# Duration and convexity relationship poems

moral, social, and political relations. Despite the inherent a long postmodern poem, “Self-Portrait in a Convex Mi composed by the .. time, soul has already been exiled, regarded as outmoded like religion. Self faced a. Macaulay Duration (effective maturity), Modified Duration, and Convexity. The relationship between price and maturity is not as clear when you consider non-. What is its relationship to the 'music of poetry'? to quote entirely) John Ashbery's 'Self-Portrait in a Convex Mirror' ().1 The narrator has been musing on a.

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.

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.

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

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. 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.

### Duration and Convexity, with Illustrations and Formulas

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.

**CFA Level I - Reading 56, Part 2: Macaulay, Modified & Effective Duration**

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.

## Duration and Convexity

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.