Premium Earning Patterns

1
Premium Earning Patterns

• Tom Struppeck
• Scottsdale, AZ
• September 14, 1999

2
Two Companies at 12/31/xx

• Company A
• Assets
• Cash 1000
• Liabilities
• UEPR 400
• Surplus 600

• Company B
• Assets
• Cash 1000
• Liabilities
• UEPR 400
• Surplus 600

3
Which of these would you rather own?

• These two companies look quite similar the
  difference, if any, is what risks the UEPR
  covers.
• Each company has written exactly one policy
• That policy expires on 12/31/xx1
• other details on the next slide

4
The two policies

• Company A
• Pays 900 if there is a storm of a certain size
  during the next year

• Company B
• Pays 900 if there is a storm of a certain size
  during the next year AND
• there is an earthquake of a certain magnitude
  during the next year

5
Now which one would you choose?

• You certainly would rather own company B since
  its policy is less likely to pay (it only pays in
  a subset of the cases that Company As policy
  pays).
• The balance sheets are misleading --- perhaps
  Company B (or Company A) has mispriced its
  product.

6
Actually, this is year 2 of a two year policy
period.

• Company As policy and Company Bs policy were
  identical except that the quake zones were
  different (and Company A got hit).
• No loss was paid (or incurred) because it is a
  dual trigger policy.
• It seems that some reserve is in order. (Is it
  premium or loss?)

7
Mark-to-Market

• Mark-to-market is a term used for valuing
  securities at current market prices.
• In fast-moving futures exchanges, positions get
  marked to market twice a day.
• How can we mark-to-market the UEPR?

8
The friendly reinsurer

• Reinsurer will accept any risk at its expected
  loss.
• Mark-to-market by computing how much the
  reinsurer would charge to assume the risk covered
  by the UEPR. This is the pure premium portion of
  the UEPR.
• Company A would get charged more than Company B.

9
Reinsurers Pricing

• Company A
• Will assume the unexpired portion of the UEPR for
  a premium of 700

• Company B
• Will assume the unexpired portion of the UEPR for
  a premium of 250

10
Resulting B/Ss

• Company A
• Assets
• Cash 1000
• Liabilities
• UEPR 700
• Surplus 300

• Company B
• Assets
• Cash 1000
• Liabilities
• UEPR 250
• Surplus 750

11
What about the income statement?

• Suppose that the original premium was 800 (the
  loss amount is only 900, so maybe this isnt
  realistic, but then most quake companies write
  more than 1 policy).
• Under the pro-rata earning method, both companies
  earned 400 last year.

12
Income Statements

• Company A
• BOY-UEPR 800
• EOY-UEPR 700
• WP 0
• So EP 100

• Company B
• BOY-UEPR 800
• EOY-UEPR 250
• WP 0
• So EP 550

13
No loss yet!

• Observe that Company A has not lost any money
  yet it simply has a much riskier book (and, as
  such, will earn much more next year).
• In all cases, the total premium earned during the
  two years will be 800, the written premium. It
  all gets earned the only thing that changes is
  the timing.