# Pension funds

Sharb Digit Pty Ltd wishes to accumulate funds to provide a retirement pension funds annuity for its Director of Marketing, Penny Peters. Penny, by contract, will retire at the end of exactly 10 years. On retirement, she is entitled to receive an annual end-of-year payment of \$35,000 for exactly 20 years. If she dies prior to the end of the 20-year period, the annual payments from the pension funds will pass to her heirs. During the 10-year ‘accumulation period’, Sharb Digit wishes to fund the pension funds annuity by making equal annual end-of-year deposits into an account earning 8% p.a. interest. Once the 20-year ‘distribution period’ begins, Sharb Digit plans to move the accumulated monies into an account earning a guaranteed 10% p.a. At the end of the distribution period the account balance will equal zero. Note that the first deposit will be made at the end of year 1 and the first distribution payment will be received at the end of year 11.

Please solve the following for the pension funds question:

1.Draw a time-line depicting all the cash flows associated with the above accumulation and distribution periods.

2.How large a sum must Sharb Digit accumulate by the end of the year 10 to provide the 20-year, \$35,000 pension funds annuity?

3. How large must Sharb’s equal annual end-of-year deposits into the account be over the 10-year accumulation period to fund Penny’s retirement pension funds annuity fully?

2) The formula you want is the “sinking fund” formula.
[1 – (1+r/n)^(-nt)]/[n/t]

In this case, r = .10, n = 1, and t = 20.
[1 – (1.1)^(-20)]/[.1] = 8.51356372 so the company needs to have 8.51356372 * 35000 = \$297,974.73 in the annuity

3) In order to accumulate that much money, they need the version of the formula for annuities
[(1 + r/n)^(nt) – 1]/[r/n]

[(1 + .08)^10 – 1]/[.08] = 14.48656247
and 297974.73/14.48656247 = \$20569.04 per year

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Money deposited like this, especially if you deposit it monthly rather than annually, accumulates quickly.