Download Measuring the Duration of Liabilities

Measuring the Duration of
Liabilities
Stephen P. D’Arcy, FCAS, MAAA, Ph.D.
University of Illinois
at Urbana-Champaign
Casualty Actuarial Society
Asset-Liability Management Session
San Diego, CA
May 20, 2002
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 liabilities – 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
An Improvement: Effective Duration
• Effective duration:
– Accommodates interest sensitive cash flows
– Can be based on any term structure
– Allows for non-parallel interest rate shifts
• Effective duration is used to value such assets as:
– Collateralized Mortgage Obligations
– Callable bonds
– And now… property-liability insurance liabilities
• Need to reflect the inflationary impact on future
loss and LAE payments of interest rate changes
A Further Refinement: Convexity
The larger the change in interest rates, the
larger the misestimate of the price change
using duration
Duration: first-order approximation
Accurate only for small changes in interest rates
Convexity: second-order approximation
Reflects the curvature of the price-yield curve
The Liabilities of PropertyLiability Insurers
• Major categories of liabilities:
– Loss reserves
– Loss adjustment expense reserves
– Unearned premium reserves
A Model for the Interest
Sensitivity of Loss Reserves
• D’Arcy and Gorvett PCAS 2000
• Divides loss reserves into “fixed” and
inflation sensitive portions
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 Adjustment Expense Reserves
• LAE on losses that have already occurred
• Primarily future expenses
• Sensitive to interest rate changes
Unearned Premium Reserves
• Statutory reserve for the unexpired portion
of premiums
• Economic value is the future losses on
current policies
• Since these losses have not occurred yet,
they would be completely interest rate
sensitive
Approach for Measuring Interest Rate
Sensitivity of Insurance Liabilities
1. Select a term structure (interest rate) model
2. Generate multiple interest rate paths based on the
selected model
3. For each path, calculate the loss and LAE
payments that will develop
4. Determine the present value of each set of cash
flows by discounting by the relevant interest rates
5. Calculate the average present value over all
interest rate paths
6. This average is PV0
Measuring Interest Rate Sensitivity
of Insurance Liabilities (2)
7. Shock the initial instantaneous interest rate by
increasing, and decreasing, it by 100 basis points
8. Repeat steps 2-5 to determine PV+ and PV9. Calculate the effective duration based on:
Effective Duration = (PV--PV+)/(2PV0)(Δr)
Results of Effective Duration
Calculations
Effective duration is less than modified duration
– Cash flows change
• Higher interest rate → higher cash flows
• Lower interest rate → lower cash flows
– Longer term interest rates don’t move as much as
short term rates with most term structure models
Illustrative Example:
Duration of Loss Reserve Liabilities
Aggregate Industry – All Lines Combined
• Based on steady-state operations and a 4% initial
short-term interest rate:
Macaulay Duration:
Modified Duration:
4.24
4.08
• Based on CIR and interest-sensitive cash flows:
Effective Duration:
1.70
Stochastic Interest Rates
• Cox-Ingersoll-Ross (CIR) term structure model
– “Equilibrium” model
– Mean-reverting, square-root diffusion process
dr  a (b  r ) dt  
a =
r =
b=
σ=
dz =
r dz
speed of reversion to long-run mean
current short-term interest rate
long-run mean of short-term interest rate
volatility factor
standard Wiener process
Assumptions Underlying Illustrative
Effective Duration Calculation
Fixed Cost Parameters
k = 0.15
m = 0.10
n = 1.00
CIR Interest Rate Parameters
a = .25
r
= .04
b = .05
σ = .08
Impact of Inflation:
Embedded inflation rate = 2.5%
Future claim inflation = 4% + .40 x short-term interest rate
Why is Duration Important?
• Corporations attempt to manage interest rate
risk by balancing the duration of assets and
liabilities
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
Implications
• Use the same approach to measure the interest
sensitivity of both assets and liabilities
• A company may choose a duration mismatch
• Need to determine if the compensation for
accepting interest rate risk is adequate