As the US Federal Reserve lays the verbal groundwork for an eventual real-world quantitative easing (QE) taper, bond prices are dropping at an accelerated rate. In order to understand the ramifications of a Federal Reserve taper on the prices of a bond or bond portfolio, what is needed is a bond convexity primer.
In the parlance of those who know calculus, convexity is the second derivative. For the layperson this is known as the rate of change in change. For convexity to make better sense, let me compare it to driving a car. When you are driving a car your speed is the rate of change in the car’s location. Want to change your speed (i.e., the rate of change)? Then you either give the car more gas with the accelerator or press down on the brakes to slow the car down. Speeding up and slowing down are the second derivative. Speeding up means that there is a positive second derivative, while slowing down means that there is a negative second derivative.
Related to the bond market, the speed of your car is called duration, while the speeding up/slowing down is known as convexity. The higher the convexity, the more dramatic the change in price given a move in interest rates. Whatever you call it, after a while, if you keep braking a car it stops. After a while, if your bond is experiencing negative convexity, it also slows down/loses value. The harder the acceleration or braking, the greater the change in your speed.
So why is the relationship between a bond’s yield and its price known as convexity? As yields change, the change in the price of the bond is not linear; it is curved in a convex fashion. To understand convexity more directly take a look at the following three graphs, all for a $1,000 par value bond, with a coupon rate of 3.452%, making payments twice per year, and with zero expectation of a yield change in the future. The only thing making these three bonds different is the number of years until maturity — 30 years, 10 years, and 1 year.
Capital Gains at Indicated Interest Rate, 30-Year Maturity
Above is the bond with a 30-year maturity. Look at how curved — i.e., how convex — the graph of the price-yield relationship is! Notice also that there are no capital gains/changes in price at the exact yield of the bond, 3.45%, where the line actually touches the horizontal axis. This means that if yields stay the same as the coupon rate there should be no change in the price of the bond.
Capital Gains at Indicated Interest Rate, 10-Year Maturity
A 10-year maturity bond is graphed above. Look at how much less convex the line is relating price to yield. Again, note that the line touches the horizontal axis at 3.45%.
Capital Gains at Indicated Interest Rate, One-Year Maturity
Last, look at this one-year maturity version of a bond. There is zero convexity/curve, just a flat line. What all of this means is that a bond’s price is sensitive to the length of its maturity. But it is also sensitive to other factors. I have created a spreadsheet that aptly demonstrates convexity for a bond under different scenarios so that you can experiment with different combinations of factors to show you the effects of convexity on a bond’s price performance. Separately, I have also attached a spreadsheet from our fixed-income valuation course that allows you to calculate duration and convexity in a numerical format if you prefer to see it quantitatively. If you are not feeling experimental, here is a summary of the factors affecting bond convexity:
- Maturity: Positive correlation; the longer the maturity the greater the convexity/price sensitivity to yield changes.
- Coupon: Negative correlation; the higher the coupon the lower the convexity/price sensitivity to yield changes.
- Yield: Negative correlation; the lower the yield the higher the convexity/price sensitivity to yield changes. To best understand this, look at the graph above for the 30-year bond. The lower the yield goes the higher the convexity/price sensitivity as compared with the higher yield portion of the curve.
I have kept things simple here. If the bond has embedded options, such as calls or puts, it will affect some of the above relationships, sometimes dramatically. Because every bond has a unique structure and issuer, it is impossible to dole out advice on the exact relationships. But because call and put options generally affect maturity you can make informed guesses as to the affect on convexity.
Another factor often not discussed in the price performance of bonds and bond portfolios is how rapidly changes in interest rates occur. The further into the future and the smaller the interest rate changes, the less damage done to a bond or portfolio today.
So, in an environment of central bank tapering, investors want low convexity bonds and bond portfolios. Such a portfolio, as rare as a golden goose, would have a short maturity, high coupon rate, and a high yield. Good luck finding that!
You can download the spreadsheets mentioned in the article here:
- Bonds: How Capital Gains Change With Interest Rate Changes, GP
- Calculation of Duration and Convexity
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All posts are the opinion of the author. As such, they should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute or the author's employer.
Photo credit: ©iStockphoto.com/Jitalia17
43 Comments
Hi Geoff,
Can u please understand how lower coupon bonds out perform in a tapering environment?
In a tapering environment, means rising yield, when yield rises value of bonds with lower coupon decreases
Hello,
Would it be possible to get the unprotected version of the spreadsheet "Bonds- how capital gains change with interest rate changes"?
Hi North,
For what purpose would you be using the unprotected version of the spreadsheet?
With smiles,
Jason
I wanted to model multiple bonds, strictly educational.
Hi North,
Thank you for your interest. I am going to decline sharing the unprotected version of the spreadsheet. If you use Excel's PV function you should be able to replicate the spreadsheet and for your interest of modeling multiple bonds.
With smiles,
Jason
Hi Sir,This was a great article...
I have a request if you will.. Can you shed some light on roll down/carry related aspect in bond holding/trading. I get confused with these terms and wonder when to slide on the curve(i understand you need to sit at the strategic point on yield curve to make any money)...but what are these concepts all about...Thanks in advance,Aj
Hi Aj,
Bond trading strategies,their explanations, their limitations, their contingencies, and their applications could fill up many pages, so my comments here have to be limited. Also, I am not sure what you already know, so I have to guess a bit.
For starters, "roll down" is a strategy that works as long as the yield curve is upward sloping, meaning that bonds of longer maturity yield more than bonds of shorter maturity. Another important consideration is that the yield curve does not shift upward while trying to take advantage of the "roll down." Put another way, the strategy is reliant on yields staying roughly the same, why? As yields rise it makes bonds with lower yields and similiar maturities and credit qualities less attractive. After all, why would I buy a 30 year maturity bond yielding 4% issued by a AAA credit, when I could buy a 30 year maturity bond just issued and yielding the current market rate of 5% for a AAA credit? Very few would purchase the 4% bond, so its price must fall. And it is the falling price that offsets the benefit of the "roll down" strategy.
So, assuming an upward sloping, stable yield curve, what is the "roll down" strategy?
For me it is helpful to think of bonds with maturities of longer than one year as multiple bonds. Say, for example, I own a 2 year maturity bond yielding 1% that pays interest once a year, this is essentially the same as owning a 1 year maturity bond paying 1% one year from today, another one year bond that won't pay me my 1% interest until the end of two years, and a zero coupon bond that will not pay me my principal amount until the end of year two. When you see bonds of greater than one year maturity in this way then you can compare long-term bonds to short-term bonds to see which one is likely to have a higher expected return given your assumptions for what the yield curve does.
To illustrate this consider the following situation, I have a $1,000 that I want to invest in a bond and I think that the current upward sloping yield curve will remain pretty much the same for the next year and I have a 1 year investment time horizon. Among my options are the following two bonds of comparable credit quality:
a) A 1 year bond yielding 0.5%
b) A 3 year bond yielding 1.5%
I could buy bond a and be paid my $5 in interest at the end of year one, as well as receive my principal of $1,000 back. Or I could buy bond b and sell it at the end of year one and earn $15 in interest at the end of year one and receive my principal back here, too, and likely the principal amount would likely be a bit higher (see below). So that 3 year bond could have been viewed as a 1 year bond by me if I think of bonds as a stream of interest payments and one principal repayment at maturity. So options a and b could be seen as two competing 1 year bonds if I wanted to think of them in that way. By buying b I have triple the return, plus a likely slightly higher principal amount received from the sale, less my trading costs for selling the bond. Obviously, I have taken on greater risk due to the duration and convexity risks in buying the second bond, but if you are confident of the shape of the yield curve this may be a more profitable strategy.
There are two opposing forces at work here. As time passes, a longer term bond becomes a shorter term bond. As a bond advances in age it tends to rise in value because its risk of not suriving to maturity decreases. Working in opposition to this, however, is the force pushing down against the value of the bond: as it advances in age it will pay less total interest so the bond's yield to maturity approaches 0% at the moment just before maturity. Why is this the case? In the 3 year example above there are 4 payments owed to the buyer of the bond when issued: 1) year 1 interest; 2) year 2 interest; 3) year 3 interest; and 4) principal repayment. At the end of year 2 any owner of the bond only receives two additional payments: year 3 interest and principal. This tends to lower the value of the bond.
So the "roll down" term comes because I can buy a longer-term bond and roll with it down the yield curve. I buy a 3 year bond, and sell it at the end of year 1 when it is now a 2 year bond.
Does this make sense? I hope so.
With smiles,
Jason
Hi Jason,
I found your explanation of roll down here to be very interesting. I have a follow up question that I've been struggling with. Suppose I bought a 10yr bond at par on an upward sloping yield curve. In one year, I want to sell that bond. Thus it would now be a 9yr and I could reap the benefits of roll-down in this situation. However, if the yield curve has shifted up during this period, what is the true impact of the roll return? The roll helps to offset some of the capital loss, how do I know exactly what impact roll has made? Would it be the difference between the 10yr and 9yr bond on the original curve, or the difference between the 10yr and 9yr bond on the T+1 curve? The total return impact would be the difference between the 10yr price on the original curve and the 9yr price on the T+1 curve, but I'm struggling with how I'd know just how much of that was driven by roll.
So in essence one would want to own floating rate securities, high yields, or low duration bonds, right?
Hi Scott,
It all depends on your forecast for interest rates and the shape of the future yield curve.
Best wishes for success!
Jason