# Coupon greater than yield

The interest from municipal bonds is not taxed by the federal government, and U. Treasury bonds, notes, and T-Bills do not incur state or local taxes. Hence, these bonds can pay a lower interest rate than a corporation with a comparable credit rating. To compare municipal bonds or Treasuries with taxable bonds, the yield is converted to a taxable equivalent yield TEY , sometimes called equivalent taxable yield.

The taxable equivalent yield is the yield that a taxable bond would have to pay to be equivalent to the tax-free bond. We can call this the federal taxable equivalent yield, but note that if you live in the municipality of the bond issuer, then the bond may be free of state and local taxes as well.

To take in consideration all taxes saved, the above formula can be extended for any tax situation by simply adding up the percentages to arrive at a composite tax bracket and use that in the above equation to get the tax-free yield. A tax-free municipal bond yielding 6. Thus, a corporate bond that is taxable by the federal, state and local government would have to pay 4.

Note, also, that U. Treasuries are considered the safest investment, so the corporate bond would have to pay a little more—even if it had the highest credit rating—than the Treasury, to compensate the investor for the additional risk. The lower the credit rating of the corporate bond, the greater the interest the corporate bond would have to pay to entice investors away from safe Treasuries, which are considered risk-free investments.

## When a Bond's Coupon Rate Is Equal to Yield to Maturity

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.

The Differences Between Coupon, Yield and Expected Return

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. 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. 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.

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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:.

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## Valuing Bonds | Boundless Finance

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.

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• A Guide for Beginning Bond Investors: Coupon vs. Yield to Maturity?
• What this yield ultimately is depends on how interest rates change over the holding period of the bond. 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. Understanding the distinct difference between coupon rates and market interest rates is an integral step on the path toward developing a comprehensive understanding of bonds and the debt security marketplace. A coupon rate can best be described as the sum, or yield, paid on the face value of the bond annual over its lifetime.

This differs from the market interest rate of a bond, which is a fluctuating value that generally reflects market sentiment.

### 4. Price & Yield

Unlike the coupon rate, the market interest rate of a bond can swing drastically during the lifetime of the bond. For example, in a scenario where experts are predicting economic inflation, the market interest rate for the bond may rise due to the fact that investors will expect more cash to offset the decrease in the value of the currency at large.

Generally speaking, if a market interest rate exceeds a coupon rate, the value of the bond will likely drop. Once a bond issuer has set a coupon rate and a face value, the bond issuer logically wishes to obtain the highest possible market price for the bond issue. Typically, private companies will hire an investment bank to underwrite the bond issue. The investment bank, or syndicate of multiple investment banks, will purchase the entire bond issue and resell the bonds to large-scale and institutional investors on the open market.

Many governmental entities, such as the United States Treasury, will sell bonds directly to large-scale investors through auctions rather than using an underwriter as a middleman. The amount paid by investors for a bond, whether purchased through a direct auction, an underwriter or from another investor is the bond's market price.

When the market price is less than face value, then the market rate, or yield, of that bond will be greater than the coupon rate. When the market price is greater than face value, then the market yield of that bond will be less than the coupon rate.