#Health Insurance

Actuaries quantify the risk inherent in insurance contracts, evaluating the premium of insurance contract to be sold (therefore covering future risk) and evaluating the actuarial reserves of existing portfolios (the liabilities in terms of benefits or claims payments due to policyholder arising from previously sold contracts). The example comes from Deshmukh12.

An insurer issues a special 3-year insurance contract to a person when the transitions among four states, 1: active, 2: disabled, 3: withdrawn, and 4: dead. The death benefit is 1000, payable at the end of the year of death. A death benefit is a payout to the beneficiary of a life insurance policy, annuity, or pension when the insured or annuitant dies. Suppose that the insured is active at the issue of policy. Insureds do not pay annual premiums when they are disabled. Suppose that the interest rate is 5 % per annum. Calculate the benefit reserve at the beginning of year 2 and 3.

benefit <- c(0, 0, 500, 1000)

transition_matrix <- matrix(c(0.5, .25, .15, .1,
                              0.4, 0.4, 0.0, 0.2,
                              0, 0, 1, 0,
                              0, 0, 0, 1), 
                            byrow = TRUE, nrow = 4)

markov_model <- new("markovchain", transitionMatrix = transition_matrix, 
                    name = "Health Insurance", states = c("active", "disable", "withdrawn", "death"))

set.seed(1000)
plot(markov_model)

The policyholders is active at T0. Therefore the expected states at T1,T2,T3 are calculated in the following.

T0 <- c(1,0,0,0)
T1 <- T0 * markov_model
T2 <- T1 * markov_model
T3 <- T2 * markov_model

paste(c("Year 0:", T0), collapse = " ")

[1] "Year 0: 1 0 0 0"

paste(c("Year 1:", T1), collapse = " ")

[1] "Year 1: 0.5 0.25 0.15 0.1"

paste(c("Year 2:", T1), collapse = " ")

[1] "Year 2: 0.5 0.25 0.15 0.1"

paste(c("Year 3:", T1), collapse = " ")

[1] "Year 3: 0.5 0.25 0.15 0.1"

The present value of future benefit (PVFB) at T0 is given by:

PVFB <- T0 %*% benefit * 1.05 ^ -0 + 
   T1 %*% benefit * 1.05 ^ -1 + 
   T2 %*% benefit * 1.05 ^ -2 + 
   T3 %*% benefit * 1.05 ^ -3

PVFB

         [,1]
[1,] 811.8454

The yearly premium payable whether the insured is alive is as follows.

P <- PVFB / (T0[1] * 1.05 ^- 0 + T1[1] * 1.05 ^ -1 + T2[1] * 1.05 ^ -2)

The reserve at the beginning of the second year, in the case of the insured being alive, is as follows.

PVFB <- T2 %*% benefit * 1.05 ^ -1 + T3 %*% benefit * 1.05 ^ -2
PVFP <- P*(T1[1] * 1.05 ^ -0 + T2[1] * 1.05 ^ -1)

PVFB - PVFP

         [,1]
[1,] 300.2528