MATT BRIGIDA
Associate Professor of Finance (SUNY Polytechnic Institute) & Financial Education Advisor, Milken Institute Center for Financial Markets
The value of a bond is the present value of the discounted expected cash flows. So to value the bond we'll need information on the cash flows and discount rate. This information is summarized in the:
For example, say a bond has a 5% coupon rate, a \$1000 par value, 6 years to maturity, makes payments annually, and lastly has a 8% $YTM$ (we'll define YTM in greater detail a little later on).
To find the value of the bond we simply sum the present value of each individual cash flow. So we sum the values in the table below. Note $1.08 = (1 + YTM)$ where the yield-to-maturity is 8%.
The sum of the present value of these cash flows is \$861.31.
The value of the bond is often written as: `$$ B_0 = \sum_{i=1}^{6} \frac{50}{1.08^i} + \frac{1000}{1.08^{6}} = \$861.31 $$`
Which can be rewritten to use the present value of an ordinary annuity formula: $B_0 = \$50\left(\frac{1-\frac{1}{1.08^{6}}}{.08}\right) + \frac{\$1000}{1.08^{6}} = \$861.31$
These calculations are equivalent, and the differences are only matters of notation. When using in a spreadsheet, the calculations are usually laid out in a form similar to the table. The annuity formula was mainly taught because prior to spreadsheets people were doing these calculations by calculator.
Most coupon bonds pay semiannual interest. Rates are always quoted annually however. So what this means is that the period of the bond is 6 months, and the coupon rate per period is $\frac{5\%}{2} = 2.5\%$ and the $YTM$ is $\frac{8\%}{2} = 4\%$.
The next slide contains an app which will allow you to value different coupon bonds. Try to value them in a spreadsheet (or whatever software you choose) and check your answer with the app.
The $YTM$ is the single discount rate for which the present value of all the bond's future cash flows equals the bond's market price.
The $YTM$ is an annual quoted rate, which means it is the rate per period multiplied by the number of periods in a year. So if you calculate the $IRR$ of a bond with semiannual payments, you must multiply it by $2$ to get the $YTM$.
You can practice calculating the $YTM$, and check your answers with the app on the following slide.
One misconception about bonds is that if you buy a bond with a 9% $YTM$ and hold it to maturity, then so long as the bond doesn't default, you will earn 9% per year.
To get the value of a bond, we discount each cash flow at the $YTM$. However, to get the $YTM$, we set the discounted cash flows equal to the market price and then solve for the discount rate ($YTM$). This, of course, is circular.
For example, two 10-year U.S. Treasury notes will have the same $YTM$, but can have vastly different prices. A 10-year note issued yesterday will trade near \$1000 because its $YTM$ is very close to its coupon. A 30-year Treasury bond issued 20 years ago (at a say 18% coupon) is now a 10-year Treasury note with a price based on its $YTM$ and coupon rate differential. The bonds will have different prices, but the same $YTM$.
Above, in the section on Semiannual Interest, we asked whether the effective semiannual rate was always lower than the effective annual rate. The answer is:
The reason is due to compounding. When we switch from annual to semiannual we are now compounding the rate---and compounding has a larger effect on higher rates. So it has a larger effect on the $YTM$ or coupon, depending on which is higher.