[MUSIC PLAYING] Now that we have walked around the concept of bonds to gain an understanding, learn their advantages, characteristics, and all the different types, it's time to look at the core financial question, what is the value of a bond.
As with all financial securities, the true value can be calculated from the net present value of all the expected cash flows. If a bond holder holds a bond to maturity, they will receive interim interest payments at the coupon rate and the face value at maturity. So the value of the bond is the present value of each of these cash flows added together.
Let's look at these cash flows on a timeline. If we're going to receive these coupon payments in the face value at maturity, what is the interest rate at which the present value equals the price of the bond today?
A quick note here about price. Price can be quoted in dollars, but it is often quoted as a percentage of face value. For example, if a bond with $1,000 face value was selling for $980, it could be said the bond is selling at 98. If it was selling at $1,050, it would be said that the bond is selling at 105.
Now if we did try to solve this through the brute strength method by directly solving the equation. Here's what we are facing. P is the price today, which we know. And CPN is the coupon payment. And the future value is the face value at maturity.
The denominators here calculate the present value for each of these cash flows. To solve for the i, interest rate, can really prove to be quite complicated. But it is important because this rate is called the yield to maturity. It is like the internal rate of return for the bond's cash flows over the market price.
Now fortunately there are tools for this, including spreadsheet software like Excel and web apps. Let's take a look. Let's use a yield calculator found on the web to calculate the yield to maturity of a bond trading at par.
Let's say the bond is face value of $1,000 with a coupon rate of 8%. And it's five years left to maturity. And the coupon payments are of course semiannually. And it currently is trading at $1,000, or at 100, or it's trading at par.
What do you think the yield of maturity would be here? Well, the yield to maturity is 8%. And it's 8%, the same as the coupon rate, because it is currently priced at par. So the yield, the internal rate of return, is 8%, the same as the coupon payments.
But what if it was selling at a discount? Let's take this same bond with $1,000 face value and the 8% coupon with five years to maturity. And it's currently selling for $940, or at 94. What do you think the yield to maturity would be here? 8%? Higher? Lower?
It would be 9.54%. Why is it higher than the coupon rate? Because the current price is low. The current price is only $940. And if we're going to still get the face value at maturity and the same coupon payments, the net present value of those is worth more to us. 954 is the yield to maturity.
Let's look at the same bond if it was selling at a premium. The face value is again 1,000 with the 8% coupon rate and five years to maturity. But its price today is $1,090, or 109. So what is the yield to maturity now? 5.90%-- less than the coupon.
Well, why is this yield less than the coupon rate? Because it's selling at a premium, and we have to pay so much for it, the yield to maturity, the return, will be less.
Let's review. A bond pays a coupon payment periodically through its life to maturity. And at maturity, the holder receives the face value, which is also fixed. With these two components being fixed, what changes in the market is the price. And the interest rate at which the discounted cash flows of the coupon and maturity payments equal the price is the yield to maturity.
In the next lesson, we'll wrap up our discussion of bonds with a look at the risk.