Rational & Mechanics of CAT Swap

Rationale and Mechanics for
Peak Natural Catastrophe
Variance Swaps in Insurance
Ivelin Zvezdov
Sebastian Rath
Igor Cizelj

Agenda
1. Motivation
2. Simulated insurance loss
3. Defining capital reserve shortage & the swap contract
4. Historical stress test
5. Using the variance swap contract
6. Convergence
7. Spatial risk metrics

Theoretical and Economic
Motivation
3
1. Geo-spatial variability of insurance risk
2. A low priority for insurance practitioners, focused on first order risk
3. Climate variability - a new significant and measurable factor in geo-
spatial risk variability
4. Support measuring of inter-dependence and clustering of insurance
risk, for risk management practitioners
5. Operational efficiencies in reserve capital management
6. Arbitrage opportunities for capital market firms

Historical simulation of known
significant events
4
Storm track & footprint
of ETC Daria, 26/26’th of
January 1990
Perturbations of
ETC Daria
Historical tracks 1999-2008
and a single perturbation of
each track

Physical intensity downscaling
and insurance loss computation
5
©2018 AIR Worldwide CONFIDENTIAL—FOR INTERNAL USE ONLY
Downscaling of intensity to 10m grid, 14 km above ground, & 6 hours of
temporal resolution

Natural catastrophe simulation
for insurance loss
6
• CRESTA zones in Holland
• Notional insurance company -
10% of industry insured
exposure by geo-zone
(business unit)
Number of simulated events
in a 10K stochastic scenarios /
years

Defining available and required
capital reserves
7
Available capital reserve 𝑄 𝑒𝑝=2%; 𝐸𝑃 2% = 𝑃 𝐿𝑜𝑠𝑠 ≧ 𝑄 𝑒𝑝=2%
Required capital reserve 𝐾𝑒𝑝=1%; 𝑃 1% = 𝑃 𝐿𝑜𝑠𝑠 ≧ 𝐾𝑒𝑝=1%
Capital reserves observe both super and sub additivity properties
𝑄 𝑒𝑝=2% > σ𝑖= 1
90
𝐵𝑈 𝑄 𝑒𝑝=2%; and 𝐾𝑒𝑝=1% < σ 𝑖=1
90
𝐵𝑈 𝐾𝑒𝑝=1%
TVaR remains strictly sub-additive
𝑇𝑉𝑎𝑅 𝑒𝑝=1% < ෍
𝑖=1
90
𝐵𝑈 𝑇𝑉𝑎𝑅 𝑒𝑝=1%

Capital reserves back-allocation
from portfolio to business unit
8
- Traditional industry practice based on contributing ratios of modeled
expected values of loss EL
- Captures inter-dependence for risk factors within the portfolio
𝐸𝐿 = ෍
𝑖=1
90
𝐵𝑈 𝐸𝐿𝑖
𝐵𝑈𝑖 𝑄 𝑒𝑝 2% = 𝑄 𝑒𝑝 2%
𝐵𝑈 𝐸𝐿 𝑖
σ 𝑖=1
90 𝐵𝑈 𝐸𝐿 𝑖
𝐵𝑈𝑖 𝐾𝑒𝑝 2% = 𝐾𝑒𝑝 2%
σ 𝑖=1
90

Defining capital reserves
shortages and rates
9
𝐵𝑈 𝑠ℎ𝑜𝑟𝑡𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 =
𝐵𝑈 𝑄 𝑒𝑝=2% − 𝐵𝑈 𝐾𝑒𝑝=1% +
𝐵𝑈 𝑖𝑛𝑠𝑢𝑟𝑒𝑑 𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒

Historical stress tests for
spatial variability
10
𝐵𝑈𝑖 𝐷𝑎𝑟𝑖𝑎 𝐿𝑜𝑠𝑠
𝐵𝑈𝑖 𝑄 𝑒𝑝=1%

Stress tested rate-on-line
11
𝑚𝑖𝑛 𝐵𝑈𝑖 𝐾𝑒𝑝=1%, max(𝐵𝑈𝑖 𝐷𝑎𝑟𝑖𝑎 𝐿𝑜𝑠𝑠 − 𝐵𝑈𝑖 𝑄 𝑒𝑝=2%, 0)
𝐵𝑈𝑖 𝐾𝑒𝑝=1%

Interpretations for practitioners
12
Available and required capital reserve bands imposed on historical
losses from Daria by BU

More interpretations for practitioners
13
- For 40 business units, historical losses breach the thresholds of available
capital reserve
- For another 30 business units historical ROL(s) exceeds a threshold of
30%, what should be considered as a very high market price
- For 8 business units the ROL is 100% and for another 4 it exceeds 80% -
historical loss breaches or approaches required capital reserves.

Inefficiencies for insurers and
opportunities for optimal outcomes
14
- Business units with extreme shortages of capital reserve post
extreme catastrophe event will require fund transfer and raised
internal cost.
- Underwriting limits are overexposed to losses in some business
units
- In other business units, lower underwriting limits may lead to
underestimating of business opportunity and market share
- Rationale for seeking both internal risk and underwriting
management optimal solutions

Adapting the Variance Swap to
Insurance Risk Transfer
15
Strike variance
𝑉𝐴𝑅 𝐾 =
1
𝑁
෍
𝐵𝑈=1
90
𝑙𝑛
𝐵𝑈 𝑄 𝑒𝑝=2%
2
= 25.97%
Realized variance
𝑉𝐴𝑅 𝑅 =
1
𝑁
෍
𝐵𝑈=1
90
𝑙𝑛
𝐿 𝐵𝑈
2
= 36.02%
Pay-outs
𝑆𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡 = 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 ∗ 25.97% − 36.02% = −10.05% ∗ 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙

Strike variance and swap rate
simulation
16
𝑠𝑤𝑎𝑝 𝑟𝑎𝑡𝑒 𝑒𝑝=1% =
1
𝑁
෍
𝐵𝑈 1
90
𝑙𝑛
𝐿 𝐵𝑈
2
− 𝑙𝑛
2
… , …
𝑠𝑤𝑎𝑝 𝑟𝑎𝑡𝑒 𝑒𝑝=5% =
1
𝑁
෍
𝐵𝑈 1
90
𝑙𝑛
𝐿 𝐵𝑈
2
− 𝑙𝑛
2

Convergence
17
Variance swap rates derived for simulated strike variances from business unit
modeled required capital reserves in the exceedance probability interval
EP5% to EP1%
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
-50.0% -40.0% -30.0% -20.0% -10.0% 0.0% 10.0% 20.0% 30.0% 40.0% 50.0%
EPofrequiredreserves
Variance swap rate

Second order risk management
18
Covariance ratio by each pair of company units
𝐶𝑂𝑉[𝑋1,𝑖: 𝑋2,𝑗]
𝑉𝐴𝑅 𝑋1,𝑖 + 𝑉𝐴𝑅 𝑋2,𝑗
Inland North
Inland Central 0.39
Inland South 0.50 0.37
Coastal South 0.49 0.49 0.41
Coastal Central
0.50 0.49 0.49 0.38
Coastal
Central
Coastal
South
Inland
South
Inland
Central
Inland
North

Covariance measure
19
Covariance percent share, by each pair of company units
𝐶𝑂𝑉[𝑋1,𝑖: 𝑋2,𝑗]
σ σ 𝐶𝑂𝑉[𝑋1…5;𝑖…𝑚: 𝑋2…4;𝑗…𝑛]
Inland North 100%
Inland Central 100% 5.99%
Inland South 100% 13.39% 6.43%
Coastal South 100% 12.35% 11.52% 5.64%
Coastal
Central 100% 12.02% 13.65% 12.74% 6.26%
Coastal
Central
Coastal
South
Inland
South
Inland
Central
Inland
North

Marginal covariance measure
20
Marginal covariance percent share for each company unit
𝐶𝑂𝑉[𝑋1,𝑖: 𝑋2,𝑗 + 𝑋3,𝑘 + 𝑋4,𝑙 + 𝑋5,𝑚]
𝑉𝐴𝑅 𝑋1,𝑖 + 𝑉𝐴𝑅 𝑋2,𝑗 + 𝑋3,𝑘 + 𝑋4,𝑙 + 𝑋5,𝑚
Inland North 0.16
Inland Central 0.35
Inland South 0.37
Coastal South 0.33
Coastal Central 0.37
All without
Coastal Central
All without
Coastal South
All without
Inland South
All without
Inland Central
All without
Inland North

Continuing work
21
Expand the effort to define coherent second order geo-spatial risk
measurement processes, and metrics for (re)insurance.
Stimulate knowledge exchange among practitioners and academics
Design of practical risk transfer and hedging strategies
Work towards consensus on underlying standards and sources of data for
modeling and pricing.

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