# Convexity and duration relationship counseling

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

Mathematically, duration is the 1st derivative of the price-yield curve, which is a line tangent to the curve at the current price-yield point. Although the effective. 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. In this article, we will cover the Duration and Convexity measure. Duration assumes a linear relationship between bond price and yield.

Indeed, interest rates may even turn negative.

## Duration & Convexity: The Price/Yield Relationship

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.

• FRM-I “Duration” Tutorial: Master the Art of Calculating Duration & Convexity
• Convexity of a Bond | Formula | Duration | Calculation
• Duration and Convexity

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.

### FRM-I “Duration” Tutorial: Master the Art of Calculating Duration & Convexity - EduPristine

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. 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. One important type of risk is the interest rate risk.

One sudden change in the interest rates can be potentially devastating to many financial institutions. This is because the value of fixed income securities like bonds depends on the interest rates i.

### Duration and Convexity, with Illustrations and Formulas

Fortunately, one can estimate to a fair degree of accuracy the change in the value of a bond given a change in the interest rate. In this article, we will cover the Duration and Convexity measure. Duration has three main variants: Here the average is a weighted average and not the simple arithmetic mean.

The weight for each coupon period is determined by dividing the present value of the payment in that period by the present value of all payments which is the bond price. The present value of the payment is nothing but the bond price.

A 1 basis point change in interest rates will lead to a change of 5 basis points in the price of a zero coupon bond maturing in 5 years! We can generalize this concept and apply it to coupon paying bonds also.

The formula is given below; The equation seems complicated and can put you off, but no need to worry! The calculation of duration as per the above formula is included below. You can also build your own Excel in attachment and will be given to students attending webinar sheet quite easily. As per figure 1. So according to definition, any change in interest rate should reflect 1.