What is Negative Convexity?
Negative convexity exists when the shape of a bond’s yield curve is concave. A bond’s convexity is the rate of change of its duration and is measured as the second derivative of the bond’s price concerning its yield. Many mortgage bonds exhibit negative convexity, and callable bonds often show negative convexity at lower yields.
Key Takeaways
- Negative convexity occurs when both the bond’s price and interest rates fall, resulting in a concave yield curve.
- Assessing a bond’s convexity provides valuable insights for measuring and managing a portfolio’s exposure to market risk.
Delving into Negative Convexity
A bond’s duration indicates the sensitivity of its price to fluctuations in interest rates. Convexity measures how this duration changes as interest rates vary. Typically, bond prices increase when interest rates decrease. However, for bonds with negative convexity, their prices may decrease as interest rates fall.
For instance, a callable bond enhances the issuer’s likelihood of calling the bond at par value as interest rates drop. Consequently, the bond’s price doesn’t increase as rapidly as a non-callable bond’s price. In some cases, the price of a callable bond can decline with rising chances of being called, leading to a negatively convex or concave price-yield curve.
Example of Convexity Calculation
Given that duration alone is an imperfect indicator of price change, calculating a bond’s convexity becomes essential. This serves as a key risk-management tool for better portfolio risk exposure assessments, allowing for more accurate price-movement predictions.
Although the actual formula for convexity can be complex, the approximation formula can be simplified as follows:
Convexity approximation = (P(+) + P(-) - 2 x P(0)) / (2 x P(0) x dy^2)
Where:
- P(+) = bond price when the interest rate decreases
- P(-) = bond price when the interest rate increases
- P(0) = current bond price
- dy = change in the interest rate in decimal form
Here’s a concrete example:
Let’s assume a bond is priced at $1,000. If interest rates fall by 1%, the bond’s price increases to $1,035. Conversely, if interest rates rise by 1%, the bond’s price drops to $970. The approximate convexity is calculated as:
Convexity approximation = ($1,035 + $970 - 2 x $1,000) / (2 x $1,000 x 0.01^2) = $5 / $0.2 = 25
To incorporate this into an estimate of the bond’s price using duration with a convexity adjustment, the formula is:
Convexity adjustment = convexity x 100 x (dy)^2
For our example:
Convexity adjustment = 25 x 100 x (0.01)^2 = 0.25
Finally, to estimate the bond’s price change for a given alteration in interest rates using duration and convexity, you can apply:
Bond price change = duration x yield change + convexity adjustment
Related Terms: duration, yield curve, callable bond, mortgage bond, convexity adjustment.
References
Get ready to put your knowledge to the test with this intriguing quiz!
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## What does negative convexity refer to in financial terms?
- [ ] The upward sloping feature of a bond's price curve
- [ ] A guarantee that bond prices will never fall
- [ ] Consistent, linear changes in bond prices
- [x] The phenomenon where the price of a bond falls as interest rates decline, but not in the same proportion
## Which type of financial instrument is most commonly associated with negative convexity?
- [ ] Corporate stocks
- [x] Mortgage-backed securities
- [ ] Floating rate bonds
- [ ] Certificates of deposit
## What happens to the price of a bond with negative convexity when interest rates drop?
- [x] It increases at a decreasing rate
- [ ] It decreases sharply
- [ ] It remains constant
- [ ] It keeps increasing at the same rate
## Why do mortgage-backed securities often exhibit negative convexity?
- [ ] Due to their floating interest rates
- [x] Because homeowners may prepay their mortgages when interest rates fall
- [ ] Because of high default rates in the market
- [ ] Due to fixed interest rate coupons
## Negative convexity is most concerning for which type of investor?
- [ ] Equity holder
- [ ] Money market investor
- [x] Fixed-income investor
- [ ] Derivatives trader
## What effect does negative convexity have on a bond's duration as interest rates change?
- [ ] It makes the duration infinite
- [ ] It has no effect on duration
- [x] It decreases bond duration when rates fall and increases bond duration when rates rise
- [ ] It makes duration increase continuously
## In the context of bond investing, what does negative convexity imply about the risk/return profile?
- [x] It implies a more complex risk/return profile due to varying sensitivity to interest rates
- [ ] It implies simple, predictive returns based solely on interest rate movement
- [ ] It implies a decrease in market risk
- [ ] It implies higher overall returns without increased risk
## Which of the following can mitigate the impacts of negative convexity in a fixed-income portfolio?
- [ ] Increasing the number of floating-rate notes
- [ ] Excluding all mortgage-backed securities
- [x] Using hedging strategies such as interest rate swaps
- [ ] Investing solely in long-duration bonds
## What characteristic of callable bonds often leads to negative convexity?
- [x] The option for the issuer to repay the bond before maturity when interest rates drop
- [ ] Their fixed interest rates
- [ ] The issuer's credit rating
- [ ] Long maturity periods
## Compared to bonds with positive convexity, what key disadvantage do bonds with negative convexity face?
- [ ] A higher yield spread
- [ ] Greater tax liabilities
- [ ] Lower coupon payments over time
- [x] Increased price volatility in response to interest rate changes
