Calculate and Interpret Bond Convexity CFA Level 1

However, duration loses accuracy in projecting price movements for assets with embedded options and bigger yield changes. Duration and convexity can be coupled to approximate the price for a given change in yield more closely. As you can see, Bond A has a higher price and a lower yield than Bond B when the market interest rate is lower than the coupon rate, and vice versa when the market interest rate is higher than the coupon rate. This is because Bond A has a positive convexity, which means that its price increases more than Bond B when the yield decreases, and decreases less than Bond B when the yield increases.

• Though they both decline as the maturity date approaches, the latter is simply a measure of the time during which the bondholder will receive coupon payments until the principal is paid.
• Of course, there are formulas that you can type in (see table below), but they aren’t easy for most people to remember and are tedious to enter.
• A bond with low duration will have a smaller price change, but the price change will be more curved and nonlinear.
• Finally, convexity is a measure of the bond or the portfolio’s interest-rate sensitivity and should be used to evaluate investment based on the risk profile of the investor.
• Therefore, portfolio managers may wish to protect (immunize) the future accumulated value of the fund at some target date, against interest rate movements.

For such bonds with negative convexity, prices do not increase significantly with a decrease in interest rates as cash flows change due to prepayment and early calls. As can be seen from the formula, Convexity is a function of the bond price, YTM (Yield to maturity), Time to maturity, and the sum of the cash flows. The number of coupon flows (cash flows) change the duration and hence the convexity of the bond. One of the most important concepts in bond investing is the relationship between bond price and yield.

Zero-coupon Bonds

In the above graph, Bond A is more convex than Bond B even though they both have the same duration, and hence Bond A is less affected by interest rate changes. The information provided herein may be obtained or compiled from public and/or third party sources that PCM has no reason to believe are unreliable. Any opinion or view herein is an expression of belief of the individual author or the indicated source (as applicable) only. PCM makes no representation or warranty that such information is accurate, complete, verified or should be relied upon as such.

As interest rates fall, the duration of a bond will increase, as the present value of the distant cash flows will become more significant. Similarly, as interest rates change, the convexity of a bond will also change, as the curvature of the price-yield relationship will vary. For example, as interest rates fall, the convexity of a callable bond will decrease, as the probability of the bond being called will increase. As interest rates rise, the convexity of a callable bond will increase, as the probability of the bond being called will decrease. Before we dive into the steps of calculating bond convexity, let us first understand what it is and why it is important. Bond convexity is a measure of how the price of a bond changes as the interest rate changes.

Institutions with future fixed obligations, such as pension funds and insurance companies, differ from banks in that they operate with an eye towards future commitments. For example, pension funds are obligated to maintain sufficient funds to provide workers with a flow of income upon retirement. As interest rates fluctuate, so do the value of the assets held by the fund and the rate at which those assets generate income.

Convexity of a bond formula

However, if the yield curve becomes steeper or flatter, the bond price change will also depend on the convexity. In general, a bond with positive convexity will have a larger price increase when interest rates fall than a price decrease when interest rates rise, and vice versa for a bond with negative convexity. This means that a bond with positive convexity will have a higher expected return and lower risk than a bond with negative convexity, all else being equal. Both duration and convexity depend on the current interest rate level, the coupon rate, the maturity, and the features of the bond. As these factors change over time, so will the duration and convexity of the bond. For example, as interest rates rise, the duration of a bond will decrease, as the present value of the distant cash flows will become less significant.

For example, duration and convexity do not account for the credit risk, liquidity risk, inflation risk, or reinvestment risk of a bond. Moreover, duration and convexity are based on certain assumptions and simplifications, such as parallel shifts in the yield curve, constant interest rate volatility, and continuous compounding. In reality, these assumptions may not hold, and the actual bond price changes may differ from the predicted ones. Therefore, bond investors and portfolio managers should use duration and convexity as guidelines, not as exact predictions, and they should also consider other factors and scenarios when evaluating bond risk and return.

Bond Convexity Calculator: How to Calculate Bond Convexity: Step by Step Tutorial

• The shape of the curve is determined by the coupon rate, maturity, and redemption value of the bond.
• Convexity is a metric for the degree of the curve between the price of bonds and their yields.
• Most mortgage-backed securities (MBS) will have negative convexity because their yield is typically higher than traditional bonds.
• If the interest rate increases by 2%, the price of Bond A should decrease by 8%, while the price of Bond B will decrease by 11%.
• Convexity demonstrates how the duration of a bond changes as the interest rate changes.

It does not have any regard to your specific investment objectives, financial situation or any of your particular needs. Accordingly, no warranty whatsoever is given and no liability whatsoever is accepted for any loss arising whether directly or indirectly as a result of your acting based convexity formula on this information.

Bond Convexity Formula

Where P is the current price of the bond, c is the annual coupon rate, F is the face value of the bond, r is the annual yield on the bond, m is the number of coupon payments per year and n is the total years to maturity. A higher convexity value indicates that a bond’s price is more sensitive to interest rate changes in a non-linear way. This means that for large rate shifts, the bond’s price deviates more from what duration alone would predict. Investors seeking to minimize risk in volatile markets often prefer bonds with higher convexity for better price protection. Several factors impact the convexity of a bond, including the bond’s coupon rate, maturity, and credit quality.

You can use either worksheet and will get the same answers because the approximation formula only cares about the prices, not the method of calculating them. By refining the modified calculation investors get a closer estimate to the actual price, which helps them reduce losses when yields rise and enhance gains when yield fall. Bank liabilities, which are primarily the deposits owed to customers, are generally short-term in nature, with low duration statistics.

This is because the YTM curve of a portfolio with positive convexity will become steeper as interest rates rise, which means that the portfolio’s overall return will increase faster than a portfolio with negative convexity. If a bond’s duration increases along with yields, this is called negative convexity. If a bond’s tenure lengthens while its yield drops, it is said to have positive convexity. Bond convexity is a measure of how sensitive the price of a bond is to changes in interest rates. It is also a measure of how much the duration of a bond changes as interest rates change.

Therefore, if a bond has negative convexity, its duration would increase, and the price would fall. Convexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change. Where duration assumes that interest rates and bond prices have a linear relationship, convexity produces a slope. The interest rate risk is a universal risk for all bondholders as all increase in interest rate would reduce the prices, and all decrease in interest rate would increase the price of the bond. This interest rate risk is measured by modified duration and is further refined by convexity.

By contrast, a bank’s assets mainly comprise outstanding commercial and consumer loans or mortgages. These assets tend to be of longer duration, and their values are more sensitive to interest rate fluctuations. In periods when interest rates spike unexpectedly, banks may suffer drastic decreases in net worth, if their assets drop further in value than their liabilities. Where P is the bond price, y is the yield, CFn is the nth cash flow of the bond, t is the time difference between time 0 and the cash flow. As you can see, the graph is curved which shows that the rate of change in price is different at different points on the graph.

How bond duration changes with a changing interest rate

Recall that a partial derivative tells you how much a function changes when one of its variables changes by a small amount. More specifically, this partial derivative will tell us how much the bond price will change when the yield changes by a small amount (say from 5% to 5.001%). When we take this change and divide it by the current price, we are simply converting the dollar change into a percentage change. So you can see that this derivative will tell us how sensitive a bond’s price will be to changes in interest rates. In the image below, the calculation of the curved line represents the change in prices, given a change in yields.

The information does not constitute, and should not be used as a substitute for tax, legal or investment advice. You may wish to obtain advice from a qualified financial adviser, pursuant to a separate engagement, before making a commitment to purchase any of the investment products mentioned herein. Where $D_m$ is the modified duration and the other variables are the same as before.

A bond with high convexity is more sensitive to changing interest rates than a bond with low convexity. That means that the more convex bond will gain value when interest rates fall and lose value when interest rates rise. The opposite is true of low convexity bonds, whose prices don’t fluctuate as much when interest rates change. When graphed on a two-dimensional plot, this relationship should generate a long-sloping U shape (hence, the term “convex”). When market interest rates increase, the price of a bond on the secondary market will fall. This is because new bonds offer higher interest rates, and sellers must accept a discount in order to sell older bonds.

In the example figure shown below, 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. Finally, convexity is a measure of the bond or the portfolio’s interest-rate sensitivity and should be used to evaluate investment based on the risk profile of the investor. Overall, the convexity of a bond portfolio is an important consideration for investors when choosing which bonds to include in their portfolio. We are now ready to find the approximate modified duration by using our approximation formula from above.