19C: Bond Pricing

19c: Bond Pricing

A bond is characterized by three sets of cash flows: initial investment, periodic coupons, and final redemption. Value of a bond is simply the present value of its cash flows. In general, there are three types of bonds: pure discount bond, level-coupon bond, and consol.

Pure Discount Bond

Pure discount bond is also known as the zero-coupon bond. It promises a single fixed payment at a fixed future date. In another word, the holder of a pure discount bond receives no cash payments until maturity, as presented in Fig. 1 below.

Fig. 1 Cash Flows for a Pure Discount Bond F

Year 0 Year 1 Year 2 F

0 1 ………………………….. Year T

Therefore, the value of a pure discount bond is the present value of its final redemption amount. The formula to price of a pure discount bond is as follows:

Value of a Pure Discount Bond = F / (1+r)T

F = the face value of the bond r = the interest rate T = years to maturity

Level-Coupon Bond

Unlike pure discount bond, level-coupon bond offer cash payments not just at maturity, but also at regular times in between, as shown in Fig. 2. below. These regular payments are referred to as coupons of the bond. Typical bonds issued by either U.S. governments or American corporations are level-coupon bonds.

Fig. 2 Cash Flows for a Level Coupon Bond F+C C C C

0 1 …………………………. Year T . Consequently, the value of a level-coupon bond is the present value of its stream of coupon payments plus the present value of its repayment of principal. The formula to value a level-coupon bond is as follows:

Value of a Level-Coupon Bond = [C/(1+r)]+[C/(1+r)2]+……+[C/(1+r)T]+[P/(1+r)T]

C = Coupon payments r = interest rate P = Principal or Face Value T = Number of years to maturity

Consol

A consol is a type of bond that perpetuates forever because it never stops paying a coupon, has no final maturity date, and therefore never matures, as shown in Fig. 3 below.

Fig. 3 Cash Flows for a Consol

C C C C

0 1 …………………………. Year T . This type of bond, a product of Bank of England, is still being offered in London today. To value a consol, a perpetuity formula would need to be used:

Value of a Consol = C/r

C = Coupon payments r = market interest rate

Basic bond valuation equation:

2 T B0 = I/(1+ry ) + I/(1+ry) + (I+M)/(1+ry)

B0 = current market price of bond or debt security ($) M = par (face, maturity) value of security ($) T = term to maturity (years) r = coupon (interest) rate (%) I = rM = annual interest ($) ry = yield to maturity (YTM, %) To truly understand bond pricing, we would need to examine some important bond concepts: 1) relationship between bond prices and interest rates and 2) yield to maturity.

Relationship between bond prices and interest rates

There is an inverse relationship between bond prices and interest rates. As interest rates rise, bond prices fall, and vice versa. For example, let’s take a two-year level-coupon bond paying 10% coupon annually and assume that the current interest rate is 10%. The bond is priced at its face value of $1,000:

Value of the level-coupon bond = [100/1.10] + [(1,000+100)/(1.10)2] = $1,000

Now, if the interest rate unexpectedly rises to 15 percent, this bond that originally issued under the 10% interest rate environment would sell at

Value of the bond = [100/1.15] + [(1,000+100)/(1.15)2] = $918.71

As it has indicated through the pricing, higher interest rates lead to lower bond prices. Since the $918.71 is below $1,000, the bond is said to sell at a discount. However, if a new bond is being issued with coupon rate of 15% in the 15% interest rate environment, the bond would be valued at $1,000. To see the opposite effect, if interest rates fall to 5%, the original bond that was issued under the 10% interest rate market would sell at

Value of the bond = [100/1.05] + [(1,000+100)/(1.05)2] = $1,092.97

Again, as interest rates fall, the value of the bond increases. Since the $1,092.97 is above $1,000, the bond is said to sell at a premium.

Yield to Maturity

In reality, there are many incidences when a set of facts of a bond, including coupon rate, market value, years to maturity, face value, etc., are given except for the interest rate. In this situation, the interest rate is referred to as yield to maturity or bond’s yield. To determine the yield to maturity, one would go through a process of trial and error or typing into a fancy calculator. For example, a two-year level-coupon bond is currently selling at $1,092.97 with 10% coupon, the return that the bondholder will receive, y, is

$1,092.97 = [100/(1+y)] + [(1,000+100)/(1+y)2] y = 5%, as it has shown in previous example.

We interpret ry as the compound annual rate of return on the investment of $B0 in the bond if it is held to maturity. 1) This is easy to see with pure coupon bonds 2) It is a bit more complicated with coupon bonds because the interpretation holds exactly only if the

bondholder can reinvest each future coupon payment at today’s ry.

 YTM – annual coupon interest: ry ≈ [I + (M-B0)/T] / [(M+2B0)/3]

 γ YTM – pure discount bonds: M = B0 (1+ry)T → Ln (1+ry) = Ln (M/B0)/T = γ, 1+ry = e Sample Questions:

1. How many different sets of cash flows is a bond characterized by? Name them. 2. How many types of bonds are there? Name them. 3. What is a pure discount bond? What is its value? 4. What is a level-coupon bond? What is its value? 5. What is a Consol? How is it valued? 6. How is annual interest calculated? 7. What are some of the important bond concepts? 8. What is the relationship between bond prices and interest rates?

Answers:

1. A bond is characterized by three sets of cash flows: initial investment, periodic coupons, and final redemption. 2. There are three types of bonds: pure discount bond, level-coupon bond and consol. 3. Pure discount bond is also known as zero-coupon bond. It promises a single fixed payment at a fixed future date. The value of a pure discount bond is the present value of its final redemption amount. 4. Level-coupon bond offer cash payments not just a maturity, but also at regular times in between. It is the present value of its stream of coupon payments plus the present value of its repayment of principal. 5. A consol is a type of bond that perpetuates forever because it never stops paying a coupon; it has no final maturity date and will never mature. To value a consol, a perpetuity formula would be needed. 6. Annual interest (I) = r*M (coupon interest rate times par value of security) 7. Some important bond concepts are: 1) relationship between bond prices and interests and 2) yield to maturity. 8. There is an inverse relationship between bond prices and interest rates. As interest rate fall, the value of the bond increases and as the interest rate rises, the value of the bond decreases.