Download Pricing Insurance Policies: The Internal Rate of Return Model

Pricing Insurance Policies: The Internal Rate of Return Model
Section 1: Introduction
There are three main motivators pushing for more accurate pricing models:
1) The time value of money: Newer models consider both the magnitudes and the dates of cash transactions
2) Competition and expected returns: Optimal price for most products is determined by production costs and
competitive constraints, but complex insurance products require an a priori analysis of expected costs and
achievable returns.
3) The rate base: Underwriting profit margin is a return on sales. Alternative bases are assets, and equity (used in most
financial pricing models)
A wide divergence between assumed profit margins and actual experience arises from regulatory disapproval of requested
rate revisions. Actuaries respond with new, more sophisticated pricing techniques. Expansion of “open competition” laws,
and the replacement of advisory rates by loss costs forces us to determine appropriate profit provisions..
IRR models have many variations. They change continually, adapting to changes in tax laws, insurance regulation, and
financial theory.
Insurance transactions can be viewed from 2 points of view:
1) Insurer & Policyholder: Policyholder pays premiums, obligating insurer to compensate for incurred losses.
Transactions occur in the product market, prices are influenced by supply and demand.
2) Equity Provider & Insurer: Shareholders invest funds. Investment provides a return. Transactions occur in the
financial market, and expected returns are influenced by the risks of insurance operations.
Since supply depends on the costs insurers pay to obtain capital, and expected returns are influenced by the risks of insurance
operations, which depends on demand, the 2 points of view are clearly related. The IRR model uses (2). Traditional
ratemaking is done from (1). Profits were related to premiums and losses, the capital structure was ignored. Financial
pricing models relate profits to assets or equity. The insurance cash flows are important only to the point that they affect the
transaction between the company and its stockholders.
The IRR model determines premium rates by comparing the IRR with the opportunity cost of capital. The decision rule is to
accept an investment opportunity which offers a rate of return in excess of the opportunity cost of capital. In capital
budgeting decisions, IRR analyses are used to value investments that require an initial outlay of capital, which promised
increased revenues in the future. In insurance, the policy issuance gives an immediate inflow of cash, but obligates the
insurer for future losses. So, from the view of the investors, the insurance operation is similar to other capital budgeting
efforts.
From the viewpoint of the investors, there’s a “cash outflow” at the inception of the policy and “cash inflows” as the policy
expires and losses are paid/
Aspects of insurance operations reflecting this perspective:
1) Part of the premium is used to pay acquisition, underwriting, and admin expenses. The rest is invested in financial
securities
2) Insurance companies “commit surplus” to support their writings, to insure the company has enough capital to
withstand unexpected losses.
Cash transactions provide an inflow of funds at inception, and an outflow as losses are paid (from the insurers perspective).
Investors must provide funds to allow the insurer to write, so there’s a net cash outflow at inception from investors. In future
years, as the policy expires, losses are paid, and the surplus is “freed”, it will be returned to the investors.
What equity is needed to support both the surplus account and other insurance operations? In insurance operations, there’s
not an intrinsic relationship between surplus and premium writings. Regulators set minimum capital levels, but they don’t
tell us the “right” surplus commitment level. The use of IRR models doesn’t depend on any particular assumptions about
market efficient, since appropriate surplus levels can be set in other ways. However, both the amount of surplus and the
timing of its commitment affect the equity flows and the IRR.
Often, surplus is allocated in proportion to loss reserves or premium writings. The procedure used is important, because the
average lag between premiums and losses is different depending on the allocation based. An association of surplus with
reserves attributes more surplus to long tailed lines than an association with premium does.
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Both the allocation of surplus to line and the IRR depend on the pattern of equity flows. In most cases, the more surplus
allocated to a policy, the lower that policies IRR.
Simple Equity Flow Example:
Premium:
Claim 1:
Claim 2:
$2,000
$1,250
$750
2.5:1
10%
1/1/2008
$2,000
$2,000
$800
$2,800
$3,080
1/1/2009
$1,250
$750
$300
$1,830
$780
$1,050
$1,155
1/1/2010
$750
$0
$0
$405
$405
$0
on Jan 1, 2008
on Jan 1, 2009
on Jan 1, 2010
Undiscounted Reserve to Surplus Ratio
Return on financial investments
Premium Collected
Reserve set up (First as UEPR, then Loss)
Required Surplus (Initial Outlay)
Combined Premium & Surplus invested
Assets at Year End
IRR Equation:
800=780v+405v^2
Time
1
2
3
Claim Paid
Loss Reserve
Required Surplus
Assets
IRR
Flows
($800)
$780
$405
v=1/(1+r)
r = IRR
35%
Compare the IRR to the Cost of Equity
Capital
Returned to Equityholders
Remaining Assets to be invested
Assets at Year End
Claim Paid
Loss Reserve
Required Surplus
Assets
Return to Equityholders
Remaining
Actual IRR models are more complex
Section 2: Cash Flows
The assumption that premium is collected at policy inception isn’t valid anymore, due to retro rated policies and other newer
products. A pricing model must examine both premium collection and loss payment patterns:
1) Industry-wide loss payment patterns by line of business may be determined from Schedule P, but industry-wide
premium collection patterns aren’t available, and few insurers even track their own.
2) Loss payment patterns are stable from year to year, while premium payment patterns vary widely, since they depend
on the payment plans offered.
We can determine premium payment patterns for each block only after discussions with the underwriting, audit, and billing
personnel of our company.
If premium is earned evenly over the year, it will be matched with an accident year loss payment pattern. If we assume that
policies are written evenly over the year, the premium collection pattern must be matched with a policy year loss payment
pattern.
Most commercial lines insurance is distributed by independent agents, who sometimes hold the premiums received for a
month or two before forwarding to the insurer. The magnitude of this delay depends on the distribution system used by the
insurer, and affects the premium collection pattern.
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The investment income earned on the “float” between premium collection and loss payment is one more reason to use a
pricing model that accounts for the time value of money.
Factors that cause the lag between occurrence and payment of claim:
1) Loss Adjustment
2) Periodic Payments
3) Delay in Reporting
4) Litigation
Most insurers collect data on loss payment patterns. Alternatively, we can use industry patterns, complied from Schedule P
data.
Loss adjustment expenses are a large part of insurance costs in several lines. Cash flow patterns for losses, ALAE, and
ULAE are different. ALAE is paid more slowly than losses. ULAE is incurred mainly when the claim is first reported. The
schedule P formula for paid ULAE uses 2 assumptions:
1) Half the ULAE is paid when the claim is reported, the other half when it is settled
2) 90% of claims are reported during the calendar year of occurred, the balance in the subsequent year
Traditional ratemaking procedures don’t focus much on expense costs. A target loss ratio is set as the complement of the
expense ratio. We should analyze expenses as closely as we do premiums and losses. The reasons we can’t:
1) Data: Few companies monitor expense payment patterns, as most keep expense data only on a calendar year basis
2) Company: Expenses levels vary widely by company. Expense levels and payment patterns depend on the
company’s distribution system
3) Risk Size: Traditional procedures assume that costs vary directly with premium, which isn’t true for overhead
expenses. In general, overhead expenses are a smaller percentage of premiums for larger risks.
4) Policy Year: Underwriting expenses, other acquisition costs, and commissions for direct writers are higher in the
first policy year than in subsequent years. Long term expense costs depend on policyholder persistency.
The expense levels and cash flow patterns should depend on company characteristics, size of the risk, and new versus
renewal underwriting.
Types of expenses:
1) State premium taxes are paid quarterly, as a percentage of the previous year’s tax liability. If the volume of business
isn’t changing, the premium ax cash flows should reflect the written premium pattern. Premium taxes and special
assessments vary by state
2) The commission payment pattern follows the premium collection pattern
3) Other acquisition expenses are difficult to model, mainly because the expenses are related to blocks of business, not
to specific policies. In addition, these expenses are incurred and paid several months before policy issuance.
4) General (Underwriting and admin) expenses have a mixed payment pattern. Many are incurred primarily when a
policy is first written, so they vary by year since inception
The cash flow pattern for “total expenses” is a weighted average (by expense level) of the patterns for each expense item.
The investment rate of return is not the same as the internal rate of return (IRR). The IRR relates to the transactions of the
equity holders, and the investment rate relates to the insurer’s transactions.
Financial holdings corresponding to the premium, loss, and LAE reserves are needed to pay claims covered by the policy.
The remaining assets (surplus) support the insurance operations. The financial holdings can’t (necessarily) be used to pay
stockholder dividends or to finance other activities.
There are two “rates of return” in the IRR model. The IRR is earned by equity holders for supplying funds to the insurance
company, while the investment rate of return is earned by the insurance company for supplying funds to the stock or bond
markets. If these returns are expressed in nominal dollars, both will vary with economic inflation. If they’re expressed in
real terms, there is still a connection. The IRR varies with the riskiness of insurance operations, because if the insurer invests
in high risk securities, the investment return will be high, but so will the cost of equity capital (the IRR demanded by equity
holders).
Federal income taxes depend on the magnitude and incidence of the firm’s earnings and on the accounting method of the
company. Relationships between tax provisions and other actuarial assumptions:
1) Revenue Offset: IRS assumes a 20% DAC expense ratio for all lines, when actual ratio varies by company and line
of business
2) Discount Rates
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3) Surplus Commitments: Were equity holders provided no return, they’d prefer to invest funds in the capital markets
directly, rather than face double taxation. To induce investment in the company, insurers use earnings form
insurance operations to compensate equity holders for tax on capital
Federal income taxes are a complex part of financial pricing models. We must consider the relationship between the tax
effects of the insurance transactions and the insurer’s overall tax situation.
Section 3: Surplus
Financial pricing models calculate a return on the capital used by a firm to furnish goods or services. The insurance contract
is a promise, a commitment to compensate policyholders if certain contingent events occur. Surplus backs this promise,
without the financial support, an insurer could not fulfill the obligation.
The surplus needed by an insurer, and its allocation to lines of business and time periods, is a theoretical number. We can
assume a relationship between surplus commitment and insurance transactions, but we might not be able to test it.
IRR models used for capital budgeting decisions examine the funds invested by a particular firm, and the returns expected by
the firm. If it invests less funds, it may generate less revenue. For the insurance pricing model, the assumption of capital
markets efficiency implies that the industry is neither over- nor under-capitalized. Any given insurer may have more or less
surplus than it needs. Insurers writing identical lines of business have diverse levels of insurance leverage.
The IRR model concentrates on the capital market, while the prices of insurance contracts are determined in the product
market, by the supply of, and demand for, insurance policies. At the industry level, the two markets are intertwined.
Insurance rates may be adequate, but a firm’s internal rate of return may be depressed by operating inefficiencies. Some
analysts determine the surplus commitments on an industry-wide practice. The surplus actually held is ignored, and replaced
with the required surplus calculated by the actuary.
Surplus exists for a company as a whole, not for individual lines. Whether it can be allocated is disputed, but it’s required for
the IRR model. The pricing actuary must make surplus allocation assumptions. Actuaries allocate surplus in proportion to
another base. The timing of the surplus allocation is equally important, when is it committed, and when is it freed? Should
the surplus allocation depend on the type of policy?
Ratio of written premium to policyholders’ surplus is a common test of surplus adequacy, that can be extended to the
allocation issue. This would imply that:
1) Required surplus varies directly with premium
2) Surplus is committed when the premium is written, and it is released when the policy expires
An alternate base is the reserves held for the block of business. This implies:
1) Required surplus varies directly with reserves
2) If the allocation base is Loss & Expense Reserves, then surplus is committed when the losses occur, and it is
released when the losses are paid
3) If the base includes the loss portion of the UEPR in addition, then surplus is committed when the policy is written,
and it is released when the losses are paid.
Increasing the surplus allocated to a line moves the internal rate of return closer to the after-tax investment yield. Increasing
the surplus decreases the internal rate of return.
Slow paying lines with large loss reserves receive a greater allocation of surplus (if reserves are used as the base). The
surplus is committed for longer periods in the slow paying lines.
Surplus protects the insurer against several types of risk:
 Asset Risk: The risk that financial assets will depreciate
 Pricing Risk: Risk that at expiration, incurred losses and expenses will be greater than expected
 Reserving Risk: Risk that loss reserves will not cover ultimate loss payments
 Asset-Liability Mismatch: Risk that changes in interest rates will affect the market value of certain assets differently
than that of liabilities
 Catastrophe Risk: Risk that unforeseen losses will depress the return realized by the insurer
 Reinsurance Risk: Risk that reinsurance recoverable will not be collected
 Credit Risk: Risk that agents will not remit premium balances or that insureds will not remit accrued retrospective
premiums
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Pricing and catastrophe risk occur during the policy period other risks continue until all losses are paid
If reserves are used as the base, what reserves? Loss and LAE reserves support claims that have already occurred, whether or
not they have been reported to the insurer. UEPR reserves (less the “equity” reflecting acquisition and u/w expenses) support
the claims anticipated over the policy term.
Surplus is committed to guard against the risks to the insurance company. It protects policyholders from adverse fluctuations
in loss payments that might threaten the insurer’s solvency. Claims made forms eliminate much of the IBNR liability, which
reduces the loss uncertainty and the surplus needed to support the risk.
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Service contracts involve no insurance risk
Retrospective Rate policies are more risky than service contracts, but less risky than a prospectively rated policy.
Risks vary by policy type, but there’s not a simple procedure to account for this. Equal treatment of all policy types
overstates the risk on retro policies and understates the risk on excess coverage. Considering only premium and reserves
which cover true insurance risk understates the risk for retrospectively rated policies, excess coverage, and large deductible
policies, since fluctuations are greater on higher layers of coverage. The ideal procedure is somewhere in the middle.
Section 4: The Cost of Equity Capital
The insurer has a financial incentive to write a policy if the IRR exceeds the opportunity cost of the equity capital. What
return do the investors demand for the use of their money?
If the market is efficient, returns achieved by investors is an indicator of the returns needed to elicit equity capital. If the
returns that investors achieved were less than what they demanded, they would withdraw capital. If the returns that the
investors achieved were greater than demanded, they would commit additional capital.
When there’s greater industry capacity, more firms, and increased competition, the market causes premium rates to fall,
underwriting standards to loosen, and overall returns to decline.
If OTHER factors are changing, the returns demanded by investors may not be equal to the returns achieved in the past.
There are 2 main models for estimating the return on stockholder supplied equity. They are:
1) Dividend Growth Model (DGM), which directly estimates the cost of equity capital, but relies on uncertain
projections about future dividend payments
2) Capital Asset Pricing Model (CAPM) which relies on observed historical data, but has “questionable theoretical
foundations”
A stock has 3 differences from a bond:
1) Stock never matures, there are periodic dividends, but no repayment of principal
2) The dividend payments are less certain
3) Bond coupons have fixed amounts; stock dividends are not fixed in nominal terms, but generally grow with
monetary inflation.
The actual future dividends are uncertain, but we can use historical experience to forecast them. If we know the current
price, and we can forecast future dividends, we can solve for the discount rate.
For simplicity, we assume that the firms’ earnings, assets, dividends, and stock price are all increasing at a constant rate.
This growth rate, together with the dividend to price ratio, determines the cost of equity capital.
The dividend growth model states:
𝐷
1+𝐺 +𝐺
𝑃
D = stockholder dividend (end of previous year)
G = anticipated (uniform) growth rate of stockholders equities
𝐾=
K = cost of equity capital
P = stock price @ beginning of year
Both parameters of the dividend growth model (the ratio of stockholder dividend to stock price and the anticipated dividend
growth rate) are calculated or projected by investment firms for the major publicly traded stock companies.
The anticipated dividend growth rate is a subjective estimate. The growth rate of the firm is often inversely related to the
dividend yield:
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

Growth stocks pay low dividends
Income stocks pay higher dividends, but grow more slowly
A sustained reduction in growth may lead to an increased dividend yield. The insurer can retain less of its earnings and pay
greater stockholder dividends. Analysts who foresee a reduction in the growth rate often anticipate an increase in the
dividend yield.
Cost of capital estimates derived from stock prices and dividends can be only done for publicly traded companies. Mutuals
and privates cannot be included in a dividend growth model analysis. If insurance risk differs by industry segment and line
of business, the opportunity cost of capital may differ as well.
The IRR model needs the cost of surplus funds, which are needed by all insurers. The cost of equity capital is needed for the
financial management of publicly traded stock companies. The IRR model assumes that the “cost of surplus funds” for
mutual and privately owned insurers is similar to the cost of equity capital of stock insurers.
If inflation accelerates suddenly, the economy enters a recession, or the firm’s book of business changes rapidly, the
Dividend Growth Model may not provide reasonable forecasts. In this case, actuaries look for a relationship between the cost
of equity capital and some steady and accessible index. The CAPM provides this relationship.
The CAPM assumes that there are two influences on common stock fluctuation:
1) Peculiar to the specific firm (firm specific, diversifiable)
2) The movement in the stock market as a whole (systematic, nondiversifiable)
CAPM assumes:
1) Expected return from a common stock is related only to the stock’s systematic risk
2) Difference between the expected return from a common stock and return on a risk free security is proportional the
the firms systematic risk
3) The systematic risk and the factor of proportionality are relatively constant over time.
𝑅 = 𝑅𝑓 + 𝛽(𝑅𝑚 − 𝑅𝑓 )
The Beta parameters reflect systematic risk, and are estimated from historical data
Section 5: A Rate Filing Illustration (Misc. Observations)
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Workers’ Compensation rates can be stated on three bases:
o Manual Premiums: Prices determined from bureau or company rate manuals
o Standard Premiums: Manual premium with adjustment for experience rating modifications
o Net Premiums: Standard Premiums after adjustment for premium discounts and retro rating
Premium taxes are state specific
Financial pricing models incorporate investment income by projecting future yields on invested assets.
The IRR model compares the IRR with the cost of equity capital. Since the cost of capital reflects the investment
strategy, so to should the IRR
Statutory reserves are greater than economically adequate reserves, since:
o The loss reserves are undiscounted
o The UEPR has no offset for DAC
o Some assets are not admitted
Characteristics of Tax liability that complicate the model
o Statutory reserves in the (NCCI) IRR model are not discounted, but IRS taxable income uses discounted
reserves
o The discount rate and loss payout pattern prescribed by the IRS don’t always match our selections
o The (NCCI) IRR model uses policy year cash flows; the IRS discounting procedures prescribe accident
year cash flows
Because federal income taxes are based on discounted reserves, but statutory income is based on undiscounted
reserves, taxes are paid before the net income is booked.
Net cash flow from underwriting = premium flow net of reserves + tax credit – expenses – dividends.
Assets support 2 items on the liability side:
o Premium & Loss Reserves
o Policyholders’ Surplus
Section 6: Potential Pitfalls in IRR analysis
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General Criticisms
IRR analyses determine the interest rate that sets the PV of cash inflows equal to the PV of cash outflows. If this rate
exceeds the cost of capital, the project may be accepted; if it doesn’t, the project should be rejected. NPV analyses use the
cost of capital to discount all cash flows to the same time. If the sum of the discounted values is positive, the project may be
accepted. These may be thought of as inverse functions. In the IRR model, the implied discount rate is a continuous function
of the NPV, and in the NPV model, the NPV is a continuous function of the discount rate. The same relationship between
discount rate and NPV underlies both models. For most decisions, the two methods give the same answer.
One goal of capital budgeting is to optimize the net worth. Some say that NPV analyses do this, while IRR doesn’t.
Cash flow patterns affect the number of positive solutions to the internal rate of return equations. If the project involves an
outflow followed by an inflow followed by another outflow, there may be two real roots. Various methods have been
proposed to solve this.
In most cases, however, sign reversals in the projected cash flows result from inaccuracies or oversimplification, not true
reversals in expected cash flows. Main oversimplifications can be inflows stemming from prepaid expenses and tax credits.
The inaccuracy due to this simplification isn’t material to the analysis as a whole, but it does cause a sign reversal. To be
more precise, we could spread the first year’s tax credit in the same proportion as the prepaid expenses.
Expected cash flows in insurance pricing models generally show a single sign reversal. Additional reversals indicate
oversimplifications. If these are material, we should reexamine our projections.
An unanticipated IBNR loss may cause a cash outflow from investors in the middle of a stream of inflows. NPV methods
can be used to both estimated expected profitability prospectively and to retrospectively determine the actual results. IRR
analyses on a single policy are better for projecting estimates than for determining results.
The IRR model derives the rate of return, whereas the NPV model determines the dollar return. Some of the issues
mentioned with regard to IRR models apply to NPV models as well.
Projects under consideration may be mutually exclusive, due to aggregate budget constraints, or if both projects do the same
thing. NPV and IRR analyses don’t always rank the projects the same way. This leads to another criticism of the IRR
method:
If project A’s IRR is 28%, and project B’s is 25%, this only means that project A is preferable if the revenue received in the
first year can be invested at 28%. In reality, the firms opportunity cost is the return that the firm’s owners can receive on
their funds. The IRR method incorrectly mixes the interest rate that equates real values of inflows and outflows with the
interest rate at which the firm can invest the funds it receives. This criticism may not apply for the insurance example:
 The model is often used to set statewide rates, not to price individual policies. As long as an insurer can grow at the
IRR, and maintain the same quality of risks, the IRR assumptions are correct.
 When the pricing model is used to determine the profit provision, we select a premium rate that equalizes the IRR
and the cost of capital.
Practical Criticisms
IRR decision rule is that if the IRR is less than the cost of equity capital, reject the project. Some regulators may say that a
low rate of return, while undesirable, is still allowing the insurer to make money.
In reality, both the NPV and the IRR analyses may show that a project is unprofitable. The former discounts the cash flows
at the cost of capital, and will show a negative NPV. The latter may show a positive, but inadequate return. Clearly, the
NPV is more convincing of the true profitability of the project in this case.
In the insurance model, as costs increase, the IRR subsides slowly, and may stay positive even as premiums become more
and more inadequate. The problem is the base used. The “required surplus” is an assumption made by the actuary. If it’s
determined by a “reserves to surplus” ratio, then as costs increase, so do surplus and investment income. The total return
may be inadequate, but the added investment income offsets some of the underwriting loss, and the IRR declines more slowly
with increasing loss.
In these cases, the implied equity flow assumptions aren’t reasonable. As costs rise but rates are depressed, equity holders
won’t provide more capital, they’ll provide less, if they provide any at all. When the contract is clearly unprofitable, we
should show the negative expected NPV and the insufficiency of net premium + investment income to fund the losses. We
may still use the IRR analysis internally.
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