The same money run twice: every payout bought back, versus the same payouts left in cash earning nothing.
Four quarters a year, three steps in each. The balance grows by the quarterly price factor, (1 + growth/100) raised to the quarter power. The fund pays a dividend of a quarter of the annual yield times that grown balance: reinvested, it joins the balance the same quarter; taken as cash, it sits in a separate pile earning nothing at all — most people would redeploy that cash, and this model gives them no credit. Then a quarter of the annual contribution is added, in both modes.
The defaults: $10,000 to start, $2,000 a year, 5% growth, a 3% yield, 20 years, 80 quarters. You contribute $50,000.00. Reinvesting ends at $144,486.58. Cash mode ends at $93,892.26 of stock plus $27,153.31 collected, a comparable $121,045.57, so reinvesting adds $23,441.01. The "Checks" line below the table is computed live, not asserted: at a 0% yield both modes land on the same $93,892.26, and contributed is start plus annual times years.
Growth and yield are constant and independent, which they are not: payouts come from earnings, and a rising price on a fixed dividend lowers the yield. Reinvestment happens at the quarter's grown value: ex-dividend dates, fractional shares and DRIP discounts are outside the model, and every payout is assumed made — no cuts. The cash pile earns nothing, which understates the cash option. Taxes are not neutral: US dividends are taxable in the year paid, reinvested or not, so the DRIP path is taxed on money you never touched.
Estimates from inputs you chose; nothing is guaranteed. Taking cash is right when you need income or want to rebalance.