Actuarial Notes

Health Insurance

Study guides: Exam ALTAM, Exam CP-321

Health Insurance, in the long-term sense of the SOA life exams, covers benefits triggered by the insured's state of health rather than by death alone: disability income insurance (an income while sick), long-term care insurance (care costs while unable to live independently) and critical illness insurance (a lump sum on diagnosis). It is modelled with a multiple state model (healthy, sick, dead) in which benefits attach to being in a state or moving between states. (Canada's public system is Health Care Insurance.)

\text{EPV(income)} = B\sum_{k=1}^{n} v^{k}\;{}_kp_x^{01}

\text{EPV(premiums)} = P\sum_{k=0}^{n-1} v^{k}\;{}_kp_x^{00}

  • {}_kp_x^{ij} is the probability that a life in state i at age x is in state j at age x+k (state 0 healthy, 1 sick, 2 dead), found from the transition matrices by the Chapman-Kolmogorov equations in discrete time or Kolmogorov's forward equations in continuous time
  • Premiums are usually waived while sick, so the premium annuity runs on {}_kp_x^{00}, not on survival; the premium follows from the equivalence principle or a profit objective
  • In continuous time a benefit on transition 0 \to 1 is valued by \int_0^n e^{-\delta t}\,{}_tp_x^{00}\,\mu_{x+t}^{01}\,dt. Recovery (1 \to 0) makes {}_tp_x^{00} larger than {}_tp_x^{\overline{00}}, the probability of staying healthy throughout
  • Policy values, reserves and profit tests are computed per state: a sick life and a healthy life of the same age hold very different reserves

Two-Year Disability Income Premium A two-year disability income policy is issued to a healthy life. A benefit of 10{,}000 is paid at the end of each year if the insured is sick at that time. A level premium is paid at the start of each year only if the insured is healthy. Annual transition probabilities (same each year): from healthy, 0.92 healthy, 0.05 sick, 0.03 dead; from sick, 0.10 healthy, 0.80 sick, 0.10 dead. With i = 5\%, find the premium.

Answer {}_1p^{01} = 0.05 and

\begin{align*}
{}_2p^{01} &= p^{00}p^{01} + p^{01}p^{11} \\
&= 0.92(0.05) + 0.05(0.80) \\
&= 0.086
\end{align*}
\begin{align*}
\text{EPV(benefits)} &= 10{,}000\left[v(0.05) + v^2(0.086)\right] \\
&= 476.19 + 780.05 \\
&= 1{,}256.24
\end{align*}
\begin{align*}
\text{EPV(premium of 1)} &= 1 + v(0.92) \\
&= 1.87619
\end{align*}
\begin{align*}
P &= 1{,}256.24 / 1.87619 \\
&= 669.57
\end{align*}

A life sick in year 2 may be one who fell sick in year 1 and stayed sick, or one who stayed healthy and fell sick, which is why {}_2p^{01} has two terms.

Critical Illness Under Constant Intensities A whole-life critical illness policy pays 100{,}000 immediately on diagnosis. A healthy life has constant intensities of diagnosis \mu^{01} = 0.004 and of death without diagnosis \mu^{02} = 0.006; \delta = 0.04. Premiums are payable continuously while healthy. Find the EPV of the benefit and the annual premium rate.

Answer Staying healthy has probability {}_tp^{00} = e^{-0.01t}.

\begin{align*}
\text{EPV} &= 100{,}000\int_0^\infty e^{-0.04t}e^{-0.01t}(0.004)\,dt \\
&= 100{,}000\left(\frac{0.004}{0.05}\right) \\
&= 8{,}000
\end{align*}

The premium annuity is \int_0^\infty e^{-0.05t}\,dt = 20, so P = 8{,}000/20 = 400 a year. With constant intensities the premium is simply 100{,}000\,\mu^{01}, the expected benefit outgo per year of healthy exposure.

How it appears on the exam

  • ALTAM gives a transition matrix or transition intensities and asks for the EPV of a disability income, long-term care or critical illness benefit, the premium by the equivalence principle, and policy values in each state, often through Thiele's differential equation.
  • ALTAM profit-analysis questions profit test a long-term health contract: the profit vector per policy in force in each state, the profit signature, NPV, internal rate of return, profit margin and discounted payback period, or the premium for a target profit.
  • Written parts ask candidates to critique the Markov assumption, for example that the chance of recovery does not depend on how long the life has been sick.

Exam traps

  • Using {}_kp_x^{\overline{00}} (healthy throughout) where the question allows recovery: the premium annuity and any healthy-state benefit need {}_kp_x^{00}, which includes lives who were sick and recovered.
  • Running the premium annuity on survival {}_kp_x^{00} + {}_kp_x^{01} when premiums are waived during sickness, which inflates the annuity and understates the premium.
  • Applying one year's transition probabilities two years ahead. {}_2p^{01} is not (p^{01})^2 or 2p^{01}; it is a matrix product over the intermediate state.
  • Confusing a benefit paid on being sick at a date (state-based) with one paid on falling sick (transition-based); at time k+1 they use {}_{k+1}p_x^{01} and {}_kp_x^{00}\,p_{x+k}^{01} respectively (ignoring recovery and re-sickness within the year).