AUTOMOBILE WARRANTY UNEARNED PREMIUMS AND
DEFERRED POLICY ACQUISITION EXPENSES
JOE S. CHENG
This paper describes one approach to calculate the unearned premium reserves of an
automobile extended warranty insurance program, test the adequacy of the calculated
reserves, and determine the allowable deferred policy acquisition expenses.
A prorata formula is commonly used to calculate unearned premium reserves in propertycasualty
insurance, but we believe that an exposure adjusted formula is more appropriate
in automobile extended warranties.
We organize data by the effective month of the manufacturer warranty and employ an
expected pure premium methodology to calculate the unearned premium reserves for an
automobile extended warranty contract.
Unearned premium reserves plus future investment income derived thereof are compared
against future claims and expenses to determine if premium deficiency exists.
Investment income is estimated from interest bearing assets, taking into account credit
risk, interest rate risk and payment pattern risk.
Automobile warranties have terms ranging from 1 year to 7 years and acquisition
expenses are large relative to the first year earned premiums. In (US and Canadian)
GAAP financial statements, insurance companies are allowed to defer policy acquisition
expenses to the extent they meet the test of recoverability.
Finally, we discuss the impact of reinsurance on a mono line warranty insurance
company's balance sheet.
A new automobile extended warranty (hereinafter called an extended warranty) is usually
defined by two limits, time and mileage. An extended warranty is expired when either
one of the two limits is reached. For example, a 5 years/60,000 miles extended warranty
means the warranty will expire either in 5 years, or when the odometer reading reaches
60,000 miles, whichever comes first. The extended warranty for new vehicles usually
does not come into effect until the coverage under the manufacturer warranty has expired.
Recently, most manufacturers have been offering 3 years/36,000 miles of full (bumper to
As the exposure of an extended warranty is measured from the registration date of the
new vehicle, the age of any extended warranty is the time elapsed between the
registration date and the valuation date. In this paper, an extended warranty is assumed
to be effective on the first day of the effective month.
2. UNEARNED PREMIUM RESERVES
The unearned premium reserves of an extended warranty can be calculated on an
exposure adjusted basis or a prorata basis. In our opinion, the exposure adjusted basis is
a better approach. Under this approach premiums are earned in proportion to the
emergence of the expected losses; when 5% of the ultimate losses are expected to be the
cumulative incurred at the end of year two, the formula should have 95% of the written
premiums as unearned premiums. As an illustration, a typical 6 years/72,000 miles
(6/72) extended warranty with an underlying three year manufacturer warranty might
have the following cumulative expected loss, earned and unearned pattern.
Time 0 12 mos 24 mos 36 mos 48 mos 60 mos 72 mos
Expected losses 0 2 5 15 45 75 100
Earned 0% 2% 5% 15% 45% 75% 100%
Unearned 100% 98% 95% 85% 55% 25% 0%
The above earned pattern, together with a proper amortization of acquisition expenses
would theoretically match the income and outgo of the 6/72 contract throughout the life
of the contract.
For a contract type l we denote the expected monthly pure premiums for month
1,2,......n, as P1,l, P2,l, P3,l, ....., Pn,l, where n is the contract term in months +1; and the
expected pure premium for contract type l as Pl.
Pl Pi l
The unearned premium ratio for a contract type l at k months is:
= - 1
, 1 =
+ = 1
For the above contract type l, we have inforce extended warranties that are 1,2.....,n-1
Let Gi,l represent the written premiums of a group of extended warranties, all with
contract type l and i months old, and Ri,l represent the unearned premium ratio for age i.
Then, the unearned premiums of these extended warranties are:
Ul Ri lGi l (2.2)
If there are m different contract types in a program, the unearned premiums of the entire
i l i l
Ul R , G ,
1 1 1
å å å-
= = =
The formulae (2.1), (2.2) and (2.3) hold true for either the prorata method or the exposure
adjusted method. In the case of the prorata method P1,l = P2,l = P3,l = ..... = Pn,l for
contract type l.
Under the prorata method, premiums are earned in proportion to the time expired on the
contract. Notwithstanding its simplicity, this method produces a severe overstatement of
premiums earned in the early part of the contract and a corresponding understatement of
earned premiums near the end of the contract.
At this moment, there is no consensus as to which method is proper. The accounting
profession has limited guidance on warranty unearned premium reserves. Under
FASB60, extended warranties are classified as short-duration contracts: "Premiums from
short-duration contracts ordinarily are recognized as revenue over the period of the
contract in proportion to the amount of insurance protection provided."1
A straight interpretation of FASB60 would suggest the following 2 approaches.
(1) Time 0 mos 12
Cumulative Earned 0 0 0 0 1/3 2/3 3/3
(2) Mileage (in
0 12,000 24,000 36,000 48,000 60,000 72,000
Cumulative Earned 0 0 0 0 1/3 2/3 3/3
The first approach presumes that no policyholder drives more than 12,000 miles per year.
We know that assumption is highly implausible. The second approach is more accurate
than the first, but it is impractical to determine the odometer readings of all policyholders
on a valuation date. The exposure adjusted method is really a blending of approach 1 and
2. When it is supported by loss experience, the exposure adjusted method is the only one
which follows the intent of FASB60.
3. DATA ORGANIZATION
As an extended warranty comes into effect when the manufacturer warranty expires, it is
convenient to track the exposure and claim payments of such an extended warranty by the
1 Summary of FASB Statement No. 60, paragraph 3 (Appendix A).
registration date of the vehicle (i.e., the effective date of its manufacturer warranty). The
sale date of an extended warranty offers less accurate information about the exposure to
the insurer because a large percentage of extended warranties are not sold on the same
date as the vehicle. Most extended warranty programs give the original owner up to 12
months to purchase an extended warranty as long as the 3 years/36,000 miles portion of
the manufacturer warranty has not expired. Claim payments are used here in lieu of
incurred claim amount because incurred claim amount might change slightly after the
valuation date (e.g. December 31, 1998). The historical data for contract type l should
look as follows:
Age of Contract 1/91 2/91 ------ ------ 10/98 11/98 12/98
1 A1,1,l A1,2,l A1,94,l A1,95,l A1,96,l
2 A2,1,l A2,2,l A2,94,l A2,95,l
3 A3,1,l A3,2,l A3,94,l
J Aj,1,l Aj,2,l
73 A73,1,l A73,2,l
Where, age of contract = valuation month/year - effective month/year of manufacturer
Ai,j,l = Claim amount from contract type l with effective month j and paid during the
month i of the contract.
A set of data for a 2 years/24,000 miles plan with a 1 year/12,000 miles manufacturer
warranty is shown in Appendix B.
4. METHODOLOGY AND ASSUMPTIONS
First, the exposures (in contract months) have to be determined. Let Ei,j,l be the number
of exposures for a specific contract type l, age (month) i and effective month j. For a
given effective month (based on manufacturer warranty effective date) and contract type,
we can project the number of exposures Ei,j,l for each month subsequent to its effective
month. We assume no lapse in our projection. For example, assume there are 1,000
contracts in a 6 years/72,000 miles program (contract type l) with effective month in July
1991, then, we would project the following exposures:
Calendar month Age in month i Exposure Ei,j,l
November 1993 29 1,000
December 1993 30 1,000
June 1997 72 1,000
July 1997 73 1,000
August 1997 74 0
The above projection assumes that after a cooling off period (usually 60 days for
consumers to reverse their impulsive decisions to purchase extended warranties), the
extended warranty count will remain the same until expiration. A small percentage of
warranties are cancelled mid-term because their underlying vehicles have been written
off in accidents. This simplification will not have a material effect on the future claim
future claim payments = pure premium x exposure in months.
The exposure term is overstated by the inclusion of cancelled extended warranties, but
the pure premium term is understated by roughly the same percentage. (The no-lapse
assumption can be removed if we keep track of exposures, not only by effective month
and contract type, but also by age of each contract.) For the balance of this paper, we
will use the no lapse assumption and drop the first subscript from Ei,j,l and use Ej,l instead.
The above projection also assumes that all contracts are effective on the first day of each
month. The extra month (73rd month) is used to capture all late payments or repairs done
in the last month of the contract.
From the data, we can estimate the monthly pure premiums by age for each contract as
LET Ni,j,l be the claim count in month i of the contract term for contract type l with
effective dates in month j.
Ej,l be the warranty count for contract type l with effective dates in month j.
Ai,j,l be the actual claim payment in month i of the contract term for contract type l
with effective dates in month j.
Pi,l be the average pure premium in month i for the contract type l,
Pi,l = claim frequency x average claim size.
= ´ å
i j l
i j l
i j l
i j l
This is usually calculated using the last 12 calendar months of data available for each age
(month i). (If it is necessary to use more than 12 months of data, some inflation
adjustment to formula (4.1) is needed.) For contracts sold recently, the data has not
reached the part of the contract term when claims are more likely to be made. Therefore,
the pure premiums have to be estimated from the more mature contracts with similar
features. In all cases, the Pi,ls should be smoothed and adjusted to the valuation date cost
level. The resultant Pi,ls become the expected monthly pure premiums for contract type l.
Using a 6 years/72,000 miles contract as an illustration, we have monthly expected pure
premiums P1 to P73. (In this illustration, only one contract type is involved. The
subscript l is dropped for simplicity.) The expected pure premium of a 6 years/72,000
miles contract with four years to expiry would be:
i å P =
Assuming there are E25 contracts that are 24 months old, the expected payments of these
contracts would be:E25´ i
i å P =
25 OR 25
7325 Pi E
= å (4.2)
Let's assume the valuation date is December 31, 1998 and there are E73 (contracts
effective in Jan. 93), ..., E25 (contracts effective in Jan. 97) .......E2 (contracts effective in
Dec. 98) in the inforce book.
There is usually some inflation in warranty repairs as very few people shop around for a
bargain when they are covered by a warranty. As Pi's from formula (4.2) are at
December 1998 cost level, they have to be adjusted for inflation after the valuation date.
If r is the monthly inflation rate, the same repair in January 1999 should cost r% more
than that in December 1998.
Therefore (4.2), the total expected payment for contracts with 4 years to expiry, becomes:
( ) 25 (4.3)
25P 1 r E
i ´ + ´
Formula (4.3) can be expanded as follows:
factor Exposure Expected Payments
25 Jan. 1999 P25 (1+r) E25 P25 x (1+r) x E25
26 Feb. 1999 P26 (1+r)2 E25 P26 x (1+r)2 x E25
27 Mar. 1999 P27 (1+r)3 E25 P27 x (1+r)3 x E25
28 Apr. 1999 P28 (1+r)4 E25 P28 x (1+r)4 x E25
72 Dec. 2002 P72 (1+r)48 E25 P72 x (1+r)48 x E25
73 Jan. 2003 P73 (1+r)49 E25 P73 x (1+r)49 x E25
There are E2 to E73 contracts with age ranging from 1 month to 72 months respectively.
The expected losses (C) of all 6/72 contracts (after the valuation date) can be estimated as
Where m = effective month of the contract
i = age of the contract( ) m
m C = P ´ + r ´ E
= = å å 73 73 1
The expected loss calculation for all 6/72 contracts can be illustrated by the following
In the above triangle the rows represent the age of the contracts and the columns
represent the effective month of the contracts. Each diagonal, however, represents a
calendar month of payments starting with January 1999.
The above triangle can be re-oriented so that each diagonal becomes a row corresponding
to the calendar month in which payments are expected. The new triangle would look as
Age Jan.93 Dec.98