Smart Contracts: Tracking stock index with cryptocurrency

Hey guys,

today I learnt about Smart Contracts, so my thoughts may be completely stupid or my ideas are already reality for years.

I learnt that you can create a Stablecoin for exampled backed with USD on Ethereum using Smart Contracts. (Of course there is a institution behind every stablecoin, which stores the USD, that you have to trust)

Isnt it possible for an institution or company to create a stablecoin pegged to a stock index like the NASDAQ then? So for every "NASDAQ-Coin" issued, the company would have to rebuild the NASDAQ index by buying the contained stocks.

The benefits are obvious: You could invest in the stock market anonymously. Also there would be tax benefits as a cryptocurrency can be traded taxless in many countries, in contrast to high taxes on trading stocks directly at exchanges.

Would this be possible?