Bonds
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:
- Coupon rate
- Par (or face) value
- Years to maturity
- Payment period (usually semiannual)
- Yield-to-Maturity (`YTM`)
Cash Flow Structure
Given the previous information, we know the cash flow structure and the rate at which to discount the cash flows. For example, say a bond has a 5% coupon rate, a $1000 par value, 10 years to maturity, makes payments annually, and lastly has a 8% `YTM` (we’ll define YTM in greater detail a little later on).
- Then the bond will pay `0.05(\$1000) = \$50` every year for 10 years, with an additional $1000 in year 10. In other words, the cash flows are $50 in years 1 through 9, and $1050 in year 10.
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| $50 | $50 | $50 | $50 | $50 | $50 | $50 | $50 | $50 | $1050 |
Bond Value
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%.
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| `\frac{\$50}{1.08}` | `\frac{\$50}{1.08^{2}}` | `\frac{\$50}{1.08^{3}}` | `\frac{\$50}{1.08^{4}}` | `\frac{\$50}{1.08^{5}}` | `\frac{\$50}{1.08^{6}}` | `\frac{\$50}{1.08^{7}}` | `\frac{\$50}{1.08^{8}}` | `\frac{\$50}{1.08^{9}}` | `\frac{\$50}{1.08^{10}}` |
The sum of the present value of these cash flows is $798.69.
Bond Value
The value of the bond is often written as: `B_0 = \sum_{i=1}^{10}\frac{50}{1.08^i} + \frac{1000}{1.08^{10}} = \$798.69`
Which can be rewritten to use the present value of an ordinary annuity formula: `B_0 = \$50\left(\frac{1-\frac{1}{1.08^{10}}}{.08}) + \frac{\$1000}{1.08^{10}} = \$798.69`
- 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.
Semiannual Interest
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 value of the bond is then: `B_0 = \sum_{i=1}^{20}\frac{\$25}{1.04^i} + \frac{\$1000}{1.04^{20}} = \$796.14`
It is important to discount the $1000 at 4% over 20 periods, and not at 8% over 10 periods. This is because the effective rate is 4% semiannually, which is not equal to 8% annually (see the time-value-of-money presentation).
Food for thought: The bond value in the semiannual case was lower than in the annual. Is this always the case? See the last slide for the answer.
Bond Valuation Interactive App
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.
- You can also use the app to see how the bond’s value is affected by changes in its inputs.
Bond Valuation Interactive App
Do We Always Earn the `YTM`?
One misconception about bonds is that if you buy a bond with a `YTM` and hold it to maturity, then so long as the bond doesn’t default, you will earn 9% per year.
This isonlytrue for zero coupon bonds.
For coupon bonds, in the `YTM` calculation it is assumed you canreinvestcoupon payments at the `YTM`. In reality, you will reinvest at whatever the rate is when you receive the coupon payment.
A Note on `YTM`
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.
The `YTM` is really a way to quote the bond’s return, so you can compare it with others. This is done because the bond’s price is not that informative.
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`.
Semiannual vs Annual Values
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:
If the bond is trading at a discount (`YTM` above the coupon rate), then switching from annual to semiannual will lower the value of the bond.
However, if the bond is trading at a premium (`YTM` below the coupon rate), then switching from annual to semiannual will raise the value of the bond.
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.
Credits and Collaboration
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