### Bonds and Bond Valuation Review

Bonds and Bond Valuation Review

Bond Terminology

A bond is a debt instrument that is commonly traded in financial markets. The borrower agrees to pay scheduled cash flows (i.e., coupon payments) to the bondholder during the life (or term) of the bond and then a lump sum payment equal to the face (or par) value of the bond when it matures. The contract that specifies the cash flows and their timing is called the bond “indenture”. Prospective bond investors evaluate the financial health of the issuer, the size and timing of the cash flows and determine a price that reflects the present value of those cash flows.

Bond Pricing Formula

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Face (Par) – Cash flow paid at bond maturity.

$CPN – (Coupon Rate* Face Value)/k

Coupon rate – (annual coupon)/(face value)

k – Number of coupon payments per year

Maturity – Number of years until the face value is paid.

t – Maturity * k

ytm – rate (APR) that makes the promised cash flows equal to the bond’s price.

- Assumes reinvestment at the YTM;
- Holding the bond until maturity;
- The probability of call is zero;
- The probability of default is zero.

Examples

Unless otherwise stated, we will always assume that the face or par value of a bond is $1,000 and that all bond cash flows occur semi-annually with semi-annual compounding.

1. Calculate the price of a semi-annual coupon bond with a coupon rate of 7.5%, 9.5 years to maturity and a yield to maturity of 8.5%.

Semi-annual Coupon = $1000*7.5%*0.5 = $37.50

Number of Periods = 9.5*2 = 19

Face Value of Bond = $1,000.00

Yield to Maturity = 8.5% (annual) or 4.25% (semi-annual)

Price = (37.50/0.0425)[1 – (1/(1+0.0425)^19)] + $1,000/((1+0.0425)^19)

= $482.22661 + $453.47650

Price = $935.70312 or it will be quoted as 93.570312

2. Calculate the yield to maturity on a bond that is selling for $1045.12 has a coupon rate of 5.25% and matures in 7 years.

$1045.12 = [(.0525/2)*$1,000][1 – (1/(1 + YTM/2)^14)]

YTM = 4.49164%

Bond Price Quotation Rules

- US Treasury bills – quoted at a discount from face value, with the discount expressed as an annual rate based on a 360 day year.

- US Government bonds – quoted as a percent of par. Trade on 32’nds.

- Corporate bonds – quoted as a percent of par. Trade on 16ths.

- Investors sell at the bid price and buy at the ask (or offered) price.

T-Bill Example:

Issue |
Bid |
Ask |
Change |
Yield |

12/3/15 |
5.08 |
5.06 |
-0.03 |
5.26 |

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Assume Face value: $10,000 and 169 days until maturity

$9,761.46 = $10,000 – $10,000((0.0508*169)/360)

9,762.46 = $10,000 – $10,000((0.0506*169)/360)

Treasury Bond Example:

Note that the dealer buys at the Bid price and sells at the Ask price.

Maturity |
Bid |
Ask |
Change |
Yield |

6 ?, 8/15/19 |
105.08 |
12 |
+3 |
5.57 |

Before considering the bond price table above, keep two things in mind:

- Bond prices are quoted as a percent of par and
- Treasury bonds trade in 32s.

So, the bid price means that the dealer is willing to pay 105 plus 8/32 percent of par for the bond issued on 8/15/11 that’s offering a coupon of 6.5%. How do we convert that into a price?

- Bond’s face value*(1.05 + (8/32)*(1/100))
- $1,000*(1.05+0.0025) = $1,052.50 = Bid Price

So, what’s the dealer’s Ask Price? The 12 indicates that the dealer will sell the bond for 105 + 12/32 of par. So,

- $1,000*(1.05+(12/32)*(1/100)) = $1,053.75 = Ask Price

Example: Calculating the realized return. What happens if you sell a bond before it matures?

3. You purchased US government 5-1/2’s of 09/15/16 at a price of 94.15 on 09/15/05. You sell the bond today at its current bid price of 100.22. (Assume today is 09/15/11). Calculate your realized annualized return. How does this realized return compare with your required return at the time you purchased the bond?

Coupon rate = 5.5% annual or 2.75% semi-annual

Coupon Payment = $27.50 (semi-annual)

Bond Price at Purchase = $941.50

Face Value of Bond = $1,000

Original Maturity of Bond = 11 years

Bond Price at Sale = $1002.20

Time till Sale = 6 years

YTM at Purchase = 6.243.11%

Realized Return = 6.73158%

Bond Types

- Treasury Bill – maturity of less than a year
- Treasury Note – maturity from 1 -10 years
- Treasury Bond – maturity from 10-30 years
- Corporate Bond – a bond issued by a corporation
- Callable Bond – a bond that can be purchased by issuer.
- Convertible Bond – the bondholder can exchange the bond for a set number

of equity shares.

- Puttable Bond – the bondholder has the option to force the issuer to buy back

the bond.

Call options and bonds

- The vast of majority of bonds with a maturity of 10 years or greater include a call option in the indenture.

- The call option gives the issuer the right the repurchase the bond at a pre-specified price after a stated number of periods (i.e., the “lockout period”).

- Is the Call option free? Who pays for the option and how do they pay?

4. Suppose you bought a 12% coupon Treasury bond with a term to maturity of exactly 10 years for $1,283.59. The government has the right to call at $1,100 in five years. What is the yield to call?