use the same 1% rate change sensitivity measured by duration but map the real-world results over every interest rate for a variety of different bond maturities. When looking at this chart, keep in mind that this is NOT the results of a backtest. It’s a fundamental equation for how bonds work.

The chart is packing a lot of data, so let’s go back to our IEF example for a moment to show how it works. Read the fund page, and the weighted average coupon is about 2.5% while the weighted average maturity is about 8.3 years. So eyeball the 2.5% interest rate on the x-axis, follow it up to just below the blue 10-year maturity curve, and follow that horizontally to the y-axis to find the capital appreciation due to a 1% drop in rates. It’s right about 7.4%, right? That matches the listed duration of the fund. You can similarly use the same chart to find the capital appreciation for all types of bonds at different interest rates.

In plain terms, bond convexity measures the curvature of the lines. Here you can see that while the line for 5-year bonds is relatively flat at all different starting interest rates, the convexity effect gets more and more pronounced the longer the maturity of the bond. 30-year bonds are especially sensitive to rate changes at the low end of the scale, and even with negative interest rates can experience massive gains with only small declines in rates.

But this chart is only for capital appreciation, so let’s combine it with the first one we discussed earlier and look at the total return including the interest payment.

To me, this is where it starts to get really interesting. All lines show the total return of a bond with a 1% drop in rates. Look at the total return of a 30-year bond at a 10% interest rate and compare it to one with a -1% interest rate. Would you have guessed that result? How might that affect how you think about the role of long term bonds in a portfolio?

Of course, with every upside comes a downside so let’s add the results for rising rates. Take a minute to really let it sink in before moving on.

This chart is one of my favorites that I’ve made in a while, as not only does it contain a lot of interesting information but I also learned a lot by making it. Here are a few of the most important takeaways:

1. At high interest rates the coupon is most important, and at low rates capital appreciation is king.

2. Short and intermediate term bonds (typically capped at about 10 years) are much less sensitive to interest rates at all levels than long term bonds.

3. Low-interest 30-year bonds are very volatile! In fact, the range of returns is similar to what you might expect from the stock market.

4. Note that the spread of total returns for long term bonds is not symmetrical. Because they are increasingly more sensitive with every drop in rates, for the same +/-1% change they actually have more upside than downside.

5. One thing that’s not obvious from the chart is that interest rate sensitivity declines as bonds age. A new 30-year bond will start on the red line. When it only has 15 years left, it has the volatility of the green line. And when it only has 5 years left it has the predictable tight range of the purple line. Just like people, bonds get less active as they mature.

But if you take only one point away from this post let it be this:

Because of convexity, bonds have way more income potential at very low or even negative rates than most people realize.

How To Apply Bond Convexity To Your Own Portfolio

So we’ve talked about how bonds make money and covered how convexity juices the returns on the low end of the interest rate spectrum. Which leads to the most important question of all — Who cares?

I get it. Too many articles like this seem to take some weird android-like joy in the curiosity of the raw math while remaining completely oblivious to what to actually do with the information. So let’s talk about actionable insights. How can bond convexity affect you?

Here are a few ideas to consider:

Rates can always go lower, and a little change goes a long way

If I had a nickel for every time I heard someone say they think bonds are poor investments because “rates have nowhere to go but up”, I’d be able to buy an island! But the simple fact is that they can always go lower, and because of convexity they really don’t have to move very far at all to turn a major profit. If you’re worried about unequal risk, go back up and read point #4 again — at the most volatile maturities the risk is asymmetrical and biased towards the upside! And always remember that investing in only one bond maturity is not required. Portfolios like the Golden Butterfly use a “barbell” approach of two very different bond maturities, in part, to capture the declining rate upside of long term bonds with the rising rate safety of short term bonds.

Low interest rates affect different portfolios in different ways

The old-school view of bonds is to use them as a trustworthy ballast to temper the volatility of risky stocks, and several portfolios are built with that in mind. For example, the Classic 60-40 and Three-Fund Portfolio both typically recommend intermediate bonds less than 10 years to maturity and ramping up the percentage as you age in order to reduce risk. From the last chart, you can see that these types of shorter bonds do tend to lose their punch as rates drop, so tempering returns expectations may be a good idea.

In contrast, a few portfolios like the Permanent Portfolio and All Seasons Portfolio built on the idea of risk parity tend to prefer long term bonds up to 30 years to maturity. These portfolios actually benefit from low interest rates, as the high volatility of an uncorrelated asset to stocks is a very desirable characteristic and is a big part of how they work. Investors in these portfolios shouldn’t be afraid of low rates. They should embrace them!

When rates go low, go long

Even if you invest in one of the aforementioned portfolios that value bond stability, the best answer to low interest rates is not necessarily to sell your bonds and put all of your money in stocks. Instead, consider mixing in some long term bonds to raise the average maturity of your bond portfolio. As you can see from the chart, doing this at low interest rates will increase both the upside and downside potential of your portfolio, so be careful about adding too much risk. But a little more income potential with a smaller percentage of bonds overall may be a very reasonable balance of risk and return suitable for a lot of people. Adding long term bonds at low rates could also be a decent idea for something like the Larry Portfolio already incorporating the idea of utilizing small percentages of highly volatile assets to generate returns.

Bond index funds sell bonds before they reach maturity for a reason

I know some investors wonder why most bond funds sell bonds before they reach maturity rather than simply holding them all the way to maturity and collecting the full principal payment. There are lots of reasons for that, but re-read point #5 and look at the sensitivity chart a little and you’ll start to understand why maintaining the right balance of maturities is important for performance especially for long term bonds. Allow bonds to age too much, and even long term bonds that used to work like gangbusters lose their ability to strongly respond to interest rate changes and deliver on the risk/return promise of the fund. For that reason, if you’re shopping for a long term bond fund, one like TLT that maintains an average weighted maturity of about 25 years might be a more desirable choice than one that holds onto the bonds a little too long and the income potential suffers as a result.

Whew — that’s a lot of info! Some of your heads may be spinning, others may have questions, and I imagine those of you in the finance field might even have spotted a few details that seem over-simplified. The truth of the matter is that bond index funds sound simple in theory but are actually kinda complicated to run in practice, and because of the specific bonds owned by a particular fund and variations in payout details their results may not perfectly track every assumption in this article. While bond fund weighted averages should map to the charts reasonably closely thanks to the very high liquidity of the funds, definitely don’t take any precise data as absolute gospel but instead think of it as opening the door to a new concept. Bonds are a lot more fascinating than even I knew before writing this post, and I hope you learned something, too.

How might you apply the idea of bond convexity to your own investing decisions? If you have any ideas I didn’t cover, please let me know!