Pricing of Bonds

1
Chapter 2

• Pricing of Bonds

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Time Value of Money (TVM)

• The price of any security equals the PV of the
  securitys expected cash flows.
• So, to price a bond we need to know
• The size and timing of the bonds expected cash
  flows.
• The required return (commensurate with the
  riskiness of the cash flows). MARKET VALUE
• You must be comfortable with TVM
• PV and FV of lump sums and annuities.
• Your text has a good review of the TVM concepts
  needed for this course.

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Two Important PV Formulas

• PV of a lump sum

• PV of an annuity (Formula 2.5, where CF A)

4
Pricing A Bond

• We begin with a simple bullet bond
• Non-callable (maturity is known with certainty)
• Coupons are paid every six months.
• The next coupon is received exactly six months
  from now.
• The interest rate at which the coupons can be
  invested is fixed for the life of the bond.
• Principal is paid at maturity (no amortizing).
• Coupon fixed for the life of the bond.

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Bond Pricing Formula

• Notation
• P price of the bond (in )
• n number of periods (maturity in years ? 2)
• C semiannual coupon (in )
• M maturity value
• The bond price is (Formulas 2.6, 2.7, 2.8)

Note All inputs to the bond pricing formula are
fixed except for r. As r changes so does P.
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Example

• Price a 20-year 10 coupon bond with a face value
  of 1,000 if the required yield on the bond is
  11.
• Formula inputs
• The coupon is 0.10 ? 1,000 100.
• The semiannual coupon, C, is 50.
• n 40
• r 0.055

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Pricing Zero-Coupon Bonds

• Zero-coupons bonds (zeros) are so called because
  they pay no coupons (i.e., C 0)
• They have only maturity value

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Example

• Price a zero that expires 15 years from today if
  its maturity value is 1,000 and the required
  yield is 9.4
• Formula inputs
• M 1,000
• n 30
• r 0.047

• An investor would pay 252.12 today and receive
  1,000 in 15 years.

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Price-Yield Relationship

• A fundamental property of bond pricing is the
  inverse relationship between bond yield and bond
  price.

Price
Yield
10
Price-Yield Relationship

• For a plain vanilla bond all bond pricing inputs
  are fixed except yield.
• Therefore, when yields change the bond price must
  change for the bond to reflect the new required
  yields.
• Example Examine the price-yield relationship on
  a 7 coupon bond.
• For r lt 7, the bond sells at a premium
• For r gt 7 the bond sells at a discount
• For r 7, the bond sells at par value

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Price-Yield Relationship

• The price-yield relationship can be summarized
• yield lt coupon rate ? bond price gt par (premium
  bond)
• yield gt coupon rate ? bond price lt par (discount
  bond)
• yield coupon rate ? bond price par (par
  bond)
• Bond prices change for the following reasons
• Discount or premium bond prices move toward par
  value as the bond approaches maturity. (Time
  Passes)
• Market factors change in yields required by the
  market.
• Issue specific factors a change in yield due to
  changes in the credit quality of the issuer.
  (Credit Spreads)

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Complications to Bond Pricing

• We have assumed the following so far
• Next coupon is due in six months.
• Cash flows are known with certainty
• We can determining the appropriate required
  yield.
• One discount rate applies to all cash flows.
• These assumptions may not be true and therefore
  complicate bond pricing.

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Complications to Bond PricingNext Coupon Due lt
6 Months

• What if the next coupon payment is less than six
  months away?
• Then the accepted method for pricing bonds is

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Complications to Bond PricingCFs May Not Be
Known

• For a non-callable bond cash flows are known with
  certainty (assuming issuer does not default)
• However, most bonds are callable.
• Interest rates then determine the cash flow
• If interest rates drop low enough below the
  coupon rate, the issuer will call the bond.
• Also, CFs on floaters and inverse floaters change
  over time and are not known (more on this later).

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Complications to Bond PricingDetermining
Required Yield

• The required yield for a bond is R rf RP
• rf is obtained from an appropriate maturity
  Treasury security.
• RP (Risk Premium) should be obtained from RPs of
  bonds of similar risk.
• This process requires some judgement.

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Complications to Bond PricingCash Flow Discount
Rates

• We have assumed that all bond cash flows should
  be discounted using one discount rate.
• However, usually we are facing an upward sloping
  yield curve
• So each cash flow should be discounted at a rate
  consistent with the timing of its occurrence.
• In other words, we can view a bond as a package
  of zero-coupon bonds
• Each cash coupon (and principal payment) is a
  separate zero-coupon bond and should be
  discounted at a rate appropriate for the
  maturity of that cash flow.

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Pricing Floaters

• Coupons for floaters depend on a floating
  reference interest rate
• coupon rate floating reference rate fixed
  spread (in bps)
• Since the reference rate is unpredictable so is
  the coupon.
• Example
• Coupon rate rate on 3-month T-bill 50bps

Reference Rate
Spread

• Floaters can have restrictions on the coupon
  rate
• Cap A maximum coupon rate.
• Floor A minimum coupon rate.

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Pricing Inverse Floaters

• An inverse floater is a bond whose coupon goes up
  when interest rates go down and vice versa.
• Inverse floaters can be created using a
  fixed-rate security (called the collateral)
• From the collateral two bonds are created (1) a
  floater, and (2) an inverse floater.
• These bonds are created so that
• Floater coupon Inverse floater coupon
  Collateral coupon
• Floater par value Inverse floater par value
  Collateral par value
• Equivalently, the bonds are structured so that
  the cash flows from the collateral bond is
  sufficient to cover the cash flows for the
  floater and inverse floater.

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Inverse Floater Example(pg. 30 text)

• Consider a 10-yr 15 coupon bond (7.5 every 6
  months).
• Suppose 100 million of bond is used to create
  two bonds
• 50 million par value floater and 50 million par
  value inverse floater.
• Assume a 6-mo coupon reset based on the formula
• Floater coupon rate reference rate 1
• Inverse coupon rate 14 - reference rate
• Notice Floater coupon rate Inverse coupon
  rate 15
• Problem if reference rate gt 14, then inverse
  floater coupon rate lt 0.
• Solution put a floor on the inverse floater
  coupon of 0.
• However, this means we must put a cap in the
  floater coupon of 15.
• The price of floaters and inverse floaters
• Collateral price Floater price Inverse
  floater price

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Price Quotes on Bonds

• We have assumed that the face value of a bond is
  1,000 and that is often true, but not always
• So, when quoting bond prices, traders quote the
  price as a percentage of par value.
• Example A quote of 100 means 100 of par value.

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Price Quotes on Bonds

• Most bond trades occur between coupon payment
  dates.
• Thus at settlement, the buyer must compensate the
  seller for coupon interest earned since the last
  coupon payment.
• This amount is called accrued interest.
• The buyer pays the seller Bond price Accrued
  Interest (often called the dirty price).
• The bond price without accrued interest is often
  called the clean price.

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Clean vs. Dirty Pricepg 31

• Suppose a bond just sold for 87.01 (based on par
  value of 100) and pays a coupon of 4 every six
  months.
• The bond paid the last coupon 120 days ago.
• What is the clean price? What is the dirty
  price?
• Clean price
• 87.01 (120/180)(4) 84.34
• Dirty price
• 87.01

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Clean vs. Dirty Bonds example
A US bond has a coupon rate of 7.2 and pays 4
times a year, on the 15th of January, April,
July, and October. It uses the 30/360 US day
count convention. A trade for 1,000 par value of
the bond settles on January 25th. The prior
coupon date was January 15th. The accrued
interest reflects ten days' interest, or 2.00
(7.2 of 1,000 (10 days/360 days)). The full
(Dirty) value of these bonds is set by the market
at 985.50