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FINC4101 Investment Analysis

Instructor: Dr. Leng Ling

Topic: Bond Pricing and Yields

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Learning objectives1. Compute the price of a zero-coupon bond.2. Compute the price of a fixed coupon bond.3. Describe the price-yield relationship of bonds. 4. Distinguish between a bond’s flat price and its invoice

price. 5. Compute different measures of bond returns. 6. Calculate how bond prices will change over time for a

given interest rate projection.7. Recognize default/ credit risk as a source of risk for

bonds. 8. Identify the determinants of bond safety and rating. 9. Understand how default risk can affect yield to maturity.

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Concept Map

Foreign Exchange

Derivatives

Market Efficiency

Fixed Income

Equity

Asset Pricing

Portfolio Theory

FI4000

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Types of fixed-income securities

Fixed-income security / bond: A security that obligates the issuer to

make specified payments to the holder over a period of time.

We focus on the pricing of two types of fixed-income securities:

1. Fixed-coupon bond (FCB)

2. Zero-coupon bond (ZCB)

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Fixed-coupon bond (FCB) 1 Firm pays a fixed amount of interest (‘coupon payment’)

to the investor every period until bond matures.

At maturity, firm pays face value of the bond to investor.

Face value also called par value. Unless otherwise

stated, always assume face value to be $1000.

Period: can be year, half-year (6 months), quarter (3

months).

Coupon rate: annual coupon payment as a fraction of

face value.

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How to ‘read’ a fixed-coupon bond: Example

A firm issues an 8% 30-year bond with annual coupon payments. Par value is $1,000.

What does the above tell us? 8%: the coupon rate. Multiply coupon rate by par value to get annual coupon

payment. Coupon= 8% x 1,000 = $80. Maturity = 30 years. Coupon of $80 is paid annually, i.e., period=annual. At maturity (end of 30 years), firm will pay $1000 to

investor. What happens if coupon is paid semi-annually?

Quarterly?

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Fixed-coupon bond (FCB) 2 FCB gives you a stream of fixed payments plus

a single payment (face value) at maturity. This cash flow stream is just an annuity plus a

single cash flow at maturity. Therefore, we calculate the price of a FCB by

finding the PV of the annuity and the single payment, using an appropriate interest rate.

We use the financial calculator to compute the price of the FCB.

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Fixed-coupon bond (FCB) 3

Reminder: The “interest rate” used to find the PV of a

bond is also known as: • Yield-to-maturity (“YTM” or “yield” for short)• Discount rate• Required rate of return• Cost of debt

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Fixed-coupon bond (FCB) 4

Price of the FCB, PFCB

Nd

N

tt

d

FCBr

F

r

CP

111

Number of periods to maturity

Fixed periodic coupon Face value

Yield to maturity (in decimals)

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Find FCB price Consider an 8%, 30-year coupon bond that pays

coupons semi-annually. Compute the bond’s price if the yield to maturity is a) 6%, b) 8%, c) 10%.

If YTM = 6%, verify that bond price = $1,276.76 If YTM = 8%, verify that bond price = $1000. If YTM = 10%, verify that bond price = $810.71

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Inverse relationship between price & interest rate

Notice that as YTM (interest rate) increases, bond price decreases. Conversely, as YTM decreases, bond price increases.

This is the inverse relationship between bond price and yield to maturity (interest rate).

This is a crucial general rule in bond pricing based on time value of money.

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The Inverse Relationship Between Bond Prices and Yields

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Figure 10.1 Prices/Yields of U.S. Treasury Bondson Aug 15, 2011

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Figure 2.3 Prices/Yields of U.S. Treasury Notespage 32

Minimum tick size: ½ of 1/32 Special case: ¼ of 1/32 =1/128Bid: ( 112+30/32)% of face value

100:18.75 37/64+1/128100:14 28/64=14/32112:30 60/64=30/32107:9.5 19/64=9.5/32…

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Treasury Bonds and Notes Treasury Notes are issued with maturities

between 1 and 10 years. Treasury bonds are issued with maturities

ranging from 10 to 30 years. Domination: $100, $1000. Both make semiannual coupon payments. Bid and ask are quoted as a percentage of

par value.

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Invoice price, Flat price, Accrued interest

when you buy or sell a bond between coupon payment dates, the invoice price must incorporate accrued interest.

Invoice price = quoted price + Accrued interest

(price quoted in the financial press)

For a semi-annual payment bond, accrued interest between two coupon payment dates =

paymentscouponseparatingdays

paymentcouponlastcedayspaymentcouponannual sin

2

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Example 10.1

Suppose the coupon rate is 8%, face value is $1000 and coupon is paid semiannually. 30 days have passed since the last coupon payment. The quoted price is $990. What is the invoice price?

990+(1000*8%/2)*(30/182)

=990+6.59

=996.59

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Exercise Suppose the coupon rate is 10%, face value is

$1000 and coupon is paid semiannually. 125 days have passed since the last coupon payment. The Ask quote is 100:11. What is the invoice price that the investor has to pay?

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Price of a maturing 8% Treasury bond

What is the quoted price one day before the maturity? $1000.

A purchaser will receive 1040 on the following day.

So he should be willing to pay an invoice price of $1040.

40 is the accrued interest since the previous coupon payment.

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Corporate Bonds Figure 10.2

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Find YTM, Coupon rate(suppose the bond is issued today)

1)A $1,000 par value bond sells for $863.05. It matures in 20 years, has a 10 percent coupon rate, and pays interest semi-annually. What is the bond’s yield to maturity on a per annum basis (to 2 decimal places)?

Verify that YTM = 11.80%2) ABC Inc. just issued a twenty-year semi-annual coupon

bond at a price of $787.39. The face value of the bond is $1,000, and the YTM is 9%. What is the annual coupon rate (in percent, to 2 decimal places)?

Verify that annual coupon rate = 6.69%What happens if bond pays coupon annually? Quarterly?

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Zero-coupon bond (ZCB) 1 Zero coupon rate, no coupon paid during bond’s life. Bond holder receives one payment at maturity, the face

value (usually $1000).

Price of a ZCB, PZCB

NZCBd

ZCBr

FP

1

F = face value of the bond

Yield to maturity (in decimals)

N = number of periods to maturity

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Zero-coupon bond (ZCB) 2 As long as interest rates are positive, the

price of a ZCB must be less than its face value.

Why? With positive interest rates, the present value of the face value (i.e., the price) has to be less than the face value.

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ZCB Problems1) Find the price of a ZCB with 20 years to maturity,

par value of $1000 and a yield to maturity of 15% p.a. Assume annual compounding.

N=20, I/Y=15, FV=1000, PMT=0. Price = $61.10

2) XYZ Corp.’s ZCB has a market price of $ 354. The bond has 16 years to maturity and its face value is $1000. What is the yield to maturity for the ZCB. Assume annual compounding.

PV=-354, FV=1000, N=16, PMT=0. YTM = 6.71% p.a.What if semi-annual compounding?

These problems are just basic TVM problems where you receive a single cash flow in the future.

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Treasury Bills (T-bills) Chapter 2, page 27

Maturities: 4,13,26, 52 weeks. Denomination: 100, 10000 Income from T-bill is taxable at the federal level, not

state or local level. The cash flow pattern looks like a ZCB. Issued at a

discount from par value, return the par value at maturity. The discount from the par (face) value is annualized

based on a 360-day year and reported as a percentage of face value.

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Figure 2.1

%0711.0)245/365(:

%0477.0000477.0:,000477.1236.995,9/000,10

215.994,9$%)057847.01(000,10$%,057847.0)360/245(%085.0:

236.995,9$%)047639.01(000,10$%,047639.0)360/245(%07.0:

HPRRAnnual

HPR

bid

ask

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Figure 2.2 (8th edition)

01212996.0)161/365(:

00535048.078.946,9/)78.946,9000,10(:

78.946,9$)00532194.01(000,10$,00532194.0)360/161(0119.0:

33.946,9$)00536667.01(000,10$,00536667.0)360/161(0120.0:

HPRRAnnual

HPR

ask

bid

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Example of T-bill

A T-bill with face value $10,000 and 87 days to maturity is selling at a bank discount ask yield of 3.4%, what is the price of the bill?

Bank discount of 87 days: 0.034 x = 0.008217

Price: $10,000 x (1 – 0.008217) = $9,917.83

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Measures of return We measure the rate of return from

investing in a bond in several ways:

1. Yield to maturity (YTM)

2. Current yield (CY)

3. Yield to call (YTC)

4. Realized compound yield (RCY)

5. Holding period return (HPR)

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Yield to maturity The discount rate that makes the present value of

a bond’s payments (coupons & par value) equal to its price.

Interpretation: it is the compound rate of return that will be earned over a bond’s life if It is bought now and held until maturity All coupons are reinvested at the same YTM.

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Annualizing YTM (1) If coupons are paid semi-annually, then the YTM

we get from the financial calculator is a six-month YTM.

We can convert this six-month YTM to an annual YTM using Simple interest => bond equivalent yield to maturity

(or bond equivalent yield for short)

OR Compound interest => effective annual yield to

maturity (or effective annual yield for short)

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Annualizing YTM (2) In general, if a coupon bond pays coupons

m times a year, then:Bond equivalent yield

= periodic YTM x mEffective annual yield

= (1 + periodic YTM)m – 1

Note: periodic YTM is the value of I/Y you get from the financial calculator.

Stated in decimals

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Annualizing YTM (3) A 20-year maturity bond with par value

$1000 makes semi-annual coupon payments at a coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond if the bond price is $950.

Verify that Bond equivalent yield = 8.53% Effective annual yield = 8.71%

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Annualizing YTM (4) Treasury bonds paying an 8% coupon rate

with semi-annual payments currently sell at par value. What coupon rate would they have to pay in order to sell at par if they paid their coupons annually ?

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Practice 6 (1)

Chapter10:

4,5,6,14,15,17,22, 27

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Homework 6 1. You are a US. Treasury bond dealer who trades a

4.75%, 3year, semi-annual coupon bond. Your required YTM is 0.92826126%. How should you quote your Asked price in percentage of par value as shown in Figure 10.1?

2. Suppose today is Oct 23, 2013. A bond with a 10% coupon paid semiannually every Feb 15 and Aug15 is listed as selling at an ask price of 102:11. if you buy the bond from a dealer today, what price will you pay for it? the coupon period has 182 days.

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Bond price, coupon rate & YTM (1) A $1,000 par value bond has coupon rate

of 5% and the coupon is paid semi-annually. The bond matures in 20 years and has a yield to maturity of 10%. Compute the current price of this bond.

FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV.

PV = -571.02. Thus, price = $571.02 < par value

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Bond price, coupon rate & YTM (2)

Go back to the bond in the last problem. Suppose annual coupon rate = 10%. Verify that price = $1000 = par value

Suppose annual coupon rate = 12% Verify that price = $1,171.59 > par value.

It turns out that the following property is true.

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Bond price, coupon rate & YTM (3)

Coupon rate < YTM Price < face value Bond is selling at a discount

Coupon rate = YTM Price = face value Bond is selling at par

Coupon rate > YTM Price > face value Bond is selling at a premium

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Apply what we learnt A 10-year annual coupon bond was issued four years

ago at par. Since then the bond’s yield to maturity decreased from 9% to 7%. Which of the following statements is true about the current market price of the bond?

A. The bond is selling at a discountB. The bond is selling at parC. The bond is selling at a premiumD. The bond is selling at book valueE. Insufficient information

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Try one more One year ago Pell Inc. sold 20-year, $1,000 par value,

annual coupon bonds at a price of $931.54 per bond. At that time the market rate (i.e., yield to maturity) was 9 percent. Today the market rate is 9.5 percent; therefore the bonds are currently selling:

A. at a discount.

B. at a premium.

C. at par.

D. above the market price.

E. not enough information.

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Current yield (1) Annual coupon payment divided by bond

price. Measures return from coupon payments. Shortcomings:

Ignores capital gains or losses from bond sale. Ignores income from reinvestment of coupon

payments.

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Current yield (2) Consider a 6%, 15-year, semi-annual

payment bond with a par value of $1000. Compute the current yield if the yield to

maturity is: (a) 7%, (b) 6%, (c) 5% Verify that

(a) Price = $908.0398, current yield = 6.61%

(b) Price = $1000, current yield = 6%

(c) Price = $1104.6515, current yield = 5.43%

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Current yield (3)

If bond is selling at… Coupon rate (%)

Current yield (%)

YTM (%)

DiscountPrice = 908.0398

6 6.61 7

ParPrice = 1000

6 6 6

PremiumPrice = 1104.6515

6 5.43 5

Observe that…

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In general, we have the following relationship between coupon rate, current yield, & YTM

If bond is selling at… Then we know that…

Discount(below par)

Coupon rate < Current yield < YTM

Par Coupon rate = Current yield = YTM

Premium(above par)

Coupon rate > Current yield > YTM

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Quick review A bond has a current yield of 9% and a

yield to maturity of 10%. Is the bond selling above or below par value?

Is the coupon rate of the bond more or less than 9%?

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Yield to call (1) Applicable only to callable bonds. What’s a callable bond?

Bond that may be repurchased by the issuer at a specified price (“call price”) before the maturity date.

Call period: The period of time during which the issuer can repurchase the bond.

Time until call: The period of time before the issuer can start repurchasing the bond.

Motive: if interest rates fall, issuer can repurchase the bonds and issue new bonds at lower coupon rate. This lowers interest payments.

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Yield to call (2) Yield to call is calculated just like the yield to

maturity, except: Time until call replaces time to maturity Call price replaces par value

Yield to call

Time until call

Fixed periodic coupon

N

tN

FCt

FCFCB r

iceCall

r

CP

1 )1(

Pr

)1(

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Yield to call (3) A 20-year maturity bond with par value $1000 makes

semi-annual coupon payments at a coupon rate of 8%. The bond is currently selling for $1,150 and is callable in 10 years at a call price of $1,100.

Compute the bond equivalent yield to call and the effective annual yield to call.

What is N? What is FV? Verify that

Bond equivalent yield to call = 6.64% Effective annual yield to call = 6.75%

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Realized compound yield (1) Compound rate of return based on coupon

payments, reinvestment income and sale price during the holding period.

Realized compound yield depends on: Reinvestment rate: interest rate at which

coupon payments are reinvested. Holding period YTM at the end of holding period

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Realized Compound Yield (2) Realized compound yield, y

Purchase price = what you paid for the bond Final proceeds = future value of coupon payments and

reinvestment income + sale price n = length of holding period (could be in years, half-

years, quarters etc).

Length of holding period

1

1

n

pricepurchase

proceedsfinaly

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Realized Compound Yield (3) Suppose you buy a 30-year, 7.5%, annual

payment coupon bond for $980 and plan to hold it for 20 years. Your forecast is that the bond’s YTM will be 8% when it is sold and that the reinvestment rate on the coupons will be 6%. Compute the annual realized compound yield.

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Realized Compound Yield (4) Five years ago, XYZ Inc issued a 5% semi-annual

payment coupon bond with a maturity of ten years. You buy the bond now at a price of $683.94 and plan to hold it for three years. You forecast that you can invest the coupon payments at a stated annual rate of 6.25% and that at the end of three years, the yield will be 7.75%.

What is the bond equivalent realized compound yield? (i.e., annualize using simple interest)

What is the effective annual realized compound yield? (i.e., annualize using compound interest)

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Yield to Maturity vs. Realized Compound Yield (1)

Consider a 7% annual payment bond with two years to maturity. The YTM is 8% right now. Compute the realized compound yield if the reinvestment rate is (a) 7%, (b) 8%, (c) 9%.

Verify that the realized compound yield is:

a) 7.97% < YTM

b) 8% = YTM

c) 8.03% > YTM

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Yield to Maturity vs. Realized Compound Yield (2)

In general, if you hold a bond to maturity, then:

RCY < YTM if reinvestment rate < YTM RCY = YTM if reinvestment rate = YTM RCY > YTM if reinvestment rate > YTM

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Holding period return (HPR) Rate of return over a single investment (holding)

period. Has two components: Price change = ending price – beginning price Coupon payment

Price change + Coupon paymentBeginning price

Ending price - Beginning price + Coupon paymentBeginning price

HPR

=

=

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HPR problem Suppose you buy a bond of General Electric at a

price of $990. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in six months’ time. You intend to hold the bond till it matures.

What is the 6-month HPR?

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HPR vs. YTM (1) When the YTM stays the same during the

holding period, HPR = YTM. If the YTM changes after you bought the

bond, then HPR will be different from the initial YTM. If YTM falls after you bought the bond,

HPR > initial YTM. If YTM rises after you bought the bond,

HPR < initial YTM.* This property is true only if the end of the holding period is NOT

the maturity date.

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HPR vs. YTM (2) Consider a 30-year bond paying an annual

coupon of $80 and selling at $1000. The bond’s initial YTM is 8%. Suppose you buy the bond now and hold it for one year. Compute the 1-year HPR if YTM at the end of the year is (a) 8%, (b) 7%, (c) 9%.

Verify that:

a) If year-end YTM is 8%, then HPR = 8%

b) If year-end YTM is 7%, then HPR = 20.28%

c) If year-end YTM is 9%, then HPR = -2.2%

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Bond prices over time Suppose the yield to maturity is the same from the time

you purchased a bond to the bond’s maturity.

If you bought the bond at ….

Bond price over time

Premium Price falls and approaches par value.

Par Price remains at par value till maturity.

Discount Price rises and approaches par value.

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Figure 9.6 Premium and Discount Bonds over Time

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Quick review (1) Consider a bond with a 10% coupon rate and

with YTM of 8%. If the bond’s YTM remains constant, then in one year, will the bond price be

a) Higher

b) Lower

c) Unchanged

d) None of the above

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Quick review (2)Which of the following statements is correct?

a) If market rates do not change, the price of a bond selling at a premium increases over time.

b) If interest rates are greater than zero, it is possible for a zero-coupon bond to sell at a premium (i.e. for more than par value).

c) If a bond’s yield to maturity is greater than its coupon rate, the bond will sell at a discount.

d) If market rates do not change, the price of a bond selling at a discount decreases over time.

e) All of the above statements are false.

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Default risk What is default?

Failure of the bond issuer to make the promised coupon &/or par value payments to the bond holder.

Default risk To the bond holder, default risk is the uncertainty in

cash flows arising from the possibility that the issuer can fail to make promised payments.

With the exception of the U.S. government, all issuers have default risk.

Some issuers have greater default risk than others.

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Measuring default risk Credit rating agencies measure default risk of:

Large corporate bond issues Municipal bond issues International sovereign bond issues

Well-known credit rating agencies: Moody’s Standard & Poor’s (S&P for short) Duff & Phelps Fitch

Each agency assigns a letter grade to reflect level of default risk.

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S&P, Moody’s credit rating scheme

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Investment vs. speculative grade

Bonds fall into one of the two broad groups: Investment grade bond:

Rated BBB and above by S&P, or Rated Baa and above by Moody’s

Speculative grade or junk bond: Rated BB or lower by S&P, or Rated Ba or lower by Moody’s or Unrated bond

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Determinants of credit rating To assign rating, credit rating agencies

look at level and trend of issuer’s financial ratios.

Key ratios:1. Coverage: times-interest-earned2. Leverage: debt-to-equity, debt-to-assets3. Liquidity: current, quick4. Profitability: ROA5. Cash flow-to-debt

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Protection Against Default Credit Default Swap (CDS)

An insurance policy on the default risk of a corporate bond. CDS buyers pay the sellers an annual premium (a percentage of each $100 of bond principal). The sellers compensate the buyers for loss of bond value in the event of a default.

Bond indenture specifies restrictions on issuer to protect bondholder against default.

Sinking fund; Subordination of further debt;

Dividend restrictions; Collateral

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Yield to Maturity and Default Risk (1)

Promised/stated YTM: YTM you get if issuer makes all promised payments. Maximum possible YTM.

Expected YTM: YTM you get if you consider the possibility of default.

Consider the following example.

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Yield to Maturity and Default Risk (2) A firm issued a 9% semi-annual payment coupon bond

20 years ago. The bond now has 10 years left until its maturity date, but the firm is having financial difficulties. Investors believe that the firm will be able to make good on the remaining interest payments but that at the maturity date, the firm will be forced into bankruptcy, and bondholders will receive only 70% of par value. The bond is selling at $750.

Calculate the promised (bond equivalent) YTM and the expected (bond equivalent) YTM.

Verify that the promised YTM is 13.7% and the expected YTM is 11.6%.

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Summary1. Compute the price of (a) fixed-coupon bond, b) zero-coupon bond.

2. Bond price is inversely related to interest rates.

3. Different measures of bond returns: a) yield to maturity,

b) current yield,

c) yield to call,

d) realized compound yield,

e) holding period return.

4. Relationships between different measures of bond returns.

5. Evolution of bond prices over time if YTM remains the same.

6. Credit/ default risk as a source of risk in bond investing.

7. Determinants of bond safety and rating.

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Practice 6 (2)

Chapter10: 7,8,9,12,16,23,28,32.