Questions: Cost of Equity and CAPM Application

Question 1

Firm A is a small mining startup with highly volatile returns — its total return variance is very high. However, its returns are completely uncorrelated with the market (beta ≈ 0). According to CAPM, what is Firm A's cost of equity?

• A. High — investors require extra compensation for the high total volatility they bear
• B. Approximately equal to the risk-free rate — uncorrelated risk earns no market premium under CAPM
• C. Equal to the market return — all equity carries the market premium regardless of correlation
• D. Undefined — CAPM cannot price assets with idiosyncratic volatility

Answer: (B)

Under CAPM, only systematic (market-correlated) risk is priced. Firm A's high total variance is entirely idiosyncratic (unsystematic) — it could be eliminated by an investor who holds a diversified portfolio. Because diversification is free in competitive markets, rational investors will not demand extra compensation for risk they can eliminate. With beta ≈ 0, the CAPM formula gives r_e ≈ r_f + 0·(r_m − r_f) = r_f. This result seems counterintuitive — a volatile startup requires no equity premium? — but the logic is rigorous: the risk premium compensates only for what cannot be diversified away.

Question 2

A utility company has beta = 0.4 and a tech startup has beta = 2.2. The risk-free rate is 4% and the equity risk premium is 6%. What is each company's cost of equity, and what does the difference reflect economically?

• A. Utility: 6.4%, Tech: 17.2% — the difference reflects total business risk
• B. Utility: 6.4%, Tech: 17.2% — the difference reflects only systematic risk exposure, not total volatility
• C. Both 10% — CAPM applies the same market return to all equity
• D. Utility: 4%, Tech: 4% — both earn the risk-free rate since equity risk can be diversified

Answer: (B)

Utility: r_e = 4% + 0.4(6%) = 4% + 2.4% = 6.4%. Tech: r_e = 4% + 2.2(6%) = 4% + 13.2% = 17.2%. The 10.8 percentage point difference reflects systematic risk only — how much each stock's returns move with the overall market. The utility's revenues are stable regardless of economic cycles (low beta); the startup's returns are highly amplified by market conditions (high beta). Both firms likely have high total volatility, but investors in the utility face little undiversifiable risk. The CAPM cost of equity measures only the compensation for that undiversifiable portion.

Question 3

Under CAPM, a stock with very high total return volatility but low beta should have a high cost of equity, because investors are exposed to substantial uncertainty.

• T. True
• F. False

Answer: False

Total volatility includes both systematic and unsystematic components. CAPM prices only the systematic component (captured by beta), because unsystematic risk can be eliminated through diversification at no cost. A stock with high total volatility but low correlation with the market has mostly unsystematic risk — the kind a diversified investor has already neutralized in their portfolio. CAPM says the market will not compensate for risk you can eliminate yourself for free. High total volatility with low beta → low CAPM cost of equity.

Question 4

Beta measures a stock's total return variability, which is why highly volatile stocks usually have high betas and high costs of equity.

• T. True
• F. False

Answer: False

Beta measures a stock's sensitivity to market movements — specifically, the covariance of the stock's returns with market returns, divided by the variance of market returns. It captures systematic (co-movement) risk, not total variability. A stock can have very high total variance (high absolute volatility) but low beta if its volatility is driven by firm-specific events uncorrelated with the market. For example, a small biotech whose returns swing wildly based on clinical trial outcomes may have a low beta if those outcomes are independent of overall market conditions.

Question 5

Why does CAPM only compensate investors for systematic risk and not for unsystematic (firm-specific) risk, and what assumption makes this true?

Model answer: CAPM assumes investors hold diversified portfolios. In a diversified portfolio, unsystematic (firm-specific) risks — individual company events, CEO changes, product failures — cancel out across many holdings. Because diversification is available to any investor, competitive markets will not reward bearing a risk that can be freely eliminated. Only systematic risk — the portion that co-moves with the entire market and cannot be diversified away even in a perfectly diversified portfolio — demands a risk premium. Beta measures exactly this undiversifiable exposure: stocks that amplify market swings (high beta) impose uncompensable risk on every investor; stocks that are insensitive to the market (low beta) do not.

The key assumption is that investors can costlessly diversify. In practice, transaction costs, illiquidity, and constraints on portfolio construction mean some investors do bear unsystematic risk. This is one reason CAPM is an idealization — real-world asset pricing shows that factors beyond beta (size, value, momentum) seem to earn premiums, suggesting the market does partially price some 'diversifiable' risks. But CAPM's core insight — that only co-movement with the market should earn a premium in a competitive, frictionless setting — remains the foundation of modern cost-of-capital estimation.