Foundational Concepts for Building Optimal Portfolios
The fundamental principle of investing is the risk-return trade-off: to achieve higher potential returns, an investor must accept a higher level of risk. Portfolio management aims to construct an optimal portfolio that provides the highest possible return for a given level of risk, considering the investor's individual risk tolerance.
Expected return is a forward-looking estimate, while historical return is the actual return earned in the past. The expected return can be broken down into the risk-free rate, expected inflation, and a risk premium.
Investors have different attitudes toward risk, which can be broadly categorized:
Prefers a certain outcome over an uncertain one with the same expected value. Most investors are risk-averse.
Indifferent between a certain and an uncertain outcome with the same expected value.
Prefers an uncertain outcome over a certain one, even if the expected value is lower.
Utility theory provides a framework for understanding investor preferences. A utility function assigns a "satisfaction" score to different risk-return combinations.
Where:
Indifference Curve:
Plots all risk-return combinations that provide the same utility level. For risk-averse investors, these curves are convex, showing that increasingly higher returns are required to take on additional risk.
The risk and return of a portfolio are not simply the weighted average of the individual assets' risk and return. The interaction between the assets, measured by covariance and correlation, is crucial.
The weighted average of the expected returns of the individual assets.
A more complex calculation involving the variance of each asset and the covariance between each pair of assets.
The Power of Diversification
The key takeaway is that by combining assets that are not perfectly correlated (correlation coefficient ρ < +1), an investor can reduce the overall portfolio risk without sacrificing expected return. This is the essence of diversification. The lower the correlation between assets, the greater the diversification benefit.
Modern Portfolio Theory (MPT) provides a framework for creating optimal portfolios.
This represents all possible combinations of risky assets an investor can hold. When plotted on a risk-return graph, it forms a specific region.
The left boundary of the opportunity set. For any given level of expected return, the portfolio on the MVF has the lowest possible risk (variance). The leftmost point on this frontier is the Global Minimum-Variance (GMV) Portfolio.
The portion of the Minimum-Variance Frontier that lies above the GMV Portfolio. Portfolios on the efficient frontier are "efficient" because they offer the highest possible expected return for a given level of risk. Rational, risk-averse investors will only choose portfolios that lie on the efficient frontier.
When a risk-free asset is combined with a portfolio of risky assets, the new set of investment opportunities is represented by a straight line called the Capital Allocation Line (CAL). The CAL connects the risk-free asset on the vertical axis to a risky portfolio on the efficient frontier.
The CAL that is tangent to the efficient frontier is the "best" possible CAL. The point of tangency represents the optimal risky portfolio—the single portfolio of risky assets that, when combined with the risk-free asset, provides the best risk-return trade-off.
According to this theorem, all investors, regardless of their risk tolerance, will hold a combination of just two "funds": the risk-free asset and this single optimal risky portfolio.
An investor's final, optimal portfolio is found at the point where their highest possible indifference curve is tangent to the best CAL. A more risk-averse investor will hold more of the risk-free asset, while a less risk-averse investor will hold more of the optimal risky portfolio (and may even borrow at the risk-free rate to invest more).