If an investor buys a bond in the secondary market and pays a price different from par value, then not only will the current yield differ from the nominal yield, but there will be a gain or loss when the bond matures and the bondholder receives the par value of the bond. Like the calculation for current yield, yield to maturity and other yields based on the purchase price of the bond in the secondary market is based on the clean bond price , excluding accrued interest.
If the investor holds the bond until maturity, he will lose money if he paid a premium for the bond, or he will earn money if it was bought at a discount. The yield-to-maturity YTM aka true yield , effective yield of a bond held to maturity accounts for the gain or loss that occurs when the par value is repaid, so it is a better measure of the investment return.
When a bond is bought at a discount, yield to maturity will always be greater than the current yield because there will be a gain when the bond matures, and the bondholder receives par value back, thus raising the true yield; when a bond is bought at a premium, the yield to maturity will always be less than the current yield because there will be a loss when par value is received, which lowers the true yield.
Because some bonds are callable, these bonds will also have a yield to call YTC , which is calculated exactly the same as yield to maturity, but the call date is substituted for the maturity date and the call price or call premium is substituted for par value. When a bond is bought at a premium, the yield to call is always the lowest yield of the bond. Some bonds are redeemed periodically by a sinking fund —also called a mandatory redemption fund —that the issuer establishes to retire debt periodically at sinking fund dates specified in the redemption schedule of the bond contract for specified sinking fund prices , which are often just par value.
Such bonds are usually selected at random for redemption on such dates, so yield to sinker is calculated as if the bond will be retired at the next sinking fund date. If the bond is retired, then the bondholder simply receives the sinking fund price, and so the yield to sinker is calculated like the yield to maturity, substituting the sinking fund date for the maturity date, and, if different, substituting the sinking fund price for the par value.
Note, however, that yield to call and yield to sinker may not be pertinent if interest rates have risen since the bonds were first issued, because these bonds will be selling for less than par value in the secondary market, and it would save the issuer money to simply buy back the bonds in the secondary market, which helps to support bond prices for bondholders who want to sell. The yield to average life calculates the yield using the average life of a sinking bond issue.
The yield to average life is also used for asset-backed securities , especially mortgage-backed securities , because their lifetime depends on prepayment speeds of the underlying asset pool. Some bonds have a put option , which allows the bondholder to receive the principal of the bond from the issuer when the bondholder exercises the put.
This yield to put would be calculated like the yield to maturity, except that the date that the put is exercised is substituted for the maturity date, because the bondholder receives the par value on the exercise date just as if the bond matured.
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Finally, there is the yield to worst , which simply calculates the bond's yield if the bond is retired at the earliest possible date allowed by the bond's indenture. The formula below shows the relationship between the bond's price in the secondary market excluding accrued interest and its yield to maturity, or other yields, depending on the maturity date chosen. In this equation, which assumes a single annual coupon payment, YTM would be the bond's yield to maturity, but this is difficult to solve, so bond traders usually read the yield to maturity from a table that can be generated from this equation, or they use a special calculator or software, such as Excel as shown further below.
Yield to call is determined in the same way, but n would equal the number of years until the call date instead of the maturity date, and P would be the call price. Similarly, the yield to put, or any of the other yields, is calculated by substituting the appropriate date when the principal will be received for the maturity date.
Note that if the bond pays a semiannual coupon, as most US bonds do, then the following formula applies:. This equation shows that the bond price is equal to the present value of all bond payments with the interest rate equal to the yield to maturity. Although it is difficult to solve for the yield using the above equation, it can be approximated by this formula:.
A good way to remember this formula is that it is simply taking the difference between the par value and the current bond price and dividing it by the remaining term of the bond. This is the profit or loss per year, which is then added to or subtracted from the annual interest payment. The resulting sum, in turn, is divided by the average of the par value and the current bond price.
Once the bond is bought, then the yield to maturity is fixed, so the current bond price is replaced with the purchase price in the above formula.
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A simplification of the YTM formula can be made if the bond has no coupon payments, since all the terms involving coupon payments become zero, and the yield to maturity reduces to the present value of the principal payment Formula 1 below :. Note that equations 1 and 2 above are the same, since the discounted bond price is the present value of the investment and the principal payment is the future value, so we can find a simple way to calculate YTM by using a basic formula for the present value and future value of money.
To find the yield to maturity, we transpose the equation for the future value of money to equal the yield to maturity. Then subtract 1 from both sides, to arrive at YTM, the yield to maturity for the discount:. Note that the above example is compounded annually. To check the result:. All dates are expressed either as quotes or as cell references e. Yield to Worst, Yield to Sinker, and Yield to Average Life can be calculated by substituting the appropriate date for the maturity date.
The realized compound yield is the yield obtained by reinvesting all coupon payments for additional interest income. It will also depend on the bond price if it is sold before maturity. What this yield ultimately is depends on how interest rates change over the holding period of the bond.
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Although future interest rates and bond prices cannot be predicted with certainty, horizon analysis is often used to forecast interest rates and bond prices over a specific time period to yield an expectation of the realized compound yield. Yield to maturity is the average yield over the term of the bond. If a bond is sold before maturity, then its actual yield will probably be different from the yield to maturity. If interest rates rise during the holding period, then the bond's sale price will be less than the purchase price, decreasing the yield, and if interest rates, decrease, then the bond's sale price will be greater.
The holding-period return is the actual yield earned during the holding period. It can be calculated using the same formula for yield to maturity, but the sale price would be substituted for the par value, and the term would equal the actual holding period.
Note that, unlike yield to maturity, the holding-period return cannot be known ahead of time because the sale price of the bond cannot be known before the sale, although it could be estimated. Money market instruments are short-term discount instruments with maturities of less than a year, so the interest is paid at maturity. Because short-term instruments are issued at a discount, their yield is often referred to as a discount yield , which is often annualized as the bond equivalent yield BEY aka investment rate yield , equivalent coupon yield , which simplifies the comparison of yields with other financial returns:.
Although the BEY is not compounded, but is simply the discount yield annualized, it can be converted directly to any compounded rate of interest by using the formula for the present and future value of a dollar. To find a compounded rate, add 1 to the discount yield and raise the result to a power equal to the number of terms in the year, then subtract the result from See how the future value of a dollar is calculated to understand the reasoning better.
Perpetuities are bonds that are not redeemable and pay only interest, but pay it indefinitely—hence the name. 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.
senjouin-kikishiro.com/images/salogusic/3502.php 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:.
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A zero-coupon bond does not make any coupon payments; instead, it is sold to investors at a discount from face value.