CFA錯題筆記10-Quantitative methods
本次做的題目是bookshelf 的R7（discounted cash flow applications）、R8（statistical concept and market return），R9（probability concepts）一共65道題，正確率是53%，下面分享15道錯題：
R7discounted cash flow applications
10.A fund receives investments at the beginning of each year and generates returns as shown in the table.
Which return measure over the three-year period isnegative?
Geometric mean return
Time-weighted rate of return
Money-weighted rate of return
考點：R73.1. Money-Weighted Rate of Return&3.2. Time-Weighted Rate of Return
【Doris 叨叨】
題目大意：
一隻基金投資每年產生的現金流及對應的收益率如下表所示，下面哪一個收益率是負的？
A.幾何平均收益率；【（1+15%）（1+14%）（1-4%）】^1/3，為正，A錯。
B.時間加權收益率；【（1+15%）（1+14%）（1-4%）】^1/3-1.為正，B錯。
C.貨幣加權收益率；由以下得，C對。
第一步，算出每年現金流3年後的終值：
第1年的現金流終值：
1000*(1+15%)(1+14%）（1-4%）=1258.56
第2年的現金流終值：
4000*（1+14%）（1-4%）=4377.6
第3年的現金流終值：
45000*（1-4%）=43200
總現金流入=1258.56+4377.6+43200=48836.16
第二步，使終值以一個貼現率貼現後的現值等於現金流以同樣的貼現率貼現後的現值加總
1000+4000/（1+r）^2+45000/（1+r）^3=48836.16/（1+r）^3
求得r=-2.08%
R8 statistical concept and market return
5.An analyst is using the data in the following table to prepare a statistical report.
Portfolio』s Deviations from Benchmark Return, 2003–2014 (%)
Thecumulative relative frequencyfor the interval?1.71% ≤ x closestto:
0.250.
0.333.
0.583.
考點：R83. SUMMARIZING DATA USING FREQUENCY DISTRIBUTIONS
Thecumulative relative frequencycumulates (adds up) the relative frequencies as we move from the first to the last interval.
【Doris 叨叨】
題目大意：
分析師正準備下表的數據以出一份統計報告，對於區間?1.71% ≤ x
第一步，先將數據從小到大排序：
-9.19，-5.11，-2.59，-1.69，-0.89，-0.55，1.33，2.48，3.04，4.72，6.84，9.47
第二步，算出最大值和最小值的差額：
9.47+9.19=18.66
第三步，根據題目算出區間差額：
2.03+1.71=3.74
第四步，算出組數：
18.66/3.74=5組
第五步，分組：
-9.19~-5.45（2）；-5.45~-1.71（1）；-1.71~2.03（4）；2.03~5.77（3）；5.77~9.51（2）
第六步，算出-1.71~2.03的累計相對概率：
2/12+1/12+4/12=58.33%
8.he following is a frequency polygon of monthly exchange rate changes in the US dollar/Japanese yen spot exchange rate from January 2010 to December 2013. A positive change represents yen appreciation (the yen buys more dollars), and a negative change represents yen depreciation (the yen buys fewer dollars).
Based on the chart, yenappreciation:
occurred more than 50% of the time.
was less frequent than yen depreciation.
in the 0.0 to 2.0 interval occurred 20% of the time.
考點：R84.1. The Histogram
【Doris 叨叨】
題目大意：下面是2010年到2013年美元對日元的即期匯率頻率直方圖，正的數值表示日元漲了（即日元可以兌換更多的美元），負的數值表示日元跌了（即日元兌換的美元變少了），基於圖表，日元升值：
A.隨著時間推移，頻率超過50%；縱軸是觀測值，橫軸是匯率漲跌，總觀測值大約=4+8+11+20+6=49日元升值的區間是0以上，由圖表可得，0以上的觀測值大約是26，頻率即26/49=53%，因此A對。
B.小於日元貶值的頻率；由圖表可得，日元升值的區域大於日元貶值的區域，B錯。
C.0~2的頻率是20%；由圖表可得，0~2的頻率大於20%，C錯。
12.A manager invests €5,000 annually in a security for four years at the prices shown in the following table.
Theaverage price paidfor the security isclosestto:
€76.48.
€77.26.
€78.00.
考點：R85.4.3. The Harmonic Mean
Equation (7)?
ˉ
ˉ
ˉ
X
H
=
n/
n∑
i=1
(
1/
X
i
)
with
X
i
>fori=1,2,…,n
【Doris 叨叨】
題目大意：
一位分析師投資了50000的股票，這隻股票每年的價格如下所示，求這隻股票平均價格：
平均值有三種：算術平均值、加權平均值、幾何平均值、調和平均值
算術平均值即簡單的加總平均
加權平均值一般用在有權重的數值
幾何平均值一般用在複利的數值
根據題目，使用調和平均值最佳。
The following information relates to Questions 13–14
The following table shows the annual MSCI World Index total returns for 2004–2013.
13.The fourth quintile return for the MSCI World Index isclosestto:
20.65%.
26.03%.
27.37%.
考點：R86.1. Quartiles, Quintiles, Deciles, and Percentiles
Equation (8)?
L
y
=
(
n+1
)
y
100
【Doris 叨叨】
題目大意：
MSCI世界指數五分之四的觀測值分布在哪個值以下？
五分之四即80%，y=80
第一步，算出這個值的位置
由公式得，L80=(10+1)*80%=8.8
第二步，將所有觀測值從小到大排序：
-40.33%，-5.02%，9.57%，10.02%，12.34%，15.25%，16.54%，20.65%，27.37%，30.79%
第三步，這個觀測值的位置是8.8，即位於20.65%~27.37%之間，到這裡，其實可以使用排除法排掉A和C，選B
不過還是講詳細點好，這個值是怎麼算出來的呢？
20.65%+（8.8-8）*（27.37%-20.65%）
14.For2009–2013,the mean absolute deviationof the MSCI World Index total returns isclosestto:
10.20%.
12.74%.
16.40%.
考點：R87.2. The Mean Absolute Deviation
【Doris 叨叨】
題目大意：
求2009-2013的標準差
這是一道非常簡單的題目！！！然鵝我還做錯了，是因為我審題不嚴謹！！！
算了2004~2013的了！！！題目問的是2009~2013！！！！
真的要氣死了。。。。寫不下去了。。。。。
16.Over the past 240 months, an investor』s portfolio had a mean monthly return of 0.79%, with a standard deviation of monthly returns of 1.16%.According to Chebyshev』s inequality, the minimum number of the 240 monthly returns that fall into the range of ?0.95% to 2.53% isclosestto:
80.
107.
133.
考點：R87.6. Chebyshev』s Inequality
According to Chebyshev』s inequality, for any distribution with finite variance, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 ? 1/k2 for all k > 1.
【Doris 叨叨】
題目大意：
經過240個月，一個投資者的投資組合的月利率是0.79%，標準差是1.16%，根據切爾雪夫不等式，240個月中有多少個月是落在-0.95%~2.53%區間的最小值範圍內？
切爾雪夫不等式：有1-1/k^2比例的觀測值落在離平均值k個標準差範圍內，k>1
最小值範圍即0.79%-（-0.95%）=1.74%，k=1.74%/1.16%=1.5
1-1/k^2=56%,因此觀測值等於240*56%=134，最接近的答案是C。
R9probability concepts
1.Suppose that 5 percent of the stocks meeting your stock-selection criteria are in the telecommunications (telecom) industry. Also,dividend-paying telecom stocks are 1 percent of the total number of stocks meeting your selection criteria. What is the probability that a stock is dividend paying, given that it is a telecom stock that has met your stock selection criteria?
0.2
考點：R92. PROBABILITY, EXPECTED VALUE, AND VARIANCE
Definition of Conditional Probability.
Equation (1)?
P(A|B) =P(AB)/P(B),P(B) ≠ 0 ?
【Doris 叨叨】
題目大意：假設有5%股票符合你挑選標準，1%股票既符合挑選標準又派發股利，求在符合挑選標準條件下，派發股利的股票概率是？
（我做錯的原因是沒有深入理解題目標黃的意思，即1%股票既符合挑選標準又派發股利，我理解成在符合挑選標準條件下，派發股利的股票概率是1%；在·····條件下，即given that）
根據公式，假設令A=pay dividend B=meet crteria
則P(A|B) =1%/5%=0.2
2.You are using the following three criteria to screen potential acquisition targets from a list of 500 companies:
If the criteria are independent, how many companies will pass the screen?
考點：R9PROBABILITY, EXPECTED VALUE, AND VARIANCE
Multiplication Rule for Independent Events.
Equation (4)?
P(AB) =P(A)P(B)
【Doris 叨叨】
題目大意：
在500個公司名單里，你正在使用下面的標準去篩選收購公司，如果每個標準都是獨立的，那有多少公司會通過篩選？
根據公式，通過的概率=0.2*0.45*0.78=7.02%
500*7.02%=35.10
5.You have developed a set of criteria for evaluating distressed credits. Companies that do not receive a passing score are classed as likely to go bankrupt within 12 months. You gathered the following information when validating the criteria:
Forty percent of the companies to which the test is administered will go bankrupt within 12 months:P(nonsurvivor) = 0.40.
Fifty-five percent of the companies to which the test is administered pass it:P(pass test) = 0.55.
The probability that a company will pass the test given that it will subsequently survive 12 months, is 0.85:P(pass test|survivor) = 0.85.
What is P(pass test | nonsurvivor)?
Using Bayes』 formula, calculate the probability that a company is a survivor, given that it passes the test; that is, calculateP(survivor | pass test).
A.0.1 B.0.927
考點：R9PROBABILITY, EXPECTED VALUE, AND VARIANCE&4.1. Bayes』 Formula
The Total Probability Rule.
Equation (5)?
P
(A)
=P
(
AS
)
+P
(
A
S
C
)
=P
(
A|S
)
P
(S)
+P
(
A
∣∣
S
C
)
P
(
S
C
)
Bayes』 Formula
【Doris 叨叨】
題目大意：
A.
P(nonsurvivor，以下簡稱ns) = 0.40
P(pass test,以下簡稱pt) = 0.55
P(pass test|survivor,以下簡稱pt| s) = 0.85
求P(pt| ns)
根據全概率公式（應用於是這個和不是這個的事件）
P(pt)=P(pt| ns)*P(ns)+P(pt| s)*P(s)
0.55=P(pt| ns)*0.4+0.85*0.6(1-0.4)
P(pt| ns)=0.1
B.P(s| pt)=[P(pt| s)*P(s)]/P(pt)=[0.85*0.6]/0.55=0.927
10.After estimating the probability that an investment manager will exceed his benchmark return in each of the next two quarters, an analyst wants to forecast the probability that the investment manager will exceed his benchmark return over the two-quarter period in total.Assuming that each quarter』s performance is independent of the other, which probability rule should the analyst select?
Addition rule
Multiplication rule
Total probability rule
考點：R9PROBABILITY, EXPECTED VALUE, AND VARIANCE
Multiplication Rule for Independent Events.
【Doris 叨叨】
題目大意：一位分析師想估計他的兩個季度的業績大於基準的概率，假設每個季度的業績都是獨立的，那麼分析師應該使用哪種概率計算方法：
由考點得知選B
17.An analyst develops the following covariance matrix of returns:
The correlation of returns between the hedge fund and the market index is closest to:
0.005.
0.073.
0.764.
考點：R93. PORTFOLIO EXPECTED RETURN AND VARIANCE OF RETURN
Cov(Ri,Rj) = ρ(Ri,Rj)σ(Ri)σ(Rj)
【Doris 叨叨】
題目大意：
求相關性，，，這道題我也不想說了，自己大意，沒有給方差開方!!!!!!!
氣得不想寫啦。。。。
19.Given a portfolio offive stocks, how many unique covariance terms, excluding variances,are required to calculate the portfolio return variance?
10
20
25
考點：R93. PORTFOLIO EXPECTED RETURN AND VARIANCE OF RETURN
Covariances are often presented in a square format called acovariance matrix.
【Doris 叨叨】
題目大意：一個投資組合有5隻股票，有多少個獨立的協方差，來計算這個組合的方差
根據考點，投資組合的協方差總是為股票數的平方，即25，除去自己和自己的協方差一共5個，還剩20個，因為矩陣是對稱的，所以是10個。
20.The probability distribution for a company』s sales is:
The standard deviation of sales isclosestto:
$9.81 million.
$12.20 million.
$32.40 million.
考點：R93. PORTFOLIO EXPECTED RETURN AND VARIANCE OF RETURN
E
(
R
p
)
=E
(
w
1
R
1
+
w
2
R
2
+…+
w
n
R
n
)
=
w
1
E
(
R
1
)
+
w
2
E
(
R
2
)
+…+
w
n
E
(
R
n
)
σ
2
(
R
p
)
=E
[
(
R
p
?E
R
p
)
2
]
【Doris 叨叨】
題目大意：公司銷售規模的概率分布圖如下：
求該公司的銷售標準差
第一步，算出均值
即0.05*70+0.7*40+0.25*25=37.75
第二步，算出方差
0.05*（70-37.75）^2+0.7*(40-37.75）^2+0.25*（25-37.75）^2=96.18
第三步，算出標準差
將方差開方即可，即9.81
22.An analyst produces the following joint probability function for a foreign index (FI) and a domestic index (DI).
The covariance of returns on the foreign index and the returns on the domestic index isclosestto:
26.39.
26.56.
28.12.
考點：R93. PORTFOLIO EXPECTED RETURN AND VARIANCE OF RETURN
Equation (18)?
Cov
(
R
A
,
R
B
)
=
∑
i
∑
j
P
(
R
A,i
,
R
B,j
)
(
R
A,i
?E
R
A
)
(
R
B,j
?E
R
B
)
【Doris 叨叨】
題目大意：
求協方差
第一步，求均值
E（RFI ）=25%*0.25+15%*0.5+10%*0.25=16.25%
E（RDI ）=30%*0.25+25%*0.5+15%*0.25=23.75%
第二步，代入公式求協方差
0.25*（25-16.25）（30-23.75）+0.5*（15-16.25）（25-23.75）+0.25*（10-16.25）（15-23.75）=26.56
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