I got this question and I don't understand how to go about it:
If someone makes a deposit of 200 dollars at the end of every month in an account earning interest $i^{(12)}=0.072$ to fund a perpetuity with monthly payment of 480 starting 1 month after his last deposit. Find the minimum number of deposits one would have to make.
I'm not sure I understand this as perpetuities last forever so how can we know the amount he has to have beforehand? any help doing this question would be great.
Best Answer
The basic principle of a perpetuity is that payments received per period never exceeds the interest accrued on the principal during that period. For example, a perpetuity that accrues interest at $i = 0.05$ per year can pay up to $5\%$ of the total principal at the end of each year. This means if the fund contains $100000$ at the beginning of the year, then at the end of the year, it will have accumulated $5000$, which can then be withdrawn, and the process repeats indefinitely so long as that interest rate is constant.
In this question, what you need to know is, assuming the nominal monthly interest rate on the perpetuity is the same $i^{(12)} = 0.072$ as the rate at which the annuitant makes deposits to fund the perpetuity, then how many deposits are needed to accrue enough accumulated value into the fund so that when $480$ is taken out at the end of every month, the principal is left intact, just like in the above situation?
To answer this, suppose the annuitant deposits $200$ at the end of every month for $n$ months. The monthly effective interest rate is $$j = \frac{i^{(12)}}{12} = 0.006.$$ Then at the end of those $n$ months, the accumulated value of the fund is $$\require{enclose} AV = 200(1+j)^{n-1} + 200(1+i)^{n-2} + \cdots + 200 = 200 s_{\enclose{actuarial}{n} j} = 200 \frac{(1+j)^n - 1}{j}.$$ Now we require this accumulated value to be sufficiently large to fund the perpetuity-immediate of $480$ per month. That is to say, the interest accrued on $AV$ after $1$ month is $AV j$ cannot be less than $480$. The minimum amount of money in the fund that allows this would be the solution to $$AV j = 480,$$ or $$AV = 80000.$$ Therefore, we must solve for $n$ in the equation $$80000 = 200 \frac{(1.006)^n - 1}{0.006}.$$ This leads us to $$n = \frac{\log 3.4}{\log 1.006} \approx 204.574,$$ and since we require an integer number of months of payment, we round up to obtain $n = 205$ months; i.e., $17$ years and $1$ month of paying $200$ in order to accumulate enough money in the fund to provide a perpetuity (interest-only withdrawals) of $480$ per month.