Bond valuation

Financial management: lecture 4

Bond valuation

The application of the present value concept


Today’s plan Review of what we have learned in the last

lecture Interest rates and compounding Some terminology about bonds Value bonds The yield curve Default risk


What have we learned in the last lecture? The present value formulas of perpetuity

and annuity The application of the PV of annuity


My solution

year Beginningbalance

Interest payment

Principlepayment

Total payment

Ending balance

0

1

2

3

$20,000

13,809

7,154

$1,500 $6,191 $7,691 $13,809

1,036 6,655

537 7,154 7,691 0

7,691 7,154


A problem John is 65 years old and wants to retire next

year. After retirement, he wants to have an annual income of $24,000 for 20 years from his retirement fund, which has an annual interest rate of 6%. Suppose John will get the first retirement income one year from now. Then• How much money should John have in his retirement

fund in the end of this year?• Suppose John started to work 19 years ago and put the

same amount of money every year in his retirement fund. How much should he put every year? ( including this year, there will be a total of 20 years)


Nominal and real interest rates Nominal interest rate

• What is it? Real interest rate

• What is it? Inflation

• What is it? Their relationship

• 1+real rate =(1+nominal rate)/(1+inflation)


Be consistent in how you handle inflation!! Use nominal interest rates to discount

nominal cash flows. Use real interest rates to discount real cash

flows. You will get the same results, whether you

use nominal or real figures

Inflation rule


Example

You own a lease that will cost you $8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease?

1 real interest rate = 1+nominal interest rate1+inflation rate


InflationExample - nominal figures

99.429,26$ 78.59708741.82=8000x1.03456.63768487.20=8000x1.033

92.68098240=8000x1.03273.727280001

10% @ PVFlowCash Year

4

3

2

10.182.87413

10.120.84872

10.182401.108000


InflationExample - real figures

Year Cash Flow [email protected]%1 = 7766.992 = 7766.99

= 7766.99= 7766.99

80001.03

7766.991.068

82401.03

8487.201.03

8741.821.03

2

3

4

7272 736809 92

3 6376 564 5970 78

26 429 99

7766 991 068

7766 991 068

7766 991 068

2

3

4

.

.

.

.

..

..

..

= $ , .


Interest

Simple interest - Interest earned only on the original investment.

Compounding interest - Interest earned on interest.

In Bus 785, we consider compounding interest rates


Simple interest

Example Simple interest is earned at a rate of 6% for five years on a principal balance of $100.


Simple interest

Today Future Years 1 2 3 4 5

Interest Earned 6 6 6 6 6Value 100 106 112 118 124 130Value at the end of Year 5 = $130


Compound interest

Example Compound interest is earned at a rate of 6% for five years on $100.

Today Future Years 1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15 7.57Value 100 106.00 112.36 119.10 126.25

133.82

Value at the end of Year 5 = $133.82


Interest compounding The interest rate is often quoted as APR, the

annual percentage rate. If the interest rate is compounded m times in

each year and the APR is r, the effective annual interest rate is

11

m

mr


Compound Interest i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183


Compound Interest

02468

1012141618

Number of Years

FV o

f $1

10% Simple

10% Compound


Interest Rates

ExampleGiven a monthly rate of 1% (interest is compounded monthly), what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?


Solution

12.00%or .12=12 x .01=APR

12.68%or .1268=1 - .01)+(1=EAR 12


Interest Rates

ExampleIf the interest rate 12% annually and interest is compounded semi-annually, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)?


Solution

APR=12% EAR=(1+0.06)2-1=12.36%


Bonds

Bond – a security or a financial instrument that obligates the issuer (borrower) to make specified payments to the bondholder during a time horizon.

Coupon - The interest payments made to the bondholder.

Face Value (Par Value, Face Value, Principal or Maturity Value) - Payment at the maturity of the bond.

Coupon Rate - Annual interest payment, as a percentage of face value.


Bonds A bond also has (legal) rights attached to

it:• if the borrower doesn’t make the required

payments, bondholders can force bankruptcy proceedings

• in the event of bankruptcy, bond holders get paid before equity holders


An example of a bond A coupon bond that pays coupon of 10%

annually, with a face value of $1000, has a discount rate of 8% and matures in three years.• The coupon payment is $100 annually• The discount rate is different from the coupon rate.• In the third year, the bondholder is supposed to get

$100 coupon payment plus the face value of $1000.• Can you visualize the cash flows pattern?


Bonds

WARNINGWARNINGThe coupon rate IS NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce. Since the coupon rate is listed as a %, this misconception is quite common.


Bond Valuation

The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

Nrcpn

rcpn

rcpnPV

)1(000,1...

)1()1( 21


Zero coupon bonds

Zero coupon bonds are the simplest type of bond (also called stripped bonds, discount bonds)

You buy a zero coupon bond today (cash outflow) and you get paid back the bond’s face value at some point in the future (called the bond’s maturity )

How much is a 10-yr zero coupon bond worth today if the face value is $1,000 and the effective annual rate is 8% ?

PV

Facevalue

Time=tTime=0


Zero coupon bonds (continue) P0=1000/1.0810=$463.2 So for the zero-coupon bond, the price is

just the present value of the face value paid at the maturity of the bond

Do you know why it is also called a discount bond?


Coupon bond

The price of a coupon bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

)()()1()1(

11

)1()(....

)1()1( 21

parPVannuityPVr

parrrr

cpn

rparcpn

rcpn

rcpnPV

tt

t


Bond Pricing

ExampleWhat is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%.


Bond Pricing

ExampleWhat is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%.

77.010,1$)056.1(

060,1)056.1(

60)056.1(

60321

PV

PV


Bond Pricing

Example (continued)What is the price of the bond if the required rate of return is 6 %?

000,1$)06.1(

060,1)06.1(

60)06.1(

60321

PV

PV


Bond Pricing

Example (continued)What is the price of the bond if the required rate of return is 15 %?

51.794$)15.1(

060,1)15.1(

60)15.1(

60321

PV

PV


Bond Pricing

Example (continued)What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually?


Bond Pricing

Example (continued)What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually?

91.010,1$)028.1(

030,1)028.1(

30...)028.1(

30)028.1(

306521

PV

PV


Bond Pricing

Example (continued)Q: How did the calculation change, given semi-

annual coupons versus annual coupon payments?


Bond Pricing

Example (continued)Q: How did the calculation change, given semi-annual

coupons versus annual coupon payments?

Time Periods

Paying coupons twice a year, instead of once

doubles the total number of cash flows to be discounted

in the PV formula.


Bond Pricing

Example (continued)Q: How did the calculation change, given semi-annual

coupons versus annual coupon payments?

Time Periods

Paying coupons twice a year, instead of once

doubles the total number of cash flows to be discounted

in the PV formula.

Discount Rate

Since the time periods are now half years, the discount rate is also

changed from the annual rate to the half year rate.


Bond Yields

Current Yield - Annual coupon payments divided by bond price.

Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s payments equal the market price of the bond.

ty

parcpn

y

cpn

y

cpnP)1(

)(....)1()1( 21


An example of a bond A coupon bond that pays coupon of 10%

annually, with a face value of $1000, has a discount rate of 8% and matures in three years. It is assumed that the market price of the bond is the same as the present value of the bond. • What is the current yield?• What is the yield to maturity.


My solution First, calculate the bond price P=100/1.08+100/1.082+1100/1.083

=$1,051.54 Current yield=100/1051.54=9.5% YTM=8%


Bond Yields

Calculating Yield to Maturity (YTM=r)If you are given the market price of a bond (P) and the coupon rate, the yield to maturity can be found by solving for r.

ty

parcpn

y

cpn

y

cpnP)1(

)(....)1()1( 21


Bond Yields

ExampleWhat is the YTM of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,010.77

77.010,1$)1(

060,1)1(

60)1(

60321

PVrrr

PV


Bond Yields

In general, there is no simple formula that can be used to calculate YTM unless for zero coupon bonds

Calculating YTM by hand can be very tedious. We don’t have this kind of problems in the quiz or exam

You may use the trial by errors approach get it.


Bond Yields (3)

Can you guess which one is the solution in the previous example?

(a) 6.6%(b) 7.1%(c) 6.0%(d) 5.6%


The bond price, coupon rates and discount rates If the coupon rate is larger than the

discount rate, the bond price is larger than the face value.

If the coupon rate is smaller than the discount rate, the bond price is smaller than the face value.


The rate of return on a bond

price bondor investmentchange price+incomeCoupon =return of Rate

Example: An 8 percent coupon bond has a price of $110 dollars with maturity of 5 years

and a face value of $100. Next year, the expected bond price will be $105. If you

hold this bond this year, what is the rate of return?

investment ofcost profit=return of Rate


My solution The expected rate of return for holing the

bond this year is (8-5)/110=2.73%• Price change =105-110=-$5• Coupon payment=100*8%=$8• The investment or the initial price=$110


The Yield Curve

Term Structure of Interest Rates - A listing of bond maturity dates and the interest rates that correspond with each date.

Yield Curve - Graph of the term structure.


The term structure of interest rates (Yield curve)


YTM for corporate and government bonds The YTM of corporate bonds is larger

than the YTM of government bonds Why does this occur?


Default Risk Default risk

• The risk associated with the failure of the borrower to make the promised payments

Default premium• The amount of the increase of your discount

rate Investment grade bonds Junk bonds


Ranking bondsStandard

Moody' 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.

Bond valuation

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