MOVING FOUR YEAR AVERAGE, AFTER EXPENSES, RATES OF RETURN

MOVING FOUR YEAR ANNUALIZED, AFTER EXPENSES, RATES OF RETURN

Morton Abramson
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      

INDEXING PENSION PAYMENTS

In accordance with the plan, retiree pensions are adjusted upwards if the average of the Fund rates of return in the four preceding years is above 6%.The pensions are never reduced,even in the event that this average is less than 6%. This is why the employer contribution to the member's money purchase account (since Jan. 1992) is 103%. The 3% is to provide a sfeguard against reduction of benefits.

AVERAGE OF THE FUND RATES OF RETURN IN THE FOUR PRECEDING YEARS

There is no formula in the plan for calculation of this Average. In practise this AVERAGE of the Fund rates of return is taken by the Actuary as the SIMPLE ARITHMETIC MEAN.
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)

ADJUSTMENT RATE TO THE PENSION PAYMENT

In the Pension Plan the adjustment rate R is given by the formula
R = (((1+A)/(1+.06))-1)x100%.

where A is theAVERAGE of the Fund rates of return in the preceding 4 years.
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

THE ADJUSTMENT RATE R IS THEN CALCULATED AS FOLLOWS:
$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.

Equivalently
          R = (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.

NEW PROPOSAL FOR CALCULATING THE FOUR YEAR AVERAGE
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.
THE NEW AVERAGE WOULD BE APPLIED TO PAYMENTS MADE ONLY AFTER APPROVAL BY THE TRUSTEES AND BOARD OF GOVERNORS.
They would not be retroactive.
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 entries


                          column 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 be


          R = ((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)

REASONS FOR NEW PROPOSAL

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.
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.