QUIZ #3
random experiment
-A group of related items -the outcome is unknowable in advance. -the result will be determined by chance.
lifetime immediate annuities
-An annuity provides a guaranteed payment for as long as its beneficiary (owner) is alive. -The present value of the annuity depends on the probability of survival. -The present value of the anuity depends on the age of the beneficiary (owner).
Insurance premiums (& how they are determined)
-Depends on probability and severity of loss. In an efficient market, this would equal the prog-weighted EV of the loss.
Main inefficiencies in the insurance market
-Moral hazard- insured person engages in more dangerous behavior b/c they have insurance to cover their losses. Ex. People who have car insurance might drive more recklessly. -Adverse selection- Customers know they are likely to incur the losses for which they are buying insurance. Ex. Sick people buying health insurance face high health care costs, charged higher premiums b/c they face chronic conditions.
True facts about life insurance premium
-The fair premium amount is the actuarial present value of the benefit amount. -The premium increases with the age of the insured. -The premium increases with the amount of the benefit. -The premium increases with the term of the policy.
Explain what Gompertz equation used for and identify/explain each parameter in the function.
-The natural log of the prob of an indiv surviving for a number of years from his or her current age. -used when a payment in the future depends on the prob of a person surviving to the age at which the payment is made (or received). ln(p(survival x,t)) = (1-e^t/b) e^(x-m)/b x = age now in years t = years of survival for which to find the prob m = modal age of death within the pop (in years) b = dispersion coefficient of human life (in years) undo the ln to solve for the prob (e^whatever val you got for ln)
In general, who needs life insurance?
-Young parents with dependent children. -Someone with a non-working spouse.
Discrete probability (explain and ex)
-the prob when the outcomes in a sample space are countable (discrete). ex. in a coin toss: p(heads) = 1/2 when 1 is prob of event occurring and 2 is # of outcomes in sample space
The probability of an event is
-the quotient of the number of outcomes that are part of the event and the number of outcomes in the sample space. -the measure of how often we expect the event is to occur in a random experiment -the prob of any event must be between 0% and 100%. -the sum of the probs of all outcomes in a sample space is 100%. -the prob of 0% indicates an imposisble event.
Multiplication rule (explain using ex)
-uses multiplication to create the product of the sample spaces of the indiv experiments. ex. if one coin toss has the sample space of (h,t) then two coin tosses will have the sample space of {(h,t) x (h,t)} = {(h,h),(h,t),(t,h),(t,t)}
Explain and identify the two main types of life insurance (identify key differences!)
1) permanent life insurance - provides a death benefit irrespective of death date as long as the policy has been paid in full, the death benefit is a future value. date received is unknown, though the amount to be received is known. 2) term life insurance - a pure insurance product; the insured pays a premium each year and the policy pays a death benefit in the event of death during the year of coverage. sold with terms of 10,20,even 30 years so the insured can continue to pay the same premium for each year of the term to maintain coverage.
Consider a simple game which involves rolling a single die. It costs $10 to play. Here are the payouts for each value of the die: Roll a 1, get $1 Roll a 2, get $3 Roll a 3, get $5 Roll a 4, get $7 Roll a 5, get $9 Roll a 6, get $20 What is the expected value for playing this game one time?
=7.50 (1/6 x 1) + ... + (1/6x20)
Deductibles
A form of risk-retention that requires the insured person to pay a portion of any losses. Ex. auto insurance has a 500 dollar deductible, you pay the first 500 in damages before insurance chips in.
trial
A single attempt an an experiment.
Consider a game of chance that involves flipping a coin twice. To win this game, you need to get the same result from the coin toss both times. Which of the following describes the event for winning this game?
E = {(t,t), (h,h)}
Are lottery tickets ever worth their cost? Make an argument using concepts of prob and expected val.
Even if you think you have a positive expected value due to the size of the jackpot being larger than the number of possible numbers, as more tickets are purchased (and the jackpot grows larger) the odds of someone else picking the winner goes up and your EV goes down. --> On average, negative EV.
event
One or more outcomes from an experiment.
Explain using an example: mortality rate, survival rate, survival prob, and actuarial pv.
Prob of a 21 year old surviving to age 80. ln(p(survival21,59)=(1-e^t/b)e^(x-m)/b = (1-e^59/11) x e^(21-83)/11 = -0.75 p(survival)=e^-0.75=0.4687 present val of 1000000 to be received in 59 years, discounted at a rate of 3% per year. PV = 1000000/(1.03^59) = 1000000/5.72 = $174825.08 actuarial pv = 46.87% x 174825.08 = 81,945.25
Consider a game of chance that involves flipping a coin twice. To win this game, you need to get the same result from the coin toss both times. Which of the following is the sample space of this game?
S = {(t,t), (t,h), (h,t), (h,h)}
deductible
The amount of any loss that is retained by the insurance purchaser.
loss
The amount paid out by an insurance policy.
Write a conceptual explanation for the premium of term life insurance. Discuss the use of the survival prob, mortality rate, and conditional mortality rate used in this calculation.
The cost of life insurance depends on the survival prob and mortality rate. The expected benefit in any year is a function of the product of the benefit amount and the conditional mortality rate at that age. premium = p(death) x death benefit amount conditional prob = one random event depends on another random event (death at age 75 depends on living from age 45 to 75)
actuarial present value
The expected value in the present of an amount to be received (or paid) in the future that depends on an uncertain event. -The Gompertz Mortality equation is used when a payment in the future depends on the prob of a person surviving to the age at which the payment is made (or received).
term
The length of time for which the insurance remains in effect.
Gompertz equation actually calculates
The natural log of the prob of an indiv surviving for a number of years from his or her current age.
death rate
The percentage of a population or cohort that dies within a certain time period.
expected value
The probability-weighted outcome of a risky process. N over sigma over e=1 . to the right of the sigma is psubscripte x payoutsubscripte = p1 x payout 1 x + ... + pn x payout n
mortality rate
The probabilty of death within in a certain time period. -are asymptotic to 1 -Benjamin Gompertz discovered within a given population, the rate of change in the natural log of mortality rates is fairly constant across all ages.
outcome
The result of a random experiment.
sample space
The set of all possible outcomes from an experiment.
All other things considered equal, an 80-year old has a greater chance of living to 100 than a 22-year-old.
True (already lived to age 80, high prob of that continuing)
Concept of insurance (explain using ex)
a risk management contract b/w an insurance company and the insured. in an insurance policy, the insured pays a premium in exchange for the insurer to reimburse specific losses. typically has a limited period of coverage.
What is an independent event? explain and ex
events that have no bearing on one another ex. the outcome of a coin toss has no impact on the outcome of rolling a die (they are independent)
cohort
in regards to survival prob, the group of people who were born in the same year
Joint probability (explain using ex)
prob of two events occurring together --> ex. find prob of drawing a red 5 out of a deck of shuffled cards. p(red 7 card) = p(red card) x p(7 card) = 1/2 x 1/13 = 1/26
survival probability
the chance of an individual not dying within a certain time period
The multiplication rule applies to calculating probabilities when
the event requires an outcome that comes from multiple domains or sets, or from repeated trials at a random experiment.
modal age of death
the most commonly occuring age of death within a population.
premium
the price that a consumer pays for an insurance policy.