Problem set

A company owns a small office building worth $500,000. Cameron is the risk manager. The building faces the risk of a fire that could completely destroy it. The probability of a fire is 5%.

Cameron is considering the following risk management options to address the risk of fire to their building:

  1. Retention

  2. Loss Reduction Program & Retention

  3. Full Insurance for a premium of $30,000

  4. Loss Reduction Program & Full Insurance (premium falls to $17,000)

The cost of the Loss Reduction Program is $10,000. It does not lower the probability of the fire. However, if a fire does occur, the loss is reduced to half ($250,000).

Retention - they keep the risk and pay for any losses

Loss matrix

retention

  • no fire - 0 dollars since they have no insurance and no fire made them lose

  • fire - 500,000 no insurance, full loss

loss reduction + retention

  • no fire - 10,000 , costs them 10k with or without fire

  • fire - 260,000, reduce loss to half so 250, but still have to pay 10k

Full insurance

  • no fire- 30k, pay premium no matter what

  • fire - 30k, pay premium

full insurance + loss reduction

  • no fire - reduced premium to 17k still add 10k for loss reduction

  • fire - same as no fire

Assume Cameronโ€™s worry value for retention (๐‘Š๐‘‰_๐‘…) is $6,500 and his worry value for retention and loss reduction program (๐‘Š๐‘‰_๐‘…+๐ฟ๐‘…) is $4,000.

What are the Expected Total Costs under each option?ย 

  • step 1 find expected value

    • .05(cost of fire) + .95(cost of no fire) =

  • step 2 add worry value

    • expected + worry

Retention

.05 ร— 500000) + (.95 ร— 0) = 25000 e.v.

add worry - 25,000+6500= 31500

loss reduction + retention

.05 ร— 10000) + (.95 ร— 260000) = 22500 e.v.

add worry - 22500+4000 = 26500

full insurance

.05 ร— 30000) + (.95 ร— 30000) = 30000 e.v.

no worry

full insurance + retention

.05 ร— 27000) + (.95 ร— 27000) = 27000 e.v.

When the loss reduction program is not introduced, what is the actuarially fair premium (AFP) for full Insurance?

AFP for full insurance is

adp = probability of loss x size of loss

.05 ร— 500000 = 25000

When the loss reduction program is introduced, what is the actuarially fair premium (AFP) for full Insurance?

AFP for full insurance after introducing loss reduction is

loss reduction halved cost so same just 250k

.05ร—250000= 12500

What would be the most Cameron willing to pay for full Insurance (his Pmax) if he does not implement any loss reduction program?

loss with no retention was 25000, add worry 6500

25000+6500= 31500

During a meeting, the Chief Risk Officer (CRO) told Cameron that the most he would pay for full Insurance was $30,000. Who is more risk-averse, the CRO or Cameron?ย 

cameron = 31500

cro = 30000

The more risk-averse person is willing to pay more to avoid risk.