{"id":5332,"date":"2013-06-25T06:55:18","date_gmt":"2013-06-25T10:55:18","guid":{"rendered":"http:\/\/www.jasonapollovoss.local\/?p=5332"},"modified":"2018-09-21T02:04:05","modified_gmt":"2018-09-21T06:04:05","slug":"a-bond-convexity-primer","status":"publish","type":"post","link":"https:\/\/jasonapollovoss.com\/web\/2013\/06\/25\/a-bond-convexity-primer\/","title":{"rendered":"A Bond Convexity Primer"},"content":{"rendered":"

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.<\/span><\/p>\n

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\u2019s 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.<\/span><\/p>\n

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.<\/span><\/p>\n

So why is the relationship between a bond\u2019s 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 \u2014 30 years, 10 years, and 1 year.<\/span><\/p>\n

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Capital Gains at Indicated Interest Rate, 30-Year Maturity<\/strong><\/span>
\n\"Capital<\/a><\/span><\/p>\n

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Above is the bond with a 30-year maturity. Look at how curved\u00a0\u2014 i.e., how convex\u00a0\u2014 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.<\/span><\/p>\n

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Capital Gains at Indicated Interest Rate, 10-Year Maturity<\/strong><\/span>
\n\"Capital<\/a><\/span><\/p>\n

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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%.<\/span><\/p>\n

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Capital Gains at Indicated Interest Rate, One-Year Maturity<\/strong><\/span>
\n\"Capital<\/a><\/span><\/p>\n

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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\u2019s 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<\/a> so that you can experiment with different combinations of factors to show you the effects of convexity on a bond\u2019s price performance. Separately, I have also attached a spreadsheet<\/a> from our fixed-income valuation course<\/a> 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:<\/span><\/p>\n