relationshi between sinking funds and annuities

Question

Question)
The future value FV of a sinking fund (starting with balance 0)
earning interest at the annual rate r after t years is given by the
formula
FV=PMT[(1+i)^n-1]/i
where PMT is the deposit made at the end of each compounding period
into the fund
i=r/m
n=mt
m=number of compounding periods per year.

Show non-algebraically that the present value of this future value FV
is same as the present value of the annuity required to fund the same
payments PMT for t years.

 

My answer)

A sinking fund is an account earning compound interest into which
periodic deposits are made.

So if PV is the present value corresponding to future value FV, this
means that depositing PV in an account is same as making periodic
deposits of PMT in the sinking fund for the same time t and and same
compound interest rate r).

Now an annuity is an account earning compound interest from which
periodic withdrawals are made. The balance that the account starts
with is known as the present value of the annuity.

Since  depositing PV in an account is same as making periodic deposits
of PMT for t years. So an account that starts with balance PV can fund
payments (withdrawals) worth PMT for t years.

So PV is the present value of the annuity required to fund payments of
PMT for t years.

Is this justification sufficient?

Answer 1

Your sinking fund formula is not quite right, it should be FV =PMT(1 +i)[(1 +i)^n -1]/i, where i is the rate of interest and n is the number of periods.