Bond Valuation

1
Bond Valuation

• Application of present value techniques to bonds
  and stocks
• Pricing and Valuation are the core issues in
  finance

2
Some standard forms of Bonds

• Ccoupon, Fface value, Tmaturity date
• Pure Discount or Zero-coupon bonds
• PVFV/(1r)T
• Level-coupon bonds
• PVC/1r C/(1r)2 C/(1r)T F/(1r)T
• Consols
• PVC/r
• Floaters
• Convertibles

3
Bond Features

• Coupon Payments Regular interest payments
• Semi annual for most US corporate bonds
• Types of Coupon payments
• Fixed Rate 8 per year
• Floating Rate 6-month Treasury bill rate 100
  basis points.

4
Bond Features

• Face or Par Value amount of money to be repaid
  at end of loan
• 1,000/bond
• Maturity number of years from issue date until
  principal is paid
• Coupon Rate annual coupon / face value

5
Features of a May Department Stores Bond

• Terms Explanations
• Amount of issue 200 million The company will
  issue 200 million worth of bonds.
• Date of issue 8/4/94 The bonds were sold on
  8/4/94.
• Maturity 8/1/24 The principal will be paid in
  30 years.
• Face Value 1,000 Denomination of the bond is
  in 1,000
• Annual coupon 8.375 Each bondholder will
  receive 83.75 per bond per year (8.375 of
  the face value).
• Offer price 100 The offer price will be 100
  of the 1,000 face value per bond.
• Coupon dates 2/1, 8/1 41.875 will be paid on
  these dates
• Security None
• Sinking Fund Annual from 8/1/05 Annual payments
  to this fund starting from the indicated
  date
• Call Provision not callable before
  8/1/04 Deferred call feature
• Call Price 104.188 initially Buy back price is
  1041.88, declining declining to 100 to
  1,000 on 8/1/1
• Rating Moodys A2 This is one of Moodys higher
  ratings. The bonds have a low probability
  of default.

6
Bond Valuation(Assuming Level Coupon Payments)

• Discounted Cash Flow Valuation
• Bond Value PV (Promised Cash Flows)
• Bond Value PV (Coupon Payments)
• PV (Face Value)
• Bond Value PV (Annuity) PV (Lump sum)

7
Yield-to-Maturity (YTM)

• Required market interest rate that makes the
  discounted cash flows of the bond equal to its
  price
• Interest rate that we will use in the bond
  valuation equation
• Does not always equal the bonds coupon rate

8
IPC issues 5-year 1,000 face value bonds with an
annual coupon of 100. What is the coupon rate
and what is the price of the bonds if the YTM on
similar bonds is 10?
9
General Expression for the Value of a Bond
Annuity Formula
10
Example 2 Pricing of a regular bond

• IPS issues a 10-year bond
• YTM 24
• Coupon Rate 8
• Face value 1,000
• What is the price of the bond at the issue date?
• What is your minimum selling price if you sell
  this bond one year before its maturity?

11
Notes on the Bond Pricing Formula

• Semi annual coupons 10-year bond with 12 coupon
  rate paid semi annually
• Halve the coupon rate and quoted YTM
• Double the number of periods
• YTMAPR!
• Risk-free market interest rate versus YTM
• YTM takes into consideration the risk of the cash
  flows
• Finding YTM trial and error, EXCEL, financial
  calculator.

12
Semi-annual couponsWhat is the price of a
1000 bond maturing in ten years with a 12
coupon that is paid semiannually if the YTM is 10
13
Discount bond example Suppose a year has gone
by and the IPC 10 annual coupon bond has 4 years
to maturity. What is the price (present value)
of the bonds if the YTM on similar bonds is 11?
14
Premium Bond ExampleSuppose in the second year
the yield-to-maturity for similar bonds decreases
to 9 instead of increasing to 11. What is the
price of the 4-year IPC 1000 par value 10
annual coupon bond?
15
Par, Discount and Premium Bonds The relation of
YTM and the coupon rate

• Par Bonds
• Price Face Value
• YTM Coupon Rate
• Discount Bonds
• Price
• YTM Coupon Rate
• Premium Bonds
• Price Face Value
• YTM

16
Interest Rate Risk of Bonds

• Risk that the bond you own will change in value
  because interest rates (e.g. YTM) have changed
• Review
• As interest rates rise, PV decreases all else
  equal
• As interest rates fall, PV increases all else
  equal
• Interest rate sensitivity depends on time to
  maturity and coupon rate

17
Determinants of the Interest Rate Risk of Bonds
Part I

• Time to Maturity
• The longer the time to maturity, the greater the
  interest rate risk, all else equal
• Higher t in formula greater compounding effect
  small changes in r, big changes in price
• 10 2-year and 15-year bonds
• 15 year bonds price will change more with a
  change in the YTM.

18
Interest Rate Changes and Bond Values Both
bonds are par bonds originally
Suppose YTM decreases to 8
Suppose YTM increases to 12
19
Interest Rate Risk and Time to Maturity (Figure
7.2)
Bond values ()
2000 1500 1000 500
1,768.62
30-year bond
Time to maturity Interest rate
1 year 30 years 5 1,047.62 1,768.62
10 1,000.00 1,000.00 15
956.52 671.70 20 916.67
502.11
1-year bond
1,047.62
916.67
502.11
Interest rates ()
5 10
15 20

Value of a Bond with a 10 Coupon Rate for
Different Interest Rates and Maturities
20
Determinants of the Interest Rate Risk of Bonds
Part II

• Coupon Rate
• The lower the coupon rate, the greater the
  interest rate risk, all else equal
• Low coupon more of the bonds value comes
  from the face amount
• 0 (pure discount) and 10 8-year bonds
• 100 of the 0 coupon bonds price comes from
  face amount
• 10 bonds price depends on eight 100 annual
  coupons plus face value

21
Which Bond has a higher interest rate risk?

• A 30-year, 10 coupon or B 15-year, 10 coupon?
• A 30-year, 10 coupon with face value of 1,000
  or B 25-year, 10 coupon, with face value of
  10,000?
• A 30-year, 11 coupon or B 30-year, 9 coupon?
• A 30-year, zero-coupon with face value of
  1,000 or B 30-year, 10 coupon, with face value
  of 10,000?
• A 30-year, 10 coupon or B 25-year, 15 coupon?
• A 30-year, 10 coupon or B 20-year, 8 coupon?

22
The Interest Rate Risk of Bonds

• Duration
• A measure of the interest rate risk of a bond
• average maturity of a bonds cash flows
• The higher the duration measure, the greater the
  interest rate risk

23
Bond pricing Theorems

• The following statements about bond pricing are
  always true.
• Bond prices and market interest rates move in
  opposite directions.
• When a bonds coupon rate is (greater than /
  equal to / less than) the markets required
  return (YTM), the bonds market value will be
  (greater than / equal to / less than) its par
  value.
• Given two bonds identical except for maturity,
  the price of the longer-term bond will change
  more than that of the shorter-term bond, for a
  given change in market interest rates.
• Given two bonds identical but for coupon, the
  price of the lower-coupon bond will change more
  than that of the higher-coupon bond, for a given
  change in market interest rates.

24
Implicit Interest on a Zero-coupon

• Suppose EIN company issues a 1,000, 5-year
  zero-coupon bond.
• Calculate the price if the YTM-15
• What is the total amount of implicit interest on
  this bond?
• What is the yearly implicit interest using
  amortization (required by law)?

25
Solution

• PV 1,000 / 1.155 497
• Total Implicit Interest 1,000 - 497 502
• Using straight-line interest expense, we have
  502/5 102.60 per year.
• Using amortization, we have for the first year
• Beginning value 497
• Ending value 1,000 / 1.154 572
• Implicit interest in year 1 572 - 497 75
• Implicit interest in year 2 (1,000 / 1.153) -
  572 86
• et cetera
• What would be preferred by the corporation?

26
Inflation and Returns

• Key issues
• What is the difference between a real and a
  nominal return?
• How can we convert from one to the other?
• Example Suppose we have 1,000, and Diet Coke
  costs 2.00 per six pack. We could buy 500 six
  packs. Now suppose the rate of inflation is 5,
  so that the price rises to 2.10 in one year.
  When we invest the 1,000 it grows to 1,100 in
  one year.
• Whats the return in dollars?
• Whats the return in six packs?

27
Reading the Wall Street Journal

• Majority of bonds is traded OTC
• Bonds are quoted as a of the face value
• Bonds are quoted as ATT 7s05
• ATT issue, maturing in 2005, 7 coupon
• Close last available price on close previous
  business day ( of F)
• Net Change
• Current Yield coupon / closing quote
• Volume

28
Treasury Bonds

• Always semi-annually
• Quoted in 32nds (smallest tick size)
• Bid price 13220 means 132 20/32 percent of
  the face value 1,329.375
• Change -46 means the price (bid or ask) fell by
  46/32, or 1.4375.
• YTM is based on ask price
• Bid-ask spread
• n indicates notes, rather than bonds

29
Inflation and Returns

• A. Dollars. Our return is
• (1100 - 1000)/1000 100/1000
  ________.
• The percentage increase in the amount of
  green stuff is 10 our dollar return is 10.
• B. Six packs. We can buy 1100/2.10 ________
  six packs, so our return is
• (523.81 - 500)/500 23.81/500 4.76
• The percentage increase in the amount of
  brown stuff is 4.76 our six-pack return is
  4.76.

30
Inflation and Returns, concluded

• The relationship between real and nominal returns
  is described by the Fisher Effect. Let
• R the nominal return
• r the real return
• h the inflation rate
• According to the Fisher Effect
• 1 R (1 r) x (1 h)
• From the example, the real return is 4.76 the
  nominal return is 10, and the inflation rate is
  5
• (1 R) 1.10
• (1 r) x (1 h) 1.0476 x 1.05 1.10