Foundations of Portfolio Theory — Risk, Return, and the Markowitz Model

1. Background Assumptions of Portfolio Theory

Before constructing an optimal portfolio, we must understand what “optimal” means. In modern portfolio theory, an optimal portfolio is one that maximizes expected return for a given level of risk, or minimizes risk for a given level of expected return.

📌 Your Portfolio Is More Than Marketable Securities

Portfolio theory assumes that all assets and liabilities should be considered together. This includes:

• Marketable securities (stocks, bonds, funds)
• Real estate
• Art, antiques, and collectibles
• Any liabilities (loans, mortgages)

This is important because returns from different assets interact, and diversification benefits arise only when we see the whole picture.

2. Understanding Risk Aversion

A core assumption of portfolio theory is that most investors are risk averse. This means:

If two investments offer the same expected return, investors prefer the one with lower risk.

📌 Evidence of Risk Aversion

• People purchase life, auto, and health insurance.
• Investors demand higher promised yields on riskier bonds (AAA → AA → A → BBB → …).
• Riskier securities must offer higher expected returns.

📌 But Not Everyone Is Always Risk Averse

Some people:

• Buy lottery tickets
• Gamble in casinos
• Participate in games where expected returns are negative

Economists such as Friedman and Savage (1948) explained this by suggesting that risk attitudes depend on the amount of money involved. Individuals may gamble with small amounts but insure against large potential losses.

Portfolio Theory’s Assumption

For people managing large investment portfolios, we assume consistent risk aversion. Thus:

Higher risk → higher expected return

3. What Is Risk in Finance?

Although economics distinguishes between risk and uncertainty, finance generally uses the terms interchangeably.

Common Definitions of Risk

• Uncertainty of future outcomes
• Probability of an unfavorable outcome

In portfolio theory, we quantify risk using statistical measures that capture how unpredictable returns may be.

4. The Markowitz Portfolio Theory

Before the 1950s, investors spoke about risk but lacked a mathematical measure. Harry Markowitz (1952, 1959) revolutionized finance by introducing:

• A way to compute expected returns
• A statistical measure of risk (variance/standard deviation)
• A formula for portfolio variance
• Guidance on how to diversify effectively

Markowitz’s Key Contribution

He showed that:

Portfolio risk is not just the sum of individual risks; it depends on how returns move together (covariance/correlation).

This insight revealed the power of diversification.

5. Assumptions of the Markowitz Model

• Investors view each investment as a probability distribution of expected returns.
• Investors maximize expected utility, which exhibits diminishing marginal utility of wealth.
• Risk is measured by the variability (variance/standard deviation) of returns.
• Investment decisions are based only on expected return and risk.
• Investors prefer higher return for a given risk, and lower risk for a given return.

Definition of an Efficient Portfolio

A portfolio is efficient if:

• No other portfolio offers higher return for the same risk, or
• Lower risk for the same return

This forms the basis of the Efficient Frontier.

6. Alternative Measures of Risk

Although standard deviation is the most widely used measure, several alternatives exist.

1. Variance / Standard Deviation

• Measures how far actual returns deviate from expected return.
• Higher dispersion = higher uncertainty = higher risk.

2. Range of Returns

• Highest return minus lowest return.
• A larger range indicates more uncertainty.

3. Downside Risk Measures

Some investors care only about the possibility of returns falling below a chosen threshold, such as:

• Below expected return → Semivariance
• Below zero → Negative semivariance
• Below risk-free rate / inflation rate / benchmark → Target semivariance

These measures assume investors primarily want to avoid downside losses, not upside deviations.

Why Use Standard Deviation?

• It is intuitive.
• It is widely recognized.
• It is used in nearly all asset pricing models (CAPM, APT).

7. Expected Return of Individual Investments

To compute expected return:

E(R_i) = Σ P_i R_i

Example:

Probability	Return	Contribution
0.35	8%	0.0280
0.30	10%	0.0300
0.20	12%	0.0240
0.15	14%	0.0210

E(R_i) = 10.3%

8. Expected Return of a Portfolio

The expected return of a portfolio is the weighted average of individual expected returns:

E(R_port) = Σ w_i R_i

Example:

Weight	Expected Return	Contribution
0.20	10%	0.0200
0.30	11%	0.0330
0.30	12%	0.0360
0.20	13%	0.0260

E(R_port) = 11.5%

9. Variance and Standard Deviation of an Individual Asset

Variance measures how much returns deviate from the expected return:

σ² = Σ (R_i - E(R_i))² P_i

Standard deviation is the square root:

σ = √Σ (R_i - E(R_i))² P_i

Example:

• Variance = 0.000451
• Standard deviation = 2.12%

Thus the asset has:

• Expected return: 10.3%
• Risk: 2.12% standard deviation

🎯 Summary of the Lecture

By the end of this lecture, we understand that:

• Investors are generally risk averse.
• Risk means uncertainty about future returns.
• Markowitz introduced variance as a meaningful risk measure.
• Diversification reduces risk—not by adding more assets, but by combining assets with different return patterns.
• Expected return of a portfolio is a simple weighted average.
• Measuring portfolio risk requires considering how assets interact.