...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)