Considering all available investable asset combinations of risky assets. We can define a minimum variance Frontier.

We can see the portfolio mix at point A and B are better than C and D respectively because they offer higher expected returns for the same level of risk or equal expected returns for lower risk. No risk averse investor will choose to invest in a portfolio to the right of the minimum variance Frontier because it offers the same return for higher risk at the same time. No investors should invest in any portfolio below the minimum variance portfolio, since they offer a lower return for the same level of risk as a result the minimum variance Frontier Narrows to the efficient Frontier the efficient Frontier contains all portfolios of risky assets that rational risk averse investors will choose choosing where on the efficient Frontier is determined by each investors risk and return objectives.

In this simplified example the expected return on a 50/50 ratio between Home Depot and Microsoft's common stock is 7.63 percent using a data table in Excel. We sensitize the weighting of Home Depot's common stock in the portfolio and recalculate the standard deviation of the portfolio measuring the risk and the expected return on the portfolio for each mix of assets a portfolio consisting of all Home Depot's common stock gives an expected return of 7.4% and a standard deviation of 5.58.

By graphing the mix of asset combinations we can lower the overall risk of the portfolio.

A 60 to 70 percent Home Depot waiting gives the portfolio the lowest standard deviation of 5.37 lower than both stocks individually.

As before the graph shows the minimum variance Frontier with the Curve above the minimum variance portfolio containing just the efficient portfolios essentially any portfolio with a Home Depot allocation of 50% or less in the real world efficient Frontier should consider more than two assets due to the increasing complexity of calculating additional assets investors. Typically utilize robust portfolio management software solutions to create the frontier rather than Excel.

Until now, we have only considered risky Assets in which the return is uncertain Markowitz did not consider the existence of the risk-free asset adding a risk-free asset allows the portfolio Theory to develop into Capital Market Theory.

The risk-free asset helps us establish what mix of expected returns and risk is optimal first. We calculate the sharp ratio, which takes the excess return on the portfolio above the risk-free rate and divides by the standard deviation. The excess return is compensation for taking the risk of investing in the portfolio, which is measured by the standard deviation. The sharp ratio allows us to measure how much return we are getting for the risk. We are taking the higher the sharp ratio the better the portfolio with the highest sharp ratio on the efficient Frontier is optimal. It has the highest reward to risk ratio among the potential portfolios anything to the right of the optimal risky portfolio on the capital allocation line is assuming adding leverage to the portfolio to amplify both return and risk to the left. The investor is lending at the risk-free rate in creating a more conservative combination of risky and risk free assets where each investor lies on the Allocation line will depend on their risk tolerance.

Using the same stocks as earlier to calculate the capital allocation line. We first sensitize the mix of Home Depot in Microsoft in the portfolio calculating the standard deviation and the expected return a third column calculates the sharp ratio, which measures the excess return versus the risk taken the optimal portfolio in this case is a 50/50 mix of Home Depot and Microsoft which gives the highest sharp ratio of 1.308.

Using the highest sharp ratio of 1.308. We can reverse engineer the expected portfolio return from the standard deviation to chart a line giving us combinations of the stock portfolio and the risk-free rate from the Left To Where the Line is tangential to the efficient Frontier the capital allocation line hits the left axis at the risk-free return where the standard deviation of the portfolio is zero where all of the portfolio is invested in risk-free assets to the right of where the line is tangential to the efficient Frontier. The investor is borrowing to amplify their returns while modern portfolio theory is very influential. It is important to be aware of the criticisms. First is that it's too simple more complex models have been created adding more factors other than risk and return. Secondly, it's backward looking analysis can have low predictive value for the future.

Third not all investors act rationally, but MPT assumes they do fourth not all investors have the ability to invest in Risky assets alone. And lastly the markets aren't always efficient.