Ch 69 - Non mathematical topics Flashcards
Reasons why design of life insurance contracts has changed recently? (5)
SVCCR = “Service credit”
- Savings or investment component added to new types of products
- Varying premiums and benefits, desired by policyholders
- Competition
- Computers - ALlowing more complex products
- Risk management techniques
Types of UW classes? (4)
Preferred
Standard
Rated
Uninsured
Applications of Profit testing? (5)
PRMSD
- Set premiums
- Set reserves
- Measure profitability
- Stress test profitability
- Determine distributable surplus
Reasons for offering pension plans? (6)
CRUTIT
- Competition for new EEs
- Reward EEs for long service
- Union pressure
- Tax-efficient compensation
- Incentive to stay with employer
- Turnover of older employees
Why Normal cost using PUC is LESS than Normal cost using TUC?
1) Under both funding approaches, contribution rate INCREASES as member acquires more service and gets closer to retirement
- Both TUC and PUC only consider service until the date of valuation
2) TUC contributions start SMALLER than PUC, then rise more steeply, ending at considerably MORE than PUC contribution
- Because TUC contributions DON’T project future salary increases
- Whereas PUC contributions are based on salary with projected future salary increases
3) So TUC contributions in any given year must reflect:
- the FULL INCREASE in salary, and
- the additional year of service now reflected in the accrued benefit at the end of the year of service (which were NOT reflected in the accrued benefit at the end of year of service)
4) The PUC contribution will NOT need to reflect the additional year of service, since it was already reflected in the accrued liability at the beginning of the year.
When to use the normal approximation?
Use when the underlying distribution is approximately normal. Ex: binomial r.v. with a high N.
When a policy pays the reserve upon death:
The reserve =
+ accumulation of premiums with interest ONLY (not survivorship)
- accumulation of benefits (other than return of reserve) with interest only (not survivorship)
= Retrospective!
Net premium definition
Premium determined by equivalence principle
Gross premium definition
Actual premium paid by policyhokder
Modified reserve
-First year Valuation premium under FPT?
bvq_x
Remember:
FPT policy = 1 year term insurance + insurance issued to a life 1 year older
Asset share, definition?
A measure of the accumulation of cash income, per surviving policy
Profit vector, definition
Profit assuming policy is inforce at BOY
Pr_t = Profit for year t for a policy inforce at BOY
Profit signature, definition
Profit assuming policy is inforce at issue (t = 0)
Rate of salary function, definition?
Index of instaneously earned salary
Salary scale, definition?
Index of salary earned during the year of age (x, x+1)
Normal cost, definition?
Portion of the cost of projected benefits allocated to the
current plan year
Difference between homogeneous and non-homogeneous markov chain
1) Homogeneous: only 1 transition matrix needed for all periods
2) Non-Homogeneous: DIFFERENT transition matrix needed for EACH period
Compare and contrast gross premium reserve and asset share
1) Gross premium reserve
- Represents the amount needed to cover future benefits and expenses, so it is prospective
- Accounting principle
2) Asset share
- The asset share is the amount of cash accumulated in the past, so it is retrospective;
- Realistic/Actual
Risks in assuming a flat yield curve?
- Influence of the yield curve on long-term contracts may not be great since the yield curve tends to flatten out
- Common in actuarial practice to use the long-term rate in traditional actuarial calculation
- Using the long-term rate OVERSTATES interest when the yield curve is RISING. This leads to actual premiums LOWER than the true premiums.
- As such, an insurer that systematically charges premiums LESS than the true price may face solvency problems in time.
Explain replicating cash flow?
the PV of any cash flow is the cost of purchasing a portfolio which exactly replicates the cash flow.
Give an example of how replicating cash flow is applied to how an insurance company prices annuities.
For a large block of traditional insurance (or annuity) policies with independent future lifetimes, if:
- Premiums are determined using the equivalence principle;
- Premiums and benefits are completely predictable;
- Net cash flows are invested at the forward rates implied by risk-free bonds;
Then the company will always have exactly what it needs to pay its outflows. Therefore, there is NO RISK.
Explain the implication of diversification of mortality risk.
Because mortality risk is diversifiable, if an insurer issues a large number of policies across all products:
-Then it can assume that the ACTUAL claims experienced will MATCH the EXPECTED.
Pension math: Accrual rate, definition?
Percentage of the benefit basis that is paid annually as a pension for each year of service.
List two types of pension plans and explain who assumes the investment risk in each plan
- Defined benefit (DB) – employer assumes the investment risk because it has to invest now for pension payments that will occur in the future
- Defined Contribution (DC) – employee assumes the investment risk because her retirement benefit depends on the performance of the retirement fund
Reversionary annuity , definition?
An annuity that makes regular payments to one status after another status has failed
Explain the difference between MDT and ADT?
1) Multiple decrement model = dependent rates
- transition probabilities depend on ALL forces
2) Associated single decrement = independent rates
- each transition probability depends only on the force of THAT transition
Explain why setting a reserve is important?
ex: setting a contract with level premiums?
- A life insurance contract may be funded by level premiums, but the cost of mortality usually increases as the insured ages.
- As a result, the INSURED OVERPAYS in the EARLIER years and UNDERPAYS in the LATER years.
- The company must save the overpayments of the early years in order to pay the benefits of the later years/
Why are expense reserves negative?
- Because expenses are usually front-loaded = paid at the BEGINNING
- Usually: First year expenses are more than the expense loading
- SO: AV(expenses) > AV(expense premiums)
Explain what the portfolio principle is?
Method to set the premium so that the probability of a future loss of the PORTFOLIO is below a certain level.
Explain equivalence principle?
Expected present value of the premiums is set equal to the expected present value of the insurance company’s payments.
Explain select mortality? give an example?
- Two people : the first one was selected randomly, whereas the second one had recently purchased a life insurance policy.
- Would the mortality rate for both of these, q60 be the same? NO.
- The second person was underwritten for a life insurance policy, which means his medical situation was reviewed. He had to satisfy certain guidelines regarding weight, blood pressure, blood lipids, family history, existing medical conditions, and possibly even driving record and credit history.
- The fact he was approved for an insurance policy implies that his MORTALITY RATE is LOWER than that of a randomly selected 60-year-old male.
explain what is a force of mortality?
Rate of mortality at age x, given survival to age x
How does UDD affect the rate of mortality within a year?
i. Assuming a uniform distribution of death, The # of lives dying in any amount of time is CONSTANT
1. Since there are fewer and fewer lives as you progress throughout the year,
2. The RATE of death INCREASES over the year
NPV definition?
- Present value of the projected emerging surplus values.
- Cash flows are discounted using the hurdle rate = risk-free rate
Discount Payback period (DPP) definition?
Break-even period = time until the insurer starts to make a profit on the contract.
Replacement ratio?
total benefit in year after retirement/salary in year prior to retirement
Note: If it’s m-thly payments, its the total of all the payments in 1 year, WITHOUT INTEREST!!!!
Zeroized reserves
-What is it, and why is it used?
Goal is to have reserves that make it so that:
- No negative profits in LATER years; and
- Allow profit to emerge as soon as possible
= INCREASE NPV!!!
Zeroization of reserves, algorithm?
- Start with the last year and solve for the reserve such that the profit for the year is zero.
- Working backwards continue this process for each year.
- If a reserve is negative, then set the reserve to zero.
- Perform profit test using these new zeroized reserves.
Explain the usefulness of FPT
- High first year acquisition expenses, aka the 1st year commission
- So the insurer indirectly adjusts the first year reserve to be lower so that he has more cash in hand to pay for the acquisition expenses.
- This way, since FPT V_1 = V_0 = 0, you don’t have the extra expense of setting up the reserve in addition to all the issue expenses
- The insurer “mechanically” adjusts the reserve to be lower to allow more funds to finance the important acquisition expenses of the first year.
True or False:
-For FPT reserve, we don’t include expenses in the calculation.
TRUE
When calculating FPT Reserve
-Do NOT include expenses, because it is a NET PREMIUM RESERVE METHOD!!!
Explain why interest rate risk is not diversifiable.
- The risk is not diversifiable because the interest scenario applies to all policies identically; all policies experience the same interest rate scenario.
- It is not diversified by increasing the number of policies.
L* = 7000 policyholders purchase 1 policy, 200 policyholders purchase 5 policies
L = 8000 policyholders purchase 1 policy
Compare the variance of L and L*
The variance of L will be SMALLER than the variance of L*
- L∗ is less diversified than L, with the same total sum insured spread around fewer policyholders. Since diversification reduces the relative uncertainty, less diversification means greater uncertainty, as confirmed by the larger 90th percentile.
- The 5 policies owned by a same person do NOT have the independent future lifetimes.
i = 6%
hurdle rate = 10%
If the pre-contract expenses increase by 10% of the premium and the commissions at the start of year 1 decrease by 10% of the premium,
-State whether the NPV will increase, decrease, or remain unchanged.
A numerical calculation of the change is not required.
The total expenses are unchanged, and the timing of the payments is effectively unchanged.
- However, for expenses deemed paid at the start of year 1 we allow for interest at 6%, and then discount at 10%.
- For pre-contract expenses there is NO INTEREST OR DISCOUNTING
- The overall effect is that moving expenses from beginning year 1 to pre-contract will DECREASE THE NPV!!
Death benefit INCREASE, based on the change from whole life to term, WITHOUT increase in premium.
-How would you account for this?
- A policyholder with compromised mortality (eg impaired health) might choose this option as a cheap way to increase the payout.
- To allow for ADVERSE SELECTION, the insurer would likely assume HIGHER MORTALITY
FPT reserve and premium
-Premiums in years 2, 3, etc. if there’s a decreasing death benefit?
- For FPT, the premiums are ONLY different in year 1
- For years after that, EVEN THOUGH the Benefit could be increasing/decreasing, the premiums are IMPLIED to be LEVEL!!
Portfolio principle and diversification
- Under the portfolio approach, ON AVERAGE, each policy makes a profit.
- Although the probability of profit for each policy is less than X, the large number of diversified policies means that, with probability X:
- The GAINS on the policies where the PROFIT is POSITIVE will be GREATER than the LOSSES on the other policies
For any policy, why would Negative profits in years 2, 3+ be a bad sign?
- It looks bad if investors want to invest in your company
- It looks like you have a product that is losing money
- Year 1 profits: it’s understandable if it’s negative bcz of acquisition expenses, UW expenses, etc.
- But by Year 2, you should generally be getting ONLY positive profits
- Reserves should be established such that policies are expected to be self financing after the initial outlay associated with acquiring the new business
Life annuity -> after a couple years, person doesn’t need the annuity. Someone asks if they can get the reserve.
-Q: Why would the insurance company NOT want to pay the full reserve?
Note: traditional life annuities CANNOT BE SURRENDERED!
- The lives whose health is bad and who therefore face higher mortality in the near future would withdraw their funds, reducing the pool available to survivors.
- the reserve formula doesn’t take into account that some people are sicker than others
- so in this situation, sick ppl withdrawing their funds would raise the cost of annuities for the healthy people
Explain what a diversifiable risk is.
A risk is diversifiable if adding more units of risk (n) reduces the standard deviation of the mean loss per contract, with a limit of zero as the number tends to infinity.
Explain what a modified reserve is (in general)/
- Reserve computed without expenses but:
- Adjusting the VALUATION premiums to allow IMPLICITLY for initial expenses.
Valuation premiums = all the premiums OTHER than the INITIAL premium
3 necessary and sufficient conditions that a survival function for a lifetime disribution must satisfy in order to be valid?
- S_x(0) = 1
- [lim t-> infinity] S_x(t) = 0
- S_x(t) must be a NON-INCREASING fct of t
- the 1st derivative must be NEGATIVE
-S_x(t) >= S_x(u) for u > t
Negative reserves -> why does the insurer want to avoid them?
positive Reserve = liability that insurer owes to insured
negative reserve = liability that insured owes to insurer
-but the customer can terminate the contract at any time (lapse) and escape paying this accrued liability
Using portfolio principle for a number “n” of policyholders.
Now: we believe that future lifetimes may not be independent.
What would happen if there is a correlation coefficient of 0.05 between each pair of future lifetimes on the minimum number of policies needed to reach a target such that
-the prob that aggregate loss is positive is less than y% of the time?
For a fixed number of policies, a positive correlation coefficient would increase the variance of the aggregate loss at issue.
Var(x+y) = Var(x) + Var(y) + 2*Cov(x,y)
Therefore, it would INCREASE the number of policies needed such that the aggregate loss at issue would be positive less than y% of the time.
For a policy with expenses, the gross net premium is 1247
-A marketing officer states that using the normal approximation, we can increase the # of policies sold to charge a lower premium of 1200 while maintaining a 5% probability of loss on the group of policies.
Why is this FALSE???
- In general, you can increase the number of policies to lower the gross annual premium due to diversification of mortality risk (deviation from expected value).
- BUT the LOWEST it can go to is the NET premium determined by the Equivalence principle.
- As n goes to infinity, the GROSS premium on a policy will tend towards the NET premium!!!
You are given that at i = 6%
- Duration of WL annuity = 7.039
- Duration of WL annuity with 10-yr guarantee = 6.977
if i = 5%, which one would be higher
REMINDER
Duration
= average horizon of an investment
=price sensitivity of an investment to interest-rate changes
if i INCREASES by 1% (so now 7%):
-> investment DECREASES in value by 7.039
if i DECREASES by 1% (so now 5%)
-> investment INCREASES in value by 7.039
so in this case, the WL annuity will have a HIGHER value. (all else equal)
Continuous Markov model
-If there are Constant transition forces/intensities
-Why is net premium reserve requrired for an insured always 0 in the healthy state?
The reserve for state 0 is zero throughout the term of the policy because
- the premium payment rate equals the expected benefit payment rate at any given time t.
- The rates are equal at any given time t because of the CONSTANT FORCES OF TRANSITION in this model
True or False:
-For Markov Model, when calculating reserves for a certain state, you can calculate them retrospectively
FALSE
-For Markov Model, the net premium reserves are ONLY PROSPECTIVE
-ONLY WAY TO CALCULATE is:
EPV(future ben) - EPV( future prem)
- REMEMBER: it’s a Markov Chain
- So everything that happens before DOESNT MATTER!!. The only thing that does matter is what state you are currently in!!!
FPT method
The FPT method adjusts NET PREMIUM reserves to APPROXIMATE GROSS PREMIUM RESERVES by implicitly assuming that the first-year premium is spent on:
- cost of insurance and
- acquisition expenses.
It reduces the initial strain on surplus from new business.
a) For a continuous Markov Chain, what is the effect of using together:
- Kolmogorov Forward equations
- and Euler’s Method ?
b) What is the point of using Euler?
a) Turns a continuous markov Chain into a discrete markov chain with time intervals of h.
b) We need to use a numerical technique because there are an infinite number of ways of going to terminal state
ex: to go from sick at age 30 to dead between ages 30.1 and 30.2. You can go directly from sick-dead, sick-healthy-dead, sick-healthy-sick-dead, sick-healthy-sick-healthy-dead, and so on
