insurance mathematics iii. lecture solvency ii – introduction solvency ii is a new regime which...

Insurance mathematics III. lecture

Solvency II – introduction

Solvency II is a new regime which changes fundamentally the insurers (and reinsurers).The insurers have to operate risk-based and it has a lot of new regulations and standards. The Solvency II. comes into force at 01.01.2016.The actuaries are affected most of all the new reserving methodology and the new SCR, MCR calculation.


Solvency II. New SCR calculationSCR

OP

Market

SLT Health

BSCR

Non-SLT Health CAT

eLifDefault

djA

IntangHealth Non-life

Spread

Equity

Interest rate

Property

Currency

Concentration

Disability morbidity

Mortality

Revision

Longevity

Lapse

Expenses

CAT

ty Disabilitymorbidi

Mortality

Revision

ityLongev

Lapse

Expenses

CAT

Premium reserve

LapsePremium reserve

Lapse


Solvency II.Reserving I.

Reserving methodology is based on the best estimate

assumptions plus additional risk margin.

The best estimate shall correspond to the probability-

weighted average of future cash-flows within the contract

boundary, taking account of the time value of money

(expected present value of future cash-flows), using the

relevant risk-free interest rate term structure.


Solvency II.Reserving II.

The risk margin shall be such as to ensure that the value of the technical provisions is equivalent to the amount that insurance and reinsurance undertakings would be expected to require in order to take over and meet the insurance and reinsurance obligations.

Contract boundary: contract shall be taking into consideration till the date when one of partners (insurer or insured) can quit from policy without any consequence. In non-life section the typical possibility to exiting from policy is 1 year, it means that usually we have to calculate premium till end of first policy year – but claims according to first policy year can be reported later.


Solvency II.Reserving III.

In non-life section we can calculate separately reserve for premium and claims. The ultimate reserve will be the sum of reserve for premium and reserve for claims.

The reserve for premium can be calculated with the next formula:

𝑃𝑟𝑒𝑚𝑅𝑒𝑠=𝑈𝑃𝑅 ∙ (1−𝑃𝑏𝐶𝑎𝑛𝑐 )−𝐶𝑙𝑃𝑎𝑦−𝐷𝐴𝐶−𝐶𝑙𝐻𝐶−𝑀𝑎𝑖𝑛𝐶

1+𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑟𝑎𝑡𝑒

Remark: if the product is profitable then the amount has negative sign.


Solvency II.Reserving IV.

whereUPR signs the Unearned Premium Reserve;

PbCanc signs the probability of cancellation;

ClPay signs the claim payment for claims which occurred before policy anniversary;

DAC signs the deferred acquisition costs;

ClHC signs the claims handling costs for claims which occurred before policy anniversary;

MainC signs the maintenance cost which are affected till policy anniversary.


Solvency II.Reserving V.

Example:

Let the total portfolio is one policy with the next data:

Beginning date: 01.10.2014

Annual premium: 50.000 Ft

Probability of cancellation: 15% yearly

Expected loss ratio: 70%

Commission: 6%

Claims handling costs: 9% of claim (in homework 0)

Maintenance costs: 10% of premium (in homework 0)

Discount rate: 5%


Solvency II.Reserving VI.

Example (continued)

We are calculating the reserve for premium at 31.12.2014.

𝑈𝑃𝑅 ∙ (1−𝑃𝑏𝐶𝑎𝑛𝑐 )=37500 ∙88,75%=33281

𝐶𝑙𝑃𝑎𝑦=𝑃𝑟𝑒𝑚∙𝐿𝑜𝑠𝑠𝑅𝑎𝑡𝑖𝑜=33281 ∙70%=23297

𝐷𝐴𝐶=𝑃𝑟𝑒𝑚 ∙𝐶𝑜𝑚𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒=37500 ∙6%=2250

𝐶𝑙𝐻𝐶=𝐶𝑙𝑃𝑎𝑦 ∙𝐶𝑙𝐻𝐶𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒=23297 ∙9%=2097

𝑀𝑎𝑖𝑛𝐶=𝑃𝑟𝑒𝑚 ∙𝑀𝑎𝑖𝑛𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒=37500 ∙10%=3750

𝑈𝑃𝑅=912∙50000=37500 𝑃𝑏𝐶𝑎𝑛𝑐=

912∙15%=11,25%


Solvency II.Reserving VII.

𝑃𝑟𝑒𝑚𝑅𝑒𝑠=h𝐶𝐹𝑤𝑖𝑡 𝑜𝑢𝑡𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡

1+𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑟𝑎𝑡𝑒=18871,05

=1797

h𝐶𝐹𝑤𝑖𝑡 𝑜𝑢𝑡𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡=33281−23297−2250−2097−3750=1887

Example (continued)

Reserve for claims

Actuaries have to estimate reported and not yet reported claims togetherplus claims handling cost in the future. It shall be applied the discount rate according to year of expected claim payment.If there is no differing information (e.g. changing of portfolio) we can use previous information for claims.


Solvency II.Reserving VIII.

One possible method is as follows:

OS reserve

1.step: Calculating the ratio of previous payments related to lagging time (year, quarter year, month).

2.step: Calculating the ratio of actual OS reserve according to occurring date (year, quarter year, month).

3.step: Calculating the real OS need based on result of earlier OS reserve (e.g. result is +10% ,then the real OS need is lower with 10%).

4.step: Estimating the payment of real OS need based on 1. and 2. step.

5.step: Discounting the result of 4. step with adequate discount factors.


Solvency II.Reserving IX.

Example:

We have 126.000.000 Ft OS reserve (according to Solvency I.) and we have to calculate Best Estimate.

1. step: we have data from past payments according to laggingas follows:

0. year 1.year 2.year 3.year

60% 30% 9% 1%

2. step: we have data about OS reserve occurring date as follows:

2011 2012 2013 2014

1.000.000 5.000.000 20.000.000 100.000.000


Solvency II.Reserving X.

3. step: Result of earlier OS reserve is +5%. It means the real OS need is as follows:

2011 2012 2013 20144761905 19047618 95238090

4. step: Payment estimation as follows according to earlier steps:


Solvency II.Reserving XI.

Occurring/Paying year

2015 2016 2017

2011 952381

2012 4761905

2013

2014

Total 94.285.620 23.333.332 2.380.952


Solvency II.Reserving XII.

5. step: The discount rates are given as follows:

1. year 2. year 3. year

5% 4% 3%

Then the reserve for OS reserve will be the next:

𝑅𝑒𝑠𝑂𝑆=942856201,05

+233333321,05 ∙1,04

+2380952

1,05 ∙1,04 ∙1,03=113.280.621


Solvency II.Reserving XIII.

IBNR

It can be calculated with classical methods (just one difference: we have to take into consideration the result of earlier IBNR) it shall be considered which part of IBNR when will be paid (according to estimation). At the end the discount factors shall be applied.


Solvency II.Reserving XIV.

Example:

Cumulated, lagging triangle

0 1 2 3

2011

2012

2013

2014

50000 55000 57000 57500

65000 70000 72000

75000 85000

85000


Solvency II.Reserving XV.

We are using chain-ladder method, we suppose that the triangle is complete.

Example (continued):

0 1 2 3

2011

2012

2013

2014

50000 55000 57000 57500

65000 70000 72000 72632

75000 85000 87720 88489

85000 93947 96954 97804

Year IBNR

2011 0

2012 632

2013 3.489

2014 12.804

Total 16.925


Solvency II.Reserving XVI.

Occurring/Paying year

2015 2016 2017

2011 0

2012 632

2013 2720

2014 8947

Total 12.299 3.775 850

Example (continued):


Solvency II.Reserving XVII.

Then the reserve for IBNR claims will be the next:

𝑅𝑒𝑠𝐼𝐵𝑁𝑅=122991,05

+3775

1,05 ∙1,04+

8501,05 ∙1,04 ∙1,03

=15927

The discount rates are given as follows:

1. year 2. year 3. year

5% 4% 3%


Solvency II.Difficulties I.

There are a lot of questions, difficulties related to Solvency II., because this is a total new regime and the technical specifications - which have to be applied - are not exactly clear in each case. I highlight just two points from these questions

1. Segmentation

In Solvency II. the target is making homogenous risk portfolio and for these groups using the specifications. In the other side (in non-life section) there is given the business lines which have to be applied. These two requirements would be controversy if one homogenous risk portfolio does not fit to the given business lines.


Solvency II.Difficulties II.

2. Claim inflation

There is not clear whether there is possible to consider claim inflation or not. And if the answer is yes then how should be calculated. (In EU there are countries in which has high inflation but other countries have no high inflation.)

insurance mathematics iii. lecture solvency ii – introduction solvency ii is a new regime which...

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