CFA level 1 Portfolio managment

According to U.S. historical asset class data (1926–2017), which investable asset class exhibited the highest average annual return and standard deviation, and which had the lowest?

Small-capitalization stocks had the highest average annual return (12.1%) and highest standard deviation (31.7%). Treasury bills had the lowest average annual return (3.4%) and lowest standard deviation (3.1%).

How is the real return approximated from the nominal return, and why are real returns preferred when comparing asset class returns over time?

Real return ≈ nominal return − inflation. Inflation varied widely (from −10.30% to +13.31% over the 92 years to 2017), so comparing nominal returns across periods is misleading; real returns adjust for this.

Why is using mean and variance alone an incomplete evaluation of investment returns according to portfolio management theory?

Returns are not normally distributed, so mean and variance don't fully describe them. Actual distributions often show negative skewness (a longer left tail of large losses) and excess kurtosis (kurtosis > 3: fat tails, so extreme outcomes are more likely than a normal distribution implies).

What does the risk–return trade-off mean, and how does 1926–2017 US data illustrate it?

A higher return can't be earned in efficient markets over long periods without accepting higher risk. US data fit this: small-company stocks had the highest return and risk, then large-company stocks, then long-term bonds, and T-bills had the lowest of both.

Over 1926–2017, what were the annual return and risk of US large-company stocks, and what was average inflation?

Large-company stocks: about 10.2% geometric mean return with 19.8% standard deviation (vs. 12.1% and 31.7% for small caps). Inflation averaged 2.9% per year over the 92 years.

Over 1926–2017, long-term US government bonds had higher risk (9.9%) than long-term corporate bonds (8.3%) but a lower return (5.5% vs. 6.1%). Does this mean government bonds had more default risk?

No. It only means government bond returns were more variable during that historical period. It is an exception to the usual risk–return pattern, not evidence of higher default risk.

In real terms (1900–2017), how did US stocks, bonds, and T-bills compare, and what explains the huge gap in ending wealth?

Real returns were about 6.5% per year for equities vs. 2.0% for bonds. USD 1 grew to about USD 1,654 in stocks, USD 10.20 in bonds, and USD 2.60 in T-bills. The modest difference in annual returns becomes enormous through compounding over 118 years.

How does the 1900–2017 risk of world stocks compare with US stocks and world-ex-US stocks, and what does it show?

US stocks: 20.0% risk; world excluding the US: 18.9%; world stocks combined: only 17.4%. Combining the two produces lower risk than either alone, showing the effect of diversification.

Besides risk and return, what market characteristic should investors weigh when choosing investments, and how does it affect trading costs?

Liquidity. Trading costs include commissions, the bid–ask spread, and price impact, and low liquidity widens the spread and increases price impact. For example, a 10-cent spread is 0.1% on a USD 100 stock but 1% on a USD 10 stock.

What does kurtosis greater than 3 mean for an asset's return distribution, and why does it matter?

The distribution has fat tails: extreme outcomes (large gains and large losses) are more likely than a normal distribution predicts. Mean and standard deviation alone understate the risk of extreme losses.

Given a choice between a certain outcome of USD 50 and a 50/50 gamble paying USD 100 or USD 0, how do risk-averse, risk-neutral, and risk-seeking investors choose?

A risk-averse investor prefers the certain outcome of USD 50. A risk-neutral investor is indifferent between the certain outcome and the gamble. A risk-seeking investor prefers the gamble over the certain payment.

What value does the risk aversion coefficient A take in U = E(r) − 0.5 × A × σ² for risk-averse, risk-neutral, and risk-seeking investors?

Risk-averse investors have A > 0 (greater A indicates higher risk aversion). Risk-neutral investors have A = 0. Risk-seeking investors have A < 0.

In the utility formula U = E(r) − 0.5 × A × σ², define each variable and state how returns and risk must be entered.

U = utility; E(r) = expected return; A = risk aversion coefficient; σ² = variance of returns. Enter E(r) and σ as decimals (e.g., 10% = 0.10, and σ = 20% gives σ² = 0.20² = 0.04).

Calculate the expected utility U for an investor with risk aversion coefficient A = 4 considering a portfolio with an expected return of 12% and a standard deviation of 20%.

Step 1) E(r) = 0.12 and σ² = 0.20² = 0.04. Step 2) U = 0.12 − 0.5 × 4 × 0.04 = 0.12 − 0.08 = 0.04.

What is the common mistake when using U = E(r) − 0.5 × A × σ² with percentage numbers?

Mixing percents and decimals (e.g., E(r) = 12 with σ² = 0.04). Use decimals throughout: U = 0.12 − 0.5 × 4 × 0.20² = 0.04. If you use whole percents, square σ in percent (20² = 400) and multiply by 0.005 instead of 0.5: U = 12 − 0.005 × 4 × 400 = 4 (i.e., 4%).

An investment has E(r) = 10% and σ = 20%, and the investor's risk aversion coefficient is A = 3. What is the utility, and what risk-free return would give the same utility?

U = 0.10 − 0.5 × 3 × 0.20² = 0.10 − 0.06 = 0.04. A risk-free asset has σ = 0, so its utility equals its return: the investor is indifferent between this risky investment and a guaranteed 4%.

Four investments have (E(r), σ) of (12%, 30%), (15%, 35%), (21%, 40%), and (24%, 45%). Which does an investor with A = 4 choose, and which does an investor with A = 2 choose?

A = 4 utilities: −0.0600, −0.0950, −0.1100, −0.1650, so choose Investment 1. A = 2 utilities: 0.0300, 0.0275, 0.0500, 0.0375, so choose Investment 3. The less risk-averse investor picks a riskier investment.

How would a risk-neutral investor (A = 0) choose among investments, and why?

With A = 0, U = E(r): the risk term drops out. A risk-neutral investor simply picks the highest expected return, regardless of standard deviation.

TRAP: "A risk-averse investor will never hold risky assets." What is wrong?

A risk-averse investor dislikes risk but will hold risky assets if the extra expected return is enough compensation for the additional risk. Given two investments with the same expected return, they choose the one with less risk.

What are the axes of an investor's indifference curve in portfolio theory, and why does an indifference curve slope upward for a risk-averse investor?

The horizontal axis (x-axis) is risk (σ or standard deviation) and the vertical axis (y-axis) is expected return E(R). It slopes upward because a risk-averse investor requires higher expected return to offset incremental risk and maintain the same level of utility.

How do the indifference curves of a highly risk-averse investor compare graphically to those of a less risk-averse investor?

The indifference curves of a highly risk-averse investor are steeper (higher slope), reflecting a demand for a larger increase in expected return for a given increase in risk.

What happens to an investor's optimal portfolio on the Capital Allocation Line (CAL) if their risk aversion coefficient A increases significantly?

The investor's indifference curves become steeper. The point of tangency with the CAL shifts down and to the left, allocating a higher percentage to the risk-free asset and a lower percentage to the risky market portfolio.

How is an investor's optimal portfolio identified using the CAL and indifference curves?

It is the point where the CAL is tangent to the highest indifference curve the investor can reach. Higher indifference curves are preferred, but those above the tangency curve are not attainable.

Compare the optimal portfolios of a more risk-averse investor and a less risk-averse investor facing the same CAL.

The less risk-averse investor has flatter indifference curves, so the tangency point lies further up and to the right on the CAL: more invested in the risky portfolio and less in the risk-free asset. The more risk-averse investor holds more of the risk-free asset.

The Lohrmanns (A = 2.5) can choose CAL portfolios with returns of 3%, 9%, 15%, and 20% and standard deviations of 0%, 12.1%, 24.1%, and 34.1%. Which portfolio is optimal?

Utility = E(R) − 0.5 × 2.5 × σ²: 3% gives 0.0300; 9% gives 0.09 − 1.25 × 0.121² = 0.0717; 15% gives 0.15 − 1.25 × 0.241² = 0.0774; 20% gives 0.20 − 1.25 × 0.341² = 0.0546. The 15% return / 24.1% risk portfolio has the highest utility, so it is optimal.

The optimal CAL is E(Rp) = 3% + 0.4978 × σp. What standard deviation does a 20% return require, and how does that compare with Asset A (20% return, 50% risk)?

0.20 = 0.03 + 0.4978 × σp, so σp = 0.17 / 0.4978 = 34.2%. The CAL portfolio earns the same 20% as Asset A with much less risk (34.2% vs. 50%), so it dominates Asset A.

A portfolio is 40% in a risky portfolio (E(R) = 12%, σ = 20%) and 60% in the risk-free asset (Rf = 4%). What are its expected return and standard deviation?

E(Rp) = 0.40 × 12% + 0.60 × 4% = 4.8% + 2.4% = 7.2%. σp = 0.40 × 20% = 8.0%, since the risk-free asset contributes no risk.

State the formula for historical sample variance s² of asset returns.

s² = Σ(Rt − R̄)² / (T − 1), where Rt = the return in period t, R̄ = the sample mean return, and T = the number of periods.

State the formula for historical sample covariance Cov1,2 between the returns of Asset 1 and Asset 2.

Cov1,2 = Σ(Rt,1 − R̄1)(Rt,2 − R̄2) / (n − 1), where Rt,1 and Rt,2 are period-t returns, R̄1 and R̄2 are the sample means, and n is the number of periods.

Asset A annual returns over 3 years are 5%, −2%, and 12%. Asset B returns are 7%, −4%, and 18%. Mean returns are R̄A = 5% and R̄B = 7%. Calculate the sample covariance CovA,B.

Products of deviations: Year 1: (5 − 5)(7 − 7) = 0. Year 2: (−2 − 5)(−4 − 7) = (−7)(−11) = 77. Year 3: (12 − 5)(18 − 7) = (7)(11) = 77. Sum = 154. Sample covariance = 154 / (3 − 1) = 77 (%²).

State the formula relating correlation coefficient ρ1,2, covariance Cov1,2, and standard deviations σ1, σ2.

ρ1,2 = Cov1,2 / (σ1 × σ2), so Cov1,2 = ρ1,2 × σ1 × σ2. Correlation is unitless and bounded by −1 and +1.

Asset A returned 5%, −2%, and 12% over three years, and Asset B returned 7%, −4%, and 18%. Calculate each asset's mean return.

Mean A = (5 − 2 + 12) / 3 = 5%. Mean B = (7 − 4 + 18) / 3 = 7%.

Asset A returned 5%, −2%, and 12% (mean 5%). Calculate its sample variance and standard deviation.

Deviations: 0, −7, 7. Squared: 0, 49, 49, sum = 98. Sample variance = 98 / (3 − 1) = 49 (%²). Standard deviation = sqrt(49) = 7%.

Asset B returned 7%, −4%, and 18% (mean 7%). Calculate its sample variance and standard deviation.

Deviations: 0, −11, 11. Squared: 0, 121, 121, sum = 242. Sample variance = 242 / 2 = 121 (%²). Standard deviation = sqrt(121) = 11%.

Assets A and B have a sample covariance of 77 (%²) and standard deviations of 7% and 11%. What is their correlation, and what does it mean?

ρ = 77 / (7 × 11) = 77 / 77 = 1.0. The returns are perfectly positively correlated: their deviations from the mean always move proportionally in the same direction.

Asset C returned 10%, 0%, and 20%, and Asset D returned 6%, 2%, and 4% over the same three years. Calculate their sample covariance and correlation.

Means: C = 10%, D = 4%. Deviations C: 0, −10, 10; D: 2, −2, 0. Products: 0, 20, 0, sum = 20, so Cov = 20 / 2 = 10 (%²). Variances: C = 200 / 2 = 100 (σ = 10%); D = 8 / 2 = 4 (σ = 2%). ρ = 10 / (10 × 2) = 0.50.

Two assets have a covariance of 0.0120 and standard deviations of 20% and 15%. What is their correlation?

ρ = Cov / (σ1 × σ2) = 0.0120 / (0.20 × 0.15) = 0.0120 / 0.03 = 0.40.

What is the difference between population variance and sample variance, and which is usually used with historical returns?

Population variance divides the sum of squared deviations from the population mean μ by T. Sample variance divides the squared deviations from the sample mean by T − 1. Analysts usually have only a sample of historical returns, so they use the sample variance.

Contrast covariance and correlation as measures of co-movement.

Covariance is an absolute measure in squared return units, and its size depends on the assets' volatilities, so it is hard to interpret alone. Correlation standardizes covariance by dividing by both standard deviations, so it is unitless and bounded between −1 and +1.

What does a correlation (or covariance) of zero between two assets' returns mean?

There is no linear relationship between the returns: knowing one asset's return for a period tells you nothing about the other's.

State the formula for the standard deviation σp of a portfolio composed of two risky assets with weights w1 and w2.

σp = sqrt(w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ1,2 × σ1 × σ2), where w1 + w2 = 1.

A portfolio is 30% invested in Stock (σS = 20%) and 70% in Bond (σB = 12%). The correlation between stock and bond returns is 0.60. Calculate the portfolio standard deviation σp.

Step 1) (0.3²)(0.2²) = 0.0036. Step 2) (0.7²)(0.12²) = 0.007056. Step 3) 2(0.3)(0.7)(0.6)(0.2)(0.12) = 0.006048. Step 4) σp² = 0.016704, so σp = sqrt(0.016704) = 12.92%.

State the expected return and the variance (using covariance) of a two-asset portfolio.

E(Rp) = w1 × E(R1) + w2 × E(R2). σp² = w1²σ1² + w2²σ2² + 2w1w2 × Cov(R1, R2), where w1 + w2 = 1. Expected return is always a weighted average; risk is not, unless ρ = +1.

A portfolio is 80% S&P 500 (E(R) = 9.93%, σ = 16.21%) and 20% emerging markets (E(R) = 18.20%, σ = 33.11%), with covariance 0.0050. Calculate its expected return and standard deviation.

E(Rp) = 0.80 × 9.93% + 0.20 × 18.20% = 11.58%. σp² = (0.80² × 0.1621²) + (0.20² × 0.3311²) + (2 × 0.80 × 0.20 × 0.0050) = 0.01682 + 0.00439 + 0.00160 = 0.02281, so σp = 15.10%.

A UK portfolio is 60% FTSE 100 (E(R) = 5.5%, σ = 13.2%) and 40% gilts (E(R) = 0.7%, σ = 4.2%), with correlation −0.01. Calculate its expected return and standard deviation.

E(Rp) = 0.6 × 5.5% + 0.4 × 0.7% = 3.58%. σp = sqrt(0.6² × 0.132² + 0.4² × 0.042² + 2 × 0.6 × 0.4 × (−0.01) × 0.132 × 0.042) = 8.08%.

Assets A (E(R) = 20%, σ = 50%) and B (E(R) = 15%, σ = 33%) have zero correlation. Calculate the return and risk of a portfolio that is 10% A and 90% B.

E(Rp) = 0.10 × 20% + 0.90 × 15% = 15.5%. With ρ = 0, the covariance term drops out: σp = sqrt(0.10² × 0.50² + 0.90² × 0.33²) = 30.12%.

For an equally weighted portfolio of N assets, how does portfolio variance relate to the average variance and the average covariance, and what happens as N grows?

σp² = (average variance) / N + [(N − 1) / N] × (average covariance). As N grows, the first term shrinks toward zero and portfolio variance approaches the average covariance, so covariance among assets drives almost all of a large portfolio's risk.

An equally weighted portfolio holds 20 assets, each with σ = 30%, and every pair has ρ = 0.20. What is the portfolio's standard deviation, and what is its limit as N becomes very large?

σp² = σ²/N + [(N − 1)/N] × ρσ² = 0.09/20 + (19/20) × 0.20 × 0.09 = 0.0045 + 0.0171 = 0.0216, so σp = 14.70%. As N → ∞, σp → sqrt(ρ) × σ = sqrt(0.20) × 30% = 13.42%.

TRAP: "Portfolio standard deviation is the weighted average of the assets' standard deviations." When is this true?

Only when ρ = +1. With any correlation below +1, portfolio standard deviation is less than the weighted average, which is the benefit of diversification. (Portfolio expected return, by contrast, is always the weighted average.)

When two risky assets are perfectly positively correlated (ρ = +1), what does the portfolio standard deviation formula simplify to?

σp = w1σ1 + w2σ2: the weighted average of the assets' standard deviations. There is no diversification benefit.

What is the effect on portfolio standard deviation σp when asset correlation ρ1,2 decreases from +1 toward −1?

Portfolio standard deviation decreases. Lower asset correlation enhances diversification benefits, reducing portfolio risk for any given expected return level.

Under what condition can portfolio risk be completely eliminated (σp = 0) using two risky assets?

When ρ1,2 = −1 and the weights are w1 = σ2 / (σ1 + σ2) and w2 = 1 − w1.

What happens to the shape of the Markowitz efficient frontier and portfolio diversification benefits if asset return correlations across all market sectors shift from 0.20 to 0.80?

Higher correlation reduces diversification benefits. The efficient frontier shifts to the right (more risk for every level of expected return) and becomes less curved.

Two stocks each have E(R) = 10% and σ = 20%, held 50/50. Calculate portfolio risk for correlations of +1, 0, and −1.

ρ = +1: σp² = 0.01 + 0.01 + 0.02 = 0.04, so σp = 20%. ρ = 0: σp² = 0.01 + 0.01 = 0.02, so σp = 14.14%. ρ = −1: σp² = 0.02 − 0.02 = 0, so σp = 0%. Expected return is 10% in every case.

In the 50/50 portfolio of two stocks (each σ = 20%), what is portfolio risk if ρ = 0.5 and if ρ = −0.5?

ρ = 0.5: σp² = 0.02 + 2 × 0.25 × 0.04 × 0.5 = 0.03, so σp = 17.32%. ρ = −0.5: σp² = 0.02 − 0.01 = 0.01, so σp = 10%.

Does changing the correlation between two assets change the portfolio's expected return?

No. Expected return is the weighted average of the assets' expected returns, whatever the correlation. Correlation affects only the portfolio's risk.

Two assets with σ1 = 20% and σ2 = 30% are perfectly negatively correlated. What weights produce a zero-risk portfolio?

w1 = σ2 / (σ1 + σ2) = 0.30 / 0.50 = 60% in Asset 1, and w2 = 40% in Asset 2. Check: σp = 0.6 × 20% − 0.4 × 30% = 0.

Why does diversification tend to work less well during financial crises?

Correlations among assets tend to rise during market turmoil (as in the 2008 credit contagion), which reduces the risk reduction that diversification normally provides.

List the avenues for diversification described in the curriculum.

Diversify across asset classes; use index funds (cheaper than buying hundreds of securities); diversify among countries; don't concentrate in your employer's stock (your human capital is already tied to it); evaluate each asset before adding it; and buy insurance-like assets (insurance, gold, puts) that are negatively correlated with your holdings.

State the rule for whether adding a new asset to a portfolio improves it.

Add the asset if (E(Rnew) − Rf) / σnew > [(E(Rp) − Rf) / σp] × ρnew,p. That is, the new asset's Sharpe ratio must exceed the current portfolio's Sharpe ratio times their correlation.

A portfolio has E(R) = 10% and σ = 15%, and Rf = 3%. A candidate asset has E(R) = 8%, σ = 20%, and correlation 0.4 with the portfolio. Should it be added?

New asset's Sharpe = (8% − 3%) / 20% = 0.25. Hurdle = portfolio Sharpe × ρ = (7% / 15%) × 0.4 = 0.467 × 0.4 = 0.187. Since 0.25 > 0.187, adding the asset improves the portfolio's risk–return trade-off.

Why might an investor rationally add an asset with a negative expected return, such as insurance or put options?

Because it is negatively correlated with the rest of the portfolio. It pays off when other assets lose value, significantly reducing risk and exposure to extreme losses, which can be worth a small negative average return.

Define the minimum-variance frontier, the global minimum-variance portfolio, and the efficient frontier of risky assets.

Minimum-variance frontier: Portfolios that minimize standard deviation for each level of expected return. Global minimum-variance portfolio: The single risky portfolio on the minimum-variance frontier with the lowest total risk. Efficient frontier: The top portion of the minimum-variance frontier lying above the global minimum-variance portfolio, offering the highest expected return for a given risk level.

What is the two-fund separation theorem?

All investors, regardless of risk preferences or wealth, hold a combination of two funds: the risk-free asset and the same optimal risky portfolio. This splits the problem into (1) the investment decision, choosing the optimal risky portfolio without regard to preferences, and (2) the financing decision, choosing how much to lend or borrow at the risk-free rate based on risk preferences.

What is the investment opportunity set?

All the portfolios (combinations of available assets) that can be formed, plotted by expected return and standard deviation. The minimum-variance frontier is its left boundary.

Why would a risk-averse investor choose only portfolios on the efficient frontier?

Every portfolio below the efficient frontier has a lower expected return than some efficient portfolio with the same risk. A risk-averse investor would never accept less return for the same risk, so non-efficient portfolios are dominated.

Once a risk-free asset is available, how is the optimal risky portfolio identified?

It is the point where a line from the risk-free rate is tangent to the efficient frontier. That CAL has the highest slope (Sharpe ratio) of any attainable CAL, so it dominates all other risky portfolio combinations.

Portfolios of Assets A and B can be combined with a 3% risk-free asset. The CAL slope is maximized at 38.20% in Asset A. What is the optimal CAL, and what does its slope represent?

E(Rp) = 0.03 + 0.4978 × σp. The slope, 0.4978, is the Sharpe ratio of the optimal risky portfolio: the maximum attainable excess return per unit of total risk.

What are the variance and standard deviation of a portfolio combining a risk-free asset (Rf) with weight (1 − wA) and a risky asset portfolio (A) with weight wA?

The risk-free asset has zero variance and zero covariance with risky assets, so σp² = wA² × σA² and σp = wA × σA. Portfolio risk is proportional to the weight in the risky portfolio.

State the expected return and standard deviation of a portfolio with weight w1 in the risk-free asset and (1 − w1) in a risky portfolio.

E(Rp) = w1 × Rf + (1 − w1) × E(Ri). σp = (1 − w1) × σi. Risk and return are both linear in the weight, so the combinations plot as a straight line (the CAL).

Rf = 5%, and the market has E(R) = 15% and σ = 20%. Find the return and risk of portfolios with 25% and 75% in the market.

25% in market: E(R) = 0.75 × 5% + 0.25 × 15% = 7.5%, σ = 0.25 × 20% = 5%. 75% in market: E(R) = 0.25 × 5% + 0.75 × 15% = 12.5%, σ = 0.75 × 20% = 15%.

Rf = 5%, and the market has E(R) = 15% and σ = 20%. An investor borrows 25%, 50%, or 100% of his capital at 5% to invest in the market. Find the return and risk of each portfolio.

Borrow 25% (w1 = −0.25): E(R) = −0.25 × 5% + 1.25 × 15% = 17.5%, σ = 1.25 × 20% = 25%. Borrow 50%: 20.0% and 30%. Borrow 100%: 25.0% and 40%. Leverage raises both expected return and risk.

Why does combining a risky portfolio with a risk-free asset improve an investor's choices?

The combinations lie on a straight line from Rf through the risky portfolio. Using the optimal (tangency) risky portfolio, this line gives a higher expected return for every level of risk than the efficient frontier of risky assets alone, except at the tangency point.

What is the Capital Allocation Line (CAL), and what are its y-intercept and slope?

The CAL shows every risk–return combination of the risk-free asset and one particular risky portfolio. Its y-intercept is Rf, and its slope is the risky portfolio's Sharpe ratio: (E(Rrisky) − Rf) / σrisky.

Write the general equation for the Capital Allocation Line (CAL) for portfolio p combined with a risk-free asset.

E(Rp) = Rf + [(E(Ri) − Rf) / σi] × σp

How does the assumption of homogeneous expectations transform an individual Capital Allocation Line (CAL) into the Capital Market Line (CML)?

Under homogeneous expectations, all investors share identical risk, return, and correlation estimates. Consequently, every investor faces the exact same efficient frontier and identifies the exact same optimal risky tangency portfolio: the Market Portfolio (M). The CAL using the Market Portfolio is the Capital Market Line (CML).

State the Capital Market Line (CML) equation and define its components.

E(Rp) = Rf + [(E(Rm) − Rf) / σm] × σp, where Rf = risk-free rate, E(Rm) = expected market return, σm = market standard deviation, and σp = portfolio standard deviation. The slope is the market's Sharpe ratio.

On the CML, what distinguishes a lending portfolio from a borrowing (leveraged) portfolio?

A lending portfolio lies to the left of Market Portfolio M (wm < 100%, wf > 0%); wealth is partially lent at Rf. A borrowing portfolio lies to the right of M (wm > 100%, wf < 0%); funds are borrowed at Rf to buy additional market portfolio assets on margin.

If the risk-free rate Rf increases while the market return E(Rm) and market volatility σm remain constant, what happens to the slope of the Capital Market Line (CML)?

The slope, (E(Rm) − Rf) / σm, falls, so the CML becomes flatter, and its intercept (Rf) moves up.

Rf = 5%, and the market has E(R) = 15% and σ = 20%. Write the CML equation, and find the risk of a CML portfolio with a 12.5% expected return.

CML: E(Rp) = 5% + [(15% − 5%) / 20%] × σp = 5% + 0.50 × σp. For 12.5%: 12.5% = 5% + 0.50 × σp, so σp = 15%, which is 75% in the market and 25% in T-bills.

How does the CML change when investors must borrow at a rate higher than the risk-free lending rate?

It becomes kinked at the market portfolio M. To the left of M (lending), the slope is (E(Rm) − Rf) / σm. To the right of M (borrowing), the slope is lower, (E(Rm) − Rb) / σm, so leveraged portfolios earn less extra return per unit of risk.

Rf = 5%, the borrowing rate is 7%, and the market has E(R) = 15% and σ = 20%. What are the return and risk if the investor borrows 25%, and if he borrows 75%?

Use Rb for borrowing. Borrow 25% (w1 = −0.25): E(R) = −0.25 × 7% + 1.25 × 15% = 17.0%, σ = 25%. Borrow 75%: E(R) = −0.75 × 7% + 1.75 × 15% = 21.0%, σ = 35%. At equal rates, borrowing 25% would have earned 17.5%.

Contrast passive and active portfolio management in the context of the CML.

Passive investors believe prices are informationally efficient: they hold an index proxy for the market portfolio plus the risk-free asset, so they stay on the CML. Active investors believe some prices are wrong: they overweight securities they think are undervalued and underweight those they think are overvalued, deviating from market weights.

What is the "market" (market portfolio) in capital market theory, and what is used in practice?

In theory, it includes all risky assets, both financial and nonfinancial, held in market-value weights. In practice, a broad index such as the S&P 500 is used as a proxy.

Contrast a CAL with the CML.

A CAL combines the risk-free asset with ANY risky portfolio, and each investor may have their own. The CML is the special CAL that uses the market portfolio, which under homogeneous expectations is the optimal risky portfolio for every investor. Its slope is the market's Sharpe ratio.

Distinguish between systematic risk and unsystematic risk, including their alternative names.

Systematic risk (market risk, non-diversifiable risk) is macro-driven and affects all assets; it cannot be eliminated by diversification. Unsystematic risk (firm-specific, unique, diversifiable risk) is unique to individual companies and can be eliminated through portfolio diversification.

Why are investors compensated in equilibrium for bearing systematic risk but not unsystematic risk?

Because diversification is costless (or very low cost), unsystematic risk can be eliminated entirely by holding a diversified portfolio. Capital market theory dictates that investors will not be compensated for bearing risk that can be eliminated for free.

Roughly how many randomly selected stocks does it take to diversify away most nonsystematic risk?

One study found about 12 to 18 stocks achieve 90% of the maximum possible diversification; another found about 30. Beyond roughly 30 stocks, portfolio risk levels off near systematic (market) risk.

Identify the trap in this statement: 'A speculative biotechnology company with extremely high standard deviation of returns must have a very high required rate of return under CAPM.'

The statement is FALSE. Standard deviation measures total risk. Under CAPM, required return depends solely on systematic risk (beta). If the biotech uncertainty is firm-specific (drug trial outcomes), unsystematic risk is high but beta may be low, resulting in a low required return.

State how total variance decomposes under the single-index (market) model.

σi² = βi² × σm² + σei². Total variance = systematic variance (βi²σm²) + nonsystematic variance (σei²). The covariance term drops out because nonmarket returns are uncorrelated with the market.

A stock has β = 1.2 and σ = 30%, and the market's σ = 20%. Calculate its systematic variance, nonsystematic variance, and the share of total variance that is systematic.

Systematic variance = 1.2² × 0.20² = 0.0576. Total variance = 0.30² = 0.09. Nonsystematic variance = 0.09 − 0.0576 = 0.0324 (σe = 18%). Systematic share = 0.0576 / 0.09 = 64%.

What are the systematic and nonsystematic risk of (1) a three-month T-bill and (2) the market portfolio?

(1) T-bill: zero systematic and zero nonsystematic risk, because it is risk-free. (2) Market portfolio: no nonsystematic risk, because it is fully diversified; all of its risk is systematic.

Asset A has twice the total risk of Asset B. Systematic risk is two-thirds of A's total risk, while all of B's risk is systematic. Which should have the higher expected return?

Asset A. Its systematic risk = (2/3) × 2 × σB = 1.33 × σB, which is one-third greater than B's. Only systematic risk is priced, so A has the higher expected return.

Give examples of firms with high and low systematic risk.

High: luxury goods makers (e.g., high-end cars and motorcycles), whose returns move closely with the economy and market. Low: utility companies, whose returns respond little to market-wide changes.

Write the general formula for a k-factor return generating model in excess return form.

E(Ri) − Rf = βi1 E(F1) + βi2 E(F2) + … + βik E(Fk) where βij is the factor loading (sensitivity) of asset i to factor j, and E(Fj) is the expected excess factor return.

What are the three factors in the Fama-French model, and what fourth factor was added by Carhart?

Fama-French 3 factors: (1) Market excess return (Rm − Rf), (2) Firm size (SMB: small minus big), (3) Book-to-market ratio (HML: high minus low). Carhart added a 4th factor: Price momentum (prior period returns).

Write the single-factor market model equation for realized asset returns.

Ri = αi + βi Rm + ei where Ri = asset return, αi = intercept, βi = slope coefficient (beta), Rm = market return, and ei = abnormal return (firm-specific error term).

What three types of factors are used in return generating models, and which type is viewed with suspicion?

Macroeconomic (e.g., GDP growth, inflation), fundamental (e.g., earnings growth, firm size, R&D), and statistical factors. Statistical factors may have no basis in theory and may reflect data mining, so they are suspect.

In the market model, how are a stock's expected return and abnormal return calculated?

Expected return = αi + βi × Rm. Abnormal (company-specific) return = ei = Ri − (αi + βi × Rm).

A regression gives Wal-Mart α = 0.0001 and β = 0.90 on daily returns. On a day the market rises 1% and Wal-Mart rises 2%, what is Wal-Mart's abnormal return?

Expected return = 0.0001 + 0.90 × 0.01 = 0.0091. Abnormal return = 0.02 − 0.0091 = 0.0109, or 1.09%.