n = number of compounding periods
FV = Future Value
PV = Present Value
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The return of a bond is largely determined by its interest rate. The interest that a bond pays depends on a number of factors, including the prevailing interest rate and the creditworthiness of the issuer, which, of course, is what is assessed by the credit rating companies, such as Standard & Poor’s and Moody’s. The higher the credit rating of the issuer, the less interest the issuer has to offer to sell its bonds. The prevailing interest rate—the cost of money—is determined by the supply and demand of money. Like virtually anything else, the greater the supply and the lower the demand, the lesser the interest rate, and vice versa. An often used measure of the prevailing interest rate is the prime rate charged by banks to their best customers.
Most bonds pay interest semi-annually until maturity, when the bondholder receives the par value of the bond back. Zero coupon bonds pay no interest, but are sold at a discount to par value, which is paid when the bond matures.
Nominal yield, or the coupon rate, is the stated rate of interest of the bond. This yield percentage is the percentage of par value—$5,000 for municipal bonds, and $1,000 for most other bonds—that is usually paid twice a year. Thus, a bond with a $1,000 par value that pays 5% interest pays $50 dollars per year in 2 semi-annual payments of $25. The return of a bond is the return/investment, or in the example just cited, $50/$1,000 = 5%.
Because bonds trade in the secondary market, they may sell for less or more than par value, which will yield an interest rate that is different from the nominal yield, called the current yield, or current return. The price of bonds moves in the opposite direction of interest rates. If rates go up, the price of bonds decrease; if the rates go down, then the bonds increase in value. To see why, consider this simple example. You buy a bond when it is issued for $1,000 that pays 8% interest. Suppose you want to sell the bond, but since you bought it, the interest rate has risen to 10%. You will have to sell your bond for less than what you paid, because why is somebody going to pay you $1,000 for a bond that pays 8% when they can buy a similar bond of equal credit rating and get 10%. So to sell your bond, you would have to sell it so that the $80 that is received per year in interest will be 10% of the selling price—in this case, $800, $200 less than what you paid for it. (Actually, the price probably wouldn’t go this low, because the yield-to-maturity is greater in such a case, since if the bondholder keeps the bond until maturity, he will receive a price appreciation which is the difference between $1,000, the bond’s par value and what he paid for it.) Bonds selling for less than par value are said to be selling at a discount. If the market interest rate of a new bond issue is lower than what you are getting, then you will be able to sell your bond for more than par value—you will be selling your bond at a premium.
| Current Yield Formula for Bonds | |
|---|---|
| Annual Interest Payment Price of Bond | = Current Yield |
| Current Yield Example | |
|---|---|
| $60 Annual Interest Payment $800 for Bond | = 8% Current Yield |
Note that if the market price for the bond is equal to its par value, then:
Current Yield = Nominal Yield
The interest from municipal bonds is not taxed by the federal government, and U.S. 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.
| Taxable Equivalent Yield (TEY) Formula for Municipal Bonds | |
|---|---|
| Muni Yield 100% - Your Federal Tax Bracket % | = Taxable Equivalent Yield (TEY) |
| Taxable Equivalent Yield (TEY) Example | |
|---|---|
| 4% Muni Yield 100% - 28% Federal Tax Bracket | = 5.5% TEY |
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.
| Taxable Equivalent Yield (TEY) Example for Municipal Bond Exempt of All Taxes | |
|---|---|
| 6.1% Muni Yield 100% - 28% federal tax - 10% state tax - 1% local tax | = 10% TEY |
To look at it from a different angle, suppose a bond pays 10%, as in the above example. That's $100 per year for a par value of $1,000. If you pay 28% of your income in federal taxes, 10% in state taxes, and 1% in local taxes, and the bond is taxable, then the federal tax will be $28, the state tax will be $10, and the local tax will be $1—that would leave you with a net of $61. A tax-free municipal bond yielding 6.1% would net you the same amount.
U.S. Treasuries do not incur state or local taxes, but federal taxes have to be paid on the interest, so the taxable equivalent yield for Treasuries is calculated using the same formula, but only the state and local tax rate is deducted from 100%.
| Taxable Equivalent Yield (TEY) Example for U.S. Treasuries | |
|---|---|
| 4% Treasury Yield 100% - 10% state tax - 1% local tax | = 4.5% TEY |
Thus, a corporate bond that is taxable by the federal, state and local government would have to pay 4.5% to net the same amount that a U.S. Treasury paying 4% would net. Note, also, that U.S. 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, and 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.
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. 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 he paid for it at a discount. The yield-to-maturity (YTM), or true yield, of a bond that is held to maturity will have to account for the gain or loss that occurs when the par value is repaid.
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, and this lowers the true yield.
Because some bonds are callable, these bonds will also have a yield to call, 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, 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, which helps to support the bond prices for investors who want to sell.
The yield to average life calculates the yield using the average life of a sinking bond issue. So for a 20-year bond with an indenture that specifies that 10% of an issue must be retired each year from the 10th year to the 20th year of the bond's term, the average life would be 15 years.
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 also 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, Y would be the bond's yield to maturity, but this is difficult to solve for Y, so usually bond traders 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.
| Formula for Calculating Yield to Maturity, Yield to Call, or Yield to Put | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B = current bond price C = coupon payment per period P = par value of bond or call premium; n = number of years until maturity Y = yield to maturity, yield to call, |
or, expressed in summation, or sigma, notation:
| B = | n ∑ k=1 | Ck | + | P |
| (1+Y)k | (1+Y)n |
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 readily solved by using Microsoft Excel, as shown below.
| Yield-to-Maturity (YTM) Formula for Bonds using Microsoft Excel |
|---|
YTM = Yield(settlement,maturity,rate,price,redemption,frequency,basis)
Note that yield to call (YTC) and yield to put (YTP) can also be calculated using this formula. To calculate the yield to call:
To calculate yield to put:
Yield to Worst, Yield to Sinker, and Yield to Average Life can be calculated by substituting the appropriate date for the maturity date. |
| Yield to Maturity (YTM) Example |
|---|
If
Then
|
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. Although future interest rates and bond prices cannot be predicted with certainty, horizon analysis is the forecasting of 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.
| When the bondholder pays... | Bond Yield Relationships |
| less than par value (discount). | Yield to Maturity > Current Yield > Nominal Yield |
| par value. | Nominal Yield = Current Yield = Yield to Maturity |
| more than par value (premium). | Nominal Yield > Current Yield > Yield to Maturity |
Original issue discount bonds are bonds that are issued with coupon rates below the prevailing interest rates or are bonds that have no coupon payments at all. Such bonds will sell at a discount to par value; the lesser the coupon payments, the greater the discount. Those bonds, such as U.S. Treasury Bills, that have no coupon payments are called zero coupon bonds (aka zeros). Most money market instruments are zeros. Although zero coupon bonds have no specified interest rate, an equivalent or effective interest rate can be found, which is the interest rate that would yield the same amount of money in the same amount of time, starting with a principal equal to the price of the discounted bond. To find this equivalent interest rate, so that discounted bonds can be compared to other investments, we use the formula for the future value of money, which is the following:
| Future Value Formula | |
|---|---|
| i = interest rate per compounding period n = number of compounding periods. | |
| Future Value = | Present Value * (1 + i)n |
To find the interest rate, i, we transpose this equation to equal i.
First, divide both sides by the Present Value:
(1 + i)n = Future Value / Present Value.
Take the nth root of both sides, then subtract 1 from both sides, to arrive at i, the interest rate for the discounted bond.
i = (Future Payment/Present Value)1/n - 1
This formula can be simplified further by simply calculating the interest rate for 1 dollar, which makes the present value equal to 1, which reduces the above equation to:
i = (Future Value per Dollar Invested)1/n
| Formula for the Effective Interest Rate of a Discounted Bond | ||
|---|---|---|
| i = | (Future Value/Present Value)1/n - 1 | |
| or | | i = interest rate per compounding period n = number of compounding periods FV = Future Value PV = Present Value |
| Calculating the Interest Rate of a Zero Coupon Bond Example |
|---|
| What interest rate is a zero coupon bond equivalent to, that costs $600 and pays $1,000 in 10 years, assuming an interest rate that compounds annually? Known:
|
| (1,000/600)1/10 - 1 = 5.24% Using a scientific calculator: 1000 ÷ 600 = x√y 10 - 1 = .052409... ≈ 5.24% |
| If we wanted to compare this zero coupon bond to a savings account that paid interest compounded daily, then we would use the same equation, but n = 3,650, the number of compounding periods in 10 years. However, the result will be the interest rate for 1 day, so we multiply the result by 365 to arrive at an equivalent annual interest rate that is compounded daily instead of annually. |
| ((1,000/600)1/3650 - 1) * 365 = 5.11% annual rate compounded daily. Using a scientific calculator: 1000 ÷ 600 = x√y 3650 - 1 = .000139962 x 365 = .051086... ≈ 5.11% |
Note that the interest rate is slightly lower when compounded daily instead of annually. This must be so, because you are only getting $400 more than you paid when the bond matures, and since an interest rate compounded daily earns a little more than one compounded annually, it must be lower to yield the same $400 after 10 years. Thus, if $600 is put in a savings account, which has an interest rate of 5.11% compounded daily, that would be equivalent to an interest rate of 5.24% compounded annually, which, at the end of 10 years, is equivalent to buying a zero coupon bond for $600, and getting $1,000 back in 10 years.
Sometimes the yields listed for short-term discount instruments have simply been annualized without compounding the interest. This simplifies the math and can be calculated using a calculator that doesn't have a root or exponential function. This uncompounded annual interest rate is simply called the annual interest rate to distinguish it from the effective annual yield, but, most often, it is called the bond equivalent yield (BEY) (aka investment rate yield, equivalent coupon yield). The simplified formula appears below:
BEY = Interest Rate per Term x Number of Terms per Year
Below is the formula relating BEY to the face value, price paid for the instrument, and days left to maturity:
| Formula for Calculating Bond Equivalent Yield (BEY) | |||
|---|---|---|---|
| Interest Rate Per Term | Number of Terms per Year | ||
| BEY = | Face Value - Price Paid ─────────────── Price Paid | x | Actual Number of Days in Year ─────────────────── Days Till Maturity |
Note that this yield is not compounded, but is the simple interest rate annualized. However, if you only have the simple interest rate of a discount instrument, then this rate can be converted directly to any compounded rate of interest by using the formula for the present and future value of a dollar. (See Calculating the Interest Rate of a Discounted Financial Instrument for more info.)
If you buy a 4-week T-bill with a face value of $1,000 for $996.50, what is the bond equivalent yield, assuming it is not a leap year?
($1,000-$996.50)/$996.50 x 365/28 = 4.58% (rounded)
Perpetuities are bonds that are not redeemable and pay only interest, but pay it indefinitely—hence the name. They do not mature and, thus, the principal is never repaid. They were first issued by the British government in the 1850's, and were called consols, and some perpetuities were issued by the U.S. Treasury, but perpetuities are very rare today.
The price of a perpetuity is equal to the present value of all future payments. While this forms an infinite series, it does have a finite limit, because successive terms become smaller and smaller, and that limit is the following:
Price of Perpetuity = Annual Coupon Payment / Nominal Interest Rate
Consequently, the yield of a perpetuity is calculated as the current yield:
Yield of Perpetuity = Current Yield = Annual Coupon Payment / Perpetuity Price
Note that because a perpetuity is not redeemable and pays no principal, a perpetuity has no yield to maturity, since it never matures.
If
Then
U.S. Treasuries are generally considered to be free of default risk, and therefore generally have the lowest yield. All other bonds have some risk of default—some more than others. To compensate investors for the greater risk, these bonds pay a higher yield. This difference in yield is known as the risk premium (aka default premium), and how the risk premium varies across different bonds and different maturities is known as the risk structure of interest rates. The greater the risk of default, the greater the risk premium.
During a recession, investors become more concerned that the risk of default is greater than in good times, since recessions can cause financial difficulties for companies. So many investors move their investments to safer bonds—a flight to quality. This causes the difference in yield between corporate bonds and riskless government bonds to increase. The default premium increases to compensate investors for the greater risk.
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