It considers several factors that affect the bond prices as compared to the linear concept of the duration. Investors can use the convexity formula to assess the sensitivity of their bond investments to interest rate changes. Similarly, if the duration of a bond decreases with a fall in the yield, it is said to have positive convexity. A bond with a positive convexity would see a greater price increase with a fall in its yield. Contrarily, a bond with a negative convexity would see a lower price change due to an increase in the yield. Convexity is a risk management tool used to define how risky a bond is as more the convexity of the bond; more is its price sensitivity to interest rate movements.
A technique called gap management is a widely used risk management tool, where banks attempt to limit the “gap” between asset and liability durations. Gap management heavily relies on adjustable-rate mortgages (ARMs) as key components in reducing the duration of bank-asset portfolios. Unlike conventional mortgages, ARMs don’t decline in value when market rates increase, because the rates they pay are tied to the current interest rate. If the duration is high, the bond’s price will move in the opposite direction to a greater degree than the change in interest rates. The higher a bond’s duration, the larger the change in its price when interest rates change and the greater its interest rate risk. If an investor believes that interest rates are going to rise, they should consider bonds with a lower duration.
To clarify, we are referring to the interest rates set by central banks like the Federal Reserve, Bank of England, and European Central Bank. We are now ready to find the approximate modified duration by using our approximation formula from above. 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. Duration is an effective analytic tool for the portfolio management of fixed-income securities because it provides an average maturity for the portfolio, which, in turn, provides a measure of interest rate sensitivity to the portfolio. This article explains duration and convexity, and it presents several formulas for calculating each, but a bond investor generally does not need to know this since most bond listings list the duration. When market interest rates increase, the price of a bond on the secondary market will fall.
Lower coupon rates lead to lower convexity formula yields, and lower yields lead to higher degrees of convexity. As convexity increases, the systemic risk to which the portfolio is exposed increases. For a fixed-income portfolio, as interest rates rise, the existing fixed-rate instruments are not as attractive. As convexity decreases, the exposure to market interest rates decreases, and the bond portfolio can be considered hedged. Typically, the higher the coupon rate or yield, the lower the convexity or market risk of a bond.
Where duration assumes that interest rates and bond prices have a linear relationship, convexity produces a slope. As we know, the bond price and the yield are inversely related, i.e., as yield increases, the price decreases. Convexity measures the curvature in this relationship, i.e., how the duration changes with a change in yield of the bond. The convexity of a bond can have important implications for bond investors and portfolio managers.
However, the relationship between bond prices and yields is typically more sloped or convex. Therefore, convexity is a better measure for assessing the impact on bond prices when there are large fluctuations in interest rates. In its most basic form, the convexity of an asset is a tool that allows us to assess the relationship between bond yields and interest rates. Note that i is the change in the term structure of interest rates and not the yield to maturity for the bond, because YTM is not valid for an option-embedded bond when the future cash flows are uncertain. Duration measures this risk, and convexity measures how duration itself changes with interest rates. So, duration estimates bond price changes with small changes in interest rates, while duration plus convexity estimates bond price changes with larger changes in interest rates.
For instance, if interest rates of the bond increase by 1% after 5 years into the bond’s life, it will lose approximately 15% of its price. Convexity tries to measure the systematic risk arising due to such changes in the prices of a bond due to any change in the market prices of the bonds. Generally, a longer duration would mean greater interest rate risk for the investors. A bond with a longer duration will take longer to recover the cash flows and will be prone to more interest rate changes.
Due to the possible change in cash flows, the convexity of the bond is negative as interest rates decrease. 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. Convexity is a measure of systemic risk as it measures the effect of change in the bond portfolio value with a larger change in the market interest rate while modified duration is enough to predict smaller changes in interest rates. Convexity is the measure of the risk arising from a change in the yield of a bond due to the changes in interest rates.
These factors make bond prices volatile, but certain bonds are more volatile than others, meaning they have greater interest rate risk. As a rule of thumb, non-callable bonds would normally have positive convexity, while many bonds that can be redeemed prior to maturity (callable bonds, i.e. those that have an embedded option) should have negative convexity. For semiannual coupon bonds, divide the final convexity value by four to maintain consistency. Whether it is better for a bond to have high or low convexity will depend on the investor’s goals and movements in market interest rates. High convexity means more potential upside if interest rates fall, and more downside if interest rates rise. Conversely, low convexity means the bond will experience smaller swings in response to interest rate changes.
For small changes in yield, it is very accurate, but for larger changes in yield, it always underestimates the resulting bond prices for non-callable, option-free bonds. 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.
Similarly, the bond price will increase by 5.25% instead of 5% for a 1% decrease in yield. Therefore, bond convexity can help bond investors and portfolio managers to better anticipate the bond price movements and adjust their strategies accordingly. As we can see, bond convexity is an important concept for bond investors, as it helps them to assess the risk and return potential of different bonds under different interest rate scenarios. By measuring bond convexity, investors can better understand how bond prices and yields are related, and how they can optimize their bond portfolio accordingly. It helps investors and portfolio managers to better estimate the price change of a bond for a given change in yield, especially for large changes.
In a nutshell, the convexity of a bond refers to the relationship between bond yields and interest rates. Where Pi is the bond price after increase in interest rate, Pd is the bond price after a decrease in interest rate, P0 is the bond price when the yield equals the coupon rate and deltaY is the change in yield. When yields are low, investors, who are risk-averse but who want to earn a higher yield, often buy bonds with longer durations since longer-term bonds pay higher interest rates. In June 2016, the 10-year German bond, called the bund, sported negative interest rates several times, when the price of the bond actually exceeded its principal. Convexity adds a term to the modified duration, increasing precision by accounting for the change in duration as the yield changes — hence, convexity is the 2nd derivative of the price-yield curve at the current price-yield point. As a result, the modified duration provides a fairly accurate estimation when there is a small change in yields and the prediction error will be relatively insignificant.
As indicated, the larger the change in interest rates, the larger the error in estimating the price change of the bond. Unfortunately, duration has limitations when used as a measure of interest rate sensitivity. While the statistic calculates a linear relationship between price and yield changes in bonds, in reality, the relationship between the changes in price and yield is convex. If the market yield graph were flat and all shifts in prices were parallel shifts, then the more convex the portfolio, the better it would perform, and there would be no place for arbitrage. However, as the yield graph is curved, for long-term bonds, the price yield curve is hump-shaped to accommodate for the lower convexity in the latter term. It shouldn’t be confused with bond maturity, though, which is simply the lifespan of a bond.
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