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Investment

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Investments team project
KUBS Investment Fund Manager
Group 14
(Step 1)
For the convenience of problem solving, we solved the problems in following orders: a–c–e–b-d.
a) The yearly expected return:
First, find the weekly return for each stock by using the formula of: rᵢ=p₁-p₀p₀ Second, for each of the stock, add all of the weekly return and then divide by the number of sample (from Oct 27, 2005 until Oct 19, 2015) to get the average of weekly return. rw=1mnί=1mnrᵢ (where mn is 521)
Third, multiply the average of weekly return with 52 weeks to get the yearly average. ry=mrw (where m is 52)
b) The standard deviation of estimate in (a):
To find the error in the mean estimate, we need to calculate its standard deviation first.
S²w=1(mn-1)i=1mn(rᵢ-r)²
S²y=mS²w (where mn is 521, m is 52)
Therefore, Sy= S²y
Next, we can find the error by using the formula stated below: σᵣ=Syn Assume that n=52152is approximately equal to 10
c) The yearly variance of the return:
S²w=1(mn-1)i=1mn(rᵢ-r)² (where mn is 521)

Note: Since we need to calculate its yearly variance, thus we need to multiply it with 52 weeks.
S²y=mS²w (where m is 52)
d) The standard deviation of estimate in (c): σs²=2S²y(mn-1) (where mn is 521)
e) The yearly standard deviation of the return:
By referring to the yearly variance in part (c) that we have calculated, we can then find the yearly standard deviation by square rooting the answer of the variance.
S²w=1(mn-1)i=1mn(rᵢ-r)²S²y=mS²w (where mn is 521, m is 52)
Therefore, Sy= S²y
f) The annualized covariance of the return of each stock with the market (S&P500):
S(r₁r₂)w=1mn-1i=1mnrᵢ1-r͞ ⁽¹⁾(rᵢ2-͞r⁽²⁾) (where mn is 521)
Note: We need to multiply the weekly covariance with 52 weeks in order to get the annualized covariance.
S(r₁r₂)y = mS(r₁r₂)w (where m is 52)
g) Beta for each stock:
To find beta, first we need to calculate its annualized sample covariance of the return of each stock with the market and then, we divide it with the yearly sample variance of the market. βs=Cov (rs,rm)Var(rm)
h) The annualized covariance between the returns of the stocks.
We need to use the same method of calculation like part (f) except, we use return of another stock instead of return of market.
Cov rs,rm= rS-rs(rm-rm) (n-1)

(Step 2)
The expected return predicted by CAPM is: ri=rf+βrm-rf The base assumption is that future expected return will be consistent with the historical data. Also, individual stocks’ expected error term, which is E(ei), is zero. Risk free rate=0.015. E(rm)=0.07188 ri=0.015+β(0.07188-0.015) Through these, we can get the following: | EBAY | ESRX | JACK | NDAQ | BBBY | S&P500 | rCAPM | 0.07849 | 0.06395 | 0.07492 | 0.09646 | 0.07159 | 0.07188 |

(Step 3)
The excess return of the market (S&P500) over the risk-free rate is the single factor in determining the excess returns of the stocks over the risk-free rate. So we can say ri-rf = ai+birm-rf+ei
We used regression tool in excel to analyze the association with 95% confidence level. Below is the result of regression analysis. | EBAY | ESRX | JACK | NDAQ | BBBY | Intercept(ai) | -0.02628 | 0.02805 | 0.095811 | 0.050805 | 0.013813 | Factor loading(bi) | 1.116276 | 0.860603 | 1.053492 | 1.432134 | 0.994884 | t-stat | -0.25855 | 0.268479 | 0.953007 | 0.527283 | 0.182876 | Beta (step 1) | 1.116276 | 0.860603 | 1.053492 | 1.432134 | 0.994884 | t-stat | -0.25855 | 0.268479 | 0.953007 | 0.527283 | 0.182876 |

Coefficient for x’s t-stat is bigger than absolute value of 2, meaning that linear relationship with market risk premium is positive. However, y coefficient, which is alpha’s t-stat’s absolute value is smaller than 2, it is rejected. It means we can say intercepts are significantly different from zero with 95% of confidence level.
Betas from direct calculation and regression tool are the same.

(Step 4)
a)
σi2=βi2σm2+Var(ei)
Where Varei=0 σi=βiσm As we suppose 521/52 is 10, yearly standard deviation of error in each CAPM estimate of mean return is σi σECAPM= βi×σMn (수정) 아래첨자 시그마 i. 위첨자 CAPM where βi=beta of each asset, σM=standard deviation of the market index(S&P500), and n=number of samples(=10(years)).
b) Due to the fact that the historical yearly return and the CAPM yearly return differs, there is a need for a different estimate. Thus, utilizing statistical knowledge to get the tilted average, we got the tilted average by using the equation, r=(P1σ12+P2σ22)×1σ12+1σ22-1 where P1=rCAPM, P2=rhistory, σ12=variance of CAPM mean, σ22=variance of historical mean.
c) According to CAPM, the mean return of EBAY should be 0.07849(7.849%). However, EBAY’s yearly average of mean return from historical data shows 0.05222(5.222%). Given the assumption that we treat the CAPM mean as the indicator of appropriate amount of return, the stock is currently overpriced. Thus, it would be wise to give up introducing the EBAY stock to the KUBS investment set.

(Step 5)
E(rp)= i, j=15wi∙ri σp2= i, j=15wi∙wj∙σij
Sharpe Ratio= Erp-rfσp
a) To compute a minimum variance portfolio(MVP),
Object : Minimize i, j=15wi∙wj∙σij, Subject to : i=15wi=1 run the Excel solver setting Wi for variables. Then, E(r) | σ | Var(MVP) | 0.07714 | 0.23979 | 0.05750 |

| EBAY | ESRX | JACK | NDAQ | BBBY | Weight | 0.18137 | 0.28105 | 0.15711 | 0.03089 | 0.34957 |

b) Assuming I. All investors are risk-averse and maximize expected utility. II. All investors invest with “mean-variance optimizer”. III. The market is Perfect Capital Market which has no market fractional factors such as transaction cost, tax, information cost, etc. Also, all investors are price-taker in perfect capital market. IV. Interest rate of risk-free asset is same for lending and borrowing. V. Investment is on single period. Under these assumptions the one-fund is a portfolio which has maximized sharpe ratio with risk free asset.
(Solution 1)
Objective : Maximize E(rp)-rfσp
Subject to : i=15wi=1 Run the Excel solver setting Wi for variables.
(Solution 2)
As i=15vi∙σki=Eri-rf, σ11σ12σ13σ14σ15σ21σ22σ23σ24σ25σ31σ32σ33σ34σ35σ41σ42σ43σ44σ45σ51σ52σ53σ54σ55∙v1v2v3v4v5=Er1-rfEr2-rfEr3-rfEr4-rfEr5-rf v1v2v3v4v5= σ11σ12σ13σ14σ15σ21σ22σ23σ24σ25σ31σ32σ33σ34σ35σ41σ42σ43σ44σ45σ51σ52σ53σ54σ55-1∙ Er1-rfEr2-rfEr3-rfEr4-rfEr5-rf Compute vi V1 | V2 | V3 | V4 | V5 | Sum | 0.09201 | 0.19087 | 0.28318 | 0.33656 | 0.17806 | 1.08070 |

Use vi to compute weight, as wj=vji=15vi , The results of (Solution 1) and (Solution 2) are same as | EBAY | ESRX | JACK | NDAQ | BBBY | Weight | 0.08514 | 0.17662 | 0.26204 | 0.31143 | 0.16477 |

E(r) | Σ | Var(one-fund) | Sharpe ratio | 0.09043 | 0.26419 | 0.06980 | 0.28551 |

c) Assume that MVP and One-fund are both efficient portfolio. Then efficient frontier can be derived with these two portfolios by Two-fund Theorem.
First, compute weekly return of MVP and One-fund by multiply the weights (which are calculated in a) and b)) on each stocks.
Second, compute covariance between MVP and One-fund weekly return.
Third, compute expected return and standard deviation of the portfolios which are combined with MVP and One-fund. Weights of MVP and One-fund are different in each portfolios.
Erp= w ∙Erone-fund+1-w∙E(rMVP) σp2 =w2∙σone-fund2+1-w2∙σMVP2+2∙w∙(1-w)∙σone-fund, MVP
Last, draw a graph with X-axis for standard deviations of the portfolios and Y-axis for corresponding expected returns.
CAL
CAL
CML
CML

(Step 6)

Sharpe ratio of the one-fund for the KUBS Fund is 0.28551. It is smaller than the market’s, 0.309581. According to portfolio theory, although one-fund has the largest sharpe-ratio, it is smaller than the sharpe-ratio of S&P500.

βp=w1β1+w2β2+…+wnβn
The beta of the one-fund is 1.133034.
By CAPM, all the assets including one-fund should be on the security market line. However, one-fund is over the SML, and it is not consistent with CAPM. This can be translated that S&P500 is providing too little market premium.
It is because first of all, one-fund is the portfolio we created using only 5 assets. It cannot reflect all the assets in the market, so it cannot be well-diversified as the market. Secondly, S&P500 can’t be the perfect market portfolio, because like Roll’s critique about CAPM, it is impossible to create market portfolio including all the assets in the market.

(Step 7)
Complete portfolio’s standard deviation on one-fund and risk-free asset combined portfolio’s CAL is σc=y2σonefund2+1-y2σrf2+2y(1-y)Cov(ronefund, rf)
Where σrf=0, Covronefund, rf=0, σc=yσonefund, where y is the portion on the risky assets and σp is the standard deviation of the one-fund, we can get σc = 0.18374 = y × 0.26419, so y = 0.69547, so the weight of each stocks and risk-free asset are y×each asset's weight=weight on complecte portfolio | EBAY | ESRX | JACK | NDAQ | BBBY | Risk-free | weight | 0.059213 | 0.122835 | 0.18224 | 0.216593 | 0.114589 | 0.304531 |

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...复旦大学管理学院院 投资学期末考试试卷 样品 课程名称:__投资学 _________ 课程代码: MANA130029.01____________ 开课院系:__管理学院财务金融系____ 考试形式:闭卷 姓 名: 学 号: 专 业: |题 号 |1 |2 |3 |4 |5 |总 分 | |得 分 | | | | | | | (以下为试卷正文) 一、选择题 (60分)Multiple choices (60 point, one point each) 1. 资本配置线可以描述为 A) 投资机会集由一个无风险资产和一个风险资产构成 B) 投资机会集由两个风险资产构成 C) 上面每个点对某个投资者来说效用都一样 D) 每个点期望收益一样但风险不一样 E) 上面一个都不对 1. The Capital Allocation Line can be described as the A) investment opportunity set formed with a risky asset and a risk-free asset. B) investment opportunity set formed with two risky assets. C) line on which lie all portfolios that offer the same utility to a particular investor. D) line on which lie all portfolios with the same expected rate of return and different standard deviations. E) none of the above. 3.无风险利率为5%,风险资产如下 Security A: E(r) = 0.15; Variance = 0.04 Security B: E(r) = 0.10; Variance = 0.0225 Security C: E(r) = 0.12; Variance = 0.01 Security D: E(r) = 0.13; Variance = 0.0625 投资者将选择哪一个资产来组成风险资产和无风险资产的组合 A) A. B) B. C) C. D) D. E) 不能决定. 3. Consider a T-bill with a rate of return of 5 percent and the following risky securities: Security A: E(r) = 0.15; Variance = 0.04 Security B: E(r) = 0.10; Variance = 0.0225 Security C: E(r) = 0.12; Variance = 0.01 Security D: E(r) = 0.13; Variance...

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