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The Actuary The magazine of the Institute & Faculty of Actuaries

Doctor Who and the story of the pensions projections

On the planet of Gallifrey a time
lord’s pension is a cash sum that
is proportional to the number of
Daleks that he has battered during
his working lifetime. This cash sum will be
paid to him, or to his estate, on his retirement
date whether or not he is alive on that date. The
task of projecting a time lord’s final Dalek total
(and therefore his pension) is an interesting
challenge to time lord actuaries, and one that
this article sets out to explore.
First I should explain the demographics of the
time lord population. All time lords start waging
war on Daleks at the same age and all retire at
the same age, so that all who live long enough
have the same working lifetime of 50 Gallifreyan
years (about 300 of our years). What
makes the problem more interesting though is
the time lords’ ability to regenerate.
For the benefit of anyone that has never seen
Doctor Who, I should explain that time lords
effectively have ten lives. Every time they are
close to death (up to a maximum of nine
times), they can regenerate into a new body,
effectively starting a new life. Nobody ever
needs to regenerate before their coming of age
the dangers that necessitate regeneration are
all due to chasing Daleks. The new body always
looks different to the last one, which is a real
pain for the BBC as it has to bring in a new actor
each time
Starting with the basics
Of course different time lords will have different
appetites for battering Daleks, and differing
Dalek-battering abilities. So our pension projections
should ideally base the future rate of Dalek
battering for a particular time lord on his past
record. One possible starting point is the basic
Dalek-rate calculation, given by the formula:
projected Daleks = Daleks to date × 50
/working Gallifreyan years used
For many years this was the standard formula
for pension projections, but many commentators
pointed out two key faults with the formula:
? The formula does not take into account the
well recognised tendency for Dalek battering
to accelerate as time lords approach retirement,
assuming instead that the Dalek-rate
stays fixed.
? The formula does not take into account the
number of regenerations that a time lord has
undergone to date. The fewer the number of
regenerations a time lord has in hand, the
fewer chances he is likely to take during his
adventures, and the lower his likely Dalekrate
(not to mention the increased probability
that he will use up all ten of his lives without
using up all his available working years).
Dalek-battering resources
At this point, a couple of innovative statisticians
announced their presence with an alternative
methodology that changed the face of time lord
pension projections forever. Their idea was to
introduce the concept of Dalek-battering
resources. At every point in his working life a
time lord has a number of outstanding working
years (y) and remaining regenerations (r). His
Dalek-battering resources are a function F(y,r)
that reflects how much more opportunity he
has in his career to batter Daleks. It is defined
in such a way that F(50,10) = 100% (so that
everybody starts their career at 100%). Also
F(0,r) = 0 for all r, and F(y,0) = 0 for all y; if a time
lord reaches retirement or uses up all ten lives,
his battering resources are zero. The new projection
formula is now:
projected Daleks = Daleks to date
/ (1 F(y,r))
I imagine that the method of construction of
the two-dimensional table F(y,r) (see table) was
as follows:
1 All historical Dalek batterings and time lord
regenerations were divided up into 500 cells,
corresponding to the number of regenerations,
r, that the corresponding time lord had
used up to that point (09) and the curtate
number of Gallifreyan years, y, that the time
lord had been working up to that point (049).
2 Regeneration rates q(y,r) were calculated for
each cell by dividing regenerations by the
corresponding exposed to risk and then
graduating the resulting rates.
At this point a time lord’s outstanding
working years and regenerations have been
modelled as a two-dimensional Markov
chain, where the process can move from (y,r)
at one point in time to either (y 1,r 1)
with probability q(y,r), or to (y 1,r) with
probability 1 q(y,r) (see figure 1).
3 Next, a battering rate b(y,r) was calculated for
each cell by dividing Dalek batterings by
exposure and then graduating.
4 The next step was to construct a table of
B(y,r), being the expected number of future
Dalek batterings for a time lord y years from
retirement and with r remaining regenerations.
The starting point was at the boundaries
where B(y,0) = 0 for all y and B(0,r) = 0
for all r. The rest of the table was filled in by
working backwards using: B(y,r) =
b(y,r) + q(y,r)B(y 1,r 1) + (1 q(y,r))B(y 1,r)
(see figure 1 again).
5 Finally the resulting table would have been
normalised using F(y,r) = B(y,r)/B(50,10) so
that F(50,10) = 100%.
And there you have it. Problem solved. Please
tell me that wasn’t difficult.
Betting on success
As well as its use in pension projections, the projection
formula can also be used in the world of
competitive Dalek battering. Imagine that the
Doctor and the Master have entered into a bet
about who will be the most successful Dalek
batterer over their careers. The Master is sent
out to batter first and scores 300 Daleks (using
up five lives) in his 50 Gallifreyan years. The
Doctor is on 150 for two after 30 Gallifreyan
years, at which point the BBC cancels his show
for reasons beyond his control and he does not
have the opportunity to use his remaining
20 Gallifreyan years.
Given that the bet was based on who was the
‘most successful’ rather than on who batters the
largest number of Daleks, the time lord actuaries
will need to be called in to determine the
winner. Using the old-fashioned Dalek-rate calculation,
the Master would win as his Dalek rate
of 300/50 = 6 exceeds the Doctor’s rate of
150/30 = 5, but this does not allow for how the
Doctor’s Dalek rate was expected to increase
over the remaining 20 Gallifreyan years, especially
with eight regenerations in hand. In order
to get a fair comparison between the time lords,
we bring in the new tables and note (from the
figures in the F(y,r) table) that the Doctor was
allowed to use only 1 F(20,8) = 47.8% of his
resources, so his 150 could be extrapolated forwards
to 150/0.478 = 314. A comfortable victory
for the Doctor.
On the other hand, what if the Doctor’s show
is unexpectedly relaunched? The bet with the
Master is on again. The problem is that he has
been off air for ten years and has only ten years
left to batter Daleks. How many Daleks should
he be aiming to batter if he is to be able to win
his bet? We know that the Doctor has already
used 47.8% of his original resources. He now
has F(10,8) = 30.8% of his original resources left,
so in total will be able to use 78.6%. So his
revised target should be 78.6% of the Master’s
300 rounded up, ie 236 Daleks. So he needs a
Dalek rate of 8.6 over the rest of his career if he
is to triumph. A challenging target (more than
one Dalek per Earth year), but with eight regenerations in hand he will have the
opportunity to take some risks.
By now many of you will have worked out
that those two innovative statisticians were
Frank Duckworth and Tony Lewis, and that
their idea has far more down-to-earth applications
than the projection of time lords’ pension