Description
different types of risks associated with fixed income securities like interest rate risk, reinvestment risk, inflation risk, credit risk, call risk, exchamge rate risk.
Risk Associated with Investment
in Fixed Income Securities
Issue
Price
Coupon
Investment
Sale
C1
C1*(1+r1)^t 1
C2
C2*(1+r2)^t 2
Major Risk Types
• Interest Rate Risk/Market Risk/Price Risk
• Reinvestment Risk
• Inflation Risk
• Credit/ Default Risk
• Call Risk
• Exchange Rate Risk
Interest Rate Risk
• The Value of Bond moves in Opposite
Direction of the Interest Rate
Approaches of Interest Rate Risk
Analysis
• Full valuation and Scenario Analysis
• Duration Convexity Approach
Duration
• Also known as Macaulay Duration
• Duration is a measure of the average life of a
bond.
• It is defined as the weighted average of the
time until each payment is made, with weights
proportional to the present value of the
payment.
Duration
P
t t PV t t PV t t PV
D
n n
* ) ( ... * ) ( * ) (
2 2 1 1
+ +
=
Duration
¿
=
× =
T
t
t
w t D
1
Where
Price Bond
) 1 /(
t
t
t
y C
w
+
=
Behaviour of Duration
• Higher the maturity higher the duration.
• Duration of a zero coupon bond is equal to its
maturity.
• Duration decreases with increase in coupon
rate.
• Duration also decreases with increase in yield
rate.
Duration and Yield rate Sensitivity
of Bond Price
DP
y y
P
+
÷ =
c
c
1
1
) 1 ( y
D
D
m
+
=
D(m) is the modified duration
P D
y
P
m
÷ =
c
c
Derivation of Formula
) 1 (
. .
) 1 (
1
) 1 (
) 1 (
1
1
1
1
y
PD
P
P
PV
y
t
y
PV
y
P
PV P
now
PV
y
t
y
tC
y
PV
y
C
PV
t
T
t
T
t
t
T
t
t
t
t
t t
t
t
t
+
÷ =
|
.
|
\
|
+
÷
=
c
c
=
c
c
=
+
÷
=
+
÷
=
c
c
+
=
¿
¿
¿
=
=
=
+
m
D
P y
P
÷ =
c
c 1
Modified duration(in absolute terms) is
equal to relative change in price in
response to change in yield rate
Duration and Interest rate sensitivity
y P D P
m
A ÷ ~ A
Duration of a Portfolio
n n p
D w D w D w D .......
2 2 1 1
+ =
Convexity
600
800
1000
1200
1400
1600
1800
2000
2200
1 2 3 4 5 6 7 8 9 10
Convexity
• Duration based valuation gives a linear
approximation of the non-linear relationship
between yield and bond price.
• Duration based valuation gives more accurate
results at proximity of current yield.
• Generally it underestimates the bond value.
Convexity
• Convexity is the rate of change of the slop of
price-yield curve.
2
2
1
y
P
P
Convexity
c
c
=
It is based on second-order derivative of the
bond price function w.r.t. yield
Convexity
| |
¿
=
+
+
=
T
t
t
t t PV
y P
Convexity
1
2
2
) (
) 1 (
1
Duration, Convexity and Bond Price
| |
2
) ( * * 2 1 y Convexity y D
P
P
m
A + A ÷ =
A
Duration
based
estimate
Adjustment
for
convexity
Duration Vs Convexity based
Estimates
500
700
900
1100
1300
1500
1700
1900
2100
2300
2500
1 2 3 4 5 6 7 8 9 10
Price
Duration Est.
conv. correction
Duration w.r.t Yield Curve
¿
=
+
+
÷
=
n
t
t
t
t
y
C t
P
QMD
1
1
) 1 (
. 1
Fisher-Weil Quasi-Modified Duration
Uses of Duration
Classical Immunization
• A bond holder becomes immune to market
(price) risk and reinvestment risk if the
investment horizon is equal to bonds duration.
doc_973233091.pptx
different types of risks associated with fixed income securities like interest rate risk, reinvestment risk, inflation risk, credit risk, call risk, exchamge rate risk.
Risk Associated with Investment
in Fixed Income Securities
Issue
Price
Coupon
Investment
Sale
C1
C1*(1+r1)^t 1
C2
C2*(1+r2)^t 2
Major Risk Types
• Interest Rate Risk/Market Risk/Price Risk
• Reinvestment Risk
• Inflation Risk
• Credit/ Default Risk
• Call Risk
• Exchange Rate Risk
Interest Rate Risk
• The Value of Bond moves in Opposite
Direction of the Interest Rate
Approaches of Interest Rate Risk
Analysis
• Full valuation and Scenario Analysis
• Duration Convexity Approach
Duration
• Also known as Macaulay Duration
• Duration is a measure of the average life of a
bond.
• It is defined as the weighted average of the
time until each payment is made, with weights
proportional to the present value of the
payment.
Duration
P
t t PV t t PV t t PV
D
n n
* ) ( ... * ) ( * ) (
2 2 1 1
+ +
=
Duration
¿
=
× =
T
t
t
w t D
1
Where
Price Bond
) 1 /(
t
t
t
y C
w
+
=
Behaviour of Duration
• Higher the maturity higher the duration.
• Duration of a zero coupon bond is equal to its
maturity.
• Duration decreases with increase in coupon
rate.
• Duration also decreases with increase in yield
rate.
Duration and Yield rate Sensitivity
of Bond Price
DP
y y
P
+
÷ =
c
c
1
1
) 1 ( y
D
D
m
+
=
D(m) is the modified duration
P D
y
P
m
÷ =
c
c
Derivation of Formula
) 1 (
. .
) 1 (
1
) 1 (
) 1 (
1
1
1
1
y
PD
P
P
PV
y
t
y
PV
y
P
PV P
now
PV
y
t
y
tC
y
PV
y
C
PV
t
T
t
T
t
t
T
t
t
t
t
t t
t
t
t
+
÷ =
|
.
|
\
|
+
÷
=
c
c
=
c
c
=
+
÷
=
+
÷
=
c
c
+
=
¿
¿
¿
=
=
=
+
m
D
P y
P
÷ =
c
c 1
Modified duration(in absolute terms) is
equal to relative change in price in
response to change in yield rate
Duration and Interest rate sensitivity
y P D P
m
A ÷ ~ A
Duration of a Portfolio
n n p
D w D w D w D .......
2 2 1 1
+ =
Convexity
600
800
1000
1200
1400
1600
1800
2000
2200
1 2 3 4 5 6 7 8 9 10
Convexity
• Duration based valuation gives a linear
approximation of the non-linear relationship
between yield and bond price.
• Duration based valuation gives more accurate
results at proximity of current yield.
• Generally it underestimates the bond value.
Convexity
• Convexity is the rate of change of the slop of
price-yield curve.
2
2
1
y
P
P
Convexity
c
c
=
It is based on second-order derivative of the
bond price function w.r.t. yield
Convexity
| |
¿
=
+
+
=
T
t
t
t t PV
y P
Convexity
1
2
2
) (
) 1 (
1
Duration, Convexity and Bond Price
| |
2
) ( * * 2 1 y Convexity y D
P
P
m
A + A ÷ =
A
Duration
based
estimate
Adjustment
for
convexity
Duration Vs Convexity based
Estimates
500
700
900
1100
1300
1500
1700
1900
2100
2300
2500
1 2 3 4 5 6 7 8 9 10
Price
Duration Est.
conv. correction
Duration w.r.t Yield Curve
¿
=
+
+
÷
=
n
t
t
t
t
y
C t
P
QMD
1
1
) 1 (
. 1
Fisher-Weil Quasi-Modified Duration
Uses of Duration
Classical Immunization
• A bond holder becomes immune to market
(price) risk and reinvestment risk if the
investment horizon is equal to bonds duration.
doc_973233091.pptx