# Convexity and duration relationship test

### Price Change of a Bond - Duration - Convexity | CFA Level 1 - AnalystPrep

For any given bond, a graph of the relationship between price and yield is convex . When used together, duration and convexity offer a better approximation of. Bond Convexity. Quiz Duration is a first approximation of a bond's price or a of the price/yield relationship in order to give a more accurate estimated price. Above is the bond with a year maturity. Look at how curved — i.e., how convex — the graph of the price-yield relationship is! Notice also that.

### Please explain Duration and Convexity | AnalystForum

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: Understanding the relationship between bond prices and interest rates

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 Rate Risk and Convexity Risk measurement for a bond involves a number of risks. These include but are not limited to: This interest rate risk is measured by modified duration and is further refined by convexity.

Convexity is a measure of systemic risk as it measures the effect of change in the bond portfolio value with larger change in the market interest rate while modified duration is enough to predict smaller changes in interest rates. As mentioned earlier convexity is positive for regular bonds but for bonds with options like callable bondsmortgage backed securities which have prepayment option the bonds have negative convexity at lower interest rates as the prepayment risk increases.

For such bonds with negative convexity prices do not increase significantly wit decrease in interest rates as cash flows change due to prepayment and early calls. As the cash flow is more spread out the convexity increases as the interest rate risk increases with more gap in between the cash flows.

## Please explain Duration and Convexity

So convexity as a measure is more useful if the coupons are more spread out and are of lesser value. If we have a zero-coupon bond and a portfolio of zero coupon bonds, the convexity are as follows: However, the convexity of this portfolio is higher than the single zero coupon bond. This is because the cash flows of the bonds in the portfolio are more dispersed than that of a single zero coupon bond.

Convexity of bonds with a put option is positive while that of a bond with call option is negative. Due to the possible change in cash flows, the convexity of the bond is negative as interest rates decrease. The measured convexity of the bond when there is no expected change in future cash flows is called modified convexity.

When there are changes expected in the future cash flows the convexity that is measured is the effective convexity. However, Treasury bonds as well as other types of fixed income investments are sensitive to interest rate risk, which refers to the possibility that a rise in interest rates will cause the value of the bonds to decline. Bond prices and interest rates move in opposite directions, so when interest rates fall, the value of fixed income investments rises, and when interest rates go up, bond prices fall in value.

If rates rise and you sell your bond prior to its maturity date the date on which your investment principal is scheduled to be returned to youyou could end up receiving less than what you paid for your bond.

Similarly, if you own a bond fund or bond exchange-traded fund ETFits net asset value will decline if interest rates rise. The degree to which values will fluctuate depends on several factors, including the maturity date and coupon rate on the bond or the bonds held by the fund or ETF. Using a bond's duration to gauge interest rate risk While no one can predict the future direction of interest rates, examining the "duration" of each bond, bond fund, or bond ETF you own provides a good estimate of how sensitive your fixed income holdings are to a potential change in interest rates.

Investment professionals rely on duration because it rolls up several bond characteristics such as maturity date, coupon payments, etc.

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

Duration is expressed in terms of years, but it is not the same thing as a bond's maturity date. That said, the maturity date of a bond is one of the key components in figuring duration, as is the bond's coupon rate. In the case of a zero-coupon bond, the bond's remaining time to its maturity date is equal to its duration.

When a coupon is added to the bond, however, the bond's duration number will always be less than the maturity date. The larger the coupon, the shorter the duration number becomes. Generally, bonds with long maturities and low coupons have the longest durations.

These bonds are more sensitive to a change in market interest rates and thus are more volatile in a changing rate environment. Conversely, bonds with shorter maturity dates or higher coupons will have shorter durations. Bonds with shorter durations are less sensitive to changing rates and thus are less volatile in a changing rate environment.

Why is this so? Because bonds with shorter maturities return investors' principal more quickly than long-term bonds do.

Therefore, they carry less long-term risk because the principal is returned, and can be reinvested, earlier. This hypothetical example is an approximation that ignores the impact of convexity; we assume the duration for the 6-month bonds and year bonds in this example to be 0. Duration measures the percentage change in price with respect to a change in yield.

FMRCo Of course, duration works both ways. If interest rates were to fall, the value of a bond with a longer duration would rise more than a bond with a shorter duration.