# Math Help - Help! Markov chain and the Equivalence principle?

1. ## Help! Markov chain and the Equivalence principle?

Hi there.

For one of my statistics courses, I have been given a question about "Equivalence principle" in regard to life insurance. For this question, I have been given a transition probability matrix/Markov chain with state space S={0,1,2,3} whereby 0 = healthy, 1 = ill, 2 = death and 3 = cancellation of insurance plan. 'a' is for the premium the person has to pay. From my understanding of Equivalence principle, this term is defined as the expected value of the premium is equal to the expected value of the pay out from an insurance company, which must be equal to zero. However, generally, how do I go about much the premium will be? I am not an actuary student, so I need someone to explain to me about how Equivalence principle works.

Cheers, your help is very much appreciated.

Edit (extra info - sorry, should have had added this in before)

'a' is for the premium the person has to pay if he is not at all sick during the year. (1-b)a is what the premium is reduced to if the person gets sick, whereby b is the fraction of the year the person is sick, and the person will get $10,000b annual payment when sick. 2. If you are interested, this is not how premiums on life contracts are actually set. Life contracts are normally long term so you need to account for the investment returns that you will make on the premiums, as well as administrative expenses and a profit margin. However, since you aren't an actuarial student you can probably forget about that. Your version of the equivalence principle says that the Premium should be equal to the expected value of claims under the contract. This means that, on average, the premiums for a large group of independant contracts will be enough to cover the total claim value. it is an application of the Law of Large numbers. To do the calculation, you need more information than you have given. Most importantly 1) How many periods the contract runs for 2) How the premium is paid (lump sum at the start? regularly while alive?) Once you know those two things, use normal methods with markov chains to calculate 1) The expected premium value at outset 2) The chance of a claim occuring during the contract = 1- P(survives to end) 3) The expected claim value at outset (= cover_level * chance_of_claim_at_any_point_in_contract) Since you have not given the information, here is an example with made up figures. 2 period markov chain. Sum assured S. Lump sum premium P, paid at outset. States = (alive,dead,cancelled) Transition matrix $A = \left( \begin{array}{l|lll} & a & d & c\\a & 0.8 & 0.1 & 0.1\\d & 0 & 0 & 1\\c &0&0&1 \end{array} \right)$ Given that the life starts in state a, the probabilities after 2 states are at time 1: P(alive) = 0.8 P(dead) = 0.1 P(cancelled) = 0.1 at time 2: P(alive) = 0.64 P(dead) = 0.18 P(cancelled) = 0.18 So P(claim) = 0.18 E(claim value) = 0.18S Equivalence Principle: E(claim value) = E(Premiums) 0.18S = P 3. Thanks for your quick response. Here are some more details about the problem: 'a' is for the premium the person has to pay if he is not at all sick during the year. (1-b)a is what the premium is reduced to if the person gets sick, whereby b is the fraction of the year the person is sick, and the person will get$10,000b annual payment when sick.

4. you will need to make (or have been told) an assumption about how often the premiums are paid (monthly?). I hope you have also been told the value of b, otherwise there are probably infinitely many solutions.

Use that to work out the Expected value of the premiums/claims under the contract.

I'll do the E(claims) since it is easier. The contract is not going to be cancelled while it is in payment, so the payments will be continued until death.
$E(claims) = S(1 + P(\text{alive 1 year after onset of sickness} + P(\text{alive 2 year after onset of sickness} + ....)$

Suppose P(ill -> dead) = 0.8
This is $E(claims|claim~happens) = S(1 + 0.2 + 0.2^2 + 0.2^3 + ...)$
$E(claims|claim~happens) = S\frac{1}{1-0.2}$

Now you just need to multiply by the chance of a claim happening (this is the chance that the he cancels or dies without getting sick first).

Note you do not have to worry about when the claim period starts, only how long it lasts when it begins. You [B]will[B] need to worry about when sickness starts when working out the expected value of the premiums

5. The question also stated that in each year the (potential) 'illness-fraction' B(n) is modelled by a positive random variable
Un. If B(n) is in state 1 then A(n) accounts for the corresponding proportion of year n where
he was ill. Otherwise, if A(n) is in any of the other states we disregard Bn. We assume
that A1; A2; : : : is a sequence of independent random variables, also being independent of
B, having constant mean E[A(1)] = E[A(2)] =. . .= 1/4.

I believe that the payments are on a yearly basis. We also haven't been told the value of b.

6. Note: i have edited my previous post as i made a mistake on it.

I assume Un is your notation for a uniform (0,1) variable.

So the expected premium in the transition year (from ill to sick) is 0.5P

so...work out the expected value of the total premiums, which is something like

Either
P*E(full years not sick) + 0.5P *P(claim occurs at some point)

Or
where
p(as) = prob(transition to sick if alive)
p(aa) = prob(transition to alive if alive)

write as the sum of two geometric progressions.

$Prem * \left(1 + P_{aa} + P_{aa}^2+ P_{aa}^3 ... - 0.5P_{as}\left[ 1 + P_{aa}+ P_{aa}^2+P_{aa}^3 ... \right] \right)$

$Prem * (1 - 0.5P_{as}) \left(1+ P_{aa} + P_{aa}^2+P_{aa}^3 ... \right)$

Interpretation: he pays the full premium for all years he remains alive. We deduct half the premium for the year he transitions from alive to sick.

7. finally, its unlikely you were given a question as complicated as this one without being told how to do it, and my methods are unlikely to be the same as your professor's.

I suggest you post the full question, un-edited, and then your attempted solution instead

8. Being an insurance agent you are negotiating with a client about a life insurance
police. He is in good heath, and it is the
beginning of the nancial year (n = 0). He agrees on the following
plan: at the end of any of the upcoming nancial years (n = 1; 2; 3; : : : ) he pays a premium
a [in $], provided he was in good health throughout the whole year. Otherwise, he receives an annual payment of b 10, 000$ from you, and his premium is reduced to (1 - b)a.
Here a is the fraction of the year when he was ill. Each year a certain proportion of your
healthy clients cancel their insurance policies.
We model the state of his insurance policy on an annual basis by a Markov chain
X = (Xt) belong to the set of natural numbers with state space S = {0, 1, 2, 3} and the following transition matrix
A = (sorry, don't know how to matrix notation in latex:

1st row 0.7 0.1 0.1 0.1
2nd row 0.6 0.2 0.2 0
third row 0 0 1 0
fourth row 0 0 0 1

where 0='good health', 1='ill during some part of the year", 2='is