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The chain-ladder or development [1] method is a prominent [2] [3] actuarial loss reserving technique. The chain-ladder method is used in both the property and casualty [1] [4] and health insurance [5] fields. Its intent is to estimate incurred but not reported claims and project ultimate loss amounts. [5]
There are two algebraically equivalent approaches to calculating the Bornhuetter–Ferguson ultimate loss. In the first approach, undeveloped reported (or paid) losses are added directly to expected losses (based on an a priori loss ratio) multiplied by an estimated percent unreported.
Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables. Traditional notation uses a halo system, where symbols are placed as superscript or subscript before or after the main letter. Example notation using the halo system can be seen below.
and the future lifetime random variable T(x) therefore follows a uniform distribution on (,). The actuarial notation for conditional probability of failure is = Pr[0 ≤ T(x) ≤ t|T(0) ≥ x]. Under de Moivre's law, the probability that (x) fails to survive to age x+t is
Finally, the conditional probability of heads on the next flip given that the first flip was heads is the conditional probability of a heads-only coin times the probability of heads for a heads-only coin plus the conditional probability of a fair coin times the probability of heads for a fair coin, or 2/3 * 1 + 1/3 * .5 = 5/6 ≈ .8333.
In actuarial science and applied probability, ruin theory (sometimes risk theory [1] or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
where F X (x) is the cumulative distribution function of the continuous age-at-death random variable, X. As Δx tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by Δx .
Year 3: $150,000 × (1.08) −3 = $119,074.84. If we sum the discounted expected claims over all years in which a claim could be experienced, we have completed the computation of Actuarial Reserves. In the above example, if there were no expected future claims after year 3, our computation would give Actuarial Reserves of $568,320.38.