DURATION & CONVEXITY Hossein Abdoh Tabrizi Maysam Radpour June 2011
Feb 21, 2016
DURATION & CONVEXITY
Hossein Abdoh TabriziMaysam Radpour
June 2011
Table of Contents
• Bonds; risk & return tradeoff
• Maturity effect; interest rate volatility risk
• Duration
• Convexity
Risk & return tradeoff
Bonds
Types of bonds based on option granted to the issuer or bondholder
Without option
Option-free bonds
Option for issuer
Callable bonds
Option for bondholder
Putable bonds
Factors effect bond return
Who is the Issuer?• issuer
How long does it take to mature?• Maturity
How easy may it be traded?• liquidity
How much is reinvestment rate of return?• Reinvestment rate
How much is tax?• Tax
Risks of return
• Default risk
Does the issuer do it’s obligations?
• Interest rate volatility risk
How much is the interest rate volatile?
• Reinvestment risk
What is the rate of periodical payments return?
• Liquidity risk
Is there an active secondary market?
Interest rate volatility risk
Maturity Effects
Price volatility in option-free bonds
There is a reverse relationship
between yield to maturity
and price .
Price
Yield to maturity
Factors affecting interest rate volatility
Coupon rate
• All other factors
constant, the lower
the coupon rate,
the greater the
price volatility.
maturity
• All other factors
constant, the
longer the maturity,
the greater the
price volatility.
Yield to maturity
• All other factors
constant, the
higher the yield
level, the lower the
price volatility.
Percentage price change for Four Hypothetical BondsInitial yield for all four bonds is 6%
Percentage price change
9% 20-year 9% 5-year 6% 20-year 6% 5-year New yield25.04 8.57 27.36 8.98 4.00%11.53 4.17 12.55 4.38 5.00%5.54 2.06 6.02 2.16 5.50%1.07 0.41 1.17 0.43 5.90%0.11 0.04 0.12 0.04 5.99%-0.11 -0.04 -0.12 -0.04 6.01%-1.06 -0.41 -1.15 -0.43 6.10%-5.13 -2.01 -5.55 -2.11 6.50%-9.89 -3.97 -10.68 -4.16 7.00%
-18.40 -7.75 -19.79 8.11 8.00%
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
Duration is a proxy that shows bond’s
percentage price change when yield
changes.
yPP
dyPdPdurationModified
Price equation of an option-free bond
P: priceC: periodical coupon interestY: yield to maturityM: maturity value (face value)N: number of periods
n21 )y1(MC
)y1(C
)y1(CP
First derivative of price equation
The first derivative of price equation shows the approximate change in
price when small change in yield occurs.
n21 )y1()MC(n
)y1(C2
)y1(C1
)y1(1
dydP
Macaulay duration, Modified duration
Percentage of
price change
Macaulay
duration
Modified
duration
n21 )y1()MC(n
)y1(C2
)y1(C1
P1
)y1(1
P1
dydP
n21 )y1()MC(n
)y1(C2
)y1(C1
P1durationMacaulay
n21 )y1()MC(n
)y1(C2
)y1(C1
P1
)y1(1durationModified
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:
PV× t Present value Cash flow Period4.3689 4.3689 4.5 18.4834 4.2417 4.5 2
12.3544 4.1181 4.5 315.9928 3.9982 4.5 419.4087 3.8817 4.5 522.6121 3.7687 4.5 625.6124 3.6589 4.5 728.4187 3.5523 4.5 831.0399 3.4489 4.5 9
777.5781 77.7578 104.5 10945.8694 112.7953 Total
8.38 Macaulay duration (in half years)
4.19 Macaulay duration (in years)
4.07 Modified duration (in years)
Example 2: Using duration to approximate price change
Duration for a 9% 20-year bond selling to yield 6% with semiannual coupon payments and face value of 100$ is:
If yields increase instantaneously from 6% to 6.1%, the percentage price change is:
If yields decrease instantaneously from 6% to 5.9%, the percentage price change is:
66.10)03.01(
98.10durationModified
98.10durationMacaulay
%066.1)001.0(66.10PP
ydurationModifiedPPdurationModified
PdP
%066.1)001.0(66.10PP
When duration does not work well?
When there are large movements in yield, duration is not adequate to approximate price reaction.
• Duration will overestimate the price change when the yield rises.
• Duration will underestimate the price change when the yield falls.
Example 3: When duration does not work well?
For the previous example, the real and approximate price change when yields change are as follows:
difference Approximate price change (based on duration)
Real price change (based on price equation)
Yield change (in percent)
0.06 -1.66 -1.60 0.10.04 +1.66 +1.70 -0.12.92 -21.32 -18.40 2.03.72 +221.32 +25.04 -2.0
Reason of duration inadequacy
Duration does not
capture the effect
of convexity of a
bond on it’s price
performance.
Price
Yield
Underestimation
Overestimation
1y2y y
Improvement in price change approximation
Taylor series for price equation:
PError)dy(
P1
dyPd
21dy
P1
dydP
PdP
ErrordyP1
dyPd
21dy
dydPdP
22
2
2
2
Convexity calculation
Convexity is the second derivative of the price
equation divided by the price.
n212
2
2
)y1()MC)(1n(n
)y1(C32
)y1(C21
P1
)y1(1
21Convexity
P1
dyd
21Convexity
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:
PV × t × (t+1) PV Cash flow Period8.7378 4.3689 4.5 125.4502 4.2417 4.5 249.4172 4.1181 4.5 379.964 3.9982 4.5 4116.451 3.8817 4.5 5
158.2854 3.7687 4.5 6204.8984 3.6589 4.5 7255.7656 3.5523 4.5 8310.401 3.4489 4.5 9
8553.358 77.7578 104.5 109762.729 112.7953 Total
40.792 Convexity (in half years )
10.198 Convexity (in years)
Example 5: Using convexity to approximate price change
Convexity for a 9% 20-year bond selling to yield 6% with semiannual coupon
payments and face value of 100$ is:
If yields increase instantaneously from 6% to 8%, the percentage price
change is:
If yields decrease instantaneously from 6% to 4%, the percentage price
change is:
053.82Convexity
%28.3)02.0(053.82)y()Convexity(PP 22
%28.3)02.0(053.82)y()Convexity(PP 22
Using duration and convexity simultaneously
Estimated percentage
price change
)y(MD
2)y(C
Example 6: Comparing approximate price change using duration and convexity and real price change
For a 9% 20-year bond selling to yield 6% with semiannual coupon payments and face value of 100$ if yield changes two
percent then we have:
difference Approximate price change (based on convexity)
Real price change (based on price
equation)
Yield change (in percent)
-0.36 -18.04 -18.40 2
0.44 +24.60 +25.04 -2
THANKS