2.5 Fixed Income Securities: Bond Components and Valuation Formula

Since the coupon cash flow of a bond constitutes an annuity, the calculation of a bond’s dollar price involves a simple solution:

[latex]^{P}Bond=\sum_{i=1}^N\frac{C_{i}}{(1+YTM/P)^{i}} + \frac{Face Value}{(1+YTM/P)^{np}}[/latex]

The above formula says that every (annuity) coupon payment and the one-time face value are discounted and aggregated to present value, which is the bond’s worth or price.  This is based on our “valuation premise.”  Here are some important terms to know:

Face Value = Maturity Value = Par Value = Principal: These phrases are synonymous and interchangeable.  Maturity value is the amount of money the investor, or bondholder, gets back when the bond matures.

Coupon Rate (“C”) = Interest Rate (“I”): This is the amount of interest that the bond pays.  (An exception would be a variable rate, but we do not deal with that here.)  The dollar amount paid is this rate times the face value.  Thus, if the rate is 10%, it will pay $100 per year for a bond with a maturity value of $1,000.  If this bond pays semi-annually, the investor will receive two $50 payments every year, one every six months.

Yield-to-Maturity (“YTM”) = Market Rate = Discount Rate: This is the market-determined rate at which the bond’s cash flows are “priced,” or discounted, to present value, given the bond’s maturity and “creditability” (based on its credit rating).  This rate will change daily depending on macroeconomic, financial market, and perhaps other conditions.  Therefore, while one may receive a fixed annuity series of coupon payments and a single maturity payment in the future, which are paid at a certain rate, the present value of the funds yet to be received shall be discounted by a market yield, or YTM.  Do not confuse coupon and market rates; they are separate and mathematically distinct.  The market yield will constantly change.

We may think of Face Value and the bond’s Dollar Price in terms of $100s or $1,000s or any multiple thereof.  Since the bond’s price is expressed in terms of 100% of the bond’s “Par” value, it doesn’t matter, for calculation purposes, how many decimal zeros we add on in an illustration or exercise.  For greater clarity on the “quoted price,” see the mathematical example to follow.

The bond trader will quote the bond in terms of a percent of par and will then ask the buyer or seller “what the size of the trade is?” or in other words how much they are dealing with.  Of course, in reality, buying or selling a thousand, or a million, dollars’ worth of bonds matters a great deal.  An example follows.