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.

## Zero Coupon Bond Effective Yield

We have provided a quick outline of what a student will need to know to understand bonds and the pricing or valuation of bonds which is the primary focus in the initial corporate finance program. More advanced finance courses will introduce students to advanced bond concepts including duration, managing bond portfolios, understanding and interpreting term structures, etc. A bond is a debt instrument that provides a periodic stream of interest payments to investors while repaying the principal sum on a specified maturity date.

The face value also known as the par value of a bond is the price at which the bond is sold to investors when first issued; it is also the price at which the bond is redeemed at maturity. In the U. Occasionally a bond is issued with a much longer maturity; for example, the Walt Disney Company issued a year bond in There have also been a few instances of bonds with an infinite maturity; these bonds are known as consols.

With a consol, interest is paid forever, but the principal is never repaid. Many bonds contain a provision that enables the issuer to buy the bond back from the bondholder at a pre-specified price prior to maturity. This price is known as the call price. A bond containing a call provision is said to be callable. This provision enables issuers to reduce their interest costs if rates fall after a bond is issued, since existing bonds can then be replaced with lower yielding bonds. Since a call provision is disadvantageous to the bond holder, the bond will offer a higher yield than an otherwise identical bond with no call provision.

Some bonds contain a provision that enables the buyer to sell the bond back to the issuer at a pre-specified price prior to maturity. This price is known as the put price. A bond containing such a provision is said to be putable. This provision enables bond holders to benefit from rising interest rates since the bond can be sold and the proceeds reinvested at a higher yield than the original bond. Since a put provision is advantageous to the bond holder, the bond will offer a lower yield than an otherwise identical bond with no put provision.

Some bonds are issued with a provision that requires the issuer to repurchase a fixed percentage of the outstanding bonds each year, regardless of the level of interest rates. A sinking fund reduces the possibility of default ; default occurs when a bond issuer is unable to make promised payments in a timely manner. Since a sinking fund reduces credit risk to bond holders, these bonds can be offered with a lower yield than an otherwise identical bond with no sinking fund. Bonds are issued by borrowers to raise funds for long-term investments; the main issuers of bonds in the U.

Treasury securities are issued by the U. These are free of default risk , which is the risk that the investor will not receive all promised payments. They are not taxed by state and local governments, but are taxed at the federal level. Another key difference between these securities is that Treasury bills are sold at a discount from their face value and redeemed at face value; Treasury notes and bonds are sold and redeemed at face value and pay semi-annual coupons to investors.

Corporations can raise funds by issuing debt in the form of corporate bonds. These bonds offer a higher promised coupon rate than Treasuries, but expose investors to default risk. The riskiest corporations offer the highest coupon rates to investors as compensation for default risk. A municipal bond is issued by a state or local government; as a result, they carry little or no default risk. Occasionally, municipalities do default on their debts; in , the city of Detroit filed for bankruptcy as a result of being unable to pay its debts.

Municipal bonds offer an extremely favorable tax treatment to investors. They are not taxed by federal, state or local governments as long as the bond holder lives in the municipality in which the bonds were issued. As a result, municipal bonds can be issued with very low yields. Foreign bonds are issued by foreign governments and corporations and are denominated in dollars.

If they are denominated in a foreign currency, they are known as eurobonds. Dollar-denominated bonds issued in the U. This formula shows that the price of a bond is the present value of its promised cash flows. The bond makes annual coupon payments. These results also demonstrate that there is an inverse relationship between yields and bond prices:. For a bond that makes semi-annual coupon payments, the following adjustments must be made to the pricing formula:. As an alternative to this pricing formula, a bond may be priced by treating the coupons as an annuity; the price is therefore equal to the present value of an annuity the coupons plus the present value of a sum the face value.

This method of valuing bonds will use the formula: The bond in the previous example can be priced using this alternate bond valuation formula as follows:. A zero-coupon bond does not make any coupon payments; instead, it is sold to investors at a discount from face value. The difference between the price paid for the bond and the face value, known as a capital gain , is the return to the investor. The pricing formula for a zero coupon bond is:.

What is the market price of this bond? In order to be consistent with coupon-bearing bonds, where coupons are typically made on a semi-annual basis, the yield will be divided by 2, and the number of periods will be multiplied by There are different types of yield measures that may be used to represent the approximate return to a bond. These include:. This equals the rate of return earned by a bond holder known as the holding period return if:.

It is impossible to solve for the yield to maturity algebraically; instead, this must be done using a financial calculator or Microsoft Excel. The variables in brackets fv, type and guess are optional values; the value of type is set to zero if it is not specified. Guess can be used to provide an initial estimate of the rate, which could potentially speed up the calculation time. Note that either pv or fv must be negative, and the other must be positive.

The negative value is considered to be a cash outflow, and the positive value is considered to be a cash inflow. Also note that entering semi-annual periods and coupon payments will produce a semi-annual yield; in order to convert this into an annual yield on a bond-equivalent basis , the semi-annual yield is doubled. For a bond that is callable, the yield to call may be used as a measure of return instead of the yield to maturity.

The process is similar to computing yield to maturity, except that the maturity date of the bond is replaced with the next call date. This is because yield to call is based on the assumption that the bond will be called on the next call date. The face value is replaced with the call price since this is the amount that the investor will receive if the bond is called.

What is the yield to call? In this case, the bond will mature in eight years, but it can be called in three years. The yield to call is computed as follows:. The current yield is simpler measure of the rate of return to a bond than the yield to maturity. The current yield is computed as:. This measure has the benefit of simplicity. It suffers from the drawback that it does not account for the time value of money. Excel contains a set of specialized bond functions that can be used to account for several complications that arise in bond pricing, such as day-count conventions.

Money market instruments e. Note that the settlement date and maturity date are represented as numerical values in Excel. The date January 1, is represented as 1; all later dates represent the number of days that have passed since January 1, As an example, suppose that a bond is sold on June 15, with a maturity date of June 15, What is the price of the bond?

Note that the price of the bond is entered as Also note that represents June 15, and represents For example,. This article is one part of a series on fixed income portfolios. Other articles in this series include:. Call us: These include: Face Value Coupon Rate Coupon Maturity Call Provisions Put Provisions Sinking Fund Provisions a Face Value The face value also known as the par value of a bond is the price at which the bond is sold to investors when first issued; it is also the price at which the bond is redeemed at maturity.

The U. Treasury Treasury securities are issued by the U. Treasuries can be classified by their maturities as follows: Treasury bills — the maturity is one year or less; the currently available maturities are 4 weeks, 13 weeks, 26 weeks and 52 weeks Treasury notes — the maturity ranges between 1 and 10 years; the currently available maturities are 2, 3, 5, 7 and 10 years Treasury bonds — the maturity ranges between 20 and 30 years; the currently available maturity is 30 years Another key difference between these securities is that Treasury bills are sold at a discount from their face value and redeemed at face value; Treasury notes and bonds are sold and redeemed at face value and pay semi-annual coupons to investors.

These results show the following important relationship: The bond in the previous example can be priced using this alternate bond valuation formula as follows: The pricing formula for a zero coupon bond is: In order to be consistent with coupon-bearing bonds, where coupons are typically made on a semi-annual basis, the yield will be divided by 2, and the number of periods will be multiplied by 2: This equals the rate of return earned by a bond holder known as the holding period return if: The yield to call is computed as follows: The yield to call is: The current yield is computed as: Other articles in this series include: Strategies, Duration, Modified Duration, Convexity.

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## Types of Bonds

The investment return of a bond is the difference between what an investor pays for a bond and what is ultimately received over the term of the bond. The bond yield is the annualized return of the bond. Thus, bond yield will depend on the purchase price of the bond, its stated interest rate — which is equal to the annual payments by the issuer to the bondholder divided by the par value of the bond — plus the amount paid at maturity. Because the stated interest rate and par value are stipulated in the bond indenture , the price of the bond will vary inversely to prevailing interest rates.

Suppose you purchase a year, zero-coupon bond with a yield to maturity of 5. You hold the bond for five years before selling it.

Would you like to her bond or politely? Would you like to the bond or politely? What is its price? What was the total return, capital return and income return? Calculate your answers as effective annual rates.

### When is a bond s coupon rate and yield to maturity the same?

A zero coupon bond, sometimes referred to as a pure discount bond or simply discount bond, is a bond that does not pay coupon payments and instead pays one lump sum at maturity. The amount paid at maturity is called the face value. The term discount bond is used to reference how it is sold originally at a discount from its face value instead of standard pricing with periodic dividend payments as seen otherwise. To find the zero coupon bond s value at its original price, the yield would be used in the formula. After the zero coupon bond is issued, the value may fluctuate as the current interest rates of the market may change. The formula would be shown as.

### Bond Yield to Maturity Calculator

We have provided a quick outline of what a student will need to know to understand bonds and the pricing or valuation of bonds which is the primary focus in the initial corporate finance program. More advanced finance courses will introduce students to advanced bond concepts including duration, managing bond portfolios, understanding and interpreting term structures, etc. A bond is a debt instrument that provides a periodic stream of interest payments to investors while repaying the principal sum on a specified maturity date. The face value also known as the par value of a bond is the price at which the bond is sold to investors when first issued; it is also the price at which the bond is redeemed at maturity. In the U. Occasionally a bond is issued with a much longer maturity; for example, the Walt Disney Company issued a year bond in There have also been a few instances of bonds with an infinite maturity; these bonds are known as consols. With a consol, interest is paid forever, but the principal is never repaid. Many bonds contain a provision that enables the issuer to buy the bond back from the bondholder at a pre-specified price prior to maturity. This price is known as the call price.

### Bond Yields

The yield to maturity YTM , book yield or redemption yield of a bond or other fixed-interest security , such as gilts , is the theoretical internal rate of return IRR, overall interest rate earned by an investor who buys the bond today at the market price, assuming that the bond is held until maturity , and that all coupon and principal payments are made on schedule. In a number of major markets such as gilts the convention is to quote annualized yields with semi-annual compounding see compound interest ; thus, for example, an annual effective yield of When the YTM is less than the expected yield of another investment, one might be tempted to swap the investments. Care should be taken to subtract any transaction costs, or taxes. What happens in the meantime? Over the remaining 20 years of the bond, the annual rate earned is not

## How to Calculate Yield to Maturity of a Zero-Coupon Bond

Only Treasury issues are insured by the U. High values for the times interest and quick ratios and a low debt to equity ratio are desirable indicators of safety. If the investment banker has appraised the market and the quality of the bond correctly, the bond will sell at or near par unless interest rates have changed very dramatically and very quickly around the time of issuance. C must be paid to the broker for the inconvenience of selling bonds between maturity dates. Accrued interest must be paid by the buyer, but is not included in the quotations page price. The buyer of a bond will buy at the asked price and will also be invoiced for any accrued interest due to the seller. Treasury note pays interest on May 30 and November 30 and is traded for settlement on August The "D" rating indicates.

On this page is a bond yield to maturity calculator , to automatically calculate the internal rate of return IRR earned on a certain bond. This calculator automatically assumes an investor holds to maturity, reinvests coupons, and all payments and coupons will be paid on time. The page also includes the approximate yield to maturity formula , and includes a discussion on how to find —or approach — the exact yield to maturity. For this particular problem, interestingly, we start with an estimate before building the actual answer. The formula for the approximate yield to maturity on a bond is:. We calculated the rate an investor would earn reinvesting every coupon payment at the current rate, then determining the present value of those cash flows. The summation looks like this:. As discussing this geometric series is a little heavy for a quick post here, let us note: For most purposes, such as quickly estimating a yield to maturity, the approximation formula should suffice.

A zero coupon bond is a fixed income security that is created from the cash flows that make up a normal bond. It can be a useful option for investors who like knowing they will receive a certain amount of set income at a certain time, without having to spend too much time thinking about their portfolio.

Zero coupon municipal bonds combine the benefits of the zero coupon instrument with those of tax-exempt municipal securities and offer the following advantages:. Further, tax-exempt zero coupon municipal bonds earn interest that, in many cases, is also free from state and local taxes. Certain out-of-state municipal bonds may be taxable at the state level. Check with your tax attorney or advisor. In contrast, taxable zero coupon bonds are taxed each year on the amount of interest that has accrued for that year even though the accrued interest is not actually paid to investors in that year. Because they are sold at a substantial discount from face amount, you can purchase more zero coupon bonds for your money than other types of bonds. The greater the number of years a zero coupon bond has until maturity, the less you have to pay for it. Zero coupon bonds allow investors to invest a modest amount of money today and know exactly how much money they will receive on a specified future date. Zero coupon municipal bonds provide investors with the opportunity to lock in a particular rate of return, without having to worry about reinvestment risk or interest rates in the future. Investors in securities that pay interest semiannually may not always achieve a total realized compounded yield equal to the quoted yield to maturity they expected when they purchased their holdings. Depending on future prevailing interest rates, their interest payments may be reinvested in lower or higher yielding vehicles. There are zero coupon municipal bonds available with maturities ranging from one to 40 years, with the majority of these bonds having maturities between eight and 20 years. The majority of zero coupon municipal bonds are rated A or better by the three major rating services:

A bond s coupon rate is equal to its yield to maturity if its purchase price is equal to its par value. The par value of a bond is its face value, or the stated value of the bond at the time of issuance, as determined by the issuing entity. Read more What is the difference between yield to maturity and the coupon rate? The par value of a bond does not dictate its market price , however. These factors include the bond s coupon rate, maturity date, prevailing interest rates and the availability of more lucrative bonds. The coupon rate of a bond is its interest rate , or the amount of money it pays the bondholder each year, expressed as a percentage of its par value. Suppose you purchase an IBM Corp. To calculate the bond s coupon rate, divide the total annual interest payments by the face value.

Zero-coupon bonds do not have re-occurring interest payments, which distinguishes yield to maturity calculations from bonds with a coupon rate. Zero-coupon bonds trade on the major exchanges. They are commonly issued by corporations, state, and local governments, and the U. The IRS calls this imputed interest. Others find the securities well suited for achieving long-term financial goals, such as college tuition. This arrangement can be most advantageous when interest rates are high and when placed in tax-sheltered retirement accounts. Most time value of money formulas require some interest rate figures for each point in time. This consequently renders the yield to maturity easier to calculate for zero-coupon bonds because there are no coupon payments to reinvest, making it equivalent to the normal rate of return on the bond. The formula for calculating the yield to maturity on a zero coupon bond is:. The formula would look as follows: When solved, this equation produces a value of 0.

**VIDEO ON THEME: Zero Coupon Bonds**

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