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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?.

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bond valuation

Bond valuation

The application of the present value concept

Financial management: lecture 4

today s plan
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
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
My solution

Ending

balance

Total

payment

Interest

payment

Principle

payment

year

Beginning

balance

0

$20,000

$1,500

$6,191

$7,691

$13,809

1

7,154

13,809

1,036

6,655

7,691

2

7,154

7,691

0

7,154

537

3

Financial management: lecture 4

a problem
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 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

inflation rule
Inflation rule
  • 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

Financial management: lecture 4

example
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?

Financial management: lecture 4

inflation
Inflation

Example - nominal figures

Financial management: lecture 4

inflation1
Inflation

Example - real figures

Financial management: lecture 4

interest
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
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 interest1
Simple interest

Today Future Years

12345

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
Compound interest

Example

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

Today Future Years

1 2 34 5

Interest Earned 6.00 6.36 6.74 7.15 7.57

Value 100 106.00 112.36 119.10 126.25 133.82

Value at the end of Year 5 = $133.82

Financial management: lecture 4

interest compounding
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

Financial management: lecture 4

compound interest1
Compound Interest

i ii iii iv v

Periods Interest Value Annually

per per APR after compounded

year 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 interest2
Compound Interest

Financial management: lecture 4

interest rates
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
Solution

Financial management: lecture 4

interest rates1
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

solution1
Solution
  • APR=12%
  • EAR=(1+0.06)2-1=12.36%

Financial management: lecture 4

bonds
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

bonds1
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
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

bonds2
Bonds

WARNING

The 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 valuation1
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.

Financial management: lecture 4

zero coupon bonds
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% ?

Face

value

PV

Time=0

Time=t

Financial management: lecture 4

zero coupon bonds continue
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
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.

Financial management: lecture 4

bond pricing
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 pricing1
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 pricing2
Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 6 %?

Financial management: lecture 4

bond pricing3
Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 15 %?

Financial management: lecture 4

bond pricing4
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 pricing5
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 pricing6
Bond Pricing

Example (continued)

Q: How did the calculation change, given semi-annual coupons versus annual coupon payments?

Financial management: lecture 4

bond pricing7
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 pricing8
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
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.

Financial management: lecture 4

an example of a bond1
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 solution1
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 yields1
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.

Financial management: lecture 4

bond yields2
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

Financial management: lecture 4

bond yields3
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
Bond Yields (3)
  • Can you guess which one is the solution in the previous example?
  • 6.6%
  • 7.1%
  • 6.0%
  • 5.6%

Financial management: lecture 4

the bond price coupon rates and discount rates
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
The rate of return on a bond

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?

Financial management: lecture 4

my solution2
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
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
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
  • 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 bonds
Ranking bonds

Financial management: lecture 4