This means that because investors will now get a yield of 3% on newly issued bonds, they won’t be interested in your 2% bonds. As such, the value of your bonds would go down as you would be required to sell them at a discount. So now that we’ve covered bond yields, we need to look at how interest rates have a major influence on the value of bonds, and in turn, its convexity.
So bonds, which are more convex, would have a lower yield as the market prices are lower risk. Convexity is a good measure for bond price changes with greater fluctuations in the interest rates. Mathematically speaking, convexity is the second derivative of the formula for change in bond prices with a change in interest rates and a first derivative of the duration equation. For bond convexity, the interest rates and bond prices move in the opposite direction. However, investor’s expectations and other factors contribute towards the final market price of the bond as well. Bond convexity is used to measure the risk for investors that arises due to a change in yield for investors.
The usual formulas for calculating Macaulay duration, modified duration, and convexity are given in the table below. There are closed-form solutions for these equations, but they are fairly long and ugly. In other words, they aren’t really suited for a spreadsheet solution without resorting to VBA.
By using both duration and convexity, they can obtain a more accurate approximation of the price change than using duration alone. For example, when yields fall bond prices rise, but due to convexity the price does increase more steeply than a linear calculation using the modified duration would estimate. On the other hand, when yields rise bond prices fall, but not as steeply as predicted by a linear model. Unlike duration, which provides a linear estimate, convexity accounts for the curvature in the price-yield relationship, offering a more precise risk assessment. Investors and portfolio managers use convexity to refine strategies and manage exposure to market fluctuations. Convexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change.
A bond with a higher convexity has a larger price change when the interest rate drops than a bond with lower convexity. Hence when two similar bonds are evaluated for investment with similar yield and duration, the one with higher convexity is preferred in stable or falling interest rate scenarios as price change is larger. In a falling interest rate scenario again, a higher convexity would be better as the price loss for an increase in interest rates would be smaller.
Duration risk would be especially large in buying bonds with negative interest rates. 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. Convexity of bonds with a put option is positive, while that of a bond with a call option is negative. This is because when a put option is in the money, then if the market goes down, you can put the bond, or if the market goes up, you preserve all the cash flows. However, for a bond with a call option, the issuer would call the bond if the market interest rate decreases, and if the market rate increases, the cash flow would be preserved.
Using the Price Function
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. Convexity of a bond is the phenomena that causes the increase in bond price due to a decrease in interest rates to be higher than the decrease in bond price owing to an increase in interest rates. From the table, we can see that Bond B has a higher convexity than Bond A, even though they have similar durations. This is because Bond B pays more frequent coupons, which increases its cash flow and reduces its interest rate risk.
Why bond convexities may differ
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%. However, using the concept of convexity, we can predict that the price change for Bond B will be less than expected based on its duration alone. This is because Bond B has a longer maturity, which means it has a higher convexity. The higher convexity of Bond B acts as a buffer against changes in interest rates, resulting in a relatively smaller price change than expected based on its duration alone. Generally, a change of 1% in interest rate annually would mean a change of 1% in bond prices.
- Simply put, a higher duration implies that the bond price is more sensitive to rate changes.
- Convexity is influenced by several factors such as coupon size and maturity and is widely used as a risk management tool and/or to enhance potential returns.
- 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.
Therefore, yield volatility, and therefore, interest rate risk, is greater for securities with more default risk, even if their durations are the same. Higher degree of convexity means that the bond is less affected by interest rate volatility relative to low convexity bonds. 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. The numerator accounts for the time-weighted impact of each cash flow, while the denominator discounts these payments to present value.
However, it actually relates to the sensitivity of the bond price in relation to a potential change of interest rates. If the convexity of a bond is positive, it means that the bond duration increases as the yield falls. We can see that Bond B has a smaller price decline and a larger price increase than Bond A, due to its higher convexity. This shows that Bond B has a more favorable risk-return profile than Bond A, as it benefits more from a decrease in yield and suffers less from an increase in yield.
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. 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.
Hence, if the duration of the bond increases with an increase in the yield, it will have negative convexity. It means that bond prices would decline more with an increase in yield than they would with a fall in the yield. Convexity is used to assess the risk for investors that comes with a rise or fall in the interest rate in the bond’s price. A bond convexity can be negative or positive with a change in yield for investors. This means that the bond price will decrease by 5.25% instead of 5% as estimated by the duration approximation.
Macaulay Duration
It is possible for a bond price to exhibit negative convexity, although this is usually an exception rather than the rule. Negative curvature typically occurs in the context of callable bonds where the issuer has the option of early repayment. Convexity is a measure of the curvature of its duration or the relationship between bond prices and yields.
If a bond has a negative convexity, it operates much like its positive counterpart, but in reverse. Ultimately, the higher the bond duration, the higher the risks for the investor. This is because the bonds will react in a more volatile manner if interest rates change.
Should I Attempt to Calculate the Bond Convexity?
It describes how the duration of a bond changes in response to changes in interest rates. Under normal market conditions, the higher the coupon rate or yield, the lower a bond’s degree of convexity. There’s less risk to the investor when the bond has a high coupon or yield since market rates would have to increase significantly to surpass the bond’s yield. A portfolio of bonds with high yields would have low convexity and subsequently less risk of existing yields becoming unattractive as interest rates rise.
- The bond price will decline by a greater rate with a rise in yields than if yields had fallen.
- 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.
- A portfolio of bonds with high yields would have low convexity and subsequently less risk of existing yields becoming unattractive as interest rates rise.
- As we know, the bond price and the yield are inversely related, i.e., as yield increases, the price decreases.
Where P(i) is the present value of coupon i, and t(i) is the future payment date. In the dynamic arena of marketing, the symphony of a campaign’s success is orchestrated by the…
Do interest rates impact the value of bonds?
On the other hand, zero-coupon bonds always exhibited the same interest rate sensitivity. One risk management strategy that is popular among financial institutions (and insurers in particular) to mitigate the potential risks that can arise from convexity is convexity hedging. Instruments such as mortgage-backed securities (MBS) are convexity formula known for having negative convexity, due to their embedded prepayment option, and can fall more rapidly when interest rates rise and achieve lower return when rates fall.
