# Relationship between convexity and duration definition

### Bond convexity - Wikipedia

It should not be surprising that there is a relationship between the change in . the effective duration of an option-embedded bond is defined as the change in. We now relate volatility to duration by examining their defining expressions. . (b ) Estimate the price of the bond using the approximation formula on page Convexity is a measure of the curvature in the relationship between bond If a bond's duration increases as yields increase, the bond is said to Some examples of bonds that exhibit negative convexity are bonds with a.

## Duration & Convexity: The Price/Yield Relationship

In the above graph Bond A is more convex than Bond B even though they both have the same duration and hence Bond A is less affected by interest rate changes. 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 bond with a higher convexity has 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 a 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.

### What is the difference between duration and convexity of a bond? - Quora

Positive and Negative Convexity Convexity can be positive or negative. A bond has positive convexity if the yield and the duration of the bond increase or decrease together, i.

The yield curve for this typically moves upward. This typical is for a bond which does not have a call option or a prepayment option. Bonds have negative convexity when the yield increases the duration decreases i.

## Convexity of a Bond | Formula | Duration | Calculation

These are typically bonds with call optionsmortgage-backed securities and those bonds which have a repayment option. If the bond with prepayment or call option has a premium to be paid for the early exit then the convexity may turn positive. The coupon payments and the periodicity of the payments of the bond contribute to the convexity of the bond. If there are more periodic coupon payments over the life of the bond then the convexity is higher making it more immune to interest rate risks as the periodic payments help in negating the effect of the change in the market interest rates.

If there is a lump sum payment then the convexity is the least making it a more risky investment. Convexity of a Bond Portfolio For a bond portfolio the convexity would measure the risk of the all the bonds put together and is the weighted average of the individual bonds with no of bonds or the market value of the bonds being used as weights.

Even though Convexity takes into account the non-linear shape of price-yield curve and adjusts for the prediction for price change there is still some error left as it is only the second derivative of the price-yield equation.

To get a more accurate price for a change in yield, adding the next derivative would give a price much closer to the actual price of the bond. Today with sophisticated computer models predicting prices, convexity is more a measure of the risk of the bond or the bond portfolio. More convex the bond or the bond portfolio less risky it is as the price change for a reduction in interest rates is less. So bond which is more convex would have a lower yield as the market prices in the lower risk.

The duration for a bond portfolio is equal to the weighted average of the duration for each type of bond in the portfolio: Minimize Duration Risk When yields are low, investors, who are risk-averse but who want to earn a higher yield, will often buy bonds with longer durations, since longer-term bonds pay higher interest rates.

But even the yields of longer-term bonds are only marginally higher than short-term bonds, because insurance companies and pension funds, who are major buyers of bonds, are restricted to investment grade bonds, so they bid up those prices, forcing the remaining bond buyers to bid up the price of junk bondsthereby diminishing their yield even though they have higher risk.

Indeed, interest rates may even turn negative. In Junethe year German bond, known as the bund, sported negative interest rates several times, when the price of the bond actually exceeded its principal.

Interest rates vary continually from high to low to high in an endless cycle, so buying long-duration bonds when yields are low increases the likelihood that bond prices will be lower if the bonds are sold before maturity.

### Convexity of a Bond | Formula | Duration | Calculation

This is sometimes called duration risk, although it is more commonly known as interest rate risk. 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.

**Quant Crash Course - Q&A: The relationship between Optionality, Convexity and Volatility**

Therefore, especially when yields are extremely low, as they were starting in and continuing even intoit is best to buy bonds with the shortest durations, especially when the difference in interest rates between long-duration portfolios and short-duration portfolios is less than the historical average. On the other hand, buying long-duration bonds make sense when interest rates are high, since you not only earn the high interest, but you may also realize capital appreciation if you sell when interest rates are lower.

Convexity Duration is only an approximation of the change in bond price. 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.