M1: Portfolio Risk and Return: Part I

What is the general historical relationship between risk and return across major asset classes?

Higher-risk asset classes have generally earned higher long-run returns. Equities have historically returned more than bonds and T-bills, but with greater volatility.

Which major asset class has historically had the highest return and volatility?

Small-cap equities.

How do equities generally compare with bonds?

Equities generally have higher expected returns and higher volatility. Bonds generally have lower returns and lower volatility.

EXAM TIP: Is historical average return the same as expected return?

No. Historical mean return describes what happened in the past; expected return is forward-looking. Historical returns may be used as an estimate but are not automatically expected returns.

What is a risk-averse investor?

An investor who prefers a certain outcome over a risky outcome with the same expected return.

What is a risk-neutral investor?

An investor who cares about expected return but is indifferent to risk.

What is a risk-seeking investor?

An investor who prefers greater risk when comparing investments with the same expected return.

What is the utility function?

U = E(R) - 0.5Aσ². Think: Utility = Expected Return - Risk Penalty.

What does A represent in the utility function?

The risk-aversion coefficient.

EXAM TIP: How do you interpret A?

A > 0 = risk-averse; A = 0 = risk-neutral; A < 0 = risk-seeking. Higher A = greater risk aversion.

What happens to willingness to accept risk when A increases?

It decreases. Higher A means greater risk aversion.

What is an indifference curve?

Combinations of expected return and risk that provide an investor with the same utility.

Why does a risk-averse investor's indifference curve slope upward?

Because the investor requires higher expected return as compensation for accepting more risk.

EXAM TIP: How do indifference curves differ for more risk-averse investors?

More risk-averse investors have steeper indifference curves because they require more additional return for additional risk.

Which indifference curves are preferred?

Curves further up and to the left represent greater utility: more expected return and/or less risk.

What is the risk premium?

Risk premium = E(R risky) - Rf. It is the additional expected return for bearing risk.

What is the Capital Allocation Line (CAL)?

The set of combinations of a risk-free asset and a particular risky portfolio.

What is the CAL equation?

E(RC) = Rf + [(E(RP) - Rf) / σP] × σC.

What is an easy way to remember the CAL equation?

Expected Return = Risk-Free Rate + (Sharpe Ratio × Risk Taken).

What is the slope of the CAL?

[E(RP) - Rf] / σP, which is the Sharpe ratio.

What is the Sharpe ratio?

Sharpe ratio = [E(RP) - Rf] / σP. It measures expected excess return per unit of risk.

What is the expected return of a combination of a risky portfolio and risk-free asset?

E(RC) = Rf + wP[E(RP) - Rf].

What is the standard deviation of a combination of a risky portfolio and risk-free asset?

σC = wP × σP.

What does wP > 1 mean?

Leverage. The investor borrows at the risk-free rate and invests more than 100% of their own capital in the risky portfolio.

What does wP = 1.4 and wf = -0.4 mean?

140% is invested in the risky portfolio and 40% is borrowed at the risk-free rate.

EXAM TIP: Does leverage change the Sharpe ratio/slope of the CAL if borrowing and lending occur at the same Rf?

No. Leverage moves the investor along the same CAL, increasing expected return and risk proportionally.

Where is an investor's optimal complete portfolio?

At the tangency between the CAL and the investor's highest attainable indifference curve.

What is the optimal risky portfolio weight formula?

w* = [E(RP) - Rf] / (AσP²).

How can you remember the optimal risky portfolio weight formula?

Risky allocation = Reward / (Risk Aversion × Variance).

EXAM TIP: What happens to optimal risky allocation when expected excess return increases?

w* increases. More reward leads the investor to allocate more to the risky portfolio.

EXAM TIP: What happens to optimal risky allocation when risk aversion A increases?

w* decreases. More risk-averse investors allocate less to the risky portfolio.

EXAM TIP: What happens to optimal risky allocation when risky portfolio variance increases?

w* decreases. More risk leads the investor to allocate less to the risky portfolio.

What is the historical arithmetic mean return formula?

R̄ = ΣRt / N.

What is the sample variance formula?

s² = Σ(Rt - R̄)² / (N - 1).

EXAM TIP: Why is N - 1 used for sample variance?

One degree of freedom is lost because the sample mean has already been estimated from the observations.

How is standard deviation calculated from variance?

s = √s².

What does standard deviation measure?

The dispersion or volatility of returns around their mean.

What is the sample covariance formula?

Cov(R1,R2) = Σ[(R1,t - R̄1)(R2,t - R̄2)] / (N - 1).

What does positive covariance mean?

The two assets tend to move in the same direction.

What does negative covariance mean?

The two assets tend to move in opposite directions.

What does zero covariance mean?

There is no linear tendency for the two asset returns to move together.

EXAM TIP: Why is covariance difficult to interpret directly?

Covariance has no fixed range and its units are returns-squared. Correlation standardizes it.

What is the correlation coefficient formula?

ρ12 = Cov(R1,R2) / (σ1σ2).

What is the possible range of correlation?

-1 ≤ ρ ≤ +1.

How do you calculate covariance from correlation?

Cov12 = ρ12 × σ1 × σ2.

EXAM TIP: Does zero correlation imply independence?

No. Zero correlation means no linear relationship. Independence is a stronger condition.

What is the expected return of a portfolio?

E(RP) = ΣwiE(Ri). Portfolio expected return is a weighted average of individual expected returns.

What must portfolio weights sum to?

Σwi = 1.

EXAM TIP: Is portfolio standard deviation simply the weighted average of individual standard deviations?

No, except when the assets are perfectly positively correlated (ρ = +1).

What is the two-asset portfolio variance formula using covariance?

σP² = w1²σ1² + w2²σ2² + 2w1w2Cov12.

What is the two-asset portfolio variance formula using correlation?

σP² = w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2.

EXAM TIP: After calculating portfolio variance, how do you get portfolio standard deviation?

σP = √σP². Do not forget the square root.

What three factors determine two-asset portfolio risk?

Portfolio weights, individual asset volatilities, and covariance/correlation between the assets.

EXAM TIP: Can adding an individually riskier asset reduce total portfolio risk?

Yes. If its correlation with the existing portfolio is sufficiently low, diversification can reduce total portfolio volatility.

What is the general N-asset portfolio variance formula?

σP² = ΣiΣj wi × wj × Cov(Ri,Rj).

Why do covariances become especially important in large portfolios?

As the number of assets increases, covariance terms dominate relative to individual variance terms, so how assets move together becomes increasingly important.

What is the central relationship between correlation and diversification?

Correlation ↓ → Diversification benefit ↑ → Portfolio standard deviation ↓, holding everything else constant.

What happens to portfolio standard deviation when ρ = +1?

σP = w1σ1 + w2σ2. Portfolio risk equals the weighted average of individual risks, so there is no diversification benefit.

What happens to portfolio standard deviation when ρ = 0?

σP = √(w1²σ1² + w2²σ2²). There is a diversification benefit.

What happens to portfolio standard deviation when ρ = -1?

σP = |w1σ1 - w2σ2|. This gives the maximum potential diversification benefit.

EXAM TIP: When do diversification benefits exist?

Whenever ρ < +1. Correlation does not need to be zero or negative.

EXAM TIP: Does ρ = +0.9 provide diversification?

Yes. Any correlation below +1 provides some diversification benefit.

EXAM TIP: Given correlations of -0.5, 0, and +0.5, which provides the greatest diversification?

-0.5. Lower/more negative correlation provides greater diversification.

What correlation gives maximum diversification potential?

ρ = -1.

What correlation gives no diversification benefit?

ρ = +1.

If two assets have ρ = -1, what weights can create a zero-variance portfolio?

w1 = σ2 / (σ1 + σ2); w2 = σ1 / (σ1 + σ2).

EXAM TIP: If two equally weighted assets have ρ = -1, is portfolio risk automatically zero?

No. With equal weights, zero risk also requires equal standard deviations. Otherwise, the weights must be adjusted.

What is the diversification ratio?

Diversification ratio = σP / (w1σ1 + w2σ2).

How do you interpret the diversification ratio?

Ratio < 1 = diversification benefit; ratio = 1 = no diversification benefit; lower ratio = greater risk reduction.

EXAM TIP: What is more important for diversification: low individual volatility or low correlation?

Low correlation. Two volatile but weakly correlated assets can diversify better than two stable but highly correlated assets.

What is the investment opportunity set?

All attainable portfolios that can be constructed from the available assets.

What is the minimum-variance frontier?

The set of portfolios providing the lowest possible variance for each level of expected return. It forms the left edge of the opportunity set.

What is the global minimum-variance (GMV) portfolio?

The single portfolio with the lowest variance among all possible portfolios of the available risky assets.

Can the GMV portfolio have lower volatility than the least volatile individual asset?

Yes. Diversification can make a combination of risky assets less volatile than the least volatile individual asset.

What is the two-asset GMV weight formula for Asset 1?

w1(GMV) = (σ2² - Cov12) / (σ1² + σ2² - 2Cov12).

How do you find the GMV weight of Asset 2?

w2(GMV) = 1 - w1(GMV).

EXAM TIP: Which asset's variance goes in the numerator when calculating the GMV weight of Asset 1?

The OTHER asset's variance: σ2² - Cov12.

EXAM TIP: Do expected returns enter the two-asset GMV weight formula?

No. The GMV portfolio is a pure variance-minimization problem.

What is the efficient frontier?

The upper portion of the minimum-variance frontier beginning at the GMV portfolio. It contains the best available risk-return combinations.

What makes a portfolio efficient?

It offers the highest expected return for a given level of risk or the lowest risk for a given expected return.

What is a dominated portfolio?

A portfolio for which another portfolio offers higher return for the same risk or lower risk for the same return.

EXAM TIP: Why are portfolios below the GMV on the minimum-variance frontier inefficient?

For the same level of risk, another portfolio on the upper branch offers a higher expected return.

What inputs are required to construct the efficient frontier?

Expected returns, variances/standard deviations, and pairwise covariances/correlations. Think: Return + Risk + Relationships.

EXAM TIP: Is investor risk aversion an input when constructing the efficient frontier?

No. Risk aversion determines which portfolio the investor chooses, not the location of the frontier.

What happens when a risk-free asset is added to the risky-asset efficient frontier?

A CAL can be drawn from Rf tangent to the risky-asset efficient frontier.

What is the tangent portfolio?

The risky portfolio where the optimal CAL touches the efficient frontier. It has the highest Sharpe ratio among the available risky portfolios.

What is the two-fund separation theorem?

All investors can hold the same tangent risky portfolio plus the risk-free asset. Risk aversion determines how much they allocate to each.

EXAM TIP: Does greater risk aversion change which tangent risky portfolio an investor holds?

No. Under the standard assumptions, investors hold the same tangent risky portfolio; risk aversion determines where they sit along the CAL.

What is the fundamental difference between portfolio return and portfolio risk?

Portfolio return is a weighted average. Portfolio risk generally is not because it also depends on covariance/correlation.

EXAM TIP: What is the fastest rule when asked which asset provides the greatest diversification benefit?

Choose the asset with the lowest correlation with the existing portfolio, not necessarily the asset with the lowest individual volatility.

What is the fastest way to remember the CAL?

Return = Rf + (Sharpe Ratio × Risk).

What is the fastest way to remember optimal risky allocation?

w* = Reward / (Risk Aversion × Variance). More reward → more risky allocation; more risk aversion or variance → less risky allocation.

What is the relationship between the GMV, minimum-variance frontier, and efficient frontier?

The GMV is the lowest-risk point on the minimum-variance frontier. The efficient frontier begins at the GMV and consists of the upper branch.

MASTER CARD: What are five high-yield relationships to know for R01?

ρ ↓ → σP ↓; A ↑ → risky allocation ↓; CAL slope = Sharpe ratio; ρ < +1 → diversification benefit; GMV = lowest-risk portfolio and efficient frontier = best risk-return portfolios.

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