Transcript MBA Finance

FINANCE
4. Bond Valuation
Professeur André Farber
Solvay Business School
Université Libre de Bruxelles
Fall 2007
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
PV 
Ct
C1
C2
CT


...


...

(1  r1 ) (1  r2 ) 2
(1  rt ) t
(1  rT )T
PV 
Ct
C1
C2
CT


...


...

(1  r ) (1  r ) 2
(1  r ) t
(1  r )T
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Review: Shortcut formulas
•
Constant perpetuity: Ct = C for all t
C
PV 
r
•
Growing perpetuity: Ct = Ct-1(1+g)
r>g
t = 1 to ∞
C1
PV 
rg
•
Constant annuity: Ct=C
•
Growing annuity: Ct = Ct-1(1+g)
t = 1 to T
t=1 to T
PV 
C
1
(1 
)
T
r
(1  r )
C1
(1  g )T
PV 
(1 
)
T
rg
(1  r )
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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
•
1
• Value of a zero-coupon bond:
PV 
(1  r )T
• Example :
• If the 1-year interest rate is 5% and is assumed to remain constant
• the zero of the previous example would sell for
PV 
1,000
 613.91
(1.05)10
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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
P0 
C
C
C
100
T


...



C

A
 100 dT
r
1  r (1  r ) 2
(1  r )T (1  r )T
• Example: If r = 5%
P0  6.5  A.505  100 d5  6.5  4.3295100 0.7835 106.49
• Note: If P0 > F: the bond is sold at a premium
•
If P0 <F: 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
C
1
• Suppose C / F > r
CF
F P F
P0 
F
0
• 1-year to maturity:
1 r
1 r
• 2-years to maturity:
• As: P1 > F
C  P1
P0 
1 r
with
P1 
CF
1 r
C
CF
F F
P0 
F
1 r
1 r
1
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A level coupon bond as a portfolio of zerocoupons
• « Cut » level coupon bond into 5 zero-coupon
•
Face value
Maturity
•
Zero 1
6.50
1
•
Zero 2
6.50
2
•
Zero 3
6.50
3
•
Zero 4
6.50
4
•
Zero 5
106.50
5
•
Total
Value
6.19
5.89
5.61
5.35
83.44
106.49
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Bond prices and interest rates
140,00
Bond prices fall with a
rise in interest rates
and rise with a fall in
interest rates
120,00
100,00
Bond price
80,00
60,00
40,00
20,00
0,00
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%
Interest rate
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Sensitivity of zero-coupons to interest rate
450,00
400,00
350,00
Bond price
300,00
250,00
5-Year
10-Year
15-Year
200,00
150,00
100,00
50,00
0,00
0%
1%
2%
3%
4%
5%
6%
7%
8%
9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%
Interest rate
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Duration for Zero-coupons
• Consider a zero-coupon with t years to maturity:
100
P
(1  r )t
• What happens if r changes?
dP
100
t
100
t
 t





P
dr
(1  r )t 1
1  r (1  r )t
1 r
• For given P, the change is proportional to the maturity.
• As a first approximation (for small change of r):
Duration = Maturity
P
t

r
P
1 r
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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 × 3+…+wt × t +…+ wT ×T
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Duration - example
• Back to our 5-year 6.50% coupon bond.
Face value
Value
Zero 1
6.50
6.19
Zero 2
6.50
5.89
Zero 3
6.50
5.61
Zero 4
6.50
5.35
Zero 5
106.50
83.44
Total
106.49
wt
5.81%
5.53%
5.27%
5.02%
78.35%
• Duration D = .0581×1 + 0.0553×2 + .0527 ×3 + .0502 ×4 + .7835 ×5
•
= 4.44
• For coupon bonds, duration < maturity
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Price change calculation based on duration
• General formula:
P
Duration

r
P
1 r
• In example: Duration = 4.44 (when r=5%)
• If Δr =+1% : Δ ×4.44 × 1% = - 4.23%
• Check: If r = 6%, P = 102.11
• ΔP/P = (102.11 – 106.49)/106.49 = - 4.11%
Difference due to
convexity
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Duration -mathematics
• If the interest rate changes:
dP dPV (C1 ) dPV (C2 )
dPV (CT )


 ... 
dr
dr
dr
dr
1
2
T

PV (C1 ) 
PV (C2 )  ... 
PV (CT )
1 r
1 r
1 r
• Divide both terms by P to calculate a percentage change:
dP 1
1
PV (C1 )
PV (C2 )
PV (CT )

(1
 2
 ...  T 
)
dr P
1 r
P
P
P
• As:
• we get:
Duration  1
PV (C1 )
PV (C2 )
PV (CT )
 2
 ...  T 
P
P
P
dP 1
Duration

dr P
1 r
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Yield to maturity
• Suppose that the bond price is known.
• Yield to maturity = implicit discount rate
C
C
CF
• Solution of following equation:
P0 


...

1  y (1  y ) 2
(1  y )T
140,00
120,00
100,00
Bond price
80,00
60,00
40,00
20,00
0,00
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%
Interest rate
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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.
A
B
C
D
1 Yield to maturity - illustration
2
3 Coupon
7%
4 Face value
100
5 Maturity
6 years
6 Price
105
7
8
0
1
2
9 Cash flows
-105
7
7
10
11 Yield to maturity 5.98% B11. =IRR(B9:H9)
E
F
3
7
G
4
7
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5
7
6
107
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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
DEquity  DAsset  ( DAsset  DLiabilities )
L
E
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Examples
SAVING BANK
Assets
Value
100
MDuration
3
Value
10
90
Equity
Liabilities
MDuration
21
1
LIFE INSURANCE COMPANY
Assets
Value
100
MDuration
15
Equity
Liabilities
Value
10
90
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-30
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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):
100
Maturity
Price
95.24 
 r1  5%
1  r1
1-year
95.24
2-year
89.85
100
89.85 
 r2  5.5%
(1  r2 ) 2
• 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
Bond
B1
B2
B3
B4
Data:
Coupon
5.00
9.00
6.50
8.00
Maturity
1
2
3
4
Price
99.06
103.70
97.54
100.36
YTM
6.00%
6.96%
7.45%
7.89%
To recover spot prices, solve:
99.06 = 105 * d1
103.70 = 9 * d1 + 109 * d2
97.54 = 6.5 * d1 + 6.5 * d2 + 106.5 * d3
100.36 = 8 * d1 +
8 * d2 +
8 * d3 + 108 * d4
Solution:
Maturity
1
2
3
4
Disc Fac. Spot rate
0.9434
6.00%
0.8734
7.00%
0.8050
7.50%
0.7350
8.00%
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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) × (1+f2) = 100 → f2 = 6%
• In general:
(1+r1)(1+f2) = (1+r2)²
• Solving for f2:
(1  r ) 2
d
f2 
• The general formula is:
2
1  r1
1 
1
d2
1
(1  rt )t
dt 1
ft 
1 
1
t 1
(1  rt 1 )
dt
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Forward rates :example
•
•
•
•
•
•
Maturity
1
2
3
4
5
Discount factor
0.9500
0.8968
0.8444
0.7951
0.7473
Spot rates
5.26
5.60
5.80
5.90
6.00
• Details of calculation:
• 3-year spot rate :
0.8444
Forward rates
5.93
6.21
6.20
6.40
1
1
1

r

(
) 3  1  5.80%
3
3
(1  r3 )
0.8444
• 1-year forward rate from 3 to 4
(1  r3 )3
d2
0.8968
f3 

1


1

 1  6.21%
2
(1  r2 )
d3
0.8444
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Term structure of interest rates
•
•
•
•
•
•
Why do spot rates for different
maturities differ ?
As
r1 < r2
if
f2 > r1
r1 = r2
if
f2 = r1
r1 > r2
if
f2 < r1
Upward sloping
Spot
rate
Flat
Downward sloping
The relationship of spot rates with
different maturities is known as the
term structure of interest rates
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 × (1+r1) × (1+E(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
100  1.113
1
100

(
1

r
)(
1

E
(
r
))
1
1
(1  r2 ) 2
(1  r1 )(1  f 2 )
• 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
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