Chapter 6 - Bonds

Introduction to Bonds

A bond is a type of security sold by the govermnent or by corporations to raise money from investors today in exchange for a promised future payment amount. In order to familiarize ourselves with bonds a little better, we will learn a couple of key vocabulary terms. The terms of the bond as well as the amounts and dates of all the payments to be made are described in a document what is referred to as a Bond Certificate. Once a bond is issued, payments on the bond are made until the maturity date, which is the final repayment date of a bond. The bond term is the time remaining until the final repayment date of the bond. One of the types of payments that bonds pay to their holders is called the face value of a bond, and it is the amount we use to compute the interest payments. It is usually due at the bond's maturity and is sometimes referred to as par value or principal amount. Along with face value, some bonds describe in their certificate additional payments called coupons. Coupons are promised interest payments of a bond, and are paid periodically until the maturity date. Each coupon payment is determined by the coupon rate, usually expressed as an APR. If you wish to calculate a bond's coupon payment, you must use the following computation formula:

Coupon Payment CPN = (Coupon Rate x Face Value) / Number of Coupon Payments per Year
For example, if you have a bond with face value of $1,000 and a coupon rate of 15% with quarterly payments, your coupon payment will be:
CPN = (.15 x 1000) / 4 = $37.50 is paid to you every 3 months.

Zero-Coupon Bonds

A zero-coupon bond is a bond that makes only one payment at maturity. This means that there is no coupon payment, hence the name zero-coupon bond. A type of zero-coupon bond is a Treasury Bill, which are issued by the U.S. governemnt with a maturity of up to one year. Zero-coupon bonds always trade at a discount, or where the trade price is less than the face value. Because this is always the case, zero-coupon bonds are also referred to as pure discount bonds.
With these types of bonds, you deal with only two cash flows. First, the initial payment of the current market price of the bond at time of purchase, then the face value we receive at the maturity date. Although the bond pays no coupon payments, you are compensated for your time value of money by purchasing the bond at a price less than the face value (discount).
Taking it a step further, we can now calculate the bond's Yield to Maturity (YTM). This is the IRR of an investment in a bond that is held to its maturity date, or the discount rate that sets the present value of the promised bond payments equal to the current market price for the bond. Essentially, this is the return you will make if you buy the bond at the current market price, hold it to maturity, and then cash it in for face value.
Hypothetical Situation: A $105,000 face value, one year bond costs you $95,000 if you purchase it today.
Equation to Calculate the bond'sYTM: 1 + YTM = (Face Value / Price)^(1/n)
where the YTM is a per-period rate of return for holding the bond until maturity date n
YTM: $95,000 = 105,000 / (1+YTM)
Rearranging the equation, you get: YTM = (105,000 / 95,000) - 1 = .10526 or 10.53%
The YTM in the above example is 10.53%, which is like saying if you invest in this bond and hold it to maturity, it is like earning 10.53% interest on an initial investment of 95,000.

A default-free zero-coupon bond that matures on date n provides a risk-free return over a period. This is guaranteed by the Law of One Price, where the risk-free interest rate equals the yield to maturity on such a bond. These default-free zero-coupon yields are sometimes referred to as spot interest rates, since the rates are offered on the spot at a particular time. In chapter 5, we got introduced to the yield curve for risk-free interest rates and different maturities. These correspond with risk-free zero coupon bonds as well; we call this the zero-coupon yield curve. It plots the yield of risk-free zero-coupon bonds as a function of the bond's maturity date. The curve has an increasing slope that gets more steady as the years increase. In general, the longer maturities show a higher yield.

Coupon Bonds

Coupon Bonds are bonds that pay regular coupon interest payments up to maturity, when the face value is also paid. Two prime examples of coupon bonds are treasury notes and treasury bonds. Treasury Notes are a type of U.S. treasury coupon security that is currently traded in financial markets having original maturities from 1 - 10 years. Treasury Bonds are similar to treasury notes, with the exception that their original maturities are more than 10 years.

Looking at the cash flow of coupon bonds, we see that the return comes from two sources. The first source is difference between the purchase price and the principal value, and the second source is the periodic coupon payments that the investor receives. For example:
If we have a 2 year note, with a par value of $100, and a coupon rate of 10% with quarterly coupon payments, what cash flows will we receive if we hold the note until maturity?
Our Coupon Payment CPN = $100 x .05 / 4 = $1.25
With this note, you receive payments of $1.25 every three months for 2 years. Your last payment will be the $100 par value plus the $1.25 coupon payment.
Once we can determine the cash flow as in the example above, we can then calculate the yield to maturity (YTM) of coupon bonds. The YTM of the bond is the single discount rate that equates the present value of the bond's remaining cash flows to its current prices. This complicates the YTM calculation for coupon bonds since there are many cash flows. The yield to maturity of a coupon bond is the interest rate y that can be calculated using the following equation:
Yield to Maturity of Coupon Bond
P = CPN x (1/y)(1-[1/(1+y)^n] + (FV / [1+y]^n)
Because this formula is very complex to solve for y, many people use a spreadsheet or an actual financial calculator to solve the YTM.

Fluctuations In Bond Prices

Changes in the price of bonds generally come from two reasons. First, with the passage of time, as the bond gets closer to the maturity date and keeping the YTM fixed, the present value of the bond's remaining cash flows will change as the time to maturity decreases. Second, the YTM and the price are always affected by changes in the market's interest rates. Bond's don't always trade at a discount, sometimes they even trade at a premium - a price at which coupon bonds trade that is greater than their face value. Sometimes, bonds will even trade at par (traded at a price equal to it's face value). In this case, the only return the investors will get comes from the coupon payments the bond offers.

When it comes to interest rates, a higher YTM has a higher discount rate being applied to the cash flows because investors demand a higher return for investment. This results in the reduction of the present value and ultimatley the bond's price. On the opposite end, when interest rates fall, investors then demand a lower YTM, ultimatly reducing the discount rate applied to the cash flow of the bonds and therefore raising the price. Simply stated, as interest rates and yields rise, bond prices will fall, and as interest rates and yields fall, bond prices will rise. Interest rates and bond prices will always move in opposite directions. Bonds with different maturities will respond to interest rates differently, each will have a unique sensitivity when responding to interest rate risks.

When comparing time and bond price, we see that the price of bonds increases to reflect the increasing present value of a cash flow as the next payment from a bond grows nearer. In essence, the price slowly rises as a coupon payment nears and then drops after the payment is made. The pattern continues for the life of a bond, creating zig-zags in a curve's general slope:


Since such fluctuations are predictable to bond traders, they don't bother to quote the price of bonds in terms of it's actual cash price (also called the dirty price or invoice price). Instead, they quote bonds in terms of what they call a clean price, or the bond's price less an adjustment for interest accrued. Clean price can be calculated by taking the cash price and subtracting the accrued interest. Accrued interest can be found by taking the coupon amount and multiplying it by the quantity of the days since last coupon payment divided by days in current coupon period. By subtracting the accrued interest from a bond's cash price and by computing the clean price, we can eliminate the zig-zag pattern in the slope and come up with a more accurate way of quoting bonds.

Bonds At The Corporate Level

Government is not the only distributor of bonds. Many corporations issue bonds as a means of raising funds without giving up ownership. With corporate bonds, however, comes a default risk, unlike with government bonds. Corporations with higher default risk will most likely pay higher coupon payments to attract investors and buyers. This default risk is referred to as a credit risk. It is often an indicator that the bond's cash flows are not known with certainty. In general, investors will pay less for bonds with credit risk than they would for an otherwise identical default-free bond. Also, because the YTM for a bond is calculated using the promised cash flows instead of the expected cash flows, the yield of bonds with credit risk will be higher than that of otherwise identical default-free bonds. This leads us to conclude that the YTM of a defaultable bond is not equal to the expected return of investing in the bond.

Bond Ratings
There are companies out there such as Standard & Poor's or Moody's that assess the creditworthiness of particular bonds and place a rating on them. The following table analyzes the rating classes each company uses. Bonds that are assumed to least likely default receive the AAA rating. The following table shows the two companies, their ratings, and the description of each rating:


Investment-grade bonds are bonds in the top four categories of creditworthiness with low risk of default. Speculative bonds (junk bonds or high-yield bonds) are bonds in one of the bottom five categories of creditworthiness that have high risk of default. These bonds promise higher yields because their risk of default is high. The difference between the yields of corporate bonds and the Treasury yields are referred to as the default spread or credit spread. The credit spreads will alter depending on how investors percieve the probability of default on bonds.