Because cash flows can change, the effective duration of an option-embedded bond is defined as the change in bond price per change in the market interest rate:. Note that i is the change in the term structure of interest rates and not the yield to maturity for the bond, because YTM is not valid for an option-embedded bond when the future cash flows are uncertain. There are several formulas for calculating the duration of specific bonds that are simpler than the above general formula.

The duration of a fixed annuity for a specified number of payments T and yield per payment y can be calculated with the following formula:. A perpetuity is a bond that does not have a maturity date, but pays interest indefinitely. Although the series of payments is infinite, the duration is finite, usually less than 15 years. The formula for the duration of a perpetuity is especially simple, since there is no principal repayment:. 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:. To better measure the interest rate exposure of a portfolio, it is better to measure the contribution of the issue or sector duration to the portfolio duration rather than just measuring the market value of that issue or sector to the value of the portfolio:.

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.

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 bonds , thereby diminishing their yield even though they have higher risk. Indeed, interest rates may even turn negative. In June , the 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.

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Duration risk would be especially large in buying bonds with negative interest rates. 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 into , it 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.

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.

## How to Calculate Convexity of a Bond | pynuzuzyvogu.cf

Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1 st derivative to the equation for the duration and the 2 nd derivative to the equation for the price-yield function. Convexity is always positive for vanilla bonds. Furthermore, the price-yield curve flattens out at higher interest rates, so convexity is usually greater on the upside than on the downside, so the absolute change in price for a given change in yield will be slightly greater when yields decline rather than increase.

Consequently, bonds with higher convexity will have greater capital gains for a given decrease in yields than the corresponding capital losses that would occur when yields increase by the same amount.

## Semi-annual coupon bond vs Annual Coupon bond present value

Note, however, that this convexity approximation formula must be used with this convexity adjustment formula, then added to the duration adjustment:. Important Note! The convexity can actually have several values depending on the convexity adjustment formula used. Many calculators on the Internet calculate convexity according to the following formula:.

Note that this formula yields double the convexity as the Convexity Approximation Formula 1. However, if this equation is used, then the convexity adjustment formula becomes:. As you can see in the Convexity Adjustment Formula 2 that the convexity is divided by 2, so using the Formula 2's together yields the same result as using the Formula 1's together. To add further to the confusion, sometimes both convexity measure formulas are calculated by multiplying the denominator by , in which case, the corresponding convexity adjustment formulas are multiplied by 10, instead of just !

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PompeyCrassus Apr 7th, am. Studying With. Is this right?

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Harrogath Apr 7th, am. GrahamDoddFisher Apr 7th, am. PompeyCrassus Apr 7th, pm. If I use the effective semi-annual rate for discounting I get the following. Smagician Apr 7th, pm. Just … wow! PompeyCrassus wrote:. Simplify the complicated side; don't complify the simplicated side.

PompeyCrassus Apr 8th, pm. Smagician Apr 8th, pm. Harrogath Apr 8th, pm. That explained it perfectly. I want to apologize for such a foolish post I made. Indeed I was wrong on my response fueled by a big confusion of mine. Skip to main content.

Be prepared with Kaplan Schweser. Twitter Facebook LinkedIn. Search form. PompeyCrassus Apr 7th, am. Studying With. Is this right? Harrogath Apr 7th, am. GrahamDoddFisher Apr 7th, am. PompeyCrassus Apr 7th, pm. If I use the effective semi-annual rate for discounting I get the following. Smagician Apr 7th, pm.

### Bond Convexity Formula

Just … wow! PompeyCrassus wrote:. Simplify the complicated side; don't complify the simplicated side.