Based on these assumptions, portfolio theory suggests that investors can construct an efficient frontier, which is a curve that shows the optimal combinations of risk and return for different portfolios. The efficient frontier is derived by using the mean-variance analysis, which calculates the expected return and the standard deviation (a measure of risk) of each portfolio. The efficient frontier is the upper part of the curve that has the highest return for a given level of risk, or the lowest risk for a given level of return. Any portfolio that lies below the efficient frontier is considered inefficient, as it has a lower return for the same risk, or a higher risk for the same return.

One of the main concepts of portfolio theory is diversification, which means spreading the investments across different assets that are not perfectly correlated, meaning they do not move in the same direction or by the same amount. Diversification can reduce the unsystematic risk, which is the risk that is specific to an individual asset or a group of assets, such as business risk, financial risk, or liquidity risk. Unsystematic risk can be eliminated by holding a large number of assets that are not affected by the same factors. However, diversification cannot eliminate the systematic risk, which is the risk that affects all assets in the market, such as inflation, interest rates, or political events. Systematic risk can only be reduced by holding less risky assets, such as bonds or cash.

Portfolio theory also introduces the concept of the capital asset pricing model (CAPM), which is a formula that calculates the expected return of an asset based on its risk and the risk-free rate. The CAPM states that the expected return of an asset is equal to the risk-free rate plus a risk premium, which is the product of the asset's beta and the market risk premium. The beta is a measure of the sensitivity of the asset's return to the market return, and it indicates how much the asset's return will change when the market return changes by one unit. The market risk premium is the difference between the expected return of the market portfolio and the risk-free rate, and it represents the extra return that investors demand for investing in the market rather than in a risk-free asset. The CAPM can be used to estimate the required return of an asset, which is the minimum return that investors expect to receive for investing in that asset. The required return can also be used to evaluate the performance of an asset, by comparing it with the actual return. If the actual return is higher than the required return, the asset is considered undervalued, and vice versa.

To apply portfolio theory to your investment decisions, you need to follow some steps, such as:

- Define your investment objectives and constraints, such as your time horizon, risk tolerance, income needs, tax situation, etc.
- Choose an asset allocation strategy, which is the proportion of your portfolio that you allocate to different asset classes, such as stocks, bonds, cash, etc. You can use the efficient frontier to find the optimal asset allocation that matches your risk-return profile, or you can use a rule of thumb, such as the 60/40 rule, which allocates 60% of your portfolio to stocks and 40% to bonds.
- Select the specific assets that you want to invest in, based on their expected returns, risks, correlations, and costs. You can use the CAPM to estimate the required returns of the assets, and compare them with their market prices, to find the best opportunities. You can also use other tools, such as fundamental analysis, technical analysis, or behavioral finance, to complement your decision-making process.
- Monitor and rebalance your portfolio periodically, to ensure that it remains aligned with your objectives and constraints, and that it maintains its optimal risk-return characteristics. You can rebalance your portfolio by selling some of the assets that have increased in value and buying some of the assets that have decreased in value, to restore your original asset allocation. You can also adjust your asset allocation if your objectives and constraints change over time, or if the market conditions change significantly.

These are some of the basic principles and steps of portfolio theory and how to apply it to your investment decisions. However, portfolio theory is not a perfect or complete theory, and it has some limitations and criticisms, such as:

- It relies on unrealistic assumptions, such as rational investors, perfect information, and normal distributions, which do not reflect the real-world behavior and situations of investors and markets.
- It does not account for other factors that may affect the returns and risks of assets, such as taxes, inflation, liquidity, transaction costs, etc.
- It does not consider the impact of human emotions, biases, and preferences, which may influence the investment decisions and outcomes of investors, sometimes irrationally or suboptimally.
- It does not capture the dynamic and complex nature of the financial markets, which may experience shocks, crises, bubbles, or anomalies, that may deviate from the expected patterns and models.

Therefore, portfolio theory should be used as a guide, not as a rule, and it should be supplemented by other theories, models, and methods, that can provide a more comprehensive and realistic view of the investment world.