Understanding Bond Convexity and Its Applications in Finance
Introduction
In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates. It represents the second derivative of the price of the bond with respect to interest rates, with duration being the first derivative. The higher the duration, the more sensitive the bond price is to changes in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance, based on the work of Hon-Fei Lai and popularized by Stanley Diller.
Calculation of Convexity
Duration is a linear measure or 1st derivative of how the price of a bond changes in response to interest rate changes. As interest rates change, the price is not likely to change linearly, but instead, it would change over some curved function of interest rates. The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity.
Convexity is a measure of the curvature or 2nd derivative of how the price of a bond varies with interest rate, i.e., how the duration of a bond changes as the interest rate changes. Specifically, one assumes that the interest rate is constant across the life of the bond and that changes in interest rates occur evenly. Using these assumptions, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question. Then the convexity would be the second derivative of the price function concerning the interest rate.
In actual markets, the assumption of constant interest rates and even changes is not correct, and more complex models are needed to price bonds. However, these simplifying assumptions allow one to quickly and easily calculate factors that describe the sensitivity of the bond prices to interest rate changes.
Convexity does not assume the relationship between Bond value and interest rates to be linear. For large fluctuations in interest rates, it is a better measure than duration.
Why Bond Convexities May Differ
The price sensitivity to parallel changes in the term structure of interest rates is highest with a zero-coupon bond and lowest with an amortizing bond (where the payments are front-loaded). Although the amortizing bond and the zero-coupon bond have different sensitivities at the same maturity, if their final maturities differ so that they have identical bond durations, then they will have identical sensitivities. That is, their prices will be affected equally by small, first-order, (and parallel) yield curve shifts. They will, however, start to change by different amounts with each further incremental parallel rate shift due to their differing payment dates and amounts.
For two bonds with the same par value, coupon, and maturity, convexity may differ depending on what point on the price yield curve they are located.
Suppose both of them have at present the same price yield (p-y) combination; also, you have to take into consideration the profile, rating, etc., of the issuers: let us suppose they are issued by different entities. Though both bonds have the same p-y combination, bond A may be located on a more elastic segment of the p-y curve compared to bond B. This means if yield increases further, the price of bond A may fall drastically, while the price of bond B won’t change; i.e., bond B holders are expecting a price rise any moment and are therefore reluctant to sell it off, while bond A holders are expecting further price-fall and are ready to dispose of it.
This means bond B has a better rating than bond A.
So the higher the rating or credibility of the issuer, the lower the convexity and the lower the gain from risk-return game or strategies. Less convexity means less price-volatility or risk; less risk means less return.
Mathematical Definition
If the flat floating interest rate is r and the bond price is B, then the convexity C is defined as:
C = (1/B) * (d²(B(r))/dr²)
Another way of expressing C is in terms of the modified duration D:
dB/dr = -DB
Therefore,
CB = d(-DB)/dr = (-D)(-DB) + (-dD/dr)(B)
Leaving:
C = D² — dD/dr
Where D is a Modified Duration
- How Bond Duration Changes with a Changing Interest Rate
Return to the standard definition of modified duration:
D = (1/(1+r)) * Σ (P(i)t(i)/B)
Where P(i) is the present value of coupon i, and t(i) is the future payment date.
As the interest rate increases, the present value of longer-dated payments declines in relation to earlier coupons (by the discount factor between the early and late payments). However, bond price also declines when interest rate increases, but changes in the present value of the sum of each coupon times timing (the numerator in the summation) are larger than changes in the bond price (the denominator in the summation). Therefore, increases in r must decrease the duration (or, in the case of zero-coupon bonds, leave the unmodified duration constant). Note that the modified duration D differs from the regular duration by the factor one over 1+r (shown above), which also decreases as r is increased.
dD/dr ≤ 0
Given the relation between convexity and duration above, conventional bond convexities must always be positive.
The positivity of convexity can also be proven analytically for basic interest rate securities. For example, under the assumption of a flat yield curve, one can write the value of a coupon-bearing bond as:
B(r) = Σ (c_i * e^(-rt_i))
Where c_i stands for the coupon paid at time t_i. Then it is easy to see that:
d²B/dr² = Σ (c_i * e^(-rt_i) * t_i²) ≥ 0
Note that this conversely implies the negativity of the derivative of duration by differentiating:
dB/dr = -DB
Application of Convexity
Convexity is a risk management figure, used similarly to the way ‘gamma’ is used in derivatives risks management; it is a number used to manage the market risk a bond portfolio is exposed to. If the combined convexity and duration of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur (parallel in the yield curve).
The second-order approximation of bond price movements due to rate changes uses the convexity:
ΔB = B * [C/2 * (Δr)² — DΔr]
Effective Convexity
For a bond with an embedded option, a yield to maturity-based calculation of convexity (and of duration) does not consider how changes in the yield curve will alter the cash flows due to option exercise. To address this, an effective convexity must be calculated numerically. Effective convexity is a discrete approximation of the second derivative of the bond’s value as a function of the interest rate:
Effective Convexity = (V_-Δy — 2V + V_+Δy) / (V_0 * Δy²)
Where V is the bond value as calculated using an option pricing model, Δy is the amount that yield changes, and V_-Δy and V_+Δy are the values that the bond will take if the yield falls by y or rises by y, respectively (a parallel shift).
These values are typically found using a tree-based model, built for the entire yield curve, and therefore capturing exercise behavior at each point in the option’s life as a function of both time and interest rates; see Lattice model (finance) § Interest rate derivatives.
Importance of Convexity in Portfolio Management
Understanding convexity is crucial for bond investors and portfolio managers, as it helps them evaluate the potential price changes in bonds due to interest rate fluctuations. By using convexity, investors can make more informed decisions about which bonds to include in their portfolios to optimize returns and minimize risk.
For instance, investors may seek bonds with higher convexity when they expect interest rates to change significantly, as these bonds tend to experience smaller price declines when rates rise and larger price increases when rates fall. Conversely, when interest rate changes are expected to be minimal, investors may opt for bonds with lower convexity, which generally offer more stable returns and less price volatility.
Convexity and Immunization
Convexity is an important factor in bond immunization strategies. Bond immunization is a technique used by fixed-income portfolio managers to minimize the risk of interest rate changes on a bond portfolio’s value by matching the portfolio’s duration with the investment horizon. In other words, immunization aims to create a portfolio that is less sensitive to interest rate fluctuations.
However, duration matching alone may not be sufficient to fully protect a portfolio from interest rate risk, especially when there are large changes in interest rates. In these cases, incorporating convexity into the immunization strategy can help further shield the portfolio from rate fluctuations. By selecting bonds with higher convexity, the portfolio will be better equipped to handle interest rate changes, as the bond prices will react more favorably to both increases and decreases in rates.
Limitations of Convexity
While convexity is a valuable tool for understanding the interest rate sensitivity of bonds, it is not without limitations. The primary limitation of convexity is that it is based on simplifying assumptions, such as constant interest rates and even changes throughout the life of the bond. In reality, interest rates are rarely constant, and changes in rates can be uneven.
Furthermore, convexity calculations are typically based on a bond’s yield to maturity, which may not accurately represent the bond’s actual cash flows if the bond has embedded options or other features that can change its cash flow structure. In these cases, the use of effective convexity can help to account for these complexities, although it requires more advanced modeling techniques and calculations.
Despite these limitations, convexity remains a fundamental and widely used concept in finance, providing valuable insights into the behavior of bonds and fixed-income securities in the face of interest rate changes.
Convexity Hedging
In addition to being a valuable tool for understanding and managing interest rate risk, convexity can also be used for hedging purposes. Convexity hedging involves taking positions in other financial instruments, such as interest rate swaps, options, or other bonds, to offset the convexity risk inherent in a bond portfolio.
For example, a portfolio manager may hold a collection of callable bonds that have negative convexity due to the issuer’s option to redeem the bonds before maturity. In this case, the manager may choose to hedge the negative convexity risk by purchasing options or other financial instruments with positive convexity. This can help to offset the potential price declines associated with the callable bonds in the event of rising interest rates.
Convexity and the Yield Curve
Convexity is also important when analyzing the shape of the yield curve. The yield curve is a graphical representation of the relationship between interest rates and the maturity of different bonds issued by the same entity. When the yield curve is steep, long-term rates are significantly higher than short-term rates, indicating that investors require a higher return for holding longer-dated bonds. In contrast, a flat yield curve suggests that long-term rates are not much higher than short-term rates.
Since convexity measures the curvature of the bond price function in relation to interest rates, it can help investors understand how changes in the yield curve will impact the value of their bond portfolios. For example, if the yield curve becomes steeper, bonds with higher convexity will generally experience larger price increases compared to bonds with lower convexity. Conversely, if the yield curve flattens, bonds with higher convexity may experience smaller price declines compared to bonds with lower convexity.
Conclusion
Bond convexity is an essential concept in finance, offering a deeper understanding of the relationship between bond prices and interest rates. By measuring the curvature or second derivative of the bond price function with respect to interest rates, convexity provides valuable insights into the sensitivity of bond prices to interest rate changes.
Investors and portfolio managers can use convexity to make informed decisions about which bonds to include in their portfolios, to optimize returns and minimize risk. Convexity also plays a crucial role in bond immunization strategies and hedging techniques, helping to protect portfolios from interest rate fluctuations.
Despite its limitations and reliance on simplifying assumptions, convexity remains a fundamental and widely used concept in the world of finance, providing critical insights into the behavior of bonds and fixed-income securities in the face of changing interest rates.
