Hossein Abdoh Tabrizi
Maysam Radpour
June 2011
Table of Contents
•
Bonds; risk & return tradeoff
•
Maturity effect; interest rate volatility risk
•
Duration
•
Convexity
Bonds
Risk & return tradeoff
Types of bonds based on option granted to the
issuer or bondholder
Without
option
Optionfree bonds
Option for
issuer
Callable
bonds
Option for
bondholder
Putable
bonds
Factors effect bond return
Risks of return
Does the issuer do it’s obligations?
• Default risk
How much is the interest rate volatile?
• Interest rate volatility risk
What is the rate of periodical payments return?
• Reinvestment risk
Is there an active secondary market?
• Liquidity risk
Maturity Effects
Interest rate volatility risk
Price volatility in option-free bonds
There is a reverse
relationship
Price
between
yield to maturity
and
price .
Yield to maturity
Factors affecting interest rate volatility
Coupon rate
• All other factors
maturity
Yield to maturity
• All other factors
• All other factors
constant, the lower
constant, the
constant, the
the coupon rate,
longer the maturity,
higher the yield
the greater the
the greater the
level, the lower the
price volatility.
price volatility.
price volatility.
Percentage price change for Four Hypothetical Bonds
Initial yield for all four bonds is 6%
Percentage price change
New yield
6% 5-year
6% 20-year
9% 5-year
9% 20-year
4.00%
5.00%
5.50%
5.90%
5.99%
6.01%
6.10%
6.50%
7.00%
8.00%
8.98
4.38
2.16
0.43
0.04
-0.04
-0.43
-2.11
-4.16
8.11
27.36
12.55
6.02
1.17
0.12
-0.12
-1.15
-5.55
-10.68
-19.79
8.57
4.17
2.06
0.41
0.04
-0.04
-0.41
-2.01
-3.97
-7.75
25.04
11.53
5.54
1.07
0.11
-0.11
-1.06
-5.13
-9.89
-18.40
Duration
Duration is a measure of interest rate volatility
risk:
• Duration is the measure of fixed income securities
price sensitivity versus interest rate changes.
• Duration encompasses the three factors (coupon,
maturity and yield level) that affects bond’s price
volatility.
Duration
Duration is a proxy for maturity:
• Duration is a proxy better than maturity and may be
considered as effective maturity of fixed income
securities.
• Duration is standardized weighted average of
bond’s term to maturity where the weights are the
present value of the cash flows.
Duration is elasticity
dP P P P
Modified duration 

dy
y
Price equation of an option-free bond
C
C
CM
P


1
2
(1  y) (1  y)
(1  y) n
P: price
C: periodical coupon interest
Y: yield to maturity
M: maturity value (face value)
N: number of periods
First derivative of price equation
The first derivative of price equation shows the approximate change in
price when small change in yield occurs.
dP
1  1C
2C
n (C  M ) 




1
2
n 
dy
(1  y)  (1  y) (1  y)
(1  y) 
Macaulay duration, Modified duration
dP 1
1 1  1C
2C
n (C  M ) 







dy P
(1  y) P  (1  y)1 (1  y) 2
(1  y) n 
Macaulayduration 
1  1C
2C
n (C  M ) 






P  (1  y)1 (1  y) 2
(1  y) n 
Modified duration 
1 1  1C
2C
n (C  M) 






(1  y) P  (1  y)1 (1  y) 2
(1  y) n 
Example 1: Duration calculation

Duration for a 9% 5-year bond selling to yield 6% with semiannual coupon payments and
face value of 100$ is:
Period
1
2
3
4
5
6
7
8
9
10
Cash flow
Present value
PV× t
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
104.5
4.3689
4.2417
4.1181
3.9982
3.8817
3.7687
3.6589
3.5523
3.4489
77.7578
112.7953
4.3689
8.4834
12.3544
15.9928
19.4087
22.6121
25.6124
28.4187
31.0399
777.5781
945.8694
Total
Macaulay duration (in half years)
8.38
Macaulay duration (in years)
4.19
Modified duration (in years)
4.07
Example 2: Using duration to approximate price change
Macaulay duration  10.98
10.98
Modified duration 
 10.66
(1  0.03)
dP
P
 Modified duration 
 Modified duration y
P
P
P
 10.66  (0.001)  1.066 %
P
P
 10.66  (0.001)  1.066 %
P
When duration does not work well?
Example 3: When duration does not work well?

For the previous example, the real and approximate
price change when yields change are as follows:
Yield change
(in percent)
0.1
-0.1
2.0
-2.0
Real price change
(based on price equation)
Approximate price change
(based on duration)
difference
-1.60
+1.70
-18.40
+25.04
-1.66
+1.66
-21.32
+221.32
0.06
0.04
2.92
3.72
Reason of duration inadequacy
Price
Underestimation
Overestimation
y2
y
y1
Yield
Improvement in price change approximation
dP
1 d2P 1
dP 
dy 
dy  Error
2
dy
2 dy P
dP dP 1
1 d 2P 1
Error
2

dy 
(dy ) 
2
P
dy P
2 dy P
P
Convexity calculation
1 d2 1
Convexity 
2 dy 2 P
1
1
1  1  2C
2  3C
n  (n  1)(C  M ) 
Convexity 




2 (1  y) 2 P  (1  y)1 (1  y) 2
(1  y) n

Example 4: convexity calculation

Convexity for a 9% 5-year bond selling to yield 6% with semiannual coupon payments and face value of
100$ is:
Period
1
2
3
4
5
6
7
8
9
10
Total
Cash flow
PV
PV × t × (t+1)
4.5
4.3689
4.2417
4.1181
3.9982
3.8817
3.7687
3.6589
3.5523
8.7378
25.4502
49.4172
79.964
116.451
158.2854
204.8984
255.7656
3.4489
77.7578
310.401
8553.358
112.7953
9762.729
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
104.5
Convexity (in half years )
40.792
Convexity (in years)
10.198
Example 5: Using convexity to approximate price change
Convexity  82.053
P
 (Convexity )  (y) 2  82.053  (0.02) 2  3.28%
P
P
 (Convexity )  (y) 2  82.053  (0.02) 2  3.28%
P
Using duration and convexity simultaneously
MD  (y)
C  (y) 2
Example 6: Comparing approximate price change using
duration and convexity and real price change
Yield change
(in percent)
Real price change
(based on price equation)
Approximate price change
(based on convexity)
difference
2
-18.40
-18.04
-0.36
-2
+25.04
+24.60
0.44
THANKS