Bonds and Bond Valuation Review

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Coupon Rate = 12% annual or 6% semi-annual

Coupon Payment = $60 semi-annually

Maturity = 10 years

Time to Call = 5 years or 10 semi-annual periods

Call Price = $1,100 or 110.00

Bond Price at Issue = $1,283.59 or 128.359

Yield to Call: $1283.59 = [60/(YTC/2)] * [1 – (1/(1 + YTC/2)^10)] + ($1,100/(1 + YTC)^10)

YTC = 6.90503%

The Yield Curve

[pic 3]

The relation between yields and maturities is called the term structure of interest and the graphical representation is the yield curve. Note that a yield curve only makes sense if all of the bonds used in its construction are risk-free or of a similar credit quality.

When we calculated the yield to maturity (i.e., ytm) in problem 2, we solved for the rate that makes the discount present value of the bond cash flows equal to the bond’s price. We used the same interest rate to discount all of the cash flows regardless of when they occurred. Notice, however, that the chart above shows that bonds with different maturities offer different yields. This suggests that we should use a maturity dependent yield to discount each bond cash flow – each coupon should be discounted at a rate that is commensurate with its occurrence.

A single coupon paying bond can be thought of as a series of zero coupon bonds. Consider a two-year Treasury note: this can be viewed as a 6-month zero, a 12-month zero, an 18-month zero, and finally a 2-year zero (the 2-year zero includes both the final coupon and the face value). If we know the various yields for each of the risk-free, zero coupon bonds we could apply them to each of the cash flows and price the bond. Note that the yields on zero coupon bonds are called spot rates.

Construction of the Theoretical Spot Rate Curve

The theoretical spot rate curve is constructed from the observed yield curve. Note that this method assumes that the observed prices of Treasury securities should be equal to the portfolio of zeros that mimic their cash flow pattern.

Consider the following chart of yields for T-Bills and T-Notes

Maturity

Coupon Rate

YTM

Price

0.5

0.00

0.05

$975.61

1.0

0.00

0.06

$942.60

1.5

0.05

0.07

$971.98

2.0

0.07

0.08

$981.85

The 6-month and 1-year T-bills are zero coupon bonds so we already know the spot rates for the 6-month and 1-year cash flows.

We’ll have to work for the 18-month spot rate. The cash flows paid to the holder of a 1.5-year maturity bond are as follows:

Term

Coupon payment

Cash flow

.05

0.05/2*$1000

= $25.00

1.0

0.05/2*$1000

= $25.00

1.5

0.05/2*$1000 + $1000

= $1025.00

Given that we know the 6 month and 1 year spots we can calculate:

(1.5-year spot rate)[pic 4]

And

(2-year spot rate)[pic 5]

These two calculations allow us to construct the following spot rate table.

Maturity

YTM

Spot Rate

0.5

0.05

0.0500

1.0

0.06

0.0600

1.5

0.07

0.0703

2.0

0.08

0.0809

We can use this spot rate data to calculate the price of any ordinary Treasury security with a maturity of less than or equal to two years.

Arbitrage and the spot rate:

What happens if the price of a coupon paying bond does not reflect the spot rates? Risk-free profit if available. If the coupon is “cheap” investors profit by purchasing coupon bonds and stripping them into zeros. Conversely, if the coupon bond is “expensive” investors profit by purchasing a portfolio of zeros in the STRIPS market and assembling them into a coupon bond. Bond stripping and reconstitution allow investors to exploit arbitrage opportunities – keeping the prices correct. Pricing coupon bonds with spot rates is appropriate precisely because investors will force, by profiting from arbitrage opportunities, the prices of coupon bonds to reflect strip values and strips to reflect coupon bond values.

5. The one-year spot rate is 11% and the two-year spot rate is 10%. Calculate the price of a two-year bond that pays an annual coupon of 6% (assume a par value of $1000 and annual coupons).