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## group_rows#Define the symbols we want to take
symbols <- c("SPY", "QQQ", "EEM", "IWM", "EFA", "TLT", "IYR", "GLD")
#Create a portfolio variable named portfolioPrices with undefined value
portfolioPrices <- NULL
#Extract data from yahoo finance and feed the data in portfolioPrices
for (symbol in symbols){
portfolioPrices <- cbind(portfolioPrices,
getSymbols.yahoo(symbol, from = '2018-01-01', to = '2022-12-31', auto.assign = FALSE)[, 6])
}
#rename the data according to symbols
colnames(portfolioPrices) <- symbols#Change data from daily to weekly
prices_weekly <- to.weekly(portfolioPrices, indexAt = "last", OHLC = FALSE)
#Change data from daily to monthly
prices_monthly <- to.monthly(portfolioPrices, indexAt = "last", OHLC = FALSE)
#Calculate daily return
asset_returns_day_xts <- na.omit(Return.calculate(portfolioPrices))
#Calculate weekly retun
asset_returns_wk_xts <- na.omit(Return.calculate(prices_weekly))
#Calculate monthly return
asset_returns_mon_xts <- na.omit(Return.calculate(prices_monthly))
head(asset_returns_day_xts)## SPY QQQ EEM IWM EFA
## 2018-01-03 0.006325147 9.716765e-03 0.009581399 0.0010396124 0.0047999683
## 2018-01-04 0.004214771 1.749640e-03 0.004951515 0.0026614054 0.0109597694
## 2018-01-05 0.006664270 1.004288e-02 0.008622577 0.0020715522 0.0055595453
## 2018-01-08 0.001828567 3.890906e-03 0.000000000 0.0015506077 -0.0002764859
## 2018-01-09 0.002263250 6.159949e-05 -0.001628514 -0.0013545769 0.0011058819
## 2018-01-10 -0.001529673 -2.337767e-03 -0.006319895 0.0001290869 -0.0019333032
## TLT IYR GLD
## 2018-01-03 0.0047812648 -0.001612653 -0.0026368517
## 2018-01-04 -0.0001583371 -0.015657899 0.0051273754
## 2018-01-05 -0.0028555972 0.001009584 -0.0010361629
## 2018-01-08 -0.0006363559 0.005675596 -0.0001596106
## 2018-01-09 -0.0133724465 -0.011537595 -0.0046284814
## 2018-01-10 -0.0012102388 -0.012179717 0.0024051631head(asset_returns_wk_xts)## SPY QQQ EEM IWM EFA
## 2018-01-12 0.016457826 0.01587213 0.007734391 0.021771565 0.015894808
## 2018-01-19 0.008959503 0.01124706 0.018784296 0.002781900 0.008571451
## 2018-01-26 0.022003724 0.02759365 0.032513830 0.006305149 0.015108797
## 2018-02-02 -0.038837524 -0.03697389 -0.058179868 -0.036152861 -0.035880529
## 2018-02-09 -0.050644406 -0.05169793 -0.053618740 -0.045894829 -0.054583118
## 2018-02-16 0.044397733 0.05675829 0.067212582 0.044763994 0.041259627
## TLT IYR GLD
## 2018-01-12 -0.009466376 -0.02938564 0.013005641
## 2018-01-19 -0.011725115 0.00649680 -0.004253316
## 2018-01-26 0.004306600 0.01691186 0.013051804
## 2018-02-02 -0.030430841 -0.02932574 -0.013117888
## 2018-02-09 -0.013714915 -0.04159034 -0.012817486
## 2018-02-16 0.006528512 0.02429009 0.025567060head(asset_returns_mon_xts)## SPY QQQ EEM IWM EFA
## 2018-02-28 -0.036360380 -0.012927725 -0.058985033 -0.038436941 -0.04834782
## 2018-03-29 -0.027410870 -0.040788170 0.005414438 0.012175598 -0.00839612
## 2018-04-30 0.005168559 0.005058039 -0.028169006 0.009813424 0.01521240
## 2018-05-31 0.024308862 0.056729381 -0.026214859 0.061635924 -0.01894275
## 2018-06-29 0.005751106 0.011450441 -0.045456886 0.006143667 -0.01584084
## 2018-07-31 0.037046359 0.027964126 0.035310534 0.016490501 0.02852021
## TLT IYR GLD
## 2018-02-28 -0.030414361 -0.066573040 -0.020759906
## 2018-03-29 0.028596251 0.037719050 0.006320008
## 2018-04-30 -0.020881294 0.002252511 -0.009539749
## 2018-05-31 0.020044490 0.033712265 -0.011959211
## 2018-06-29 0.006457591 0.040594503 -0.036149440
## 2018-07-31 -0.014368398 0.008314927 -0.022418912Try to find the GMVP for these eight assets using the derived asset_returns data. —- Note that you have to compute optimal weights and GMVP returns using weekly and monthly asset returns.
#Find the covariance matrix
cov_w <- cov(asset_returns_wk_xts)
#Create the 8x1 matrix with value = 1
one <- rep(1,8)
one_81 <- matrix(one,ncol=1) #8x1
#According to the formula to calculate GMVP, we denote a_w as the numerator and b_w as the denominator
a_w <- inv(cov_w)%*%one_81 # 8x8 x 8x1 = 8x1
b_w <- t(one_81)%*%inv(cov_w)%*%one_81 # 1x8 x 8x8 x 8x1 = 1x1
#Calculate GMVP
mvp_w <- a_w/as.vector(b_w)
#Sum of weight
sum(mvp_w)## [1] 1#Format the table
colnames(mvp_w) <- "Weight"
mvp_w <- data.frame(mvp_w) %>%
kbl(format = "html", caption = "Global Minimum Variance Portfolio (weekly return)") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
mvp_w| Weight | |
|---|---|
| SPY | 0.9920161 |
| QQQ | -0.4594994 |
| EEM | 0.1668419 |
| IWM | -0.0741905 |
| EFA | -0.1213648 |
| TLT | 0.4855622 |
| IYR | -0.2326813 |
| GLD | 0.2433158 |
#Find the covariance matrix
cov_m <- cov(asset_returns_mon_xts)
#According to the formula to calculate GMVP, we denote a_w as the numerator and b_w as the denominator
a_m <- inv(cov_m)%*%one_81 # 8x8 x 8x1 = 8x1
b_m <- t(one_81)%*%inv(cov_m)%*%one_81 # 1x8 x 8x8 x 8x1 = 1x1
#Calculate GMVP
mvp_m <- a_m/as.vector(b_m)
#Sum of weight
sum(mvp_m)## [1] 1#Format the table
colnames(mvp_m) <- "Weight"
mvp_m <- data.frame(mvp_m) %>%
kbl(format = "html", caption = "Global Minimum Variance Portfolio (monthly return)") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
mvp_m| Weight | |
|---|---|
| SPY | 1.1924604 |
| QQQ | -0.7673292 |
| EEM | 0.0189775 |
| IWM | 0.1903891 |
| EFA | -0.1510422 |
| TLT | 0.5703838 |
| IYR | -0.3099006 |
| GLD | 0.2560611 |
Given the portfolio return is specified as 0.045 annual rate. Try to find the optimal weights for the minimum variance portfolio. You should use weekly and monthly asset returns to derive the solutions.
#Calculate the mean return according to symbol
mean_w <-colMeans(asset_returns_wk_xts) #1x8
#We use Lagrangian function to solve the problem. Finally, we have the function A*x=b0, with x is the portfolio weight, A and b0 is denoted as:
A_w <-rbind(2*cov_w,mean_w,t(one_81))
A_w <- cbind(A_w, rbind(t(tail(A_w, 2)), matrix(0, 2, 2))) #8x8
b0_w <- c(rep(0, ncol(asset_returns_wk_xts)), 0.045/52, 1) #1x8
#0.045/52 is the weekly return
#Calculate the portfolio weight
x_w <- solve(A_w,b0_w)
port_w <- as.matrix(x_w[1:8])
#Sum of weight
sum(port_w)## [1] 1#Format the table
colnames(port_w) <- "Weight"
port_w <- data.frame(port_w) %>%
kbl(format = "html", caption = "Portfolio with minimum risk and annual return at 4.5% (weekly return)") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
port_w| Weight | |
|---|---|
| SPY | 1.0758978 |
| QQQ | -0.4015881 |
| EEM | 0.0884025 |
| IWM | -0.0962366 |
| EFA | -0.1668216 |
| TLT | 0.4265153 |
| IYR | -0.2435862 |
| GLD | 0.3174169 |
#Calculate the mean return according to symbol
mean_m <-colMeans(asset_returns_mon_xts) #1x8
#We use Lagrangian function to solve the problem. Finally, we have the function A*x=b0, with x is the portfolio weight, A and b0 is denoted as:
A_m <-rbind(2*cov_m,mean_m,t(one_81))
A_m <- cbind(A_m, rbind(t(tail(A_m, 2)), matrix(0, 2, 2))) #8x8
b0_m <- c(rep(0, ncol(asset_returns_mon_xts)), 0.045/12, 1) #1x8
#0.045/12 is the monthly return
#Calculate the portfolio weight
x_m <- solve(A_m,b0_m)
port_m <- as.matrix(x_m[1:8])
#Sum of weight
sum(port_m)## [1] 1#Format the table
colnames(port_m) <- "Weight"
port_m <- data.frame(port_m) %>%
kbl(format = "html", caption = "Portfolio with minimum risk and annual return at 4.5% (monthly return)") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
port_m| Weight | |
|---|---|
| SPY | 1.3335777 |
| QQQ | -0.7064553 |
| EEM | -0.0449000 |
| IWM | 0.1495389 |
| EFA | -0.2471838 |
| TLT | 0.4885673 |
| IYR | -0.3156009 |
| GLD | 0.3424562 |
Find the tangency portfolio based on Q2. Risk-free rate is assumed to be zero. You should compute the tangency portfolio weights and its returns based on weekly monthly returns.
#The tangency portfolio is the portfolio that has largest sharpe ratio, by using Lagrangian function we have the formula following:
nume_t_w <- inv(cov_w)%*%mean_w # 8x8 x 8x1 = 8x1
deno_t_w <- t(one_81)%*%nume_t_w # 1x8 x 8x1
#Calculate the tangency portfolio
tangency_w <- nume_t_w/as.vector(deno_t_w)
#Sum of weight
sum(tangency_w)## [1] 1#Format the table
colnames(tangency_w) <- "Weight"
tangency_w <- data.frame(tangency_w) %>%
kbl(format = "html", caption = "Tangency Portfolio with weekly data") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
tangency_w| Weight | |
|---|---|
| SPY | 2.8905973 |
| QQQ | 0.8512665 |
| EEM | -1.6085566 |
| IWM | -0.5731809 |
| EFA | -1.1502360 |
| TLT | -0.8509063 |
| IYR | -0.4795046 |
| GLD | 1.9205207 |
#The tangency portfolio is the portfolio that has largest sharpe ratio, by using Lagrangian function we have the formula following:
nume_t_m <- inv(cov_m)%*%mean_m # 8x8 x 8x1 = 8x1
deno_t_m <- t(one_81)%*%nume_t_m # 1x8 x 8x1
#Calculate the tangency portfolio
tangency_m <- nume_t_m/as.vector(deno_t_m)
#Sum of weight
sum(tangency_m)## [1] 1#Format the table
colnames(tangency_m) <- "Weight"
tangency_m <- data.frame(tangency_m) %>%
kbl(format = "html", caption = "Tangency Portfolio with weekly data") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
tangency_m| Weight | |
|---|---|
| SPY | 4.2186546 |
| QQQ | 0.5380816 |
| EEM | -1.3508473 |
| IWM | -0.6856247 |
| EFA | -2.2127539 |
| TLT | -1.1841342 |
| IYR | -0.4321397 |
| GLD | 2.1087637 |
The attached file “berndt.xslx” contains monthly returns of 15 stocks, market index returns (MARKET), and risk-free monthly rate (RKFREE) for 120 months. Compute GMVP (global minimum variance portfolio) weights and its return by using the covariance derived from single index model.
berndt <- read_excel("berndt.xlsx")
ber_port <- berndt %>% select(, -MARKET, -RKFREE)
ber_market <- berndt %>% select(MARKET, RKFREE) %>% mutate(Market = MARKET - RKFREE)
ber_market <- ber_market%>% select(Market)We have the single index model: Ri = a + B*X + ei where Ri is the stock return a is the expected excess return B is the sensitivity of the stock to the market X is the market excess return (Rm-rf) ei is the firm specific surprise Our goal is to find covariance matrix of return. To achieve the goal, we will define the beta (B) and the covarience of ei
#According to the formula in the class, we define Y= BX while Y is the portfolio_return matrix (120x15 - 15 assets in 120 months), B is the matrix that consists of a in the first column and B in the second column:
one_120 <- rep(1,120)
X <- as.matrix(cbind(one_120, ber_market)) # 120x2
Y <- as.matrix(ber_port) # 120x15
#We calculate the b_hat:
b_hat = solve(t(X)%*%X)%*%t(X) %*% Y #2x15 : first row- alpha, second row - beta
#The e_hat is equal the difference between actual return and return derived from the formula:
E_hat = Y - X %*% b_hat #120x15
head(E_hat,6) ## CITCRP CONED CONTIL DATGEN DEC DELTA
## [1,] -0.08906182 -0.0924325479 -0.08644938 -0.03294113 -0.07166573 -0.01200381
## [2,] -0.02947098 -0.0213220221 0.03962148 -0.10233449 -0.08098502 -0.04367536
## [3,] 0.02222799 0.0001459493 -0.02322229 0.01692842 -0.04144488 0.04005781
## [4,] 0.08137923 -0.0280423719 0.14407347 0.11922277 0.10229784 0.11357564
## [5,] -0.04312626 -0.0373788404 0.02232575 -0.01165793 -0.02788962 -0.06925976
## [6,] -0.00126376 0.0159743906 -0.05395791 -0.02491578 -0.03617703 0.01394154
## GENMIL GERBER IBM MOBIL PANAM
## [1,] -0.100413553 -0.028965238 -0.01280576 -0.021675972 0.06312884
## [2,] 0.001973487 0.144900665 -0.05166925 -0.031704502 -0.07498803
## [3,] -0.049582534 -0.075756917 -0.08963000 0.006102306 0.15303258
## [4,] 0.015865982 -0.044052752 0.09732727 0.024616945 0.04828272
## [5,] 0.031860393 -0.004401685 -0.05238371 -0.066068798 0.03852208
## [6,] -0.007140637 0.014969963 -0.01116196 -0.055338490 0.01944396
## PSNH TANDY TEXACO WEYER
## [1,] 0.008151819 -0.04037246 -0.03121181 -0.0790751380
## [2,] -0.020345243 -0.02681935 -0.02066688 -0.1429243732
## [3,] 0.022349580 0.05968252 -0.01983395 0.0436777086
## [4,] -0.014444627 -0.02327918 -0.04296475 0.0927777034
## [5,] 0.011764204 0.09376762 -0.07426696 -0.0853085904
## [6,] 0.037351739 -0.03333676 -0.03363874 -0.0002054953# Excluding constant term and keep only beta in b_hat
b_hat = as.matrix(b_hat[-1,])
#Calculate the covariance of ei and put them in diagonal matrix
diagD_hat = diag(t(E_hat) %*% E_hat)/(120-2)
diag(diagD_hat) ## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.004526617 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [2,] 0.000000000 0.002511069 0.00000000 0.00000000 0.000000000 0.000000000
## [3,] 0.000000000 0.000000000 0.02039461 0.00000000 0.000000000 0.000000000
## [4,] 0.000000000 0.000000000 0.00000000 0.01140139 0.000000000 0.000000000
## [5,] 0.000000000 0.000000000 0.00000000 0.00000000 0.006537662 0.000000000
## [6,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.008161795
## [7,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [8,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [9,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [10,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [11,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [12,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [13,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [14,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [15,] 0.000000000 0.000000000 0.00000000 0.00000000 0.000000000 0.000000000
## [,7] [,8] [,9] [,10] [,11] [,12]
## [1,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [2,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [3,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [4,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [5,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [6,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [7,] 0.00393017 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [8,] 0.00000000 0.005930261 0.000000000 0.000000000 0.00000000 0.00000000
## [9,] 0.00000000 0.000000000 0.002540858 0.000000000 0.00000000 0.00000000
## [10,] 0.00000000 0.000000000 0.000000000 0.004107716 0.00000000 0.00000000
## [11,] 0.00000000 0.000000000 0.000000000 0.000000000 0.01498815 0.00000000
## [12,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.01187757
## [13,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [14,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [15,] 0.00000000 0.000000000 0.000000000 0.000000000 0.00000000 0.00000000
## [,13] [,14] [,15]
## [1,] 0.00000000 0.00000000 0.000000000
## [2,] 0.00000000 0.00000000 0.000000000
## [3,] 0.00000000 0.00000000 0.000000000
## [4,] 0.00000000 0.00000000 0.000000000
## [5,] 0.00000000 0.00000000 0.000000000
## [6,] 0.00000000 0.00000000 0.000000000
## [7,] 0.00000000 0.00000000 0.000000000
## [8,] 0.00000000 0.00000000 0.000000000
## [9,] 0.00000000 0.00000000 0.000000000
## [10,] 0.00000000 0.00000000 0.000000000
## [11,] 0.00000000 0.00000000 0.000000000
## [12,] 0.00000000 0.00000000 0.000000000
## [13,] 0.01121942 0.00000000 0.000000000
## [14,] 0.00000000 0.00464578 0.000000000
## [15,] 0.00000000 0.00000000 0.004133323# Covariance matrix by single factor model
cov_sfm <- as.numeric(var(ber_market))*b_hat%*%t(b_hat) + diag(diagD_hat) #15x15#Create the 15x1 matrix with value = 1
one_15 <- rep(1,15)
one_15_1 <- matrix(one_15,ncol=1)
#According to the formula to calculate GMVP, we denote a_sfm as the numerator and b_sfm as the denominator
a_sfm <- inv(cov_sfm)%*%one_15_1 # 15x15 x 15x1 = 15x1
b_sfm <- t(one_15_1)%*%inv(cov_sfm)%*%one_15_1 # 1x1 5x 15x15 x 15x1 = 1x1
#Calculate the GMVP
mvp_sfm <- a_sfm/as.vector(b_sfm)
#Sum of weight
sum(mvp_sfm)## [1] 1#Format the table
colnames(mvp_sfm) <- "Weight"
mvp_sfm <- data.frame(mvp_sfm) %>%
kbl(format = "html", caption = "GMVP from single index model") %>%
row_spec(row = 0, bold = TRUE, color = "black", background = "#F9EBEA") %>%
kable_styling(bootstrap_options = "striped", full_width = F, position = "center")
mvp_sfm| Weight | |
|---|---|
| CITCRP | 0.0443133 |
| CONED | 0.3755073 |
| CONTIL | 0.0057706 |
| DATGEN | -0.0236873 |
| DEC | -0.0050185 |
| DELTA | 0.0526752 |
| GENMIL | 0.1816517 |
| GERBER | 0.0428599 |
| IBM | 0.1859635 |
| MOBIL | 0.0338142 |
| PANAM | 0.0075571 |
| PSNH | 0.0662851 |
| TANDY | -0.0262830 |
| TEXACO | 0.0581522 |
| WEYER | 0.0004386 |