Dear Dr. Don, Is it ever a good idea to put money in zero-coupon bonds? Dear Ellen, I own some municipal zeros, so I certainly hope so! Because no interest is paid but interest is earned, the bond sells at a discount to its stated face value. A Treasury bill is a form of zero-coupon security. No interest payments are made.

## The ABCs of Zero Coupon Bonds

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 โ the maturity is one year or less; the currently available maturities are 4 weeks, 13 weeks, 26 weeks and 52 weeks Treasury notes โ the maturity ranges between 1 and 10 years; the currently available maturities are 2, 3, 5, 7 and 10 years Treasury bonds โ 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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## Calculators

Click any one of these buttons to calculate the associated parameter given the values of the other three parameters. The discount rate R is stated in percentage points so that 5. The N button enters N, but recalculates the present value PV. That is, you can set N to any positive value, but this calculator does not solve for N. There is a limit of periods. The TEXT button restores this text after you produce a table.

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Our yield to maturity YTM calculator measures the annual return an investor would receive if a particular bond is held until maturity. To calculate a bond s yield to maturity, enter the face value also known as " par value " , the coupon rate, the number of years to maturity, the frequency of payments and the current price of the bond. To learn more about yield to maturity or any of the above terms, click on them. That ll take you to our Financial Dictionary , where we ll show you what each term means and exactly why it matters. If you buy this bond today, you will earn

### Converting from par rates

Zero coupon bonds are predominantly issued by the federal government, and typically, they are issued with maturities of years. Zero coupon bonds are traded on recognized financial markets and exchanges, which may offer investors liquidity in the event they choose not to hold them to maturity. One of the biggest risks of zero coupon bonds is their sensitivity to swings in interest rates. In a rising interest rate environment, their value is likely to fall more than other bonds. For individuals, zero coupon bonds may serve several investment purposes. By placing funds in a U.

### Zero Coupon Bond Yield Calculator

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 To sell to a new investor the bond must be priced for a current yield of 5. Then continuing by trial and error, a bond gain of 5.

### Zero Coupon Bond Effective Yield

By Amruth Sundarkumar 4 Comments. Fixed Income Tutorials. They are:. The price of a bond is essentially a function of the above. Combining them all, for a bond paying coupons annually and YTM annually compounded we get:.

## Yield to maturity

Bond value equals the present value of the bond cash flows i. There is an inverse relationship between the bond value and market interest rates. If the rates increase, the bond value drops and vice versa. A bond whose coupon rate is lower than the market discount rate is traded at a discount i. This is because in a high market interest rate environment, bonds with lower coupon rates are not attractive and their prices drop. The value of a conventional bond i. Where c is the coupon rate, F is the face value, m is the number of coupon payments per year, n is the number of years till maturity and i is the yield to maturity on the bond, i. The price determined above is the clean price of the bond. To find the full price i. If the market discount rate is not available, a bond can be valued using matrix pricing , an approach in which yield to maturity on comparable bonds is used to value a bond.

Use our financial calculators to compare loans, calculate different indicators, for example CAGR. Also, you can evaluate how profitable the Deposit in the Bank, or compare different deposits. For those who are engaged in investing, the dividend discount calculator is perfect,or the Gordon model calculator. With this calculator you can find out the rate of growth of dividends. This can be useful when comparing companies and their dividends. A2 Finance. Sign in Sign up. Sign up Sign in. Calculators Main Page Calculators. Corporate and business.

Most notes, bonds, and other debt that companies offer to investors pay interest on a regular basis. However, some debt issues are structured as non-interest-bearing notes or zero-coupon bonds.

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. This site was designed for educational purposes. The user should use information provided by any tools or material at his or her own discretion, as no warranty is provided. When considering this site as a source for academic reasons, please remember that this site is not subject to the same rigor as academic journals, course materials, and similar publications. Feel Free to Enjoy! Contact us at: Contact FinanceFormulas. New to Finance? Zero Coupon Bond Calculator Your browser does not support iframes.

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If we know the par yield, we can calculate both the zero coupon yield and the forward yield for the same maturities and risk class. Example 1: Converting from par rates to zero coupon rates. The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:. This means that 2. In addition, the original investment will also be repaid at Time 2. The one period par rate p represents the identical deal to the one period zero coupon rate z The conversion of par rates to zero coupon rates is sometimes known as bootstrapping , because the results of calculating each earlier zero coupon rate are used successively to calculate the next-longer maturity zero coupon rate. Example 2: Converting from zero coupon rates to forward rates. Given the calculated zero coupon rates z , the forward rates f can also be calculated in turn. The calculation of forward rates from zero coupon rates is explained in more detail on the page Converting from zero coupon rates.

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