Financial Risk Management of
Insurance Enterprises
Duration and Convexity – Part 2
Applications of Duration
• Remember, ALM evaluates the interaction
of asset and liability movements
• Insurers attempt to equate interest
sensitivity of assets and liabilities so that
surplus is unaffected
– Surplus is “immunized” against interest rate risk
• Immunization is the technique of matching
asset duration and liability duration
Why Worry About Interest Rate Risk?
• The 1970s Savings & Loan industry didn’t
– Asset-liability “mismatch”
• Interest rates can and do fluctuate substantially
• Examples of 7 Year U.S. T-bond interest rates:
r at
r at
t
t-12 months
t
i
March 1980
July 1981
Oct 1982
May 1984
April 1986
Dec 1995
9.15%
9.84
15.33
10.30
11.34
7.80
13.00%
14.49
10.88
13.34
7.16
5.63
3.85%
4.65
- 4.45
3.04
- 4.18
- 2.17
Assumptions Underlying Macaulay
and Modified Duration
• Cash flows do not change with interest rates
This does not hold for:
– Collateralized Mortgage Obligations (CMOs)
– Callable bonds
– P-L loss reserves – due to inflation-interest rate correlation
• Flat yield curve
Generally, yield curves are upward-sloping
• Interest rates shift in parallel fashion
Short term interest rates tend to be more volatile
than longer term rates
Assuming Parallel Shifts
• The assumption of parallel shifts in the
yield curve is not plausible
• In reality, short-term rates move more than
long-term rates
• Also, it is possible that the yield curve
“twists”
– Short-term and long-term rates move in
opposite directions
An Illustration
• There are two cash flows, 100 at the end of
year 1 and 100 at the end of the second year
– The interest rate is a flat 5%
• Calculating modified duration
100 100
P

 185.94
2
1.05 1.05
1
1 100 2  100 
D  


 1.42
2
3 
1.05  185.94
 1.05
Partial Duration
• Each term in the calculation tells us
something about interest rate sensitivity
– It is the sensitivity of the cash flow to that
interest rate
• In this example, define two “partial”
durations
– One for each cash flow period
1 100
1
D1 

 0.49
2
1.05 185.94
2  100
1
D2 

 0.93
3
1.05
185.94
Interpreting Partial Duration
• Note that the sum of the partial durations is
equal to the original duration calculation
• Using partial duration, we can determine the
interest rate sensitivity to any non-parallel
shift in the yield curve
• We can use partial duration to predict price
changes
Percentage change in bond value
 - D1  r1  D2  r2
Example
• From our two period cash flow, what is the
change in value if the one year rate goes to
4% and the two year rate goes to 6%
Predicted price change  -0.49  (-0.01) - 0.93  0.01
 - 0.44%
Predicted price  185.12
Actual price  185.15
Key Rates
• Interest rates of “similar” maturities move
in the same fashion
– The 10 year rate and the 10½ year rate move
similarly
• Therefore, partial durations can be based on
a few points on the yield curve
• These are called key rates
– Partial durations are sometimes referred to as
key rate durations
Typical Key Rates
• Popular key rates are:
6 o
m
o
1
yr
2
yr
3
yr
5
yr
7
y
10 r
y
30 r
yr
m
3
– 3 month and 6 month
rate
– 1 year
– 2 years
– 3 years
– 5 years
– 7 years
– 10 years
– 30 years
Key Rates
Applications of Key Rate
Durations
• Key rate durations are very useful for
hedging purposes
• Because multiple partial durations provide
more information than a single duration
number, insurers can determine their
sensitivity to interest rates based on various
parts of the yield curve
• If the insurer is not immunized, it can use
interest rate derivatives to hedge the risk
Cash Flows Change with Interest
Rates
• Effective Duration
• Effective Convexity
Effective Duration
PV - - PV +
Effective duration = ED0 =
2PV0 ( r)
Effective Convexity
(Note – Fabozzi includes a 2 in denominator)
PV  PV  2 PV0
Effective convexity=
2
PV0 (r )
Calculation of the Change in
Economic Value of a Cash Flow
V = (-1)(Effective Duration)(r)
+ (1/2)(Convexity)(r)2
(Note: if using Fabozzi convexity calculation, omit the (1/2) in
the second term.)
Example – Fixed Cash Flow
Cash flow of $1000 in 10 years
No interest rate sensitivity
Current interest rate = 10%
Macaulay Duration = 10
Modified Duration = 9.0909
Convexity = 90.909
Example – Variable Cash Flow
Cash flow of $1000 occurs at x years if r = x%
Current r = 10%, cash flow at year 10
PV = 1000/(1.10)^10 = 385.5433
∆r = 50 basis points
PV_= 422.2463
PV+= 350.5065
Effective Duration = 18.6075
Effective Convexity = 172.8719
Example – Variable Cash Flow 2
Cash flow of $1000 occurs at 10 years if r = 10%
Cash flow changes at ½ the percentage change that
interest rates change (from 10%)
If interest rates rise to 10.5%, cash flow is $1025
If interest rates fall to 9.5%, cash flow is $975.
PV = 1000/(1.10)^10 = 385.5433
∆r = 50 basis points
PV_= 393.4263
PV+= 377.6601
Effective Duration = 4.0894
Effective Convexity = -0.0169
Estimated Impact of Change in
Economic Value for
100 Basis Point Rise in Interest Rate
V = (-1)(Effective Duration)(r)
+ (1/2)(Convexity)(r)2
Fixed cash flow
Variable cash flow 1
Variable cash flow 2
-8.6%
-17.7%
-4.1%
Surplus Duration
• Sensitivity of an insurer’s surplus to changes
in interest rates
D S S = DA A - D L L
DS = (DA - DL)(A/S) + DL
where
D = duration
S = surplus
A = assets
L = liabilities
Surplus Duration and
Asset-Liability Management
• To “immunize” surplus from interest rate risk,
set DS = 0
• Then, asset duration should be:
DA = DL L / A
• Thus, an accurate estimate of the duration of
liabilities is critical for ALM
Are Property-Liability Insurers
Exposed to Interest Rate Risk?
• Absolutely!!
• Long-term liabilities
– Medical malpractice
– Workers’ compensation
– General liability
• Assets
– Significant portion of assets invested in long
term bonds
The Liabilities of Property-Liability
Insurers
• Major categories of liabilities:
– Loss reserves
– Loss adjustment expense reserves
– Unearned premium reserves
Loss Reserves
• Major categories:
– In the process of being paid
– Value of loss is determined, negotiating over
share of loss to be paid
– Damage is yet to be discovered
– Continuing to develop: some of loss has been
fixed, remainder is yet to be determined
• Inflation, which is correlated with interest
rates, will affect each category of loss
reserves differently.
What Portion of the Loss Reserve is
Affected by Future Inflation (and
Interest Rates)?
• If the damage has not yet occurred, then the
future loss payments will fully reflect future
inflation
• If the loss is continuing to develop, then a
portion of the future loss payments will be
affected by future inflation (and another
portion will be “fixed” relative to inflation)
How to Reflect “Fixed” Costs?
• “Fixed” here means that portion of damages
which, although not yet paid, will not be
impacted by future inflation
• Tangible versus intangible damages
• Determining when a cost is “fixed” could
require
– Understanding the mindset of jurors
– Lots and lots of data
A Possible “Fixed” Cost Formula
Proportion of loss reserves fixed in value as of time t:
f(t) = k + [(1 - k - m) (t / T) n]
k = portion of losses fixed at time of loss
m = portion of losses fixed at time of settlement
T = time from date of loss to date of payment
Proportion
of Ultimate
Payments
Fixed
k
1
m
n<1
n=1
n>1
0
1
0
Proportion of Payment Period
“Fixed” Cost Formula Parameters
• Examples of loss costs that might go into k
– Medical treatment immediately after the loss occurs
– Wage loss component of an injury claim
– Property damage
• Examples of loss costs that might go into m
– Medical evaluations performed immediately prior to
determining the settlement offer
– General damages to the extent they are based on the
cost of living at the time of settlement
– Loss adjustment expenses connected with settling
the claim
Loss Reserve Duration Example
For the values:
k = .15
m = .10
r = 5%
rr,i = 0.40
Exposure growth rate = 10%
Automobile
Insurance
Macaulay duration: 1.52
Modified duration: 1.44
Effective duration: 1.09
Convexity
Effective convexity
5.75
1.99
n = 1.0
Workers’
Compensation
4.49
4.27
3.16
50.77
16.04
Example of ALM for a Hypothetical WC
Insurer
Loss & LAE Reserve
UPR
Other liabilities
Total liabilities
710
Dollar
Modified
Effective
Value
Duration
Duration
590
4.271
3.158
30
3.621
1.325
90
0.952
0.952
3.823
2.801
Total assets
1,000
Asset duration to immunize surplus:
2.714
1.989
Conclusion
• Asset-liability management depends upon
appropriate measures of effective duration (and
convexity)
• Potentially significant differences between effective
and modified duration values
• Critical factors and parameters
–
–
–
–
Line of business
Payment pattern
Correlation between interest rates and inflation
Interest rate model (?)
Next
•
•
•
•
Review for first exam
First exam – February 27, 2008
An introduction to stochastic processes
The use of stochastic movements in
modeling interest rates
• Using interest rate models to calculate
duration and convexity