COMM 467 · Practice exam · Ch 1, 4, 5, 7

Practice Midterm期中模拟考试

Original practice questions written to match the coverage and style of the Midterm Review slides: definitions, theory and calculations in multiple choice, then numerical and short-answer problems. Not past exam questions.

按老师 Midterm Review 课件的范围和题型自编的练习题，不是历年真题。

CoverageCh 1 (lecture slides), Ch 4, Ch 5, Ch 7 up to estimating and testing the SIM and MM
Not coveredBlume, Vasicek, fundamental betas, Bloomberg / WRDS betas
Test rulesCritical t = ±1.96; residual correlation rule of thumb ±2/√T
ConventionsR ~ (mean, variance); population formulas (÷T) unless stated
Score 0 / 0 answered · 72 questions
80:00

Part A · Multiple choice第一部分：选择题

Click an answer to check it. The explanation appears right away. 点选答案后立即显示对错和解析。

Part B · Numerical and short answer第二部分：计算与简答

Work each problem on paper first, then open the solution. 先在纸上做，再看答案。

B1Two-period consumption (Ch 1)两期消费

You earn $50,000 this year and $40,000 next year and can borrow and lend at 5%.

  1. What is the maximum you can consume this year? Next year?
  2. Write the equation of the opportunity set.
  3. If you consume $60,000 this year, how much can you consume next year?
  4. Why do indifference curves for a person who prefers more to less have a negative slope and a convex shape?
Show solution 查看答案
(a) Max C₀ = 50,000 + 40,000/1.05 = 88,095.24 Max C₁ = 50,000(1.05) + 40,000 = 92,500 (b) C₁ = 40,000 + (50,000 − C₀)(1.05) = 92,500 − 1.05C₀ (c) C₁ = 92,500 − 1.05(60,000) = 29,500 (borrow 10,000 today, repay 10,500)

(d) Negative slope: giving up consumption in one period must be compensated by more in the other to stay equally satisfied. Convex: the less you have of one period’s consumption, the more of the other you require to give up another unit (diminishing marginal rate of substitution).

向下倾斜：少一期就要多另一期补偿；凸向原点：边际替代率递减。

B2Two stocks, probability distribution (Ch 4)概率分布下的两只股票

StateProbabilityRX (%)RY (%)
Boom0.25208
Normal0.50104
Recession0.25−410
  1. Expected return, variance and standard deviation of X and Y.
  2. Covariance and correlation between X and Y.
  3. Mean and standard deviation of a portfolio with 40% in X and 60% in Y.
  4. Composition, mean and standard deviation of the minimum-variance portfolio (no short sales).
Show solution 查看答案
(a) E[X] = .25(20) + .5(10) + .25(−4) = 9.0 E[Y] = .25(8) + .5(4) + .25(10) = 6.5 deviations X: 11, 1, −13 Y: 1.5, −2.5, 3.5 V[X] = .25(121) + .5(1) + .25(169) = 73.00 σX = 8.544 V[Y] = .25(2.25) + .5(6.25) + .25(12.25) = 6.75 σY = 2.598 (b) Cov = .25(11)(1.5) + .5(1)(−2.5) + .25(−13)(3.5) = 4.125 − 1.25 − 11.375 = −8.50 ρ = −8.50 / (8.544 × 2.598) = −0.383 (c) E[Rp] = .4(9) + .6(6.5) = 7.50 V[Rp] = .4²(73) + .6²(6.75) + 2(.4)(.6)(−8.5) = 11.68 + 2.43 − 4.08 = 10.03 σp = 3.167 (d) X_X = (σY² − σXY)/(σX² + σY² − 2σXY) = (6.75 + 8.5)/(73 + 6.75 + 17) = 0.158 E = 0.158(9) + 0.842(6.5) = 6.89 σ = 2.085

The negative correlation lets the 40/60 portfolio have far less risk (3.17) than X alone (8.54). 负相关带来明显的分散化效果。

B3Returns from prices and dividends (Ch 4)由价格和股利计算收益

Month-end prices for a stock are 50, 52, 51, 54.50, 53. A dividend of $0.75 is received at the end of month 3 and month 5 (the 51 and 53 observations).

  1. Compute the four monthly holding-period returns.
  2. Compute the mean return and the variance (population, ÷T). What would the sample variance be?
Show solution 查看答案
R_t = (P_t − P_t−1 + D_t) / P_t−1 R₁ = (52 − 50)/50 = 4.00% R₂ = (51 − 52 + 0.75)/52 = −0.48% R₃ = (54.50 − 51)/51 = 6.86% R₄ = (53 − 54.50 + 0.75)/54.50 = −1.38% Mean = (4.00 − 0.48 + 6.86 − 1.38)/4 = 2.25% Population variance = Σ(R − 2.25)²/4 = 11.24 Sample variance (÷3) = 14.98

B4Naive diversification (Ch 4)等权分散化

For a population of stocks, the average variance is 52 and the average covariance is 14.

  1. What is the variance of an equally weighted portfolio of 10 stocks? Of 25 stocks?
  2. What is the minimum variance attainable by naive diversification?
  3. How many stocks are needed for a portfolio variance of at most 15?
Show solution 查看答案
σp² = (1/N)(avg var) + ((N−1)/N)(avg cov) = (1/N)(52 − 14) + 14 = 38/N + 14 (a) N = 10: 17.80 N = 25: 15.52 (b) N → ∞: σp² → 14 (the average covariance) (c) 38/N + 14 ≤ 15 → N ≥ 38

B5IOS under ρ = +1, −1 and 0 (Ch 5)三种相关系数下的 IOS

A ~ (mean 8%, σ 12%) and B ~ (mean 14%, σ 20%). Fully invested, no short sales, no borrowing or lending.

  1. For ρ = +1, −1 and 0, find the minimum-variance portfolio (XA, mean, σ).
  2. Write the IOS equation (E[RP] as a function of σP) for ρ = +1 and for ρ = −1.
  3. For each case, describe the efficient frontier.
Show solution 查看答案
X_A(MVP) = (σB² − ρσAσB) / (σA² + σB² − 2ρσAσB) ρ = +1: formula gives X_A = 2.5 (needs short sales) → no SS: X_A = 1, E = 8.00, σ = 12.00 ρ = −1: X_A = σB/(σA + σB) = 20/32 = 0.625, E = 10.25, σ = 0 ρ = 0: X_A = 400/544 = 0.735, E = 9.59, σ = 10.29 IOS, ρ = +1: straight line through A and B: E[Rp] = −1 + 0.75σp (12 ≤ σp ≤ 20) IOS, ρ = −1: two lines from (0, 10.25): E[Rp] = 10.25 ± 0.1875σp

(c) ρ = +1: the whole line from A to B is efficient (MVP is A). ρ = −1: the upper line from (0, 10.25) to B is efficient; the lower line to A is not. ρ = 0: the curve from the MVP (10.29, 9.59) up to B is efficient.

ρ = +1 整条直线有效；ρ = −1 上面那条线有效；ρ = 0 从 MVP 到 B 的曲线有效。

B6Short sales (Ch 5)允许卖空

Use A and B from B5 with ρ = 0.2.

  1. Find the minimum-variance portfolio.
  2. Short sales are now allowed. Compute the mean and σ of (i) XA = 1.5, XB = −0.5 and (ii) XA = −0.5, XB = 1.5.
  3. Which of the two is on the efficient frontier, and why?
Show solution 查看答案
σAB = 0.2(12)(20) = 48 (a) X_A = (400 − 48)/(144 + 400 − 96) = 0.786 E = 9.29 σ = 11.11 (b)(i) E = 1.5(8) − 0.5(14) = 5.00 σ² = 2.25(144) + 0.25(400) + 2(1.5)(−0.5)(48) = 352 → σ = 18.76 (ii) E = −0.5(8) + 1.5(14) = 17.00 σ² = 0.25(144) + 2.25(400) + 2(−0.5)(1.5)(48) = 864 → σ = 29.39

(c) (ii). Efficient portfolios lie on the upper half of the curve, above the MVP return of 9.29%. Portfolio (i) earns only 5%, below the MVP, so it is inefficient. Shorting the low-return asset (A) to buy more of B extends the efficient frontier beyond B.

(ii) 有效：收益高于 MVP 的上半支才有效；(i) 收益 5% 低于 MVP。

B7Unlimited borrowing and lending at RF (Ch 5)无限借贷

RF = 4%. Three risky portfolios are available: K (10%, σ 12%), L (13%, σ 20%), M (7%, σ 5%).

  1. Which portfolio would every risk-averse investor combine with RF? Justify.
  2. An investor wants σP = 9%. How should they invest, and what is E[RP]?
  3. Another investor wants σP = 3%. How should they invest, and what is E[RP]?
  4. Which theorem says every investor holds the same risky portfolio?
Show solution 查看答案
Slope = (E[R] − RF)/σ: K = 6/12 = 0.50 L = 9/20 = 0.45 M = 3/5 = 0.60 → M (b) σP = X_M σM → X_M = 9/5 = 1.8 (borrow 0.8 at RF) E = 4 + 1.8(3) = 9.40% (c) X_M = 3/5 = 0.6 (lend 0.4 at RF) E = 4 + 0.6(3) = 5.80%

(d) The separation theorem: the choice of the risky portfolio (M) is separate from the investor’s risk preferences, which only determine how much to lend or borrow.

分离定理：风险组合的选择与个人风险偏好无关，偏好只决定借贷多少。

B8Lend at RF, borrow at a higher R′F (Ch 5)借贷利率不同

Investors can lend at RF = 4% but borrow only at R′F = 7%. The tangency portfolio for lending is T₁ (10%, σ 15%) and for borrowing is T₂ (14%, σ 20%).

  1. Describe the efficient frontier.
  2. An investor wants σ = 7.5%. What do they hold and what is E[R]?
  3. An investor wants σ = 24%. What do they hold and what is E[R]?
  4. Does the separation theorem hold here?
Show solution 查看答案

(a) Three pieces: the straight line from RF to T₁ (lending), the curved risky frontier between T₁ and T₂, and the straight line from T₂ outward using borrowing at R′F.

三段：RF 到 T₁ 的直线、T₁ 到 T₂ 之间的曲线、从 T₂ 向外的借款直线。

(b) X_T1 = 7.5/15 = 0.5, lend 0.5 at 4%: E = 0.5(10) + 0.5(4) = 7.0% (c) X_T2 = 24/20 = 1.2, borrow 0.2 at 7%: E = 1.2(14) − 0.2(7) = 15.4%

(d) No. Investors on different parts of the frontier hold different risky portfolios (T₁, T₂, or a portfolio on the curve between them), so the risky choice depends on preferences.

不成立：不同投资者持有不同的风险组合。

B9Reading an Excel regression output (Ch 7)读回归输出

Monthly returns of stock Z were regressed on the S&P/TSX (T = 60). The variance of the market return is 18.

CoefficientsStandard Error
Intercept0.420.61
RM1.350.21
R Square0.41
  1. Write out the estimated model.
  2. Test H₀: α = 0. Interpret.
  3. Test H₀: β = 0. Interpret.
  4. Test H₀: β = 1. Interpret.
  5. Interpret R².
  6. Compute the systematic, total and firm-specific variance of Z.
  7. In one month RM = 3.2% and Z returned 5.9%. Find the predicted return and the residual.
Show solution 查看答案
(a) R̂_Z = 0.42 + 1.35 R_M (b) t = (0.42 − 0)/0.61 = 0.69 → −1.96 < t < 1.96: fail to reject. α not significantly different from 0. (c) t = (1.35 − 0)/0.21 = 6.43 → reject. Z is significantly related to the market. (d) t = (1.35 − 1)/0.21 = 1.67 → fail to reject. Z’s beta is not significantly different from the market’s. (f) Systematic = β²σM² = 1.35²(18) = 32.81 Total = 32.81 / 0.41 = 80.01 Firm-specific = 80.01 − 32.81 = 47.21 (g) Predicted = 0.42 + 1.35(3.2) = 4.74% Residual e = 5.90 − 4.74 = 1.16%

(e) 41% of the variance of Z’s returns is explained by the market (systematic risk); 59% is firm-specific and could be diversified away. 41% 由市场解释，59% 为公司特有风险。

B10SIM vs Market Model inputs (Ch 7)SIM 与 MM

RM ~ (10, 16). Single-index model estimates:

Stockαβσ²e
A1.51.220
B−0.50.89
C2.01.530
  1. Mean and variance of each stock.
  2. Covariance of each pair under the SIM.
  3. αP, βP, mean and variance of an equally weighted portfolio under the SIM.
  4. Under the Market Model you are told Cov(eA, eB) = 3 (other residual covariances 0). Recompute Cov(A, B) and the portfolio variance.
Show solution 查看答案
(a) E[Ri] = αi + βi(10): A 13.5 B 7.5 C 17.0 V[Ri] = βi²(16) + σ²ei: A 23.04 + 20 = 43.04 B 10.24 + 9 = 19.24 C 36 + 30 = 66.00 (b) Cov = βiβj(16): AB 15.36 AC 28.80 BC 19.20 (c) αP = (1.5 − 0.5 + 2)/3 = 1.00 βP = (1.2 + 0.8 + 1.5)/3 = 1.167 E[RP] = 1.00 + 1.167(10) = 12.67 V[RP] = βP²(16) + (1/3)²(20 + 9 + 30) = 21.78 + 6.56 = 28.33 σP = 5.32 (d) MM: Cov(A,B) = βAβBσM² + Cov(eA,eB) = 15.36 + 3 = 18.36 V[RP] = 28.33 + 2(1/3)(1/3)(3) = 29.00

B11Estimate beta by hand (Ch 7)手算 β

Month12345
RM24−150
Ri36−271
  1. Compute the means, Var(RM), Var(Ri) and Cov(Ri, RM) (÷T).
  2. Estimate β and α.
  3. Compute the correlation and R².
  4. Compute the residual for month 3.
Show solution 查看答案
(a) R̄M = 10/5 = 2 R̄i = 15/5 = 3 dev RM: 0, 2, −3, 3, −2 dev Ri: 0, 3, −5, 4, −2 Var(RM) = (0 + 4 + 9 + 9 + 4)/5 = 5.2 Var(Ri) = (0 + 9 + 25 + 16 + 4)/5 = 10.8 Cov = (0 + 6 + 15 + 12 + 4)/5 = 7.4 (b) β = 7.4/5.2 = 1.423 α = 3 − 1.423(2) = 0.154 (c) ρ = 7.4/√(5.2 × 10.8) = 0.987 R² = ρ² = 0.975 (d) predicted = 0.154 + 1.423(−1) = −1.269 e₃ = −2 − (−1.269) = −0.731

B12Testing residual correlations (Ch 7)残差相关检验

Regressions of stocks A, B and C on the market used T = 25 months. Correlations of the residuals (and with RM):

eAeBeC
eB−0.12
eC0.470.31
RM0.000.000.00
  1. State the hypotheses and the critical value.
  2. Which pairs violate the SIM assumption?
  3. Why are all correlations with RM exactly zero?
  4. Which model, SIM or MM, better describes the covariance of A and C?
Show solution 查看答案
(a) H₀: Corr(ei, ej) = 0 vs Hₐ: Corr(ei, ej) ≠ 0 critical = ±2/√25 = ±0.40 (b) A–B: −0.12 → fail to reject B–C: 0.31 → fail to reject A–C: 0.47 > 0.40 → reject

(c) OLS forces residuals to be uncorrelated with the regressor RM by construction. (d) For A and C the residuals are significantly correlated, so the Market Model, which keeps Cov(eA, eC), describes their covariance better; the SIM would set it to zero.

只有 A–C 拒绝原假设，对这一对 MM 更合适；与 RM 的相关为 0 是 OLS 的构造性质。