FINANCE 4' Bond Valuation

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FINANCE4. Bond Valuation

• Professeur André Farber
• Solvay Business School
• Université Libre de Bruxelles
• Fall 2007

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Review present value calculations

• Cash flows C1, C2, C3, ,Ct, CT
• Discount factors DF1, DF2, ,DFt, , DFT
• Present value PV C1 DF1 C2 DF2 CT
  DFT

If r1 r2 ...r
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Review Shortcut formulas

• Constant perpetuity Ct C for all t
• Growing perpetuity Ct Ct-1(1g)
• rgtg t 1 to 8
• Constant annuity CtC t1 to T
• Growing annuity Ct Ct-1(1g)
• t 1 to T

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Bond Valuation

• Objectives for this session
• 1.Introduce the main categories of bonds
• 2.Understand bond valuation
• 3.Analyse the link between interest rates and
  bond prices
• 4.Introduce the term structure of interest rates
• 5.Examine why interest rates might vary according
  to maturity

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Zero-coupon bond

• Pure discount bond - Bullet bond
• The bondholder has a right to receive
• one future payment (the face value) F
• at a future date (the maturity) T
• Example a 10-year zero-coupon bond with face
  value 1,000
• Value of a zero-coupon bond
• Example
• If the 1-year interest rate is 5 and is assumed
  to remain constant
• the zero of the previous example would sell for

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Level-coupon bond

• Periodic interest payments (coupons)
• Europe most often once a year
• US every 6 months
• Coupon usually expressed as of principal
• At maturity, repayment of principal
• Example Government bond issued on March 31,2000
• Coupon 6.50
• Face value 100
• Final maturity 2005
• 2000 2001 2002 2003 2004 2005
• 6.50 6.50 6.50 6.50 106.50

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Valuing a level coupon bond

• Example If r 5
• Note If P0 gt F the bond is sold at a premium
• If P0 ltF the bond is sold at a
  discount
• Expected price one year later P1 105.32
• Expected return 6.50 (105.32
  106.49)/106.49 5

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When does a bond sell at a premium?

• Notations C coupon, F face value, P price
• Suppose C / F gt r
• 1-year to maturity
• 2-years to maturity
• As P1 gt F

with
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A level coupon bond as a portfolio of zero-coupons

• Cut level coupon bond into 5 zero-coupon
• Face value Maturity Value
• Zero 1 6.50 1 6.19
• Zero 2 6.50 2 5.89
• Zero 3 6.50 3 5.61
• Zero 4 6.50 4 5.35
• Zero 5 106.50 5 83.44
• Total 106.49

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Bond prices and interest rates
Bond prices fall with a rise in interest rates
and rise with a fall in interest rates
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Sensitivity of zero-coupons to interest rate
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Duration for Zero-coupons

• Consider a zero-coupon with t years to maturity
• What happens if r changes?
• For given P, the change is proportional to the
  maturity.
• As a first approximation (for small change of r)

Duration Maturity
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Duration for coupon bonds

• Consider now a bond with cash flows C1, ...,CT
• View as a portfolio of T zero-coupons.
• The value of the bond is P PV(C1) PV(C2)
  ... PV(CT)
• Fraction invested in zero-coupon t wt PV(Ct) /
  P
• Duration weighted average maturity of
  zero-coupons
• D w1 1 w2 2 w3 3wt t wT T

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Duration - example

• Back to our 5-year 6.50 coupon bond.
• Face value Value wt
• Zero 1 6.50 6.19 5.81
• Zero 2 6.50 5.89 5.53
• Zero 3 6.50 5.61 5.27
• Zero 4 6.50 5.35 5.02
• Zero 5 106.50 83.44 78.35
• Total 106.49
• Duration D .05811 0.05532 .0527 3
  .0502 4 .7835 5
• 4.44
• For coupon bonds, duration lt maturity

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Price change calculation based on duration

• General formula
• In example Duration 4.44 (when r5)
• If ?r 1 ? 4.44 1 - 4.23
• Check If r 6, P 102.11
• ?P/P (102.11 106.49)/106.49 - 4.11

Difference due to convexity
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Duration -mathematics

• If the interest rate changes
• Divide both terms by P to calculate a percentage
  change
• As
• we get

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Yield to maturity

• Suppose that the bond price is known.
• Yield to maturity implicit discount rate
• Solution of following equation

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Yield to maturity vs IRR
The yield to maturity is the internal rate of
return (IRR) for an investment in a bond.
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Asset Liability Management

• Balance sheet of financial institution (mkt
  values)
• Assets Equity Liabilities ? ?A ?E ?L
• As ?P -D P ?r
  (D modified duration)
• -DAsset A ?r -DEquity E ?r -
  DLiabilities L ?r
• DAsset A DEquity E DLiabilities L

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Examples
SAVING BANK
LIFE INSURANCE COMPANY
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• Immunization DEquity 0
• As DAsset A DEquity
  E DLiabilities L
• DEquity 0 ? DAsset A DLiabilities L

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Spot rates

• Spot rate yield to maturity of zero coupon
• Consider the following prices for zero-coupons
  (Face value 100)
• Maturity Price
• 1-year 95.24
• 2-year 89.85
• The one-year spot rate is obtained by solving
• The two-year spot rate is calculated as follow
• Buying a 2-year zero coupon means that you invest
  for two years at an average rate of 5.5

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Measuring spot rate
Data
To recover spot prices, solve
99.06 105 d1103.70 9 d1 109
d2 97.54 6.5 d1 6.5 d2 106.5
d3100.36 8 d1 8 d2
8 d3 108 d4
Solution
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Forward rates

• You know that the 1-year rate is 5.
• What rate do you lock in for the second year ?
• This rate is called the forward rate
• It is calculated as follow
• 89.85 (1.05) (1f2) 100 ? f2 6
• In general
• (1r1)(1f2) (1r2)²
• Solving for f2
• The general formula is

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Forward rates example

• Maturity Discount factor Spot rates Forward
  rates
• 1 0.9500 5.26
• 2 0.8968 5.60 5.93
• 3 0.8444 5.80 6.21
• 4 0.7951 5.90 6.20
• 5 0.7473 6.00 6.40
• Details of calculation
• 3-year spot rate
• 1-year forward rate from 3 to 4

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Term structure of interest rates

• Why do spot rates for different maturities differ
  ?
• As
• r1 lt r2 if f2 gt r1
• r1 r2 if f2 r1
• r1 gt r2 if f2 lt r1
• The relationship of spot rates with different
  maturities is known as the term structure of
  interest rates

Upward sloping
Spotrate
Flat
Downward sloping
Time to maturity
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Forward rates and expected future spot rates

• Assume risk neutrality
• 1-year spot rate r1 5, 2-year spot rate r2
  5.5
• Suppose that the expected 1-year spot rate in 1
  year E(r1) 6
• STRATEGY 1 ROLLOVER
• Expected future value of rollover strategy
• (100) invested for 2 years
• 111.3 100 1.05 1.06 100 (1r1)
  (1E(r1))
• STRATEGY 2 Buy 1.113 2-year zero coupon, face
  value 100

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Equilibrium forward rate

• Both strategies lead to the same future expected
  cash flow
• ? their costs should be identical
• In this simple setting, the foward rate is equal
  to the expected future spot rate
• f2 E(r1)
• Forward rates contain information about the
  evolution of future spot rates