Thorie Financire 2' Valeur actuelle

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Théorie Financière2. Valeur actuelle

• Professeur André Farber

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Present Value general formula

• Cash flows C1, C2, C3, ,Ct, CT
• Discount factors v1, v2, ,vt, , vT
• Present value PV C1 v1 C2 v2 CT
  vT
• An example
• Year 0 1 2 3
• Cash flow -100 40 60 30
• Discount factor 1.000 0.9803 0.9465 0.9044
• Present value -100 39.21 56.79 27.13
• NPV - 100 123.13 23.13

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Using prices of U.S. Treasury STRIPS

• Separate Trading of Registered Interest and
  Principal of Securities
• Prices of zero-coupons
• Example Suppose you observe the following prices
• Maturity Price for 100 face value
• 1 98.03
• 2 94.65
• 3 90.44
• 4 86.48
• 5 80.00
• The market price of 1 in 5 years is DF5 0.80
• NPV - 100 150 0.80 - 100 120 20

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Present value and discounting

• How much would an investor pay today to receive
  Ct in t years given market interest rate rt?
• We know that 1 0 gt (1rt)t t
• Hence PV ? (1rt)t Ct gt PV Ct/(1rt)t
  Ct ? vt
• The process of calculating the present value of
  future cash flows is called discounting.
• The present value of a future cash flow is
  obtained by multiplying this cash flow by a
  discount factor (or present value factor) vt
• The general formula for the t-year discount
  factor is

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Spot interest rates

• Back to STRIPS. Suppose that the price of a
  5-year zero-coupon with face value equal to 100
  is 75.
• What is the underlying interest rate?
• The yield-to-maturity on a zero-coupon is the
  discount rate such that the market value is equal
  to the present value of future cash flows.
• We know that 75 100 v5 and v5
  1/(1r5)5
• The YTM r5 is the solution of
• The solution is
• This is the 5-year spot interest rate

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Term structure of interest rate

• Relationship between spot interest rate and
  maturity.
• Example
• Maturity Price for 100 face value YTM (Spot
  rate)
• 1 98.03 r1 2.00
• 2 94.65 r2 2.79
• 3 90.44 r3 3.41
• 4 86.48 r4 3.70
• 5 80.00 r5 4.56
• Term structure is
• Upward sloping if rt gt rt-1 for all t
• Flat if rt rt-1 for all t
• Downward sloping (or inverted) if rt lt rt-1 for
  all t

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Zero coupon yield curve Euro 5-aug-2005Sourcehtt
p//epp.eurostat.cec.eu.int
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Using one single discount rate

• When analyzing risk-free cash flows, it is
  important to capture the current term structure
  of interest rates discount rates should vary
  with maturity.
• When dealing with risky cash flows, the term
  structure is often ignored.
• Present value are calculated using a single
  discount rate r, the same for all maturities.
• Remember this discount rate represents the
  expected return.
• Risk-free interest rate Risk premium
• This simplifying assumption leads to a few useful
  formulas for
• Perpetuities (constant or growing at a constant
  rate)
• Annuities (constant or growing at a constant
  rate)

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Constant perpetuity
Proof PV C d C d² C d3 PV(1r) C C
d C d² PV(1r) PV C PV C/r

• Ct C for t 1, 2, 3, .....
• Examples Preferred stock (Stock paying a fixed
  dividend)
• Suppose r 10 Yearly dividend 50
• Market value P0?
• Note expected price next year
• Expected return

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Growing perpetuity

• Ct C1 (1g)t-1 for t1, 2, 3, .....
  rgtg
• Example Stock valuation based on
• Next dividend div1, long term growth of dividend
  g
• If r 10, div1 50, g 5
• Note expected price next year
• Expected return

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Constant annuity

• A level stream of cash flows for a fixed numbers
  of periods
• C1 C2 CT C
• Examples
• Equal-payment house mortgage
• Installment credit agreements
• PV C v1 C v2 C vT
• C v1 v2 vT
• C Annuity Factor
• Annuity Factor present value of 1 paid at the
  end of each T periods.

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Constant Annuity

• Ct C for t 1, 2, ,T
• Difference between two annuities
• Starting at t 1 PVC/r
• Starting at t T1 PV C/r 1/(1r)T
• Example 20-year mortgage
• Annual payment 25,000
• Borrowing rate 10
• PV ( 25,000/0.10)1-1/(1.10)20 25,000 10
  (1 0.1486)
• 25,000 8.5136
• 212,839

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Growing annuity

• Ct C1 (1g)t-1 for t 1, 2, , T r ? g
• This is again the difference between two growing
  annuities
• Starting at t 1, first cash flow C1
• Starting at t T1 with first cash flow C1
  (1g)T
• Example What is the NPV of the following project
  if r 10?
• Initial investment 100, C1 20, g 8, T 10
• NPV 100 20/(10 - 8)1 (1.08/1.10)10
• 100 167.64
• 67.64

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Useful formulas summary

• 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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Compounding interval

• Up to now, interest paid annually
• If n payments per year, compounded value after 1
  year
• Example Monthly payment
• r 12, n 12
• Compounded value after 1 year (1 0.12/12)12
  1.1268
• Effective Annual Interest Rate 12.68
• Continuous compounding
• 1(r/n)n?er (e 2.7183)
• Example r 12 e12 1.1275
• Effective Annual Interest Rate 12.75

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Juggling with compounding intervals

• The effective annual interest rate is 10
• Consider a perpetuity with annual cash flow C
  12
• If this cash flow is paid once a year PV 12 /
  0.10 120
• Suppose know that the cash flow is paid once a
  month (the monthly cash flow is 12/12 1 each
  month). What is the present value?
• Solution 1
• Calculate the monthly interest rate (keeping EAR
  constant)
• (1rmonthly)12 1.10 ? rmonthly 0.7974
• Use perpetuity formula
• PV 1 / 0.007974 125.40
• Solution 2
• Calculate stated annual interest rate 0.7974
  12 9.568
• Use perpetuity formula PV 12 / 0.09568
  125.40

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Interest rates and inflation real interest rate

• Nominal interest rate 10 Date 0 Date 1
• Individual invests 1,000
• Individual receives 1,100
• Hamburger sells for 1 1.06
• Inflation rate 6
• Purchasing power ( hamburgers) H1,000 H1,038
• Real interest rate 3.8
• (1Nominal interest rate) (1Real interest
  rate)(1Inflation rate)
• Approximation
• Real interest rate Nominal interest rate -
  Inflation rate

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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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Law of one price

• Suppose that you observe the following data

What are the underlying discount factors?
Bootstrap method
100.97 v1 104105.72 v1 7 v2
107101.56 v1 5.5 v2 5.5 v3
105.5
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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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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