Duration and Convexity, with Illustrations and Formulas

Higher convexity indicates that the bond’s price will experience smaller fluctuations in response to interest rate shifts, providing a cushion against rate changes. Bonds with higher convexity provide a volatility buffer against interest rate changes, as their prices are less sensitive to rate fluctuations. Duration and convexity let investors quantify this uncertainty, helping them manage their fixed-income portfolios. 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.

1. Duration and convexity let investors quantify this uncertainty, helping them manage their fixed-income portfolios.
2. By monitoring convexity levels across various bonds and adjusting their holdings accordingly, these managers can exploit opportunities to enhance returns and manage risk more effectively.
3. Therefore, portfolio managers may wish to protect (immunize) the future accumulated value of the fund at some target date, against interest rate movements.
4. Generally, most bonds have positive convexity, because they have fixed coupon payments and a fixed maturity date.

While duration estimates the linear price-yield relationship, convexity accounts for the non-linear effects. This is crucial in scenarios where interest rates change, as convexity helps refine the estimate of bond price changes. New bond issues must also have higher rates to satisfy investor demand for lending the issuer their money. 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, most likely newer issues, paying higher yields. 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.

Portfolio managers will use convexity as a risk-management tool, to measure and manage the portfolio’s exposure to interest rate risk. Bonds with higher convexity experience larger price increases in response to falling interest rates, providing the potential for greater capital gains. Active bond portfolio managers can use convexity to capitalize on interest rate trends and convexity formula excel market inefficiencies. Incorporating convexity into bond portfolio management helps investors diversify their interest rate risk. Their convexity increases with their time to maturity, making them more sensitive to interest rate changes as their maturity extends. This makes them more attractive to investors seeking to minimize interest rate risk in their bond portfolios.

Examples of Convexity Formula (With Excel Template)

Treasuries have lower coupon rates and current yields than corporate bonds of similar maturities because of the difference in default risk. Therefore, U.S. Treasuries should have higher durations than corporate bonds, and, therefore, change in price more when market interest rates change. However, changes in perception of the risk of default may also change bond prices, blunting or augmenting what duration would predict. 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. However, the relationship between bond prices and interest rates is non-linear, leading to inaccuracies in duration-based price change estimates.

We have written this article to help you understand what bond convexity is and how to apply the bond effective convexity formula. We will also show some bond convexity examples to help you understand the metric. This means that the bond price will change more when the interest rate is low than when it is high. For example, if the interest rate goes from 5% to 4%, your bond price will increase more than if it goes from 5% to 6%.

On the other hand, if long-term bonds are held to maturity, then you may incur an opportunity cost, earning low yields when interest rates are higher. If a bond’s duration rises and yields fall, the bond is said to have positive convexity. As yields fall, bond prices rise by a greater rate or duration than if yields rise. If a bond has positive convexity, it would typically experience price increases as yields fall, compared to price decreases when yields increase.

Understanding Bond Convexity in Excel Formulas

Note that modified duration is always slightly less than duration, since the modified duration is the duration divided by 1 plus the yield per payment period. Therefore, the convexity of the bond has changed from 13.39 to 49.44 with the change in the frequency of coupon payment from annual to semi-annual. Let us take the example of the same bond while changing the number of payments to 2 i.e. semi-annual coupon payment.

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. It is important to note that all these functions apply to all bond types supported by QuantLib that include fixed rate bonds, floating notes (both Libor and CMS) and inflation bonds. Let’s consider a 5-year bond with a face value of $1,000, a coupon rate of 5%, and a yield of 4%. Therefore, yield volatility, and therefore, interest rate risk, is greater for securities with more default risk, even if their durations are the same.

Convexity and Bond Price Volatility

Using the concept of duration, we can calculate that Bond A has a duration of 4 years while Bond B has a duration of 5.5 years. 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%. To calculate convexity in Excel, begin by designating a different pair of cells for each of the variables identified in the formula. The first cell acts as the title (P+, P-, Po and Effective Duration), and the second carries the price, which is information you have to gather or calculate from another source. To calculate convexity in Excel, begin by designating a different pair of cells for each of the variables identified in the formula.

Convexity relates to the interaction between a bond’s price and its yield as it experiences changes in interest rates. The effective duration is a metric used to assess the interest rate risk of bond, just like the effective convexity. However, instead of measuring the linear effect of interest rate changes, the metric focuses on the non-linear effects. Bond convexity is important for investors and portfolio managers, because it helps them assess the risk and return of different bonds and bond portfolios.

Therefore, portfolio managers may wish to protect (immunize) the future accumulated value of the fund at some target date, against interest rate movements. In other words, immunization safeguards duration-matched assets and liabilities, so a bank can meet its obligations, regardless of interest rate movements. Bank liabilities, which are primarily the deposits owed to customers, are generally short-term in nature, with low duration statistics. By contrast, a bank’s assets mainly comprise outstanding commercial and consumer loans or mortgages.

However, bond convexity is an approximation and relies on certain assumptions, such as a constant yield curve and small interest rate changes. It should be used in conjunction with effective duration and other risk measures for a comprehensive assessment of a bond’s interest rate risk. Convexity is the second derivative of the bond’s price with respect to the yield. A higher convexity implies a greater price change in response to interest https://personal-accounting.org/ rate fluctuations. Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function. Consequently, bonds with higher convexity will have greater capital gains for a given decrease in yields than the corresponding capital losses that would occur when yields increase by the same amount.

For example, a payment that is due in one year will be less affected by the interest rate than a payment that is due in 10 years, because the interest rate has more time to compound in the latter case. A higher result means that the price is more sensitive to changes in interest rates. Increasing convexity means the systemic risk a portfolio is exposed to increases.