financial management: lecture 4 bond valuation the application of the present value concept
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Financial management: lecture 4
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
Financial management: lecture 4
What have we learned in the last lecture?
The present value formulas of perpetuity and annuity
The application of the PV of annuity
Financial management: lecture 4
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
Financial management: lecture 4
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)
Financial management: lecture 4
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)
Financial management: lecture 4
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
Financial management: lecture 4
Example
You own a lease that will cost you $8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease?
1 real interest rate = 1+nominal interest rate1+inflation rate
Financial management: lecture 4
Inflation
Example - nominal figures
99.429,26$
78.59708741.82=8000x1.034
56.63768487.20=8000x1.033
92.68098240=8000x1.032
73.727280001
10% @ PVFlowCash Year
4
3
2
10.182.87413
10.120.84872
10.18240
1.108000
Financial management: lecture 4
Inflation
Example - real figures
Year Cash Flow [email protected]%
1 = 7766.99
2 = 7766.99
= 7766.99
= 7766.99
80001.03
7766.991.068
82401.03
8487.201.03
8741.821.03
2
3
4
7272 73
6809 92
3 6376 56
4 5970 78
26 429 99
7766 991 068
7766 991 068
7766 991 068
2
3
4
.
.
.
.
..
..
..
= $ , .
Financial management: lecture 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
Financial management: lecture 4
Simple interest
Example
Simple interest is earned at a rate of 6% for five years on a principal balance of $100.
Financial management: lecture 4
Simple interest
Today Future Years
1 2 3 4 5
Interest Earned 6 6 6 6 6
Value 100 106 112 118 124 130
Value at the end of Year 5 = $130
Financial management: lecture 4
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 7.57
Value 100 106.00 112.36 119.10 126.25133.82
Value at the end of Year 5 = $133.82
Financial management: lecture 4
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
Financial management: lecture 4
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
Financial management: lecture 4
Compound Interest
02468
1012141618
Number of Years
FV
of
$1
10% Simple
10% Compound
Financial management: lecture 4
Interest Rates
Example
Given a monthly rate of 1% (interest is compounded monthly), what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
Financial management: lecture 4
Solution
12.00%or .12=12 x .01=APR
12.68%or .1268=1 - .01)+(1=EAR 12
Financial management: lecture 4
Interest Rates
Example
If 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)?
Financial management: lecture 4
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.
Financial management: lecture 4
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
Financial management: lecture 4
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?
Financial management: lecture 4
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.
Financial management: lecture 4
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.
Nr
cpn
r
cpn
r
cpnPV
)1(
000,1...
)1()1( 21
Financial management: lecture 4
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
Financial management: lecture 4
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?
Financial management: lecture 4
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
par
rrrcpn
r
parcpn
r
cpn
r
cpnPV
tt
t
Financial management: lecture 4
Bond Pricing
Example
What 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%.
Financial management: lecture 4
Bond Pricing
Example
What 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
Financial management: lecture 4
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
Financial management: lecture 4
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
Financial management: lecture 4
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?
Financial management: lecture 4
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
Financial management: lecture 4
Bond Pricing
Example (continued)
Q: How did the calculation change, given semi-annual coupons versus annual coupon payments?
Financial management: lecture 4
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.
Financial management: lecture 4
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.
Financial management: lecture 4
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
Financial management: lecture 4
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.
Financial management: lecture 4
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%
Financial management: lecture 4
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
Financial management: lecture 4
Bond Yields
Example
What 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
PV
rrrPV
Financial management: lecture 4
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.
Financial management: lecture 4
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%
Financial management: lecture 4
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.
Financial management: lecture 4
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
Financial management: lecture 4
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
Financial management: lecture 4
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.
Financial management: lecture 4
YTM for corporate and government bonds
The YTM of corporate bonds is larger than the YTM of government bonds
Why does this occur?
Financial management: lecture 4
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
Financial management: lecture 4
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.