Feeding the trolls

Scientists have long admired a 'neat trick', meaning an ingenious idea for overcoming an obstacle, and definitely not trying to mislead anyone. Just mentioning a 'trick' can mean trouble, though, if it gets the attention of internet trolls. Climate scientist Phil Jones found this out the hard way when he mentioned "Mike's Nature trick" in a private email that was later hacked (Pearce 2010, Chapter 14). The trolls deliberately twisted the meaning of the word 'trick' and Professor Jones's life was changed forever.

There are several 'neat tricks' in the world of pensions and insurance, all perfectly above board, which have yet to be attacked by trolls (but you never know).

The first the student meets deals with valuing increasing cashflows. If cashflows starting at \(X\)  grow at compound rate \(g\) per year then the present value, at compound interest \(i\) per year, of a cashflow \(X \, (1+g)^n\) payable \(n\) years in the future, is:

\[ \frac{(1+g)^n}{(1+i)^n} X = \left( \frac{1+g}{1+i} \right)^n X \]

just as if the future cashflow was \(X\) and the interest rate was \((1+i)/(1+g) - 1\). Valuing a level series of future cashflows at a lower single rate of interest is, of course, an easier calculation. This 'trick'  can be useful for inflation-linked quantities, where it is well understood as the 'real'' interest rate. Perhaps being 'well-understood' has kept the trolls at bay (so far).

The same technique used to work in the with-profits world. A compound reversionary bonus (remember those?) of \(b\) per year worked on the benefits in just the same way as \(g\) worked on cashflows above, so they could be valued simply at rate of interest \((1+i)/(1+b) - 1\) by assuming no future bonuses. If one did a somewhat similar thing with the premiums, the net premium method resulted and there was no need to mention a bonus rate \(b\) at all. This was helpful, because it meant the printing-press trolls of the day couldn't treat \(b\) as a promise and hold the actuary to ransom.

At a more advanced level, no-arbitrage valuation in finance means replacing the probability distribution of an asset's future price, usually denoted by \(P\), with an alternative 'risk-neutral' distribution, usually denoted by \(Q\).  The idea is now so pervasive that '\(P\)-measure' and '\(Q\)-measure' are becoming part of the language. Paradoxically, '\(P\)-measure' means 'real-world' probabilities, which sounds right but is wrong.

How about mixing interest and probabilities? That can also work. In a continuous-time setting, suppose the rate of interest is \(r\), the mortality hazard rate is \(\mu_{x+t}\), and we want to add a competing decrement (such as a lapse rate) with hazard rate \(\nu\). Given the right contract design, we can assume no lapses, but use a rate of interest \(r - k \, \nu\) where \(0 \le  k \le 1\). The case \(k=0\) is known to European actuaries as Cantelli's Theorem (Cantelli 1914), the idea being that if the cash outgo on lapsing is 100% of the reserve then being held, lapses can be ignored in  calculating that reserve. This gets round a circular definition, where benefits depend on reserves, but reserves also depend on benefits.

Reserving correctly for future benefits by assuming the wrong probabilities, or that increases are never paid, or that the benefits are never paid at all, are certainly some very neat tricks. Just don't tell the trolls.