CFA students and MBA students that specialize in corporate finance will learn about managing fixed income or bond portfolios. This article is one part of a series on fixed income portfolios. The management of bond portfolios or fixed income portfolios introduces several unique challenges; among the most important is the ability to determine the risk associated with fixed income instruments. Three of the most important measures of interest rate risk are known as:. These measures also called interest rate risk measures quantify the bond portfolio risk.

- Zero Coupon Bond Effective Yield
- Duration: Understanding the relationship between bond prices and interest rates
- Duration and Convexity
- Advanced Bond Concepts: Duration
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- Bond duration
- Convexity of a Bond | Formula | Duration | Calculation
- Modified Duration
- Zero Coupon Bond
- Effective duration of a zero

## Zero Coupon Bond Effective Yield

By Sunita Sethi Leave a Comment. Zero Coupon Bond Also known as Pure Discount Bond or Accrual Bond refers to those bonds which are issued at a discount to its Par Value and makes no periodic interest payment unlike a normal coupon bearing Bond. In other words, its annual implied interest payment is included in its Face Value which is paid at the maturity of such bond.

Therefore this Bond is the one where the sole return is the payment of the nominal value on maturity. The difference between the current price of the bond i. These Bonds avoid the risk of Reinvestment of Coupon Bonds as Interest Rates keep changing with the passage of time which impacts the Yield to Maturity of such coupon bearing Bonds. Since there are no interim cash flows, the investor is assured of a fixed rate of return. Usually, these Bonds are issued for a longer time frame which can be used by a potential investor to align with their life goals such as Marriage, Children Education, and retirement and so on.

Thus a smart investor based on their time horizon can invest in different maturity Zero-coupon Bonds by paying a smaller amount initially as Zero-coupon Bonds are issued at deep discounts one can buy more with lesser amount and stagger them as per their career and life goals without getting impacted by the volatility. Not all Zero-coupon Bonds have a ready secondary market which results in illiquidity.

Furthermore, in case of any urgent need funds, it is difficult to liquidate the same without getting a major haircut in value. They have a single cash inflow for the Investor which happens at the maturity and as such these bonds have the greatest Duration which results in Interest Rate Risk. Further, These are issued with call provisions which allow the issuer of such Bonds to redeem the bonds prior to their maturity at dates and prices which are predetermined at the time of issue of such Bonds.

In such cases, the Investor is left with the risk of reinvesting the proceeds at the rates available at the time of redemption which will obviously be less than the earlier slated yield on the redeemed bonds. Furthermore one has to pay tax on the accrued interest on such bonds every year. However, it is pertinent to note here that there are certain categories of Zero Coupon Bonds which can overcome the taxation problem.

This has been a guide to what is Zero Coupon Bond. You can learn more about from the following articles —. Filed Under: Your email address will not be published. Save my name, email, and website in this browser for the next time I comment. Free Excel Course. Free Accounting Course. By continuing above step, you agree to our Terms of Use and Privacy Policy. Free Valuation Course.

Download Colgate s Financial Model. Download Colgate Ratio Analysis Template. Free Investment Banking Course. Login details for this Free course will be emailed to you. What is Zero Coupon Bond? Such bonds possess the greatest duration which is equivalent to the maturity of such bonds and as such are subject to the greatest level of Interest Rate Risk. Since the Interest accrued is discounted from the Par value of such Bonds at purchase which effectively enables Investors of Zero Coupon Bonds to buy a greater number of such bonds compared to any other Coupon Bearing Bond.

Zero Coupon Bond Formula. Popular Course in this category. Leave a Reply Cancel reply Your email address will not be published. It refers to fixed Income security which involves regular payment in the form of coupons and may be issued at a discount or premium depending upon market dynamism. Involves the greatest level of Interest Rate Risk due to the high duration of the Bond. There is no Reinvestment Risk in a Zero-coupon Bond as there are no cash flows during the life of the Bond.

Suffers from Reinvestment Risk due to regular cash flow in the form of coupon payments during the life of the Bond.

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

Remember that interest rates and bond prices move in opposite directions: When interest rates rise, bond prices fall, and vice versa. Zero-Coupon Bond. Duration is equal to its time to maturity. Plain Vanilla Bond.

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The zero coupon bond effective yield formula is used to calculate the periodic return for a zero coupon bond, or sometimes referred to as a discount bond. A zero coupon bond is a bond that does not pay dividends coupons per period, but instead is sold at a discount from the face value. Although no coupons are paid periodically, the investor will receive the return upon sell assuming that the rates remain constant or upon maturity. The zero coupon bond effective yield formula shown up top takes into consideration the effect of compounding. For example, suppose that a discount bond has five years until maturity.

### Duration and Convexity

By Dheeraj Vaidya Leave a Comment. Fixed Income Tutorials. Duration of a bond is the linear relationship between the bond price and interest rates where, as interest rates increase bond price decreases. Simply put, a higher duration implies that the bond price is more sensitive to rate changes. For a small and sudden change in bond yield duration is a good measure of the sensitivity of the bond price.

### Advanced Bond Concepts: Duration

Duration is a measure of a bond s sensitivity to interest rate changes. The higher the bond s duration, the greater its sensitivity to changes in interest rates also known as volatility and vice versa. The formula is complicated, but what it boils down to is: By using the present value formula, we can find PV of Cash Flows for each period. And so on until we make up the following table. Notice in the table above that we first weighted the cash flows by the periods in which the occurred and then calculated the present value of each of these weighted cash flows. As mentioned earlier, duration can help investors understand how sensitive a bond is to changes in prevailing interest rates. By multiplying a bond s duration by the change, the investor can estimate the percentage price change for the bond. For example, consider the Company XYZ bonds with a duration of 5. Why would the bond be selling for a discount?

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## Bond duration

Our forum contains hundreds of questions about duration. Duration often vexes new candidates, in part because there are several types. Duration estimates changes in the bond price assuming that variables other than the yield-to-maturity or benchmark rates are held constant. Most importantly, the time-to-maturity is unchanged. Therefore, duration measures the instantaneous or, at least, same-day change in the bond price. The accrued interest is the same, so it is the flat price that goes up or down when the full price changes. Duration is a useful measure because it represents the approximate amount of time a bond would have to be held for the market discount rate at purchase to be realized if there is a single change in interest rate. If the bond is held for the duration period, an increase from reinvesting coupons is offset by a decrease in price if interest rates increase and a decrease from reinvesting coupons is offset by an increase in price if interest rates decrease. There are several types of bond duration. In general, these can be divided into yield duration and curve duration.

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

Enter your email to subscribe to DW Simpson weekly actuarial job updates. Duration of Zero Coupon Bonds. It seems that a basic duration rule that many authors use is that the duration of a zero-coupon bond equals its time to maturity example: Investments p. But this is not immediately obvious to me. For example: Can someone help clarify my confusion? Does the issue have to do with the mechanics of our approximation formula, or am I confusing different types of duration? Enough Exams Already. At is supposed to be a subscript.

The Macaulay duration of a zero-coupon bond is equal to the time to maturity of the bond. The Macaulay duration can be viewed as the economic balance point of a group of cash flows.

### Modified Duration

In finance , the duration of a financial asset that consists of fixed cash flows , for example a bond , is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield , duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields. The dual use of the word "duration", as both the weighted average time until repayment and as the percentage change in price, often causes confusion. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. Both measures are termed "duration" and have the same or close to the same numerical value, but it is important to keep in mind the conceptual distinctions between them. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond. Modified duration, on the other hand, is a mathematical derivative rate of change of price and measures the percentage rate of change of price with respect to yield. Price sensitivity with respect to yields can also be measured in absolute dollar or euro , etc. The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration. Modified duration is used more often than Macaulay duration in modern finance. For every-day use, the equality or near-equality of the values for Macaulay and modified duration can be a useful aid to intuition.

### Zero Coupon Bond

Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds. One method of measuring interest rate risk due to changes in market interest rates is by the full valuation approach , which simply calculates what bond prices will be if the interest rate changed by specific amounts. The full valuation approach is based on the fact that the price of a bond is equal to the sum of the present value of each coupon payment plus the present value of the principal payment. That the present value of a future payment depends on the interest rate is what causes bond prices to vary with the interest rate, as well. Another method to measure interest rate risk, which is less computationally intensive, is by calculating the duration of a bond, which is the weighted average of the present value of the bond s payments. Consequently, duration is sometimes referred to as the average maturity or the effective maturity. The longer the duration, the longer is the average maturity, and, therefore, the greater the sensitivity to interest rate changes.

## Effective duration of a zero

By Sunita Sethi Leave a Comment. Zero Coupon Bond Also known as Pure Discount Bond or Accrual Bond refers to those bonds which are issued at a discount to its Par Value and makes no periodic interest payment unlike a normal coupon bearing Bond. In other words, its annual implied interest payment is included in its Face Value which is paid at the maturity of such bond. Therefore this Bond is the one where the sole return is the payment of the nominal value on maturity. The difference between the current price of the bond i. These Bonds avoid the risk of Reinvestment of Coupon Bonds as Interest Rates keep changing with the passage of time which impacts the Yield to Maturity of such coupon bearing Bonds. Since there are no interim cash flows, the investor is assured of a fixed rate of return. Usually, these Bonds are issued for a longer time frame which can be used by a potential investor to align with their life goals such as Marriage, Children Education, and retirement and so on. Thus a smart investor based on their time horizon can invest in different maturity Zero-coupon Bonds by paying a smaller amount initially as Zero-coupon Bonds are issued at deep discounts one can buy more with lesser amount and stagger them as per their career and life goals without getting impacted by the volatility. Not all Zero-coupon Bonds have a ready secondary market which results in illiquidity. Furthermore, in case of any urgent need funds, it is difficult to liquidate the same without getting a major haircut in value. They have a single cash inflow for the Investor which happens at the maturity and as such these bonds have the greatest Duration which results in Interest Rate Risk.

**VIDEO ON THEME: What is Duration & Modified Duration? - Macauley Duration & Modified Duration calculations**

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