Bond valuation

1
Bond valuation

• The application of the present value concept

2
Todays 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

3
What have we learned in the last lecture?

• The present value formulas of perpetuity and
  annuity
• The application of the PV of annuity

4
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
5
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)

6
Nominal and real interest rates

• Nominal interest rate
• What is it?
• Real interest rate
• What is it?
• Inflation
• What is it?
• Their relationship
• 1real rate (1nominal rate)/(1inflation)

7
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

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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?

9
Inflation

• Example - nominal figures

10
Inflation

• Example - real figures

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Interest

• Simple interest - Interest earned only on the
  original investment.
• Compounding interest - Interest earned on
  interest.
• In Bus 785, we consider compounding interest rates

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Simple interest

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

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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

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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 100 106.00 112.36 119.10 126.25 133.82
• Value at the end of Year 5 133.82

15
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

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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
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Compound Interest
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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)?

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Solution

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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)?

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Solution

• APR12
• EAR(10.06)2-112.36

22
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.

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Bonds

• A bond also has (legal) rights attached to it
• if the borrower doesnt make the required
  payments, bondholders can force bankruptcy
  proceedings
• in the event of bankruptcy, bond holders get paid
  before equity holders

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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?

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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.

26
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.

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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 bonds face value at
  some point in the future (called the bonds
  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
Time0
Timet
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Zero coupon bonds (continue)

• P01000/1.0810463.2
• 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?

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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.

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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.

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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.

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Bond Pricing

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

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Bond Pricing

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

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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?

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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?

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Bond Pricing

• Example (continued)
• Q How did the calculation change, given
  semi-annual coupons versus annual coupon
  payments?

37
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.
38
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.
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Bond Yields

• Current Yield - Annual coupon payments divided by
  bond price.
• Yield To Maturity (YTM)- Interest rate for which
  the present value of the bonds payments equal
  the market price of the bond.

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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.

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My solution

• First, calculate the bond price
• P100/1.08100/1.0821100/1.083
• 1,051.54
• Current yield100/1051.549.5
• YTM8

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Bond Yields

• Calculating Yield to Maturity (YTMr)
• 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.

43
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

44
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
  dont have this kind of problems in the quiz or
  exam
• You may use the trial by errors approach get it.

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Bond Yields (3)

• Can you guess which one is the solution in the
  previous example?
• 6.6
• 7.1
• 6.0
• 5.6

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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.

47
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?
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My solution

• The expected rate of return for holing the bond
  this year is (8-5)/1102.73
• Price change 105-110-5
• Coupon payment10088
• The investment or the initial price110

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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.

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The term structure of interest rates (Yield curve)
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YTM for corporate and government bonds

• The YTM of corporate bonds is larger than the YTM
  of government bonds
• Why does this occur?

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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

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Ranking bonds