Principles of Finance/Section 1/Chapter 4/Bonds/Valuation

The valuation of a bond can be broken down into two basic tasks: the valuation of the stream of coupon payments, and the valuation of the repayment of the face value of the bond.

Valuing the stream of coupon payments is no different than valuing any other basic annuity.

PVcoupons = $$C\left[\frac{1}{r}-\frac{1}{r\left(1+r\right)^t}\right]$$

where C = coupon payment, t = years to maturity, and r = required rate of return.

Valuing the principle is even simpler, just use a basic present value formula:

PVprinciple = $$\frac{F}{(1+r)^t}$$

where F is the face value, r is the discount rate, and t is the number of years to maturity.

And so, to find the present value of the entire bond, we put these two formulae together to get:

PVbond = $$C\left[\frac{1}{r}-\frac{1}{r\left(1+r\right)^t}\right] + \frac{F}{(1+r)^t}$$

Now, let's suppose we are given the following problem: ''A $1000 8% annual bond matures in 7 years and is yielding 5%. Find the price of the bond.''

The first thing we see is $1000. This is the face value of the bond, which is the amount of money the issuing party will pay on maturity. The next thing we see is 8% annual. This means that the company is paying 8% of $1000 every year as interest, which comes out to $80. Therefore, we shall use 80 in place of C in our equation. The time to maturity is 7, which is self explanatory. Finally, we see that the bond is "yielding 5%". This means that we should use 5% as the discount rate in our equation. So, plugging in this information we see that:

$$80\left[\frac{1}{.05}-\frac{1}{.05\left(1.05\right)^7}\right] + \frac{1000}{(1.05)^7} = $1,173.59$$

It is important to note that the price of this bond, $1,173.59, is higher than the face value of $1000. This means that the bond is selling at a Premium. If the price were lower than the face value, the bond would sell at a Discount, and if the bond were selling for exactly the face value, the bond is said to be selling at Par. You can determine if a bond is selling at a discount, a premium, or par without actually valuing the bond. If the coupon rate is higher than the yield, the bond will sell for a premium. If the coupon rate is lower than the yield, the bond will sell for a discount. And, of course, if the yield equals the coupon, the bond will sell at par.

It is also possible to determine the price of a bond using a financial calculator, such as the Texas Instruments BA-II.

By pressing CPT followed by PV, the calculator will compute the price of the bond. Given any 4 of the TVM variables, the calculator can compute the remaining variable. Note that in order to make the calculation work, the PV must be negative (since purchasing the bond is a negative cash flow to the buyer).

Most bonds in the real world pay semi-annual coupons, and this must be taken into account when valuing a bond. In the example above, let's suppose the bond paid semi-annual interest. In order to calculate the correct price, we divide the coupon payment in half to get $40, and double the time to maturity, to get 14. In order to find the appropriate yield, we must use the following formula:

$$Effective Rate\ =\ \left[1+\frac{Annual Rate}{m}\ \right]^m-1$$

Where m is the number of compounding periods per year. In this case, it is two. So:

$$0.05\ =\ \left[1+\frac{x}{2}\ \right]^2-1$$

In this case, x = 0.04939, and so by dividing this in two we can get the effective per period interest rate of 2.4695%.

Then we simply plug in the values into the bond valuation formula:

$$40\left[\frac{1}{.024695}-\frac{1}{.024695\left(1.024695\right)^{14}}\right] + \frac{1000}{(1.024695)^{14}} = $1,179.31$$

All other things being equal, a bond will be worth more the more times per year interest is paid.