Aggregate XS Loss Cost Estimation (Fisher)

Question 1

Contrast vertical and horizontal slices approaches

Answer 1

Vertical approach is:
+ more intuitive
+ quicker if just 1 entry ratio
- deals with each risk individually

But horizontal approach is:
+ faster if we need to calculate for multiple limits.

Question 2

Define entry ratios

Answer 2

If risks we are pricing are not exactly the same as underlying distribution, we need to make an adjustment otherwise dollars of expected losses will be off.

To deal with this, we simply rescale vertical axis by dividing actual losses by expected losses.

r = A/E

Question 3

Without rescaling, tot area under curve = ?

Answer 3

E(Loss)

Question 4

Once we use r instead, tot area under curve = ?

Answer 4

1

Question 5

Explain insurance charge

Answer 5

E(XS)/E = E(Losses above LG)/E

Area between curve and rG

Question 6

Explain insurance savings

Answer 6

E(shortfall)/E = E(shortfall below LH)/E

Area between rH and curve

Question 7

Calculate Net Insurance Charge

Answer 7

Insurance charge - Insurance savings = phi(rG) - psi(rH)

Question 8

Define Table M

Answer 8

Collection of related aggregate XS loss factors and related savings.

It ignores the impact of per-occurrence limit.

Question 9

Explain how to create a Table M

Answer 9

1. Calculate r for each claim
2. Create one row for each ri and one for r=0
3. Add a column for # risks at each ri
4. Add a column for # risks above ri
5. Add a column for % risks above ri
6. Add a column for Charge(ri)
   Charge(rmax) = 0
   Charge(ri) = Charge(ri+1)+(ri+1 - ri)*(% risks above ri)
7. Make sure Charge(r0) = 1, o/w rescale

Question 10

How to get savings from charge

Answer 10

Savings = Charge + r - 1

Question 11

Why tables defined in terms of r and phi are more desirable

Answer 11

Tables are less vulnerable to inflation since as risk increases due to inflation, you can simply map to different existing Table M charge column instead of recalculating.

Question 12

What are the 4 properties of insurance charge

Answer 12

1. charge(0) = 1
2. charge(inf) = 0
3. First derivative is negative (charge decreases as r increases)
4. Second derivative is positive (charge decreases at an increasing rate)

Question 13

What are the 4 properties of insurance savings

Answer 13

1. savings(0) = 0
2. savings(inf) = inf
3. First derivative is positive (savings increase as r increases)
4. Second derivative is positive (savings increase at an increasing rate)

Question 14

Discuss what happens to insurance charge when changing r and risk size

Answer 14

As risk size increases, var(r) decreases so charge curve flatten

As risk size decreases, charge goes to 1

As r increases, charge decreases

Question 15

Discuss what happens to insurance savings when changing r and risk size

Answer 15

As risk size decreases, savings go to r (since savings = charge + r - 1 = 1 + r - 1 = r)

As r increases, savings increase

Question 16

Discuss how error in estimation of expected losses impact insurance charge

Answer 16

If E too high, we will be using wrong Table M and we will have a flatter aggregate loss distribution so charge will be underestimated.

On the other hand, we will have a smaller entry ratio which will result in higher charge.

The 2 impacts are partially offsetting so final impact on insurance charge is uncertain.

Question 17

How error in estimation of E changes based on r

Answer 17

At lower r, curve for largest risks has steepest negative slope so using slightly incorrect r will result in biggest absolute error in Table M charge.

When results multiplied by E(A), error in $ is very large.

At higher r, curve for largest risks is flattest so using slightly incorrect r will result in smaller absolute error in Table M charge.

However, since charges for high entry ratios of large risks are so low, even a small absolute error results in large % error.

In a nutshell,
For high r, % error in charge is greatest for large risks
For low r, $ error is greater for large risks

Question 18

Calculate Adjusted Expected Losses

Answer 18

Some policies of same size are riskier than other policies.

To account for this, adjust E to match those of a risk with different size but similar variance in aggregate loss distribution.

E adjusted = E * State HG Differential

Since 2019, NCCI uses Expected Count Group instead of Expected Loss Group to determine appropriate Table M column for risk.

Question 19

List 2 things State HG Differential adjusts for

Answer 19

1. Riskiness: some classes have higher proportion of larger claims
2. Location: differences in medical care cost, benefit, court verdicts, etc.

Question 20

What is the idea behind using E(N) to determine which table to use instead of Adjusted Expected Losses

Answer 20

A risk with higher severity state will have lower E(N) implying greater volatility.

This approach works well is sev distribution differs in scale but not in shape, otherwise we also need to adjust for sev distr as well.

Question 21

Calculate Basic Premium (B) for Retro policy

Answer 21

B = e - (c-1)E + cI

I = E*(phi(rG) - psi(rH))

Question 22

What are the 2 Table M Balance Equations

Answer 22

Diff in charges = (e+E)T - H / cET
Diff in r = (G-H) / cET

Same denominator!!!!

Question 23

Discuss how variance of limited aggregate distribution is reduced by occurrence limit

Answer 23

Occurrence limit reduces variance of aggregate loss distribution (flattens curve) because variance of underlying severity distribution is reduced by occurrence limit.

The smaller the occurrence limit, the less variance in severity distribution so limited aggregate loss distribution has less variance as well

Question 24

Why are Table M no longer appropriate if per-occurrence limit is applicable

Answer 24

They do not recognize overlap between occurrence limit and aggregate limit.

Question 25

What are the 2 options if per-occurrence limit applicable (instead of Table M)

Answer 25

1. We can estimate the per-occ XS losses separately from limited aggregate XS losses.
   Limited aggregate XS losses will be estimated using Limited Table M.
2. We can estimate the per-occ XS losses and aggregate XS losses simultaneously.
   Using a Table L which includes a charge for both per-occ XS losses as well as limited aggregate XS losses.

Question 26

Explain how to construct a Limited Table M

Answer 26

Identical to Table M build but using limited loss data (Ad)

r* = Ad/E(Ad)

R* = (B_LM + cAd)T

B_LM = e - (c-1)E + c(I_LM + kE)
k = 1 - E(Ad)/E

Diff in charges = (e+E)T - H / cE(Ad)T

Diff in ratios = (G - H) / cE(Ad)T

Question 27

Explain how to construct a Table L

Answer 27

r* = Ad/E

phi(r*max) = k instead of 0

R* = (B_L + cAd)T

E(R) = (B_L + c(E - I_L))T

E(Ld) = E - I_L

I_L = E(phi(rG) - psi(r*H))

B_L = e - (c-1)E + cI_L

Diff in charges = (e+E)T - H / cET

Diff in ratios = (G-H) / cET

Same denom as Table M!!!!!

Question 28

Discuss the purpose and thinking behind the ICRLL Approximation

Answer 28

Since presence of occurrence limit would normally require entirely new tables, an approximation can be used to simulate a Limited Table M by adjusting the column used for regular Table M.

The thinking behind the approximation is that occurrence limit reduces variance of limited aggregate loss distribution compared to unlimited aggregate loss distribution, which is the same thing that occurs as we move to larger risk sizes.

NCCI used Insurance Charge Reflecting Loss Limitation (ICRLL) procedure to change Table M column used for a risk with occurrence limit to be the column that would normally be used for larger risk in absence of occurrence limit.

Question 29

Calculate adjusted E under ICRLL Approx

Answer 29

Adjusted E = E * State HG Differential * (1+0.8k)/(1-k)

k = (LGAF-1)/(0.8+LGAF)

If you use this procedure, since you simulate Limited Table M, you need to use the corresponding balance equations in performing Table M search.

Question 30

Is ICRLL Approximation still used?

Answer 30

Nowadays, NCCI publishes Limited Table M directly so ICRLL procedure no longer needed.

Question 31

Explain how (2) to prevent overlap between per-occurrence limit and aggregate limit

Answer 31

1. Apply per-occurrence limit first to not increase Table M charge and prevent overlap.
2. Use ICRLL to select from different set of Table M charges to approximate Limited Table M charge.

Question 32

Contrast Expected XS Agg Loss using Table L and Limited Table M and Insurer expected retained loss

Answer 32

Table L:
E(XS Agg Loss) = Charge*E
Insurer retained = E - XS

Limited Table M:
E(XS Agg Loss) = ChargeE(Ad) + kE
Insurer retained = E - XS

Question 33

True or False?
Table L avoids double-counting

Answer 33

True, by limiting distribution.

Question 34

Explain the impact on charge for given r if E(A) underestimated

Answer 34

E(A) underestimated so charge underestimated, so charge will increase