The Modern Portfolio Theory (MPT) is a theoretical framework to create an optimal portfolio for an investors with certain required return and risk aversion. The MPT also has certain assumptions needed to create optimal portfolios.

If an investor wishes to implement the MPT framework to create an optimal portfolio, which one of the statements below preserves the assumptions and/or the methodology for the framework?

The MPT assumes there are no frictions in the capital markets to consider when constructing an optimal portfolio.

In the real world, an investor does not need to consider either taxes or transaction costs when buying or selling investments.

The MPT framework requires expected returns, volatility and covariance of each investments (listed and unlisted) in the capital market.

In the real world, historical returns of each investment provides correct estimates of expected returns, volatility and covariance of each investment.

The MPT assumes that there must be many investors whose trades will make the markets efficient and there are no large investors who are able to control the market by artificially increase, or decrease asset prices.

In the real world, an investor is able to buy, or sell, investments at the correct price at all times when constructing the optimal portfolio.

The MPT assumes that the borrowing rates and lending rates should be the same in capital markets to construct an optimal portfolio.

In the real world, an investor is able to borrow to leverage the portfolio, or is able to lend at the borrowing rate.

The MPT framework selects optimal portfolios (those with the lowest total risk for a given level of return) using returns, volatility and covariance matrix for each investment as an input

In the real world optimal portfolio can be constructed using an optimization process (such as Solver in MS Excel), if expected returns, volatility and covariance matrix for each investment is provided as an input.