# Convexity and duration relationship

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

We can derive the relationship between a change in the yield to maturity and the change in the market value of a standard fixed-income bond using a bit of. Duration & Convexity: The Price/Yield Relationship. Investors who own fixed income securities should be aware of the relationship between interest rates and a. In finance, bond convexity is a measure of the non-linear relationship of bond prices to.

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. 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.

## Duration and Convexity

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. 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.

### Duration & Convexity - Fixed Income Bond Basics | Raymond James

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. 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.

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.

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.

This is because duration is a tangent line to the price-yield curve at the calculated point, and the difference between the duration tangent line and the price-yield curve increases as the yield moves farther away in either direction from the point of tangency.

A diagram of the convexity of 2 representative bond portfolios, showing the general relationship between the percentage change in the value of bond portfolios to a change in yield.

## Duration & Convexity: The Price/Yield Relationship

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.

Convexity is always positive for vanilla bonds. Furthermore, the price-yield curve flattens out at higher interest rates, so convexity is usually greater on the upside than on the downside, so the absolute change in price for a given change in yield will be slightly greater when yields decline rather than increase.