...I couldn't find this equation anywhere else to verify.
See the footnote in
Nords' blog (shows a derivation without existing assets) and
Heart of Tin's post (shows an algebraically identical version of the final equation) for a couple of sources.
Repeating variable definitions here:
A = Asset amount currently invested in funds you will draw upon in retirement.
E = Total (including taxes) annual expenses in retirement
i = Real return on invested retirement funds, e.g., 3% (conservative - we hope...)
S = Annual amount invested in funds you will draw upon in retirement.
WR = Withdrawal Rate planned for retirement, using Trinity Study definitions (e.g., 4%)
To derive the whole equation, start with E/WR as the future value you need to declare success, according to the
Trinity Study folks.
The
future value of a series of equal investments (aka an annuity) is S * [ (1 + i)^n – 1] / i
The
future value of current assets is A * (1 + i)^n
Combine the above to get:
E/WR = S *[ (1 + i)^n – 1] / i + A * (1 + i)^n
Do a little algebra to get:
E/WR + S/i = S/i * (1 + i)^n + A * (1 + i)^n
A little more algebra:
E/WR + S/i = (S/i + A) * (1 + i)^n
Getting closer:
(E/WR + S/i) / (S /i + A) = (1 + i)^n
Rearrange a little:
(S + i*E/WR) / (S + i*A) = (1 + i)^n
And finally solve for n:
n = ln((S + i*E/WR) / (S + i*A)) / ln(1 + i)