A drunkard's walk through allocated pensions

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					A drunkard's walk through allocated pensions
  Garth Price, 


    •   Current arguments for investing in Australian equities are based on the performance of the
        last thirty years. This time has been atypically bullish compared with the total history. A
        sixty year analysis, which better matches the full history, reveals hidden risks.
    •   In the long term, the ASX index corrected for inflation, shows no appreciable gain. It is a
        directionless set of random numbers – a drunkard's walk.
    •   Direction is found if dividends are included, but even then 50 years of data are needed to
        confirm if this accumulation index goes up or down. The average time in retirement is of
        order 25 years.
    •   For a self-funded retiree, there is a good expectation of not funding a retirement by investing
        in equities alone.
    •   Lowering long term volatility well below that of equities is essential to provide a retirement
    •   3% above inflation is a reasonable expectation as income from an allocated pension before
    •   The finance industry charges at least 1% of capital which is about 30% of a self funded
        retirees income. Value given is negligible compared with the effort to accumulate the capital.
        The solution is to reform inept legislation and to change benchmarks to percent of income,
        not capital.

  I have retired with my own superannuation fund which is now in the pension phase. I draw on
an allocated pension under the so called 'Simple Super' brought in in 2006. With a technical, but
not a financial background, I have been able to analyse the risks and rewards in a way I have found
useful and which might be of use to others. Please take it all with a grain of salt. It has not been
checked by others and I would like to get it right – feedback is most welcome. Although there is
nothing new in the conclusions, I hope there is value in a perspective dominated by risk, not gain.
And even if much of my analysis deals with allocated pensions, the conclusions are generally
applicable to managing investment in retirement.
   For a self funded retiree, risk is the problem. You must grow the initial stake through good and
bad times while drawing down a salary. Pension rules rip fixed percentages from the untaxed haven
of your super fund into the tax-exposed outside world. Retiring at 65, you are expected by the
finance industry and government to die at 82 if male and 87 if female. You must plan for 100, not
only to spite them, but knowing that the two most expensive years of your life are likely to be the
two before your death.
   When I worked out the risks and rewards, I found quite a gap between my results and the happy
illusions of the financial press. There is agreement with a common rule-of-thumb that a good return                                 1/24                                    10:50 24/01/11
is 3 % above inflation1 2 3; but just how difficult that is over the long term is not obvious. It's a jump
to the often claimed 9% to 13% return on the Australian Stock Exchange.
  There are two parts to this paper. The first is a non-technical account of the risks and rewards of
investing in equities with illustrations of risk reduction using bonds and fixed interest. The results
apply to a suitably managed allocated pension or to ordinary investments before tax. The second
part gives the detail on which these conclusions are based. Only reasonable high-school
mathematics and statistics are needed.

Walking randomly with equities4



           All Ords

                        100                                                             Exponential
                                                                                        Regression for


                                 1884 1894 1904 1914 1924 1934 1944 1954 1964 1974 1984 1994 2004
                              1879 1889 1899 1909 1919 1929 1939 1949 1959 1969 1979 1989 1999
      Fig 1: All ordinaries 1875 to 2009. Drift = 5.1%, Volatility =16.0%

   The argument for equity investment is often presented as a plot of an index over a long period of
time (Fig.1). A regression line of best fit is passed through the individual points, and over a hundred
years this line appears to go eternally upward on the same course, guaranteeing prosperity for all
fortunate investors. The line is not disturbed by the odd world war or generational financial
disaster; the line is a prophecy to which the market must always return. The argument is nonsense
for the retiree or even the buy-and-hold investor. Volatility and inflation that can give negative
returns over decades.
  We will use as a proxy for equity investment, the all ordinaries index over the last 60 years
shown in Fig 2. It has approximately the same growth rate and variation as the 135 years of Fig 1,
but we have the benefit of a more detailed knowledge of inflation in this period.

1 and see other citations in
2 Cooley, Philip L., Carl M. Hubbard, and Daniel T. Walz. 1998. "Retirement Savings: Choosing a Withdrawal Rate
  That Is Sustainable." AAII Journal 10, 3: 16–21
3 citing: Ashton, Michael, Maximizing Personal
  Surplus: Liability-Driven Investment for Individuals (October 28, 2010). Available at SSRN:
4 The inspiration for using random walk analysis came from M.F.M. Osbourne, The Stockmarket and Finance from a
  Physicist's Viewpoint (Crossgar, 1977)                                            2/24                                         10:50 24/01/11


  1000                                                                                                      AO*div(accum)
                                                                                                            AO index


        1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006
     1949 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008

         Fig 2: All ordinaries 1949 to 2009: From the top: accumulation – with dividends, the index, index corrected
         for dividends and inflation, index corrected for inflation alone. Yearly index is mean of monthly.

    The 2nd curve (blue) of Fig 2 is the bare index. After inflation is removed, all that remains is the
 bottom (yellow) curve, giving a ~ 1% return/year over the last 60 years. Fortunately, if we add in
 reinvestment of dividends, then we obtain the 3rd green line which has about a 5% return/year. The
 top (red) curve, the uncorrected accumulation index, is that used by the finance industry to argue for
 equity investment.
   We shall be concerned mainly with the 3rd green curve. Although over the 60 years it grows by
 5% per year, there is great variation and the risk is very large, even for a normal investor who is not
 drawing an income.
   Risk is most easily seen by putting oneself in the positions of eight people who invests a lump
 sum for 25 years (an average time of retirement) with their starting year 5 years apart and
 beginning in 1950. The last investor spans 1985 to 2009. We simply cut the 3rd curve of Fig 2 into
 overlapping pieces, normalise with the same starting value of 1.0 and display on a log plot.                                           3/24                                          10:50 24/01/11
     Fig 3: Performance of equal equity investments at different times over 25 years. Drift and volatility curves
     for the all ordinaries index are also shown. Log(yearly gain) vs years after initial investment.



        1                                                                    1975
                     h.20 – δ.√20                                            -δ
                     Expectation of 1/6 at 20 years                          -2 δ

            0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425

    The Australian stockmarket has performed better than most markets in this post war period and
yet volatility can ruin investors. For example, investing in 1975 was excellent while 1970 was a
disaster. The growth5 of the all ordinaries corrected for inflation and assuming dividends and
reinvestment is 5% and the volatility is 17% as determined statistically. Consequently the
probability of variation of the growth each year is much greater than the growth itself . Large
changes in fortune can result in one or two years.
   The data curves of in the figures are governed by random walk statistics. From just two
parameters, the growth or drift, and the volatility, the probability of future gains and losses can be
deduced. These are shown by the δ curves in Fig 3. We expect 2/3rd of the points on the chart to
lie between - δ and + δ and only 1/20th to lie outside the -2δ and +2δ lines. Of most concern is the
downside: a useful measure is that there is 1 in 6 chance of falling below - δ in some year, and a 1
in 40 chance of falling below -2 δ.
  From now on we will use the δ and mean lines to characterise investments. They can be worked
out on a calculator using a simple formula6.
                                            ln  g i =h i±  i .                                                  (1)
where gi is the gain, ln is the natural logarithm, h is the expected mean gain, i the years since
investment and δ is the volatility. For the corrected All Ordinaries over the 60 years, h = 0.0468 and
δ = 0.1658.
   For example, if you invest $100,000 in equities when you are 65, you may like to know the risk

5 The growth of the logarithm of the gain is known as drift.
6 See technical appendix for derivation.                                          4/24                                           10:50 24/01/11
and return in 20 years when you turn 85. The expectation of gain is given by
                                  ln  g i =h×i=0.0468×20=0.936
                                             ∴ g i =e0.936 =2.55

Consequently you are counting on having $255,000, after inflation, at 85. But there is a 1 in 6
chance of being at or below the – δ line. The gain there is given by

                ln  g i =h×20−×  20=0.0468×20−0.1658×4.472=0.1945
                                 ∴ g i = e 0.1945 = 1.2147

and you would only have $121,000. And perhaps nothing at all if you had counted on an income of
4% of capital and had been drawing down while the investment fell in value!

   It is easy to see from the figure why the term random walk is applied to equity movements.
Extensive analysis for over eighty years in many markets has shown that stock prices are random
and can be modelled as if they are independent of one another. Each of the lines in Fig 3 is a
random walk with no segment dependent on any other. This is certainly true for yearly data that we
are using; there is more debate as the time scale reduces to daily intervals7. Sharemarket gains are
accurately modelled by randomly assigning numbers that fit a lognormal distribution to consecutive
  This means that past history can give the expected increase per year h, and volatility δ, but the
actual price movement is not predictable and what happens one month is independent of what
happens the next.
   A drunkards walk is named after the statistical problem of a drunk starting off from a lampost and
lurching around with no particular influence to go in a certain direction. The question is what is the
probability that the drunk will go a certain distance from the lamp. The all ordinaries index
corrected for inflation only and shown as the bottom curve of Fig 2 is a good example of a
drunkards walk because there is no direction (see Fig 15). The accumulation index corrected for
inflation (the 3rd curve), is an example of a random walk where there is some expectation of
movement away from the lampost given by h although the drunken lurches δ are much greater than
this movement. The drunk is on a slight incline as he carries away your wealth.

A measure of risk: the difference between investors and pensioners
   This picture conflicts in another important way with the sales pitch given by financiers when
presenting the 1st curve of Fig 2. When prices drop, they say “don't worry, they always bounce back
again!” In the brilliant bull run we have had since 1980, this has been often true; but it is not
something that can be relied upon for investing, and certainly not if you are also drawing down an
income. The ability to get back to the original expectation is a major distinction between investors
and pensioners.
   In the Table 1, the passive investor begins with wealth of 100 and is content to watch as his
capital undergoes some growth, a sudden recession, a bounce back to his original stake, and then
resumption of steady growth. The active investor, at the end of each year, adds 4 units to her
investment: e.g. in year 3, she finished with 119, started year 4 with 123, experienced a loss of
70/110 (the market is the passive investor) resulting in 78 at year end. The pensioner subtracts 4

7 Athough share data can give a “thickening” of the distribution tail on the high side, the assumption made here of its
  absence is conservative. Given that growth expectation needs 50 years of data to determine its sign (19), it is not
  clear whether there is sufficient data to determine the presence of the higher distribution moments.                                         5/24                                           10:50 24/01/11
instead of adding but otherwise the calculation is the same. The bounce back is not seen by the


                              Passive                  Investment             160
   Years   Active investor                Pensioner
                             Investor                 or Drawdown
    1           100            100           100            4                 140
    2           109            105           101            4
    3           119            110           101            4                 120
    4           78              70           62             4

    5           94              80           66             4                 100
    6           110             90           70             4                                                      investor
    7           127            100           73             4                  80                                   Passive
    8           136            104           72             4                                                      Investor
    9           148            110           72             4                  60                                  Pensioner
    10          159            115           71             4
    11          171            121           71             4                  40
    12          184            127           70             4
Table 1: Comparison of investor and pensioner                                   0
                                                                                    1 2 3 4 5 6 7 8 9 10 11 12

                                                                    Fig 4: A plot of Table 1

   The problem for the pensioner is that he wants to live. He needs no less than 4 units of wealth
each year. But if his total wealth suddenly halves, then his deduction doubles as a percentage and
this value is greater than the expected growth rate of the market as a whole: he will continue to fall
behind. He must allow for a calamity at the beginning by drawing down a pension perhaps half that
of the expected growth of the market, or gamble and hope to double his stake before a recession.
   In Fig 3, we see that a useful measure of this “bouncability” and hence of investment risk, is how
long the – δ curve takes to get back to the original value of 1.0. From the figure it is about 12 years
if you have suffered a 1 in 6 chance of being on this curve. It gives a measure of the number of
years needed to get back into the black after a share market drop. Of course this is not a prediction
of how long it will take for any given situation but it provides a useful comparison as we try to
improve the investment outcomes. The time iR, to go positive is just when gi is 1.0 again or when
ln(gi) = 0:

                                          ln  g i =i R h− i R =0
                                          or i R=     

    For our h = 0.0468 and δ = 0.1658, iR = 12.5 years as in Fig 3. Risk is lowered by playing off
gain and volatility in a portfolio. If we can halve the volatility, we quarter the risk. When an income
is drawn down, this risk increases markedly. The effect of taking a salary is to reduce h but not to
alter δ which is a property of the market. Calling the salary expectation r,

                                        ln  g i =i R h−r −  i R =0
                                        or i R=     

  If h = r, then iR = ∞ , and recovery is not impossible, but unlikely.                                              6/24                                               10:50 24/01/11
Fig 5: Comparison of investment with 5% return (i) supplemented by 5% income; (ii) the investment alone; (iii) a
retiree drawing a pension of 5%.
 4.00                                  4.00                                  4.00

 2.00                                  2.00                                  2.00

                                          +2 σ                                  +2 σ                               +2 σ
                                          +1σ                                   +1σ                                +1σ
                                          mean                                  mean                               mean
 1.00                                  1.00                                  1.00
                                          -1 σ                                  -1 σ                               -1 σ
                                          -2 σ                                  -2 σ                               -2 σ

 0.50                                  0.50                                  0.50
         2 6 10 14 18 22 26 30 34 38           2 6 10 14 18 22 26 30 34 38           2 6 10 14 18 22 26 30 34 38
        0 4 8 12 16 20 24 28 32 36            0 4 8 12 16 20 24 28 32 36            0 4 8 12 16 20 24 28 32 36
                years                                 years                                 years

   Fig 5 compares two investors with a retiree. The first contributes income equal to a 5% return
resulting in a 10% per annum increase; iR = 3.3 years ( =0.1658). The second adds no more to the
investment: iR = 11.5 years. The retiree lives off income alone, aiming to maintain capital: iR = ∞ .
Returns of less than 3% give iR > 25 years, an expected time of retirement.
   If you are an investor, starting at age 40, and putting away a good fraction of your salary for
retirement, then it is likely that you will easily match or exceed h as in (i). The effect of volatility on
your investments will be minor and you may expect a couple of years of negative growth which is
easy to shrug off.
   The retiree with an allocated pension or one who expected to preserve capital and spend
dividends, will expect to draw down 4 to 5% and will not be in a position to let time compensate for
market drops. There is no longer any mechanism to rejoin the original line of expectation. The line
is level where ever the market falls: if it decreased by 40% as in the GFC, then it will stay depressed
for life.
   Fig 5 also demonstrates the falsity of the financiers argument using the uncorrected
accumulation index as a guarantee of equity investment success. This has an uncorrected
expectation of h > 10%, like the the first plot. Bounceback is under 3 years. This is why a straight
regression line seems like a magnet for the investment. In reality, a retiree with dividends is
described by the last plot and has no ability to recover. Having spend the dividends, her capital
follows the last curve of Fig 2 – the drunkard's walk.
   In the next sections, results are given based on these considerations for an allocated pension.
However, the total wealth calculations are only marginally different for a pensioner in a simpler tax
free environment or where allowance is made for tax in the expected rate of return.

The allocated pension model
  There are three parts to this allocated pension model:
    •    The first is the superannuation pension fund. The model begins at age 65 where, unless you
         pass the work test, you can no longer contribute to the fund – it is frozen and you must
         deduct a fixed percentage of the balance starting with 5%. Thus the fund is a market
         determined source of cash.                                              7/24                                           10:50 24/01/11
    •   The second is the external money not needed for salary that comes from the compulsory
        deduction from the super fund. This is assumed to be saved and invested along similar
        market lines to the super fund. Returns are taxed which is estimated by a tax factor. I ignore
        non-concessional contributions up to 75 which allow escape from tax, since as will be seen,
        tax is of little effect up to this age.
    •   The third is the salary that you take from the external and super funds.
   The two most important quantities are your total wealth at the end of each year (30th June) which
is labelled wn for the nth year, and how much you take to live on, called salary and labelled zn. When
you are 65, you must draw down 5% from your tax free pension fund. This percentage increases as
you age. The wealth of this pension fund at the end of the year is labelled sn . It may be that you
don't need to use it all and so you invest it outside the pension fund in other investments, called
collectively the external fund, whose wealth is tn . Both funds are assumed to have the same
investment performance. The external fund is taxed, but this turns out to be negligible.
   Each year starts with the total wealth w n−1=s n−1t n−1 of the year before. It then undergoes a
market defined increase or decrease to arrive at an intermediate total wealth w n=  n  n before
                                                                                 s t
you make any withdrawal . Just before this time the compulsory amount dn is transferred from the
pension fund to the external fund but this does not affect the total wealth. Then as the last action of
the year, a salary z n=w n−wn =max×w n is taken where max is a fraction of the intermediate
                                        

 Fig 6: Schematic of an allocated pension
                                                      z(n-1)                                    z(n
                                             Ŧ(n-1)                                        Ŧ(n) )
         t(n-1)                                           t(n-1)                                      t(n)
                                gf(n-1)                                  gf(n)

                           d(n-1)                                     d(n)

                               g(n-1)                                        g(n)
         s(n-2)                                           s(n-1)                            š(n)

   The mechanics of an allocated pension are shown in the schematic for two arbitrary years n-1
and n. There are two streams: the super fund (s) and the external fund (t). On July 1 in year n the
amounts in the two streams are s n−1 and t n−1 . At this point the total wealth is
  w n−1=s n−1t n−1 . Through the year they make gains g n and g nf where f takes into
account the taxation of the external fund. At the beginning of the last week of the financial year in
June, a compulsory fraction d n of s n−1 , the balance on the previous July 1, is transferred from
the super fund to the external fund: the total wealth is w n= g n s n−1 g nf t n−1 =  n n . After this
                                                                                     s t
point, and in the last week, the year's salary z n is taken. The salary is taken first from the
external fund to limit tax and topped up from the super fund if necessary. After this deduction the
total wealth to begin the next financial year is w n=s nt n .
   The reason for defining an intermediate stage before the end of the year and before salary
deduction, is to enable the amount of the salary z n to be dependent on total wealth w n . After
trying a number of models, the simple deduction of a constant percentage of the total wealth in any
year was found to be stable. If, for example, the requirement is for a fixed percentage of the initial                                   8/24                                     10:50 24/01/11
stake, then an expectation is imposed on the performance of the fund which, since it is a random
walk, cannot be met – your expectations are irrelevant: in bad times the fund can be too rapidly
depleted to be able to recover. On the other hand, with the assumed model, as the fund drops you
might have to go back to work but the fund will be healthy in the long term even if you are not.
Hence the salary is given by:
          s 
    z n=  n t n −s nt n= wn −w n=max× w n
                                            
where max is a constant fraction determined by an analytical model described later. In the three
sets of results, the fraction max is chosen to make the expectation of growth (h) and income (r) the
same – with expectation of constant wealth; or h > r for increasing wealth, or h< r for decreasing
   More detailed analysis which includes and analytical version of the model is given in the
technical appendix.

Equity results
   Fig 7 shows the results for r = h in an approximation of constant wealth of 1.0. The salary figure
has a linear ordinate with units of % of initial stake at i=0. Against the background of mean and δ
curves are plotted 3 lines determined by sequences of gains from the dividend and inflation
corrected 1949-2009 AORD data. They are 83-2009, 1970-2009, 1949-1987; labelled by their first
year. If viewing in black and white, the legend order follows the position of the data points. The
following figures are similar.
  Overall characteristics are as expected. The super fund decays gently into old age as the
compulsory extractions bite. The external fund has a slow start because salaries are close to the
super fund contributions in the early years. As designed with h = r, the total wealth which is just the
sum of the super and external funds, is flat. The salary mean is also flat since it is a constant fraction
max of the wealth. The linear ordinate results in asymmetric δ curves.
   The +δ to –δ area encompasses 2/3 of the probability. The differences between these extremes are
not minor. To be at or above +δ with a chance of 1/6 ensures a carefree retirement. To drop below
-δ can be very uncomfortable; below -2δ, with a 1/40 chance, is disastrous.                                   9/24                                     10:50 24/01/11
                                                       salary                                                                                                                wealth
       10%                                                                                                              8.00

        7%                                                                                                              2.00
                                                                                                                                                                                                                                    +2 σ
        6%                                                                                                                                                                                                                          83
                                                                                                             +1σ        1.00
        5%                                                                                                   mean
                                                                                                             49         0.50
        4%                                                                                                                                                                                                                          49
        3%                                                                                                   -1 σ       0.25                                                                                                        -1 σ
                                                                                                             -2 σ
        2%                                                                                                                                                                                                                          -2 σ

        0%                                                                                                              0.06
                  2       6       10        14        18        22        26        30        34        38                              2       6       10        14        18        22        26        30        34        38
              0       4       8        12        16        20        24        28        32        36                           0           4       8        12        16        20        24        28        32        36
                                             years                                                                                                                 years

                                                 super fund                                                                                                       external fund
       4.00                                                                                                             10.00


                                                                                                             +2 σ        1.00                                                                                                       +2 σ
       0.50                                                                                                  83                                                                                                                     83
                                                                                                             70                                                                                                                     70
                                                                                                             49                                                                                                                     49
                                                                                                             +1σ                                                                                                                    +1σ
                                                                                                             mean        0.10                                                                                                       mean
       0.06                                                                                                  -1 σ                                                                                                                   -1 σ
                                                                                                             -2 σ                                                                                                                   -2 σ

       0.02                                                                                                              0.01
                  2       6       10        14        18        22        26        30        34        38                              2       6       10        14        18        22        26        30        34        38
              0       4       8        12        16        20        24        28        32        36                               0       4       8        12        16        20        24        28        32        36
                                             years                                                                                                                 years

    Fig 7: Results for r=h=0.0468 & δ = 0.1658 (the 49-09 AORD), with max=0.0457 for constant wealth.

   The curves show what a wonderful period we have had from '83 to now, even including the '87
crash and the '08 GFC. This has conditioned the expectations of the baby boomers who are now
retiring. The other two data curves are in strong contrast and yet have identical expectation and
volatility. Both the '70 and '49 demonstrate the return to expectation, the '49 twice with the salary
curve showing periods of half rations; you are not well off in your 90s. The '70 is miserable, falling
early on below -2δ and spending the 30 years to death inching back to parity.
    Perhaps we should be more careful and try to grow our wealth in retirement. Fig 8 shows the
position for r < h stemming from the requirement to keep wealth above 0.2 at -2δ after 35 years.
The '83 retiree is dampened but the salary still races off the graph – happy days! Not much has
changed for the other two. The slumps in the market are far too great to be overcome by slight
thrift.                                                                                                    10/24                                                                                                          10:50 24/01/11
  Fig 8: Salary and wealth for gain of 0.2 after 35 years at -2δ.

                                                      salary                                                                                                                  wealth
     12%                                                                                                               16.00

     10%                                                                                                                8.00

        8%                                                                                                                                                                                                    +2 σ
                                                                                                             +1σ        2.00
        6%                                                                                                   mean
                                                                                                             49         1.00                                                                                  49
        4%                                                                                                                                                                                                    70
                                                                                                             -1 σ       0.50                                                                                  -1 σ
                                                                                                             -2 σ
                                                                                                                                                                                                              -2 σ
        2%                                                                                                              0.25

        0%                                                                                                              0.13
                 2       6       10        14        18        22        26        30        34        38                                2       6        10        14        18        22     26 30 34 38
             0       4       8        12        16        20        24        28        32        36                                 0       4        8        12        16        20        24 28 32 36
                                            years                                                                                                                    years

   The last alternative is to spend capital which is one of the intentions of allocated pensions, but
hopefully not to die at the expected age which is the other design assumption. Fig 9 has r > h, with
– δ replacing -2δ in pinning wealth above 0.2 at 35 years. The mean wealth decreases to about 0.5.
We still don't need to worry about the '83 retiree who follows the +δ line which is flat for this r: a
max = 6% characterises this period. The '49 does better for the first 10 years and about the same as
Fig 9 before year 26 by drawing a higher salary with only a marginal disadvantage that year. The
'70 is also better off earlier and marginally below Fig 9 at the end. There is for both the option of
cashing in the wealth and spending it all in the last 10 years at ~5% of the original stake per annum.
 Fig 9: Salary and wealth for gain of 0.2 after 35 years at -δ.

                                                 s alary                                                                                                                 wealth
  12%                                                                                                               4 .0 0

  10%                                                                                                               2 .0 0

                                                                                                                    1 .0 0
   8%                                                                                                                                                                                                        +2 σ
                                                                                                            +1 σ    0 .5 0
                                                                                                                                                                                                             +1 σ
   6%                                                                                                       m ean
                                                                                                                                                                                                             m ea n
                                                                                                            49      0 .2 5                                                                                   49
   4%                                                                                                                                                                                                        70
                                                                                                            -1 σ    0 .1 3                                                                                   -1 σ
                                                                                                            -2 σ
                                                                                                                                                                                                             -2 σ
   2%                                                                                                               0 .0 6

   0%                                                                                                               0 .0 3
             2       6    10 14 18 22 26 30 34 38                                                                                2           6       1 0 1 4 18 2 2 2 6 3 0 3 4 3 8
        0        4       8 1 2 1 6 2 0 2 4 28 3 2 3 6                                                                        0           4       8      12 16 20 24 28 32 36
                                      ye a rs                                                                                                                   ye a rs

   It is dangerous to draw conclusions from specific examples. The conclusions are made with
hindsight. If you know your investment will do badly, then of course you should pull your money
out which is the equivalent of taking a very high salary. You will then do better. But do we face the
heaven of '83 or the purgatory and too-late redemption of '70? Can it be even more depressed? An
interesting question is whether the market would exist if it followed the -2δ for 35 years.                                                                                                    11/24                                                                                     10:50 24/01/11
   There are two problems with equities for the retiree which stem from their high volatility. One is
the real probability that equities will not deliver their expected gain over the retirement period. The
second is that the withdrawal must be kept low enough to keep much of the wealth until close to
death. If your date of death was known accurately, then well before you could cash up and spend the
capital. With equities, to assure your future, you never get to spend your wealth.
   There is no effective way of greatly reducing the systemic volatility in this asset class without
derivatives which themselves increase risk. The partitioning into different equity classes by the
financial industry is a pretence at volatility reduction through diversity. The real aim is to create
multiple industry winners with a confused market. This in turn provides reasons for advisors and
products to exist, with rebalancing between multiple packages of the same product as a mechanism
to extract income.

A proxy portfolio 
  So far I have used the All Ordinaries index as a proxy for equity investment. To examine lower
volatilities, the zero risk line connecting the All Ordinaries index with Treasury capital indexed
bonds will be used as the proxy for a retiree's investment universe. The points along this line give




                                                                                 ZR line


                                               0   0.05    0.1      0.15   0.2
                             Fig 10: Zero risk line for AORDS. (0.168,0.048) to (0,

returns and volatilities of portfolios that mix equities and bonds.
   The justification for this choice is not that it represents optimum portfolios, but that it is
representative of what a retiree might expect from diverse investments. Index-following funds
historically fall within the upper quartile of funds and can represent the “efficient frontier” of
portfolio analysis. The line to treasury bonds then gives the range of investment portfolios that can
be achieved by a retiree with a mix of investments.

Combined fixed & equity8
   Treasury capital index bonds pay a coupon interest on a face value of $100 each. This face value
is indexed and hence the quarterly interest payments and value at maturity are also indexed. I
assume that they have no correlation with equities and that volatility is zero if kept to maturity. The
real yield depends on the initial market value and is about 2.45% for the 2025 today, assuming
8 Compare with US analysis: and: Ashton,
  Michael, Maximizing Personal Surplus: Liability-Driven Investment for Individuals (October 28, 2010). Available at
  SSRN:                                            12/24                                   10:50 24/01/11
constant reinvestment. A portfolio ranging from zero risk to high risk depends on the ratio of bonds
to equities. Because of the payment of indexed face value at maturity, the income from a bond is
uneven. Here we will assume that we are dealing with a number of bonds at different maturities,
managed to provide a uniform yield9.
  An example of a 50:50 combination of fixed and equity investments is sufficient to demonstrate.
The portfolio gain and volatility are given by:
                                                 g e g f         e h eh  e          f

                                         h p=ln           =ln           
                                                      2               2                                                                        (6)
                                          p = e / 22  f /22= e /2

where subscripts p, e and f refer to the combined portfolio, equities and fixed rate investments; x is
the proportion of equities, g is the gain. The zero volatility of treasury bonds reduces the portfolio
volatility to the equity proportion. For x = 0.5, hp = 0.0356 and δp = 0.0829 bringing the equity
values down from 0.0468 and 0.1658 respectively. (iR = 5.5 cf 12.5 yrs).

Fig 11: Salary and wealth for r=h=0.0356 for an equal mix of bonds and equities.





                                                                      +2 σ 1.00
                                                                      mean 0.50
    4%                                                                49
    3%                                                                -1 σ 0.25
                                                                      -2 σ


          1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
         0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38                            1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
                                                                                          0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

   Comparing Fig 11 with Fig 7 shows that the risk has halved as expected but with the cost of
reducing income from 4.57% to 3.49% – a reduction of a quarter. The salary of our mournful friend
of '70 did not go below 2% and the '49 now looks acceptable with a salary above 3% for most of his
life. '83 booms on.

Duplex fixed & equity­­ consuming the fixed
   The problem with the combined portfolio is assurance: risk makes you keep your capital. The
lower risk does not fully compensate the reduced return. As an alternative, we can take advantage
of the predictability of capital indexed bonds, to consume their capital entirely without risk if they
9 This is an artificial assumption but even if the investment is made through a fund, the volatility is much smaller
  than that of equities and correlations can be usefully negative when taken over periods of three years or more.                                                   13/24                                                         10:50 24/01/11
are dealt with separately from finite risk investments.

  Beginning with a capital indexed stake p and gain g, we look for v, the fraction of p, that we can
draw down over n periods, leaving nothing remaining.
                                  if  pg−vp g−vp g −vp −vp=0
                                  then pg n = pv  g n−1g n−2 g1
                                             n−1                                                                                 (7)
                                                               1−1 /g 
                                  and v=1/ ∑ 11/ g i =
                                             i=1               1−1/ g n
   For g =1.0245 and n = 35, v = 0.04185. This is a risk free return comparable to that of equities
although in exchange for losing all the wealth at age 100.

Fig 12: Salary and wealth for a 50:50 portfolio of equities and risk free capital indexed bonds. Equity h = r = 0.468.
Bond income is 0.04185 of initial stake. Compare with Fig 7 and Fig 11.

                           salary duplex                                                       wealth duplex
 10.0%                                                                8.00


  7.0%                                                                2.00
                                                               +2 σ                                                              +2 σ
                                                               +1σ                                                               83
                                                               83     1.00                                                       +1σ
  5.0%                                                         mean                                                              mean
                                                               49                                                                49
  4.0%                                                         70                                                                70
                                                               -1 σ                                                              -1 σ
  3.0%                                                         -2 σ   0.25                                                       -2 σ

  0.0%                                                                0.06
          1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35                    1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
         0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36                 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

   Fig 12 shows the portfolio with equities kept at constant wealth and bonds that are depleted after
35 years. The wealth expectation moves to 0.5 and the -2σ risk increases as a proportion of the
original total capital as the bonds decrease. However the salary risk is much decreased. Even our
'70 retiree has done well and the '83 is doing better when compared with the combination portfolio
because the bonds are providing more of the income. The combination is better at preserving
wealth, but the duplex is much better at preserving salary and still providing opportunity to increase
salary in good times.

  High gain investments carry such high risks that capital can not be consumed safely. Low risk
investments can be consumed but offer no flexibility to spend fractions of the capital in
emergencies. They also have low returns if not consumed. A solution is to mix the high and low risk
and consume the low.
  The result is a good chance of a 3% return on capital, leaving about ¼ of your wealth at death.
This low return makes it exceptionally important to reduce fees and charges.                                                14/24                                                 10:50 24/01/11
Charges & expenses
   Perhaps the greatest scandal of the financial industry is the gouging of investors by the alliance of
financial sector greed and government incompetence. It results in the widely used benchmark that
the financial industry deserves an income of 1% of your capital each year. This is the standard
commission for running an allocated pension. It is also the standard for running an SMSF which is
in pension mode.
   When you look at the curves above, it is clear that you are likely to draw down no more than 6%
of what you have saved as salary and there is a good chance that for extended periods this will be
reduced to 2% and average 3% overall. Thus the financial industry will take 30% of your income
and hence 30% of the use of your own capital until death. You have spent 45 years of your life
building this capital. Their few days or hours of administration each year adds no value to you nor
to the country.
   The only principle for charging must be expressed as a percentage of your income, not of your
capital. The larceny of the current system would then be apparent. The only reason we cannot
escape from the financial industry is that legislation makes what should be now extremely simple,
extremely complex and we must delegate to specialists. But as well as adding no value to us, it
adds no value to the government through taxes. There is no reason why administration and taxation
of a super fund in pension mode should not be simpler than a domestic electronic tax return.

  Reduction of expenses is critical. You cannot survive anything other than the '83 bonanza with a
1% expense. You will have nothing for periods of years if that extra 1% deduction was put in.
   The underlying value of the stockmarket is close to constant and its return compensates for
inflation over the long term but nothing more. Note that it does not protect against inflation: the
'70s curve shows this. Dividends provide real income.
  The volatility of the index means that there is a reasonable probability of not having your
expected retirement prosperity if investing wholly in equities. When evaluating investments, the
past volatility is as important to know as the drift, and to compare both with the index.
   Lowering risk using fixed interest investments and risk free consumption of capital can improve
retirement salaries.                                 15/24                                    10:50 24/01/11
Technical appendix

Random Walks10
   A random walk analysis of equities characterises the past with just two parameters, and uses
them to deduce risk and reward for future possibilities. The two parameters are measures of
expected advance in a certain direction and the risk of deviation. The drunkard's walk is the special
case of zero expectation of advance which can approximate some stockmarket histories. The drunk
starts from a lamp post, staggers around with no purposeful direction and where he gets to
determines our future wealth.
   Take a game where two coins are flipped and if you get 2 heads (HH) you advance 2 spaces, if a
head and a tail (HT) you advance 1 space and if two tails you don't advance at all. Your expectation
of making a step is the sum of each of the steps multiplied by their probability. The probabilities are
¼ for HH and TT, and ½ for HT since there are two ways they can come up. Consequently the
expectation and variance are:
                                            1        1   1
                                   E  s= ⋅2 ⋅1 ⋅0=1
                                            4        2   4
                                                       1   1
                            s ²=E  s² − E  s ²= ⋅4 ⋅1−1=0.5
                                                       4   2
With the dispersion σs = 0.7 and less than the expected advance, players are likely to stay within a
competitive range of one another.
  In roulette, the wheel has 36 numbers and one star. If betting odds and evens or on colours, the
house has 1/37 chance each play. With dispersion much greater than expected advance –
                                        18        18   1     1
                                E  s =    ⋅1− ⋅1− ⋅1=−
                                        37        37   37    37
                                                    18    18
                         s ²=E  s² − E  s ²= ⋅1² ⋅−1 ²−1=1
                                                    37    37
frequent wins mask your slow decline.
   In general we can think of the position or price after i steps. Starting with p0 , the initial stake, p(i)
is the sum of the steps after i periods which will be years in our application.
                                            p i= p0 ∑ s j                                                 (10)

and where each step can have a different expectation                E  s j =h j and variance

                                       j 2=E  s j2  –  E  s j 2                                     (11)

   We assume that the steps have independent probability and so the expected value of i steps is the
sum of the expectations of each step; the expectation of the product of two different step lengths is
the product of their expectations. Applying these assumption to (10) —

10 The statistics and examples come from M.F.M. Osbourne, The Stockmarket and Finance from a Physicist's
   Viewpoint (Crossgar, 1977). Wikipedia gives excellent explanations of all the terms used.                                         16/24                                    10:50 24/01/11
                                                                   i                 i                                     i
                        E  pi = p0 E ∑ s j = p 0∑ E  s j = p0 ∑ h j                                                                         (12)
                                                                   j=1          j =1                                       j=1

                       E  p i            = E  p0 ∑ h j  ²
                                                    i                                    i                  i          i                             (13)
                                             = ∑ h j  p 0 ²2 p 0 ∑ h j ∑
                                                                                                                    ∑            h k hl
                                                    j=1                              j =1               k =1  l ≠k l =1


                                  = E  p 0∑ s j  ²
                   E  pi 
                                                              j =1
                                               i                                 i                      i          i
                                  = E  ∑ s j  p 0 ²2 p 0 ∑ s j ∑                                             ∑
                                                                                                                            sk sl                   (14)
                                               j=1                              j=1                 k =1  l ≠k l =1
                                         i                                                    i               i       i
                                  = ∑ h j 2 j 2 p 0 ²2 p 0 ∑ h j ∑                                                  ∑        h k hl
                                         j=1                                                 j =1               k =1  l ≠k l =1

where the expectations of                s j ² have been resolved by the substitution of (11) . Subtracting (13)
from (14) gives

                                           p ²=E  p i ²− E  pi  ²=∑  j ²

 (12) and (15) are classic statistics – the expected value of a sum of independent variables is the
sum of the expected values; the variance of the sum of independent variables is the sum of the
  With no information of the future, we will assume that each step has the same expected value h
and dispersion  . We find the simple result –
                                                     p = i  , E  pi = p0 i h

  With this equation, the relationship between the random walk and Fig 1 is shown in Fig 13. Year
zero is the present. The risk of the past is known and is given by the standard deviations lines
running parallel to the mean. The uncertainty of the future increases by the square root of the time.                                                               17/24                                                         10:50 24/01/11



                               6                                                            +2δ


                                     -58 -52 -46 -40 -34 -28 -22 -16 -10 -4 2 8 14 20
                                   -61 -55 -49 -43 -37 -31 -25 -19 -13 -7 -1 5 11 17
                   Fig 13: The ln(gain) of share data against time. Year zero is the present. Linear
                   regression determines the volatility of the past; random walk analysis applies
                   this volatility to determine the probabilities of random walks in the future .

Walking randomly with shares
   We need to choose a probability distribution that fits the stockmarket. (16) has general
application to distributions provided the assumptions hold. When we model the sharemarket we are
interested in gain. If the index or share price for successive periods is p 0, p 1, … , p n then the gain
for each period is:
                                         p1 p2    pi
                                           , ,…,       = g 1 , g 2 ,… , g i                                    (17)
                                         p0 p1   p i−1

and the total gain after i periods is the product g 1⋅g 2 , ... ,⋅g i . Therefore the distribution we need
must be able to model probability products. However the normal or Gaussian distribution applies
to additive probabilities like randomly throwing balls into many buckets and though useful for very
high gains, does not apply for those near zero: it does not manage the product of gains giving results
unlimited above unity but constricted by zero for those less than one.
   The lognormal distribution is the best known for our purpose. This distribution simply transforms
(17) into an additive normal distribution by taking logs (remember ln a⋅b=ln aln b and
  ln a/b =ln  a−ln b ). The log of gain after i periods is ln  g 1…ln  g i  .
  For this distribution,  is known as the volatility and h, in this logarithmic average, as the drift.
Thus the expected gain is
                                                 ln  g i =h i±  i .                                        (18)
  (Note the equals sign is not really an equals sign because the “equation” refers to three separate
solutions: the mean and the two standard deviation curves.)                                                18/24                                  10:50 24/01/11
   As an example, take the uncorrected share data from the Australian all ordinaries for the last 30
 years (Table 2). The third, fourth and fifth columns are unnecessary but are included for clarity. The
 Δln(AORD) column is computed directly from the AORD data and its standard deviation and mean
 gives h and .
    Normally the two parameters are determined by the previous history and so would apply from
 2010 onward. The AORD data plot thus has no significance other than as a check on whether the
 mean and  lines are sensible. We see that the last point on the mean corresponds with the sum of
 the data as it must – ln(4065/418) = 2.28. The mean of the random walk is simply the straight line
 between the initial and last point.
    This was the walk of the past, but the future can be almost anything although we don't expect
 either the market to collapse to zero or have a gain of 10 in any one year. The probability to exceed
   ± is 1 in 3 and ±2  is 1in 20 but of most interest is the risk of loss which is 1 in 6 to
 exceed − and 1 in 40 for −2  . The peaks were in 1987 and 2008. If we had started the
 data set a year later, all the points would be dropped by 0.4. Gains at the beginning pad out the
 future; losses are hard to claw back.

                          h         0.073                                                     Figure for Table 1
                          δ         0.159
    Date       AORD      GAIN      Ln(GAIN)     Ln(AORD) Δln(AORD)          4
        1979       418         1            0         6.04         0
        1980       624      1.49          0.4         6.44       0.4      3.5
        1981       650      1.04         0.04         6.48      0.04
        1982       496      0.76        -0.27         6.21     -0.27        3
        1983       627      1.26         0.23         6.44      0.23
        1984       730      1.17         0.15         6.59      0.15      2.5
        1985       896      1.23          0.2          6.8       0.2                                                           2δ
        1986      1207      1.35          0.3          7.1       0.3
                                                                            2                                                  δ
        1987      1732      1.44         0.36         7.46      0.36
        1988      1475      0.85        -0.16          7.3     -0.16                                                           mean
        1989      1581      1.07         0.07         7.37      0.07
        1990      1496      0.95        -0.06         7.31     -0.06
        1991      1508      1.01         0.01         7.32      0.01        1                                                  -2 δ
        1992      1570      1.04         0.04         7.36      0.04                                                           All ORDS
        1993      1808      1.15         0.14          7.5      0.14      0.5
        1994      2066      1.14         0.13         7.63      0.13
        1995      2037      0.99        -0.01         7.62     -0.01        0
        1996      2274      1.12         0.11         7.73      0.11
        1997      2553      1.12         0.12         7.85      0.12           1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
                                                                         -0.5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
        1998      2673      1.05         0.05         7.89      0.05
        1999      2977      1.11         0.11            8      0.11
        2000      3175      1.07         0.06         8.06      0.06
        2001      3229      1.02         0.02         8.08      0.02                           years
        2002      3176      0.98        -0.02         8.06     -0.02
        2003      3055      0.96        -0.04         8.02     -0.04
        2004      3354       1.1         0.09         8.12      0.09    Fig 14: ln(gi) vs i – equ. (7), for h and δ from the table with
        2005      4278      1.28         0.24         8.36      0.24
        2006      5081      1.19         0.17         8.53      0.17
                                                                        the AORD superimposed as the random walk.
        2007      6230      1.23          0.2         8.74       0.2
        2008      5020      0.81        -0.22         8.52     -0.22
        2009      4065      0.81        -0.21         8.31     -0.21

Table 2: Uncorrected AORD index data as calculation example.
h and δ are the mean and standard deviation of columns 4 or 6.

 The use of the chart
    For regression of known data such as Fig 1, the  and h are known. The random walk begins
 where this leaves off; the future is marked by increasing uncertainty indicated by the increasing 
 with time (i) in (16) and Fig 13. The random walk view is that there are uncountable different
 futures, ranging from those with positive gains every year through to the highly negative every year
 for decades. It gives the probabilities for each of these. An extreme outcome can rewrite the
 regression curves.                                               19/24                                            10:50 24/01/11
   One use of the random walk is to specify these probabilities at the outset of the investment and to
move it forward on the past-future border. A second use is to apply it at the outset but keep it fixed
and plot outcomes on it. This is not a rational application since the walk can only begin from the
last known point – you are denying the existence of the past. Best practice is to shift the origin to
the present market position and observe the probability of regaining your initial stake.

   The data set from which h and  is derived is critical. That of Fig 2 is monthly data from 1948
to now, corrected for inflation and for dividends. It is chosen because I could cobble together the
data and because it includes much of the dramatic variation needed to model risk. The drift and
volatility are close to the total history shown in Fig 1.
    One of the scandals of Australian finance is that the standard benchmark of the ASX300
accumulation index, or any accumulation index, is not available freely to investors and is held by an
overseas company. In the past I have managed to get dividend information and the some of the
ASX200 Accumulation index and found that the market averages about 3.8% dividend yield. It
would be greater in an allocated pension fund, with tax-free, fully franked income. Because of the
variation between portfolios that might include non-franked shares such as overseas and
speculative, I have left it at 3.8%.
   The data shown in Fig 2 is summarised in Table 3. Financiers spruik the top two curves. The
bottom one is never mentioned. Over the 60 years, the market went up by 50 times, inflation by 25
times giving the index growth rate of 2 times, or 1.2% return per annum. Shares as a whole do not
intrinsically increase in value; dividends provide wealth. Capital growth is a drunkard's walk.
   This conclusion is supported by Fig 1 – the drift is only 5% since 1875 and is likely to be
accounted fully by inflation. MFMO11 makes the same point about the American stock exchange. It
is in fact quite difficult to tell, given a run of data, whether a stock or index is going up at all in the
long term. A scientific criterion is accuracy ≥2  or, similar to our risk measure of (4)

                                         hi−2  i =0
                                     2         2

                                     =4   
                                        0.0468      
                                                 =50 years of data!

   for the inflation and dividend corrected (3rd curve) of Fig 2. But it is 957 years for
the fourth curve or Fig 15!

                                                   1949-2009                          1984-2009
       AORD                                     h              δ                h                  δ
       Index                                 0.0623         0.1544           0.0718             0.1401
       Corrected for inflation               0.0107         0.1655           0.0344             0.1384
       Dividends and inflation               0.0468         0.1658           0.0717             0.1384
       Table 3: Drift and volatility for two AORD periods corresponding to Fig 2. Derived from monthly data.

11 M.F.M. Osbourne, The Stockmarket and Finance from a Physicist's Viewpoint (Crossgar, 1977).                                        20/24                                         10:50 24/01/11
Fig 15: Detail of the bottom curve of Fig 2. Log of the all ordinaries index corrected for inflation from 1950 to 2010. A
10 year moving linear regression and standard deviation difference lines are superimposed with no lag correction.



                                                                                                                                         10yr lin reg
   4.5                                                                                                                                   -δ
























































   It might be that the slight upward movement (0.0107) is merely a discrepancy between the
commodity of companies' values and those in the inflation statistician's basket of goods. A good
first approximation is that the inflation corrected index is a directionless set of random numbers. A
curious result for the metric of the financial world.
   Fig 2 shows there is a bull run from '52 to '70, a collapse from '70 to '82, and a strong bull from
'82 to '08. There are 12 years of negative return from '70 to '82 even including dividends. The total
period includes many of the elements needed to examine risk.
   A rule-of-thumb for yield in retirement is 3% above inflation, often assumed as 4% in the long
term, and therefore requiring 7% return. A balanced portfolio of cash, bonds, property and equities
would pull the 4.8% above inflation here, easily down to 3% but with an important reduction in
volatility. There is no obvious contradiction. But Fig 2 and equation (19) show that such rules-of-
thumb may average out only over a time greater than from retirement to death.

What is inflation?
   In Australia, inflation is measured by the consumer price index or CPI and this is used here. The
problem is whether it fits a retiree's experience of inflation. The ABS measures an employee to have
an inflation rate 110% of CPI and a self employed retiree 97% of CPI. But this does not measure
the constraints of income. The employee and retiree may well be spending up to the limits of
income and risk (and beyond with credit), but the CPI measures the change in the cost of things
they buy, and not how much. Thus the other side of the equation is average weekly earnings which
gives another measure of inflation and which can be up to CPI+2%.
   A social measure that combines the two rates and applies to retirees is the inflation rate used by
three separate defined pensions: parliamentarians (the rulers), old-age pensions (those who can
influence the rulers) and normal defined pensions such as the armed services and public servants
(the ruled). The first two have pensions which are indexed according to the higher of the CPI or
average weekly earnings. The third is the CPI alone.
   Over the last twenty years, the ruler's pension has gone up 140%, those with influence 130%, and
the ruled by 70%.                                                                21/24                                                   10:50 24/01/11
  The use of the CPI to index income means that over 25 years of retirement on the indexed
salaries calculated here, you will have half the buying power of those indexed to wages.
  If you can expect an income from your investment of 3% above the CPI in retirement, then for
$30,000 income you need an initial stake of $1,000,000. In 25 years, the value of your income will
have fallen below that of the old age pension.

The allocated pension model

                                                          z(n-1)                                z(n)
                                                 Ŧ(n-1)                                  Ŧ(n)
         t(n-2)                     gf(n-1)                  t(n-1)            gf(n)               t(n)

                                d(n-1)                                   d(n)

                                   g(n-1)                                       g(n)
         s(n-2)                                              s(n-1)                    š(n)
Fig 16: Schematic of an allocated pension

  The equations used in computing the figures are:
                            s s
   0=t 0=t 0=d 0 ;1=w 0=s 0=  1=  0
     n= g n−d n s n−1
     n=t n−1 g nf d n s n−1
    w n=s nt n
    t n=  n – max×wn
         t                  if  n≥max×wn , otherwise 0
    s n=  n− max w n−  n−t n  if
         s                t                          t
                                              max×wn  n −t n , otherwise       s
    z n=  n  n −s nt n= wn −w n=max× w n
          s t                               
   The factor f is taken to be 0.90. This is consistent with data given by Vanguard for the effect of
taxation on its indexed funds, assuming 30% marginal tax. In any case, because it takes a decade or
more to establish the external fund comparable to the super fund, the sensitivity to taxation is low.

How much salary can be taken?
   These relations give an accurate description of the model and are the basis of the figures. But we
can find a much simpler analytic description that is sufficiently accurate for our purpose, answers
this question and gives a good sense of the playoffs between the various factors.
  At the beginning of the year i, the wealth is wi-1 which changes with expectation ĥi and volatility                                     22/24                                10:50 24/01/11
δ to ŵ ( ĥi is not h nor necessarily constant in time because of the tax-affected external fund).
Salary is deducted to bring the wealth to wn; this deduction can be characterised by a fixed
expectation r which has zero volatility – we determine it. Thus the gain in each year is

                      gi           wi wi
                                               wi
                                                         wi
              ln          =ln      ⋅ =ln                             
                                                    ln  = hi −r    hi −r ±                (20)
                     g i−1        wi−1 wi
                                              wi−1       wi

and since ŵ(1-max)=wi, r = ln(ŵ/wi ) = ln(1-max):
                                      max=1−e−r .                                                   (21)
 Since both external and super funds are subject to the same market volatility, the only unknown is
ĥi. Writing hf as the expectation of the external fund, we look for an approximation:

                                  g s g f t     eh s i−1e h t i −1 h
                                  = i i−1 i i−1 =                    ≃e     i
                            wi −1     s i−1t i−1    s i−1t i−1

   A comparison of the super and external funds in Fig 7 shows that the external fund is much
smaller than the super fund for the first 25 years and so the difference in their gains due to tax can
be neglected. The figures have been calculated with an f=0.9, and it is only from years 25 to 35 that
there is a noticeable tax effect and it remains within 3% of wealth. We can therefore make the
simple approximation that ĥi = h. For the first 20 years it is accurate to within 1% of w.
  In this approximation we can write the wealth as
                                      ln wi =h−r  i±  i .                                     (23)


  Wealth and salary are constant when ln(wi) =0 or h = r. From (21), max = 1-e-h = 0.0457 for the
preferred h=0.0468 of Table 3.


   A criterion is to ensure that the salary is not so large that it completely depletes our wealth before
we die. Inserting a definite negative volatility into (23) and rearranging, the general expression for
the salary expectation is:
                                                   wi  k 
                                        r =h−ln         −                                           (24)
                                                    i     i
   where k is a constant determining probability in units of δ and where (i, wi, kδ) fix the probability
of the minimum gain at a certain future time for a distribution (h, δ).
  The -2δ line has a probability of 1 in 40 of being exceeded and a conservative choice is to keep
wi above 0.2 w 0 over the next 35 years. Substituting i=35 , w i=0.2 in (24),
                                    r =h−0.33806 0.04598                                          (25)
 showing the playoff between h, r and δ. You must increase h to increase r or decrease δ. For our
preferred h and δ of 0.0468 and 0.1658, r = 0.0367 or max=0.0361. If you had the misfortune to
reach this (i,wi) point, your salary would only be 0.0072 of your original stake. But at age 100 it
may be time to be reckless and consume your 0.2 capital over the next five years.                                     23/24                                 10:50 24/01/11
      years i       wi, gain at i         kδ               r               max

         35             0.14              2δ               h             0.04572           Fig 7

         35              0.2              2δ            0.03673          0.03607           Fig 8

         35              0.2              δ             0.64760          0.06271           Fig 9

Table 4: Parameters calculate from (24); h and δ are the drift and volatility from Table 3 for
AORD 1949-2009 corrected for dividends and inflation.                                24/24                                   10:50 24/01/11

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