Financial Economics Mid Term
Barovier Allybose
04/09/2020
Question 1. Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following forecast information:
RiskReturn <- matrix(c(0.67,2.37,1.25,2.95),ncol=2,byrow=TRUE)
colnames(RiskReturn) <- c("Expected Monthly Returns","Standard Deviation of Monthly Returns")
rownames(RiskReturn) <- c("Original Portfolio","ABC Company")
RiskReturn <- as.table(RiskReturn)
RiskReturn## Expected Monthly Returns
## Original Portfolio 0.67
## ABC Company 1.25
## Standard Deviation of Monthly Returns
## Original Portfolio 2.37
## ABC Company 2.95The correlation coefficient of ABC stock returns with the original portfolio returns is 0.40.
- The inheritance changes Grace’s overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the:
I. Expected return of her new portfolio which includes the ABC stock.
OrigionalProportion <- ((900000)/(900000+100000))
ABCProportion <- ((100000)/(900000+100000))
ExpectedReturn <- ((0.67*OrigionalProportion)+(1.25*ABCProportion))
ExpectedReturn## [1] 0.728- Covariance of ABC stock returns with the original portfolio returns.
0.40*0.0237*0.0295## [1] 0.00027966- Standard deviation of her new portfolio, which includes the ABC stock.
sqrt(((0.9^2)*(.0237^2))+((0.1^2)*(0.0295^2))+(2*0.1*0.9*0.00027966))## [1] 0.02267179- If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the:
Expected return of her new portfolio, which includes the government securities.
ExpectedReturnriskfree <- ((0.0067*0.90)+(.0042*0.10))
ExpectedReturnriskfree## [1] 0.00645Covariance of the government security returns with the original portfolio returns.
correlation <- 0
sdgovernmentsecurity <- 0
correlation*sdgovernmentsecurity*0.0295## [1] 0Standard deviation of her new portfolio, which includes the government securities.
sd <- sqrt(((0.9^2)*(.0237^2))+((0.1^2)*(0^2))+(2*0.1*0.9*0))
sd## [1] 0.02133Determine whether the systematic risk of her new portfolio, which includes the government securities, will be higher or lower than that of her original portfolio. Since a less risky security is added to her portfolio this should reduce the overall risk (and potential return) of the portfolio and of course, the systematic risk. Therefore the systematic risk of the portfolio with the government security will be less than the systematic risk of the origional portfolio. This should be evidence in a newly reduced beta.
On the basis of conversations with her husband, Grace is considering selling the $100,000 of ABC stock and acquiring $100,000 of XYZ Company common stock instead. XYZ stock has the same expected return and standard deviation as ABC stock. Her husband comments, “It doesn’t matter whether you keep all of the ABC stock or replace it with $100,000 of XYZ stock.” State whether her husband’s comment is correct or incorrect. Justify your response. Both stocks may not be perfectly correlated. In fact, if they are negatively correlated then replacing one for the other may make a significant difference in risk/return. Therefore the husbands suggestion is unjustified and incorrect.
In a recent discussion with her financial adviser, Grace commented, “If I just don’t lose money in my portfolio, I will be satisfied.” She went on to say, “I am more afraid of losing money than I am concerned about achieving high returns.”
I. The investor is more concerned with the tails of the distribution of returns, which standard deviation does a poor job at describing. Instead standard deviation works better for a distribution that is assumed to be normal and so discrimination of negative vs positive price movements is not evident with standard deviation.
- Skewness does a better job because it tells whether the distribution leans negatively or positively (measure of asymmetry).
Question 2a. A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
MicroForecasts <- matrix(c(20,18,17,12,1.3,1.8,0.7,1.0,58,71,60,55),ncol=3,nrow=4)
colnames(MicroForecasts) <- c("Expected Return (%)","Beta","Residual Standard Deviation (%)")
rownames(MicroForecasts) <- c("Stock A","Stock B","Stock C","Stock D")
MicroForecasts <- as.table(MicroForecasts)
MicroForecasts## Expected Return (%) Beta Residual Standard Deviation (%)
## Stock A 20.0 1.3 58.0
## Stock B 18.0 1.8 71.0
## Stock C 17.0 0.7 60.0
## Stock D 12.0 1.0 55.0Micro Forecast
MacroForecast <- matrix(c(8,0,16,23),ncol=2,byrow=TRUE)
colnames(MacroForecast) <- c("Expected Return (%)", " Standard Deviation (%)")
rownames(MacroForecast) <- c("T Bills","Passive Equity Portfolio")
MacroForecast <- as.table(MacroForecast)
MacroForecast## Expected Return (%) Standard Deviation (%)
## T Bills 8 0
## Passive Equity Portfolio 16 23Macro Forecast
- Calculate expected excess returns, alpha values, and residual variances for these stocks.
StockA_excess <-0.20-0.08
StockB_excess <-0.18-0.08
Stockc_excess <-0.17-0.08
StockD_excess <-0.12-0.08
StockA_Alpha <- (0.20-(1.3*(0.16-0.08))-0.08)
StockB_Alpha <- (0.18-(1.8*(0.16-0.08))-0.08)
StockC_Alpha <- (0.17-(0.7*(0.16-0.08))-0.08)
StockD_Alpha <- (0.12-(1*(0.16-0.08))-0.08)
StockA_ResidualVariance <- (58)^2
StockB_ResidualVariance <- (71)^2
StockC_ResidualVariance <- (60)^2
StockD_ResidualVariance <- (55)^2
DesStat <- matrix(c(StockA_excess,StockB_excess,Stockc_excess,StockD_excess,StockA_Alpha,StockB_Alpha,StockC_Alpha,StockD_Alpha,StockA_ResidualVariance,StockB_ResidualVariance,StockC_ResidualVariance,StockD_ResidualVariance),ncol=3,nrow=4)
colnames(DesStat) <- c("Excess Returns","Alpha","Residual Variance")
rownames(DesStat) <- c("StockA","StockB","StockC","StockD")
DesStat <- as.table(DesStat)
DesStat## Excess Returns Alpha Residual Variance
## StockA 0.120 0.016 3364.000
## StockB 0.100 -0.044 5041.000
## StockC 0.090 0.034 3600.000
## StockD 0.040 -0.040 3025.000- Construct the optimal risky portfolio.
numerator_StockA <-StockA_Alpha/(StockA_ResidualVariance/100)
numerator_StockB <- StockB_Alpha/(StockB_ResidualVariance/100)
numerator_StockC <- StockC_Alpha/(StockC_ResidualVariance/100)
numerator_StockD <- StockD_Alpha/(StockD_ResidualVariance/100)
Portfolio_ResidualDeviance <- (numerator_StockA )+(numerator_StockB)+(numerator_StockC)+(numerator_StockD)
Weight_StockA <- numerator_StockA/Portfolio_ResidualDeviance
Weight_StockB <- numerator_StockB/Portfolio_ResidualDeviance
Weight_StockC <- numerator_StockC/Portfolio_ResidualDeviance
Weight_StockD <- numerator_StockD/Portfolio_ResidualDeviance
ActivePortfolio <- matrix(c(numerator_StockA,numerator_StockB,numerator_StockC,numerator_StockD,Weight_StockA,Weight_StockB,Weight_StockC,Weight_StockD),ncol=2,nrow=4)
colnames(ActivePortfolio) <- c("Alpha/ResidualVariance","Weight Of Each Stock")
rownames(ActivePortfolio) <- c("StockA","StockB","StockC","StockD")
ActivePortfolio <- as.table(ActivePortfolio)
ActivePortfolio## Alpha/ResidualVariance Weight Of Each Stock
## StockA 0.0004756243 -0.6136390105
## StockB -0.0008728427 1.1261207074
## StockC 0.0009444444 -1.2184995740
## StockD -0.0013223140 1.7060178771PortfolioAlpha <-(StockA_Alpha*Weight_StockA)+(StockB_Alpha*Weight_StockB)+(StockC_Alpha*Weight_StockC)+(StockD_Alpha*Weight_StockD)
PortfolioAlpha## [1] -0.1690372PortfolioBeta <-(1.3*Weight_StockA)+(1.8*Weight_StockB)+(0.7*Weight_StockC)+(1.0*Weight_StockD)
PortfolioBeta ## [1] 2.082355PortfolioVariance<- (StockA_ResidualVariance)*((Weight_StockA)^2)+(StockB_ResidualVariance)*((Weight_StockB)^2)+(StockC_ResidualVariance)*((Weight_StockC)^2)+(StockD_ResidualVariance)*((Weight_StockD)^2)
PortfolioVariance## [1] 21808.78PortfolioSd <- sqrt(PortfolioVariance)
PortfolioSd## [1] 147.678- What is Sharpe’s measure for the optimal portfolio and how much of it is contributed by the active portfolio? (Compared with Market Sharpe)
Informationratio_A <- (PortfolioAlpha/PortfolioSd)*100
Informationratio_A## [1] -0.1144634SharpeMeasure <- sqrt(((8/23)^2)+(Informationratio_A)^2)
SharpeMeasure## [1] 0.366176Comparison <- SharpeMeasure-(8/23)
Comparison## [1] 0.01834991- What should be the exact makeup of the complete portfolio for an investor with a coefficient of risk aversion of 2.8?
B_P <- 1.0486+((-0.0486)*2.08)
B_P## [1] 0.947512Mean_ExcessReturn_P <- ((-0.0486)*(-16.90))+(0.95*8)
Mean_ExcessReturn_P## [1] 8.42134Variance_P <- (0.95*23)^2+((-0.0486)^2)*(21809.6)
Variance_P## [1] 528.9359SD_P <- sqrt(Variance_P)
SD_P## [1] 22.99861Optimalposition_Y <- 8.42/(0.01*2.8*528.94)
Optimalposition_Y## [1] 0.5685225PassiveStrategy_Y <- Y <- 8/(0.01*2.8*(23)^2)
PassiveStrategy_Y ## [1] 0.5401026Difference_RiskyPortfolio_Market <- Optimalposition_Y-PassiveStrategy_Y
Difference_RiskyPortfolio_Market## [1] 0.02841987Bills <- 1-0.5685
M <- 0.5685*0.0486
A <- 0.5685*(-0.0486*-0.6142)
B <- 0.5685*(-0.0486*1.1265)
C <- 0.5685*(-0.0486*-1.12181)
D <- 0.5685*(-0.0486*1.7058)
Final_Positions <- matrix(c(Bills,M,A,B,C,D),ncol=2,nrow=6)
colnames(Final_Positions) <- c("","Weights")
rownames(Final_Positions) <- c("Bills","M","StockA","StockB","StockC","StockD")
Final_Positions <- as.table(Final_Positions)
Final_Positions## Weights
## Bills 0.43150000 0.43150000
## M 0.02762910 0.02762910
## StockA 0.01696979 0.01696979
## StockB -0.03112418 -0.03112418
## StockC 0.03099460 0.03099460
## StockD -0.04712972 -0.04712972Final Positions
Question 2b.Recalculate Question 2a for a portfolio manager who is not allowed to short sell securities.(Exclude negative alphas, not able to short sell)
StockA_excess <-0.20-0.08
Stockc_excess <-0.17-0.08
StockA_Alpha <- (0.20-(1.3*(0.16-0.08))-0.08)
StockC_Alpha <- (0.17-(0.7*(0.16-0.08))-0.08)
StockA_ResidualVariance <- (58)^2
StockC_ResidualVariance <- (60)^2
DesStat <- matrix(c(StockA_excess,Stockc_excess,StockA_Alpha,StockC_Alpha,StockA_ResidualVariance,StockC_ResidualVariance),ncol=3,nrow=2)
colnames(DesStat) <- c("Excess Returns","Alpha","Residual Variance")
rownames(DesStat) <- c("StockA","StockC")
DesStat <- as.table(DesStat)
DesStat## Excess Returns Alpha Residual Variance
## StockA 0.120 0.016 3364.000
## StockC 0.090 0.034 3600.000- What is the cost of the restriction in terms of Sharpe’s measure?
numerator_StockA2 <- StockA_Alpha/(StockA_ResidualVariance/100)
numerator_StockC2 <- StockC_Alpha/(StockC_ResidualVariance/100)
PortfolioAlpha2 <-(StockA_Alpha*Weight_StockA)+(StockC_Alpha*Weight_StockC)
Portfolio_ResidualDeviance2 <- (numerator_StockA2)+(numerator_StockC2)
Weight_StockA <- numerator_StockA/Portfolio_ResidualDeviance2
Weight_StockC <- numerator_StockC/Portfolio_ResidualDeviance2
PortfolioAlpha2## [1] -0.05124721PortfolioBeta2 <-(1.3*Weight_StockA)+(0.7*Weight_StockC)
PortfolioBeta2 ## [1] 0.9009583PortfolioVariance2<- (StockA_ResidualVariance)*((Weight_StockA)^2)+(StockC_ResidualVariance)*((Weight_StockC)^2)
PortfolioVariance2## [1] 1969.711PortfolioSd2 <- sqrt(PortfolioVariance2)
PortfolioSd2## [1] 44.38143ActivePortfolio_weight <- (2.8/1969.711)/(8/(23)^2)
ActivePortfolio_weight## [1] 0.09399856RiskyPortfolio_Proportion <- 0.094/(1+((1-0.90)*0.094))
Informationratio2 <- 2.80/44.37
Informationratio2## [1] 0.0631057SharpeMeasure2 <- sqrt((8/23)^2)+(0.0631^2)
SharpeMeasure2## [1] 0.3518077Note: The new sharpe ratio due to the restriction (on short selling) is 0.3518077 which is less than the origional sharpe ratio of 0.366176 when short selling was allowed.
- What is the utility loss to the investor (A = 2.8) given his new complete portfolio? Utility = E(r)-0.5AVar
Strategies <- matrix(c(4.79,4.46,4.32,170.95,159.36,154.31,10.40,10.23,10.16),ncol=3,nrow=3)
colnames(Strategies) <- c("E(R)","Variance","Utility")
rownames(Strategies) <- c("Unconstrained","Constrained","Passive")
Strategies <- as.table(Strategies)
Strategies## E(R) Variance Utility
## Unconstrained 4.79 170.95 10.40
## Constrained 4.46 159.36 10.23
## Passive 4.32 154.31 10.16UtilityLost <- 10.40-10.23
UtilityLost## [1] 0.17Question 3. Suppose you are the manager of a large pension fund. Assume that the risk-free rate equals zero. You wish to create an optimal portfolio that consists of two stocks. One stock is General Motors (GM) and the other stock is Apple Inc (AAPL). You have complied relevant data in an EXCEL spreadsheet file (please see accompanying EXCEL file). The data is MONTHLY and for the period indicated in the EXCEL file. Please search and download the corresponding data for Apple Inc, and merge it with the EXCEL file. For this purpose, let GM = “Asset 1”, let the Market Index = “Index 1”, and let AAPL = “Asset 2”. The data you need for the portfolio can be extracted from a regression of each stock’s returns on the returns of the Market Index (Index 1) using the history of monthly returns. You have projected that Index 1, during this period, has an expected return and an estimated standard deviation of return equal to the historical average return and historical standard deviation as shown on the (given) EXCEL spreadsheet.
library(readxl)
Midterm_Spreadsheet_Financial_Economics_2 <- read_excel("Midterm Spreadsheet_Financial Economics-2.xlsx")
head(Midterm_Spreadsheet_Financial_Economics_2)## # A tibble: 6 x 5
## `Monthly Return` Date `Market Index` GM `AAPL (Please paste here)`
## <dbl> <chr> <dbl> <dbl> <dbl>
## 1 0 2011-01 0.0178 -0.0114 0.0519
## 2 1 2011-02 0.0371 -0.0822 0.0409
## 3 2 2011-03 0.00277 -0.0752 -0.0133
## 4 3 2011-04 0.0284 0.0339 0.00465
## 5 4 2011-05 -0.0151 -0.00893 -0.00657
## 6 5 2011-06 -0.0186 -0.0458 -0.0350StockStatistics <- matrix(c(0.0082631,0.00302843333333334,0.0189567800929, 0.0348158719798692,0.085823210064499,0.0684799695735718,-0.084873,-0.176502,-0.14408924468494,0.113882,0.280871,0.188310730910731,0.198755,0.457373,0.33239997559567),ncol=5,nrow=3)
colnames(StockStatistics) <- c("AVG","SD","Min","MAX","Max-Min")
rownames(StockStatistics) <- c("Market Index","GM","AAPL")
StockStatistics <- as.table(StockStatistics)
StockStatistics## AVG SD Min MAX Max-Min
## Market Index 0.008263100 0.034815872 -0.084873000 0.113882000 0.198755000
## GM 0.003028433 0.085823210 -0.176502000 0.280871000 0.457373000
## AAPL 0.018956780 0.068479970 -0.144089245 0.188310731 0.332399976SingleIndexModelGM <- lm(Midterm_Spreadsheet_Financial_Economics_2$`Market Index`~ Midterm_Spreadsheet_Financial_Economics_2$GM)
SingleIndexModelAAPL <- lm(Midterm_Spreadsheet_Financial_Economics_2$`Market Index`~ Midterm_Spreadsheet_Financial_Economics_2$`AAPL (Please paste here)`)Please do the following: a. Create two scatter diagrams, one for GM (Asset 1) versus Index 1, and one for AAPL (Asset 2) versus Index 1. [Hint: see Figure 8.1 of the textbook].
plot(Midterm_Spreadsheet_Financial_Economics_2$GM, Midterm_Spreadsheet_Financial_Economics_2$`Market Index`, main="Scatter Diagram for GM against Market Index",
xlab="GM", ylab="Market Index", pch=19)
abline(SingleIndexModelGM, col="blue")
plot(Midterm_Spreadsheet_Financial_Economics_2$`AAPL (Please paste here)`, Midterm_Spreadsheet_Financial_Economics_2$`Market Index`, main="Scatter Diagram for AAPL against Market Index", xlab="AAPL", ylab="Market Index", pch=19)
abline(SingleIndexModelAAPL, col="blue")
- Create two tables, one that shows the results of your regression of the returns of Asset 1 versus Index 1 and the other, the results of your regression of the returns of Asset 2 versus Index 1.
library(stargazer)##
## Please cite as:## Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.## R package version 5.2.2. https://CRAN.R-project.org/package=stargazerstargazer(SingleIndexModelGM, type="text",
dep.var.labels=c("Market Index"), covariate.labels=c("GM"), out="models.txt")##
## ===============================================
## Dependent variable:
## ---------------------------
## Market Index
## -----------------------------------------------
## GM 0.273***
## (0.039)
##
## Constant 0.007**
## (0.003)
##
## -----------------------------------------------
## Observations 60
## R2 0.454
## Adjusted R2 0.445
## Residual Std. Error 0.026 (df = 58)
## F Statistic 48.260*** (df = 1; 58)
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01stargazer(SingleIndexModelAAPL, type="text",
dep.var.labels=c("Market Index"), covariate.labels=c("AAPL"), out="models.txt")##
## ===============================================
## Dependent variable:
## ---------------------------
## Market Index
## -----------------------------------------------
## AAPL 0.205***
## (0.061)
##
## Constant 0.004
## (0.004)
##
## -----------------------------------------------
## Observations 60
## R2 0.162
## Adjusted R2 0.148
## Residual Std. Error 0.032 (df = 58)
## F Statistic 11.216*** (df = 1; 58)
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01Now, using the appropriate Single-Index formulas and data from your regressions, calculate the following in same order!!
Expected return of Asset 1 (GM).
Expected return of Asset 2 (AAPL).
ERmMarket = mean(Midterm_Spreadsheet_Financial_Economics_2$`Market Index`)
Rf=0
ERGM = Rf + 0.273*(ERmMarket-Rf)
ERGM## [1] 0.002255826ERAAPL = Rf + 0.205*(ERmMarket-Rf)
ERAAPL## [1] 0.001693935Variance of returns of Asset 1. sdmarketindex^2*beta^2+standard error^2
Variance of returns of Asset 2.
Standard deviation of returns of Asset 1.
Standard deviation of returns of Asset 2.
Coefficient of Variation of Asset 1. **(Please not that this is the formula that I am using for coefficient of variation, the values seem large: (sd/mean)100
Coefficient of Variation of Asset 2.
SDMarketIndex <- 0.034815872
VarGM <- ((SDMarketIndex)^2)*((0.273)^2)+(0.026)^2
VarGM ## [1] 0.00076634VarAAPL <- ((SDMarketIndex)^2)*((0.205)^2)+(0.032)^2
VarAAPL## [1] 0.00107494SDGM <- sqrt(VarGM)
SDAAPL <- sqrt(VarAAPL)
sd(Midterm_Spreadsheet_Financial_Economics_2$GM, na.rm=TRUE)/ mean(Midterm_Spreadsheet_Financial_Economics_2$GM, na.rm=TRUE)*100## [1] 2833.914sd(Midterm_Spreadsheet_Financial_Economics_2$`AAPL (Please paste here)`, na.rm=TRUE)/ mean(Midterm_Spreadsheet_Financial_Economics_2$`AAPL (Please paste here)`, na.rm=TRUE)*100## [1] 361.2426- Expected return of a portfolio that combines the assets in the following proportions: 65% in Asset 1 and 35% in Asset 2.
ExpReturn1 <- (0.65*0.002255826)+(0.35*0.001693935)
ExpReturn1## [1] 0.002059164Variance of returns of a portfolio that combines the funds in the following proportions: 65% in Asset 1 and 35% in Asset 2.
Standard deviation of returns of a portfolio that combines the funds in the following proportions: 65% in Asset 1 and 35% in Asset 2.
Coefficient of Variation of a portfolio that combines the funds in the following proportions: 65% in Asset 1 and 35% in Asset 2.
Covariance between the returns of Asset 1 and Asset 2.
Correlation coefficient between the returns of Asset 1 and Asset 2.
correlation <- cor(Midterm_Spreadsheet_Financial_Economics_2$GM,Midterm_Spreadsheet_Financial_Economics_2$`AAPL (Please paste here)`)
Var1=((0.65)^2)*((SDGM)^2)+ ((0.35)^2)*((SDAAPL)^2) + 2*(0.65)*(0.35)*(correlation)*(SDGM)*(SDAAPL)
Var1## [1] 0.0005776478P_SD <- sqrt(Var1)
P_Mean <- (.65*0.003028433)+(.35*0.018956780)
P_Coefficient_Variation <- (P_SD/ P_Mean)*100
P_Coefficient_Variation## [1] 279.3597Covariance <- cov(Midterm_Spreadsheet_Financial_Economics_2$GM,Midterm_Spreadsheet_Financial_Economics_2$`AAPL (Please paste here)`)
Covariance## [1] 0.001738945correlation## [1] 0.2958813- Suppose you were to use this model to create a portfolio of 9000 stocks. Demonstrate the benefits of using this model instead of the full-covariance version of Markowitz model. (Hint: Use Calculations as done in class. No formula provided here!!)
The Markowitz model is one way to estimate the most efficient portfolio based on risk and return. In order to optimize such portfolio, one needs the following estimates: ((n)^2)-n)/2. And so for a portfolio of 9000 that would be:
(((9000)^2)-9000)/2## [1] 40495500That is 40495500 estimates in order to find teh optimum portfolio.
On the other hand, the Single Index Model is much simpler: Ri=alpha+B*(Rm-Rf)+ei. In this model there would only be as many estimates as follows:
9000+9000+9000+2## [1] 27002That is: 9000 estimates of non market premium (alpha) 9000 estimates of sensitivity coefficient (Beta) 9000 estimates of firm specific variance (ei) 1 estimate of market risk premium 1 estimate of variance of common macroeconomic factor (market variance)
In conclusion the single index model reduces the number of estimates from 40495500 to 27002
- What are the drawbacks of this model compared with the Markowitz model?
-The Markowuz model takes more factors into consideration, so may be more robust and flexible -The Single index model relies on certain assumptions, which may lead to error if violated -A common assumption is that return residuals are uncorrelated. This assumption will be incorrect if the index used omits a significant risk factor.
- Using your results, construct the Optimal Risky Portfolio. Please state ALL your assumptions (for example, what Coefficient of Risk Aversion did you use and why?).
Assuming that the investor is a risk averse, I will use a risk aversion coefficient of 1
ER_GM = 0.002255826
WeightGM2 <- (ER_GM)/(4*VarGM)
WeightGM2## [1] 0.735909WeightAAPL2 <- (1-WeightGM2)
WeightAAPL2## [1] 0.264091Weight for GM: 0.735909, therefore the weight for AAPL is 0.264091
Question 4. Write a (maximum of) one-page report, comparing and contrasting the Optimal Risky Portfolio you obtained in Question 3 with the Optimal Risky Portfolio you would obtain if you used the Full-Covariance (Markowitz’s Efficient Frontier) Approach of Chapter 7 of the textbook.
ERGM = Rf + 0.273*(ERmMarket-Rf)
ERAAPL = Rf + 0.205*(ERmMarket-Rf)
VarGM <- ((SDMarketIndex)^2)*((0.273)^2)+(0.026)^2
VarAAPL <- ((SDMarketIndex)^2)*((0.205)^2)+(0.032)^2
SDGM <- sqrt(VarGM)
SDAAPL <- sqrt(VarAAPL)
correlation ## [1] 0.2958813WeightGM <- (((0.002255826)*(0.00107494))-(0.001693935*0.2958813*0.02768285*0.03278628))/((0.002255826*0.03278628)+((0.001693935)*0.00076634)-((0.002255826+0.001693935)*0.2958813*0.02768285*0.03278628))
WeightAAPL <- (1-WeightGM)
WeightGM## [1] 0.02655041WeightAAPL## [1] 0.9734496Optimal Risky Portfolio: WeightGM: 0.02655041 & WeightAAPL: 0.9734496
The solution in Question 3 uses the individual investor’s degree of risk aversion, A, to calculate the optimal proportions of the complete portfolio. Note that in this case the investor is very risk averse, with a risk aversion coefficient of 4. Of course, the weight for each asset is very different in each approach: taking into consideration the investor’s risk aversion coefficient, 73.6% (26.4 for AAPL) of his portfolio will be dedicated to GM, while only 2.67% (97.34 for AAPL) of the total portfolio will be dedicated to AAPL using the Full-Covariance (Markowitz’s Efficient Frontier) Approach. It is important to note that since the investor is willing to take less risk, they tend towards the less riskier asset which is GM (the asset with the lower variance), which underscores the reverse in the weighting for either stock’s position in each portfolio. On the other hand, as the investor seeks to optimize their portfolio return, they are willing to take greater risk according to the Full-Covariance approach. In this portfolio the investor is compensated for bearing risk and, in theory, the higher the Risk, the higher the Return. Moreover, understanding that both assets have a positive correlation [0.2958813], this is not a very diversified portfolio, as prices are more likely to move in the same direction. The approach in Question 3, helps to hedge against risk, by dedicating less resources to the riskier asset. The risk sensitive portfolio therefore accounts for less risk, at a relatively greater return.