Policyholder

Study guides: Exam P-1, Exam ILA-201I, Exam ILA-201U, Exam INV-201

A policyholder is the person or organisation that buys an insurance policy, pays its premium and holds the rights under it. The insurer pays covered claims to the policyholder or on its behalf. On Exam P, the policyholder keeps whatever part of a loss $X$ the insurer's payment $Y$ does not cover.

$X = Y + (X - Y)$

$Y = \min\!\big(c\,(X - d)^+,\ u\big)$

$X$ is the ground-up loss, $Y$ the insurer's payment, and $X - Y$ the amount the policyholder retains. $d$ is the deductible, $c$ the insurer's coinsurance percentage and $u$ the benefit limit (the maximum payment). Check the policy information for the order in which the terms apply. With coinsurance, capping the loss at $u$ and capping the payment at $u$ give different answers.
The policyholder retains three things: the deductible, its coinsurance share $(1-c)(X-d)$ , and anything above the limit. P-21-05 gives the reasons for a deductible: it saves the expense of processing small claims, lowers the premium, and gives the policyholder an economic incentive to prevent losses (moral hazard).
Inflation is leveraged. With a fixed $d$ , the deductible part of the policyholder's share doesn't grow with the loss, so the insurer's payment grows faster than the loss does. Once the limit binds, the policyholder absorbs all further growth (Inflation).
Policyholder, beneficiary and claimant can be different people. The insurer pays the policyholder or a designated beneficiary, and the claimant, who makes the demand for payment, can be an insured or a third party alleging injuries or damages that the policy covers. The person paid need not be the policyholder.

Splitting Two Losses Before and After Inflation A policy has a $\$500$ deductible, $80\%$ coinsurance and a maximum payment of $\$8{,}000$ . This year the policyholder has losses of $\$6{,}000$ and $\$12{,}000$ . Next year every loss is $10\%$ larger.

Split each loss between insurer and policyholder in both years.
Answer The $\$6{,}000$ loss:
 $\begin{align*} Y_{\text{now}} &= 0.80 \times (6{,}000 - 500) \\ &= 4{,}400 \\[4pt] Y_{\text{next}} &= 0.80 \times (6{,}600 - 500) \\ &= 4{,}880 \end{align*}$ 
The policyholder keeps $\$1{,}600$ this year and $\$1{,}720$ next year. The insurer's payment rises $10.9\%$ and the policyholder's share rises $7.5\%$ .

The $\$12{,}000$ loss:
 $\begin{align*} Y_{\text{now}} &= \min(0.80 \times 11{,}500,\ 8{,}000) \\ &= 8{,}000 \\[4pt] Y_{\text{next}} &= \min(0.80 \times 12{,}700,\ 8{,}000) \\ &= 8{,}000 \end{align*}$ 
The policyholder keeps $\$4{,}000$ , then $\$5{,}200$ , a $30\%$ increase. Below the limit, inflation lands mostly on the insurer. Above it, all of the inflation lands on the policyholder.

Expected Amount the Policyholder Retains A policyholder's loss is exponential with mean $\$1{,}000$ . The policy has a $\$250$ deductible and no other terms.

Find the insurer's expected payment and the policyholder's expected retained loss.
Answer
 $\begin{align*} E[Y] &= \int_{250}^{\infty} (x - 250)\,\frac{1}{1000}\,e^{-x/1000}\,dx \\ &= 1000\,e^{-0.25} \\ &= 778.80 \end{align*}$ 
 $\begin{align*} E[X - Y] &= E[X] - E[Y] \\ &= 1000 - 778.80 \\ &= 221.20 \end{align*}$ 
This is $E[X \wedge 250]$ , the limited expected value at the deductible. The deductible removes $22.1\%$ of expected loss. A share $1 - e^{-0.25} = 22.1\%$ of all losses produce no claim payment at all.