Level 1 — RecognitionPortfolio Theory

Portfolio Theory

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Chapter: 5.5 Portfolio Theory Difficulty Level: 1 — Recognition (MCQ, Matching, True/False with justification) Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each, 10 marks)

Choose the single best answer.

Q1. Diversification primarily reduces which type of risk? A) Systematic risk B) Unsystematic (specific) risk C) Market risk D) Interest-rate risk

Q2. The correlation coefficient between two assets can take values in the range: A) $0$ to $1$ B) $-1$ to $0$ C) $-1$ to $+1$ D) $-\infty$ to $+\infty$

Q3. In the CAPM equation $E(R_i) = R_f + \beta_i\,(E(R_m) - R_f)$ , the term $(E(R_m) - R_f)$ is called the: A) Risk-free rate B) Market risk premium C) Alpha D) Beta premium

Q4. The Sharpe ratio is defined as: A) $\dfrac{E(R_p) - R_f}{\beta_p}$ B) $\dfrac{E(R_p) - R_f}{\sigma_p}$ C) $\dfrac{E(R_p)}{\sigma_p}$ D) $\dfrac{E(R_p) - R_m}{\sigma_p}$

Q5. A stock with $\beta = 1.0$ is expected to: A) Move independently of the market B) Move in the opposite direction to the market C) Move in line with the market D) Have zero total risk

Q6. The efficient frontier represents portfolios that offer: A) The lowest return for a given risk B) The maximum return for a given level of risk C) Zero risk at all return levels D) Only fully diversified index funds

Q7. Which measure uses downside deviation rather than total standard deviation in its denominator? A) Sharpe ratio B) Treynor ratio C) Sortino ratio D) Alpha

Q8. The Treynor ratio measures excess return per unit of: A) Total risk B) Systematic risk (beta) C) Downside risk D) Tracking error

Q9. A positive Jensen's alpha indicates that a portfolio has: A) Underperformed its CAPM-predicted return B) Exactly matched its CAPM-predicted return C) Outperformed its CAPM-predicted return D) Zero beta

Q10. The Security Market Line (SML) plots expected return against: A) Standard deviation B) Beta C) Variance D) Sharpe ratio

Section B — Matching (1 mark each, 8 marks)

Match each term (Q11–Q18) to its correct description (i–viii).

Term		Description
Q11. Covariance		(i) Reward per unit of total risk
Q12. Systematic risk		(ii) Measures how two assets move together (unstandardised)
Q13. Sharpe ratio		(iii) Cannot be diversified away
Q14. Beta		(iv) The set of optimal risk-return portfolios
Q15. Efficient frontier		(v) Sensitivity of an asset's return to the market
Q16. Unsystematic risk		(vi) Excess return over CAPM prediction
Q17. Alpha		(vii) Firm-specific, diversifiable risk
Q18. Risk-free rate		(viii) Return with (theoretically) zero risk

Section C — True / False with Justification (2 marks each, 12 marks)

State True or False (1 mark) and give a one-line justification (1 mark).

Q19. "Combining two assets with a correlation of $+1$ produces diversification benefits."

Q20. "According to CAPM, in equilibrium a stock plotting above the SML is underpriced (offers more return than required)."

Q21. "The Markowitz mean-variance model assumes investors care only about expected return and variance of returns."

Q22. "A portfolio's total variance equals the simple weighted average of the individual asset variances, regardless of correlation."

Q23. "The Sortino ratio will always be lower than the Sharpe ratio for the same portfolio."

Q24. "A well-diversified portfolio can eliminate systematic risk entirely."

Answer keyMark scheme & solutions

Section A (10 marks)

Q1 — B. Diversification cancels firm-specific shocks; systematic risk remains. (1)

Q2 — C. Correlation is standardised covariance, bounded in $[-1, +1]$ . (1)

Q3 — B. $(E(R_m)-R_f)$ is the market risk premium; multiplied by $\beta$ gives the asset's premium. (1)

Q4 — B. Sharpe = excess return $\div$ total risk $\sigma_p$ . (1)

Q5 — C. $\beta=1$ means the stock moves one-for-one with the market. (1)

Q6 — B. Efficient frontier = max return per unit risk (upper edge of feasible set). (1)

Q7 — C. Sortino uses downside deviation only. (1)

Q8 — B. Treynor divides excess return by beta (systematic risk). (1)

Q9 — C. Positive alpha = return exceeds CAPM benchmark → outperformance. (1)

Q10 — B. SML: expected return vs. beta (unlike CML, which uses $\sigma$ ). (1)

Section B (8 marks — 1 each)

Q	Answer
Q11 Covariance	(ii) unstandardised co-movement
Q12 Systematic risk	(iii) cannot be diversified away
Q13 Sharpe ratio	(i) reward per unit total risk
Q14 Beta	(v) sensitivity to market
Q15 Efficient frontier	(iv) set of optimal portfolios
Q16 Unsystematic risk	(vii) firm-specific, diversifiable
Q17 Alpha	(vi) excess return over CAPM
Q18 Risk-free rate	(viii) zero-risk return

Section C (12 marks)

Q19 — False (1). Justification (1): With $\rho=+1$ assets move perfectly together; portfolio risk is just the weighted average of standard deviations — no reduction, hence no diversification benefit.

Q20 — True (1). Justification (1): Above the SML means actual expected return > required return for its beta, so the stock is undervalued and will be bid up.

Q21 — True (1). Justification (1): Markowitz mean-variance framework characterises each portfolio solely by mean (return) and variance (risk).

Q22 — False (1). Justification (1): Portfolio variance $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2$ includes the covariance term; only equals the weighted average of variances in a degenerate case.

Q23 — False (1). Justification (1): Sortino's denominator (downside deviation) is usually smaller than total $\sigma$ , so Sortino is typically higher, not lower.

Q24 — False (1). Justification (1): Diversification removes unsystematic risk only; systematic (market) risk cannot be eliminated.

[
  {"claim":"Two assets rho=+1, w=0.5 each: portfolio sigma equals weighted average of sigmas (no diversification)", "code":"w1,w2=Rational(1,2),Rational(1,2)\ns1,s2=Rational(20,100),Rational(30,100)\nrho=1\nvar=w1**2*s1**2+w2**2*s2**2+2*w1*w2*rho*s1*s2\nsig=sqrt(var)\nweighted_avg=w1*s1+w2*s2\nresult=simplify(sig-weighted_avg)==0"},
  {"claim":"Portfolio variance with rho<1 is less than weighted-average-of-variances degenerate assumption (Q22 false)", "code":"w1,w2=Rational(1,2),Rational(1,2)\ns1,s2=Rational(20,100),Rational(30,100)\nrho=Rational(1,5)\nvar_true=w1**2*s1**2+w2**2*s2**2+2*w1*w2*rho*s1*s2\nvar_wrong=w1*s1**2+w2*s2**2\nresult=var_true!=var_wrong"},
  {"claim":"CAPM: Rf=5%, beta=1.2, market return=12% gives expected return 13.4%", "code":"Rf=Rational(5,100)\nbeta=Rational(12,10)\nRm=Rational(12,100)\nEr=Rf+beta*(Rm-Rf)\nresult=Er==Rational(134,1000)"},
  {"claim":"Sharpe ratio: Rp=15%, Rf=5%, sigma=20% equals 0.5", "code":"Rp=Rational(15,100)\nRf=Rational(5,100)\nsigma=Rational(20,100)\nsharpe=(Rp-Rf)/sigma\nresult=sharpe==Rational(1,2)"}
]