Chapter 14

Bond Prices and Yields

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Provisions of Bonds

• Secured or unsecured• Call provision• Convertible provision• Put provision (putable bonds)• Floating rate bonds• Sinking funds

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Bond Pricing

P Cr

ParValuer

B tT

t

T

TT

( ) ( )1 11

PB = Price of the bond

Ct = interest or coupon payments

T = number of periods to maturity

r = semi-annual discount rate or the semi-annual yield to maturity

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Price of 8%, 10-yr. with yield at 6%

77.148,1

)03.1(

11000

)03.1(

140 20

20

1

P

P

B

ttB

Coupon = 4%*1,000 = 40 (Semiannual)

Discount Rate = 3% (Semiannual

Maturity = 10 years or 20 periods

Par Value = 1,000

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Bond Prices and Yields

Prices and Yields (required rates of return) have an inverse relationship

• When yields get very high the value of the bond will be very low

• When yields approach zero, the value of the bond approaches the sum of the cash flows

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Prices and Coupon Rates

Price

Yield

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Alternative Measures of Yield

• Current Yield• Yield to Call

• Call price replaces par• Call date replaces maturity

• Holding Period Yield• Considers actual reinvestment of coupons• Considers any change in price if the bond is

held less than its maturity

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Premium and Discount Bonds

• Premium Bond• Coupon rate exceeds yield to maturity• Bond price will decline to par over its

maturity

• Discount Bond• Yield to maturity exceeds coupon rate• Bond price will increase to par over its

maturity

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Types of Bonds

• High Yield vs Investment grades• Example • AAA 5% with .2% historical default• B, 9% with 4% historical default rate• 40% recovery rate on defaults• Return = (1 – default rate) * interest rate –

default rate * (1-recovery rate)• Return for A, .998 * .05 - .002*.6 = 4.87%.• Return for B, .96 * .09 - .04 * .6 = 6.24%

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Duration• A measure of the effective maturity of a bond• The weighted average of the times until each payment is

received, with the weights proportional to the present value of the payment

• Duration is shorter than maturity for all bonds except zero coupon bonds

• Duration is equal to maturity for zero coupon bonds

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Duration: Calculation

t tt

w CF y ice ( )1 Pr

D t wt

T

t

1

CF CashFlow for period tt

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Duration Calculation

8%Bond

Timeyears

Payment PV of CF(10%)

Weight C1 XC4

1 80 72.727 .0765 .0765

2 80 66.116 .0690 .1392

Sum

3 1080 811.420

950.263

.8539

1.0000

2.5617

2.7774

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Uses of Duration

• Summary measure of length or effective maturity for a portfolio

• Immunization of interest rate risk (passive management)• Net worth immunization• Target date immunization

• Measure of price sensitivity for changes in interest rate

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Duration/Price Relationship

Price change is proportional to duration and not to maturity

P/P = -D x [(1+y) / (1+y)

D* = modified duration

D* = D / (1+y)

P/P = - D* x y

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Pricing Error from Convexity

Price

Yield

Duration

Pricing Error from

Convexity

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.

Correction for Convexity

)(21 2yConvexityyD

P

P

Modify the pricing equation:

Convexity is Equal to:

N

tt

t tty

CFP 1

22 )1(y)(1

1

Where: CFt is the cashflow (interest and/or principal) at time t.