Long-time readers may recall that I wrote an article last year, with year notes at 2. The sell-off was primarily driven by the fact that the Fed had abused the hell out of the bond market and pushed it to unsustainable levels. Indeed, the Fed is actually claiming credit for the fact that the selloff was only bps. How wonderfully serendipitous it is that even the most egregious failures of the Federal Reserve turn out to benefit society in heretofore unexpected ways. You will recall that one of the main reasons given by the Federal Reserve to purchase mortgages in the first place was to help unfreeze the mortgage market, and to provoke additional mortgage origination.

## Convexity of a Bond | Formula | Duration | Calculation

Here I take a high-level look at what convexity hedging is and how it affects the UST market. The asset class most affected by convexity hedging is agency mortgage-backed securities MBS. These securities are common to such institutions as money managers, insurance companies, commercial banks, and hedge funds, among others. Although some participants do not need to hedge the interest rate risk of holding agency MBS, a large percentage of buyers do.

The difficulty of hedging agency MBS lies in the fact that the bonds exhibit negative convexity. That is, all else being equal, an increase in interest rates will lengthen the bond as prepayments slow down, but a decrease in interest rates will shorten the average life of the bond as homeowners refinance prepay into a lower rate. As one might imagine, as UST rates have fallen precipitously over the past few years, prepayments have accelerated on high-coupon mortgages. The implications of falling mortgage rates combined with the massive intervention by the Fed have created interesting dynamics in the MBS world.

Naturally, with refinancing and new mortgage origination, the new, lower-coupon bonds have a lot more convexity. The following table shows the duration of a year 3. As the table illustrates, the duration of the year 3. If rates rise, the duration of the MBS bond rises as well. How this relates to UST selling is obvious: If rates were to rise and MBS holders needed to adjust their hedge ratios, incremental UST selling would occur, potentially exacerbating a selloff.

The effects would extend beyond U. Mortgage REITs can be forced sellers of mortgages if rates rise prepayments slow and prices fall enough. As shown in the table, duration could rise on a year 3. Please note that the content of this site should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute. It is really a nice and useful piece of info. I am glad that you just shared this useful information with us.

Please stay us up to date like this. Thanks for sharing. Looking at your post today, not only what you wrote easy to understand, but it foretold what had happened with rates in the last three weeks, especially the last week. Your email address will not be published. Notify me of follow-up comments by email.

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## What Is Bond Convexity?

Bond convexity is closely associated with duration but takes the concept one step further. While duration estimates how a bond s price can be expected to react to changes in market interest rates , convexity measures how the bond s duration—and by implication, its price—will change depending on how much interest rates change. Taken together, both duration and convexity show how a bond or bond portfolio can be expected to perform when interest rates change. This helps investors understand the price risk of owning fixed-income securities under different interest rate scenarios.

Duration is a linear measure, meaning it assumes that for a certain percentage of change in interest rates, an equal percentage change in price will occur.

By Dheeraj Vaidya Leave a Comment. Fixed Income Tutorials. Duration of a bond is the linear relationship between the bond price and interest rates where, as interest rates increase bond price decreases. Simply put, a higher duration implies that the bond price is more sensitive to rate changes.

### Bond convexity

Can somebody please explain why this is the case? There are two callable bonds callable immediately. Which bond is likely the higher coupon and the lower coupon? Why are higher coupon bonds more likely to be negative convex? Lower coupons result in higher duration and convexity holding all else constant.

### Duration and Convexity

Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds. One method of measuring interest rate risk due to changes in market interest rates is by the full valuation approach , which simply calculates what bond prices will be if the interest rate changed by specific amounts. The full valuation approach is based on the fact that the price of a bond is equal to the sum of the present value of each coupon payment plus the present value of the principal payment. That the present value of a future payment depends on the interest rate is what causes bond prices to vary with the interest rate, as well. Another method to measure interest rate risk, which is less computationally intensive, is by calculating the duration of a bond, which is the weighted average of the present value of the bond s payments. Consequently, duration is sometimes referred to as the average maturity or the effective maturity. The longer the duration, the longer is the average maturity, and, therefore, the greater the sensitivity to interest rate changes. Graphically, the duration of a bond can be envisioned as a seesaw where the fulcrum is placed so as to balance the weights of the present values of the payments and the principal payment. Mathematically, duration is the 1 st derivative of the price-yield curve, which is a line tangent to the curve at the current price-yield point.

### Advanced Bond Concepts: Convexity

Negative convexity refers to the shape of a bond s yield curve and the extent to which a bond s price is sensitive to changing interest rates. The degree to which a bond s price changes when interest rates change is called duration , which often is represented visually by a yield curve. Convexity describes how much a bond s duration changes when interest rates change, meaning that investors can learn a lot not just from the direction of the yield curve but the curviness of the yield curve. Accordingly, convexity helps investors anticipate what will happen to the price of a particular bond if market interest rates change. Generally, when interest rates fall, bond prices rise. But a bond with negative convexity loses value when interest rates fall. This is often the case for mortgage-backed securities MBS because they rely on underlying mortgage loans , which are typically refinanced and thus paid off early when interest rates fall.

## Convexity Hedging: What Is It, and Why Does It Matter?

Convexity is a measure of the curvature in the relationship between bond prices and bond yields that demonstrates how the duration of a bond changes as the interest rate changes. Convexity is used as a risk-management tool, which helps measure and manage the amount of market risk to which a portfolio of bonds is exposed. Before explaining convexity, it s important to know how bond prices and interest rates relate to each other. As interest rates fall, bond prices rise while conversely, rising interest rates lead to falling bond prices. In the example figure shown above, Bond A has a higher convexity than Bond B, which indicates that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall. As interest rates rise, bond yields rise as new bonds coming on the market are issued at the new, higher rates. Also, as rates are rising, investors demand a higher yield from bonds.

### Negative Convexity: Definition & Examples

Long-time readers may recall that I wrote an article last year, with year notes at 2. The sell-off was primarily driven by the fact that the Fed had abused the hell out of the bond market and pushed it to unsustainable levels. Indeed, the Fed is actually claiming credit for the fact that the selloff was only bps. How wonderfully serendipitous it is that even the most egregious failures of the Federal Reserve turn out to benefit society in heretofore unexpected ways. You will recall that one of the main reasons given by the Federal Reserve to purchase mortgages in the first place was to help unfreeze the mortgage market, and to provoke additional mortgage origination. In that, it evidently failed, for if it had succeeded then the total amount of negative convexity in public hands would not have changed very dramatically. In fact, it would have been worse since the new origination would have been current coupons and replacing higher coupons. A negative-convexity selloff has two parts:

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### Duration & Convexity: The Price/Yield Relationship

If you have been watching the yield curve for U. As Treasury yields go north, so do mortgage rates. In fact, those have risen by 76 basis points in about a month. With suppressed opportunities for refinancing mortgages and other loan servicing, security duration increases and holders are exposed to risk. To mitigate the risk, investors may seek to sell longer-dated Treasuries and mortgage bonds, or alternatively to make interest rate swaps or options on their bonds, putting further upwards pressure on yields and increasing spreads. In other words, convexity hedging will increase; the only question is by how much. With the Fed holding so much of the mortgage paper, it really knocks down the amount of mortgage hedging needed when yields rise. Dismissing the inherent risk is overestimating the power of the Fed, not to mention missing the channel via which convexity hedging is most prevalent: Mortgage-backed securities. Convexity hedging affects mortgage-backed securities most of all asset classes, and poses certain challenges, seeing as these bonds exhibit negative convexity, meaning that ceteris paribus, an increase in the interest rate, such as what has been observed for U. Treasuries, will lengthen the bond as prepayments slow down and vice versa; when interest rates decrease, prepayment accelerates on high-coupon mortgages and shortens the bond. Since interest rates have been decreasing over the past couple of years in response to monetary stimulus, the new, lower-coupon bonds come with a lot more convexity, which is why even the modest increase in U. Treasuries has made some investors and observers of the market call for caution.

As a general rule, the price of a bond moves inversely to changes in interest rates: M odified duration attempts to estimate how the price of a bond will change in response to a change in interest rates and is stated in terms of a percentage change in price. These two measurements can provide insight into how a bond is expected to perform should interest rates change and can help investors understand the price risk of fixed income securities in different interest rate environments. In simple terms, modified duration gives an idea of how the price of a bond will be affected should interest rates change. A higher duration implies greater price sensitivity upwards downwards should rates move down up. Duration is quoted as the percentage change in price for each given percent change in interest rates. For example, the price of a bond with a duration of 2 would be expected to increase decline by about 2. The duration of a bond is primarily affected by its coupon rate, yield, and remaining time to maturity. The duration of a bond will be higher the lower its coupon, lower its yield, and longer the time left to maturity.

In finance , bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates , the second derivative of the price of the bond with respect to interest rates duration is the first derivative. In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance. Duration is a linear measure or 1st derivative of how the price of a bond changes in response to interest rate changes. As interest rates change, the price is not likely to change linearly, but instead it would change over some curved function of interest rates. The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity. Convexity is a measure of the curvature or 2nd derivative of how the price of a bond varies with interest rate, i. Specifically, one assumes that the interest rate is constant across the life of the bond and that changes in interest rates occur evenly. Using these assumptions, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question. Then the convexity would be the second derivative of the price function with respect to the interest rate.

**VIDEO ON THEME: Bond convexity**

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