The price-yield function is the relationship between the bond price and the yield to maturity, which reflects the present value of the bond’s future cash flows. The second derivative of this function measures how the slope of the price-yield curve changes as the yield changes, which indicates the degree of curvature or convexity of the curve. The higher the convexity, the more sensitive the bond price is to changes in interest rates. Convexity is a mathematical concept in fixed income portfolio management that is used to compare a bond’s upside price potential with its downside risk.

Example: Calculating Modified Duration using Microsoft Excel

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. We hope that by reading our guide from start to finish, you now have a firm understanding of what the convexity of a bond is. As we have discussed in great depth, you will first need to get to grips with three key metrics in order to understand convexity. This includes the bond yield, bond duration, and how the movement of government interest rates can impact the value of a bond. In short, if you’re an everyday retail trader buying and selling bonds on a DIY basis, it’s highly unlikely that you will have the necessary tools to calculate the bond convexity.

Similarly, if the yield increases and the duration falls, its convexity is negative. In the equations, $P$ is the bond price, $CF_t$ is the cash flow in period $t$, and $i$ is the per period yield to maturity. It is important to note that these equations work only on an interest payment date. The approximation technique that we will show will work on any date as long as you use the Price function. Duration and convexity are important numbers in bond portfolio management, and duration is pretty simple in Excel because there are built-in functions. 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.

Convexity of a Bond Portfolio

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. Bond investors can use convexity to their advantage by managing their bond portfolios to take advantage of changes in interest rates.

If there is a lump sum payment, then the convexity is the least, making it a more risky investment. If the bonds have a negative convexity, then the duration of the bond increases as the yield rises. Although bonds are generally safer investments than stocks, bonds do have risks. Interest rates change because the supply and demand of loanable funds changes, because inflation or expected inflation changes, or the central bank changes the target interest rate, which affects all other interest rates. For instance, in the United States, the Federal Reserve influences interest rates by setting the federal funds rate, which is the rate charged in interbank lending, and which affects all other interest rates in the United States.

  • This book will show you how, and it will show real examples of how this works and how much you can potentially profit, and how bonds, at times, can even be better than stocks.
  • Because duration depends on the weighted average of the present value of the bond’s cash flows, a simple calculation for duration is not valid if the change in yield could change cash flow.
  • You will get 10.90 years for modified duration, and 11.23 years for Macaulay duration.
  • This explains why premium bonds (trading above face value) and discount bonds (trading below face value) can exhibit different convexity behaviors despite similar durations.

By adding or subtracting bonds with different convexities, they can increase or decrease the overall convexity of their portfolios, which affects their sensitivity to interest rate changes. Bond managers often want to know how much the market value of a bond portfolio changes when interest rates change by 1 basis point. It can easily be seen that modified duration changes as the yield changes because it is obvious that the slope of the line changes with different yields. The gap between the modified duration and the convex price-yield curve is the convexity adjustment, which — as can be easily seen — is greater on the upside than on the downside. Before 1938, it was well known that the maturity of a bond affected its interest rate sensitivity, but it was also known that bonds with the same maturity could differ widely in price changes with changes to yield.

How bond prices change with interest rates and the concept of duration?

The effective duration (aka option-adjusted duration) is the change in bond prices per change in yield when the change in yield can cause different cash flows. For instance, for a callable bond, the bond will not rise above the call price when interest rates decline because the issuer can call the bond back for the call price, and will probably do so if rates drop. As you can see, modified duration gives a better estimate of the new price than Macaulay duration since it is closer to the price modified by discounted cash flows. Of course, interest rates usually change in small steps, so duration measures interest rate sensitivity effectively. With coupon bonds, investors rely on a metric known as duration to measure a bond’s price sensitivity to changes in interest rates.

Do interest rates impact the value of bonds?

As a result, it would take a significant rise in yields to make an existing holder of an MBS have a lower yield, or less attractive, than the current market. 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. Convexity demonstrates how the duration of a bond changes as the interest rate changes. Portfolio managers will use convexity as a risk management tool, to measure and manage the portfolio’s exposure to interest rate risk.

These are typically bonds with call options, mortgage-backed securities, and those bonds which have a repayment option. If the bond with prepayment or call option has a premium to be paid for the early exit, then the convexity may turn positive. Bond convexity is a measure of how the price of a bond changes as the interest rate changes. It is an extension of the concept of bond duration, which captures the linear relationship between price and yield.

Because a coupon bond makes a series of payments over its lifetime, fixed-income investors need ways to measure the average maturity of a bond’s promised cash flow, to serve as a summary statistic of the bond’s effective maturity. The duration accomplishes this, letting fixed-income investors more effectively gauge uncertainty when managing their portfolios. 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. Even though Convexity takes into account the non-linear shape of the price-yield curve and adjusts for the prediction for convexity formula price change, there is still some error left as it is only the second derivative of the price-yield equation.

How Does Bond Convexity Work?

However, the convexity goes one step further by exploring how both the bond price and bond duration will change when interest rates go up or down. In doing so, this allows financial institutions to gauge how much risk their government bond portfolios present. Both bonds have the same maturity and coupon rate, but bond B has a higher initial yield and a lower price than bond A.

  • Convexity relates to the interaction between a bond’s price and its yield as it experiences changes in interest rates.
  • This is typically used to achieve optimal interest rate risk management and better capital adequacy.
  • This means that for every 1% change in interest rates, Bond A’s price will change by 4% while Bond B’s price will change by 5.5%.
  • Where $P_+$ is the bond price when the yield increases by $\Delta y$, $P_-$ is the bond price when the yield decreases by $\Delta y$, and $P_0$ is the bond price at the original yield.
  • Convexity is usually a positive term regardless of whether the yield is rising or falling, hence, it is positive convexity.

To get a more accurate price for a change in yield, adding the next derivative would give a price much closer to the actual price of the bond. Today with sophisticated computer models predicting prices, convexity is more a measure of the risk of the bond or the bond portfolio. More convex the bond or the bond portfolio less risky; it is as the price change for a reduction in interest rates is less.

The price of bonds returning less than that rate will fall, as there would be very little demand for them as bondholders will look to sell their existing bonds and opt for bonds with higher yields. Eventually, the price of these bonds with the lower coupon rates will drop to a level where the rate of return is equal to the prevailing market interest rates. The bond yield is the earnings or returns an investor can expect to make by buying and holding that particular security. The bond price depends on several characteristics, including the market interest rate, and can change regularly. Convexity is the curvature in the relationship between bond prices and interest rates. It reflects the rate at which the duration of a bond changes as interest rates change.

Yes, there is a direct relationship between the market value of government bonds and the central bank interest rate. In the vast majority of cases, the value of a bond will go down as interest rates go up. So, there is really no need to memorize the complicated exact formulas for these bond risk measures. Note that these functions return the modified duration and Macaulay duration in years, so there is no need to divide by the payment frequency.

The bond convexity goes one step further than simply measuring the relationship between interest rates and yields, as it also takes into account the bond’s duration. The duration measures the sensitivity of an asset in relation to external market forces, such as interest rates. We know how to calculate the price of a bond using Excel, so the problem here is how to calculate that partial derivative. 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%).