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ALM – Managing Interest Rate Risk
Wei Hao FSA, MAAA
September 2010
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Outline
Background
Duration and convexity
Duration management
Liability Duration
Assumptions
Summary of results
Asset Duration Target
Effect of interest rate changes
Surplus protection
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Background
Variations of Duration
Macaulay Duration
Modified Duration
Effective Duration (option
(option-adjusted)
adjusted)
Key Rate Duration
Interpretations of Duration
Weighted average time where CFs from a FIS are
received (Macaulay D)
First derivative of P-Y relationship of FIS (Modified D)
Measure of sensitivity of bond price to small changes
in parallel yield curve shift (Effective D)
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Variations on Duration (I)
•
Macaulay Duration: weighted average timetime
to-maturity of the cash flows of a bond. (the
weight of each cash flow is based on its
discounted present value)
n
n
t 1
t 1
Mac D   wt * t  
PVt
*t
Pr ice
P
r
% P 
  Mac D *
P
1 r
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Variations on Duration (II)
• Modified Duration: an adjusted measure of
Macaulay duration that produces a more
accurate estimate of bond price sensitivity
Mac D
Modified Duration (MD)=
(
)  r
1  
 m
m is
s the
t e#o
of co
compounding
pou d g pe
period
od pe
per yea
year
P
P
  % P   M D * r
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Variations of Duration (III)
• A duration/convexity measure that includes the
effect of embedded options on a bond’s price
behavior
P  P
• Effective Duration D E 
2 P0  r


P
P
2
P


0
• Effective Convexity C E 
2
2 P0 r
• Bond price to interest rate change:
P
1
%P 
  DE r  CE r 2
P
2
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Effective Duration and Convexity
• Duration calculation is valid for small changes
g in
interest rate. It is less accurate for large
changes.
• The duration approximation – a straight line
relating change in bond price to change in
i t
interest
t rate
t – always
l
understates
d t t the
th price
i off
the bond; it underestimates the increase in bond
price when rates fall, and overestimates the fall
i price
in
i when
h rates rise.
i
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Price
Convexity Adjustment
Pricing Error from Convexity
Duration
Interest rate
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Duration Management
introduction
• An immunized portfolio is largely
g y protected from
fluctuations in market interest rates.
• seldom possible to eliminate interest rate risk
completely
l t l
• a portfolio’s immunization can wear out,
requiring managerial action to reinstate
• continually immunizing a portfolio can be timeconsuming
g and costly
y
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Duration Management (cont’d)
(cont d)
duration matching
g
• Duration matchingg selects a level of duration that
minimizes the combined effects of reinvestment rate
and interest rate risk
• Two versions of duration matching
• bullet immunization (target date immunization)
• bank immunization (surplus immunization)
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Duration Management (cont’d)
(cont d)
bullet immunization
• Seeks to ensure that a predetermined sum of
money is available at a specific time in the
future regardless of interest rate movement
• Obj
Objective
i is
i to get the
h effects
ff
off interest
i
rate
and reinvestment rate risk to offset
• If interest rates rise,
rise coupon proceeds can
be reinvested at a higher rate
• If interest rates fall, coupon proceeds can
be reinvested at a lower rate
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Duration Management (cont’d)
(cont d)
surplus immunization
• Addresses the problem that occurs if interestsensitive liabilities are included in the portfolio
• Interest rate changes cause changes in both
assets and
d liliabilities
bili i and
dh
hence iin surplus.
l
Can we eliminate this effect?
• Yes: match dollar duration of assets and
liabilities!
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Duration Management (cont’d)
(cont d)
surplus immunization
• To immunize surplus, must reorganize
balance sheet such that:
$A DA  $L DL
where
$A,L = dollar value of assets or liabilities
DA,A L = dollar
dollar-weighted
weighted duration of assets or
liabilities
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Duration Management
g
((cont’d))
surplus immunization
Price
Price
Surplus
Assets
S  0
Surplus
Liabilities
change of interest rate
change of interest rate
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Duration Management (cont’d)
(cont d)
Assets
Liabilities
surplus immunization
1
A   DA  A  y  C A  A  (y ) 2
2
1
L   DL  L  y  C L  L  (y ) 2
2
Surplus change
(assuming asset convexity C A  0 )
1
S  A  L   DA  A  y  DL  L  y  C L  L  (y ) 2
2
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Duration Management (cont’d)
(cont d)
surplus immunization
Surplus percentage change
recognizing A  S  L
A
1
L
S
2
 ( DL  DA )   y  DL  y  C L   (y )
S
S
2
S
S
For a given
g
surplus
p
p
percentage
g change
g
S
S
1
L
2

 DL  y  C L   (y )
2
S
DA  DL  S
A
 y
S
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Liability Duration Calculation
assumptions
p
• Corporate model used as the baseline.
• Due to low interest rate environment,, a shift of 10bp
p
is used.
• Durations are calculated for each line of business
g
and theyy are dollar-weighted.
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Liability Duration Calculation
summary of results
Left Blank Intentionally
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Interest Rate Risk on Surplus
A
1
L
S
2
 ( DL  DA )   y  DL  y  C L   (y )
S
S
2
S
Given currently
DA  5 , DL  5.5
rate change
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
Surplus % change
‐1.7%
‐4.1%
‐7.3%
‐11.2%
‐15.9%
‐21.4%
‐27.6%
27.6%
‐34.5%
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Asset Duration Target
S
For a given surplus percentage change , and interest rate
S
change y
S
1
L

 DL y 
CL 
 ( y )2
S
2
S
DA  DL 
A
y
S