You plan to retire in 9 years with X dollars and withdraw $92,000 per year for 21 years at 4.31% annual return, with the first withdrawal in 10 years; what is X?

What the timeline is telling you

You retire at the end of year 9, so you need the account balance at time 9. The first withdrawal is at time 10, which is one year after retirement. That makes the withdrawals an ordinary annuity as of time 9 (payments at 10, 11,..., 30).

Present value at retirement (time 9)

Use the present value of an ordinary annuity formula with:

• Payment $PMT = 92{,}000$
• Interest rate $r = 0.0431$
• Number of payments $n = 21$

$$ X = PV_9 = PMT\times\frac{1-(1+r)^{-n}}{r} $$

Substitute: $$ X = 92{,}000\times\frac{1-(1.0431)^{-21}}{0.0431} $$

Compute the discount factor: $$ (1.0431)^{21}\approx 2.4227\quad\Rightarrow\quad (1.0431)^{-21}\approx 0.4127 $$

So: $$ X \approx 92{,}000\times\frac{1-0.4127}{0.0431} =92{,}000\times 13.631 \approx 1{,}254{,}052 $$

Quick reasonableness check

If returns were 0%, you would need $92{,}000\times 21=1{,}932{,}000$. With a positive return (4.31%), the required amount should be lower, so about $1.25$ million makes sense.