When market rates value, they use the yield to find the bond's price.
The appropriate yield on a bond is a function of cer go up, bond prices go down and the risk-free rate of return.
Key issue and issuer characteristics, like years to maturity and move up and down at the same speed because they don't move in the issue's bond rating, don't affect bond prices.
The required straight line is formed by these forces.
Bond prices will rise at a discount rate in the bond valuation process if the required yields and return are not good.
When prices go up than you'll lose when prices go down, you're dealing with an annuity of coupon payments for a specified number of make more money when prices go up than you'll lose when prices go down.
We will demonstrate the bond valuation process in two different ways.
Because of its computational simplicity, we will assume that we are dealing with coupons that are paid once a year.
Bond valuation under conditions of semiannual compounding is the way most bonds actually pay their interest.
Remaining to maturity is PArt FoUr I InvEstInG.
A 4.5% bond is priced to yield 5%.
The bond pays an annual coupon of 4.5%, has 20 years left to maturity, and has a yield to maturity of 5%, which is the current market rate on bonds of this type.
The bond's price can be found using Equation 11.3.
We have an annuity of coupon payments of $45 a year for 20 years, plus a single cash flow of $1,000 at the end of 20 years, because this is a coupon-bearing bond.
We add the present value of the coupon annuity to the present value of the recovery of principal at maturity.
If you earn 5% on your money, you should be willing to pay almost $938 for this bond.
The coupon rate is below the market's required return so it trades at a discount.
You can link the size of the discount on this bond to the present value of the coupons that it pays and the coupons that would be required if the bond matched the market's 5% required return.
For a bond that trades at a premium, the size of that premium is the difference between the coupon that the bond pays and the coupon that the market requires.
Bond issuers generally set the bond's coupon rate equal or close to the market's required return at the time the bonds are issued, so bonds initially sell for a price close to par value.
If the market interest rate changes during the life of the bond, the bond's price will change to reflect the difference between the coupon rate and the market interest rate.
As the maturity date arrives, bond prices will converge to par value, despite the fact that bonds can sell at premiums or discounts over their lives.
The principal to be repaid at maturity is becoming an ever bigger portion of the bond because there are fewer interest payments remaining.
The bond's price can be calculated using the following spreadsheet.
The way most bonds are valued in the marketplace is different from using annual compounding.
It is appropriate to use semiannual compounding to value bonds because they pay interest every six months.
Cut the annual interest income and required rate of return in half and double the number of periods until maturity and you're good to go.
There are two 6-month periods per year instead of one compounding and payment interval per year.
Pricing a bond using annual compounding is similar to finding the price of a bond under conditions of semian nual compounding.
In the previous example, you priced a 20-year bond to yield 5%, assuming annual interest payments of $45.
The bond could make semiannual interest payments.
You adjust the semiannual return to 2.5% and the number of periods to 40 with semiannual payments of $22.50
The price of the bond is slightly less than what we got with annual compounding.
In cell B8 the required annual return is divided by coupon payment Frequency to find the required rate of return per 6-month period, and the number of years to maturity is divided by the coupon pay ment Frequency to find the total number of 6-month periods remaining until maturity.
You can trade bonds whenever the market is open, even if they pay interest every six months.
If you own a bond, it will make interest payments on January 15 and July 15 each year.
If you sell the bond on October 15 it will be roughly halfway between the two payment dates.
If you sell the bond before the coupon payment, you won't sacrifice any interest that you earned.
When you sell a bond in between coupon dates, the bond buyer adds accrued interest to the bond's price, which is calculated using Equation 11.3 or 11.4 depending on whether coupons arrive annually or semiannually.
If you purchase a $1,000 par value bond, it will pay a 6% coupon in semiannual installments of $30.
You received a coupon payment two months ago and are ready to sell the bond.
You will get both the market price and accrued interest if you sell the bond.
The total cash that you receive in exchange for your bond is $1,020 because you receive accrued interest of $10 between the last coupon payment and the next one.
The price of a bond is referred to as either clean or dirty by traders in the bond market.
Bond price quotations that you can find in financial periodicals or online are usually clean prices.
The clean price is $1,010 and the dirty price is $1,020.
The yield to maturity of a bond is the most important factor in the bond market.
The yield to maturity helps determine the price at which a bond trades, but it also measures the rate of return on the bond.
If you can observe the price of a bond that is trading in the market, you can reverse the bond valuation process to solve for the bond's yield to maturity rather than its price.
If you bought the bond at its current market price, that gives you a pretty good idea of the return that you could earn.
The yield to call, the current yield, and the yield to maturity are used to assess the return on a bond.
A bond's annual interest income is the only source of return this measure looks at.
It shows the amount of current income a bond has relative to its market price.
For every $1,000 of principal, an 8% bond pays $80 per year in interest.
The current yield measures a bond's annual interest income, so it is of interest to investors seeking high levels of current income.
It looks at the bond's interest income and any gain or loss that comes from the difference between the price that an investor pays for a bond and the par value that the investor receives at maturity.
The cash flow received over a bond's life is taken into account.
The calculation assumes that the investor can reinvested the coupon payments at an interest rate equal to the bond's yield to maturity.
This "rein vestment assumption" plays a vital role in the Ytm, which we will discuss in more detail later in this chapter.
The yield to maturity is used to gauge the return on a single issue but also to track the behavior of the market in general.
Market interest rates are a reflection of the average promised yields that exist in the market.
The yield to maturity provides valuable insights into an issue's investment merits that investors can use to assess the attractiveness of different bonds.
The more attractive an issue is, the higher the promised yield.
The best and most accu rate procedure is derived from the bond valuation model.
You can use Equations 11.3 and 11.4 to determine the YTP for a bond.
The difference is that now you know the price of the bond, you know the discount rate that will equate the present value of the bond's cash flow to its current market price, and you are trying to find it.
This procedure may be familiar to you.
It is similar to the internal rate of return measure described in the text.
The internal rate of return on a bond is called the Ytm.
You have the bond's yield to maturity when you find that.
It is a matter of trial and error to find yield to maturity.
You try different values until you find a solution to the equation.
Let's say you want to find a 7.5% annual coupon-paying bond that has 15 years left to maturity and is currently trading in the market at $820.
Notice that the bond sells at a discount.
When the required return is higher than the coupon rate, bonds sell at a discount.
We might initially try a discount rate of 8% or 9%, since it sells at a discount, any value above the bond's coupon.
The present value of the bond's cash flows is very close to the bond's market price if we try a discount rate of 10%.
We can say that 10% is the approximate yield to maturity on this bond because the computed price is close to the market price.
10% is the discount rate that leads to a computed bond price that's very close to the bond's current market price.
You would expect to earn close to 10% if you held the bond to maturity.
Doing trial and error by hand can be time consuming, so you can use a handheld calculator or computer software.
It's possible to find the YTM using semiannual compounding, given some fairly simple modifications.
We halved the annual coupon and discount rate and doubled the number of periods to maturity.
Let's see what happens when you use Equation 11.4 and try an initial discount rate of 10%.
A 5% discount rate results in a computed bond value that is less than the market price.
If you need a higher price, you have to try a lower Ytm.
The semiannual yield on this bond has to be less than 5%.
The yield to maturity on this bond is just a shade under 5% per half year.
The yield is expressed over a period of 6 months.
The market convention states that the annual yield is twice the semiannual yield.
You know that the issue has a semiannual yield of about 4%.
That is 9.97 percent.
The 9.98% figure in the previous paragraph is due to the calculator's more precise rounding.
The yield on a semiannual bond can be calculated as shown in the following spreadsheet.
In addition to holding the bond to maturity, there are other critical assumptions embedded in any yield to maturity figure.
The promised yield measure is based on present value concepts and contains important reinvestment assumptions.
The Ytm calculation assumes that when each coupon payment arrives, you can invest it for the rest of the bond's life at a rate that is equal to the Ytm.
The return that you earn over a bond's life is the same as the Ytm.
The yield to maturity figure is the return promised only if the issuer meets all interest and principal obligations on a timely basis and the investor reinvested all interest income at a rate equal to the com puted yield.
If you reinvested each of the coupon payments, you would earn a 10% return on those reinvested funds.
Failure to do so will result in a realized yield of less than 10%.
If you reinvested the coupons, you would earn a realized yield over the 15-year investment horizon of just over 6.5%--far short of the 10% promised return.
The actual yield on your bond over the 15 years would be higher if you reinvested coupons at a rate greater than 10%.
A significant portion of the bond's total return over time comes from reinvested coupons, unless you are dealing with a zero-coupon bond.
There are three components of return when we use present value-based measures of return.
Interest on interest is a measure of what you do with the profits from an investment.
M12_SMAR3988_13_GE_C11.indd 472 has the required minimum reinvestment rate.
If you put your investment profits to work at this rate, you'll earn a rate of return equal to Ytm.
If there's an annual or semiannual flow of interest income, the reinvestment of that income and interest on interest are matters that you must deal with.
The bigger the coupon and the longer the maturity, the more important the reinvestment assumption is.
For long-term, high-coupon bond investments, interest alone can account for half the cash flow.
You can use the procedures described above to find the yield to maturity on a zero-coupon bond.
You can ignore the coupon portion of the equation because it will equal zero.
You could buy a 15-year zero-coupon bond for $315 today.
An annual compound return of 8% is paid by the zero-coupon bond.
We'd use the same equation except for substituting 30 for 15 because there are 30 semiannual periods in 15 years.
The yield would change to 3.93% per half year.
That is 6.985%.
Bonds can either be noncallable or callable.
The issuer of a noncallable bond cannot call the bond before it matures.
The standard yield to maturity can be used to value such issues.
A callable bond gives the issuer the right to retire the bond before its maturity date, so the issue may not remain outstanding to maturity.
If you purchase a callable bond, you may not always get a good measure of the return.
The impact of the bond being called away should be considered.
The YTC is used with bonds that have deferred-call provisions.
After a call defer ment period of 5 to 10 years, such issues become freely callable.
If the deferred-call bond is retired at the end of the deferment period, the YTC will measure the expected yield on the bond.
There are two simple modifications you can make to the standard Ytm equation to find the YTC.
Second, instead of using the bond's par value, use the bond's call price, which is stated in the indenture and is frequently greater than the bond's par value.
For example, if you want to find the YTC on a 20-year, 10.5% deferred-call bond that is currently trading in the market at $1,206 but has five years to go to first call, you can do so at which time it can be called.
Instead of using the bond's maturity of 20 years in the equation, you use the number of years to first call, five years, and the issue's call price, $1,085.
You can still use the bond's coupon and market price.
At a discount rate of 7%, the present value of the future cash flows will be exactly the same as the bond's current market price.
This bond has a YTC of 7%.
The bond's Ytm is 8.31%.
In practice, bond investors compute both Ytm and Ytc for deferred-call bonds that are trading at a premium.
The yield to call is used to value the premium bond.
The assumption is that because interest rates have dropped so much, it will be called in the first chance the issuer gets.
When this or any bond trades at a discount, the situation is completely different.
The YTP will always be less than the YTP on any discount bond.
The YTC is only used with premium bonds.
You can use the keystrokes shown in the margin.
Some investors prefer to trade in and out of bonds over a short period of time.
Measures such as yield to maturity and yield to call are meaningless, other than as indicators of the rate of return used to price the bond.
These investors need an alternative measure of return that they can use to assess the investment appeal of the bonds they intend to trade.
It shows the rate of return an investor can expect by holding a bond for less than the life of the issue.
The expected return doesn't have the precision of the yield to maturity because the major cash flow variables are the product of investor estimates.
The future selling price of the bond and the length of the holding period are pure estimates and subject to uncer tainty.
You can use the same procedure to find a bond's realized yield as you did to find the promised yield.
The following equation can be used to find the expected return on a bond.
The future price of the bond must be determined when the expected return is calculated.
The standard bond price formula is used to do this.
Coming up with future market interest rates that you feel will exist when the bond is sold is the most difficult part of getting a reliable future price.
If you want to calculate the bond's future price, you can use current and expected market interest rate conditions to estimate the yield and then use that yield to calculate the bond's future price.
We have a 7.5%, 15-year bond.
Let's assume that the price of the bond will rise as interest rates fall over the next few years.
Assume the bond is currently priced at $810 and you will hold it for three years.
You think market rates will fall to 8%.
If you know that the bond will have 12 remaining coupon payments in three years, you can use Equation 11.3 to estimate the bond's price at $960 in three years.
If you buy the bond today at a market price of $810 and sell it three years later at a price of $960, you'll make a profit.
The expected return on this bond is 14.6%, which is the discount rate in the following equation that will produce a current market price of $820.
The return on this investment is fairly substantial, but keep in mind that this is only an estimate.
It is subject to variation if things don't go as planned, particularly with regard to the market yield at the end of the holding period.
This example uses annual compounding, but you could just as easily have used semiannual compounding, which would have resulted in an expected yield of 14.4% rather than the 14.6% found with annual compounding.
That is 14.21%.
The bond equivalent yield is shown in the spreadsheet.