The value of any asset is the present value of its cash flows. Therefore, we need to know two things:. We have already identified the cash flows above.
Take a look at the time line and see if you can identify the two types of cash flows. Using the principle of value additivity , we know that we can find the total present value by first calculating the present value of the interest payments and then the present value of the face value. Adding those together gives us the total present value of the bond.
We don't have to value the bond in two steps, however.
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The PV function can handle this calculation as we will see in the next example:. Assuming that your required return for the bond is 9. We can calculate the present value of the cash flows using the PV function, but we first need to set up our worksheet.
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Open a new workbook, and then duplicate the worksheet presented below:. Note that I have set up the data using annual values for the coupon rate, required return, and term to maturity. I have also included a cell B6 that provides a place to specify the number of payments per year.
This way, we can set up the formula without making assumptions regarding the payment frequency, which adds some flexibility since not all bonds pay semiannually. To calculate the value of the bond, in B8, we use the PV function:. Take notice of the "-" in front of the function.
If I didn't put that there, then the function would have returned a negative value. Technically, that would be correct because you would have to pay a cash outflow that amount. However, we tend to think in terms of positive dollars, not negative. Also note that the required return and annual payment are converted in the function to semiannual values by dividing by the payment frequency.
Similarly, the number of years to maturity is converted to the number of semiannual periods by multiplying by the payment frequency.
Notice that the bond is currently selling at a discount i. This discount must eventually disappear as the bond approaches its maturity date. A bond selling at a premium to its face value will slowly decline as maturity approaches. In the chart below, the blue line shows the price of our example bond as time passes. The red line shows how a bond that is trading at a premium will change in price over time. Both lines assume that market interest rates stay constant. In either case, at maturity a bond will be worth exactly its face value.
Keep this in mind as it will be a key fact in the next section. In the previous section we saw that it is very easy to find the value of a bond on a coupon payment date. However, calculating the value of a bond between coupon payment dates is more complex.
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As we'll see, the reason is that interest does not compound between payment dates. That means that you cannot get the correct answer by entering fractional periods e. We are going to go through the whole process here, but you can jump directly to the section that uses the Price function if you don't care about the details. Let's start by using the same bond, but we will now assume that 6 months have passed.
That is, today is now the end of period 1. What is the value of the bond at this point? To figure this out, note that there are now 5 periods remaining until maturity, but nothing else has changed. Therefore, simply scroll up to B5 and change the value to 2. Notice that the value of the bond has increased a little bit since period 0. As noted previously, this is because the discount must eventually vanish as the maturity date approaches.
Now, is there another way that we might arrive at that period 1 value? First reset B5 to 3. Remember that your required return is 4. Therefore, the value of the bond must increase by that amount each period. Put this formula in a blank cell to prove it:. That's not the same answer. However, remember that this is the total value of your holdings at the end of period 1.
If we subtract that, you can see that we do get the same result:.
This is one of the key points that you must understand to value a bond between coupon payment dates. Let me recap what we just did: We wanted to know the value of the bond at the end of period 1. So, we calculated the value as of the previous coupon payment date, and then calculated the future value of that price. Then, we subtracted the amount of accrued interest to get to the quoted price of the bond. Using the same bond as above, what will the value be after 3 months have passed in the current period? Assume that interest rates have not changed.
So, we are now looking for the value of the bond as of period 0. Unfortunately, the PV function can only help us with this for the first step.
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Recall that we first need to calculate the PV of the cash flows as of the previous payment date period 0. So, in cell A12 put the label "Fraction of Period Elapsed", and then enter 0. Remember that this gives us the "dirty" price of the bond it includes the accrued interest. The YTM takes into account both the interest income and this capital gain over the life of the bond. There is no formula that can be used to calculate the exact yield to maturity for a bond except for trivial cases. Instead, the calculation must be done on a trial-and-error basis.
This can be tedious to do by hand. Fortunately, the Rate function in Excel can do the calculation quite easily. Technically, you could also use the IRR function, but there is no need to do that when the Rate function is easier and will give the same answer. But wait a minute! That just doesn't make any sense.
You need to remember that the bond pays interest semiannually, and we entered Nper as the number of semiannual periods 6 and Pmt as the semiannual payment amount So, when you solve for the Rate the answer is a semiannual yield. Since the YTM is always stated as an annual rate, we need to double this answer.
In this case, then, the YTM is 9. Change your formula in B14 to:. So, always remember to adjust the answer you get from Rate back to an annual YTM by multiplying by the number of payment periods per year. Many bonds but certainly not all , whether Treasury bonds, corporate bonds, or municipal bonds are callable. That is, the issuer has the right to force the redemption of the bonds before they mature. This is similar to the way that a homeowner might choose to refinance call a mortgage when interest rates decline.
If you wish, you can jump ahead to see how to use the Yield function to calculate the YTC on any date. Given a choice of callable or otherwise equivalent non-callable bonds, investors would choose the non-callable bonds because they offer more certainty and potentially higher returns if interest rates decline. Therefore, bond issuers usually offer a sweetener, in the form of a call premium , to make callable bonds more attractive to investors.
A call premium is an extra amount in excess of the face value that must be paid in the event that the bond is called before maturity. Notice that the call schedule shows that the bond is callable once per year, and that the call premium declines as each call date passes without a call. It should be obvious that if the bond is called then the investor's rate of return will be different than the promised YTM. That is why we calculate the yield to call YTC for callable bonds.
The yield to call is identical, in concept, to the yield to maturity, except that we assume that the bond will be called at the next call date, and we add the call premium to the face value. Let's return to our example:. What is the YTC for the bond?
I have already entered this additional information into the spreadsheet pictured above. Remember that we are multiplying the result of the Rate function by the payment frequency B8 because otherwise we would get a semiannual YTC. Note that the yield to call on this bond is Now, ask yourself which is more advantageous to the issuer: Obviously, it doesn't make sense to expect that the bond will be called as of now since it is cheaper for the company to pay the current interest rate.
As noted above, a major shortcoming of the Rate function is that it assumes that the cash flows are equally distributed over time say, every 6 months. However, bonds only pay interest twice a year, so there are only 2 days per year that the Rate function will give the correct answer. On any other date, you need to use the Yield function. Note that this function as was the case with the Price function in the bond valuation tutorial is built into Excel YIELD settlement , maturity , rate , pr , redemption , frequency ,basis.
Note that the dates must be valid Excel dates, but they can be formatted any way you wish. Also, both pr and redemption are percentages entered in decimal form. Our worksheet needs a little more information to use the Yield function, so set up a new worksheet that looks like the one in the picture below:. Note that I've had to add exact dates for the settlement date and the maturity date , rather than just entering a number of years as we did before. Also, since industry practice which the Yield function uses is to quote prices as a percentage of the face value, I have added for the redemption value in B3.
Finally, I have added a row B11 to specify the day count basis. With that additional information, using the Yield function to calculate the yield to maturity on any date is simple.