Many bonds make periodic interest payments during the life of the bond, and then when the bond matures, the principal is returned. Instead, investors purchase the zero-coupon bond for less than its face value, and when the bond matures, they receive the face value. On the open market, investors pay higher prices for zero-coupon bonds when they require a lower rate of return and lower prices when a higher rate of return is required. The higher the risk the bond issuer will go bankrupt and not repay the bond holders, the higher the interest rate you need to compensate for that risk. Also, remember to consider the length of time until the bond matures. The lower the price you pay for the zero-coupon bond, the higher your rate of return will be.

## Zero Coupon Bond Value

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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## Bond Calculator: Introduction

Are you a student? Did you know that Amazon is offering 6 months of Amazon Prime - free two-day shipping, free movies, and other benefits - to students? Click here to learn more. One of the key variables in choosing any investment is the expected rate of return. We try to find assets that have the best combination of risk and return.

Bonds can prove extremely helpful to anyone concerned about capital preservation and income generation. Bonds also may help partially offset the risk that comes with equity investing and often are recommended as part of a diversified portfolio.

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.

### Bond Yield to Maturity Calculator

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. This means an investor knows his return in advance if he holds a zero coupon to maturity. The yield of a zero coupon bond is different than the yield of a normal bond of the same issuer. The individual coupons are the semi-annual interest payments due on the bond prior to maturity.

### Advanced Bond Concepts: Bond Pricing

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.

### Bond Pricing Formula

Compounding bonds are known by a number of different names:. Compounding or zero-coupon bonds became available in Canada in In the years prior to the use of computers for maintaining records, bond certificates were issued to bondholders. Attached to each certificate were coupons that represented all the future periodic interest payment to be paid to the investor. To collect the periodic payments, a bondholder would need to clip the individual coupon for the payment period, from the bond certificate and present it for payment. Therefore, each bond certificate was viewed as having two distinct components:.

## Compounding Bonds

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. Also, the bond gain and the bond price add up to For bonds with multiple coupons, it is not generally possible to solve for yield in terms of price algebraically.

### Zero Coupon Bond Calculator

Log in or sign up to add this lesson to a Custom Course. Log in or Sign up. Most bonds make periodic interest payments to pay back bondholders for borrowing money. For some bond investors, these regular payments are an annoyance; they d rather receive one big payday later on. Tom is looking for an investment that fits this goal and has come across zero coupon bonds. Let s take a look at what s different about these bonds and how Tom can calculate what s a good purchase price to meet his desired returns. A zero coupon bond is a type of bond that doesn t make a periodic interest payment. In bond investing, the term coupon refers to the interest rate repaid periodically to the bondholder. When Tom buys the bond, it will have a face value , which represents how much money he ll receive from the bond issuer at maturity. Since Tom won t be receiving any periodic interest payments, the only time he ll receive payment from the issuer is when the bond matures.

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.

### How to Calculate the Price of a Zero Coupon Bond

Problem 13QP from Chapter Calculating Returns. You purchased a zero-coupon bond one ye The market interest rate is now 9 percent. If the bond had 20 years to maturity when you originally purchased it, what was your total return for the past year? In this case, Zero coupon bonds have maturity of 20 years. But one year is already completed. Hence, the remaining years to maturity is years. For calculating total return, we must compute the current price of zero coupon bonds. Zero coupon bonds are compounded semiannually.

### Bond YTM Calculator

Most bonds make periodic interest payments to their owners as a return on investment and a reward for taking the risk inherent in the bond. These payments are known as coupons, because many years ago bonds would actually come with coupons that you could mail in to prompt the payment of interest. A "zero-coupon" bond, however, does not make interest payments. Instead, the bond holder is rewarded with an increase in the value of the bond over time. You can calculate the present value of a zero coupon bond using a formula involving the stated yield return , the par or face value, and the time until maturity when the bond s par or face value will be paid out to the bond holder. To create this article, volunteer authors worked to edit and improve it over time.

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**VIDEO ON THEME: FRM: TI BA II+ to compute bond yield (YTM)**

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