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 interest rate risk is a universal risk for all bond holders as all increase in interest rate would reduce the prices and all decrease in interest rate would increase the price of the bond.
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 bonds , mortgage 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.
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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. 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:. 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.
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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. Convexity arises due to the shape of the price-yield curve. If the market yield graph were flat and all shifts in prices were parallel shifts then more convex the portfolio, the better it would perform and there would be no place for arbitrage.
However as the yield graph is curved, for long-term bonds, the price yield curve is hump-shaped to accommodate for the lower convexity in the later term. Your email address will not be published. Save my name, email, and website in this browser for the next time I comment.
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The Coupon Effect on Yield to Maturity
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