• Valuation Definition -- The value of any financial asset/security is equal to the present value of all cash flows which that asset/security will generate over its lifetime discounted back to today at an appropriate discount rate.
• The primary difference among different assets/securities (such as stocks vs bonds vs corporate projects) is that each asset/security will have a different cash flow stream and a different discount rate depending on the riskiness of that cash flow stream.

Three-Step Valuation Process

We can use the same process to value all stocks, bonds, or other investment opportunities. The process is as follows

1. Forecast all cash flows which that asset/security is expected to generate over its lifetime.
2. Determine an appropriate discount rate
3. Solve for PV

What changes as we deal with different stocks, bonds, and other investment opportunities is not the process, but how we apply the process. For instance, estimating the cash flows for bonds is simple as the cash flow is generally fixed in size and for a specific time period. On the other hand, estimating the cash flow stream for stocks is trickier because the cash flows are usually variable and infinite. Stocks tend to be riskier than bonds so a higher discount is usually used. Finally, solving for PV is a straightforward application of the 5-key approach for bonds while it typically involves formulas and the cash-flow worksheet for stocks.

Bond Pricing

As stated above, the value of a bond is equal to the present value of the cash flows that the particular bond will pay. Bonds pay cash flows in two different ways. First, a coupon payment. Every six months the bondholder receives a coupon payment determined by the stated coupon rate. When bonds are issued, they state the coupon rate which typically (in this class we will assume always) remains fixed over the life of the bond. This is the percentage of par value that the the bondholder will receive in annual interest payments (while technically different bonds may have different par values, most corporate bonds have a standard par value of $1000 which we will use as the standard for this class...always assume the par value is $1000 unless specifically stated otherwise). To calculate the amount the bondholder will receive every six months, just take the annual payment and divide by two. In addition to the coupon payment, at the end of the bonds life the bondholder will receive the par value ($1000). Sometimes this par value is referred to as maturity value or face value. Thus, to find the price (or value) of a bond (B0), we want to find the present value of the coupon payments and the par value.

Consider the following example which I'll walk through using the financial calculator. I want to purchase a bond that pays a 6% coupon and want to earn an 8% return on my investment. The bond has 20 years remaining until maturity. The first thing I need to do is figure out what my coupon payment is going to be.

Annual Coupon ==> 0.06*$1000 = $60
Semi-Annual (every 6 months) Coupon ==> $60/2 = $30

This tells me that I am going to receive an interest payment of $30 twice per year for each of the next 20 years plus at the end of the 20th year, I will receive $1000. This is my cash flow stream which must be discounted back to today at the 8% required return that I want to receive. Because I am receiving 2 coupon payments each year, I must be careful to set my calculator to 2 periods per year (and remember that now N represents the number of periods (40) instead of the number of years (20). This is done as follows:

Financial Calculator
2 P/Y
40 N
8 I/YR
30 PMT
1000 FV
Compute PV = $802.07

Note that the PMT and the FV are both positive. They could also both be negative. The key is to recognize that they should both be the same sign. Why is this? >From the perspective of the bondholder, we will RECEIVE both an annual cash flow (the coupon payment ==> PMT) and a single cash flow at maturity (the par value ==> FV). Since we receive these, they should be positive. If we took the perspective of the bond ISSUER, both of these would be payments and thus be negative. The key is that both the PMT and the FV are flowing in the same direction and thus must have the same sign

One thing that can be confusing with bonds is that there are two "rates" that are mentioned in bond pricing. The first is the coupon rate and the second is the discount rate. Note that the discount rate can take on many names -- market rate of interest, interest rate, rate of return, required return and yield-to-maturity -- they all mean the same thing. In order to avoid confusion, try to think of the coupon rate as a cash flow rather than a rate. The coupon rate tells us what our yearly payment will be. It is not a rate of return and it doesn't change over time. The discount rate (market rate of interest, interest rate, rate of return, required return, and/or yield-to-maturity) tells us what rate of return we want to earn on our investment in this bond. It can (and will) change over time -- sometimes increasing and sometimes decreasing -- depending on market conditions.

Yield-to-Maturity (YTM)
The YTM represents the EXPECTED return on the bond if it was purchased at the current price AND held until maturity. We can also use the YTM to tell us what the current required return is for the market. We solve for this by using the same approach we used to solve for interest rates (or discount rates, rates of return, growth rates) in Chapter Five -- by solving for the I/Y with the 5-key approach on our financial calculator. We know the bond price (B0 ==> PV), coupon rate (provides PMT), number of years until maturity (provides N), and maturity (par) value (provides FV of $1000). The only thing we don't know is the I/Y (which is the yield to maturity). In order to get the YTM, we are solving for the rate of return that makes the PV of cash flows (coupon payments and par value) equal to the current bond price (B0). Again, lets work through a brief example.

Assume that I am considering buying a bond that pays a 7.5% coupon and can purchase this bond for $1095. If this bond has 10 years remaining until maturity, what is my YTM? Again, the first step is to find the coupon payment.

Now I know that I can buy this bond for $1095 today and in return I will receive cash flows of $37.50 twice per year for each of the next 10 years PLUS $1000 at the end of the 10th year. Remember, we are trying to find the discount rate where the PV of the cash flows is equal to $1095.

Financial Calculator
2 P/Y
20 N
-1095 PV
37.50 PMT
1000 FV -- Note that the PMT and FV must both be the same sign and opposite of the PV
Compute I/YR = 6.21%

Bond Prices and Interest Rates

1. As market rates of interest rise, bond prices will fall. Alternatively as market rates of interest fall, bond prices will rise. Bond prices and the market rate of interest are inversely related. This is because the cash flow stream you receive from the bond is fixed. As market rates of interest go up, you are discounting that fixed cash flow stream back at a higher rate which makes it less valuable. As market rates of interest go down, you are discounting that fixed cash flow stream back at a lower rate which makes it more valuable. The market rate of interest is the rate of interest available on similar risk securities purchased today. Also, we can think of the YTM as the market rate of interest. Keep in mind when we say interest rates went up or down, we are talking about the YTM, NOT the coupon rate. The coupon rate does not change.
2. This relationship is stronger for bonds with a longer time until maturity. Therefore, a 20-year bond will have a higher premium than a similar 3-year bond after interest rates have declined. The rationale for this relates again to the time value of money. A bond with a greater time to maturity will have a longer fixed cash flow stream which means it will be affected to a greater degree by changes in interest rates. As a side note, this is a little bit of a simplification. The sensitivity to interest rates is technically affected by a combination of time to maturity and the coupon rate (duration), but that is beyond the scope of this class. For our purposes, it will suffice to know that the greater the time to maturity, the more sensitive the bond is to changes in interest rates.
3. This relationship is also stronger for bonds with lower coupon rates. Everything else being equal, the lower the coupon rate of a bond the more sensitive it will be (in terms of percentage changes in the bond price) to interest rate changes. Therefore, a 15-year 4% coupon bond will see a greater percentage price increase if interest rates decline than a similar 15-year 10% coupon bond.

Call Provisions

A Call Provision is a provision included in the bond indenture that gives the company that issued the bond the right, at their discretion, to purchase (call) the bond back from investors before it matures for a pre-set price. Usually the call provision does not start immediately, but becomes effective after a 5-10 year time period. Also, the pre-set price is typically a small premium to the $1000 maturity value.

To calculate the Yield-to-Call (YTC) we approach the problem in a similar manner as the YTM, except for two differences. First, the number of years until the first call date is used as opposed to the number of years until maturity. Second, the call price (which usually includes a small premium over par value) is used instead of the maturity value. For example, consider a bond that has a current price of $925, a coupon rate of 5.5%, and is callable in 5 years at $1050. Find the YTC for this bond. To do this, follow the same procedure outlined above for calculating the YTM, but now the FV is $1050 instead of $1000 and the number of years is 5. As a check figure, you should arrive at a YTC of 8.19%.

Zero Coupon Bonds

Zero coupon bonds are really quite easy. Essentially, as the name implies, they are bonds that pay no coupon payments. Thus, you buy them today and at maturity you receive the maturity value (typically $1000). Since you get no annual payments, these bonds will always be purchased for less than maturity value. Therefore, your return comes from the difference between what you pay for the bond and the maturity value you will receive when the bond matures. This difference is referred to as the discount. If interest rates do not change between the time you purchase the bond and the time the bond matures, the value of the bond will gradually increase as it approaches maturity. However, interest rates rarely remain unchanged which will cause the bond value to increase or decrease over the life of the bond, but the long-term trend will always be upwards since you will receive $1000 upon maturity (assuming the bond doesn't default).

Pricing and finding the YTM for a zero coupon bond is quite simple. Just use the process that we used for pricing normal bonds and plug in a zero for the coupon payment.

Zero coupon bonds are even more sensitive than ordinary bonds to interest rate changes. To verify this consider two bonds with 30 years to maturity. Let one be an 8% coupon bond and one be a zero-coupon bond. Then calculate their prices when the current market rate of interest is 8%. You should get $95.06 (even though there are no coupon payments, we keep the semi-annual -- 2 periods per year -- discounting for consistency) for the zero and $1000 for the 8% coupon. Now, how much will prices fall if the market rate of interest falls to 7%. This time you should get $126.93 for the zero and $1124.72 for the 8% coupon. Note that the decline in interest rates caused the zero to increase in value by 33.5% [($126.93 - $95.06)/$95.06] while the 8% coupon bond only increased by 12.5% [($1124.72 - $1000)/$1000]. Don't take my word for these numbers, do them yourself.

Many of you likely own now or at one time previously owned a form of zero-coupon bond. Most US Savings Bonds are zero-coupon in nature. The big difference is that most zero-coupon bonds mature for $1000 while US Savings Bonds earn interest for 30 years and may mature for much more than their stated value. If you have some US Savings Bonds and you are curious as to what they are worth, there is a link on the main page that you can use to find out.

Other Key Bond Terms

Make sure you are familiar with the terms from the Key Bond Terms Handout (available in class or on Blackboard).