Bond Prices and Yields

Chapter 14

A bond is a security that is issued in connection with a borrowing arrangement. The borrower issues (i.e. sells) a bond to the lender for some amount of cash. The arrangement obligates the issuer to make specified payments to the bondholder on specified dates.

When the bond matures, the issuer repays the debt by paying the bondholder the bond’s face value or par value – usually $1,000.

Bond Characteristics

Bond Characteristics

The coupon rate of the bond serves to determine the interest payment: the annual payment is the coupon rate times the bond’s par value. Coupons are usually paid semiannually.

Bonds usually are issued with coupon rates set high enough to induce investors to pay par value to buy the bond.

Sometimes, however, Zero coupon bonds are issued that make no coupon payments. In this case, investors receive par value at the maturity date but receive no interest payments until then: the bond has a coupon rate of zero. These bonds are issued at prices considerably below par value, and the investor’s return comes solely from the difference between issue price and the payment of par value at maturity.

The coupon rate, maturity date and par value of the bond are part of the bond indenture, which is the contract between the issuer and the bondholder.

Accrued Interest and Quoted Bond Prices

The bond prices that are quoted in the financial pages are sometimes not actually the prices that investors pay for the bond. This is because the quoted price does not include the interest that accrues between coupon payment dates.

If a bond is purchased between coupon payments, the buyer must pay the seller for accrued interest, the share of the upcoming semiannual coupon.

For example, if 40 days have passed since the last coupon payment and there are 182 days in the semiannual coupon period, the seller is entitled to payment of accrued interest of 40/182 of the semiannual coupon. The invoice price of the bond would equal the stated price plus the accrued interest.

Accrued Interest and Quoted Bond Prices

Suppose that the coupon rate is 8%. Then the

semiannual coupon payment is $40. Because

40 days have passed since the last coupon

payment, the accrued interest on the bond is

$40 * (40/182) = $8.79. If the quoted price of

the bond is $990, then the invoice price will

be $990 + $8.79 = $998.79

Call provision: The call provision allows the issuer to repurchase the bond at a specified call price before the maturity date. For example, if a company issues a bond with a high coupon rate when market interest rates are high, and interest rates later fall, the firm might like to retire the high coupon debt and issue new bonds at a lower coupon rate to reduce interest payments. This is called refunding.

The option to call the bond is valuable to the firm, allowing it to buy back the bonds and refinance at lower interest rates when market rate falls. Of course, the firm’s benefit is the bondholder’s burden – risky for investors. To compensate investors for this risk, callable bonds are issued with higher coupons and promised yields to maturity than noncallable bonds.

Provisions of Bonds

Provisions of Bonds

Convertibility provision: Convertible bonds give bondholders an option to exchange each bond for a specified number of shares of common stock of the firm.

The conversion ratio gives the number of shares for which each bond may be exchanged. Suppose a convertible bond that is issued at par value of $1,000 is convertible into 40 shares of a firm’s stock. The current stock price is $20 per share, so the option to convert is not profitable now. Should the stock price later rise to $30, each bond may be converted profitably into $1,200 worth of stock.

Provisions of Bonds

The market conversion value is the current value of the shares for which the bonds may be exchanged. At the $20 stock price, for example, the bond’s conversion value is $800. The conversion premium is the excess of the bond value over its conversion value. If the bond were selling currently for $950, its premium would be $150.

Convertible bonds give their holders the ability to share in price appreciation of the company’s stock. The benefit comes at a price. Convertible bonds offer lower coupon rates and stated or promised yields to maturity than do nonconvertible bonds.

Provisions of Bonds

Put Provision: While the callable bond gives the issuer the option to extend or retire the bond at a specified call price, the extendable or put bondgives this option to the bondholder.

If the bond’s coupon rate exceeds current market yields, the bondholder will choose to extend the bond’s life. If the bond’s coupon rate is too low, it will be optimal not to extend; the bondholder instead reclaims principal, which can be invested at current yields.

1 (1 )(1 )

T

Tt

t

BFVCP

rr

𝑃𝐵 = Price of the bond

𝐶 = interest or coupon payments

𝑇 = number of periods to maturity

𝑟 = required rate of return or yield to maturity

FV = Face value or Par value of the bond

Bond Pricing

Bond Pricing

Using annuity formula:

𝑃𝐵 =𝐶

𝑟1 −

1

1+𝑟 𝑇 +𝐹𝑉

1+𝑟 𝑇

𝐶 = $40 (SA)

𝐹𝑉 = $1000

𝑇 = 20 periods

𝑟 = 3% (SA)

Solving for Price: 10-yr, 8% Coupon Bond,Face Value = $1,000, Semiannual coupon

20

20

1

20 20

1 100040

1.03 (1.03)

40 1 10001

0.03 1 (1.03)

$1,148.77

t

t

P

Pr

P

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.

Bond Prices and Yields

Bond Prices and Yields

Yield to Maturity (YTM)

Interest rate that makes the present value of the

bond’s payments equal to its price.

Investors earn the YTM if the bond is held to

maturity and all coupons are reinvested at YTM.

Solve the bond formula for r

1 (1 )(1 )

T

Tt

t

BFVCP

rr

Yield to Maturity Example

)1(

1000

)1(

35950

20

1 rrT

t

t

10 yr. Maturity Coupon Rate = 7%

Price = $950

Solve for 𝒓 = semiannual rate

𝒓 = 3.8635%

Yield to Maturity Example

Approximate YTM formula:

Approximate YTM =𝐶+

𝐹𝑉−𝑃

𝑇𝐹𝑉+2𝑃

3

=35+

1000−950

201000+2(950)

3

=37.5

966.67= 3.8793%

Yield Measures

Suppose semiannual YTM = 3.86%

Annual Bond Equivalent Yield

3.86% × 2 = 7.72%

Effective Annual Yield

(1.0386)2 - 1 = 7.88%

Current Yield

Annual Coupon Payment / Market Price

$70 / $950 = 7.37 %

Yield to Maturity for Zero Coupon Bonds

1

1TFV

rP

Assume a zero coupon bond sells at $920 and will mature

in 5 years. What is its YTM?

YTM = 1.67%

Use the YTM to check the price. The price should be $920

Bond Price vs. Par value and Coupon rate vs YTM

If 𝑃𝐵 = 𝐹𝑉, then 𝑐 = 𝑟

If 𝑃𝐵 < 𝐹𝑉, then 𝑐 < 𝑟

If 𝑃𝐵 > 𝐹𝑉, then 𝑐 > 𝑟

Holding-Period Return: Single Period

𝐻𝑃𝑅 = [ 𝐶 + (𝑃1− 𝑃0)]/𝑃0where

𝐶 = Coupon payment

𝑃1 = price in period 1

𝑃0 = purchase price

Holding-Period Example

𝐶𝑅 = 8%, 𝑌𝑇𝑀 = 8%, 𝑁=10 years, Semiannual Compounding,

→ 𝑃0 = Par Value = $1000

If after 6 months, YTM is still 8%, then the HPR will also be

8%. So, if YTM doesn’t change, then as time passes, HPR

remains equal to YTM (if 𝑃0 = 𝐹𝑉). If 𝑃0 ≠ 𝐹𝑉, then if YTM

remains unchanged, then as time passes, the bond price will

approach par value.

In 6 months, if the rate falls to 7%, then price increases to:

𝑃1 = $1,068.55

𝐻𝑃𝑅 = [40 + ( 1068.55 - 1000)] / 1000

𝐻𝑃𝑅 = 10.86% (semiannual)

→ Bond Equivalent HPR (annual) = 21.72%

Practice Problems

Chapter 14:

5, 6, 8, 13 (assume annual coupon payments),

14, 15