Long-Term Actuarial Mathematics I
Chapter 1 –
Introduction to Life Insurance
MATH 3630 | Fall 2020
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Background
❑ Primary responsibility of the insurance company actuary is to maintain the solvency and profitability of the
insurer. They’ve been depended on since the early eighteenth century. Responsibilities include:
▪ Identifying and managing risk
▪ Setting appropriate premiums given the risks involved
▪ Determining the level of assets to set aside to meet future obligations (i.e. quantify liabilities)
▪ Assist in managing assets
❑ The modeling of mortality and other contingent risks (morbidity, etc.) is a critical aspect of modeling insurance
liabilities and mortality modeling will be an important area of focus throughout this course
❑ Important terms:
▪ Premium – amount paid by the policyholder to the insurer, typically at the beginning of the period
▪ Sum Insured – amount paid, contingent on the death of the insured life, to the policyholder or his/her
estate
▪ Assessmentism – matching income (policyholder premiums) and outgo (expected death claims) annually
without any attempt to balance the premiums
• This led to increasing premiums (due to increasing mortality), thereby discouraging policyholder
renewals
• This methodology continues to be used for group life insurance
▪ Insurable Interest – requires a policyholder to face financial loss when the insured life dies (i.e. the
beneficiary should not be better off if the insured dies)
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Traditional Life Insurance Contracts
❑ There are three traditional forms of life insurance, all of which will be heavily covered in this
course –
▪ Term life insurance
▪ Whole life insurance
▪ Endowment Insurance
❑ These dominated insurance markets until the 1980s
❑ These three remain popular today in various situations, but the design of insurance contracts
has greatly expanded and increased in complexity since these products dominated
❑ Technical advances and knowledgeable customers have promoted sophisticated products
which combine investment savings with insurance needs
▪ (i.e. universal life insurance with different indices and variable sub-accounts, etc.)
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Term Insurance
❑ Pays a lump sum benefit on the death of a policyholder, provided death occurs before the end of a specified
term. Coverage ends if premium is not paid as scheduled. Provides limited flexibility to the policyholder.
❑ Premiums tend to be small relative to the sum insured given that the probability of payout is lower
▪ Consider a 10-year term insurance contract, issued to a 40-year old male. The probability that a benefit is
paid is equivalent to the probability that the 40-year old dies within 10 years (small)
❑ Term insurance most often used for family protection
▪ Protect against financial hardship: replace lost income, pay off a mortgage, complete college funding, etc.
❑ Term also used by companies for key personnel whose death would lead to financial harm (Key Person/COLI)
❑ The most common type of term policy offers a level face amount with level premiums for a specific term period
(10/15/20/30 years) followed by annually increasing premiums
▪ Jump in premium following level term period is significant
❑ Other types of term insurance –
▪ Decreasing – death benefit decreases over term of policy
▪ Renewable – gives policyholder the option to renew their policy at end of original term without further
underwriting (subject to maximum age)
▪ Convertible – gives policyholder the option to convert to permanent insurance with no additional
underwriting (subject to maximum age and duration)
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Whole Life Insurance
❑ Pays a lump sum benefit on the death of the policyholder when it occurs
▪ Coverage ends if premium is not paid as scheduled. Limited flexibility.
❑ Premiums are payable to some maximum age (typically between 80-90 years old)
❑ Whole life premiums > Term life premiums (why?)
❑ If a whole life policyholder decides to discontinue the policy after an initial period (i.e.
lapse/surrender their policy), they receive a cash value/surrender benefit, representing the
investment part of the accumulated premiums (Standard Non-Forfeiture Law)
❑ Life insurance combines premiums and investment income (earned through premium
investment), such that, by the time the policyholder dies, the premiums + investment income
can fund the sum insured due at death, cover expenses, and return a profit
▪ As the duration of a contract increases, the investment risk does as well
▪ Requires strong assumption setting for investment returns and mortality
▪ The difference between the interest assumption used in pricing and the interest earned
by the insurer is known as the interest spread – contributes to profit/loss experience
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Participating Insurance
❑ Also known as ‘par insurance’ and ‘with-profit’ insurance
❑ Charge policyholders a higher premium, but promise to pay back the policyholder some
share of the profits if actual experience (mortality, expense, investment income) is better than
assumed in pricing.
❑ The share of profits that is distributed to policyholders is known as dividends or bonuses
▪ Dividends – distributed as cash or cash equivalents (i.e. premium reduction)
▪ Bonuses – distributed as additional insurance (i.e. increase face amount)
❑ Many different variants – common for policyholders to be given some choice about the
distribution
❑ Reversionary Bonus – applied to contracts in force, increasing the benefits by a specific
percentage (simple, compound, super-compound)
❑ Terminal Bonus – used to top up the sum insured when the benefit is finally paid
❑ Separating profit shares into these two types of bonus gives insurers more flexibility and
allows them to exercise caution with distributing unrealized capital gains
❑ Cannot be any negative dividends/bonuses – only profits are shared, never losses
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Endowment Insurance
❑ Pays a lump sum benefit either on the death of the policyholder, or at the end of a specified
term, whichever occurs first
▪ Hybrid of term insurance and a fixed term investment (i.e. zero-coupon bond)
▪ Guarantees a payout (assuming no lapses)
❑ No longer offered throughout North America & UK
❑ Main purpose was as an investment, but lack of flexibility and low returns deem it
uncompetitive with other investment products
❑ Evolved into modern insurance/investment products such as Universal Life insurance
❑ Increasing in popularity in developing countries, particularly in the area of microinsurance
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Variations on Traditional Insurance
❑ Joint life insurance – two lives insured and benefits payable on first death or second death
❑ Multiple life insurance – offers benefit to be payable on first death or on each death
❑ Guaranteed cash values – allows policyholder to lock in guaranteed cash values
❑ Policy loans – allows policyholder to borrow from insurer and use cash value as a form of collateral
▪ Commonly used to pay premiums as a method of funding. During claim status, benefit is reduced
by outstanding loan balance (principal and interest).
❑ Accelerated death benefit – pays benefit early if policyholder has terminal illness
❑ Accidental death benefit – increases sum insured if the cause of death is accidental in exchange for
additional premium (Double Indemnity – 1944)
❑ Premium waiver on disability – allows policyholders to suspend premiums in case of disability or
severe illness
❑ Family income benefit – allows specified amount to be paid at regular intervals between the
policyholder’s death and the end of the original contract term
❑ Critical illness insurance – benefit is paid on diagnosis of one of a specific set of critical illnesses or
disabilities (cancer, stroke, and heart disease)
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Modern Insurance Contracts
❑ Modern Insurance vs. traditional policies
▪ More complexity and options
▪ More transparent (costs & benefits)
▪ More flexibility of premiums and variability with benefits
❑ Insurers are constantly innovating product designs due to
▪ Competition with mutual funds and banks for policyholders’ savings
▪ Accommodate the evolving needs of policyholders
• Financial planning needs, tax law changes, risk appetite
▪ Developments in technology, financial management tools
▪ Increased policyholder sophistication
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Universal Life Insurance
❑ Generally issued as a permanent product with transparent cash values, allowing policyholders to view
their contract as a form of savings account with built-in insurance.
❑ Policyholders choose a level of death benefit, which may be fixed or increase with cash value growth
or additional premium payments
❑ Typically, premiums are flexible, not fixed (unlike traditional insurance contracts). Coverage
continues as long as the account value remains positive.
❑ Premiums are deposited into a notional account (held within the company’s general account, unless it
is a variable product)
❑ The account value (the notional account) at any time is equal to:
▪ the premiums paid plus credited interest, reduced by monthly deductions related to mortality and
expenses
▪ If the policy is surrendered prior to the end of the surrender charge period, the AV is reduced by
applicable surrender charges to recover initial issue costs incurred by the insurer
❑ Credited interest may be based on rates declared by the insurer that typically involve guaranteed
minimums (Fixed UL) or based on the performance of various equity/fixed income sub accounts
(Variable UL).
❑ Insurer profit reflects the level of mortality, investment, expense margins
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Unitized with Profit
❑ Popular in the UK and Australia during the 1980s and 1990s. Similar to Universal Life.
❑ Policyholder funds are expressed as units (as opposed to account values)
▪ Units represent shares in a notional asset portfolio and increase in value based on the
performance of the underlying investments
❑ Bad publicity led to this product being withdrawn in early 2000s, but companies will need to
continue holding reserves for these products for the foreseeable future
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Equity Linked Insurance
❑ Has an endowment insurance structure, with benefits payable at the earlier of the
policyholder’s death and the end of the contract term
❑ Death, surrender, and maturity benefits linked to performance of an actual specified fund
(separate account assets), not a notional fund
❑ Equity linked insurance benefits vary over time (can increase or decrease) in line with the
fund’s performance
❑ In the United States, variable annuities fall under this category
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Distribution Methods
❑ Brokers are individuals who connect individuals to an appropriate insurance product
▪ Given how complex insurance is, there needs to be an intermediary between the
prospective policyholder and insurance product
▪ ‘insurance is sold not bought’
❑ Paid on commission, typically specified as a percentage of the premium paid
▪ Typically, there is a higher percentage paid on the first premium than on subsequent
premiums – referred to as front-end load
▪ Agents can also be paid based on a fixed fee system or on a salary basis (less common)
❑ Direct marketing – insurance sold through television advertising or online sales
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Underwriting
❑ Underwriting – process of collecting and evaluating information that classifies a policyholder into a
specific risk category
❑ Life insurance applicants typically categorized by age/gender/smoking status
❑ Life insurance applicants will typically be categorized into one of the following sub-groups
▪ Preferred lives – good health status, good family history, low mortality risk
▪ Normal lives – insurable at standard rates with some higher rated risk factors
▪ Rated lives – unacceptable at standard rates, but can be insured at higher premiums
▪ Uninsurable lives – significant risk and insurer will likely not engage
❑ Adverse selection – arises when policyholders use information abut their own individual risk profile
to make choices that will benefit them at the expense of the insurer
▪ Underwriting helps to prevent adverse selection
▪ There are tradeoffs between the amount of adverse selection and the degree of underwriting
❑ Distribution method normally impacts product features available and type/level of underwriting
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Premiums
❑ Fixed (Traditional) or Flexible (UL)
❑ Single premium or periodic
❑ May have an end date
❑ Premium modality (annual, semi-annual, quarterly, monthly)
❑ Premiums paid at the start of each period
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Life Annuities
❑ Annuity contracts – regular series of payments that are typically used as retirement
instruments to combat longevity risk
▪ In the life insurance industry, these types of annuities are referred to as ‘payout
annuities’
❑ Life annuities depend on the survival of the recipient
❑ Whole life annuity – annuity payments continue until death of the annuitant
❑ Term life annuity – annuity payments continue for a specified period, or until death of the
annuitant, whichever is earlier
❑ Annuities cannot be surrendered as there is no cash value once the annuity payments
commence
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Types of Annuities
❑ Single Premium Deferred Annuity (SPDA) – a single premium is paid in exchange for a
series of annuity payments starting at some future date
❑ Single Premium Immediate Annuity (SPIA) – a single premium is paid in exchange for a
series of annuity payments that begin immediately
❑ Regular Premium Deferred Annuity (RPDA) – regular premiums are paid during the
deferral period in exchange for a series of annuity payments that begin following the deferral
period
❑ Joint Life Annuity – annuity issued on two lives which continues payments while both
individuals are alive and ceases at the first death
❑ Last Survivor Annuity – annuity issued on two lives which continues payments while at
least one individual is alive and ceases when both lives die
❑ Reversionary Annuity – annuity issued on two lives, where one is designated as the insured
and one as the annuitant. No benefit is paid while both individuals are alive; the annuitant
receives their annuity benefits when the insured dies
❑ Guaranteed Annuity – paid for a minimum period, regardless of the survival/death status of
the annuitant. Following the guaranteed period, the annuity is paid for the remainder of the
annuitant’s life
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Long Term Coverage in Health Insurance
❑ Disability Income Insurance – designed to replace income for individuals who cannot work
at full capacity due to sickness or disability
▪ Premiums paid at regular intervals but typically suspended during periods of disability
▪ Benefits typically capped at 50-70% of replaced salary to discourage abuse
▪ Benefits start after a defined waiting/elimination period
❑ Long-Term Care Insurance – designed to help cover the cost of assisted living
▪ Payment is triggered when policyholder requires assistance to perform two or more
activities of daily living (ADLs)
❑ Critical Illness Insurance – pays a lump sum benefit on diagnosis of one of a list of specified
diseases and conditions
▪ Once the claim arises, the benefit is paid and the policy expires
❑ Chronic Illness Insurance – pays a benefit on diagnosis of a chronic illness, defined as one
from which the policyholder will not recover (illness does not need to be terminal)
▪ Illness needs to be sufficiently severe such that policyholder cannot perform two or
more ADLs
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Types of Insurers
❑ Mutual Insurance Company – no shareholders; owned by with-profit policyholders
▪ Profits are distributed to the with-profit policyholders through dividends/bonuses
❑ Proprietary Insurance Company – has shareholders that own the company
▪ Also has with-profit policyholders (not owners but do have some profit sharing)
❑ There have been trends towards demutualization (i.e. transitioning the mutual insurer to a
proprietary insurer)
❑ Demutualizing to a proprietary insurance company allows the insurer to more easily raise
capital, improve their efficiency, and provide a clearer corporate structure
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Other Life Contingent Contracts
❑ Continuing care retirement communities (CCRCs) – residential facilities for seniors that
provide different levels of support designed to adapt to the residents as they age
▪ There are various forms of funding options, mainly through the combination of an entry
fee and a monthly charge
❑ Structured Settlement – payment schedule agreed between an injured party and the
responsible person
▪ This is typically arranged through lawyers or through an insurer when the responsible
party’s liability is covered by an insurance policy
▪ Often used for payments under workers’ compensation
▪ Often involves assessment of impaired life mortality
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Pensions
❑ Pensions plans have similar risks to life insurance contracts and require the expertise of an
actuary for plan design, valuation and risk management
❑ Defined Benefit pension plans – provides retiree with lifetime retirement income which is
formulaically based on years of service and salary
❑ Defined Contribution pension plans – works like a bank account where contributions
(percentage of salary) are paid by an employee and employer into a fund which grows with
interest