这个作业是使用资本资产定价模型(CAPM)来确定的资产收益率

QBUS2810: Statistical Modelling for Business
Assignment Task #3

1.财务中使用资本资产定价模型(CAPM)来确定理论上适当的
资产的要求收益率,将资产添加到已经充分分散的资产中
鉴于该资产的不可分散风险。传统上,CAPM的应用程序仅使用
一个变量来描述投资组合或股票的收益与整个市场的收益:
rstock-rf =αstock+βstock(rm-rf)+ ut
相比之下,Fama-French模型使用三个变量:
rstock-rf =αstock+βstock(rm-rf)+β2SMB+β3HML+εt
rstock是股票的收益率,rf是无风险收益率,rm是收益率
整个股市。参数αstock是股票的“ alpha”。它衡量多少库存
在CAPM下的表现优于其“理论”的预期收益,而βstock就是该股票的“ beta”,
衡量股票在整个市场中的敞口。不同的股票会有不同的
参数。
Fama-French模型包含两个额外的因素来解释股票收益。小盘市值减去大盘(SMB)衡量的是小盘股相对大盘的历史超额收益。
较高的市销率(BtM)减去较低的市销率(HML)衡量了历史
价值股票(BtM比率较小)相对于增长股票(BtM比率较高)的超额收益。这些因素
是根据由排名股票(BtM排名,资本化排名)和可用的历史市场数据组成的投资组合的组合进行计算的。肯尼思(Kenneth)上有历史价值
法国针对美国股票的网页。
本练习中使用的变量如下:
rBHP =在澳大利亚证券交易所观察到的必和必拓股票的月收益。
rm =市场每月收益指数,这里是所有普通股指数(AOI)。
SMB =小市值减去大市值因子。
HML =高市销率减去低市销率
您假设每月无风险利率rf = 0.005。您的任务是估算Fama-French
使用给定数据的三因素模型。并确定它是否更好地解释了
必和必拓的股票收益率与所有普通股指数给出的市场超额收益率相比。
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(a)写下五位数摘要以及均值,标准差,偏度和峰度
rBHP系数。
(b)随时间绘制并评论rBHP系列。
(c)生成两个新变量rBHP-rf和rm-rf并估计单因素CAPM模型:
rstock − rf =β0+β1(rm − rf)+ ut
将回归输出复制并粘贴到答案纸中。写下拟合的回归
方程。
(d)评论估计系数β1的符号,并说明这是否是您所期望的。
(e)测试超额市场收益是否解释了必和必拓股票在
α= 0.05水平。
(f)测试BHP的“β”值是否在α= 0.05时大于1。
(g)估算Fama-French三要素CAPM模型:
rstock − rf =β0+β1(rm − rf)+β2SMB+β3HML+εt
将回归输出复制并粘贴到答案纸中。写下拟合的回归
方程。
(h)建立一般线性假设,以测试Fama-French 3-Factor CAPM
该模型比单因素CAPM模型更好地解释了股票收益;即确定L,
β,对于H0为c:Lβ= c。
(i)对(h)部分进行假设检验。
(j)一位财务分析师认为,账面市值(HML)对股票收益的影响为
是市值(SMB)的两倍。提出适当的假设
测试并使用重新参数化将其转换为简单的t检验以测试断言。执行
所需的回归并在α= 0.05的水平上陈述您的结论。
(k)获得回归模型中参数估计量的方差-协方差矩阵
(g)部分。利用(g)部分中的回归结果和方差-协方差矩阵重复
(j)部分中的假设检验通过简单的t检验进行。
2.一家生产针对儿童的新谷物的公司的市场经理要检查
盒子徽标形状对谷物批准等级的影响。他组合了4种颜色
和2种形状,以产生总共8种设计。每个徽标分为2个不同的组(共
(共16组儿童),并记录每组的认可度,如下所示。
颜色
形状红色绿色蓝色黄色
圈52、44 67、61 36、44 45、41
平方34,36 56,58 36,31 21,25
(a)这个实验有几个因素?确定因素并说明多少级别
每个因素都有。
(b)如果比较所有组合,则实验中有多少种不同的处理(细胞)?什么是响应变量?
(c)考虑以下回归模型:
Y =β0+β1C+β2R+β3G+β4B+β5CR+β6CG+β7CB+ε
如果形状=圆形,则C = 1;否则为0。如果颜色=红色,则R = 1;否则,否则为0。如果颜色是G = 1
=绿色;否则为0。如果颜色=蓝色,则B = 1;否则,否则为0。
使用
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Colour
Shape Red Green Blue Yellow
Circle µ11 = β0 + β1 + β2 + β5
Square
(d) The factor effects model is Yijk = µ.. + αi + βj + (αβ)ij + εijk where µ..
is a constant. αi are
constants subject to the restriction Pαi = 0. βj are constants subject to the restriction Pβj
= 0. (αβ)ij are constants subject to the restrictions P
i
P
j
(αβ)ij = 0. εijk are independent
N(0, σ
2
), i = 1, 2, …, a; j = 1, 2, …, b; k = 1, 2, …, n.
Why are the constraints Pαi =
Pβj =
P(αβ)ij = 0 required? What is the advantage of this
model?
(e) Refer to Part (d). Modify the factor effects model to apply to this study with a = 2 and b = 4.
(f) Set up the Y, X, and β matrices for the factor effects regression model.
(g) Refer to part (e). Obtain the fitted regression function.
(h) Plot the residuals against the fitted values and the QQ-plot of the residuals. Use these two
residual plots to check if the assumptions of two-way ANOVA are justifiable. Briefly explain.
(i) Plot an interaction plot. What does this plot suggest?
(j) Fill in the blanks in the following ANOVA table.
Source of Variation SS df MS
Between treatments
Factor A
Factor B
AB Interactions
Error
Total
(k) Test if the two factors interact.
(l) Is it meaningful here to test for main factor effects? If so, test if the main effects for color and
shape are present.
(m) All pairwise comparisons among the color group level means via Tukey procedure with a 95
percent family confidence coefficient are constructed below:
Treatment Difference Lower 95% limit Upper 95% bound
Red Green -19.00 -35.8696 -2.1304
Red Blue 4.75 -12.1196 21.6196
Red Yellow 8.50 -8.3696 25.3696
Green Blue 23.75 6.8804 40.6196
Green Yellow 27.50 10.6304 44.3696
Blue Yellow 3.75 -13.1196 20.6196
Determine which means differ using Tukey’s multiple comparison test.
(n) Based on the above analysis, what combination of color and shape should be used for the logo
design?
(o) Suppose that in the shape population, 60 percent are circle, and 40 percent are square. Construct a 95% percent confidence interval for the mean overall rating in the shape population.
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3. A person’s muscle mass is expected to decrease with age. To explore this relationship in women,
a nutritionist randomly selected 4 women from each 10-year age group, beginning with age 40 and
ending with age 79. X is age, and Y is a measure of muscle mass.
(a) Below is a scatter plot of the data with muscle mass on the y axis and age on the x axis.
Based on the plot, does it seem reasonable that there are two different (but connected) regression functions – one when age ≤ 60 and one when age > 60?
(b) The nutritionist conjectures that the regression of muscle mass on age follows a two-piece linear
relation, with the slope changing at age 60 without discontinuity. State the regression model
that applies if the nutritionist’s conjecture is correct.
(c) Refer to part (b). What are respective response functions when age is 60 or less and when age
is over 60?
(d) Explain whether or not the model specified in part (b) violates the principle of marginality.
Also, discuss and show whether or not this model is continuous at X = 60. Is continuity or
marginality more important here and why?
(e) Estimate the regression model specified in part (b). Copy and paste the regression output into
your answer sheet. Write down the fitted regression equation.
(f) Test whether a two-piece linear regression function is needed at α = 0.05.
(g) Refer to part (e). What is the estimated regression function for muscle mass whose age ≤ 60?
for muscle mass whose age > 60?
(h) Based on your estimated regression function, what is the predicted muscle mass when age =
50? When age = 70?
(i) Do you get the same prediction for age = 60 regardless of which estimated regression function
in part (e) you use?
(j) Modify the regression model in part (b) with the slope changing at age 60 without continuity.
(k) Specify the regression model for the case where the slope changes at age 40 and again at age
60 with no discontinuities.


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