The annuity function is basically the sum of the first n terms, e.g. n = 45, of a geometric series of the form
a ( 1 + b + b2 + b3 + b4 + ……..)
where a and b are constants. The first n terms can be reduced to a compact or closed form by multiplying this series of terms by the quantity ( b-1) which cancels out all powers of b except for the nth power and dividing the result by ( b-1) so that the sum of the first n terms is preserved. That is
a( 1 + b + b2 + b3 + b4 + …. + bn-1 )( b-1) / (b-1) = a ( bn – 1) / (b-1)
In an annuity account an amount p is invested each year for n years and earns an annual compound rate of return r. In the first year the account value is p, in the second year it is p + p( 1+r), and in the third year it is p + p(1+r) + p(1+r)2, etc. So it can be seen that the account value in the nth year is the sum of n terms where the nth term is p(1+r)n-1 and that the sum in compact form is
p [ (1+r)n – 1 ] / [ (1+r) - 1] = p [ (1+r)n – 1 ] / r = p[ M(r,n) ]
where the multiplier function
M(r,n) = [(1+r)n - 1] /r
is the not-so-well-known annuity function.
The inflated annuity function is a little more complicated and a little more realistic in that it allows for inflation. So let i be the annual inflation rate, taken to be constant, let r be the real rate of return and a = r + i the apparent rate of return. Let p be the initial invested amount, p(1+i) the amount invested in year 2, p(1+i)2 the amount in year 3, etc., so that p(1+i)n-1 is the amount invested in year n.
In order to see how the inflated annuity function is derived, the account values for the first 4 years are tabulated.
year account value
1 p
2 p(1+i) + p(1+a)
3 p(1+i)2 + p(1+i)(1+a) + p(1+a)2
4 p(1+i)3 + p(1+i)2 (1+ a) + p(1+i)(1+a)2 + p(1+a)3
or
p(1+i)3 [ 1 + (1+a) / (1+i) + (1+a)2 / (1+i)2 + (1+a)3 / (1+i)3 ]
Noting that (1+a)/(1+i) = (1+ r + i)/(1+i) = 1 + r/(1+i) = 1 + re
where re is the effective rate of return, the account value in year n is
p(1+i)n-1 [(1+re)n – 1] / [(1+re) - 1] = p(1+i)n-1 [(1+re)n – 1 ] / [ re ]
= p(1+i)n-1 M(re ,n)
where in the case of interest p = .04W.
The derivation of the maximum withdrawal rate taking into account an
assumed constant annual inflation rate i is similar to the
derivation of the inflated annuity. In the derivation, r is the real rate of
return, a = r + i is the apparent return rate, and re
= r / (1+i) is the effective return rate. To counter the effect of
inflation, we take p % of the initial account value A as the first
withdrawal,
In order to obtain the expression for account balance at the end of n years, the details for the first 4 years are tabulated as follows:
year withdrawal account value at end of year
1 Ap A(1-p)(1+a)
2 Ap(1+i) A((1-p)(1+a)2 – p(1+i)(1+a))
3 Ap(1+i)2 A((1-p)(1+a)3 - p(1+i)(1+a)2- p(1+i)2(1+a))
4 Ap(1+i)3 A((1-p)(1+a)4 - p(1+i)(1+a)3 – p(1+i)2(1+a)2 – p(1+i)3(1+a))
or A((1-p)(1+a)4 - p(1+i)4((1+a)3 / (1+i)3 + (1+a2 / (1+i)2 + (1+a) / (1+i))
Noting again that (1+a) / (1+i)) = (1+r+i) / (1+i) = 1 + r / (1+i) = 1 + re
And that, in solving for a maximum value of p, we require that the account value at end of year n be Ap(1+i)n in order to have one withdrawal left, we have:
A[(1-p)(1+a)n – p(1+i)n[(1+re)n-1+(1+re)n-2+…..+(1+re)]] = Ap(1+i)n
or
A(1-p)(1+a)n = Ap(1+i)n [(1+re)n-1+(1+re)n-2+…..+(1+re)+1]
= Ap(1+i)n M(re , n).
Dividing both sides by A(1+i)n and using the ratio of (1+a) to (1+i) as noted above
we have:
(1-p)(1+re)n = pM(re,n)
or
p = [(1+re)n] / [(1+re)n + M(re,n)]
Note that re is derived from knowledge of the gain rate a and the inflation rate i.
Table 1. below shows the data derived from the EOY ( end-of year) values of the NYSE Composite Index from 1939 to 2004 and used to analyze the gain in the Index over multi-year periods. The tabulated data are specified by the last two digits of each year from 1940 through 2003, e.g. 40 for 1940 , 00 for 2000. The Index data for 2004 were referenced to a different base than that used for all the other data and were therefore not used. The data in the first column to the right of a year-column are the corresponding EOY or Index values which are a weighted average of the Index high, low, and closing values. The high and low values were each weighted ¼ and the closing values were weighted ½. For example the Index value for year 40 is 6.52 and the Index value for year 61 is 36.64. This weighted averaging was necessary to get a reasonable EOY value because of the large differences within the high, low, and closing values.
| Yr | Index | Gain | Yr | Index | Gain | Yr | Index | Gain | ||
|---|---|---|---|---|---|---|---|---|---|---|
| 40 | 6.52 | 1.000 | 62 | 33.46 | .9132 | 84 | 94.00 | 1.0168 | ||
| 41 | 5.61 | .8604 | 63 | 38.54 | 1.1518 | 85 | 114.92 | 1.2226 | ||
| 42 | 5.61 | 1.000 | 64 | 44.57 | 1.1565 | 86 | 135.17 | 1.1762 | ||
| 43 | 6.98 | 1.2442 | 65 | 48.41 | 1.0862 | 87 | 147.59 | 1.0919 | ||
| 44 | 7.88 | 1.1289 | 66 | 44.52 | .9196 | 88 | 152.17 | 1.0310 | ||
| 45 | 10.11 | 1.2830 | 67 | 51.39 | 1.1543 | 89 | 186.09 | 1.2229 | ||
| 46 | 9.91 | .9802 | 68 | 56.94 | 1.1080 | 90 | 181.08 | .9731 | ||
| 47 | 9.19 | .9273 | 69 | 52.92 | .9294 | 91 | 214.82 | 1.1863 | ||
| 48 | 9.17 | .9978 | 70 | 47.63 | .9000 | 92 | 235.11 | 1.0945 | ||
| 49 | 9.49 | 1.0349 | 71 | 55.06 | 1.1560 | 93 | 253.76 | 1.0793 | ||
| 50 | 11.47 | 1.2086 | 72 | 62.58 | 1.1366 | 94 | 253.18 | .9977 | ||
| 51 | 13.34 | 1.1630 | 73 | 54.54 | .8715 | 95 | 310.23 | 1.2253 | ||
| 52 | 14.20 | 1.0645 | 74 | 39.24 | .7195 | 96 | 376.22 | 1.2127 | ||
| 53 | 13.65 | .9613 | 75 | 45.90 | 1.1697 | 97 | 481.54 | 1.2799 | ||
| 54 | 17.98 | 1.3172 | 76 | 55.42 | 1.2074 | 98 | 567.39 | 1.1783 | ||
| 55 | 22.55 | 1.2542 | 77 | 53.12 | .9585 | 99 | 634.97 | 1.1191 | ||
| 56 | 24.29 | 1.0772 | 78 | 54.00 | 1.0166 | 00 | 641.94 | 1.0110 | ||
| 57 | 22.36 | .9205 | 79 | 60.29 | 1.1165 | 01 | 587.60 | .9154 | ||
| 58 | 27.00 | 1.2075 | 80 | 73.01 | 1.2110 | 02 | 494.09 | .8409 | ||
| 59 | 31.41 | 1.1633 | 81 | 71.58 | .9804 | 03 | 536.34 | 1.0855 | ||
| 60 | 30.56 | .9729 | 82 | 75.80 | 1.0590 | |||||
| 61 | 36.64 | 1.1990 | 83 | 92.45 | 1.2197 |
The values in the column to the right of the column of Index values are the corresponding annual gain factors (1+a) which were computed from the Index values. For example, the gain factor for year 41 is found by dividing the Index value for year 41 by that for year 40. The gain factors are used together with the corresponding annual inflation factors (1+i) from Table 2 to compute the corresponding values of effective gain factor (1+re) = (1+a) / (1+i). These effective annual gain factors (gain after inflation) were used to construct Table 3. which is like Table 1 except that the gain values are smaller due to inflation. Note here that the annual gain factors show a pronounced cyclical or roller coaster behavior over time. Remember that the percentage gain for a year is found by subtracting 1 from the gain factor and multiplying the result by 100. For example, the percentage gain for year 59 is 16.33% and the percentage gain for year 60 is -2.71%. When the gain factor is less than 1, as in year 60, subtract it from 1 and multiply by 100 to get the negative percentage gain or the percentage loss.
The period gain for a multi-year period is found by dividing the Index value for the end year of that period by that of the beginning year of the period. Then if n years is the duration of the period, the nth root of the period gain is the geometric mean or geometric average annual gain factor for the period. For example, the period gain for the 20-year period from 48 to 68 is 56.94 / 9.17 or 6.2094 and the 20th root of 6.2094 is 1.0956. (This is readily computed using a hand-held calculator that computes logarithms and anti-logarithms. You simply divide the logarithm of 6.2094 by 20 and take the anti-logarithm of the result.) In this example, the geometric average annual percentage gain for the 20-year period was 9.56%. It is of interest to note the variation in the values of the 20 annual gain factors in the period and to see how they varied about the average value of 9.56%, i.e. the values for years 49, 50,..., 68. The high value for the period was +31.72% in year 54 and the low value was -8.68% in 62.
There are two more examples of period gain
and the corresponding geometric average annual percentage gain that are
important to note at this point. These are for the periods 68 to 81 and
81 to 01. The period gain for the 13-year period from 68 to 81 is 71.58
/ 56.94 or 1.2571 and the geometric average annual gain factor was
1.0178, so the average annual percentage gain for
this 13-year period was only 1.78%. This was really quite low but
wait until inflation is factored in! That will be considered in a
later example. The period gain for the 20-year period from 81 to 01 is
The Consumer Price Index inflation rates,
produced by The U.S. Bureau of Labor Statistics,
were used to determine the inflation factors 1+i (using the Index Ave-Ave
values) corresponding to the market gain factors 1+a contained in Table 1.
Using the inflation rates for the years 41 through 03, the cumulative
inflation values relative to the year 40 (1940) were computed. Table 2.
below lists the inflation rate data as the inflation factors 1+i under the
heading Factor and the cumulative inflation values under the heading Index.
As with the gain factors, the inflation rate for the year is found by
subtracting 1 from the inflation factor and multiplying the result by 100 to
get the rate as a percentage.(The rate for 40 is not shown because 40 is
taken as the base or reference year.) For example, the inflation factor for
the year 47 is 1.144 so the inflation rate for that year was 14.4%. The
cumulative inflation value, or index value, for year 47 is 1.5926
which means it would take about $1.59 to buy what you could buy in year 40
for $1.The inflation factor for year 49 is .988 so the inflation rate for
that year was
Inflation gain (increase in index value) over a period of years can be computed in the same manner as market gain. To compute the inflation gain, the cumulative inflation value for the end year of the period is divided by that for the first year of the period. For example, the inflation gain for the period 68 to 81 is 6.4885 / 2.4856 or 2.6104. Then the geometric average annual inflation factor and corresponding inflation rate can be computed as was done above with period gain. For the period 68 to 81 the average annual inflation factor was 1.0766 so the corresponding inflation rate was 7.66%. Now using the cumulative inflation value for 2003 of 13.1463 it may be seen that the average inflation rate for the entire 63 year period was 4.17%. Further if we compute the average inflation rates for the periods 1940-1968 and 1981-2003 we find the average inflation rate for both periods was 3.3% ! Therefore it may be seen that the period 1968-1981 was not only a period of a severe stock market depression, but also a period of extreme inflation! Note also that the inflation rate in 1980 was 13.5%, the highest annual rate in the entire 63 year period.
It is of interest to note that as the ratio 1/(1+i) gives the amount of purchasing power left in a dollar after a year with an inflation rate of i%, that the percentage loss of purchasing power is approximately equal in magnitude to the inflation rate expressed as a percentage. For example, if i = 3%, 1/1.03 = .9709 so the percentage loss in purchasing power is approximately 3%. If i = 4%, 1/(1.04) = .9615. Works for low inflation rates.
| Yr | Factor | Index | Yr | Factor | Index | Yr | Factor | Index | ||
|---|---|---|---|---|---|---|---|---|---|---|
| 40 | 1.000 | 62 | 1.010 | 2.1567 | 84 | 1.043 | 7.4171 | |||
| 41 | 1.050 | 1.050 | 63 | 1.013 | 2.1847 | 85 | 1.036 | 7.6841 | ||
| 42 | 1.109 | 1.1645 | 64 | 1.013 | 2.2131 | 86 | 1.019 | 7.8301 | ||
| 43 | 1.061 | 1.2355 | 65 | 1.016 | 2.2485 | 87 | 1.036 | 8.1120 | ||
| 44 | 1.017 | 1.2565 | 66 | 1.029 | 2.3137 | 88 | 1.041 | 8.4446 | ||
| 45 | 1.023 | 1.2854 | 67 | 1.031 | 2.3854 | 89 | 1.048 | 8.8499 | ||
| 46 | 1.083 | 1.3921 | 68 | 1.042 | 2.4856 | 90 | 1.054 | 9.3278 | ||
| 47 | 1.144 | 1.5926 | 69 | 1.055 | 2.6223 | 91 | 1.042 | 9.7196 | ||
| 48 | 1.081 | 1.7216 | 70 | 1.057 | 2.7718 | 92 | 1.030 | 10.0112 | ||
| 49 | .988 | 1.7009 | 71 | 1.044 | 2.8938 | 93 | 1.030 | 10.3115 | ||
| 50 | 1.013 | 1.7230 | 72 | 1.032 | 2.9864 | 94 | 1.026 | 10.5796 | ||
| 51 | 1.079 | 1.8591 | 73 | 1.062 | 3.1716 | 95 | 1.028 | 10.8758 | ||
| 52 | 1.019 | 1.8944 | 74 | 1.110 | 3.5205 | 96 | 1.030 | 11.2021 | ||
| 53 | 1.008 | 1.9096 | 75 | 1.091 | 3.8409 | 97 | 1.023 | 11.4597 | ||
| 54 | 1.007 | 1.9230 | 76 | 1.058 | 4.0637 | 98 | 1.016 | 11.6431 | ||
| 55 | .996 | 1.9153 | 77 | 1.065 | 4.3278 | 99 | 1.022 | 11.8992 | ||
| 56 | 1.015 | 1.9440 | 78 | 1.076 | 4.6567 | 00 | 1.034 | 12.3038 | ||
| 57 | 1.033 | 2.0082 | 79 | 1.113 | 5.1829 | 01 | 1.028 | 12.6483 | ||
| 58 | 1.028 | 2.0644 | 80 | 1.135 | 5.8826 | 02 | 1.016 | 12.8507 | ||
| 59 | 1.007 | 2.0789 | 81 | 1.103 | 6.4885 | 03 | 1.023 | 13.1463 | ||
| 60 | 1.017 | 2.1142 | 82 | 1.062 | 6.8908 | |||||
| 61 | 1.010 | 2.1353 | 83 | 1.032 | 7.1113 |
Table 3. lists the effective gain factor 1+ re = (1+a) / (1+i) and the cumulative effective market gain or index value for the years 1940 through 2003. The years are listed by the last two digits of the year, e.g.. 40 for 1940, under the heading Yr, the effective market gain factors under the heading Factor, and the cumulative effective market gain values, relative to the year 1940, are listed under the heading Index. For example, the effective gain factor for year 50 indicates the effective rate of market return was over 19%(19.31%). The cumulative effective gain factor for year 50 was only 1.0209 and it may be seen that year 50 was the first year for which a positive cumulative effective gain was sustained in subsequent years. The effective gain for multi-year periods and the geometric average annual effective gain factors can be computed for such periods in the same way as for the previous tables. Although the average annual effective market gain are tabulated in the SOLUTION page for a number of multiyear periods, it is worthwhile to compute a few values here. Noting that the cumulative gains for years 48 and 68 are .8169 and 3.5129 respectively, the cumulative gain for this 20 year period is 3.5129/.8169 = 4.3003 and the average annual effective market gain was 7.57%. That's a very good average value! Scanning down the column of actual gain factors for this period we see gain of 31% for year 54, 26% for year 55, 19% for year 61 and 14% for years 63 and 64. So an average value of 7.6% looks reasonable. Computing the average value for the period 68-81 in the same manner, we get an average value of -5.5%. That's a tragic value and it is -5.5% for a 13 year period! There are two other startling points to note in the table. The gain factor for year 74 is .6482 which means the gain was -35% ! That is, the cumulative effective market gain was reduced by 35% in that one year! The second point to note is that the cumulative gain was 3.5129 in 68 and this value was not surpassed in subsequent years until 1992. This means that for a 24 year period the cumulative market gain was essentially zero!
| Yr | Factor | Index | Yr | Factor | Index | Yr | Factor | Index | ||
|---|---|---|---|---|---|---|---|---|---|---|
| 40 | 1.000 | 1.000 | 62 | .9042 | 2.3792 | 84 | .9749 | 1.9439 | ||
| 41 | .8194 | .8194 | 63 | 1.1370 | 2.7052 | 85 | 1.1801 | 2.2940 | ||
| 42 | .9017 | .7389 | 64 | 1.1417 | 3.0885 | 86 | 1.1543 | 2.6480 | ||
| 43 | 1.1727 | .8665 | 65 | 1.0691 | 3.3019 | 87 | 1.0540 | 2.7910 | ||
| 44 | 1.1100 | .9618 | 66 | .8937 | 2.9509 | 88 | .9904 | 2.7642 | ||
| 45 | 1.2542 | 1.2063 | 67 | 1.1196 | 3.3038 | 89 | 1.1669 | 3.2255 | ||
| 46 | .9051 | 1.0918 | 68 | 1.0633 | 3.5129 | 90 | .9232 | 2.9778 | ||
| 47 | .8106 | .8850 | 69 | .8809 | 3.0945 | 91 | 1.1385 | 3.3902 | ||
| 48 | .9230 | .8169 | 70 | .8515 | 2.6350 | 92 | 1.0626 | 3.6024 | ||
| 49 | 1.0475 | .8557 | 71 | 1.1073 | 2.9177 | 93 | 1.0479 | 3.7750 | ||
| 50 | 1.1931 | 1.0209 | 72 | 1.1014 | 3.2136 | 94 | .9724 | 3.6708 | ||
| 51 | 1.0778 | 1.1003 | 73 | .8206 | 2.6371 | 95 | 1.1919 | 4.3752 | ||
| 52 | 1.0447 | 1.1495 | 74 | .6482 | 1.7094 | 96 | 1.1774 | 5.1514 | ||
| 53 | .9537 | 1.0963 | 75 | 1.0721 | 1.8326 | 97 | 1.2511 | 6.4449 | ||
| 54 | 1.3080 | 1.4340 | 76 | 1.1412 | 2.0914 | 98 | 1.1597 | 7.4742 | ||
| 55 | 1.2592 | 1.8057 | 77 | .9000 | 1.8823 | 99 | 1.0950 | 8.1843 | ||
| 56 | 1.0613 | 1.9164 | 78 | .9448 | 1.7784 | 00 | .9778 | 8.0026 | ||
| 57 | .8911 | 1.7077 | 79 | 1.0031 | 1.7839 | 01 | .8905 | 7.1263 | ||
| 58 | 1.1746 | 2.0059 | 80 | 1.0670 | 1.9034 | 02 | .8277 | 5.8985 | ||
| 59 | 1.1552 | 2.3172 | 81 | .8888 | 1.6917 | 03 | 1.0611 | 6.2589 | ||
| 60 | .9566 | 2.2166 | 82 | .9972 | 1.6870 | |||||
| 61 | 1.1871 | 2.6313 | 83 | 1.1819 | 1.9939 |