### DEFINITION of YORK UNIVERSITY PENSION FUND RATE

Mort Abramson
Department of Mathematics and Statistics. 416-736 5250
York University

## Actuarial Definition of York Pension Fund Rate Of Interest

In the Annual Actuarial Evaluation of The York University Pension for 1993 Appendix B page B-3 the FUND RATE OF INTEREST is defined as follows:
The rate of interest credited to the Fund balances at each year end is determined each year according to the formula below where the "net gain" is defined to be the excess of all realized gains and losses and investment income of the Fund over the expenses associated with the investment of the Fund:

### Fund Rate =

(2 x Net Gain)/(sum of market value of Fund at beginnining and end of year - Net Gain)

To understand this, we use the following terms:
Fund Rate denotes the fund interest rate for the year.
Beginning Value denotes the market value of fund at beginning of year.
End Value denotes the market value of fund at end of year.
Contributions denotes the value of the total contributions (including transfers) made into fund during the year.
Withdrawals denotes the value to the total withdrawals (including distributions) made from the fund during the year.
Net Gain denotes the net gain of the fund for the year.
(Net External Growth is equal to Contributions less Withdrawals)
It is assumed all income earned by the fund is included in the End Value. Then, the following relations hold.

Net Gain = End Value - (Beginning Value + Contributions - Withdrawals)
or, equivalently
End Value = Beginning Value + Contributions - Withdrawals + Net Gain
Assuming contributions and withdrawals are made in the middle of the year (this is an assumption for calculation purposes) and the Fund Rate is a simple interest rate we have

End Value = Beginning Value + Fund Rate x Beginning Value + Contributions - Withdrawals + (1/2)(Fund Rate) x (Contributions - Withdrawals).
The above formula can be written compactly as

End Value = Beginning Value x (1 + Fund Rate) + (Contributions - Withdrawals) x (1 + (1/2)(Fund Rate).
The above formula, which defines the fund rate implicitly is equivalent to the following explicit formula:

### Fund Rate = (2 x Net Gain)/((sum of market value of Fund at beginnining and end of year) - Net Gain)

in agreement with the actuarial definition given above.

YORK PENSION FUND RATE FOR 1995 (before investment expenses). According to the SEI FUNDS EVALUATION SERVICES, a firm hired to evaluate the performance of the York Pension Fund,
the market value (Beginning Value) of the fund at 1/1/95 was \$553,659,000,

the Net external growth (Contributions - Withdrawals) was - \$11,954,000 and

the Market Value (End Value) as of 12/31/95 was \$630,958,000.

Income received was \$24,406,000 and the Gain was \$64,848,000.

Using our definition above we obtain ,
Net Gain is \$ 630,958,000 - (\$553,659,000 - \$11,954,000 ) = \$89,253,000.
This is in agreement with the SEI figures if we add the income of \$24,406,000 to the Gain of 64,848,000 to obtain a Net Gain of \$89,254,000 the difference apparently due to rounding off. Using our preceding formula (assuming all contributions and withdrawals are made in the middle of the year) it follows that the

Fund Rate = 2x89253000/(553659000+630958000-89253000) =.162965005 or 16.2965005%

This is in agreement with the figure of 16.3% given by SEI. If we assume all contributions and withdrawals are evenly made over 12 months, then using the formula described in the section at the end, we obtain the Fund Rate to be 16.2816933%.

ACTUAL FUND RATE AFTER INVESTMENT EXPENSES for 1995
Subtracting expenses the actuary has determined that the actual Fund Rate for 1995 is 15.8261%, see (click)

### The calculation of the 1995 rate of return after expenses

. This means that If a person had \$100000 in the fund at the beginning of the year and total contributions were \$4800 for the person with no withdrawals, then (assuming contributions and withdrawals are made in middle of year) for that person the
End Value = 100000(1 + .158261) + 4800(1 + (1/2)(.158261)) = \$121,005.93,
and the Net Gain is \$16205.93. The final figures may be somewhat lower as there may be additional expenses.

### Another Way to Calculate the Fund Rate

HERE WE ASSUME Contributions and Withdrawals ARE MADE MONTHLY.
i = Fund Rate of interest earned for the year
BV = Market Value of fund at beginning of year.
EV = Market Value of fund at end of year.
C = total Contributions or inflows into fund during year.
C(k) = total Contributions made at end of kth month, k = 1,2,3,...,12.
W = total Withdrawals or outflows from fund during year.
W(k) = total Withdrawals made at the end kth month, k= 1,2,3,...,12.
N = Net Gain in the fund for the year.

C = C(1) +C(2) +C(3) +...+C(12).
W = W(1) +W(2) +W(3) +...+W(12).
N = EV - (BV +C-W)
EV = BV + C - W + N
EV = BV(1 + i ) + Sum from k=1 to 12 over (C(k)-W(k))(1+((12-k)/12)i).
i = N/(BV+Sum from k=1 to 12 over (C(k)-W(k))(12-k)/12)

#### Pension Fund Rate for 1995 Before Expenses Assumming Contributions and Withdrawals are Calculated Monthly

The Beginning Value is \$553,659,000,
the End Value is \$630,958,000,
theContributions - Withdrawals = -\$11,954,000/12 per month.
Then the Net Gain is \$630,958,000 - (\$553,659,000 - \$11,954,000) = \$89,253,000.
Using the immediately preceding formula it follows that the Fund Rate (befiore any investment expense) is
i = 89253000/(553659000-(11954000/12)(11/12)-(11954000/12)(10/12) - ... - (11954000/12)(1/12))
i = .162816933 or 16.2816933%.