email address: abramson@mathstat.yorku.ca

Department of Mathematics and Statistics 416 736 5250.

York University

A B C D E F AVERAGES AVERAGES ADJUSTMENTS ADJUSTMENTS UNDER ACTUALLY TO PAYMENTS TO PAYMENTS NEW USED (in foll. period) (in following period) PROPOSAL UNDER ACTUALLY NEW USED Annualized PROPOSAL equivalent what would constant have been Accumulated effective (for comparison value with rate per only) interest at period to end of end of Arithmetic Actual period of period of $1 average After $1 invested invested at of actual Expenses at beginning beginning returns Adjusted Adjusted Period Pension of 4th of 4th for the increase increase number Period Fund preceding preceding 4 year to pension to pension Rate period. period period payments payments 0 1 Jul171toJun301972 10.4000% 2 Jul172toJun301973 -2.9350% 3 Jul173toJun301974 -3.8130% 4 Jul174toJun301975 10.5590% $1.14 3.3203% 3.5528% 0.0000% 0.0000% 5 Jul175toJun301976 5.5420% $1.09 2.1644% 2.3383% 0.0000% 0.0000% 6 Jul176toJun301977 7.2980% $1.20 4.7567% 4.8965% 0.0000% 0.0000% 7 Jul177toJun301978 10.8602% $1.39 8.5417% 8.5648% 2.3978% 2.4196% 8 Jul178toJun301979 21.1175% $1.52 11.0452% 11.2044% 4.7596% 4.9098% 9 Jul179toJun301980 15.1890% $1.66 13.5001% 13.6162% 7.0756% 7.1851% 10 Jul180toJun301981 11.8784% $1.73 14.6925% 14.7613% 8.2005% 8.2654% 11 Jul181toJun301982 -5.6220% $1.47 10.1688% 10.6407% 3.9329% 4.3780% 12 Jul182toDec311982 25.8010% $1.53 11.2188% 11.8116% 4.9234% 5.4826% 13 Jan183toDec311983 19.5973% $1.59 12.2679% 12.9137% 5.9131% 6.5223% 14 Jan184toDec311984 7.8471% $1.53 11.2426% 11.9059% 4.9459% 5.5716% 15 Jan185toDec311985 23.5085% $2.00 18.9810% 19.1885% 12.2462% 12.4420% 16 Jan186toDec311986 14.1153% $1.82 16.1162% 16.2671% 9.5435% 9.6859% 17 Jan187toDec311987 4.7558% $1.59 12.3329% 12.5567% 5.9744% 6.1855% 18 Jan188toDec311988 10.9729% $1.64 13.1381% 13.3381% 6.7341% 6.9228% 19 Jan189toDec311989 14.7895% $1.52 11.0862% 11.1584% 4.7983% 4.8664% 20 Jan190toDec311990 0.0725% $1.34 7.4986% 7.6477% 1.4138% 1.5544% 21 Jan191toDec311991 17.8581% $1.50 10.7129% 10.9233% 4.4462% 4.6446% 22 Jan192toDec311992 7.3571% $1.45 9.7999% 10.0193% 3.5848% 3.7918% 23 Jan193toDec311993 21.1404% $1.53 11.2880% 11.6070% 4.9887% 5.2896% 24 Jan194toDec311994 -1.1628% $1.51 10.9430% 11.2982% 4.6632% 4.9983% 25 Jan195toDec311995 15.8261% $1.49 10.4617% 10.7902% 4.2091% 4.5191% A B C D E F

For example, for the 4 years from Jan.1 1992 to Dec. 31 1995, the actual rates of return (column A) are

column A 1992 7.3571% 1993 21.1404% 1994 -1.1628% 1995 15.8261%

The Arithmetic mean is then (7.3571+21.1404-1.1628+15.8261)/4 = 10.7902% (See column D)

where A is the

To understand how this formula is obtained consider the following:

Suppose a person is receiving pension payments of $1000 monthly and suppose at the end of a year the total lump sum value of a person's pension fund is $100,000. It is assumed the fund will increase at a rate of 6% per year. Therefore,

Assumed value of person's fund at end of year

is $100,000 + 6% of $100,000 = $(1.06)x100,000 = $106,000

Suppose the 4 year average return is 10.7902% for the 4 preceding years.

Then the value of person's fund at end of year is taken to be

$100,000 +10.7902% of $100,000 = $(1.107902)x100,000 = $110,790.20

$110,790.20 = $106,000 + R% of $106,000 = $(1+R)x106,000 (R in dec. form)

Solving the above equation for R, we obtain

R = (110790.20 - 106000)/106000 = 0.045190566 or 4.5190566% or R = ((1+.107902)/(1+.06))-1 = 0.045190566 or 4.5190566% or R = (.107902-.06)/1.06 = 0.045190566 or 4.5190566%

This means that if a person's pension payments have been $1000 per month and the adjustment rate is A = 10.7902% as in the above example, the payments would then be adjusted to $1000+4.5190566% of $1000 = $1045.19.

If we replace the average .107902 or 10.7902% by any other average rate, say A, which is greater than 6%, then the middle formula above becomes

R = ((1+A)/(1+.06))-1, where R is in decimal form or R = (((1+A)/(1+.06))-1)x100%, where R is % form,

which corresonds to the formula given in the Pension Plan.

EquivalentlyR = (A-.06)/1.06 where R is in decimal form or R = ((A-.06)/1.06)x100% where R is in % form

In general,pension payments are increases by R% if the 4 year average A is greater than 6%, and not changed otherwise.

## Instead of using the arithmetic mean of the fund rates of return for the preceding four years for the average A as described above, i.e. the rates given in column D in the table above, the actuary for the plan is proposing that the annualized four year rates for the immediately preceding four years be used.These annualized rates are given in column C in the table above.

NEW PROPOSAL FOR CALCULATING THE FOUR YEAR AVERAGE## They would not be retroactive.

THE NEW AVERAGE WOULD BE APPLIED TO PAYMENTS MADE ONLY AFTER APPROVAL BY THE TRUSTEES AND BOARD OF GOVERNORS.For example, from the table above actual rates for 1992-1995 are: column A 1992 7.3571% 1993 21.1404% 1994 -1.1628% 1995 15.8261%

If $1 is invested in the fund at the beginning of 1992 then its value at the end of 1995 is equal to

(1+.073571)(1+.211404)(1-.011628)(1+.158261) = $1.49 (Column B)

Let A be the annualized or equivalent rate for each of the four years that also gives an accumulated value of $1.49 at the end of 1995 for $1 invested in the fund at the beginning of 1992.i.e.suppose the rate of return was A% for each of the four years: i.e. suppose we had the entriescolumn A 1992 A% 1993 A% 1994 A% 1995 A%

Then the accumulated value is (1+A)(1+A)(1+A)(1+A) = $1.49 Hence (1+A)^4 = $1.49 (^ means "raised to the power") and A = ((1.49)^(.25))-1 = .104617 or 10.4617% (column C) or using our previous calculation,we also have A = ((1+.073571)(1+.211404)(1-.011628)(1+.158261))^.25-1 =.104617 or 10.4617%

This means that if a person's pension payments have been $1000 per month and the average rate used is A = 10.4617% , the adjustment rate would beR = ((1+.104617)/(1+.06))-1 = .042091509 or 4.2091509% The payments would then be adjusted to $1000+4.2091509% of $1000 = $1042.09.

In general, if x,y,w,z are the actual rates of return for four years, the annualized rate of return,denoted by A, for the four years is given by the formula

A = ((1+x)(1+y)(1+w)(1+z))^(.25)-1, (column C in table above)

## In the opinion of the Actuary, the current method of calculating the averages and resulting upward adjustments to pension payments leads to somewhat higher adjusted pension payments than warranted by the Fund.These slight additional extra payments over time may, in total, accumulate to a relatively significant shortfall in the variable Pension Fund.

REASONS FOR NEW PROPOSAL

Using mathematics (see below)it can be shown that the proposed "annualized rates of return" for the preceding four years to be used as the Average is always the same or slightly less than the Arithmetic Average now used. Hence the adjustment rate to the payments will always be the same or slightly less.(The rates are the same if the actual return in the Fund is exactly the same in each of the four years.) See columns E and F in the table above to see what the adjustment rates have actually been and what they would have been under the proposal.

Another comparison example is the following situation: Suppose a person was receiving a pension payment of $1000 per month in 1986.Then,the accumulated value at the end of 1996 of all payments received by the person since the beginning of 1987, using the current method would be $164,890.99, and under the proposed change would have been $163,244.29 This is a reduction of .9987% from $164,890.99

MATHEMATICAL COMPARISON OF THE TWO AVERAGES

Suppose x,y,w,z are the actual rates of return for four cosecutive years. For the four years,

The Arithmetic Average is (x+y+w+z)/4 (column D)

and the Annualized rate is ((1+x)(1+y)(1+w)(1+z))^(.25)-1, (column C)

A well known result in mathematics is Cauchy's inequality,which can be stated as "the arithmetic mean is greater than or equal to the geometric mean of a number of positive numbers." Applying this result, we obtain the inequality

((1+x)+(1+y)+(1+w)+(1+z))/4 is > or = ((1+x)(1+y)(1+w)(1+z))^(.25)

The above inequality implies that

(x+y+w+z)/4 is > or = ((1+x)(1+y)(1+w)(1+z))^(.25)-1

Hence, the Arithmetic Average is > or = the Annualized rate, which is the proposed new Average to be used and only when greater than 6%.

When x=y=w=z then the Arithmetic Average = Annualized Rate.