Auto Warranty Premiums - Unearned Premiums and Deferred Expenses for Auto Warranty and RV Warranty Premiums

Warranty Premiums - Unearned Premiums and Deferred Expenses
Warranty Premiums - Unearned Premiums and Deferred Expenses
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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.

1. INTRODUCTION

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

bumper) coverage.

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.

Then,

å=

=

n

i

Pl Pi l

1

, .

The unearned premium ratio for a contract type l at k months is:

l

k

i

i l

k l

P

P

R

å=

= - 1

,

, 1 =

l

n

i k

i l

P

P å

+ = 1

,

(2.1)

For the above contract type l, we have inforce extended warranties that are 1,2.....,n-1

months old.

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:

å-

=

=

1

1

, ,

n

i

Ul Ri lGi l (2.2)

If there are m different contract types in a program, the unearned premiums of the entire

program are:

i l i l

n

i

m

l

m

l

Ul R , G ,

1

1 1 1

å å å-

= = =

= (2.3)

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

mos

24

mos

36

mos

48

mos

60

mos

72

mos

Cumulative Earned 0 0 0 0 1/3 2/3 3/3

(2) Mileage (in

miles)

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:

Effective Month

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

warranty +1

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

projection because

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

follows:

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

å

j

j l

j

i j l

i l

E

N

P

,

, ,

, å

å

j

i j l

j

i j l

N

A

, ,

, ,

å

å

=

j

j l

j

i j l

i l

E

A

P

,

, ,

, (4.1)

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

i å P =

73

25

Assuming there are E25 contracts that are 24 months old, the expected payments of these

contracts would be:

E25´ i

i å P =

73

25 OR 25

73

25 Pi E

i

´

= 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)

73 24

25

P 1 r E

i

i

i ´ + ´

-

= å

Formula (4.3) can be expanded as follows:

Age of

Claim

Payment

Month

Expected

Pure

Premium

Inflation

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

follows:

(4.4)

Where m = effective month of the contract

i = age of the contract

( ) m

i m

i m

i

m C = P ´ + r ´ E

- +

= = å å 73 73 1

2 1

The expected loss calculation for all 6/72 contracts can be illustrated by the following

diagram:

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

follows:

Age Jan.93 Dec.98

1

2

Expected

(future)

payments

73

Effective month

.................................

 

 


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