This is the case of most market cap indices. There is no problem with that unless the history is revised. You could replicate the same performance in your own stock account. (Not sure what you mean by "without realising losses"?)

Dow Jones 'performance' is based on a price-weighted average. If a $2 per share stock goes to $1, $200 per share stock going to $201 offsets the loss.

If $400 were allocated equally among the two stocks, an actual investor would lose 25% ($100) while the Dow showed no movement.

Yes, so you should not allocate the $400 equally across the shares to match the performance of the index. You should buy an equal number of shares of each stock. It may not be a greatly constructed index by modern standards but to my understanding you can in theory get the same performance yourself.

Buying an equal number of shares does not track the DJIA either. Apple closed at 126.6 the day it replaced ATT. ATT closed at 33.48.

I struggle to see why that is a problem? You will need to rebalance the portfolio, yes.

The DJIA replaced each share of ATT with a share of Apple on a ledger. Without additional investment or buying and selling other assets, there's no way to replicate that in a portfolio due to the difference in share price. Essentially, the DJIA pumped in an ~300% increase in equity value on the ATT shares...or divided among thirty companies a 10% profit.

If this was a new rule change your point would be valid. It's fair to consider that but you'd need to adjust all the previous data points to reflect those swaps as well.Why not just compare a larger market pool?

Since the first time one company replaced the next, it has always been the case that the performance of the Dow Jones Industrial Average cannot be replicated by an individual investor. While it may be a useful model, it is actively engineered to keep going higher.

I think you are wrong, replicating its performance is very easy, you just hold the exact same thing it does.

The point being raised is that to "hold the exact same thing it does" actively costs money on an ongoing basis (even before transaction costs!) as you need to sell cheaper stocks to buy more expensive ones.

Surely this is (ignoring transaction costs) reflected in the index performance already. If you sell and buy at the time (and price) of the index you will match the performance. This is not unique to DJIA by the way.

There are indexes that can be matched. DJIA is not one of them.

If that is the case, how do you explain that there are ETFs tracking the index? E.g. http://www.etf.com/DIA

They suffer a loss when a more expensive stock replaces a less expensive stock. Portfolios have risks, indexes don't.