Bonds Valuation
PERTEMUAN 17-18
Bond Valuation
• Objectives for this session :
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–
–
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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
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:
1
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
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

6.50
6.50
2003
6.50
2004
6.50
2005
106.50
Valuing a level coupon bond
P0 
C
C
C
100


...


 C  ArT  100  dT
2
T
T
1  r (1  r )
(1  r )
(1  r )
• Example: If r = 5%
P0  6.5  A.505  100  d 5  6.5  4.3295  100  0.7835  106.49
• Note: If P0 >: 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%
A level coupon bond as a portfolio of zerocoupons
• « 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
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
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
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):
P
t

r
P
1 r
Duration = Maturity
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
Duration - example
• Back to our 5-year 6.50% coupon bond.
Face value
6.50
6.50
6.50
6.50
106.50
Zero 1
Zero 2
Zero 3
Zero 4
Zero 5
Total
Value
6.19
5.89
5.61
5.35
83.44
106.49
wt
5.81%
5.53%
5.27%
5.02%
78.35%
• Duration = .0581×1 + .0553×2 + .0527 ×3 + .0502 ×4 + .7835 ×5
•
= 4.44
• For coupon bonds, duration < maturity
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
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
Yield to maturity
• Suppose that the bond price is known.
• Yield to maturity = implicit discount rate
C
C
CF
P



...

• Solution of following equation:
0
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%
Interest rate
12% 13%
14%
15% 16%
17%
18% 19%
20%
Spot rates
• Consider the following prices for zero-coupons (Face value = 100):
Maturity
1-year
2-year
Price
95.24
89.85
• The one-year spot rate is obtained by solving:
95.24 
• The two-year spot rate is calculated as follow:
89.85 
100
 r1  5%
1  r1
100
 r2  5.5%
(1  r2 ) 2
• Buying a 2-year zero coupon means that you invest for two years at
an average rate of 5.5%
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
(1  r2 ) 2
d
f2 
1  1 1
1  r1
d2
• Solving for f2:
• The general formula is:
(1  rt )t
d t 1
ft 
1 
1
t 1
(1  rt 1 )
dt
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
1
1
0.8444 

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%
(1  r2 ) 2
d3
0.8444
Term structure of interest rates
•
Why do spot rates for different
maturities differ ?
•
As
•
r1 < r2
if
•
r1 = r2
if
f2 = r1
•
r1 > r2
if
f2 < r1
•
The relationship of spot rates with
different maturities is known as the
f2 > r1
term structure of interest rates
Upward sloping
Spot
rate
Flat
Downward sloping
Time to maturity
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
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