金融工程习题课（一）

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练习

现代投资组合理论

  1. Considering the world economic outlook for the coming year and estimates of sales and earning for the pharmaceutical industry, you expect the rate of return for Lauren Labs common stock to range between −20 percent and +40 percent with the following probabilities.

    Probability Possible Returns
    0.10 -0.20
    0.15 -0.05
    0.20 0.10
    0.25 0.15
    0.20 0.20
    0.10 0.40



  1. Given the following market values of stocks in your portfolio and their expected rates of return, what is the expected rate of return for your common stock portfolio?

    Stock Market Value ($ Mil.) E(Ri)E(R_i)
    Disney $15,000 0.14
    Starbucks 17,000 −0.04
    Harley Davidson 32,000 0.18
    Intel 23,000 0.16
    Walgreens 7,000 0.12



  1. The following are the monthly rates of return for Madison Cookies and for Sophie Electric during a six-month period.

    Month Madison Cookies Sophie Electric
    1 −0.04 0.07
    2 0.06 −0.02
    3 −0.07 −0.10
    4 0.12 0.15
    5 −0.02 −0.06
    6 0.05 0.02

    Compute the following.
    a. Average monthly rate of return Rˉi\bar{R}_i for each stock
    b. Standard deviation of returns for each stock
    c. Covariance between the rates of return
    d. The correlation coefficient between the rates of return
    What level of correlation did you expect? How did your expectations compare with the computed correlation? Would these two stocks be good choices for diversification? Why or why not?


  2. You are considering two assets with the following characteristics.
    E(R1)=0.15E(σ1)=0.10w1=0.5E(R2)=0.20E(σ2)=0.20w2=0.5\begin{align} E(R_1 )=0.15&&E(σ_1 )=0.10&&w_1=0.5\nonumber\\ E(R_2 )=0.20&&E(σ_2 )=0.20&&w_2=0.5\nonumber \end{align}

    Compute the mean and standard deviation of two portfolios if r_1,2=0.40 and −0.60, respectively. Plot the two portfolios on a risk–return graph and briefly explain the results.


  3. Given:
    E(R1)=0.10E(σ1)=0.03E(R2)=0.15E(σ2)=0.05\begin{align} E(R_1 )=0.10&&E(σ_1 )=0.03\nonumber\\ E(R_2 )=0.15&&E(σ_2 )=0.05\nonumber \end{align}

    Calculate the expected returns and expected standard deviations of a two-stock portfolio in which Stock 1 has a weight of 60 percent under the following conditions.
    a. r1,2=1.00r_{1,2}=1.00
    b. r1,2=0.75r_{1,2}=0.75
    c. r1,2=0.25r_{1,2}=0.25
    d. r1,2=0.00r_{1,2}=0.00
    e. r1,2=−0.25r_{1,2}=-0.25
    f. r1,2=−0.75r_{1,2}=-0.75
    g. r1,2=−1.00r_{1,2}=-1.00


  4. Given:
    E(R1)=0.12E(σ1)=0.04E(R2)=0.16E(σ2)=0.06\begin{align} E(R_1 )=0.12&&E(σ_1 )=0.04\nonumber\\ E(R_2 )=0.16&&E(σ_2 )=0.06\nonumber \end{align}

    Calculate the expected returns and expected standard deviations of a two-stock portfolio having a correlation coefficient of 0.70 under the following conditions.
    a. w1=1.00w_1=1.00
    b. w1=0.75w_1=0.75
    c. w1=0.50w_1=0.50
    d. w1=0.25w_1=0.25
    e. w1=0.05w_1=0.05


  5. The following are monthly percentage price changes for four market indexes.

    Month DJIA S&P 500 Russell 2000 Nikkei
    1 0.03 0.02 0.04 0.04
    2 0.07 0.06 0.10 −0.02
    3 −0.02 −0.01 −0.04 0.07
    4 0.01 0.03 0.03 0.02
    5 0.05 0.04 0.11 0.02
    6 −0.06 −0.04 −0.08 0.06

    Compute the following.
    a. Average monthly rate of return for each index
    b. Standard deviation for each index
    c. Covariance between the rates of return for the following indexes:
    DJIA–S&P 500
    S&P 500–Russell 2000
    S&P 500–Nikkei
    Russell 2000–Nikkei
    d. The correlation coefficients for the same four combinations
    e. Using the answers from parts (a), (b), and (d), calculate the expected return and standard deviation of a portfolio consisting of equal parts of (1) the S&P and the Russell2000 and (2) the S&P and the Nikkei. Discuss the two portfolios.


  6. 已知市场无风险利率为 rf=6%r_f = 6\% , 两种风险资产的收益率和方差分别为 (26% , 0.6) 和 (16% , 0.2) ，两种风险资产收益率的相关 系数为3/3\sqrt{3}/3
    a. 如果某组合投资于两种风险资产的比例分别为 0.4 和 0.6 ， 求该组合收益率的均值和方差。
    b. 如果存在无风险资产，求两种风险资产的最优组合 M 。
    c. 如果某投资者要求的收益率水平为 36%，求其最优的资产 组合。


资本资产定价模型

  1. Assume that you expect the economy's rate of inflation to be 3 percent, giving an RFRR F R of 6 percent and a market return (RM)\left(R_{\mathrm{M}}\right) of 12 percent.
    a. Draw the SML under these assumptions.
    b. Subsequently, you expect the rate of inflation to increase from 3 percent to 6 percent. What effect would this have on the RFR and the RMR_{\mathrm{M}} ? Draw another SML on the graph from Part a.
    c. Draw an SML on the same graph to reflect an RFRR F R of 9 percent and an RMR_{\mathrm{M}} of 17 percent. How does this SML differ from that derived in Part b? Explain what has transpired.


  2. a. You expect an RFRR F R of 10 percent and the market return (RM)\left(R_{\mathrm{M}}\right) of 14 percent. Compute the expected return for the following stocks, and plot them on an SML graph.

    Stock Beta E(Ri)E(R_i)
    U\mathrm{U} 0.850.85
     N\mathrm{~N} 1.251.25
    D\mathrm{D} −0.20-0.20

    b. You ask a stockbroker what the firm's research department expects for these three stocks. The broker responds with the following information:

    Stock Current Price Expected Price Expected Dividend
    U 22 24 0.750.75
    N 48 51 2.002.00
    D 37 40 1.251.25

    Plot your estimated returns on the graph from Part a and indicate what actions you would take with regard to these stocks. Explain your decisions.


  3. You are evaluating various investment opportunities currently available and you have calculated expected returns and standard deviations for five different well-diversified portfolios of risky assets:

    Portfolio Expected Return Standard Deviation
    Q 7.8%7.8 \% 10.5%10.5 \%
    R 10.010.0 14.014.0
    S 4.64.6 5.05.0
    T 11.711.7 18.518.5
    U 6.26.2 7.57.5

    a. For each portfolio, calculate the risk premium per unit of risk that you expect to receive ([E(R)−RFR]/σ)([E(R)-R F R] / \sigma). Assume that the risk-free rate is 3.03.0 percent.
    b. Using your computations in Part a, explain which of these five portfolios is most likely to be the market portfolio. Use your calculations to draw the capital market line (CML).
    c. If you are only willing to make an investment with σ=7.0%\sigma=7.0 \%, is it possible for you to earn a return of 7.07.0 percent?
    d. What is the minimum level of risk that would be necessary for an investment to earn 7.07.0 percent? What is the composition of the portfolio along the CML that will generate that expected return?
    e. Suppose you are now willing to make an investment with σ=18.2%\sigma=18.2 \%. What would be the investment proportions in the riskless asset and the market portfolio for this portfolio? What is the expected return for this portfolio?


  4. You are an analyst for a large public pension fund and you have been assigned the task of evaluating two different external portfolio managers ( Y\mathrm{Y} and Z\mathrm{Z} ). You consider the following historical average return, standard deviation, and CAPM beta estimates for these two managers over the past five years:

    Portfolio Actual Avg. Return Standard Deviation Beta
    Manager Y 10.20%10.20 \% 12.00%12.00 \% 1.201.20
    Manager Z 8.808.80 9.909.90 0.800.80

    Additionally, your estimate for the risk premium for the market portfolio is 5.005.00 percent and the risk-free rate is currently 4.504.50 percent.
    a. For both Manager Y\mathrm{Y} and Manager Z, calculate the expected return using the CAPM. Express your answers to the nearest basis point (i.e., xx.xx%).
    b. Calculate each fund manager's average "alpha" (i.e., actual return minus expected return) over the five-year holding period. Show graphically where these alpha statistics would plot on the security market line (SML).
    c. Explain whether you can conclude from the information in Part b if: (1) either manager outperformed the other on a risk-adjusted basis, and (2) either manager outperformed market expectations in general.


  5. Based on five years of monthly data, you derive the following information for the companies listed:

    Company ai(a_i( Intercept) σi\sigma_i riMr_{i \mathrm{M}}
    Intel 0.220.22 12.10%12.10 \% 0.720.72
    Ford 0.100.10 14.6014.60 0.330.33
    Anheuser Busch 0.170.17 7.607.60 0.550.55
    Merck 0.050.05 10.2010.20 0.600.60
    S&P 500 0.000.00 5.505.50 1.001.00

    a. Compute the beta coefficient for each stock.
    b. Assuming a risk-free rate of 8 percent and an expected return for the market portfolio of 15 percent, compute the expected (required) return for all the stocks and plot them on the SML.
    c. Plot the following estimated returns for the next year on the SML and indicate which stocks are undervalued or overvalued.

    • Intel-20 percent
    • Ford-15 percent
    • Anheuser Busch-19 percent
    • Merck-10 percent


  6. The following are the historic returns for the Chelle Computer Company:

    Year Chelle Computer General Index
    1 37 15
    2 9 13
    3 −11-11 14
    4 8 -9
    5 11 12
    6 4 9

    Based on this information, compute the following:
    a. The correlation coefficient between Chelle Computer and the General Index.
    b. The standard deviation for the company and the index.
    c. The beta for the Chelle Computer Company.


  7. The following information describes the expected return and risk relationship for the stocks of two of WAH's competitors.

    Expected Return Standard Deviation Beta Return
    Stock X 12.0%12.0 \% 20%20 \% 1.31.3
    Stock Y 9.09.0 15 0.70.7
    Market Index 10.010.0 12 1.01.0
    Risk-free rate 5.05.0

    Using only the data shown in the preceding table:
    a. Draw and label a graph showing the security market line, and position Stocks X\mathrm{X} and Y\mathrm{Y} relative to it.
    b. Compute the alphas both for Stock X\mathrm{X} and for Stock Y. Show your work.
    c. Assume that the risk-free rate increases to 7 percent, with the other data in the preceding matrix remaining unchanged. Select the stock providing the higher expected riskadjusted return and justify your selection. Show your calculations.


  8. As an equity analyst, you have developed the following return forecasts and risk estimates for two different stock mutual funds (Fund T\mathrm{T} and Fund U):

    Forecasted Return CAPM Beta
    Fund T\mathrm{T} 9.0%9.0 \% 1.201.20
    Fund U\mathrm{U} 10.010.0 0.800.80

    a. If the risk-free rate is 3.93.9 percent and the expected market risk premium (i.e., E(RM)−E\left(R_{\mathrm{M}}\right)- RFR)R F R) is 6.16.1 percent, calculate the expected return for each mutual fund according to the CAPM.
    b. Using the estimated expected returns from Part a along with your own return forecasts, demonstrate whether Fund T\mathrm{T} and Fund U\mathrm{U} are currently priced to fall directly on the security market line (SML), above the SML, or below the SML.
    c. According to your analysis, are Funds T\mathrm{T} and U\mathrm{U} overvalued, undervalued, or properly valued?


  9. Given the following results, indicate what will happen to the beta for Sophie Fashion Co., relative to the market proxy, compared to the beta relative to the true market portfolio:

    Year YEARLY RATES OF RETURN
    Sophie Fashion (%) Market Proxy (%) True Market (%)
    1 10 8 6
    2 20 14 11
    3 -14 -10 -7
    4 -20 -18 -12
    5 15 12 10

    Discuss the reason for the differences in the measured betas for Sophie Fashion Co. Does the suggested relationship appear reasonable? Why or why not?


  10. Draw the security market line for each of the following conditions:
    a. (1) RFR=0.08;RMR F R=0.08 ; R_{\mathrm{M}} (proxy) =0.12=0.12
    (2) Rz=0.06;RM(R_z=0.06 ; R_{\mathrm{M}}( true )=0.15)=0.15
    b. Rader Tire has the following results for the last six periods. Calculate and compare the betas using each index.

    Period RATES OF RETURN
    Rader Tire (%) Proxy Specific Index (%) True General Index (%)
    1 29 12 15
    2 12 10 13
    3 -12 -9 -8
    4 17 14 18
    5 20 25 28
    6 -5 -10 0

    c. If the current period return for the market is 12 percent and for Rader Tire it is 11 percent, are superior results being obtained for either index beta?


  11. 某公司拟对高科技股票、写字楼、企业债券和货币市场基金四种资产进行组合投资，四种资产的β\beta系数分别为 2、1.2、1和0.5 ，市场上国库券的收益率为 6% ，市场平均风险资产的投资 回报率为 10% 。
    a. 计算各自的预期投资报酬率。
    b. 若该公司以 4:3:3 投资于前三种资产，计算该组合的 预期报酬率和β\beta系数。
    c. 若该公司以 4:3:3 投资于1、2、4三种资产，计算该组 合的预期报酬率和β\beta系数。
    d. 若该公司想提高报酬率，应选择以上哪种组合？


套利定价理论与多因素模型

  1. Consider the following data for two risk factors ( 1 and 2)) and two securities (J(\mathrm{J} and L)\mathrm{L}) :
    λ0=0.05bJ1=0.80λ1=0.02bJ2=1.40λ2=0.04bL1=1.60bL2=2.25\begin{array}{ll} \lambda_0=0.05 & b_{\mathrm{J} 1}=0.80 \\ \lambda_1=0.02 & b_{\mathrm{J} 2}=1.40 \\ \lambda_2=0.04 & b_{\mathrm{L} 1}=1.60 \\ & b_{\mathrm{L} 2}=2.25 \end{array}
    a. Compute the expected returns for both securities.
    b. Suppose that Security J\mathrm{J} is currently priced at $22.50\$ 22.50 while the price of Security L is $15.00\$ 15.00. Further, it is expected that both securities will pay a dividend of $0.75\$ 0.75 during the coming year. What is the expected price of each security one year from now?


  2. Exhibit 9.89.8 demonstrated how the Fama-French three-factor and four-factor models could be used to estimate the expected excess returns for three stocks (MSFT, CSX, and XRX). Specifically, using return data from 2005-2009, the following equations were estimated:
    Three-Factor Model:
     MSFT: [E(R)−RFR]=(0.966)(λM)+(−0.018)(λSMB)+(−0.388)(λHML) CSX: [E(R)−RFR]=(1.042)(λM)+(−0.043)(λSMB)+(0.370)(λHML) XRX: [E(R)−RFR]=(1.178)(λM)+(0.526)(λSMB)+(0.517)(λHML)\begin{array}{ll} \text { MSFT: } & {[E(R)-R F R]=(0.966)\left(\lambda_{\mathrm{M}}\right)+(-0.018)\left(\lambda_{S M B}\right)+(-0.388)\left(\lambda_{H M L}\right)} \\ \text { CSX: } & {[E(R)-R F R]=(1.042)\left(\lambda_{\mathrm{M}}\right)+(-0.043)\left(\lambda_{S M B}\right)+(0.370)\left(\lambda_{H M L}\right)} \\ \text { XRX: } & {[E(R)-R F R]=(1.178)\left(\lambda_{\mathrm{M}}\right)+(0.526)\left(\lambda_{S M B}\right)+(0.517)\left(\lambda_{H M L}\right)} \end{array}
    Four-Factor Model:
     MSFT: [E(R)−RFR]=(1.001)(λM)+(−0.012)(λSMB)+(−0.341)(λHML)+(0.073)(λMOM) CSX: [E(R)−RFR]=(1.122)(λM)+(−0.031)(λSMB)+(0.478)(λHML)+(0.166)(λMOM) XRX: [E(R)−RFR]=(1.041)(λM)+(0.505)(λSMB)+(0.335)(λHML)+(−0.283)(λMOM)\begin{array}{ll} \text { MSFT: } & {[E(R)-R F R]=(1.001)\left(\lambda_{\mathrm{M}}\right)+(-0.012)\left(\lambda_{S M B}\right)+(-0.341)\left(\lambda_{H M L}\right)+(0.073)\left(\lambda_{M O M}\right)} \\ \text { CSX: } & {[E(R)-R F R]=(1.122)\left(\lambda_{\mathrm{M}}\right)+(-0.031)\left(\lambda_{S M B}\right)+(0.478)\left(\lambda_{H M L}\right)+(0.166)\left(\lambda_{M O M}\right)} \\ \text { XRX: } & {[E(R)-R F R]=(1.041)\left(\lambda_{\mathrm{M}}\right)+(0.505)\left(\lambda_{S M B}\right)+(0.335)\left(\lambda_{H M L}\right)+(-0.283)\left(\lambda_{M O M}\right)} \end{array}
    Using the estimated factor risk premia of λM=7.23%,λSMB=2.00%,λHML=4.41%\lambda_{\mathrm{M}}=7.23 \%, \lambda_{S M B}=2.00 \%, \lambda_{H M L}=4.41 \% and λMOM=\lambda_{M O M}= 4.91%4.91 \%, it was then shown that the expected excess returns for the three stocks were 5.24%,9.08%5.24 \%, 9.08 \%, and 11.86%11.86 \% (three-factor model) or 6.07%,10.98%6.07 \%, 10.98 \%, and 8.63%8.63 \% (four-factor model), respectively.

Exhibit 9.8 Estimates for Risk Factor Premia, Factor Sensitivities, and Expected Returns

A. Risk Factor Premium Estimates Using Historical and Hypothetical Data
Risk Factor 1995-2009 1980-2009 1927-2009 Hypothetical Forcaset
Market 7.23% 7.11% 7.92% 7.00%
SMB 2.00 1.50 3.61 2.50
HML 4.41 5.28 5.02 4.00
MOM 4.91 7.99 9.79 5.00


B. Estimates of Factor Sensitivities and Expected Risk Premia: Three-Factor Model
MSFT CSX XRX
Factor: Market 0.966 1.042 1.178
SMB -0.018 -0.043 0.562
HML -0.388 0.370 0.517
E(Risk Prem): 1995-2009 5.24% 9.08% 11.86%
Hypothetical 5.17 8.67 11.63


C. Estimates of Factor Sensitivities and Expected Risk Premia: Four-Factor Model
MSFT CSX XRX
Factor: Market 1.001 1.122 1.041
SMB -0.012 -0.031 0.505
HML -0.341 0.478 0.335
MOM 0.073 0.166 -0.283
E(Risk Prem): 1995-2009 6.07% 10.98% 8.63%
Hypothetical 5.98 10.52 8.48

a. Exhibit 9.89.8 also lists historical factor risk prices from two different time frames: (1) 1980−2009(λM=7.11%,λSMB=1.50%1980-2009\left(\lambda_{\mathrm{M}}=7.11 \%, \lambda_{S M B}=1.50 \%\right., and λHML=5.28%)\left.\lambda_{H M L}=5.28 \%\right), and (2) 1927-2009 (λM=\left(\lambda_{\mathrm{M}}=\right. 7.92%,λSMB=3.61%7.92 \%, \lambda_{S M B}=3.61 \%, and λHML=5.02%)\left.\lambda_{H M L}=5.02 \%\right). Calculate the expected excess returns for MSFT, CSX, and XRX using both of these alternative sets of factor risk premia in conjunction with the three-factor risk model.
b. Exhibit 9.89.8 also lists historical estimates for the MOM risk factor: (i) λMOM=7.99%\lambda_{M O M}=7.99 \% (1980-2009), and (2) λMOM=9.79%\lambda_{M O M}=9.79 \% (1927-2009). Using this additional information, calculate the expected excess returns for MSFT, CSX, and XRX in conjunction with the four-factor risk model.
c. Do all of the expected excess returns you calculated in Part a and Part b make sense? If not, identify which ones seem inconsistent with asset pricing theory and discuss why.
d. Would you expect the factor betas to remain constant over time? Discuss how and why these coefficients might change in response to changing market conditions.


  1. You have been assigned the task of estimating the expected returns for three different stocks: QRS, TUV, and WXY. Your preliminary analysis has established the historical risk premiums associated with three risk factors that could potentially be included in your calculations: the excess return on a proxy for the market portfolio (MKT), and two variables capturing general macroeconomic exposures (MACRO1 and MACRO2). These values are: λMKT=7.5%,λMACRO1=−0.3%\lambda_{\mathrm{MKT}}=7.5 \%, \lambda_{\mathrm{MACRO1}}=-0.3 \%, and λMACRO2=0.6%\lambda_{\mathrm{MACRO} 2}=0.6 \%. You have also estimated the following factor betas (i.e., loadings) for all three stocks with respect to each of these potential risk factors:

    Stock FACTOR LOADING
    MKT MACRO1 MACRO2
    QRS 1.24 -0.42 0.00
    TUV 0.91 0.54 0.23
    WXY 1.03 -0.09 0.00

    a. Calculate expected returns for the three stocks using just the MKT risk factor. Assume a risk-free rate of 4.5%4.5 \%.
    b. Calculate the expected returns for the three stocks using all three risk factors and the same 4.5% risk-free rate.
    c. Discuss the differences between the expected return estimates from the single-factor model and those from the multifactor model. Which estimates are most likely to be more useful in practice?
    d. What sort of exposure might MACRO2 represent? Given the estimated factor betas, is it really reasonable to consider it a common (i.e., systematic) risk factor?


  2. Consider the following information about two stocks (D and E) and two common risk factors (1 and 2):

    Stock bi1b_{i1} bi2b_{i2} E(Ri)E(R_i)
    D 1.2 3.4 13.1%
    E 2.6 2.6 15.4%

    a. Assuming that the risk-free rate is 5.0%5.0 \%, calculate the levels of the factor risk premia that are consistent with the reported values for the factor betas and the expected returns for the two stocks.
    b. You expect that in one year the prices for Stocks D and E will be $55\$ 55 and $36\$ 36, respectively. Also, neither stock is expected to pay a dividend over the next year. What should the price of each stock be today to be consistent with the expected return levels listed at the beginning of the problem?
    c. Suppose now that the risk premium for Factor 1 that you calculated in Part a suddenly increases by 0.25%0.25 \% (i.e., from x%x \% to (x+0.25)%(x+0.25) \%, where xx is the value established in Part a. What are the new expected returns for Stocks D and E?
    d. If the increase in the Factor 1 risk premium in Part c\mathrm{c} does not cause you to change your opinion about what the stock prices will be in one year, what adjustment will be necessary in the current (i.e., today's) prices?


  3. Suppose that three stocks (A, B, and C) and two common risk factors ( 1 and 2 ) have the following relationship:
    E(RA)=(1.1)λ1+(0.8)λ2E(RB)=(0.7)λ1+(0.6)λ2E(RC)=(0.3)λ1+(0.4)λ2\begin{aligned} &E\left(R_{\mathrm{A}}\right)=(1.1) \lambda_1+(0.8) \lambda_2 \\ &E\left(R_{\mathrm{B}}\right)=(0.7) \lambda_1+(0.6) \lambda_2 \\ &E\left(R_{\mathrm{C}}\right)=(0.3) \lambda_1+(0.4) \lambda_2 \end{aligned}
    a. If λ1=4%\lambda_1=4 \% and λ2=2%\lambda_2=2 \%, what are the prices expected next year for each of the stocks? Assume that all three stocks currently sell for $30\$ 30 and will not pay a dividend in the next year.
    b. Suppose that you know that next year the prices for Stocks A, B, and C will actually be $31.50,$35.00\$ 31.50, \$ 35.00, and $30.50\$ 30.50. Create and demonstrate a riskless, arbitrage investment to take advantage of these mispriced securities. What is the profit from your investment? You may assume that you can use the proceeds from any necessary short sale.

Problems 6 and 7 refer to the data contained in Exhibit 9.12, which lists 30 monthly excess returns to two different actively managed stock portfolios ( A\mathrm{A} and B\mathrm{B} ) and three different common risk factors (1,2, and 3). (Note: You may find it useful to use a computer spreadsheet program such as Microsoft Excel to calculate your answers.)

Exhibit 9.12 Monthly Excess Return Data for Two Portfolios and Three Risk Factors

Period Portfolio A Portfolio B Factor 1 Factor 2 Factor 3
1 1.08% 0.00% 0.01% −1.01% −1.67%
2 7.58 6.62 6.89 0.29 −1.23
3 5.03 6.01 4.75 −1.45 1.92
4 1.16 0.36 0.66 0.41 0.22
5 −1.98 −1.58 −2.95 −3.62 4.29
6 4.26 2.39 2.86 −3.40 −1.54
7 −0.75 −2.47 −2.72 −4.51 −1.79
8 −15.49 −15.46 −16.11 −5.92 5.69
9 6.05 4.06 5.95 0.02 −3.76
10 7.70 6.75 7.11 −3.36 −2.85
11 7.76 5.52 5.86 1.36 −3.68
12 9.62 4.89 5.94 −0.31 −4.95
13 5.25 2.73 3.47 1.15 −6.16
14 −3.19 −0.55 −4.15 −5.59 1.66
15 5.40 2.59 3.32 −3.82 −3.04
16 2.39 7.26 4.47 2.89 2.80
17 −2.87 0.10 −2.39 3.46 3.08
18 6.52 3.66 4.72 3.42 −4.33
19 −3.37 −0.60 −3.45 2.01 0.70
20 −1.24 −4.06 −1.35 −1.16 −1.26
21 −1.48 0.15 −2.68 3.23 −3.18
22 6.01 5.29 5.80 −6.53 −3.19
23 2.05 2.28 3.20 7.71 −8.09
24 7.20 7.09 7.83 6.98 −9.05
25 −4.81 −2.79 −4.43 4.08 −0.16
26 1.00 −2.04 2.55 21.49 −12.03
27 9.05 5.25 5.13 −16.69 7.81
28 −4.31 −2.96 −6.24 −7.53 8.59
29 −3.36 −0.63 −4.27 −5.86 5.38
30 3.86 1.80 4.67 13.31 −8.78



  1. a. Compute the average monthly return and monthly standard return deviation for each portfolio and all three risk factors. Also state these values on an annualized basis. (Hint: Monthly returns can be annualized by multiplying them by 12 , while monthly standard deviations can be annualized by multiplying them by the square root of 12 .)
    b. Based on the return and standard deviation calculations for the two portfolios from Part a, is it clear whether one portfolio outperformed the other over this time period?
    c. Calculate the correlation coefficients between each pair of the common risk factors (i.e., 1&2,1&31 \& 2,1 \& 3, and 2&32 \& 3 ).
    d. In theory, what should be the value of the correlation coefficient between the common risk factors? Explain why.
    e. How close do the estimates from Part b come to satisfying this theoretical condition? What conceptual problem(s) is created by a deviation of the estimated factor correlation coefficients from their theoretical levels?


  2. a. Using regression analysis, calculate the factor betas of each stock associated with each of the common risk factors. Which of these coefficients are statistically significant?
    b. How well does the factor model explain the variation in portfolio returns? On what basis can you make an evaluation of this nature?
    c. Suppose you are now told that the three factors in Exhibit 9.129.12 represent the risk exposures in the Fama-French characteristic-based model (i.e., excess market, SMBS M B, and HML)H M L). Based on your regression results, which one of these factors is the most likely to be the market factor? Explain why.
    d. Suppose it is further revealed that Factor 3 is the HML factor. Which of the two portfolios is most likely to be a growth-oriented fund and which is a value-oriented fund? Explain why.

有效市场假说

  1. Compute the abnormal rates of return for the following stocks during period tt (ignore differential systematic risk):

    Stock Rit\boldsymbol{R}_{i t} Rmt\boldsymbol{R}_{\boldsymbol{m} t}
    B\mathrm{B} 11.5%11.5 \% 4.0%4.0 \%
    F\mathrm{F} 10.010.0 8.58.5
    T\mathrm{T} 14.014.0 9.69.6
    C\mathrm{C} 12.012.0 15.315.3
    E\mathrm{E} 15.915.9 12.412.4

    Rit=R_{i t}= return for stock ii during period tt
    Rmt=R_{m t}= return for the aggregate market during period tt


  2. Compute the abnormal rates of return for the five stocks in Problem 1 assuming the following systematic risk measures (betas):

    Stock βi\beta_i
    B 0.950.95
    F 1.251.25
    T 1.451.45
    C 0.700.70
    E −0.30-0.30



  1. Compare the abnormal returns in Problems 1 and 2 and discuss the reason for the difference in each case.


  2. Look up the daily trading volume for the following stocks during a recent five-day period:

    • Merck
    • Caterpillar
    • Intel
    • McDonald's
    • General Electric

    Randomly select five stocks from the NYSE, and examine their daily trading volume for the same five days.
    a. What are the average volumes for the two samples?
    b. Would you expect this difference to have an impact on the efficiency of the markets for the two samples? Why or why not?