Specific question about investment returns that needs someone smart to answer it!

Question

OK, so I'm trying to work out the answer to a very specific question and for me it's complex maths. I'm sure for you mathematicians out there it's fairly everyday. If anyone can help I'd be very grateful:

I start with $200,000 and want it to last exactly 15 years whilst making equal drawdowns each year. The funds continue to make 5% returns each year during drawdown. What is the exact equal annual drawdown so the pot is at zero at the end of year 15 taking into account the continue capital returns of 5%?

I'd be interested in the answer and also an idiots guide to how it's calculated!

Many thanks.

Answer 1

Deposit of $a_0$ must fund a stream of withdrawals $d$ in the end of year $i\in [1,N]$ whose present value is $d(1+r)^{-i}$. Summing that stream in present terms we get: $$a_0=d(1-r)^{-1}\frac {1-(1+r)^{-N}}{1-(1+r)^{-1}}$$ hence $$d=\frac {a_0r} {1-(1+r)^{-N}}\approx \frac {a_0r}{1-e^{-Nr}}$$ which tells you by how much you can increase your withdrawal compared to forever lasting annuity (when $N=\infty$ and withdrawals are $a_0r$). The last approximation is an exact answer if interest accumulation and withdrawals are done continuously.

Answer 2

While you're doing this, suppose your friend has an account that starts with a certain amount $A$, but makes no withdrawals. At the rate of $5 \%$ per year, the friend's account will grow to $1.05^{15} A$ in those $15$ years while yours is dwindling to $0$. Now let's say your withdrawal happens to be just enough to cover the interest on the two accounts together. Then the total of the two accounts will stay constant. It starts at $200,000 + A$ and ends with $0 + 1.05^{15} A = 2.078928179 A $. Therefore $200,000 + A = 2.078928179 A$, which we solve to get $A = 185,369.15$.

The yearly interest on $200,000 + A = \$385,369.15$ is $.05 \times 385,369.15 = \$ 19,268.46$. So that's your yearly drawdown.