Insurance mathematics II. lecture Reserving – introduction I. One of the most important function of actuaries. Every year-end closure there is a requirement.

Insurance mathematics II. lecture

Reserving – introduction I.

One of the most important function of actuaries. Every year-end closure there is a requirement to calculate reserves for future liabilities and grant the realistic P&L.Now there is a lot of type of reserves with special calculation rules.In the future (from 2016) there will be a change in this viewpoint: it will be just one (two) type(s) of reserve.


Reserving – introduction II.

Legal regulation: 8/2001 Ministry of Finance RegulationRequirement: all reserves have to calculate per business lines (not per products)Reserves are important because of measure also (total reserves are about 2.000 billion HUF at 2013).


Unearned premium reserve I.Example: we take out a household policy at 01.12.2014 with 24.000 HUF yearly premium. We agree yearly payment frequency and we pay total premium in December. What is the realistic P&L situation at 31.12.2014?

2.000 HUF

22.000 HUF2.000 HUF

22.000 HUF2.000 HUF

01.12.2014 31.12.201430.11.2015


Unearned premium reserve II.

Suppose that no any claim related to this policy. What is the real P&L figure in 2014 and 2015?

- If we would book whole premium for 2014 then we would not book any premium for 2015, but we are in risk during 11 months! But we got really the whole premium in 2014.

The solution:

1. We book whole premium for 2014.

2. We calculate unearned premium reserve for 2014. It means that this part of premium will be the offset of risk in 2015. The value of UPR is:

HUF 000.22000.2412

11


Unearned premium reserve III.

Generalizing:

Let it be

d the premium due to payment frequency;

T the duration of the payment which continues into next year;

K the duration till date of year-closure, then:

T

KTdUPR


Unearned premium reserve IV.

Open problems with this definition:-the lengths of months are not equal (in the example the real UPR is not 21.962 HUF ?);- the risk can be not equal in the whole period (example: fleets);- based on Hungarian regulation it should not reserve UPR more then 1 year;- difference between Hungarian and international regulation. The base of UPR is the yearly premium in international standard, but the payment regarding frequency in Hungarian regulation.


Mathematical reserve

Now we are dealing with mathematical reserve in non-life insurance, i.e. related to liability insurance and accident insurance. (Apart from these types there are mathematical reserve regarding life insurance and health insurance also.)


Mathematical reserve in liability insurance I.

This type is used generally if the insurer has to pay annuity based on liability insurance (typically in Hungary MTPL). The total future annuity payments are estimated with the next formula:

))1(()1(

11

0

bdSil

lMR kk

n

k x

kx

where:- x is an age of annuitant;- lx comes from mortality table (number of x ages);- n is the unexpired years regarding annuity;- Sk is the yearly annuity in the k-th year (with

taxes);


Mathematical reserve in liability insurance II.

- i is the technical interest (the yield which the insurer will reach till end of annuity with guarantee; now based on the regulation the maximum of technical interest is 0 according to liability insurance);

- d,b are cost factors; typically one of them is 0.

If there is an L limit in the policy related to annuity the formula will change as follows:

),min(* MRLMR


Mathematical reserve in liability insurance III.

The formula is uncomplicated, but the estimation of parameters is not easy:

- in the most cases the insurer knows just the initial annuity. In the long run it can be a lot of change regarding the health status of annuitant, inflation, etc.;

- the mortality of annuitant is different comparing the total mortality rate (but there are no separate mortality table of annuitant). It can cause unexpected profit or loss.

Because of above reasons the insurer usually tends to pay lump sum (typically 50-60% of virtual mathematical reserve).


Mathematical reserve in accidental insurance

The formula is similar as in liability insurance, but the change of annuity is not so frequent – because usually the measure of annuity is exact in the policy.


Claims reserve I.

Reasons:- lag in reporting of claims;- lag in payment of claims.

There are two types of claims reserve:- If the insurer has known the claims but the claims are

not totally payed, there are Outstanding Claims Reserve (OS Reserve);

- If the insurer has not yet known the claims, it can be used IBNR reserve (incurred but not reported).


Claims reserve II.There are two different possible approach:- separate assumption for OS and IBNR reserve or- together estimating with statistical methods.

Now in Hungary it is used the separate approach generally, but because of Solvency II. in the future the second approach will come into view.

The measure of lag is characteristic for products, for example the CASCO and accident products have usually higher speed run-off, and MTPL and other liability products have usually slower run-off.


Claims reserve III.

For assumption it can consider inflation and the yield of reserves also.

It is interesting and generally unanswerable question how is the most useful splitting of portfolio for the assumption:

The target is to find homogenous groups of risks. It can be per products or per business lines or sometimes in one product there is useful to further splitting (for example, in MTPL splitting between annuities and non-annuities, or splitting big claims and non-big claims).


Outstanding claims reserve and claims handling reserve

In separate OS reserve assumption methods the actuaries have no a lot of tasks. The claim experts have experience how much can be the ‘best estimate’ of the different claim event.

The actuaries have just one task: calculate claims handling reserve with the next formula:

CLRCP

CHPCHR where

CHR – claims handling reserve;

CHP – claims handling payment in current year;

CP – claims payment in current year;

CLR – claims reserve (OS or IBNR).


IBNR reserve I.

There are a lot of different algorithm to evaluate IBNR reserve. Before the detailed description of these methods it can be useful to define several basic ideas.

Run-off triangles

Accident year

Reporting/payment year

1,1X nX ,1

1,nX

jiX ,

…………..

……

……

.. jiX , means the total amount of claims which are occurred in i-th year and reported/paid in j-th year


IBNR reserve II.

Lagging triangles

…………..

……

……

.. jiX , means the total amount of claims which are occurred in i-th year and reported/paid in (i+j-1)-th year

Accident year

Development year

1,nX

jiX ,

1,1X nX ,1


IBNR reserve III.

Cumulated triangles

…………..

……

……

.. jiX , means the total amount of claims which are occurred in i-th year and reported/paid till (i+j-1)-th year

Accident year

Development year

1,nX

jiX ,

1,1X nX ,1


IBNR reserve IV.

The cumulated triangle is complete, if there is no any reporting/paying after n-th year (difficult to say).

It is possible to make run-off triangles for number of claims also.

Hungarian regulation requires for IBNR calculation just using run-off triangles.

For assumption the cumulated triangle will be the basic usually.


IBNR reserve V.

Denote niX , the claims which are reported/paid after n-th

year regarding claims occurred in i-th year.

Our target is estimating the next formula (for each i):

ininii XXIBNR 1,,

Our best estimate is as follows:

)1,(ˆ,,, nvuXXEX vunini

Our problem is that in practice usually we do not know covariance and common distribution of claims. That is why we simplify in the methods which we are using for calculation of IBNR.


Methods of IBNR calculation Grossing Up method I.

Lagging triangle

Accident year

Development yearExample:

2009

2010

2011

2012

2013

1 2 3 4 5

223; 311; 252; 127; 29

254; 378; 249; 153

312; 411; 276

359; 435

384


Methods of IBNR calculation Grossing Up method II.

Cumulated triangle

Accident year

Development yearExample:

2009

2010

2011

2012

2013

1 2 3 4 5

223; 534; 786; 913; 942

254; 632; 881; 1034

312; 723; 999

359; 794

384


Methods of IBNR calculation Grossing Up method III.

Is the cumulated triangle complete? No, we have data from earlier years as follows:

Example:

Year Claims until 5th year

Total Ratio (5th year/Total)

2005 780 830 93,98%

2006 810 890 91,01%

2007 800 860 93,02%

Total 2390 2580 92,64%


Methods of IBNR calculation Grossing Up method IV.

Assumption for 2009:

Example:

1. year 2. year 3. year 4. year 5. year Total

2009 223 534 786 913 942 1017

Ratio 21,9% 52,5% 77,3% 89,8% 92,64% 100%

10179264,0

942

The base of assumption will be 2009 as next table shows:

We assume that the run-off of next years will be equal as 2009.


Methods of IBNR calculation Grossing Up method V.

Occurring

year

Example:

2009

2010

2011

2012

2013

1 2 3 4 5

223; 534; 786; 913; 942

254; 632; 881; 1034

312; 723; 999

359; 794

384

Total IBNR

1017 75

1152 118

1292 293

1512 718

1751 1367

Total 2571


Methods of IBNR calculation Grossing Up method VI.

Generalizing:

1. Based on earlier year we estimate

n

nn X

Xd

,1

,1

If we have no any data from earlier year we can use data from similar products or OS reserves.

2. Further factors:

n

nn

X

Xd

,1

1,11 ˆ

n

nn

X

Xd

,1

2,12 ˆ

….

nX

Xd

,1

1,11 ˆ

3. Ultimate payment estimation:

1

1,2,2

ˆ

n

nn d

XX

2

2,3,3

ˆ

n

nn d

XX ….

1

1,,

ˆd

XX n

nn


Methods of IBNR calculation Grossing Up method VII.

4. Reserve assumption:

niXXV ininii 1 ;ˆ1,,

n

iiVV

1

The above calculation is the basic version, but there are some modified possibility of this method.


Methods of IBNR calculation Modified Grossing Up methods I.

1. version: If we have data related to earlier year splitting per year then we can calculate more exact the d factors.

Let andniX

Xd

n

ii

1 ;ˆ

)1(,1

1,1 nikddd iii 1 );(),...,2(),1(

the other experience d factors. Then the ultimate used factors:

nik

kddddd iiii

i

1 ;

1

)(...)2()1()1(


Methods of IBNR calculation Modified Grossing Up methods II.

Example:


2005Ratio (%)

17020,5%

42250,8%

66079,5%

75090,4%

78094%

830100%

2006Ratio (%)

14015,7%

43548,9%

68076,4%

78287,9%

81091%

890100%

2007Ratio (%)

13215,3%

42849,8%

67077,9%

78090,7%

80393%

860100%

2009Ratio (%)

22321,9%

53452,5%

78677,3%

91389,8%

94292,7%

1017100%

Averageratio

18,4% 50,5% 77,8% 89,7% 92,7% 100%


Methods of IBNR calculation Modified Grossing Up methods III.

Example: Year Total (ultimate) payment

IBNR

2009 1017 75

2010 1153 119

2011 1284 285

2012 1572 778

2013 2090 1706

Total 2964


Methods of IBNR calculation Modified Grossing Up methods IV.

2. version: If we have data related to earlier year splitting per year then we can calculate more exact the d factors.

Let andniX

Xd

n

ii

1 ;ˆ

)1(,1

1,1 nikddd iii 1 );(),...,2(),1(

the other experience d factors. Then the ultimate used factors:

nikddddd iiiii 1 ));();...;2();1();1(min(


Methods of IBNR calculation Modified Grossing Up methods V.

Example:


2005Ratio (%)

17020,5%

42250,8%

66079,5%

75090,4%

78094%

830100%

2006Ratio (%)

14015,7%

43548,9%

68076,4%

78287,9%

81091%

890100%

2007Ratio (%)

13215,3%

42849,8%

67077,9%

78090,7%

80393%

860100%

2009Ratio (%)

22321,9%

53452,5%

78677,3%

91389,8%

94292,7%

1017100%

Minimum ratio

15,3% 48,9% 76,4% 87,9% 91% 100%


Methods of IBNR calculation Modified Grossing Up methods VI.


IBNR

2009 1035 93

2010 1177 143

2011 1308 309

2012 1625 831

2013 2502 2118

Total 3493


Methods of IBNR calculation Modified Grossing Up methods VII.

3. version: We estimate nidX in 1 );1( ;ˆ,1 as in 1. version. After

it we judge ultimate payment for 2.year:

With this result we define the d factors and estimate ultimate payment as follows:

)1(ˆ

1

1,2,2

n

nn d

XX

nn

nn

X

Xd

X

Xd

,2

1,21

,2

2,22 ˆ

)2( ...; ;ˆ

)2(

2

)2()1( 222

nn

n

ddd

2

2,3,3

ˆ

n

nn d

XX

After it we continue this process till each d factors and payments will be calculated.


Methods of IBNR calculation Modified Grossing Up methods VIII.

Example:


2009Ratio (%)

22321,9%

53452,5%

78677,3%

91389,8%

94292,6%

1017100%

2010Ratio (%)

25422,1%

63254,9%

88176,5%

103489,8%

1152100%

2011Ratio (%)

31224%

72355,6%

99976,9%

1299100%

2012Ratio (%)

35924,6%

79454,3%

1461100%

2013Ratio (%)

38423,1%

1659100%


Methods of IBNR calculation Modified Grossing Up methods IX.


IBNR

2009 1017 75

2010 1152 118

2011 1299 300

2012 1461 667

2013 1659 1275

Total 2434


Methods of IBNR calculation Modified Grossing Up methods X.

4. version: We estimate nidX in 1 );1( ;ˆ,1 as in 1. version. After

it we judge ultimate payment for 2.year:

With this result we define the d factors and estimate ultimate payment as follows:

)1(ˆ

1

1,2,2

n

nn d

XX

nn

nn

X

Xd

X

Xd

,2

1,21

,2

2,22 ˆ

)2( ...; ;ˆ

)2(

))2();1(min( 222 nnn ddd

2

2,3,3

ˆ

n

nn d

XX

After it we continue this process till each d factors and payments will be calculated.


Methods of IBNR calculation Modified Grossing Up methods XI.

Example:


2009Ratio (%)

22321,9%

53452,5%

78677,3%

91389,8%

94292,6%

1017100%

2010Ratio (%)

25422,1%

63254,9%

88176,5%

103489,8%

1152100%

2011Ratio (%)

31224%

72355,6%

99976,5%

1306100%

2012Ratio (%)

35924,6%

79452,5%

1512100%

2013Ratio (%)

38421,9%

1753100%


Methods of IBNR calculation Modified Grossing Up methods XII.


IBNR

2009 1017 75

2010 1152 118

2011 1306 307

2012 1512 718

2013 1753 1369

Total 2587


Methods of IBNR calculation Link ratio methods I.

We suppose thatji

jij X

Xic

,

1,)(

1. Determiningni

ni

nn X

X

dc

,

,1

2. Ultimate payment and IBNR reserve estimation:

nnn XcX ,1,1ˆ nnnn XcXXV ,1,1,11 )1(ˆ

ratio does not depend on significantly for i

is similar then in the Grossing Up method.

(with experience of earlier years or OS reserve). Other jc factors will be

defined as function of actual )(ic j . With choosing different function will

be defined the different version of link ratio method.


Methods of IBNR calculation Link ratio methods II.

2. Ultimate payment and IBNR reserve estimation:

1,21,2ˆ

nnnn XccX 1,211,2,22 )1(ˆ nnnnn XccXXV

1,11, ...ˆnnnnn XcccX 1,111,, )1...(ˆ

nnnnnnn XcccXXV

… …

Basic version: 11 );1( njcc jj

1. modification: 11 ;)(...)2()1(

nj

jn

jncccc jjj

j


Methods of IBNR calculation Link ratio methods III.

11 ;)(...)2()1( ,,2,1

njjn

jncccc jjjnjjjj

j

3. modification:

11 ));();...2();1(max( njjncccc jjjj2. modification:

In 3. modification with special α factors we will get the most popular ‘chain-ladder’ method.


Methods of IBNR calculation Chain ladder method I.

11 ;...

...

...

)(...)2()1(

,,2,1

1,1,21,1

,,2,1

,,2,1

njXXX

XXX

XXX

jncXcXcXc

jjnjj

jjnjj

jjnjj

jjjnjjjjj

This is the most popular process for IBNR estimation.


Methods of IBNR calculation Chain ladder method II.


IBNR

2009 1017 75

2010 1152 118

2011 1300 301

2012 1458 664

2013 1648 1264

Total 2420


Methods of IBNR calculation Naive loss ratio method

iniiii XPV 1,)1(

In the next methods we are using premium data also (not just claim data).

We suppose that the ultimate loss of i-th year will be the

-th part of the premium.

Then the reserve can be calculated as follows:

i1

n

iiVV

1

, where signs the earned premium of i-th year.iP

The disadvantage of this method is that IBNR reserve is independent of actual claim data. Starting company without any own claim data can use this method.


Methods of IBNR calculation Bornhuetter-Ferguson method I.

)1(ˆ, iini PX

This method combines the naive loss ratio and grossing up (or link ratio) methods.

1. We calculate the ultimate loss payment with naive claim ratio methods:

2. For calculating development factors we are using grossing up (or link ratio method):

ni

i

nnni

ni

nnn

ni

ni

nn X

X

cccd

X

X

ccd

X

X

cd

,

1,

111

,

1,

11

,

,

...

1 ;...

1 ;

1


Methods of IBNR calculation Bornhuetter-Ferguson method II.

3. The reserves will be estimated as follows:

nn

nn Xc

XdV ,1,11ˆ)

11(ˆ)1(

nnn

nn Xcc

XdV ,21

,212ˆ)

11(ˆ)1(

….

nnnn

nnn Xccc

XdV ,11

,1ˆ)

...

11(ˆ)1(

n

iiVV

1


Methods of IBNR calculation Separation method I.

ji

In this method we do not use cumulated triangles, but we are using lagging triangles. The used triangles have to be complete.

We suppose thatnjirncX jijiji ,1 ;,

where signs the number of claims in the i-th year (known),

signs an inflation, c is the average claim amount.in

There are two types of this method:

- arithmetic;

- geometric.

Now we consider detailed the arithmetic version.


Methods of IBNR calculation Separation method II.

1k

We suppose that1

1

n

jjr

It means that signs the expected proportion of claims development year j without inflation effect. In this case

is the inflation factor according to first year.

jr

Then we define the next formula:

jiji

jiji rc

n

XB ,

,

And we use the lagging triangle for these new elements as follows


Methods of IBNR calculation Separation method III.

...

……

…..

After we define diagonal sums as follows:

Accident year

Development year

nrc 1

11 rc nnrc 22 rc

21 rc 32 rc ...nnrc 1


Methods of IBNR calculation Separation method IV.

….

111101111101 )1()...(... nnnnnnnnn rcrrrcrcrcrcd

00 rcdo

11011101 )( rrcrcrcd

22102221202 )( rrrcrcrcrcd

nnnnnnnn crrrcrcrcrcd )...(... 1010


Methods of IBNR calculation Separation method V.

n

jjnjn Bc

11,

From the last equation we will get the first assumptions:

n

nn

Br

ˆ ,1

After that we could calculate recursively the next factors as follows:

n

nn r

dc

ˆ1ˆ 1

1

1

1,21,11 ˆˆ

ˆ

nn

nnn

cc

BBr

1

22 ˆˆ1

ˆ

nn

nn rr

dc

21

2,32,22,12 ˆˆˆ

ˆ

nnn

nnnn

ccc

BBBr

…

nc


Methods of IBNR calculation Separation method VI.

nnn 221ˆ;...;ˆ;ˆ

Then we will assume the future inflations:

After we fill the remaining part of the lagging triangle:

2 ;ˆˆˆ, njirnX jijiji

The assumption for the IBNR reserve will be the next:

n

injjijii rnV

1

ˆˆ

n

iiVV

1


Methods of IBNR calculation Separation method VII.

The geometric separation method will be calculated similar, just the beginning assumption is different:

n

iir

1

1

In the estimator formulas we use the products of diagonal instead of sums of diagonal.


Methods of IBNR calculation Which method do will use in practice?

There are no one ‘true’ method which is the most useful in each case

(products, business lines). What can the actuary do in practice?

The most useful possibility is calculating IBNR reserve with methods as much as possible, back-testing the results, and year by year fining the method. There is important to consider result of earlier IBNR as follows:

ClIBNRClOSntClaimPaymeOpIBNRRIBNR


Bonus reserve I.

There are some products in which insurer pays back a part of premium if the policy has few claims or the loss ratio of policy is low. For these cases actuaries have to reserve the expected refund. The description of general method follows:

- insurer guarantees if policy has k claims per policy year then insurer will pay back -th part of premium,

- insurer guarantees if policy has loss ratio between and per policy year, then insurer will pay back -th part of premium,

Let ξ the original variate of risk, then the new variate (with refunds) will be as follows:

𝑐 𝑗− 1𝑐 𝑗− 1


Bonus reserve II.

ζ=ξ+𝑑 ∙¿where d signs the original premium,

η signs the number of claims.

For reserving we suppose that till date of closure the number of reported claims are and the paid amount is Based on Hungarian regulation the actuaries have to reserve the pro-rata part of expected premium refund.Letis the length between date of closure and beginning date of policy,is the length between end date of policy and date of closure,is the original risk between date of closure and beginning date of policy, is the original risk between end date of policy and date of closure,is the number of claims between date of closure and beginning date of policy, is the number of claims between end date of policy and date of closure.


Bonus reserve III.

If the premium refund is affected when the policy is claim-free then the formula will be quieter as follows:

𝑉=𝑡1

𝑡1+𝑡 2

∙𝑑 ∙¿

))

Then we will get for bonus reserve:

𝑉={ 𝑡 1

𝑡1+𝑡 2

∙𝑑 ∙𝛼 ∙𝑃 ( η2=0|η1=0 ) ,𝑖𝑓 η1=0 ;

0 , 𝑖𝑓 η1>0

η1=0


Other reserves I.

This reserve is affected just in life insurance (it has to be calculated in case of plus yield return to policyholder). Equalization reserve

This reserve can be calculated for those business line which are profitable (it has to be calculated, but the measure of reserve can be 0).If the result of business line is negative then the reserve has to be reduced with the measure of negative result.

Large claim reserve

This reserve has to be calculated in case of huge risks (for example: nuclear power station).

Bonus reserve for life insurance


Other reserves II.

Based on own experience it has to be reserved the expected cancelled part of premium in the future. It is important to take into consideration the unearned premium reserve (it is prohibited to reserve for unearned part of premium) and in life insurance the saving part of the premium (it is prohibited to reserve for the saving part of the premium). Other reserve

If the insurer will lose related to a policy (or a product) and there is no any possibility to close or change this policy (or product) then this type of reserve can be calculated. The formula of this reserve is as follows:

Cancellation reserve


Solvency capital and security capital I.

The reserves are calculated generally based on the expected value. For the worse scenario the insurer has to have solvency capital which is a part of own fund. Security capital

This capital need for the fundamental operation of insurer. If the own fund is lower then the security capital the authority will act immediately.

Solvency capital


Solvency capital and security capital II.

Minimum solvency capital requirement (based on Solvency I., in non-life section) where𝑃𝑆𝐶𝑅=max (𝑟𝑎𝑡 𝑒𝑜 𝑓 𝑟𝑒𝑖𝑛𝑠

;0,5)∙¿

+max (0 ;𝑐𝑜𝑟𝑟𝑝𝑟 −61300000𝑒𝑢𝑟 ) ∙0,16 ¿

𝐶𝑙𝑆𝐶𝑅=max (𝑟𝑎𝑡 𝑒𝑜 𝑓 𝑟𝑒𝑖𝑛𝑠;0,5)∙¿

+max (0 ;𝑐𝑜𝑟𝑟¿ − 42900000𝑒𝑢𝑟) ∙0,23¿

Minimum security capital requirement (based on Solvency I., in non-life section) where if the insurer has no liability business or if the insurer has liability business.

Insurance mathematics II. lecture Reserving – introduction I. One of the most important function of actuaries. Every year-end closure there is a requirement.

Documents

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