Deductibles

FREQUENCY AND SEVERITY WITH COVERAGE MODIFICATIONS

8.2 Deductibles

Insurance policies are often sold with a per-loss deductible of𝑑. When the loss,𝑥, is at or below𝑑, the insurance pays nothing. When the loss is above𝑑, the insurance pays𝑥−𝑑. In the language of Chapter 3, such a deductible can be defined as follows.

Definition 8.1 Anordinary deductiblemodifies a random variable into either the excess loss or left censored and shifted variable(see Definition3.3). The difference depends on whether the result of applying the deductible is to be per payment or per loss, respectively.

This concept has already been introduced along with formulas for determining its moments. The per-payment variable is

𝑌^𝑃 = {

undefined, 𝑋 ≤𝑑, 𝑋−𝑑, 𝑋 > 𝑑, while the per-loss variable is

𝑌^𝐿= {

0, 𝑋≤𝑑, 𝑋−𝑑, 𝑋 > 𝑑.

Note that the per-payment variable𝑌^𝑃 =𝑌^𝐿|𝑌^𝐿>0. That is, the per-payment variable is the per-loss variable conditioned on the loss being positive. For the excess loss/per-payment variable, the density function is

𝑓_𝑌𝑃(𝑦) = 𝑓_𝑋(𝑦+𝑑)

𝑆_𝑋(𝑑) , 𝑦 >0, (8.1) noting that for a discrete distribution, the density function need only be replaced by the probability function. Other key functions are

𝑆_𝑌𝑃(𝑦) = 𝑆_𝑋(𝑦+𝑑) 𝑆_𝑋(𝑑) , 𝐹_𝑌𝑃(𝑦) = 𝐹_𝑋(𝑦+𝑑) −𝐹_𝑋(𝑑)

1 −𝐹_𝑋(𝑑) , ℎ_𝑌𝑃(𝑦) = 𝑓_𝑋(𝑦+𝑑)

𝑆_𝑋(𝑦+𝑑) =ℎ_𝑋(𝑦+𝑑).

Note that as a per-payment variable, the excess loss variable places no probability at zero.

The left censored and shifted variable has discrete probability at zero of 𝐹_𝑋(𝑑), representing the probability that a payment of zero is made because the loss did not exceed 𝑑. Above zero, the density function is

𝑓_𝑌𝐿(𝑦) =𝑓_𝑋(𝑦+𝑑), 𝑦 >0, (8.2)

while the other key functions are¹(for𝑦≥0)

𝑆_𝑌𝐿(𝑦) =𝑆_𝑋(𝑦+𝑑), 𝐹_𝑌𝐿(𝑦) =𝐹_𝑋(𝑦+𝑑).

It is important to recognize that when counting claims on a per-payment basis, changing the deductible will change the frequency with which payments are made (while the frequency of losses will be unchanged). The nature of these changes is discussed in Section 8.6.

EXAMPLE 8.1

Determine the previously discussed functions for a Pareto distribution with𝛼= 3and 𝜃= 2,000for an ordinary deductible of 500.

Using the preceding formulas, for the excess loss variable, 𝑓_𝑌𝑃(𝑦) = 3(2,000)³(2,000 +𝑦+ 500)⁻⁴

(2,000)³(2,000 + 500)⁻³ = 3(2,500)³ (2,500 +𝑦)⁴, 𝑆_𝑌𝑃(𝑦) =

( 2,500 2,500 +𝑦

)3

,

𝐹_𝑌𝑃(𝑦) = 1 −

( 2,500 2,500 +𝑦

)3

, ℎ_𝑌𝑃(𝑦) = 3

2,500 +𝑦.

Note that this is a Pareto distribution with𝛼= 3and𝜃= 2,500. For the left censored and shifted variable,

𝑓_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0.488, 𝑦= 0, 3(2,000)³

(2,500 +𝑦)⁴, 𝑦 >0,

𝑆_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0.512, 𝑦= 0, (2,000)³

(2,500 +𝑦)³, 𝑦 >0,

𝐹_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0.488, 𝑦= 0, 1 − (2,000)³

(2,500 +𝑦)³, 𝑦 >0,

ℎ_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

undefined, 𝑦= 0, 3

2,500 +𝑦, 𝑦 >0.

1The hazard rate function is not presented because it is not defined at zero, making it of limited value. Note that for the excess loss variable, the hazard rate function is simply shifted.

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

0 200 400 600 800 1,000

x

f(x) Original

Excess loss Left censored/shifted 0.488 probability at zero for the

left censored/shifted distribution

Figure 8.1 The densities for Example 8.1.

Figure 8.1 contains a plot of the densities. The modified densities are created as follows. For the excess loss variable, take the portion of the original density from 500 and above. Then, shift it to start at zero and multiply it by a constant so that the area under it is still 1. The left censored and shifted variable also takes the original density function above 500 and shifts it to the origin, but then leaves it alone. The remaining probability is concentrated at zero, rather than spread out. □ An alternative to the ordinary deductible is the franchise deductible. This deductible differs from the ordinary deductible in that, when the loss exceeds the deductible, the loss is paid in full. One example is in disability insurance where, for example, if a disability lasts seven or fewer days, no benefits are paid. However, if the disability lasts more than seven days, daily benefits are paid retroactively to the onset of the disability.

Definition 8.2 A franchise deductible modifies the ordinary deductible by adding the deductible when there is a positive amount paid.

The terms left censored and shifted and excess loss are not used here. Because this modification is unique to insurance applications, we use per-payment and per-loss terminology. The per-loss variable is

𝑌^𝐿= {

0, 𝑋≤𝑑, 𝑋, 𝑋 > 𝑑, while the per-payment variable is

𝑌^𝑃 = {

undefined, 𝑋 ≤𝑑, 𝑋, 𝑋 > 𝑑.

Note that, as usual, the per-payment variable is a conditional random variable. The related

functions are now

𝑓_𝑌𝐿(𝑦) =

{𝐹_𝑋(𝑑), 𝑦= 0, 𝑓_𝑋(𝑦), 𝑦 > 𝑑, 𝑆_𝑌𝐿(𝑦) =

{𝑆_𝑋(𝑑), 0≤𝑦≤𝑑, 𝑆_𝑋(𝑦), 𝑦 > 𝑑, 𝐹_𝑌𝐿(𝑦) =

{𝐹_𝑋(𝑑), 0≤𝑦≤𝑑, 𝐹_𝑋(𝑦), 𝑦 > 𝑑, ℎ_𝑌𝐿(𝑦) =

{

0, 0< 𝑦 < 𝑑, ℎ_𝑋(𝑦), 𝑦 > 𝑑, for the per-loss variable and

𝑓_𝑌𝑃(𝑦) = 𝑓_𝑋(𝑦)

𝑆_𝑋(𝑑), 𝑦 > 𝑑, 𝑆_𝑌𝑃(𝑦) =

⎧⎪

⎨⎪

⎩

1, 0≤𝑦≤𝑑, 𝑆_𝑋(𝑦)

𝑆_𝑋(𝑑), 𝑦 > 𝑑,

𝐹_𝑌𝑃(𝑦) =

⎧⎪

⎨⎪

⎩

0, 0≤𝑦≤𝑑,

𝐹_𝑋(𝑦) −𝐹_𝑋(𝑑)

1 −𝐹_𝑋(𝑑) , 𝑦 > 𝑑, ℎ_𝑌𝑃(𝑦) =

{

0, 0< 𝑦 < 𝑑, ℎ_𝑋(𝑦), 𝑦 > 𝑑,

for the per-payment variable.

EXAMPLE 8.2

Repeat Example 8.1 for a franchise deductible.

Using the preceding formulas for the per-payment variable, for𝑦 >500, 𝑓_𝑌𝑃(𝑦) = 3(2,000)³(2,000 +𝑦)⁻⁴

(2,000)³(2,000 + 500)⁻³ = 3(2,500)³ (2,000 +𝑦)⁴, 𝑆_𝑌𝑃(𝑦) =

( 2,500 2,000 +𝑦

)3

, 𝐹_𝑌𝑃(𝑦) = 1 −

( 2,500 2,000 +𝑦

)3

, ℎ_𝑌𝑃(𝑦) = 3

2,000 +𝑦.

For the per-loss variable,

𝑓_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0.488, 𝑦= 0, 3(2,000)³

(2,000 +𝑦)⁴, 𝑦 >500, 𝑆_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0.512, 0≤𝑦≤500, (2,000)³

(2,000 +𝑦)³, 𝑦 >500, 𝐹_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0.488, 0≤𝑦≤500, 1 − (2,000)³

(2,000 +𝑦)³, 𝑦 >500, ℎ_𝑌𝐿(𝑦) =

⎧⎪

⎨⎪

⎩

0, 0< 𝑦 <500, 3

2,000 +𝑦, 𝑦 >500. □

Expected costs for the two types of deductible may also be calculated.

Theorem 8.3 For an ordinary deductible, the expected cost per loss is 𝐸(𝑋) −𝐸(𝑋∧𝑑)

and the expected cost per payment is

𝐸(𝑋) −𝐸(𝑋∧𝑑) 1 −𝐹(𝑑) . For a franchise deductible the expected cost per loss is

𝐸(𝑋) −𝐸(𝑋∧𝑑) +𝑑[1 −𝐹(𝑑)]

and the expected cost per payment is

𝐸(𝑋) −𝐸(𝑋∧𝑑) 1 −𝐹(𝑑) +𝑑.

Proof: For the per-loss expectation with an ordinary deductible, we have, from (3.7) and (3.10), that the expectation is E(𝑋) −E(𝑋∧𝑑). From (8.1) and (8.2) we see that, to change to a per-payment basis, division by1 −𝐹(𝑑)is required. The adjustments for the franchise deductible come from the fact that when there is a payment, it will exceed that for the

ordinary deductible by𝑑. □

EXAMPLE 8.3

Determine the four expectations for the Pareto distribution from Examples 8.1 and 8.2, using a deductible of 500.

Expectations could be derived directly from the density functions obtained in Examples 8.1 and 8.2. Using Theorem 8.3 and recognizing that we have a Pareto distribution, we can also look up the required values (the formulas are in Appendix A). That is,

𝐹(500) = 1 −

( 2,000 2,000 + 500

)3

= 0.488, E(𝑋∧ 500) = 2,000

2 [

1 −

( 2,000 2,000 + 500

)2]

= 360.

With E(𝑋) = 1,000, for the ordinary deductible, the expected cost per loss is 1,000 − 360 = 640, while the expected cost per payment is 640∕0.512 = 1,250.

For the franchise deductible, the expectations are640 + 500(1 − 0.488) = 896and

1,250 + 500 = 1,750. □

8.2.1 Exercises

8.1 Perform the calculations in Example 8.1 for the following distribution, using an ordinary deductible of 5,000:

𝐹4(𝑥) = {

0, 𝑥 <0,

1 − 0.3𝑒^{−0.00001𝑥}, 𝑥≥0. 8.2 Repeat Exercise 8.1 for a franchise deductible.

8.3 Repeat Example 8.3 for the model in Exercise 8.1 and a 5,000 deductible.

8.4 (*) Risk 1 has a Pareto distribution with parameters𝛼 >2and𝜃. Risk 2 has a Pareto distribution with parameters0.8𝛼and𝜃. Each risk is covered by a separate policy, each with an ordinary deductible of𝑘. Determine the expected cost per loss for risk 1. Determine the limit as 𝑘goes to infinity of the ratio of the expected cost per loss for risk 2 to the expected cost per loss for risk 1.

8.5 (*) Losses (prior to any deductibles being applied) have a distribution as reflected in Table 8.1. There is a per-loss ordinary deductible of 10,000. The deductible is then raised so that half the number of losses exceed the new deductible as exceeded the old deductible.

Determine the percentage change in the expected cost per payment when the deductible is raised.

Table 8.1 The data for Exercise 8.5.

𝑥 𝐹(𝑥) E(𝑋∧𝑥)

10,000 0.60 6,000

15,000 0.70 7,700

22,500 0.80 9,500

32,500 0.90 11,000

∞ 1.00 20,000

8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary