Bonds and Their Valuation

Chapter 7

Chapter 7 Topic Overview

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Bond Characteristics Annual and Semi-Annual Bond Valuation Reading Bond Quotes Finding Returns on Bonds Bond Risk and Other Important Bond Valuation

Relationships

Bond Characteristics

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Par (or Face) Value (M) = stated face value that is the amount the issuer must repay, usually $1,000

Coupon Interest RateCoupon (INT) = Coupon Rate x Face ValueMaturity Date = when the face value is repaid.A legal contract called the bond indenture

specifies these values.This makes a bond’s cash flows look like this:

Bond ValuationDiscount the bond’s cash flows at the

investor’s required rate of return.the coupon payment (INT) stream (an annuity).the par (M) value payment (a lump sum).V = INT (PVA r, n) + M /(1+r)n

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00 1 1 2 . . . 2 . . . nn

INT INT INTINT INT+MINT+M

Bond Valuation Example #1

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Duff’s Beer has $1,000 par value bonds outstanding that make annual coupon payments. These bonds have a 7.5% annual coupon rate and 15 years left to maturity. Bonds with similar risk have a required return of 9%, and Moe Szyslak thinks this required return is reasonable.

What’s the most that Moe is willing to pay for a Duff’s Beer bond?

0 1 2 3 . . . 15

1000 ? 75 75 75 . . . 75

r = 9%

Let’s Play with Example #1

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Homer Simpson is interested in buying a Duff Beer bond but demands an 7.5 percent required return.

What is the most Homer would pay for this bond?

0 1 2 3 . . . 15

1000 ? 75 75 75 . . . 75

r = 7.5%

Let’s Play with Example #1 some more.

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Barney (belch!) Gumble is interested in buying a Duff Beer bond and demands on a 6 percent required return.

What is the most Barney (belch!) would pay for this bond?

0 1 2 3 . . . 15

1000 ? 75 75 75 . . . 75

r = 6%

Lesson from Example 1: Bond Prices and Interest Rates have an inverse relationship!

Bond Values for 7.5% Annual Coupon Bond

0200400600800

100012001400

0% 2% 4% 6% 8% 10%

Required Return

($)M

ark

et V

alu

e

15-yr Bond

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Another Example 1 Lesson: Bond Premiums and Discounts

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What happens to bond values if required return is not equal to the coupon rate?

The bond's price will differ from its par value

P0 < par valuer > Coupon Interest Rate DISCOUNT =

P0 > par valuer < Coupon Interest Rate PREMIUM =

P0 = par valuer = Coupon Interest Rate PAR =

Bonds with Semiannual Coupons

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Double the number of years, and divide required return and annual coupon by 2.

VVdd = = INTINT/2/2(PVA(PVAr/2,2nr/2,2n) + M ) + M /(1+r/2)2n

Semiannual Example

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Kwickee-Mart has an $1000 par value bond with an annual coupon rate of 6% that pays coupons semiannually with 20 years left to maturity. What is the most you would be willing to pay for this bond if your required return is 7% APR?

Semiannual coupon = 6%/2($1000) = $30Semiannual coupon = 6%/2($1000) = $30 20x2 = 40 remaining coupons20x2 = 40 remaining coupons

0 1 2 3 . . . 40

1000 30 30 30 . . . 30

Bond Yields

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Current Yield - Annual coupon payments divided by bond price.

Yield To Maturity - Interest rate for which the present value of the bond’s payments equal the price. Also known as the market’s required rate of return.

Yield To Maturity = total expected return = current yield + expected capital gains yield (change in price)

Yield Definitions

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CGY

Expected

CY

Expected YTM return total Expected

price Beginningprice in Change

(CGY) yieldgains Capital

priceCurrent payment coupon Annual

(CY) eldCurrent yi

Bond Yields

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Calculating Yield to Maturity (YTM=r)

If you are given the price of a bond (PV) and the coupon rate, the yield to maturity can be found by solving for r.

PVcpn

r

cpn

r

cpn par

r t

( ) ( )

....( )

( )1 1 11 2

Yield to Maturity Example

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Burns Power $1000 face value bond with a 5% coupon rate paid annually with 10 years left to maturity sells for $890.

What is this bond’s yield to maturity?

0 1 2 3 . . . 10

1000-890 50 50 50 . . . 50

What is bond’s YTM?

What is the bond’s current yield?

U.S. Treasury Bond Quotations

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RATEMATURITY

MO/YRBID ASKED CHG

ASK

YLD

Government Bonds & Notes8.50 Feb 20 136:30 136:31 +42 4.67

Rate Coupon rate of 8.5%

Bid pricesAsk prices

(percentage of par value)

Bid price: the price traders receive if they sell a bond to the dealer.

Quoted in increments of 32nds of a %age of par

Ask price: the price traders pay to the dealer to buy a bond

Bid-ask spread: difference between ask and bid prices.

Ask Yield Yield to maturity on the ask price

Verifying the T-bond’s YTM (Ask Yld)

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Par Value = $1000, semi-annual couponsAsk Price = 136.969%($1000) = $1,369.69INT/2 = 8.5%($1000)/2 = $42.50N = 2020-2007 = 13, 2N = 26

Causes of Bond Price Changes

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Since a bond’s cash flows are fixed:

1. Changes in interest rates, and

2. Passage of time.

cause changes in a bond’s price.

Bond Value Changes Over Time

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Returning to the Duff’s Beer original example #1, where r = 9%, N = 15, C (PMT) = $75, par (FV) = $1000, & PV = $879.09.

What is bond value one year later when N = 14 and r is still = 9%?

Bond values over time

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At maturity, the value of any bond must equal its par value.

If rd remains constant:The value of a premium bond would decrease

over time, until it reached $1,000.The value of a discount bond would increase over

time, until it reached $1,000.A value of a par bond stays at $1,000.

Changes in Bond Value over TimeWhat would happen to the value of these three

bonds is bond if its required rate of return remained at 10%:

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Years to Maturity

1,184

1,000

816

10 5 0

13% coupon rate

7% coupon rate

10% coupon rate.

VB

What about this?

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What if Barney buys Duff Bond at $1145.68 with n =15 and sells at r = 7% with n = 14. What is his return?

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Interest Rate Risk

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Measures Bond Price Sensitivity to changes in interest rates.

In general, long-term bonds have more interest rate risk than short-term bonds.

Also, for bonds with same time to maturity, lower coupon bonds have more interest rate risk than higher coupon bonds.

Interest Rate Risk Example

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Recall from our earlier example (#1), the 15-year, 7.5% annual coupon bond has the following values at kd = 6%, 7.5%, & 9%. Let’s compare with a 2-yr, 7.5% annual coupon bond.

15-year bond 2-year bondr=6%: PV = $1,145.68 PV = $1,027.50r=7.5%: PV = $1,000 PV = $1,000r=9%: PV = $879.09 PV = $973.61

Interest Rate Risk: Bond Price Sensitivity Graph

Bond Values for 7.5% Annual Coupon Bonds

400650900

115014001650190021502400

2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%

2-yr Bond

15-yr Bond

30-yr Bond

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Reinvestment rate risk

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Reinvestment rate risk is the concern that rd will fall, and future CFs will have to be reinvested at lower rates, hence reducing income.

EXAMPLE: Suppose you just won

$500,000 playing the lottery. You

intend to invest the money and

live off the interest.

Reinvestment rate risk example

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You may invest in either a 10-year bond or a series of ten 1-year bonds. Both 10-year and 1-year bonds currently yield 10%.

If you choose the 1-year bond strategy:After Year 1, you receive $50,000 in income and

have $500,000 to reinvest. But, if 1-year rates fall to 3%, your annual income would fall to $15,000.

If you choose the 10-year bond strategy:You can lock in a 10% interest rate, and $50,000

annual income.

Conclusions about interest rate and reinvestment rate risk

Short-term AND/OR High coupon bonds

Long-term AND/OR Low coupon bonds

Interest

rate riskLow High

Reinvestment rate risk

High Low

CONCLUSION: Nothing is riskless!

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Default Risk

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Credit riskDefault premiumInvestment gradeSpeculative grade (Junk bonds)

Default Risk

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StandardMoody' s & Poor's Safety

Aaa AAA The strongest rating; ability to repay interest and principalis very strong.

Aa AA Very strong likelihood that interest and principal will berepaid

A A Strong ability to repay, but some vulnerability to changes incircumstances

Baa BBB Adequate capacity to repay; more vulnerability to changesin economic circumstances

Ba BB Considerable uncertainty about ability to repay.B B Likelihood of interest and principal payments over

sustained periods is questionable.Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already beCa CC in default or in danger of imminent defaultC C C-rated bonds offer little prospect for interest or principal

on the debt ever to be repaid.

Callable Bonds & the effect of a call provision

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Allows issuer to refund the bond issue if rates decline (helps the issuer, but hurts the investor).

Borrowers are willing to pay more, and lenders require more, for callable bonds.

Most bonds have a deferred call and a declining call premium.

A 10-year, 10% semiannual coupon bond selling for $1,135.90 can be called in 4 years for $1,050, what is its yield to call (YTC)?

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The bond’s yield to maturity can be determined to be 8%. Solving for the YTC is identical to solving for YTM, except the time to call is used for N and the call premium is FV.

INPUTS

OUTPUT

N I/YR PMTPV FV

8

3.568

50 1050- 1135.90

Yield to call

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3.568% represents the periodic semiannual yield to call.

YTCNOM = rNOM = 3.568% x 2 = 7.137% is the rate that a broker would quote.

The effective yield to call can be calculatedYTCEFF = (1.03568)2 – 1 = 7.26%

If you bought these callable bonds, would you be more likely to earn the YTM or YTC?

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The coupon rate = 10% compared to YTC = 7.137%. The firm could raise money by selling new bonds which pay 7.137%.

Could replace bonds paying $100 per year with bonds paying only $71.37 per year.

Investors should expect a call, and to earn the YTC of 7.137%, rather than the YTM of 8%.

When is a call more likely to occur?

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In general, if a bond sells at a premium, then (1) coupon rate > rd, so (2) a call is more likely.

So, expect to earn:YTC on premium bonds.YTM on par & discount bonds.

Sinking Fund

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Provision to pay off a loan over its life rather than all at maturity.

Similar to amortization on a term loan.Reduces risk to investor, shortens

average maturity.But not good for investors if rates decline

after issuance.

How are sinking funds executed?

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Call x% of the issue at par, for sinking fund purposes.Likely to be used if rd is below the coupon rate and

the bond sells at a premium.Buy bonds in the open market.

Likely to be used if rd is above the coupon rate and the bond sells at a discount.

Other types (features) of bonds

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Convertible bond – may be exchanged for common stock of the firm, at the holder’s option.

Warrant – long-term option to buy a stated number of shares of common stock at a specified price.

Putable bond – allows holder to sell the bond back to the company prior to maturity.

Income bond – pays interest only when interest is earned by the firm.

Indexed bond – interest rate paid is based upon the rate of inflation.