### Default risk and the duration of zero coupon bonds

The formula for convexity approximation is as follows:. As can be seen from the formula Convexity is a function of the bond price, YTM Yield to maturity , Time 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.

## Zero Coupon Bond (Definition, Formula, Examples, Calculations)

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. 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. 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.

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 options , mortgage-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. 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.

If there is a lump sum payment then the convexity is the least making it a more risky investment. 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.

Locate a bond ETF's duration from either the Snapshot page or Key Statistics, where the duration of the specific ETF can be compared to the asset class median duration.

Locate a bond's duration under each bond's Bond Details page. Compare the duration of two bonds. As you review potential bond investments, you can easily compare duration and other characteristics between two bonds using this tool. Select from a variety of individual bond and bond funds that may meet your investing needs. Bonds or bond funds are fixed income investments that generally pay a set rate of interest over a fixed time period.

In general, the bond market is volatile, and fixed income securities carry interest rate risk. As interest rates rise, bond prices usually fall, and vice versa. This effect is usually more pronounced for longer-term securities. Fixed income securities also carry inflation risk, liquidity risk, call risk, and credit and default risks for both issuers and counterparties. Any fixed income security sold or redeemed prior to maturity may be subject to loss.

A bond ladder, depending on the types and amount of securities within it, may not ensure adequate diversification of your investment portfolio. While diversification does not ensure a profit or guarantee against loss, a lack of diversification may result in heightened volatility of your portfolio value.

You must perform your own evaluation as to whether a bond ladder and the securities held within it are consistent with your investment objectives, risk tolerance, and financial circumstances. To learn more about diversification and its effects on your portfolio, contact a representative. Votes are submitted voluntarily by individuals and reflect their own opinion of the article's helpfulness.

### Bond Risk Basics

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## Duration: Understanding the relationship between bond prices and interest rates

This material was developed and produced by FMG Suite to provide information on a topic that may be of interest. The opinions expressed and material provided are for general information, and should not be considered a solicitation for the purchase or sale of any security. Copyright FMG Suite.