# Duration and convexity relationship help

Duration assumes linear relationship between bond price and interest rate changes. What this means is that for a given change in the interest rate in either. 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. Macaulay Duration (effective maturity), Modified Duration, and Convexity. Each of these The relationship between price and maturity is not as clear when you consider non-zero coupon bonds. .. They are relative risk measures that help.

If only modified duration is used: This difference of 1. Convexity Approximation Formula As seen in the convexity calculation can be quite tedious and long especially f the bond is long term and has numerous cash flows. The formula for convexity approximation is as follows: Convexity and Risk Management As can be seen from the formula Convexity is a function of the bond price, YTM Yield to maturityTime to maturity and the sum of the cash flows.

The number of coupon flows cash flows change the duration and hence the convexity of the bond. The duration of a zero bond is equal to its time to maturity but as there still exists a convex relationship between its price and yield, zero coupon bonds have the highest convexity and its prices most sensitive to changes in yield.

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.

- Duration & Convexity: The Price/Yield Relationship

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.

**Duration and Convexity**

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.

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

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. Portfolio Duration Duration is an effective analytic tool for the portfolio management of fixed-income securities because it provides an average maturity for the portfolio, which, in turn, provides a measure of interest rate risk to the portfolio.

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.

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

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.

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.

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

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.