Risk Management and Insurance Review
C
Risk Management and Insurance Review, 2012, Vol. 15, No. 2, 129-152
DOI: 10.1111/j.1540-6296.2012.01214.x
FEATURE ARTICLES
PRICING FOR MULTILINE INSURER: FRICTIONAL COSTS,
INSOLVENCY, AND ASSET ALLOCATION
Li Zhang
Norma Nielson
ABSTRACT
This article examines multiline insurance pricing based on the contingent claim
approach in a limited liability and frictional costs environment. Capital allocation is based on the value of the default option, which satisfies the realistic
assumption that each distinct line undertakes a pro rata share of deficit caused
by insurer insolvency. Premium levels, available assets, and default risk interact
with each other and reach equilibrium at the fair premium. The assets available to pay for liabilities are not predetermined or given; instead, the premium
income and investment income jointly influence the available assets. The results show that equity allocation does not influence the overall fair premium.
For a given expected loss, the premium-to-expected-loss ratio for firms offering
multiple lines is higher than that for firms only offering a single line, due to
the reduced risk achieved through diversification. Premium-to-expected-loss
ratio and equity-to-expected-loss ratio vary across lines. Lines having a higher
possibility or claim amount not being paid in full exhibit lower premium-toexpected-loss ratio and higher equity-to-expected-loss ratio. Positive correlation
among lines of business results in lower premium-to-expected-loss ratio than
when independent losses are assumed. Positive correlation between investment
return and losses reduces the insolvency risk and leads to a higher premiumto-expected-loss ratio.
INTRODUCTION
Setting a fair or competitive premium plays an important role in the insurance industry.
Capital is invested or retained in the insurance industry only if the return provided by
the insurance industry is comparable to that offered by other industries. Determining
an appropriate insurance premium has been the subject of extensive scrutiny over the
last several decades among both academia and industry practitioners. Starting from
the earliest attempt to determine the fair premium—the Target Underwriting Profit
Li Zhang is an Assistant Professor at G.R. Herberger College of Business, St. Cloud State University, 720 Fourth Avenue, South St. Cloud, MN 56301; phone: 320-308-3876; fax: 320-308-4973;
e-mail: [email protected]. Norma Nielson holds the Chair in Insurance and Risk Management at Haskayne School of Business, University of Calgary, 2500 University Drive N.W.,
Calgary, AB T2N 1N4, Canada. This article was subject to double-blind peer review.
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Margin promulgated by the National Convention of Insurance Commissioners in 1921—
a variety of insurance pricing models have been proposed and applied, including the
capital asset pricing model (e.g., Fairley, 1979; Hill, 1979; Cummins and Harrington,
1985; Hill and Modigliani, 1987), the internal rate of return approach (e.g., Cummins,
1990), the discounted cash flow approach (e.g., Myers and Cohn, 1987; Cummins, 1990;
D’Arcy and Garven, 1990), the arbitrage pricing model (e.g., Kraus and Ross, 1982;
Urrutia, 1987), and the option pricing model (e.g., Doherty and Garven, 1986; D’Arcy
and Garven, 1990; Phillips et al., 1998; Sherris, 2006; Ibragimov et al., 2010). Such financial
insurance pricing models have the strength that they incorporate the capital market into
insurance pricing and could provide nonarbitrage insurance pricing.
D’Arcy and Garven (1990) compared the major property–liability insurance pricing
models, including target underwriting profit margin method, total rate of return model,
capital asset pricing model (CAPM), and option pricing model (OPM), over the 60-year
period from 1926 through 1985. Their results showed that the total rate of return model
and option pricing model usually produced a better fit, but the relative goodness of fit
of the these models was not stable over time. Their results also found that the option
pricing model was particularly sensitive to changes in tax-related parameters, making it
a good tool to carefully examine the effects of taxation on underwriting profit margin and
insurance premium. Garven (1992) concluded several important practical advantages of
the option pricing model. OPM can explicitly quantify the value of insolvency risk and
the effects of underutilized tax shields.
Since the 1970s, the financial field has witnessed tremendous growth in the application
of the OPM (Campbell et al., 1997; McNeil et al., 2005). Unexceptionally OPM has received increasing attention among both insurance academia and industry practitioners
(e.g., Doherty and Garven, 1986; Cummins, 1988; Derrig, 1989; D’Arcy and Garven,
1990; Garven, 1992; Wang, 2000; Sherris 2006; Ibragimov et al., 2010). The rationale for
applying OPM in insurance pricing is that insurance policies can be viewed as a package of contingent payments depending on the insurer’s underwriting and investment
performance, and the value of the contingent payments can be estimated within the
framework of OPM.
In early insurance applications of the Black–Scholes model, many studies assumed that
insurers provide only one line of business (or viewed the total business as one single
line). For example, Merton (1977) applied the OPM to estimate the pricing of loan guarantees and deposit insurance. Doherty and Garven (1986) modeled the contingent claims
to shareholders, policyholders, and tax authorities by using European options to estimate the insurance premium and underwriting profit margin. Sommer (1996) applied
the OPM framework to measure insolvency risk and derived that insurance price was the
present value of loss claims minus the value of an insolvency put option that captured
the insolvency risk of insurer. The empirical results from his regression model supported
the hypotheses derived from the theoretical framework that insolvency risk was negatively related to insurance price.1
1
Motivated by the problems caused by the flat rate guarantee fund premium scheme, Cummins
(1988) developed a risk-based premium estimation technique for insurance guaranty funds. The
value of the insurance guaranty fund was modeled using a put option with the value of the
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More recent research in insurance pricing develops pricing models for the multiline
insurer. Yow and Sherris (2007) evaluated optimal capitalization and pricing strategies
in a single-period model of a multiline insurer, incorporating frictional costs, imperfectly competitive demand, and policyholders’ preferences for financial quality, based
on value at risk (VaR) and proportional capital allocation to line of business. They found
that VaR-based methods for determining capital and prices are not consistent with enterprise value added maximization (i.e., shareholder value maximization). In order to
be consistent with value-maximizing pricing, the effect of insurance demand elasticity
should be considered. Based on total firm value (i.e., sum of shareholders’ value and
policyholders’ value) maximization criteria, mutual insurer structure is optimal, and
mutual insurers exhibit lower default risk, higher business volume, and lower premiums. Yow and Sherris (2008) developed a single-period economic model of a multiline
insurer in an imperfect market with imperfectly elastic insurance demand, frictional cost,
and policyholder preferences for financial quality. Based on this model that analyzed the
impact of frictional costs on optimal capital, pricing, and enterprise risk management,
they found that holding an optimal level of capital reduces frictional costs and allows
profit-maximizing sales of policies due to policyholder preferences for financial quality.
Sherris (2006) and Ibragimov et al. (2010) considered the links between solvency, capital
allocation, and fair rate of return in a single-period model. Both papers allocated the
assets (or equity) based on an ex post pro rata sharing rule, i.e., based on the value of
a default option. Both papers implicitly assumed that the total asset value at the end
of the period is predetermined and not influenced by premium income. However, the
available asset at the end of the period depends on both premium income and investment
return. The premium, available assets, and default option interact with each other, and
reach equilibrium at the fair premium. At a given risk level, a high level of initial assets
reduces insolvency risk and allows an insurer to charge higher premiums; premium
income and investment income in turn determine the assets available to pay claims
and the insolvency risk. Frictional costs, such as expenses and corporate taxes, are also
important factors influencing the premium.
The purpose of this article is to study multiline insurance pricing in a way that considers
tax liability, expenses, insolvency risk, and capital allocation. Using a multiline model
developed on the basis of the single-line framework of Doherty and Garven (1986), the
interactions among available assets, premium levels, insolvency option, and tax liability
are considered. Capital is allocated in the model based on the value of default option;
the financial claims of shareholders, policyholders, and tax authorities are modeled as
European options written on the income generated by the insurer’s asset and liability
portfolio.
The remainder of the article is organized as follows. The next section develops the
theoretical model. Then a discrete state model is used to illustrate the results. Concluding
remarks follow.
insurer’s total liability being the exercise price and the insurer’s total assets being the underlying
security.
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THEORETICAL MODEL
In this section, the authors extend Doherty and Garven’s (1986) model from considering
a single line of business into a multiline model and take into account expenses and taxes
as well. The model for aggregate lines is first introduced and then followed by that for
distinct lines.
For Aggregate Lines
In the single-period model, S0 denotes shareholders’ initial capital investment, and P0
denotes the premium received from the policyholders. Net of expense, the initial cash
flow, Y0 , is expressed as:
Y0 = S0 + P0 × (1 − e),
(1)
where e represents expenses2 and is expressed as a proportion of premium.
Assume claims and corporate income taxes are paid at the end of period and investment
income is generated at rate r̃a . Before claims payment and corporate income tax, the
terminal cash flow, Ỹ1 , is:
Ỹ1 = (S0 + P0 × (1 − e)) × (1 + r̃a ).
(2)
The value of Ỹ1 is allocated to claimholders, i.e., policyholders, governments, and shareholders, in a set of payments having the characteristics of put and call options. Under
the usual bankruptcy constraint (i.e., limited liability for shareholders), the aggregate
payment to policyholders is the minimum of the total claims and the insurer’s total
assets. Assuming Ỹ1 will not be negative, the payment to policyholders H̃1 is:
H̃1 = min(Ỹ1 , L̃) = L̃ − max( L̃ − Ỹ1 , 0),
(3)
where, L̃ is the insurer’s underwriting claims cost including claims adjustment expenses.3
Corporate income taxes are assumed to be paid to the government if the insurer has
positive profit with the insurer paying zero corporate income tax if its profit is zero or
negative. The corporate income tax paid to the government, T̃1 , can be expressed as:
T̃1 = max(tC I × (Ỹ1 − S0 − L̃), 0) lim ,
δx→0
(4)
where tC I is the corporate income tax rate.
2
The expense variable here also includes taxes other than the corporate income tax such as
premium taxes, fire taxes, property taxes, etc.
3
Interestingly, the same transformation seen in Equation (3) also was used in the literature on
robust option pricing via bounds for min(Y,L) in inventory problem studies (e.g., Scarf, 1958;
Lo, 1987).
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The present values of claim payments H̃1 and tax payments T̃1 at the beginning of the
period, H0 and T0 , can be expressed by the values of put and call options:
H0 = V( L̃) − PUT[Ỹ1 , L̃],
(5)
T0 = tC I × C[Ỹ1 − S0 , L̃],
(6)
where, V(•) means the present value; PUT[A,B] is the current market value of a European
put option based on underlying assets having a terminal value of A and exercise price
of B. C[A,B] is the current market value of a European call option based on underlying
assets having a terminal value of A and exercise price of B. PUT[Ỹ1 , L̃] is the put option
value of insurer default. The value of the claim payments to policyholders is the present
value of the claim payments minus the value of default put option, since not all claims
will be honored with 100 percent certainty.
Shareholders own the residual claim, i.e., the difference between the market value of
insurer’s total assets, Ỹ1 , and the values of claims to policyholders H0 and governments
T0 . Shareholders’ value Ve can be expressed as:
Ve = V(Ỹ1 ) − H0 − T0 = V(Ỹ1 ) − (V( L̃) − PUT[Ỹ1 , L̃]) − tC I × C[Ỹ1 − S0 , L].
(7)
If the insurance premium is set at a level such that a “fair” return is delivered to shareholders, then the current market value of the shareholders’ value Ve must be equal to
the initial capital investment, S0 . The “fair” return is the internal rate of return implied
in the equilibrium relationship in the competitive capital markets. Ỹ1 is a function of P0 .
The fair premium, P0 , satisfies:
Ve = S0 .
(8)
The fair premium P0 , available assets before claim Ỹ1 , default option PUT[Ỹ1 , L̃], and
L] are intercorrelated, and depend on the investment return
tax liability tC I × C[Ỹ1 − S0 , r̃a and claim loss L̃, as well as the correlation between investment return and claim loss.
By incorporating the frictional costs, the fair premium is not just the present value of
claim loss minus the value of default put option. Instead, the fair premium is the present
value of the loss, adjusted by the value of default option and its share of expense and
tax liability.
For Distinct Lines of Insurance
Except for the historical model used at Lloyd’s of London, insurers globally are organized as corporations that are subject to limited liability. For multiline insurers, each
line of business has equal priority in the event of default. If the premium and accumulated investment income of one particular business line is insufficient to cover the
liabilities/claims from this business line, part or all of firm’s equity may be used to
make up the deficiency. However, if the total equity is not sufficient to cover the total
shortfall, the insurer defaults on the remaining loss payments. In the event of default,
the liabilities to policyholders of all business lines are ranked equally, and the amounts
that the policyholders can expect to receive are proportional to the value of the claims
134 RISK MANAGEMENT
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that they hold against the insurer, i.e., policyholders in jth business line become entitled
to a share of ω j = L̃ j/L̃ of the total assets of the insurer; where, L̃ j is the outstanding
claim amount of policyholders in jth business line and L̃ is the total outstanding claim
J
of all business lines, i.e., L̃ = i=0 L̃ j .
The value of the policyholders’ claims on the jth business line equals the amount of the
jth business line’s claims if the insurer’s total assets are greater than the total claims, or
equals a pro rata share of the total assets if the insurer’s total asset is less than the total
claims. This relationship is presented by the following formula:
⎧
⎪
⎨ L̃ j
H j1 = L̃ j
⎪
⎩
· Ỹ1
L̃
if
Ỹ1 ≥ L̃
if
Ỹ1 < L̃,
i.e.,
L̃ j
H j1 = min L̃ j ,
× Ỹ1
L̃
L̃ j
= L̃ j − max L̃ j −
× Ỹ1 , 0 .
L̃
Hence, the value of the policyholders’ claim on the jth business line at the beginning of
the period is expressed as follow:
L̃ j
H j0 = V L̃ j − PUT
× Ỹ1 , L̃ j .
L̃
L̃
The value of jth line’s default option PUT[ L̃j × Ỹ1 , L̃ j ] also depends on the correlation
between jth business line’s claim loss and the firm’s total losses.
Similarly, the corporate income tax paid by each distinct line depends on the insurer’s
total profit.
If the insurer’s total profit is positive, no matter the profit of the jth business line is
positive or negative, its tax contribution is proportional to the jth business line’s profit.
Each line’s tax liability can be positive or negative. If the insurer’s total profit is negative,
even though profit of the jth business line is positive, no tax is paid. Thus, the value of
the tax payments of the jth business line to government is:
Tj1 =
tC I × (Ỹj1 − S j0 − L̃ j )
if
Ỹ1 − S0 − L̃ ≥ 0
0
if
Ỹ1 − S0 − L̃ < 0
= tC I ×
Ỹj1 − S j0 − L̃ j
× max((Ỹ1 − S0 − L̃), 0),
Ỹ1 − S0 − L̃
where Ỹj1 is jth business line’s asset before claim and tax, and S j0 is the allocated initial
equity for jth business line.
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Ỹ −S − L̃
Let ν̃ j = Ỹj1 −Sj0− L̃ j so that the present value of the tax payments of the jth business line
1
0
to government at the beginning of the period is: Tj0 = tC I × V(ν̃ j ) × C[(Ỹ1 − S0 ), L̃].
The present value of the payment to shareholders by the jth business line is the present
value of the jth line’s assets minus the present value of the claims to the policyholders
and the taxes paid to government, i.e., V(Ỹj1 ) − H j0 − Tj0 .
At equilibrium, the “fair” premium of the jth business line, P j0 , should satisfy the
condition that the market value of jth business line to shareholders should equal the
initial capital investment in it, i.e.,
S j0 = V(Ỹj1 ) − H j0 − Tj0 .
Equity Allocation
In order to estimate the “fair” premium of the jth business line, P j0 , the initial equity to
each distinct line of business needs to be virtually allocated for the purpose of analysis.
The virtual allocation of initial equity does not affect the solvency risk of each distinct
line, which depends only on the insurer’s total assets and liabilities. Furthermore, it
should not influence the total fair premium for all lines combined. But the virtual equity
allocation does influence the fair premium, the premium-to-expected-loss ratio, and the
equity-to-expected-loss ratio for each distinct line. It is expected that those lines that
are more likely to have higher losses at a time when the firm is insolvent should have
greater equity support.
The allocation of initial equity is not unique. However, the equity allocation should
satisfy the following sharing rules: (1) the sum of the equity allocated to distinct lines
should equal the firm’s total equity; (2) in case of default, distinct lines share the deficit in
assets proportionally based on the claims; and (3) no-claim lines do not receive payment.
Based on an ex post sharing rule, an equity allocation based on the value of a default
option satisfies all the aforementioned sharing rules (as in Sherris, 2006). Ibragimov et al.
(2010) also concluded and proved that based on an ex post sharing rule, equity allocation
based on the value of a default option is the only capital allocation rule that does not lead
to redistribution between new and old insureds with marginal expansions of insurance
lines.
Upon an insurer’s insolvency, all lines become insolvent and the claims on all lines are
ranked equally; every line has the same insolvency probability and partially default on
claim payments by the same percentage. Based on the ex post deficit sharing fact, the
equity is allocated based on the value of the default option for each line.
The total claims for all lines is H̃1 = min(Ỹ1 , L̃) = L̃ − max( L̃ − Ỹ1 , 0). The present value
of the claim payment is H0 = V( L̃) − PUT[Ỹ1 , L̃], where PUT[Ỹ1 , L̃] is the value of
default option.
L̃
L̃
The claim for jth line can be written as H̃ j1 = L̃j × min(Ỹ1 , L̃) = L̃ j − L̃j × max
( L̃ − Ỹ1 , 0). The present value of the jth line claim payment is H j0 = V( L̃ j ) − PUT
L̃
L̃
[ L̃j × Ỹ1 , L̃ j ], where PUT[ L̃j × Ỹ1 , L̃ j ], is the value of default option for the jth line.
It can be seen that the values of the default options depend on factors influencing Ỹ1
(including initial equity, total premium, investment return, etc.), the total losses, the
136 RISK MANAGEMENT
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losses for each distinct line, the correlation between investment return and total losses,
the correlation between the losses of a distinct line and total losses, the total premium.
The proportion of initial equity allocated to the jth line, αj , is the ratio of the default option
L̃
value for jth line to the default option value for all lines combined, i.e.,
and αj = 1.
PUT[ L̃j ×Ỹ1 , L̃ j ]
,
PUT[Ỹ1 , L̃]
From the analysis above, the analytical results can be summarized as follows:
Implication 1: Considering frictional costs, the fair premium is not just the value of the claim
payments to policyholders (which is the present value of loss minus the value of default option).
The fair premium should also cover the loading for expense and income tax liability.
Implication 2: Initial equity allocation does not influence the total fair premium or the firm’s
overall default risk; instead, it influences the fair premium, the equity-to-expected-loss ratio and
the premium-to-expected-loss ratio for each line. These ratios are not constant across distinct
lines. The sum of premiums in distinct lines’ premiums equals the total premium collected in the
aggregate.
Implication 3: Lines of business with higher possibility and claim amount not being paid
in full have lower premium-to-expected-loss ratio, and need more capital support (i.e., higher
equity-to-expected-loss ratio) in order to reflect the higher risk that the claims may not be paid in
full.
Implication 4: At any given expected loss, the premium-to-expected-loss ratio for firms offering
multiple lines is higher than that of firms offering only a single line since the multiline operation
provides diversification at the firm level even if the risks are positively correlated (so long as
the losses from different lines are not perfectly positively correlated) thus lowering the firm’s
insolvency risk.
Implication 5: Positive correlation among losses from distinct lines leads to lower premiumto-expected-loss ratios than when losses across lines are independent. The higher insolvency risk
increases the default option value thereby reducing the premium that an insurer can charge.
Implication 6: Positive correlation between investment return and losses reduces the insolvency
risk and leads to a higher premium-to-expected-loss ratio.
PRICING EXAMPLES
To illustrate these results, we provide several simple examples. In examples I and IV, we
assume a single risky asset and single business line; in examples II and III, we assume
two lines of business. For illustrative purposes, assume:
• expense ratio = 33 percent;
• initial equity S0 = $2000;
• risk free return = 5 percent;
• corporate income tax = 34 percent.
These assumed numbers are not meant to be realistic, but to demonstrate the key results.
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TABLE 1
Investment Return and the Probability
State
Return
P-probability
Q-probability
1
−10%
0.2
0.4
2
15%
0.8
0.6
TABLE 2
Loss and its Probability
State
Loss
Q-probability
1
0
0.9
2
$10,000
0.1
E(loss) = 1,000
Example I: Single Line, Independent Investment Return and Loss
Assume the investment return has only two states as shown in Table 1, i.e., its real-world
probability, i.e. P-probability, and risk-neutral equivalent probability, i.e., Q-probability.
Similarly the loss is assumed to have only two states distributed as shown in Table
2. Loss, investment return, joint probability, and the payoffs at each state are as presented in Table 3. In state 1, the loss is zero and investment return is −10 percent; in
state 2, loss is $10,000 and investment return is −10 percent. State 3 represents zero losses
and 15 percent investment return, and state 4 represents $10,000 losses and 15 percent
investment return.
The present value of S1 after tax should equal to $2,000 based on the equilibrium condition, which produces a solution for P of $575.76. The resulting premium-to-expected-loss
ratio is 575.76/1,000 = 0.576. Here we observe that the value of the payments to policyholders, i.e., the expected loss minus the value of default option (calculated in the
table below) at 232.57, is not the same as the premium. The difference is caused by
the frictional costs, viz. expense and tax liability. The indicated fair premium should be
enough to cover not only the claims but also an insurer’s expenses and tax liability.
Based on the premium derived, the default claim payment in each state is:
State
Default Claim
1
0
2
10,000 − (1,800 + 0.603P) = 7852.81
3
0
4
10,000 − (2,300 + 0.7705P) = 7256.38
0
10,000
3
4
if Y1 < 10,000
otherwise
2,300 + 0.7705P − 10,000
otherwise
if Y1 < 10,000
0.06
0
2,300 + 0.7705P
1,800 + 0.603P − 10,000
1,800 + 0.603P
S 1 Before Tax
15%
0.54
0.04
−10%
15%
0.36
−10%
0b
(2,300 + 0.7705P − 2,000) × 0.34
0b
(1,800 + 0.603P − 2,000) × 0.34
Tax Liability
(2,000 + 0.67P) × 1.15 = 2,300 + 0.7705P
(2,000 + 0.67P) × 1.15 = 2,300 + 0.7705P
(2,000 + 0.67P) × 0.9 = 1,800 + 0.603P
(2,000 + 0.67P) × 0.9 = 1,800 + 0.603P
Y1a
0
otherwise
S 1 After Tax
otherwise
0b
2,198 + 0.50853P
0b
1,868 + 0.39798P
10,000
2,300 + 0.7705P if Y1 < 10,000
10,000
1,800 + 0.603P if Y1 < 10,000
0
Claim Payment
b
Y1 is total assets before claim payment; P is the fair premium.
Because P is expected to be around the total expected loss of 1000, S1 before taxes in state 2 and state 4 are likely to be 0. The calculated results
confirm this. For simplicity, the table displays 0 here for both the tax liability and S1 after tax.
a
4
3
2
1
0
10,000
2
0
1
Q-probability
Return
AND INSURANCE
State
L
State
TABLE 3
Investment Return, Loss, Their Joint Probability, and the Payoffs to Stakeholders for Example I (Single Line, Independent Investment
Return, and Loss)
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TABLE 4
Losses and Their Probabilities For Example II Two Independent Business Lines, Independent
Investment Return, and Losses
State
Line 1
Q-probability
1
0
0.9
2
$4,000
0.1
Line 2
Q-probability
1
0
0.95
2
$12,000
0.05
E(line 1 loss) = 400
State
E(line 2 loss) = 600
In the example, the value of the default option is 713.81.
Example II: Two Independent Business Lines, Independent Investment Return and Losses
Continue with the basic assumptions used in example I, but consider two business
lines that have expected losses that total $1,000. Further assume the losses from these
two lines are independent. The losses are assumed to each have only two states with
distributions as shown in Table 4. Losses, investment return, joint probability, and the
payoffs to stakeholders at each state are presented in Table 5.
The equilibrium condition implies that the present value of S1 after tax is equal to $2000.
From this we solve the total premium, (P1 + P2 ), to be $840.26. The resulting premiumto-loss ratio is 840.26/1000 = 0.840. In both examples I and II, the total expected loss is
$1,000; however, the total premium for the multiline case is higher. This result occurs
because the multiline operation provides diversification at the firm level and lowers the
risk of default on claim payments. The lower default risk is, in turn, reflected in the
higher premium.
To estimate the premiums for line 1 and line 2, initial equity needs to be virtually divided.
Because allocating initial equity based on the default option value satisfies the equity
sharing rules described earlier, we allocate the initial equity to each line in proportion
to the line’s default option value. Based on the total premium derived, the default
claim payments in each state for the total default and default for each line are provided
in Table 6. The default option value and the allocated initial capital are presented in
Table 7.
It is observed that the sum of default option value for line 1 and line 2 equals the firm’s
default option value. Also, the total default option for example II, 580.75, is lower than
its counterpart for example I, 713.81. This reduction occurs in example II because the
multiline insurer is more likely to pay claims and/or larger portion of claims if insolvent
because of diversification at the firm level. It is this reduced default risk that results in
the higher premium level.
0
0
4000
4000
0
0
4000
4000
2
3
4
5
6
7
8
Loss 1
1
State
12,000
0
12,000
0
12,000
0.018
0.038
0.002
−10%
−10%
−10%
15%
15%
15%
0.003
0.057
0.027
0.513
0.342
−10%
15%
Q-prob
Return
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1800 + .603(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603(P1 + P2 )
Y1 a
otherwise
otherwise
0
otherwise
otherwise
otherwise
16,000
(Continued)
otherwise
2,300 + 0.7705(P1 + P2 ) if Y1 < 16,000
4,000
2,300 + 0.7705(P1 + P2 ) if Y1 < 4,000
12,000
2,300 + 0.7705(P1 + P2 ) if Y1 < 12,000
16,000
1,800 + 0.603(P1 + P2 ) if Y1 < 16,000
4,000
1,800 + 0.603(P1 + P2 ) if Y1 < 4,000
12,000
1,800 + 0.603(P1 + P2 ) if Y1 < 12,000
0
Claim Payment
AND INSURANCE
0
12,000
0
Loss 2
TABLE 5
Losses, Investment Return, Joint Probability, and Payoffs to Stakeholders for Example II
Two Independent Business Lines, Independent Investment Return, and Losses
140 RISK MANAGEMENT
REVIEW
0
2, 300 +
4
5
b
b
8
otherwise
otherwise
1,800 + 0.603 (P1 + P2 )
0b
0b
0b
2,300 + 0.7705 (P1 + P2 )
otherwise
2,198 + 0.50853 (P1 + P2 ) if S1 > 2,000
0b
0b
0b
if S1 > 2,000
1,868 + 0.39798 (P1 + P2 )
S1 After Tax
b
Y1 is the total asset before claim payments; P1 and P2 are the fair premium for line 1 and line 2.
The sum of P1 and P2 is expected to be around the total expected loss of 1,000. Therefore, S1 before tax in state 2–4 and state 6–8 are likely to be 0.
Similarly the tax liability is expected to be zero. The calculated results confirm this. For simplicity, the table displays 0 directly rather than using
the complete expression.
a
0b
0b
7
0
0b
0b
0
otherwise
(300 + 0.7705 (P1 + P2 ) ) 0.34 if S1 > 2,000
0
6
0.7705 (P1 + P2 )
b
b
0
0b
0b
3
0b
0
0b
(−200 + 0.603 (P1 + P2 ) ) 0.34 if S1 > 2,000
Tax Liability
2
1,800 + 0.603 (P1 + P2 )
S1 Before Tax
1
State
TABLE 5
(Continued)
PRICING
FOR
MULTILINE INSURER 141
142 RISK MANAGEMENT
AND INSURANCE
REVIEW
TABLE 6
Claim Payments and the Defaults on Claim Payments for Example II
Aggregate
State
Line 1
Line 2
Claim
Default in
Claim
Default in
Claim
Default in
Payment
Payment
Payment
Payment
Payment
Payment
1
0
0
0
0
0
0
2
2306.68
9693.32
0
0
2306.68
9693.32
3
2306.68
1693.32
2306.68
1693.32
0
0
4
2306.68
13693.32
576.67
3423.33
1730.01
10269.99
5
0
0
0
0
0
0
6
2947.42
9052.58
0
0
2947.42
9052.58
7
2947.42
1052.58
2947.42
1052.58
0
0
8
2947.42
13052.58
736.86
3263.14
2210.56
9789.44
TABLE 7
Value of Default Options and Initial Capital Allocation for Example II
Value of the default option
Initial equity
Aggregate
Line 1
Line 2
580.75
134.27
446.48
2000
462.39
1537.61
The premiums for lines 1 and 2 can be derived based on the allocated initial equity and
total premium. Assets before claim (Y1 ), claim payment, equity before tax, tax liability,
and equity after tax for lines 1 and 2 are shown in Tables 8 and 9, respectively.
Based on the equilibrium condition, P1 is solved as 532.32. The premium-to-expectedloss ratio is 1.33 and the equity-to-expected-loss ratio is 1.16.
Based on the equilibrium condition, P2 is solved as 308.17. The premium-to-expectedloss ratio is 0.5136 and the equity-to-expected-loss ratio is 2.563. As shown in Table
6, line 2 will experience default on higher amounts of claims upon insolvency. This
higher default risk produces a higher default option value and implies the need for
more equity support. The higher default risk is reflected in the lower premium-toexpected-loss ratio and a higher equity-to-expected-loss ratio than was observed for
line 1.4
4
The difference between the sum of the line 1 and line 2 premiums, 840.49, and the total premium
calculated previously, 840.26, is due to rounding error.
PRICING
FOR
MULTILINE INSURER 143
TABLE 8
Premium Calculation for Line 1 in Example II
Claim
Payment
S1 Before Tax
Tax Liability
S1 After Tax
Y1 for Line 1
for Line 1
for Line 1
for Line 1
for Line 1
1
416.15 + 0.603P1
0
416.15 + 0.603P1
−16.06 + 0.20502P1
2
416.15 + 0.603P1
0
416.15 + 0.603P1
No taxa
416.15 + 0.603P1
3
416.15 + 0.603P1
2306.68
−1890.53 + 0.603P1
No tax
−1890.53 + 0.603P1
4
416.15 + 0.603P1
567.67
−160.52 + 0.603P1
No tax
−160.52 + 0.603P1
5
531.75 + 0.7705P1
0
531.75 + 0.7705P1
23.58 + 0.26197P1
6
531.75 + 0.7705P1
0
531.75 + 0.7705P1
No taxa
531.75 + 0.7705P1
7
531.75 + 0.7705P1
2947.42
−2415.7 + 0.7705P1
No tax
−2415.7 + 0.7705P1
8
531.75 + 0.7705P1
736.86
−205.11 + 0.7705P1
No tax
−205.11 + 0.7705P1
State
432.21 + 0.39798P1
508.17 + 0.50853P1
a
Even though in states 2 and 6, the equity before tax is higher than the initial allocated equity as
a result of line 1 generating a profit, there is still no tax liability because the firm’s total profit is
negative.
TABLE 9
Premium Calculation for Line 2 in Example II
Claim
State
Y1 for Line 2
Payment
S1 Before Tax
Tax Liability
S1 After Tax
for Line 2
for Line 2
for Line 2
for Line 2
1
1383.85 + 0.603P2
0
1383.85 + 0.603P2
−51.60 + 0.20502P2
2
1383.85 + 0.603P2
2306.68
−922.83 + 0.603P2
No tax
−922.83 + 0.603P2
1435.45 + 0.39798P2
3
1383.85 + 0.603P2
0
1383.85 + 0.603P2
No taxa
1383.85 + 0.603P2
4
1383.85 + 0.603P2
1730.01
−346.16 + 0.603P2
No tax
−346.16 + 0.603P2
5
1768.25 + 0.7705P2
0
1768.25 + 0.7705P2
78.42 + 0.26197P2
6
1768.25 + 0.7705P2
2947.42
−1179.11 + 0.7705P2
No tax
−1179.11 + 0.7705P2
7
1768.25 + 0.7705P2
0
1768.25 + 0.7705P2
No taxa
1768.25 + 0.7705P2
8
1768.25 + 0.7705P2
2210.56
−442.31 + 0.7705P2
No tax
−442.31 + 0.7705P2
1689.83 + 0.50853P2
a
Even though in states 3 and 7, the equity before tax is higher than the initial allocated equity as
a result of line 2 generating a profit, there is still no tax liability because the firm’s total profit is
negative.
Example III: Two Positively Related Business Lines With Correlation = 0.2294,
Independent Investment Return and Losses
Continue example II, except that the losses from line 1 and line 2 are positively correlated
with correlation of 0.2294. All other conditions are held the same. The expected total
loss from these two lines is kept at $1,000. Losses, investment return, joint probability,
and the payoffs to stakeholders at each state are presented in Table 10. In states 1–4, the
investment return is −10 percent. State 1 is with zero loss from both line 1 and line 2;
state 2 has $12,000 in losses from line 2 only, state 3 has $4,000 in losses from line 1 only;
0
0
4000
4000
0
0
4000
4000
1
2
3
4
5
6
7
8
0.032
0.008
−10%
12,000 −10%
12,000
0
12,000
0
0
15%
15%
15%
0.012
0.048
0.018
0.522
0.012
12,000 −10%
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705 (P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 1.15 = 2,300 + 0.7705 (P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603(P1 + P2 )
(2,000 + (P1 + P2 ) 0.67) 0.9 = 1,800 + 0.603 (P1 + P2 )
Y1 a
2,300 + 0.7705 (P1 + P2 )
if Y1 < 16,000
otherwise
2,300 + 0.7705 (P1 + P2 )
16,000
(Continued)
otherwise
4,000
if Y1 < 4,000
otherwise
if Y1 < 12,000
otherwise
16,000
0
otherwise
if Y1 < 16,000
4,000
1,800 + 0.603 (P1 + P2 )
1,800 + 0.603 (P1 + P2 )
otherwise
if Y1 < 4,000
12,000
if Y1 < 12,000
0
1,800 + 0.603 (P1 + P2 )
12,000
2,300 + 0.7705 (P1 + P2 )
Claim Payment
AND INSURANCE
15%
0.348
0
−10%
State Loss 1 Loss 2 Return probability
Q-
TABLE 10
Losses, Investment Return, Probability, and the Payoffs for Example III
Two Business Lines With Correlation = 0.2294, Independent Investment Return and Losses
144 RISK MANAGEMENT
REVIEW
0b
0b
2,300 + 0.7705 (P1 + P2 )
0b
0b
0b
5
6
7
8
if S1 > 2,000
otherwise
(300 + 0.7705 (P1 + P2 )) 0.34
0
otherwise
1,800 + 0.603 (P1 + P2 )
otherwise
2,300 + 0.7705 (P1 + P2 )
0b
0b
0b
if S1 > 2,000
2,198 + 0.50853 (P1 + P2 )
0b
0b
0b
if S1 > 2,000
1,868 + 0.39798 (P1 + P2 )
S1 After Tax
b
Y1 is the total asset before claim payments; P1 and P2 are the fair premium for line 1 and line 2.
The sum of P1 and P2 is expected to be around the total expected loss of 1000. So S1 before tax in states 2–4 and states 6–8 are likely to be 0. This
results in a tax liability of 0 and also a value of 0 for S1 after tax. The calculated results confirm this. For simplicity, the table displays 0 directly
rather than using the complete expression.
a
0b
0
4
0
b
b
0b
0b
3
otherwise
0b
0
0b
if S1 > 2,000
(−200 + 0.603 (P1 + P2 )) 0.34
Tax Liability
2
1,800 + 0.603 (P1 + P2 )
S1 Before Tax
1
State
TABLE 10
(Continued)
PRICING
FOR
MULTILINE INSURER 145
146 RISK MANAGEMENT
AND INSURANCE
REVIEW
TABLE 11
Claim Payments and Defaults on Claim Payments for Example III
Total
State
Line 1
Line 2
Claim
Total
Claim
Default on
Claim
Default on
Payment
Defaults
Payment
Line 1
Payment
Line 2
1
0
0
0
0
0
0
2
2251.68
9748.32
0
0
2251.68
9748.32
3
2251.68
1748.32
2251.68
1748.32
0
0
4
2251.68
13748.32
562.92
3437.08
1688.76
10311.24
5
0
0
0
0
0
0
6
2877.14
9122.86
0
0
2877.14
9122.86
7
2877.14
1122.86
2877.14
1122.86
0
0
8
2877.14
13122.86
719.29
3280.71
2157.85
9842.15
TABLE 12
Value of Default Option and Initial Capital Allocation for Example III
Value of the default option
Initial equity
Total
Line 1
Line 2
627.14
168.29
458.85
2000
536.70
1463.30
and state 4 represents $4,000 in losses from line 1 plus $12,000 in losses from line 2. The
joint loss distributions in states 5–8 are the same as those in states 1–4 correspondingly,
while the investment return is 15 percent.
Based on the equilibrium condition, the total premium, (P1 + P2 ), is solved as $749.05.
This translates into a premium-to-expected-loss ratio of 749.05/1,000 = 0.74905. This
total premium for example III (the positively correlated situation) is lower than was
generated for example II with no correlation. The positive loss correlation increases the
risk of default on claim payments, which is reflected in a lower premium charged for a
given expected loss. Compared to example I, the single risk situation, the premium for
example III is still higher, since the risk of default is reduced below that of a single risks,
i.e., some diversification effect presents even when risks are positively (but imperfectly)
correlated.
Similarly, based on the total premium derived, the default claim payment in each state
for the total default and default for each line are provided in Table 11. The default option
value and the allocated initial capital are presented in Table 12.
The total value of the default option for example III, 627.14, is higher than that of example
II with its two independent lines (580.75) and is lower than that of example I with its
single line of business (713.81). Even with positive correlation, diversification at the firm
PRICING
FOR
MULTILINE INSURER 147
TABLE 13
Premium Calculation for Line 1 in Example III
Claim
Payment
S1 Before Tax
Tax Liability
S1 After Tax
Y1 for Line 1
for Line 1
for Line 1
for Line 1
for Line 1
1
483.03 + 0.603P1
0
483.03 + 0.603P1
−18.25 + 0.20502P1
2
483.03 + 0.603P1
0
483.03 + 0.603P1
No taxa
483.03 + 0.603P1
3
483.03 + 0.603P1
2251.68
−1768.65 + 0.603P1
No tax
−1768.65 + 0.603P1
4
483.03 + 0.603P1
562.92
−79.89 + 0.603P1
No tax
−79.89 + 0.603P1
5
617.21 + 0.7705P1
0
617.21 + 0.7705P1
27.37 + 0.26197P1
589.84 + 0.50853P1
6
617.21 + 0.7705P1
0
617.21 + 0.7705P1
No taxa
617.21 + 0.7705P1
7
617.21 + 0.7705P1
2877.14
−2259.93 + 0.7705P1
No tax
−2259.93 + 0.7705P1
8
617.21 + 0.7705P1
719.29
−102.08 + 0.7705P1
No tax
−102.08 + 0.7705P1
State
501.28 + 0.39798P1
a
Even though in state 2 and 6, the equity before tax is higher than the initial allocated equity,
which means line 1 generates profit in state 2 and 6, there is still not tax liability since the firm’s
total profit is negative and thus does not have tax liability in these states.
TABLE 14
Premium Calculation for Line 2 in Example III
Claim
Payment
S1 Before Tax
Tax Liability
S1 After Tax
Y1 for Line 2
for Line 2
for Line 2
for Line 2
for Line 2
1
1316.97 + 0.603P2
0
1316.97 + 0.603P2
−49.75 + 0.20502P2
2
1316.97 + 0.603P2
2251.68
−934.71 + 0.603P2
No tax
3
1316.97 + 0.603P2
0
1316.97 + 0.603P2
No taxa
1316.97 + 0.603P2
4
1316.97 + 0.603P2
1688.76
−371.79 + 0.603P2
No tax
−371.79 + 0.603P2
5
1682.80 + 0.7705P2
0
6
1682.80 + 0.7705P2
2877.14
−1194.34 + 0.7705P2
No tax
7
1682.80 + 0.7705P2
0
1682.80 + 0.7705P2
No taxa
1682.80 + 0.7705P2
8
1682.80 + 0.7705P2
2157.86
−475.06 + 0.7705P2
No tax
−475.06 + 0.7705P2
State
1682.80 + 0.7705P2
74.63 + 0.26197P2
1366.72 + 0.39798P2
−934.71 + 0.603P2
1608.17 + 0.50853P2
−1194.34 + 0.7705P2
a
In states 3 and 7, the equity before tax is higher than the initial allocated equity. Even though
line 2 generates profit in those states, the firm’s total profit is negative and thus does not have tax
liability in these states.
level reduces the default risk though that reduction is not as great as when lines are
independent.
The premiums for line 1 and line 2 are derived based on the allocated initial equity and
total premium. Assets before claims (Y1 ), claim payments, equity before tax, tax liability,
and equity after tax for line 1 and 2 are shown in Tables 13 and 14, respectively.
Based on equilibrium condition, P1 is solved as 466.74. The premium-to-expected-loss
ratio is 1.16 and the equity-to-expected-loss ratio is 1.34.
0.08
(2,000 + 0.67P) × 1.15 = 2300 + 0.7705P
(2,000 + 0.67P) × 1.15 = 2300 + 0.7705P
if Y1 < 10,000
otherwise
0
2,300 + 0.7705P − 10,000
2300 + 0.7705P
otherwise
1,800 + 0.603P − 10,000
0b
(2300 + 0.7705P − 2000) × 0.34
0b
1,800 + 0.603P
10,000
0
otherwise
if Y1 < 10,000
otherwise
if Y1 < 10,000
0b
2198 + 0.50853P
0b
1,868 + 0.39798P
S1 After Tax
2,300 + 0.770P
10,000
0
Claim Payment
b
Y1 is the total asset before claim payments; P is the fair premium.
P is expected to be around the total expected loss of 1000. So S1 before tax in state 2 and state 4 are likely to be 0. As a result the tax liability and
S1 after tax are also 0. The calculated results confirm this. For simplicity, the table displays 0 directly rather than using the complete expression.
a
4
3
2
if Y1 < 10,000
0
(1,800 + 0.603P − 2,000) × 0.34
15%
0.52
1,800 + 0.603P
10,000
4
15%
(2000 + 0.67P) × 0.9 = 1800 + 0.603P
1
0
3
0.02
−10%
(2,000 + 0.67P) × 0.9 = 1,800 + 0.603P
Tax Liability
10,000
2
0.38
−10%
Y1 a
S1 Before Tax
0
1
Qprobability
Return
AND INSURANCE
State
L
State
TABLE 15
Investment Return and Loss, Joint Probability, and Payoffs to Stakeholders for Example IV
Single Line, Positively Related Investment Return, and Loss With Correlation = 0.272
148 RISK MANAGEMENT
REVIEW
PRICING
FOR
MULTILINE INSURER 149
TABLE 16
Default Claim Payments for Example IV
State
Default Claim
1
0
2
10,000 − (1,800 + 0.603P) = 7841.39
3
0
4
10,000 − (2,300 + 0.7705P) = 7241.78
Based on the equilibrium condition, P2 is solved as 282.29. The premium-to-expectedloss ratio is 0.47 and the equity-to-expected-loss ratio is 2.44. Line 2 has higher claim
amounts that will be subject to default upon insolvency as shown in Table 11. This higher
default risk results in a higher default option value and implies a need for more equity
support. This is reflected in the lower premium-to-expected-loss ratio and higher equityto-expected-loss ratio than those for line 1. The sum of the line 1 and line 2 premiums,
749.04, is the same as the aggregate premium as calculated in Table 10.
Example IV: Single Line, Positively Correlated Investment Return and With Correlation =
0.272
The correlation between investment return and claims is also an important factor influencing the fair premium. Positive correlation between investment return and losses
reduces the chance of insolvency, thus leading to a lower premium-to-expected-loss
ratio. To reflect the relationship, here we use a single line as our example. Continue
example I, except that loss and investment return are positively correlated, with correlation equals to 0.272. Loss, investment return, joint probability, and the payoffs at each
state is presented in Table 15.
Based on the equilibrium condition, P is solved to be 594.71. The premium-to-loss ratio
is 594.71/1000 = 0.595. Since positive correlation between investment return and losses
reduces insolvency risk, an insurer in such an environment is able to charge higher
premium than where these factors are independent (example I); there the comparable
premium is 575.76. Based on the premium derived, the default claim payment in each
state is shown in Table 16. The value of the default option is 701.11, which is lower than
the value of the default option in example I, e.g., 713.81.
Similarly, Ibragimov et al. (2008) found that the benefit of diversification can be achieved
only in a market with a large number of risks, where risks are relatively homogeneous,
and where risks are not highly correlated. A multiline industry structure is optimal
in such a market, as shown in our examples. However, in a market that has limited
number of risks and risks are heavy tailed and correlated, such as catastrophe line, a
monoline industry structure is more efficient despite the fact that it exhibits a higher
default option value. The intuition is that a multiline insurer with readily diversifiable
risks will only accept a catastrophe line if the capital loading on it is high enough to
ensure the insurer’s expected default rate remains unchanged. This implies charging a
relatively high premium rate on the catastrophe line. In this case, the insured seeking
coverage in the catastrophe line may be better off to obtain that coverage from monoline
150 RISK MANAGEMENT
AND INSURANCE
REVIEW
TABLE 17
Results Summary
Example II
Example III
Example
Example
I
Expected loss
Aggregate Line 1 Line 2 Aggregate Line 1 Line 2
400
600
1000
400
600
IV
1000
1000
Premium
575.76
840.26
532.32 308.17
749.05
466.74 282.29
594.71
1000
Default option
713.81
580.75
134.27 446.48
627.14
168.29 458.85
701.11
0.58
0.84
1.33
0.51
0.75
1.16
0.47
0.6
2
2
1.16
2.56
2
1.34
2.44
2
value
Premium/expected
loss
Equity/expected
loss
Notes:
Example I: Single line, independent investment return and loss.
Example II: Two independent lines, independent investment return and losses.
Example III: Two positively correlated lines with correlation = 0.2294, independent investment
return and loss.
Example IV: Single line, positively correlated investment return and loss with correlation = 0.272.
insurer that holders less capital and offers lower premium rate, albeit with a higher
default risk.
CONCLUSION
This article has extended the single-line framework of Doherty and Garven (1986) to
multiline insurance pricing and to consider tax liability, expenses, and capital allocation.
Considering the frictional costs (i.e., expenses and tax liability), the fair premium is not
just the amount of the difference between the value of expected loss and the value of the
default option; instead it should be higher than this by an amount sufficient to also cover
the insurer’s expense and tax liability. At a given initial equity, premium level, available
assets, and insolvency risk are inter-correlated and should be considered concurrently
in the pricing. The available assets should not be simply assumed as given, since they
depend on the premium income.
The theoretical and analytical results display the interaction between available assets,
premium, insolvency option, and tax liability as well as the capital allocated to a line of
business based on the value of the default risk modeled as European options. Table 17
summarizes the results in the four examples. Those results show that:
1. Equity allocation does not influence the overall fair premium; the sum of distinct
lines’ premiums equals the total premium calculated aggregately.
2. Premium-to-expected-loss ratio and equity-to-expected-loss ratio vary across lines
and the capital is allocated according to the line’s share of default option to reflect
the line’s equal status in insurer’s default. Lines with higher possibility/claim
PRICING
FOR
MULTILINE INSURER 151
amount not being paid in full have lower premium-to-expected-loss ratio and
higher equity-to-expected-loss ratio.
3. At the given expected loss, the premium-to-expected-loss ratio for firms offering
multiple lines is higher than for firms only offering a single line due to the reduced
risk through diversification.
4. Positive correlation of losses in different lines of business results in a lower
premium-to-expected-loss ratio than when losses are independent across lines.
5. Positive correlation between investment return and losses reduces the insolvency
risk and leads to a higher premium-to-expected-loss ratio.
Conditional on the correlation among losses not being extremely high, our results show
conclusively that diversification across business lines reduces default risk and supports a
higher fair premium in the marketplace. Correlation between the claims across multiples
lines of business or correlation between claims and investment returns also influences
the value of the respective options, the fair premium, and affects capital requirements.
The success of any financial model, including ours, relies on the accuracy of the estimations regarding the target parameters. Further study is needed to estimate the distributions of the important parameters in insurance pricing and to better understand the
sensitivity of the premium to variables such as the loss distribution, correlation between
losses, correlation between losses and investment return, and investment return, etc.
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