Bond Pricing (1)

8/11/2019 Bond Pricing (1)

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FINC4101Investment 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 bonds 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

ForeignExchange

Derivatives

MarketEfficiency

Fixed Income

Equity

AssetPricing

PortfolioTheory

FI4000


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Types of fixed-income securities

Fixed-income security / bond:

A security that obligates the issuer tomake specified payments to the holderover a period of time.

We focus on the pricing of two types offixed-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 everyperioduntil 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 afixed-coupon bond: Example

A firm issues an 8% 30-year bond with annualcoupon 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 toinvestor.

What happens if coupon is paid semi-annually?Quarterly?


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FCB gives you a stream of fixed payments plusa 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 singlepayment, using an appropriate interest rate.

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


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Reminder:

The interest rate used to find the PV of abond 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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Price of the FCB, PFCB

Nd

N

t

t

d

FCB

r

F

r

CP

111

Number of

periods to

maturity

Fixed periodic

couponFace value

Yield to maturity (in decimals)


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Find FCB price

Consider an 8%, 30-year coupon bond that payscoupons semi-annually. Compute the bondsprice 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 betweenprice & 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 (interestrate).

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


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The Inverse Relationship BetweenBond 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

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Minimum tick size: of 1/32Special case: 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 maturitiesbetween 1 and 10 years.

Treasury bonds are issued with maturitiesranging 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 couponpayment dates, the invoice price mustincorporate 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%, facevalue is $1000 and coupon is paidsemiannually. 30 days have passed since

the last coupon payment. The quoted priceis $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. 125days have passed since the last coupon

payment. The Ask quote is 100:11. What is theinvoice 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 thematurity? $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 previouscoupon 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 in20 years, has a 10 percent coupon rate, and paysinterest semi-annually. What is the bonds yield tomaturity 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 couponrate (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 bonds life.

Bond holder receives one payment at maturity, the facevalue (usually $1000).

Price of a ZCB, PZCB

NZCBdZCB

r

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, theprice of a ZCB must be less than its facevalue.

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

These problems are just basic TVM problems where you receive a


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ZCB Problems

1) 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 = $61.102) 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 asingle 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, notstate or local level.

The cash flow pattern looks like a ZCB. Issued at adiscount from par value, return the par value at maturity.

The discount from the par (face) value is annualizedbased on a 360-day year and reported as a percentageof face value.

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Figure 2.1

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%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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Example of T-bill

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

Bank discount of 87 days: 0.034 x87 days

360 days= 0.008217

Price: $10,000 x (10.008217) = $9,917.83

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Measures of return

We measure the rate of return frominvesting 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 ofa bonds payments (coupons & par value) equalto its price.

Interpretation: it is the compound rate of returnthat will be earned over a bonds life if

It is bought now and held until maturity

All coupons are reinvestedat the same YTM.


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Annualizing YTM (1)

If coupons are paid semi-annually, then the YTMwe get from the financial calculator is a six-month YTM.

We can convert this six-month YTM to an annualYTM 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 couponsm times a year, then:

Bond equivalent yield

= periodic YTM x mEffective annual yield

= (1 + periodic YTM)m1

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 couponpayments at a coupon rate of 8%. Find

the bond equivalent and effective annualyield to maturity of the bond if the bondprice 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 ratewith semi-annualpayments currently sellat par value. What coupon rate would they

have to pay in order to sell at par if theypaid 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 a4.75%, 3year, semi-annual coupon bond. Your requiredYTM 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 islisted 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 rateof 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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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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Coupon rate < YTM Price < face value Bond is sel l ingat a discoun t

Coupon rate = YTM Price = face value Bond issel lingat par

Coupon rate > YTM Price > face value Bond is sel l ingat a prem ium


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

ago at par. Since then the bonds yield to maturitydecreased from 9% to 7%. Which of the followingstatements 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 premium

D. 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 bondprice.

Measures return from coupon payments.

Shortcomings:

Ignores capital gains or losses from bond sale.

Ignores income from reinvestment of couponpayments.


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Current yield (2)

Consider a 6%, 15-year, semi-annualpayment bond with a par value of $1000.

Compute the current yield if the yield tomaturity 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 Couponrate (%)

Currentyield (%)

YTM (%)

Discount

Price = 908.0398

6 6.61 7

Par

Price = 1000

6 6 6

Premium

Price = 1104.6515

6 5.43 5

Observe that


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In general, we have the following relationshipbetween 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 ayield to maturity of 10%. Is the bondselling above or below par value?

Is the coupon rate of the bond more orless than 9%?


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Yield to call (1)

Applicable only to callable bonds.

Whats 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 beforethe issuer can start

repurchasing the bond.

Motive: if interest rates fall, issuer can repurchase thebonds 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 tomaturity, 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

C

P 1 )1(

Pr

)1(


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Yield to call (3)

A 20-year maturity bond with par value $1000 makessemi-annual coupon payments at a coupon rate of 8%.The bond is currently selling for $1,150 and is callable in10 years at a call price of $1,100.

Compute the bond equivalent yield to call and theeffective 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 couponpayments, reinvestment income and saleprice 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%, annualpayment coupon bond for $980 and planto hold it for 20 years. Your forecast is that

the bonds YTM will be 8% when it is soldand that the reinvestment rate on thecoupons 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-annualpayment coupon bond with a maturity of ten years. Youbuy the bond now at a price of $683.94 and plan to holdit for three years. You forecast that you can invest the

coupon payments at a stated annual rate of 6.25% andthat at the end of three years, the yield will be 7.75%.

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

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

Yi ld t M t it


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Yield to Maturity vs.Realized Compound Yield (1)

Consider a 7% annual payment bond with twoyears 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

Yi ld t M t it


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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 pricebeginning price

Coupon payment

P rice change + Coupon payment

Beginning priceEnding price - Beginning price + Coupon payment

Beginning price

HPR

=

=


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HPR problem

Suppose you buy a bond of General Electric at aprice 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 sixmonths time. You intend to hold the bond till itmatures.

What is the 6-month HPR?


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HPR vs. YTM (1)

When the YTM stays the same during theholding period, HPR = YTM.

If the YTM changes after you bought the

bond, then HPR will be different from theinitial 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 annualcoupon of $80 and selling at $1000. The bondsinitial 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 timeyou purchased a bond to the bonds 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.

Figure 9 6 Premium and Discount


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Figure 9.6 Premium and DiscountBonds over Time


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Quick review (1)

Consider a bond with a 10% coupon rate andwith YTM of 8%. If the bonds YTM remainsconstant, then in one year, will the bond price

bea) 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 sellingat a premium increases over time.

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

c) If a bonds 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 sellingat 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 uncertaintyin

cash flowsarising from thepossibilitythat the issuer

can fail to make promised payments.

With the exception of the U.S. government, all issuershave 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: Moodys Standard & Poors(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, Moodys 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 Moodys

Speculative grade or junk bond:

Rated BB or lower by S&P, or

Rated Ba or lower by Moodys or

Unrated bond


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Determinants of credit rating

To assign rating, credit rating agencieslook at level and trend of issuers financialratios.

Key ratios:1. Coverage: times-interest-earned

2. Leverage: debt-to-equity, debt-to-assets

3. Liquidity: current, quick

4. Profitability: ROA

5. Cash flow-to-debt


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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 bond20 years ago. The bond now has 10 years left until itsmaturity date, but the firm is having financial difficulties.Investors believe that the firm will be able to make goodon the remaining interest payments but that at thematurity date, the firm will be forced into bankruptcy, andbondholders will receive only 70% of par value. Thebond 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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Summary

1. 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.

Documents

Bond Pricing (1)