**(T.J.
Fairclough C.Math., MIMA**.)

The clock was striking the appointed hour on a
drizzly December morning as David Copperfield tapped lightly on the
counting-house window. Above his head
and below the three brass balls a painted wooden sign proclaimed "
Wickfield & Heep - Independent
Financial Advisers". Inside Uriah
Heep jumped down from his stool and thrust aside the great fat book in which
his lank forefinger had been tracing each line as he read. Hurrying over to the door he unctuously
admitted his visitor with a grinding of palms and his customary two bows and a
scrape.

'Good morning Master Copperfield my dear Sir and
most welcome I'm sure to my er... I should say of course *our* umble establishment. It
is indeed the mark of a gentleman to be so punctual especially in such
inclement weather if I may take the liberty of saying.

'Good Morning Heep, Mr. Micawber', said Copperfield, nodding to
each as he entered the chamber, removing his stove-pipe hat to shake the
raindrops from it and sitting down as he spoke.
'I have an appointment with Mr. Wickfield, please be so good as to tell
him that I've arrived.' 'Bless you
sir' replied Heep 'But unfortunately Mr. Wickfield is indisposed today and has
asked me to stand in for him unless of course you have any objections Master
Copperfield? - I should say Mister, but the other comes so natural,' said Uriah

'I suppose not' replied David
unenthusiastically. 'Very well then
let's get down to business straight away shall we? I've requested this meeting
because I want some financial advice. Thanks to the generosity shown to me in
the will of a distant relative, the late Mrs. Lirriper, I have recently
inherited a modest lump sum. It is my
present intention to invest this unexpected benefaction at 4% per annum with
the Bank of England in a quantity sufficient to provide me a regular pension of
forty pounds a year. Such an income,
although considerably more than the seven shillings that was once my weekly
stipend from Murdstone and Grinby, may yet be insufficient to meet all my
future needs and I am interested to discuss ways of increasing the income. I'm willing to consider any prudent
suggestions you may care to recommend.'

Mr. Micawber, sitting on the high stool in his
capacity as confidential clerk to Uriah Heep, had been listening to the
proceedings with interest. He now jumped
down to say 'My dear Copperfield, I'm
delighted to hear of your windfall and please accept my sincere commiseration
on the sad circumstances that gave rise to it.
With regard to such matters as income and expenditure I hold resolute
opinions and my advice is easily stated^{1} "*Annual income forty pounds, annual expenditure thirty-nine pounds
nineteen and six, result happiness. Annual income forty pounds, annual
expenditure forty pounds ought and six, result misery. The blossom is blighted,
the leaf is withered, the god of day goes down upon the dreary scene, and - and
in short you are forever floored. As I
am*!"

Copperfield weighed these words of wisdom carefully
before venturing to remark 'That's very clear Mr. Micawber, I will definitely
bear those sensible precepts in mind and thank you most kindly. What do you say Heep?'

'Of course Master Copperfield, most excellent advice
it is if I may be allowed to say.
Particularly suitable advice it is on the *expenditure* side of the equation certainly but perhaps a trifle
more guidance could yet be supplied on the *income*
side of the equation sir if I may make so bold?
Why, any fortunate gentleman blessed with such a considerable degree of
capital might well increase the amount of income he draws from it. Of course the precise amount of income taken
would depend on the particular circumstances of his age and health and his desire
or otherwise to leave a fortune behind him when the time eventually comes for
him to become a partaker of glory.'

'A trifle more guidance to be supplied Sir?'
spluttered Mr. Micawber at last and with some asperity. 'Pray explicate to the
present company on the specific shortcomings of my advice that you so evidently
perceive and to which you have so kindly alluded!'

' Goodness gracious me Micawber', said Uriah Heep,
'A thousand apologies if my clumsy manner of speaking has led you to mistakenly
believe I thought there are any shortcomings in your clear assessment of the
situation I'm sure.'

Mr. Micawber, seemingly mollified, went on to
briskly say 'Perhaps then Mr. Heep you would be good enough to explain how it
is possible to take more than 4% per annum from an investment paying 4% per
annum without, in short sir, ending up in the workhouse!?'

'By all means sir, if you will kindly allow me the
opportunity I will attempt to clarify my meaning.' said Heep, hurriedly
continuing. 'It is clear that the lump sum in question must be £1000 for that
is the amount required to furnish £40 yearly interest when invested at 4% per
annum. Obviously if interest rates
remain fixed at 4% per annum and if only the £40 interest is spent each year as
income then the £1000 capital will of course survive intact, continuing to earn
£40 a year for ever and eventually pass into the hands of the estate
beneficiaries. Indeed for this reason
such an income producing investment may be termed a *perpetuity*.'

'If Master Copperfield wishes however, he could
increase the annual income over and above £40 by simply taking out instead
say £50 or even £100 a year. Of course taking the extra income would eat
into the original £1000 capital. The
capital could even disappear altogether if Master Copperfield outlives it. That would leave nothing for the estate and
more to the point, nothing more for Master Copperfield. Another, perhaps more cautious, way to
increase the income is to sign over the entire £1000 to a Mutual Society in
return for what is termed an *annuity*. This would provide a guaranteed income for
the whole of Master Copperfield's lifetime.
It should be appreciated however that unless special terms have been
negotiated at the outset, annuity income generally dies with the recipient
leaving nothing for the heirs'

David Copperfield
nodded as he leaned back in his chair.
He said 'I must admit that the notion of slowly running down the fund by
drawing a trifle more income over and above the £40 interest has already
occurred to me. Unfortunately I encountered some arithmetical difficulties
when attempting to calculate the number of years the fund would last for the
various hypothetical amounts of income.
Ideally of course I would like to arrange matters so that the fund and
myself expire together for if the fund expires first then I doubt I can be long
after. It seems that on average a
person of my age can expect to live for about 22 more years and I certainly
wouldn't wish the last 5 or 10 of them to be passed in the workhouse as Mr.
Micawber has speculated. Perhaps Heep,
you or Mr. Wickfield may have some further advice to help me out of this
difficulty?'

Uriah Heep held his hands clasped together in front
of his chest and raised up his eyes before saying *'How
much have I to be thankful for in living all these years with Mr.
Wickfield! It is Mr. Wickfield's kindly
intention that gave me my articles, which would otherwise not have been within
the umble means of mother and self!*
Indeed it was also Mr. Wickfield as kindly showed me the way to
calculate the number of years a fund might last depending on the income taken
from it. Mr. Wickfield effects it by
means of a formula that I now propose to write on the blackboard, if you will
only be so kind as to permit me?'

Hearing no objections raised he walked to the
blackboard on the counting house wall and wrote there in bright yellow chalk

_{}. (1)

David Copperfield stared with bafflement at the simple
rule written so neatly on the dark counting house blackboard and he fell
speechless for a minute or two. At last
he broke the silence with "You may say, Heep, that it is 'easily shown'
but I must admit it's not so easy for me.
I assume that *T* is the number
of years that the fund will last and most probably the symbol *R* signifies the interest rate as the
number of percent per annum, or 4 in this case?
However I'm at a loss to know what the symbol_{} might be and for the life of me I can't even guess how 'ln'
is pronounced let alone what that stands for, if anything. Some additional explanation would be welcome
please Heep."

'Bless you kindly Sir' said Heep, 'Why it's
simplicity itself. "ln" is pronounced "l" "n" and
stands for nothing more nor less than the natural logarithms listed in this
book of tables. _{} we might term the
"income multiple" for want of a better description and is defined to
be the ratio of the annual income taken at the end of each year to the interest
earned in the first year.

In the
calculations it is assumed that after the initial investment of £1000 no
further debits or credit transactions are allowed on the account, except that
at the end of every year the interest for that year is credited to the account
and the income is paid out. The income
is considered fixed in that the same absolute amount of income will be taken at
the end of every year. The interest
earned for the year, however, will vary and each year it will be R % of whatever
the balance was throughout the year. At
4% per annum the interest earned for the first year would be £40 and if we
decided to draw out more income than the interest earned, let us say for
example £50 every year, then the income multiple would be _{} or 1.25. This means that immediately after the end of
the first year the account balance would be
£1000 + £40 interest - £50 income
= £990. We can see that by taking such
a large income we are effectively spending some of the capital. The interest
for the next year would be 4% of £990 or £39 12s 0d. The amount of interest earned in subsequent
years will be even less and will continue to decrease as the balance remaining
to earn further interest is correspondingly reduced. As time passes therefore
each £50 income payment would have to be made up of a growing proportion of
capital and a shrinking proportion of interest.

Let us now continue with the case when £50 annual
income is taken; substituting the value _{} into formula (1) we
obtain _{}. In round numbers sir
it means the annual income payments could be maintained for a useful 40 years.'

'Marvellous and crystal clear' said
Copperfield. 'This means that for my
£1000 I can take £50 income every year for 40 years! That is much more like it and my
congratulations to Mr. Wickfield on his handsome little formula.'

'I should
point out ' said Heep 'that formula (1) is only an approximation to the true
position, albeit a good approximation, and generally good enough for most
practical purposes. However if a more
accurate figure should be required it can be obtained using a version free of
approximation'. Another formula now
joined the first on the blackboard:

_{} (2)

' This more accurate version shows that the £50
income can actually be taken for 41.0 years, slightly longer than the 40 years
previously calculated.'

Heep continued on.
'With Micawber's permission we might now analyse the situation referred to
earlier where £40 0s 6d is the income is
taken each year from the account. When
compared to the £40 interest earned in the first year we find that _{} and the approximate
formula (1) tells us that _{}. This is about 184 years(or 188.1 years using the more
accurate formula). As Micawber has
foreseen, misery will inevitably follow somewhere around Master Copperfield's
240th birthday.' At this sly dig Mr.
Micawber cast an irritable glance in Heep's direction but refrained from
comment.

Heep went on 'The new Babbage Mark IV can be easily
programmed to produce a table showing how the value of *T* reduces as the income multiple is increased.' (First three
columns of Table 1)

Income (£ p.a.) |
Income multiple |
Flat income not linked to
inflation |
Income payments increased
in line with inflation at 2.5 % p.a. |

£40 0s 6d |
1601/1600 |
188.1 |
32.3 |

£ 50 |
1.25 |
41.0 |
24.6 |

£ 60 |
1.50 |
28.0 |
19.8 |

£ 70 |
1.75 |
21.6 |
16.6 |

£ 80 |
2.00 |
17.7 |
14.3 |

£ 100 |
2.50 |
13.0 |
11.2 |

Table 1

R = 4 % p.a., The £1000
starting capital is exhausted in *T*
years

David
Copperfield relaxed back into his chair and studied the table handed to him by
Heep. Thank you kindly Heep. Clearly I am a little better set than I
thought, providing always of course that I don't break any longevity
records. In short it seems that with
interest rates fixed at 4% per annum I could take £40 a year in perpetuity or _{}for more than 180 years, or £50 for 41 years or even £80 for
nearly 18 years. Your formula is clearly
very useful for finding *T *when we are
given ε but might it not also be useful to have a formula the other way
round as it were so that I can choose *T*
and use it to calculate ε?

Heep considered the request and pointed to the
formulae on the wall saying 'The two
formulae you see for *T* can both be
easily rearranged to give formulae for ε.
Formula (1) rearranges to give

_{} (3)

and formula (2) tells us that

_{}.' (4)

Heep wrote up these two formulae alongside their
counterparts on the board. He then
walked over to Copperfield and handed him the sheet of paper he had taken from
his desk drawer, saying 'I have taken the liberty of drawing up a chart using
the more accurate formula (4). The
chart allows the desired value of *T*
to be selected and the value of ε can then be read off for the given value
of *R*.
For example if you want the fund to last for *T* = 25 years at 4% per annum then the chart (better still, use
formula (4) directly) shows that ε = 1.60 and the corresponding income would
thus be £64 per annum.

T = 25 yrs. T=20 yrs. T=15 yrs. T = 35 yrs. T = 30 yrs. T=40 yrs.

Figure 1

A chart to show how the
income multiple varies with R for some values of *T*

'That's very useful Heep and much appreciated' said
Copperfield, slipping the chart into his pocket. 'There is however still one small fly in the
ointment. It would be most inconvenient
for me to have to wait a whole year before picking up any income at all. How
would I manage in the meantime? I would much prefer to spread the income out
over the year by receiving it more frequently, say in monthly or even weekly
instalments. What would become of your
marvellous formulae then I wonder?'

'Why, that would present no insurmountable
difficulty Master Copperfield' replied Heep. 'In fact a simple calculation
leads to yet another expression that will prove useful.' Another formula joined the others on the
blackboard:

_{} (5)

Heep continued 'This formula (5) shows how the
successive periodic account balances, _{}, will reduce over time if the annual income is paid out in *N *equal instalments over the year
instead of just a single payment at the end of the year. In (5) _{}may be termed the periodic interest rate or the interest rate
per interest period and the integer _{} is used to label the
interest periods with _{} representing the £1000
starting capital. Assume now that at
the end of each period we take a fixed amount of income equal to _{} times the interest
earned in the first period but now each period does not have to be a whole
year, it can be a month or a week or whatever we choose. The amount of each income instalment will be _{} and there will be *N* such payments spread equally over each
year. The interest is paid in and the
income is taken out at the end of each period.
By making _{} a bigger multiple we
could obtain a bigger income but run out of money sooner.

We can use equation (5) to find out how many
payments can be made before the fund is completely exhausted. We see that P* _{k} *becomes zero
when

_{} (6)

so that with *N*
payments per year the fund will last for *T*
years, where now

_{} (7)

By setting *N*
= 1 in (7) we recover our earlier situation where there was just one payment a
year and obtain equation number (2) as expected.

If on the other hand in (5) we increase *N* without limit corresponding to the
continuous compounding of interest and the continuous payment of income we find
that after *t* years the amount of
Principal remaining, *P*(*t*) may now be found using

_{} (8)

Equation (8) may be used to determine the length of
time, *T*, before the fund becomes
exhausted (i.e. set_{}) and we find that *T* comes out to be the same as our old
friend that was written in (1).'

David Copperfield leaned back in the chair looking
interested and suitably impressed. 'I've
just had a disturbing thought gentlemen.
Inflation! Would it not be
sensible to increase the income as time passes to make some allowance for
inflation? Is there any formula to deal
with that?

Uriah Heep smiled as he said 'An excellent strategy
Sir, and a very prudent course of action if I may say so. To allow for inflation we can make use of
formula (9) instead of formula (7).
First we choose the amount of income that we wish to take as our first
income payment and use it to calculate the value of ε exactly as before. Then we assume an appropriate (constant)
inflation rate, let this be *I *% per
annum. Finally we substitute our value
of _{} together with the assumed
values for *R*,* N* and *I* into formula (9). After the first income payment all
subsequent income payments are increased to rise in line with the assumed
inflation and formula (9) tells us how long the funds will last.

_{} (9)

Formula (9) is really quite similar to formula
(7). In fact (9) can actually be
obtained from (7): it is as if we have
used in (7) an 'effective' income multiple*
*_{} instead of _{}and an
'effective' interest rate _{}* * instead of *R.*

Inflation has a very detrimental effect on the
situation. We have already seen that
when no allowance is made for inflation, *R*
= 4% p.a. and with annual payments

(*N *= 1) the fund provides a flat annual
income of £50 a year for 41 years (ε = 1.25). If we want to increase this same £50 income
in line with inflation at the rate of

I* *= 2.5%
per annum formula (9) predicts that the inflation linked equivalent income can
only be supported for *T* = 24 years
before the money is all gone.' (Some
further examples are given in column 4 of table 1.)

Mr. Micawber had been listening to the proceedings
with increasing agitation.

He now jumped to his feet and said 'My dear Copperfield^{2}, *I am older than you; a man of some
experience in life, and - and of some experience, in short, in difficulties,
generally speaking. At present, and
until something turns up (which I am, I may say, hourly expecting), I have
nothing to bestow but advice. Still my
advice is so far worth taking, that - in short, that I have never taken it
myself, and am the' - here Mr. Micawber, who had been beaming and smiling, all
over his head and face, up to the present moment, checked himself and frowned -
'the miserable wretch you behold.' *'To
continue Sir, my advice in short is to avoid such a hazardous strategy that in
all probability could lead you into poverty and a miserable old age. A man in your prime physical condition could
easily outlive his fortune if it is consumed at such rapid rates as have been
mooted.

Were Mr. Wickfield himself to
be present at this meeting I am convinced he would strongly encourage us to
look to an annuity as the best solution. It is possible I may be of some assistance in that regard. As you know *I have entered into arrangements,
by virtue of which I stand pledged and contracted to our friend Heep, to assist
and serve him in the capacity of - and to be - his confidential clerk. *It
is in that capacity that I have recently had occasion to investigate the
subject of annuities, bringing to bear *such address and intelligence as I chance to
possess.* As a result I have managed to turn up a little formula of my own that may be of
interest. Taking into account the 'bonus'
of mortality cross-subsidy, whereby the people
who die early after taking out an annuity subsidise their fellow annuitants who
live longer, my formula estimates the annual income, *I*(*x*)
to be expected when a Principal sum _{} (£1000 in this case)
is used to purchase a continuous life annuity by a person aged '*x*' years at its commencement. I stress that such income of course would be **guaranteed **for the lifetime of the
annuitant.' Mr. Micawber took the chalk
in his hand and with a firm hand wrote on the board in green chalk

_{} (10)

This was followed by a stunned silence from David
Copperfield who eventually said 'Steady
on please Mr. Micawber, have some
pity. Now I'm completely floored and I
no longer know if I'm coming or going.
What is all this about and what am I supposed to make of these new
hieroglyphs and what can possibly be the link between my pension and that great
green snake you have drawn on the board?'

Mr. Micawber beamed and smiled. 'My dear Copperfield, one thing you could
make of it indeed is that a male aged 57 taking out an annuity with £1000 would
expect to receive no less than £72 9s 9d income every year for however long he
lives. That very same income with Mr.
Heep's method would completely exhaust the £1000 fund in just over 20** **years so with an annuity you would be
'ahead' as it were if you live beyond the age of 77 years.

As to the symbols used; the function *L*(*x*) is the survival function obtained using the
life expectancy tables^{3} for males.
Out of 100,000 live births *L*(*x*) is the number surviving to age *x *years.
For instance my mortality tables tell me that for males *L*(57) = 90705, this being the number of
males per 100000 live births expected to be still alive on their 57th
birthday. The symbol _{} and the time *t* is measured in years. As far as
inflation goes one must bear in mind that nobody knows what the future rates of
inflation will be and so it is impossible for any company with finite resources
to guarantee an annuity income that will rise in line with inflation. However we can approximately take into
account an assumed steady inflation rate at *I
*% per annum by simply using the effective interest rate _{}in place of _{}in formula (10). I
have evaluated the integral (10) by means of a cubic spline fitted to the data
and the results are here in this table', he said, handing Table 2 to David
Copperfield.

Age when taking up the
annuity ( |
Flat annuity income (£
p.a.) not rising with inflation |
Number of years, |
Annuity income payments (£ p.a.)
increased in line with inflation at 2.5 % p.a. . |

50 |
£ 62 |
26 |
£ 45 |

55 |
£ 69 |
22 |
£ 52 |

57 |
£ 72 |
20 |
£ 55 |

60 |
£ 78 |
18 |
£ 61 |

65 |
£ 91 |
15 |
£ 74 |

70 |
£ 110 |
12 |
£ 92 |

75 |
£ 137 |
9 |
£ 118 |

Table 2

Annuity income (ρ = 4 % p.a.) for a £1000
annuity, with and without allowance for inflation (Figures rounded to the nearest £ and nearest
year)

Copperfield took
Table 2 into his hand and studied it.
'This all looks very interesting, Mr. Micawber and many thanks to you for all your hard work
and for sharing this with me. It
certainly seems from these figures that mortality cross-subsidy is not to be
sneezed at'

Rising to his feet Copperfield took one more look at
the formulae on the board and said 'Well gentlemen, thank you again for your
most interesting contributions. You have
certainly given me plenty to think about and I shall need some time to do so. I think the rain has stopped so I will bid
you good day and contact you again at a later date.'

Uriah Heep escorted David Copperfield to the
counting house door. At the door and out
of the earshot of Mr. Micawber he said
'You do realise Master Copperfield I hope that there's really no
necessity for you to draw down your fund or to heed Micawber's call to purchase
an annuity if you don't wish it? There
are so many other interesting opportunities that we can talk about on your next
visit. Why there are Precipice Bonds, Ostrich Farms, Time-Shares and Shipping
Containers to name just a few of them so please don't hesitate to let me guide
you through all these important investment decisions. You know full well you can always trust your
sincere and umble servant Uriah Heep and rely on him to show you the right
path. After all, if you can't trust your
I.F.A., just who can you trust eh?

** **

References

1. "David Copperfield" by Charles Dickens, Chapter 12. (Where the income is £1 and not £40)

2.
A few sentences have been shamelessly purloined verbatim from Reference 1
above. Some, but not all, are italicized
but in any case the reader will no doubt identify them by their superior
construction.

3.
http://www.gad.gov.uk/Life_Tables/Interim_life_tables.htm

Interim
Life Tables, 1999 - 2001 United Kingdom Males

(*x *, *L*(*x*))

(50,
94283), (51, 93887),
(52, 93475), (53,
93029), (54, 92534),

(55, 91987),
(56, 91385), (57,
90705), (58, 89946),
(59, 89137),

(60,
88239), (61, 87239),
(62, 86161), (63,
84989), (64, 83725),

(65,
82377), (66, 80865),
(67, 79216), (68,
77410), (69, 75493),

(70,
73386), (71, 71117),
(72, 68661), (73,
66025), (74, 63206),

(75,
60234), (76, 57107),
(77, 53871), (78,
50489), (79, 47006),

(80,
43489), (81, 40002),
(82, 36488), (83,
32913), (84, 29258),

(85,
25714), (86, 22291),
(87, 19114), (88,
16160), (89, 13387),

(90,
10917), (91, 8784) ,
(92, 6975), (93,
5390), (94, 4074),

(95,
3008), (96, 2162),
(97, 1523), (98,
1022), (99, 674),

(100, 435).

For the purposes of integration in (10) the survival
function *L*(*x*) was approximated by a cubic spline fitted to the above data for _{}. Visual inspection of
the extrapolated spline beyond* x *=* *100 revealed that it continued its
downward descent to hit zero at about *x*
= 102. In the absence of other data (and
with the greatest respects to all 102+ year-olds) and without any justification
apart from convenience the survival function has been approximated by the
fitted spline in the extrapolated domain _{}and taken to be zero for *x*
> 102.