Bond Prices and Yields

CHAPTER 10

Bond Prices and YieldsBond Prices and YieldsBond Prices and YieldsBond Prices and YieldsObjectives:

1. Analyze the relationship between bond prices and bond yields.

2. Calculate how bond prices will change over time for a given interest-rate projection.

3. Identify the determinants of bond safety and rating.

4. Analyze how callable, convertible, and sinking fund provisions will affect a bond's equilibrium yield to maturity.

5. Define the yield curve and study its properties

Bond Characteristics

Long-term debt contract Fixed interest payment is paid throughout the life of bond Entire principal payment is paid at maturity date Coupon rate: determines the fixed interest payment Yield to maturity: the average return per year that the

investors (or the market) require on the bond if they buy and hold the bond until maturity

Coupon rate is fixed, determined by the issuing firm YTM can fluctuate, depending on the investors in the

market Zero-coupon bond

zero coupon payment par at maturity date

Treasury Notes and Bonds

T Note maturities range up to 10 years T bond maturities range from 10 – 30 years Bid and ask price

Quoted in points and as a percent of par Accrued interest

Quoted price does not include interest accrued Example: if the coupon payments are made on May 1

and Nov 1, you buy a bond on June 11. Assume the price on June 11 is 990, how much you have to pay in order to buy the bond. (assume there are 40 days from May 1-June 11

Figure 10.1 Listing of Treasury Issues

Corporate Bonds Most bonds are traded over the counter Registered Bearer bonds Secured and unsecured

Secured: Collateral Mortgage

Unsecured Debentures Notes

Call provisions: allows issuer to buy back bond before maturity date at a specific call price Why the company wants to call the bond? Call provision is in favor of the issuer. So if everything else is the same,

callable bond would have to give higher yield, higher coupon to investors than regular bonds

Corporate Bonds Convertible bond: a bond with option allowing the

bondholder to exchange the bond for a specific number of shares of common stock in the firm When bondholder want to convert bond into stocks?

Puttable bond: gives the option to bondholder to either exchange the bond for par value at some date or to extend for a given number of year When bondholders want to exchange for par value before

maturity When bondholders want to extend the bond for a given number

of year after maturity Floating rate bond

coupon rate is tied to current market rates Preferred stocks

Figure 10.2 Investment Grade Bonds

Other Domestic Issuers Federal Home Loan Bank Board Farm Credit Agencies Ginnie Mae Fannie Mae Freddie Mac

Innovations in the Bond Market Reverse floaters

Reverse of floating rate bond Asset-backed bonds

backed by assets of the firm Pay-in-kind bonds

issuers may choose to pay interest either in cash or in additional bon Catastrophe bonds

issued by insurance company, give high yield In the event of catastrophe, the obligation to pay interest and principal

can be delayed or forgiven Indexed bonds

payments are tied to a general price index or price of a particular commodity

TIPS (Treasury Inflation Protected Securities)

Innovations in the Bond Market

TIPS: adjust for inflation

Example: n = 3 years, annual coupon, par 1000, coupon 4%

Time Inflation Par coupon principaltotal

payment payment

0 1000

1 2% 1020 40.80 040.80

2 3% 1050.60 42.02 042.02

3 1% 1061.11 42.44 1061.11 1103.55

Bond Pricing

Price of bond = present value of all future coupon payments +present value of the par value

PB = Price of the bond

Ct = interest or coupon paymentsT = number of periods to maturityr = semi-annual discount rate or the semi-annual yield to maturity

P Cr

Par Valuer

B tT

t

T

TT

( ) ( )1 11

T

T

B r

par

rr

CP)1(

)1(1

1

Example

1) 8% coupon, pay annually, 10 years to maturity, par = 1000, YTM = 6%

•Using formula

•Using calculator

•PMT = 80, FV = 1000, n = 10, I/Y = 6

2) The same information, but the bond is paying interest semi-annually. What is the price of the bond?

Yield to MaturityYield to MaturityYield to MaturityYield to Maturity

Yield to maturity is a measure of the average rate of return that will be earned if the bond is held to maturity.

YTM is the discount rate that makes the present value of a bond’s payments equal to its price

YTM is the solution of :

T

T

tt

t

YTM

par

YTM

coupon

)1()1( Price Bond

1

8% coupon, 30-year bond selling at $1,276.76, what is the yield to maturity?

Bond Prices and Yields Prices and Yields (required rates of return) have an

inverse relationship If YTM increase, then the price decreases and vice versa

MarketMaturity Coupon Interest Bond Value Rate Rate Price

$ 1,000 8% 8% $1,000.00

$ 1,000 8% 10% $ 810.71

$ 1,000 8% 6% $1,276.75

Figure 10.3 The Inverse Relationship Between Bond Prices and Yields

rd 1-year Change 10-year Change

5% $1,048 $1,386

10% 1,000 4.8% 1,000 38.6%

15% 956 4.4% 749 25.1%

Interest rate risk: change in rd causes bond’s price to change.

Bond Prices and Yields

Longer time to maturity, higher change in price when interest rate changes (or higher interest rate risk)

0

500

1,000

1,500

0% 5% 10% 15%

1-year

10-year

rd

Value

Bond Prices and Yields

Alternative Measures of Yield Yield to Call

Call price replaces par Call date replaces maturity Example: 8% coupon, semi-annual, 30 years to

maturity, current price = 1150, callable in 10 years, call price = 1100. What is the yield to call and yield to maturity

Holding Period Return

yieldgain capital yieldcurrent

P

)P-(P

P

Coupon

P

)P-(PCoupon

price beginnning

(loss)gain capital incomeinterest HPR

t

t1t

t

t

t

t1tt

WherePt = Bond Price at time tPt+1= Bond Price at time t+1

Definitions

Current yield =

Capital gains yield =

= YTM = +

Annual coupon pmtCurrent price

Change in priceBeginning price

Expected totalreturn

Expected Curr yld

Expected capgains yld

Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 and YTM = 10.91%.

Current yield =

= 0.1015 = 10.15%.

$90 $887

YTM = Current yield + Capital gains yield.

Cap gains yield = YTM - Current yield = 10.91% - 10.15% = 0.76%.

Could also find values in Years 1 and 2,get difference, and divide by value inYear 1. Same answer.

BOND PRICES OVER TIME

Premium and Discount Bonds Premium Bond

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

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

Bond selling at par price = par Yield to maturity = coupon rate bond price is constant throughout the life of bond

Suppose the bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return or the YTM remained at 10%, or at 13%, or at 7%?

BOND PRICES OVER TIME

M

Bond Value ($)

Years remaining to Maturity

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

rd = 7%.

rd = 13%.

rd = 10%.

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

The value of a premium bond would decrease to $1,000.

The value of a discount bond would increase to $1,000.

A par bond stays at $1,000 if rd (YTM) remains constant.

BOND PRICES OVER TIME

Figure 10.7 The Price of a Zero-Coupon Bond over Time

DEFAULT RISK AND BOND PRICING

Default Risk and Ratings Rating companies

Moody’s Investor ServiceStandard & Poor’sFitch

Rating Categories Investment gradeSpeculative grade

Figure 10.8 Definitions of Each Bond Rating Class

Bond Ratings Provide One Measureof Default Risk

Investment Grade Junk Bonds

Moody’s Aaa Aa A Baa Ba B Caa C

S&P AAA AA A BBB BB B CCC D

Factors Used by Rating Companies

Coverage ratios Leverage ratios Liquidity ratios Profitability ratios Cash flow to debt

Protection Against Default Sinking funds

A bond that calls for the issuer to periodically repurchase some proportion of the outstanding bonds prior to maturity

Subordination of future debt Restrictions on additional borrowing that stipulates that senior

bondholders will be paid first in the event of bankruptcy Dividend restrictions

Limit dividend payout to protect bondholders Collateral

Uses assets to back up bonds: mortgage bond, collateral trust bond, equipment obligation bond.

Collaterals are secured bonds Unsecured bond: debentures

THE YIELD CURVE

Term Structure of Interest Rates Relationship between yields to maturity

and maturity Yield curve - a graph of the yields on

bonds relative to the number of years to maturityUsually Treasury BondsHave to be similar risk or other factors

would be influencing yields

Figure 10.11 Treasury Yield Curves

Theories of Term Structure Expectations

Long term rates are a function of expected future short term rates

Upward slope means that the market is expecting higher future short term rates

Downward slope means that the market is expecting lower future short term rates

Liquidity Preference Upward bias over expectations The observed long-term rate includes a risk premium

Combination (Synthesis)

Expectation hypothesis (when short-term interest rate is expected to increase)

Current interest on 1-year bond = 8% (r1=8%) Everyone in the market believes that the interest one 1-

year bond next year will rise to 10% (E(r2) = 10%) Investment A: year 1: buy 1-year bond

year 2: buy another 1-year bond Investment B: year 1: buy 2-year bond In order for investment B to be competitive with

investment A, B has to offer an average annual compound return = average annual compound return of A

What is the average annual compound return of B or YTM of the 2-year bond in B?

You will find the YTM of 2-year bond > YTM of 1-year bond at time = 0, this is due to E(r2) > r1 ( the expected short-term rate increases in the future)

Figure 10.12 Returns to Two 2-year Investment Strategies

Expectation hypothesis (when short-term interest rate is expected to decrease)

Current interest on 1-year bond = 8% (r1=8%) Everyone in the market believes that the interest one 1-

year bond next year will decrease to 6% (E(r2) = 6%) Investment A: year 1: buy 1-year bond

year 2: buy another 1-year bond Investment B: year 1: buy 2-year bond In order for investment B to be competitive with

investment A, B has to offer an average annual compound return = average annual compound return of A

What is the average annual compound return of B or YTM of the 2-year bond in B?

You will find the YTM of 2-year bond < YTM of 1-year bond at time = 0, this is due to E(r2) < r1 ( the expected short-term rate decreases in the future)

The Expectation HypothesisThe Expectation Hypothesis•In practice, we don’t observe directly the expectation of next year’s rate, but we can observe the yields on bonds of different maturities. So from that, using the yields on bonds of different maturities, we can calculate the expected short-term interest rate (or the expected YTM of 1-year bond) in the future •According to expectation hypothesis, the forward rate = expected short term rate. •Forward rate is the inferred short-term rate of interest for a future period that makes the expected total return of a long-term bond = that of rolling over short-term bonds.

Example: Suppose that 1-year bonds offer yields to maturity of 8%, and 2-year bonds have yields of 8.995%. What is the expected one-period rate or forward rate for the third year?

Forward Rates Implied in the Yield Curve

)0902.1()06.1()07.1(

)1()1()1(23

1

1

fyy nnn

nn

For example, using a 1-yr and 2-yr rates

Longer term rate, y(n) = 7%

Shorter term rate, y(n-1) = 6%

Forward rate, a one-year rate in one year = 9.02%

Example: Suppose that 2-year bonds offer yields to maturity of 6%, and 3-year bonds have yields of 7%. What is the expected one-period rate for the third year?

Liquidity preference theory

According to expectation theory, long-term yield depends on expected short-term yield. If expected short-term increases, then long-term increases, and vice versa

This theory does not take into account risk. Risk of long-term > risk of short-term Liquidity preference theory:

investors demand of risk premium (liquidity premium) for holding long-term bonds

fn = E(rn) + Liquidity premium

Forward rate in year n = expected short-term rate in year n + liquidity premium

Predict upward sloping yield curve even in the case of expected short-term rates are unchanged.

Liquidity preference theory

Suppose the short-term rate of interest rate is currently 8%, and investors expect it to remain at 8% next year.

In the absence of liquidity premium, with no expectation of a change in yields, the YTM on two-year bonds would be 8% (according to expectation hypothesis), and the yield curve would be flat, the forward rate would be 8%

However, what if investors demand a risk premium to invest in two-year rather than one-year bonds? If the liquidity premium = 1%, then according to liquidity theory, the forward rate would be = 8%+1% = 9% and the YTM of two-year bond would be(1+y2)2 = 1.08×1.09 = 1.7772

therefore y2=0.085 = 8.5%

YTM of 2-year bond (8.5%) > YTM of 1-year bond (8%) solely due to the liquidity premium.

A Synthesis

Figure 10.14 Term Spread

SummarySummarySummarySummary

•Inverse relationship between bond prices and bond yields

•Premium and discount bonds•Corporate bonds and default risk•Term structure of interest rates

Expectations theoryLiquidity preference theorySynthesis

•Next Class: Managing Bond Portfolios