A dividend is a cash disbursement from the company. The value of the company goes down the same amount of the dividend, so it is analogous to having money in a savings account and taking a withdrawal every month. Obviously you are going to have less in the end than if you just kept the money in the account.
suppose that I own 10 different stocks, and don't reinvest dividends, but keep them on account, and each month or two, as I add more money to invest, either in one of my existing stocks, or perhaps something new, I add whichever dividend amount is currently available in cash to my new purchase, would this strategy provide the same results?
Roughly, yes. Reinvesting dividends is essentially buying more stock at the lower price, which is a net zero effect in total balance. So if you invested in the same stocks, yes you'd be in the same place. If you invested in different stocks, then you would have a performance difference depending on what you invested in.
The risk is the temptation to take the cash dividend and not reinvest it, but take it in cash, thereby reducing your earning power.
That is, is there some particular reason that the brokers are recommending automatically reinvesting dividends as opposed to reinvesting them manually, perhaps not always in the same item?
I'd like to think that they're looking after your best interest (and they might be), but the cynical part of me thinks that they're either trying to keep your business by increasing your returns, or there's some UK regulation I'm not aware of that requires them to disclose the effect of reinvesting dividends.
£100 invested in the UK stock market since 1899 would have grown into just £177 after adjusting for inflation.
This figure seems ludicrous to me. I haven't actually measured what the historical returns on the "UK market" are, but that would mean an annualized return (adjusted for inflation) of just 0.5%. Either UK stocks pay a ridiculous amount of dividends or there's something wrong with the math.
I still have not found a definitive source for the real UK market return, but according to this inflation calculator, £100 in 1899 would equate to almost £12,000 today, for an average inflation rate of 4.14 percent, which would put the CAGR of the UK market at about 4.9%, which seems reasonable. The CAGR with dividend reinvestment would then be about 9.1%, making dividend reinvestment a no-brainer in the UK market at least.