Actuarial calculators

The narrative. A peso today is not worth the same as a peso ten years from now, and all of actuarial science lives on that difference. Discounting and accumulating cash flows is the first building block: insurance premiums, pension reserves and bond valuations all rest on it. The annuity (a series of equal payments) is the piece that recurs most often. Its present value \(a_{\overline{n}|}\) answers the most frequent question in the profession: what is a promise of future payments worth today? Switching the annuity from immediate to due (paying at the end or at the start of each period) multiplies it by \((1+i)\), a detail that moves millions in pensions.

The narrative. Life insurance is a symmetric bet against mortality: the insurer collects today and pays on an uncertain date. The equivalence principle sets the premium by equating, in expected present value, what comes in and what goes out. The net single premium for whole life insurance is \(A_x\); spread over annual payments for as long as the insured is alive, it becomes \(P=A_x/\ddot a_x\). Because more is collected in the early years than the risk requires, a reserve builds up: the insurer's debt to the insured, which grows and is later released. That reserve is the heart of a life company's balance sheet and of any defined benefit pension plan.

The narrative. The average does not bankrupt an insurer; the tail does. That is why actuaries measure risk by its extremes. Value at Risk (VaR) is the quantile: the loss that will not be exceeded except with probability \(1-p\). It is intuitive but blind to what happens beyond that point. Tail Value at Risk (TVaR, or CTE) fixes that: it is the average of losses once the VaR has been crossed, so it is always larger and does distinguish a heavy tail from a thin one. Solvency II and Mexico's risk-based capital approach are built on these measures, typically at 99.5%.

The narrative. In property and casualty insurance, claims are not paid on the day they occur: they take months or years to be reported and settled. The insurer knows what it has already paid, but it must estimate what is still to be paid on accidents that have already happened: the IBNR (incurred but not reported). The Chain-Ladder method is the workhorse: it arranges payments in a triangle by accident year and development year, calculates how much the typical row grows from one column to the next (the development factors) and projects each year forward to its ultimate value. The difference between that ultimate and what has already been paid is the reserve. Edit the triangle and watch how the factors and the IBNR change.

The narrative. A client had few claims last year. Are they genuinely a good risk, or were they lucky? Credibility answers by placing a weight \(Z\) between 0 and 1 on their individual experience, and \(1-Z\) on the behavior of the group. With little information, \(Z\) is small and we rely on the group; with plenty, \(Z\) tends to 1 and we let the client's own data speak. The classical version (limited fluctuation) sets how many claims are needed for full credibility. Bühlmann's version is more elegant: \(Z=n/(n+K)\), where \(K\) measures how much variance comes from chance as opposed to real differences between risks. It is essentially the same shrinkage idea that regularizes machine learning models today.