library(quantmod)
library(xts)
library(PerformanceAnalytics)
library(quadprog)
library(knitr)
library(ggplot2)

1 Part I: Computer Questions (40%)

1.1 Question 1: Download ETF Daily Data

Download ETF daily data from Yahoo Finance for SPY, QQQ, EEM, IWM, EFA, TLT, IYR, GLD from 2010 to 2025. We use the quantmod package and adjusted prices.

tickers <- c("SPY", "QQQ", "EEM", "IWM", "EFA", "TLT", "IYR", "GLD")

# Download data from Yahoo Finance
getSymbols(tickers, src = "yahoo", from = "2010-01-01", to = "2025-12-31",
           auto.assign = TRUE)

## [1] "SPY" "QQQ" "EEM" "IWM" "EFA" "TLT" "IYR" "GLD"

# Extract adjusted prices into a single xts object
prices <- do.call(merge, lapply(tickers, function(t) Ad(get(t))))
colnames(prices) <- tickers

head(prices)

##                 SPY      QQQ      EEM      IWM      EFA      TLT      IYR
## 2010-01-04 84.79639 40.29079 30.35151 51.36656 35.12843 56.13519 26.76812
## 2010-01-05 85.02081 40.29079 30.57181 51.18994 35.15940 56.49770 26.83238
## 2010-01-06 85.08071 40.04776 30.63577 51.14177 35.30803 55.74139 26.82070
## 2010-01-07 85.43985 40.07380 30.45811 51.51910 35.17178 55.83516 27.06027
## 2010-01-08 85.72419 40.40363 30.69973 51.80011 35.45043 55.81016 26.87913
## 2010-01-11 85.84389 40.23871 30.63577 51.59137 35.74147 55.50389 27.00769
##               GLD
## 2010-01-04 109.80
## 2010-01-05 109.70
## 2010-01-06 111.51
## 2010-01-07 110.82
## 2010-01-08 111.37
## 2010-01-11 112.85

tail(prices)

##                 SPY      QQQ   EEM      IWM   EFA      TLT      IYR    GLD
## 2025-12-22 682.9648 618.4302 54.01 253.1297 95.70 86.39350 93.34799 408.23
## 2025-12-23 686.0863 621.3265 54.31 251.6324 96.29 86.53194 93.27818 413.64
## 2025-12-24 688.4997 623.1443 54.42 252.2613 96.41 87.05608 93.95628 411.93
## 2025-12-26 688.4299 623.1043 54.80 250.9736 96.57 86.76929 94.05600 416.74
## 2025-12-29 685.9766 620.0881 54.66 249.4363 96.28 87.09564 94.23550 398.60
## 2025-12-30 685.1389 618.6499 54.88 247.5896 96.44 86.88797 94.44491 398.89

cat("Dimensions:", dim(prices), "\n")

## Dimensions: 4023 8

cat("Date range:", as.character(index(prices)[1]), "to",
    as.character(index(prices)[nrow(prices)]), "\n")

## Date range: 2010-01-04 to 2025-12-30

# Normalize prices to 100 for comparison
prices_norm <- prices / rep(as.numeric(prices[1, ]), each = nrow(prices)) * 100

plot.zoo(prices_norm, plot.type = "single", col = 1:8, lwd = 1.5,
         main = "Normalized ETF Prices (Base = 100)",
         xlab = "Date", ylab = "Normalized Price")
legend("topleft", legend = tickers, col = 1:8, lty = 1, lwd = 1.5,
       cex = 0.8, ncol = 2)

1.2 Question 2: Calculate Monthly Returns (Discrete)

We convert daily adjusted prices to monthly and compute discrete returns: \[R_t = \frac{P_t - P_{t-1}}{P_{t-1}}\]

# Convert to monthly prices (end of month)
monthly_prices <- to.monthly(prices, indexAt = "lastof", OHLC = FALSE)

# Calculate discrete monthly returns
monthly_returns <- Return.calculate(monthly_prices, method = "discrete")
monthly_returns <- monthly_returns[-1, ]  # Remove first NA row

head(monthly_returns)

##                    SPY         QQQ          EEM         IWM          EFA
## 2010-02-28  0.03119489  0.04603816  0.017763916  0.04475129  0.002667856
## 2010-03-31  0.06087928  0.07710931  0.081108991  0.08230640  0.063853720
## 2010-04-30  0.01547042  0.02242555 -0.001662258  0.05678563 -0.028045231
## 2010-05-31 -0.07945481 -0.07392383 -0.093935581 -0.07536719 -0.111928362
## 2010-06-30 -0.05174074 -0.05975669 -0.013986639 -0.07743339 -0.020619098
## 2010-07-31  0.06830056  0.07258302  0.109325154  0.06730871  0.116103833
##                     TLT         IYR          GLD
## 2010-02-28 -0.003424700  0.05457065  0.032748219
## 2010-03-31 -0.020572438  0.09748516 -0.004386396
## 2010-04-30  0.033217261  0.06388069  0.058834363
## 2010-05-31  0.051084654 -0.05683524  0.030513147
## 2010-06-30  0.057978096 -0.04670136  0.023553189
## 2010-07-31 -0.009463754  0.09404862 -0.050871157

tail(monthly_returns)

##                    SPY          QQQ          EEM          IWM          EFA
## 2025-07-31 0.023031543  0.024236743  0.006633454 0.0166829854 -0.020919628
## 2025-08-31 0.020519538  0.009539756  0.026770994 0.0719267492  0.045246876
## 2025-09-30 0.035620413  0.053762268  0.070998795 0.0317910868  0.020660248
## 2025-10-31 0.023837414  0.047803919  0.035580494 0.0176475324  0.011995297
## 2025-11-30 0.001949944 -0.015610490 -0.017721479 0.0102343041  0.007408238
## 2025-12-31 0.008267742  0.001579450  0.024767431 0.0004492333  0.031490000
##                      TLT          IYR          GLD
## 2025-07-31 -0.0113965353  0.001160569 -0.006134551
## 2025-08-31  0.0001267691  0.029089429  0.049874625
## 2025-09-30  0.0359099467  0.000554634  0.117584158
## 2025-10-31  0.0138108172 -0.024927912  0.035586671
## 2025-11-30  0.0027233086  0.023663708  0.053678176
## 2025-12-31 -0.0187683309 -0.013658495  0.028385092

cat("Monthly returns dimensions:", dim(monthly_returns), "\n")

## Monthly returns dimensions: 191 8

cat("Date range:", as.character(index(monthly_returns)[1]), "to",
    as.character(index(monthly_returns)[nrow(monthly_returns)]), "\n")

## Date range: 2010-02-28 to 2025-12-31

# Summary statistics
stats <- data.frame(
  Mean = round(colMeans(monthly_returns) * 100, 3),
  SD = round(apply(monthly_returns, 2, sd) * 100, 3),
  Min = round(apply(monthly_returns, 2, min) * 100, 3),
  Max = round(apply(monthly_returns, 2, max) * 100, 3)
)
kable(stats, caption = "Monthly Return Statistics (%)")

Monthly Return Statistics (%)
	Mean	SD	Min	Max
SPY	1.214	4.134	-12.487	12.698
QQQ	1.606	4.967	-13.596	14.974
EEM	0.500	5.152	-17.895	16.268
IWM	1.019	5.644	-21.477	18.244
EFA	0.671	4.477	-14.107	14.269
TLT	0.291	3.941	-9.424	13.206
IYR	0.807	4.817	-19.632	13.190
GLD	0.799	4.552	-11.062	12.275

1.3 Question 3: Download Fama-French 3 Factors

We download the Fama-French 3-factor data (Mkt-RF, SMB, HML) from Ken French’s website and convert from percentage to decimal.

# Download FF3 factors from Ken French's website
ff3_url <- "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Research_Data_Factors_CSV.zip"
temp <- tempfile()
download.file(ff3_url, temp, mode = "wb")
unzip(temp, exdir = tempdir())

# Read the CSV file
ff3_file <- list.files(tempdir(), pattern = "F-F_Research_Data_Factors", full.names = TRUE)
ff3_file <- ff3_file[grepl("\\.CSV$", ff3_file, ignore.case = TRUE)]

# Read the file, skipping header rows
ff3_raw <- read.csv(ff3_file[1], skip = 3, header = FALSE, stringsAsFactors = FALSE)
colnames(ff3_raw) <- c("Date", "Mkt_RF", "SMB", "HML", "RF")

# Keep only monthly data (6-digit dates: YYYYMM)
ff3_raw <- ff3_raw[nchar(trimws(ff3_raw$Date)) == 6, ]
ff3_raw <- ff3_raw[!is.na(as.numeric(ff3_raw$Mkt_RF)), ]

# Convert to numeric
ff3_raw$Mkt_RF <- as.numeric(ff3_raw$Mkt_RF)
ff3_raw$SMB <- as.numeric(ff3_raw$SMB)
ff3_raw$HML <- as.numeric(ff3_raw$HML)
ff3_raw$RF <- as.numeric(ff3_raw$RF)

# Convert from percentage to decimal
ff3_raw$Mkt_RF <- ff3_raw$Mkt_RF / 100
ff3_raw$SMB <- ff3_raw$SMB / 100
ff3_raw$HML <- ff3_raw$HML / 100
ff3_raw$RF <- ff3_raw$RF / 100

# Create date column (end of month)
ff3_raw$DateParsed <- as.Date(paste0(ff3_raw$Date, "01"), format = "%Y%m%d")
ff3_raw$DateParsed <- as.Date(as.yearmon(ff3_raw$DateParsed), frac = 1)

# Convert to xts
ff3 <- xts(ff3_raw[, c("Mkt_RF", "SMB", "HML", "RF")],
           order.by = ff3_raw$DateParsed)

# Filter to our period
ff3 <- ff3["2010/2025"]

head(ff3)

##             Mkt_RF     SMB     HML    RF
## 2010-01-31 -0.0335  0.0043  0.0033 0e+00
## 2010-02-28  0.0339  0.0118  0.0318 0e+00
## 2010-03-31  0.0630  0.0146  0.0219 1e-04
## 2010-04-30  0.0199  0.0484  0.0296 1e-04
## 2010-05-31 -0.0790  0.0013 -0.0248 1e-04
## 2010-06-30 -0.0556 -0.0179 -0.0473 1e-04

tail(ff3)

##             Mkt_RF     SMB     HML     RF
## 2025-07-31  0.0198  0.0027 -0.0127 0.0034
## 2025-08-31  0.0184  0.0387  0.0442 0.0038
## 2025-09-30  0.0339 -0.0185 -0.0105 0.0033
## 2025-10-31  0.0196 -0.0055 -0.0310 0.0037
## 2025-11-30 -0.0013  0.0038  0.0376 0.0030
## 2025-12-31 -0.0036 -0.0106  0.0242 0.0034

cat("FF3 dimensions:", dim(ff3), "\n")

## FF3 dimensions: 192 4

1.4 Question 4: Merge Monthly Returns and FF3 Data

library(zoo)

# Convert both to zoo with yearmon index for proper merging
ret_zoo <- zoo(coredata(monthly_returns), order.by = as.yearmon(index(monthly_returns)))
ff3_zoo <- zoo(coredata(ff3), order.by = as.yearmon(index(ff3)))

# Merge by yearmon
merged_zoo <- merge(ret_zoo, ff3_zoo, all = FALSE)

# Convert back to xts
merged_data <- as.xts(merged_zoo)
colnames(merged_data) <- c(tickers, "Mkt_RF", "SMB", "HML", "RF")

head(merged_data)

##                  SPY         QQQ          EEM         IWM          EFA
## Feb 2010  0.03119489  0.04603816  0.017763916  0.04475129  0.002667856
## Mar 2010  0.06087928  0.07710931  0.081108991  0.08230640  0.063853720
## Apr 2010  0.01547042  0.02242555 -0.001662258  0.05678563 -0.028045231
## May 2010 -0.07945481 -0.07392383 -0.093935581 -0.07536719 -0.111928362
## Jun 2010 -0.05174074 -0.05975669 -0.013986639 -0.07743339 -0.020619098
## Jul 2010  0.06830056  0.07258302  0.109325154  0.06730871  0.116103833
##                   TLT         IYR          GLD  Mkt_RF     SMB     HML    RF
## Feb 2010 -0.003424700  0.05457065  0.032748219  0.0339  0.0118  0.0318 0e+00
## Mar 2010 -0.020572438  0.09748516 -0.004386396  0.0630  0.0146  0.0219 1e-04
## Apr 2010  0.033217261  0.06388069  0.058834363  0.0199  0.0484  0.0296 1e-04
## May 2010  0.051084654 -0.05683524  0.030513147 -0.0790  0.0013 -0.0248 1e-04
## Jun 2010  0.057978096 -0.04670136  0.023553189 -0.0556 -0.0179 -0.0473 1e-04
## Jul 2010 -0.009463754  0.09404862 -0.050871157  0.0692  0.0022 -0.0050 1e-04

tail(merged_data)

##                  SPY          QQQ          EEM          IWM          EFA
## Jul 2025 0.023031543  0.024236743  0.006633454 0.0166829854 -0.020919628
## Aug 2025 0.020519538  0.009539756  0.026770994 0.0719267492  0.045246876
## Sep 2025 0.035620413  0.053762268  0.070998795 0.0317910868  0.020660248
## Oct 2025 0.023837414  0.047803919  0.035580494 0.0176475324  0.011995297
## Nov 2025 0.001949944 -0.015610490 -0.017721479 0.0102343041  0.007408238
## Dec 2025 0.008267742  0.001579450  0.024767431 0.0004492333  0.031490000
##                    TLT          IYR          GLD  Mkt_RF     SMB     HML     RF
## Jul 2025 -0.0113965353  0.001160569 -0.006134551  0.0198  0.0027 -0.0127 0.0034
## Aug 2025  0.0001267691  0.029089429  0.049874625  0.0184  0.0387  0.0442 0.0038
## Sep 2025  0.0359099467  0.000554634  0.117584158  0.0339 -0.0185 -0.0105 0.0033
## Oct 2025  0.0138108172 -0.024927912  0.035586671  0.0196 -0.0055 -0.0310 0.0037
## Nov 2025  0.0027233086  0.023663708  0.053678176 -0.0013  0.0038  0.0376 0.0030
## Dec 2025 -0.0187683309 -0.013658495  0.028385092 -0.0036 -0.0106  0.0242 0.0034

cat("Merged data dimensions:", dim(merged_data), "\n")

## Merged data dimensions: 191 12

cat("Date range:", as.character(index(merged_data)[1]), "to",
    as.character(index(merged_data)[nrow(merged_data)]), "\n")

## Date range: Feb 2010 to Dec 2025

1.5 Question 5: CAPM-based MVP (Covariance Matrix from 2020/03 - 2025/02)

Using the CAPM model, we estimate the covariance matrix for the 8-asset portfolio from past 60-month returns (2020/03 - 2025/02), then compute the Minimum Variance Portfolio (MVP).

CAPM Model: $R_i - R_f = \alpha_i + \beta_i (R_m - R_f) + \epsilon_i$

The CAPM-based covariance matrix: $\Sigma = \beta \beta' \sigma_m^2 + D$ where $D$ is the diagonal matrix of residual variances.

# Define the 60-month estimation window
est_start <- "2020-03"
est_end <- "2025-02"

# Subset merged data for estimation period
est_data <- merged_data[paste0(est_start, "/", est_end)]
cat("Estimation window observations:", nrow(est_data), "\n")

## Estimation window observations: 60

# Extract ETF excess returns and market factor
rf <- est_data[, "RF"]
mkt_rf <- est_data[, "Mkt_RF"]

# Calculate excess returns for each ETF
excess_returns <- est_data[, tickers] - as.numeric(rf)

# CAPM regression for each ETF
capm_betas <- numeric(length(tickers))
capm_alphas <- numeric(length(tickers))
capm_residual_var <- numeric(length(tickers))
names(capm_betas) <- names(capm_alphas) <- names(capm_residual_var) <- tickers

for (i in seq_along(tickers)) {
  fit <- lm(as.numeric(excess_returns[, i]) ~ as.numeric(mkt_rf))
  capm_alphas[i] <- coef(fit)[1]
  capm_betas[i] <- coef(fit)[2]
  capm_residual_var[i] <- var(residuals(fit))
}

capm_results <- data.frame(
  Alpha = round(capm_alphas * 100, 4),
  Beta = round(capm_betas, 4),
  Residual_Var = round(capm_residual_var * 10000, 4)
)
kable(capm_results, caption = "CAPM Regression Results")

CAPM Regression Results
	Alpha	Beta	Residual_Var
SPY	0.0623	0.9552	0.3287
QQQ	0.2636	1.0634	5.8119
EEM	-0.6243	0.6963	13.3578
IWM	-0.6455	1.1858	9.9075
EFA	-0.3805	0.8243	6.8931
TLT	-1.1563	0.3310	16.2335
IYR	-0.8011	1.0036	10.2892
GLD	0.6293	0.1746	16.4671

# CAPM-based covariance matrix
sigma2_mkt <- as.numeric(var(mkt_rf))
beta_vec <- matrix(capm_betas, ncol = 1)
D_capm <- diag(capm_residual_var)
Sigma_capm <- beta_vec %*% t(beta_vec) * sigma2_mkt + D_capm
colnames(Sigma_capm) <- rownames(Sigma_capm) <- tickers

cat("\nCAPM-based Covariance Matrix:\n")

## 
## CAPM-based Covariance Matrix:

round(Sigma_capm * 10000, 4)

##         SPY     QQQ     EEM     IWM     EFA     TLT     IYR     GLD
## SPY 26.2336 28.8373 18.8819 32.1577 22.3550  8.9775 27.2157  4.7343
## QQQ 28.8373 37.9135 21.0193 35.7979 24.8856  9.9938 30.2964  5.2702
## EEM 18.8819 21.0193 27.1207 23.4395 16.2945  6.5437 19.8373  3.4508
## IWM 32.1577 35.7979 23.4395 49.8272 27.7510 11.1445 33.7848  5.8770
## EFA 22.3550 24.8856 16.2945 27.7510 26.1848  7.7473 23.4862  4.0855
## TLT  8.9775  9.9938  6.5437 11.1445  7.7473 19.3447  9.4318  1.6407
## IYR 27.2157 30.2964 19.8373 33.7848 23.4862  9.4318 38.8819  4.9738
## GLD  4.7343  5.2702  3.4508  5.8770  4.0855  1.6407  4.9738 17.3323

# Compute MVP weights using quadratic programming
# MVP: minimize w'Σw subject to sum(w) = 1
# solve.QP: min(-d'b + 1/2 b'Db) subject to A'b >= b0

n_assets <- length(tickers)
Dmat <- 2 * Sigma_capm
dvec <- rep(0, n_assets)

# --- Unconstrained MVP (short selling allowed) ---
Amat_unc <- matrix(1, n_assets, 1)
bvec_unc <- 1
mvp_capm_unc <- solve.QP(Dmat, dvec, Amat_unc, bvec_unc, meq = 1)
w_capm_unc <- mvp_capm_unc$solution
names(w_capm_unc) <- tickers

cat("Unconstrained CAPM MVP Weights (short selling allowed):\n")

## Unconstrained CAPM MVP Weights (short selling allowed):

print(round(w_capm_unc, 4))

##     SPY     QQQ     EEM     IWM     EFA     TLT     IYR     GLD 
##  0.2744 -0.1429  0.1719 -0.1891  0.1748  0.3330 -0.0312  0.4092

# --- Constrained MVP (long-only, no short selling) ---
Amat <- cbind(rep(1, n_assets), diag(n_assets))
bvec <- c(1, rep(0, n_assets))
meq <- 1

mvp_capm <- solve.QP(Dmat, dvec, Amat, bvec, meq)
w_capm <- mvp_capm$solution
names(w_capm) <- tickers

cat("\nConstrained CAPM MVP Weights (long-only):\n")

## 
## Constrained CAPM MVP Weights (long-only):

mvp_weights_capm <- data.frame(
  ETF = tickers,
  Weight = round(w_capm, 6),
  Weight_Pct = paste0(round(w_capm * 100, 2), "%")
)
kable(mvp_weights_capm, caption = "CAPM-based MVP Optimal Weights")

CAPM-based MVP Optimal Weights
	ETF	Weight	Weight_Pct
SPY	SPY	0.000000	0%
QQQ	QQQ	0.000000	0%
EEM	EEM	0.140092	14.01%
IWM	IWM	0.000000	0%
EFA	EFA	0.083844	8.38%
TLT	TLT	0.342481	34.25%
IYR	IYR	0.000000	0%
GLD	GLD	0.433583	43.36%

# MVP expected return and risk
mvp_ret_capm <- sum(w_capm * colMeans(est_data[, tickers]))
mvp_sd_capm <- sqrt(t(w_capm) %*% Sigma_capm %*% w_capm)

cat("CAPM MVP Monthly Expected Return:", round(mvp_ret_capm * 100, 4), "%\n")

## CAPM MVP Monthly Expected Return: 0.3927 %

cat("CAPM MVP Monthly Std Dev:", round(mvp_sd_capm * 100, 4), "%\n")

## CAPM MVP Monthly Std Dev: 2.9838 %

1.6 Question 6: FF 3-Factor Model MVP (2020/03 - 2025/02)

FF3 Model: $R_i - R_f = \alpha_i + \beta_{i1}(Mkt-RF) + \beta_{i2}(SMB) + \beta_{i3}(HML) + \epsilon_i$

The factor model covariance matrix: $\Sigma = B \Sigma_f B' + D$

# FF3 regression for each ETF
ff3_betas <- matrix(0, nrow = length(tickers), ncol = 3)
ff3_alphas <- numeric(length(tickers))
ff3_residual_var <- numeric(length(tickers))
rownames(ff3_betas) <- tickers
colnames(ff3_betas) <- c("Mkt_RF", "SMB", "HML")

smb <- est_data[, "SMB"]
hml <- est_data[, "HML"]

for (i in seq_along(tickers)) {
  fit <- lm(as.numeric(excess_returns[, i]) ~
              as.numeric(mkt_rf) + as.numeric(smb) + as.numeric(hml))
  ff3_alphas[i] <- coef(fit)[1]
  ff3_betas[i, ] <- coef(fit)[2:4]
  ff3_residual_var[i] <- var(residuals(fit))
}

ff3_results <- data.frame(
  Alpha = round(ff3_alphas * 100, 4),
  Beta_Mkt = round(ff3_betas[, 1], 4),
  Beta_SMB = round(ff3_betas[, 2], 4),
  Beta_HML = round(ff3_betas[, 3], 4),
  Residual_Var = round(ff3_residual_var * 10000, 4)
)
kable(ff3_results, caption = "FF 3-Factor Regression Results")

FF 3-Factor Regression Results
	Alpha	Beta_Mkt	Beta_SMB	Beta_HML	Residual_Var
SPY	-0.0106	0.9853	-0.1487	0.0194	0.1213
QQQ	0.3220	1.0813	-0.0890	-0.3994	2.2366
EEM	-0.6228	0.6794	0.0834	0.1476	12.8013
IWM	-0.3042	1.0058	0.8895	0.2660	0.6215
EFA	-0.4871	0.8477	-0.1152	0.2169	5.8029
TLT	-1.1213	0.3443	-0.0658	-0.2622	14.6810
IYR	-0.8329	0.9953	0.0409	0.2032	9.3686
GLD	0.4817	0.2420	-0.3330	-0.0197	15.4200

# Factor covariance matrix
factors <- cbind(as.numeric(mkt_rf), as.numeric(smb), as.numeric(hml))
Sigma_f <- cov(factors)

# FF3-based covariance matrix: B * Sigma_f * B' + D
B <- ff3_betas
D_ff3 <- diag(ff3_residual_var)
Sigma_ff3 <- B %*% Sigma_f %*% t(B) + D_ff3
colnames(Sigma_ff3) <- rownames(Sigma_ff3) <- tickers

cat("\nFF3-based Covariance Matrix:\n")

## 
## FF3-based Covariance Matrix:

round(Sigma_ff3 * 10000, 4)

##         SPY     QQQ     EEM     IWM     EFA     TLT     IYR     GLD
## SPY 26.2336 28.8464 18.8100 31.0236 22.5723  8.9935 27.2170  5.1812
## QQQ 28.8464 37.9135 19.6376 32.4269 23.1390 12.3494 28.4823  5.8384
## EEM 18.8100 19.6376 27.1207 25.1136 16.8963  5.6300 20.5364  3.0842
## IWM 31.0236 32.4269 25.1136 49.8272 28.1948  8.8694 35.4628  2.9239
## EFA 22.5723 23.1390 16.8963 28.1948 26.1848  6.6072 24.3788  4.2840
## TLT  8.9935 12.3494  5.6300  8.8694  6.6072 19.3447  8.2368  2.0365
## IYR 27.2170 28.4823 20.5364 35.4628 24.3788  8.2368 38.8819  4.6983
## GLD  5.1812  5.8384  3.0842  2.9239  4.2840  2.0365  4.6983 17.3323

# Compute MVP weights using FF3 covariance matrix
Dmat_ff3 <- 2 * Sigma_ff3

# --- Unconstrained FF3 MVP ---
mvp_ff3_unc <- solve.QP(Dmat_ff3, dvec, Amat_unc, bvec_unc, meq = 1)
w_ff3_unc <- mvp_ff3_unc$solution
names(w_ff3_unc) <- tickers

cat("Unconstrained FF3 MVP Weights (short selling allowed):\n")

## Unconstrained FF3 MVP Weights (short selling allowed):

print(round(w_ff3_unc, 4))

##     SPY     QQQ     EEM     IWM     EFA     TLT     IYR     GLD 
##  0.1399 -0.2280  0.1988 -0.0563  0.1810  0.3777 -0.0138  0.4007

# --- Constrained FF3 MVP ---
mvp_ff3 <- solve.QP(Dmat_ff3, dvec, Amat, bvec, meq)
w_ff3 <- mvp_ff3$solution
names(w_ff3) <- tickers

cat("\nConstrained FF3 MVP Weights (long-only):\n")

## 
## Constrained FF3 MVP Weights (long-only):

mvp_weights_ff3 <- data.frame(
  ETF = tickers,
  Weight = round(w_ff3, 6),
  Weight_Pct = paste0(round(w_ff3 * 100, 2), "%")
)
kable(mvp_weights_ff3, caption = "FF3-based MVP Optimal Weights")

FF3-based MVP Optimal Weights
	ETF	Weight	Weight_Pct
SPY	SPY	0.000000	0%
QQQ	QQQ	0.000000	0%
EEM	EEM	0.156532	15.65%
IWM	IWM	0.000000	0%
EFA	EFA	0.082084	8.21%
TLT	TLT	0.339123	33.91%
IYR	IYR	0.000000	0%
GLD	GLD	0.422261	42.23%

# MVP expected return and risk
mvp_ret_ff3 <- sum(w_ff3 * colMeans(est_data[, tickers]))
mvp_sd_ff3 <- sqrt(t(w_ff3) %*% Sigma_ff3 %*% w_ff3)

cat("FF3 MVP Monthly Expected Return:", round(mvp_ret_ff3 * 100, 4), "%\n")

## FF3 MVP Monthly Expected Return: 0.3883 %

cat("FF3 MVP Monthly Std Dev:", round(mvp_sd_ff3 * 100, 4), "%\n")

## FF3 MVP Monthly Std Dev: 2.9739 %

1.7 Question 7: Realized Portfolio Returns in March 2025

We invest in the 8-asset portfolio in 2025/03 using optimal MVP weights from Q5 (CAPM) and Q6 (FF3).

# Get March 2025 actual returns
march_2025 <- merged_data["2025-03", tickers]

if (nrow(march_2025) > 0) {
  march_ret <- as.numeric(march_2025[1, ])
  names(march_ret) <- tickers

  # Realized portfolio return using CAPM weights
  realized_capm_march <- sum(w_capm * march_ret)

  # Realized portfolio return using FF3 weights
  realized_ff3_march <- sum(w_ff3 * march_ret)

  cat("Individual ETF Returns in March 2025:\n")
  march_df <- data.frame(
    ETF = tickers,
    Return_Pct = round(march_ret * 100, 4),
    CAPM_Weight = round(w_capm, 4),
    FF3_Weight = round(w_ff3, 4),
    CAPM_Contrib = round(w_capm * march_ret * 100, 4),
    FF3_Contrib = round(w_ff3 * march_ret * 100, 4)
  )
  kable(march_df, caption = "March 2025 Realized Returns and Contributions")

  cat("\n\nRealized MVP Portfolio Return (March 2025):\n")
  cat("  CAPM-based MVP:", round(realized_capm_march * 100, 4), "%\n")
  cat("  FF3-based MVP:", round(realized_ff3_march * 100, 4), "%\n")
} else {
  cat("March 2025 data not yet available in the dataset.\n")
  cat("Please ensure data download includes March 2025.\n")
}

## Individual ETF Returns in March 2025:
## 
## 
## Realized MVP Portfolio Return (March 2025):
##   CAPM-based MVP: 3.8576 %
##   FF3-based MVP: 3.773 %

1.8 Question 8: Realized MVP Portfolio Return in April 2025

We shift the 60-month window by one month (2020/04 - 2025/03), re-estimate the MVP, and compute April 2025 realized return.

# New 60-month estimation window
est_start2 <- "2020-04"
est_end2 <- "2025-03"

est_data2 <- merged_data[paste0(est_start2, "/", est_end2)]
cat("New estimation window observations:", nrow(est_data2), "\n")

## New estimation window observations: 60

if (nrow(est_data2) >= 55) {
  # Extract factors
  rf2 <- est_data2[, "RF"]
  mkt_rf2 <- est_data2[, "Mkt_RF"]
  smb2 <- est_data2[, "SMB"]
  hml2 <- est_data2[, "HML"]
  excess_returns2 <- est_data2[, tickers] - as.numeric(rf2)

  # --- CAPM for new window ---
  capm_betas2 <- numeric(length(tickers))
  capm_resvar2 <- numeric(length(tickers))
  for (i in seq_along(tickers)) {
    fit <- lm(as.numeric(excess_returns2[, i]) ~ as.numeric(mkt_rf2))
    capm_betas2[i] <- coef(fit)[2]
    capm_resvar2[i] <- var(residuals(fit))
  }
  sigma2_mkt2 <- as.numeric(var(mkt_rf2))
  Sigma_capm2 <- matrix(capm_betas2, ncol=1) %*% matrix(capm_betas2, nrow=1) * sigma2_mkt2 + diag(capm_resvar2)
  mvp_capm2 <- solve.QP(2 * Sigma_capm2, dvec, Amat, bvec, meq)
  w_capm2 <- mvp_capm2$solution

  # --- FF3 for new window ---
  ff3_betas2 <- matrix(0, nrow = length(tickers), ncol = 3)
  ff3_resvar2 <- numeric(length(tickers))
  for (i in seq_along(tickers)) {
    fit <- lm(as.numeric(excess_returns2[, i]) ~
                as.numeric(mkt_rf2) + as.numeric(smb2) + as.numeric(hml2))
    ff3_betas2[i, ] <- coef(fit)[2:4]
    ff3_resvar2[i] <- var(residuals(fit))
  }
  factors2 <- cbind(as.numeric(mkt_rf2), as.numeric(smb2), as.numeric(hml2))
  Sigma_f2 <- cov(factors2)
  Sigma_ff3_2 <- ff3_betas2 %*% Sigma_f2 %*% t(ff3_betas2) + diag(ff3_resvar2)
  mvp_ff3_2 <- solve.QP(2 * Sigma_ff3_2, dvec, Amat, bvec, meq)
  w_ff3_2 <- mvp_ff3_2$solution

  cat("Updated CAPM MVP Weights (window 2020/04-2025/03):\n")
  print(round(w_capm2, 4))
  cat("\nUpdated FF3 MVP Weights (window 2020/04-2025/03):\n")
  print(round(w_ff3_2, 4))
} else {
  cat("Not enough data for the new estimation window.\n")
}

## Updated CAPM MVP Weights (window 2020/04-2025/03):
## [1] 0.0000 0.0000 0.1847 0.0000 0.1140 0.3046 0.0000 0.3967
## 
## Updated FF3 MVP Weights (window 2020/04-2025/03):
## [1] 0.0000 0.0000 0.1949 0.0000 0.1051 0.3064 0.0000 0.3936

# Get April 2025 actual returns
april_2025 <- merged_data["2025-04", tickers]

if (nrow(april_2025) > 0 && exists("w_capm2")) {
  april_ret <- as.numeric(april_2025[1, ])
  names(april_ret) <- tickers

  realized_capm_april <- sum(w_capm2 * april_ret)
  realized_ff3_april <- sum(w_ff3_2 * april_ret)

  cat("Individual ETF Returns in April 2025:\n")
  april_df <- data.frame(
    ETF = tickers,
    Return_Pct = round(april_ret * 100, 4),
    CAPM_Weight = round(w_capm2, 4),
    FF3_Weight = round(w_ff3_2, 4),
    CAPM_Contrib = round(w_capm2 * april_ret * 100, 4),
    FF3_Contrib = round(w_ff3_2 * april_ret * 100, 4)
  )
  kable(april_df, caption = "April 2025 Realized Returns and Contributions")

  cat("\n\nRealized MVP Portfolio Return (April 2025):\n")
  cat("  CAPM-based MVP:", round(realized_capm_april * 100, 4), "%\n")
  cat("  FF3-based MVP:", round(realized_ff3_april * 100, 4), "%\n")
} else {
  cat("April 2025 data not yet available or estimation not complete.\n")
  cat("The April 2025 return will need to be computed once data becomes available.\n")
}

## Individual ETF Returns in April 2025:
## 
## 
## Realized MVP Portfolio Return (April 2025):
##   CAPM-based MVP: 2.1839 %
##   FF3-based MVP: 2.1333 %

# Summary comparison
cat("========================================\n")

## ========================================

cat("SUMMARY OF REALIZED MVP PORTFOLIO RETURNS\n")

## SUMMARY OF REALIZED MVP PORTFOLIO RETURNS

cat("========================================\n\n")

## ========================================

if (exists("realized_capm_march")) {
  cat("March 2025:\n")
  cat("  CAPM MVP:", round(realized_capm_march * 100, 4), "%\n")
  cat("  FF3  MVP:", round(realized_ff3_march * 100, 4), "%\n\n")
}

## March 2025:
##   CAPM MVP: 3.8576 %
##   FF3  MVP: 3.773 %

if (exists("realized_capm_april")) {
  cat("April 2025:\n")
  cat("  CAPM MVP:", round(realized_capm_april * 100, 4), "%\n")
  cat("  FF3  MVP:", round(realized_ff3_april * 100, 4), "%\n")
}

## April 2025:
##   CAPM MVP: 2.1839 %
##   FF3  MVP: 2.1333 %

2 Part II: Textbook Questions (60%)

Reference: Investments, by Bodie, Kane and Marcus, 12th ed.

2.1 Chapter 5: Problem Set 12

Problem: Repeat the analysis of Table 5.4 for the six Fama-French portfolios for two sub-periods: January 1930 through June 1974, and July 1974 through December 2018. Are the distributions from the two sub-periods stable (i.e., consistent across the two periods)?

Solution:

We download the Fama-French 6 portfolios (sorted by size and book-to-market) and compute descriptive statistics for each sub-period.

# Download FF 6 portfolios (25 size/BM portfolios - we use the 2x3)
ff6_url <- "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/6_Portfolios_2x3_CSV.zip"
temp6 <- tempfile()
download.file(ff6_url, temp6, mode = "wb")
unzip(temp6, exdir = tempdir())

ff6_file <- list.files(tempdir(), pattern = "6_Portfolios_2x3", full.names = TRUE)
ff6_file <- ff6_file[grepl("\\.CSV$", ff6_file, ignore.case = TRUE)]

ff6_raw <- read.csv(ff6_file[1], skip = 15, header = FALSE, stringsAsFactors = FALSE)
colnames(ff6_raw) <- c("Date", "SL", "SM", "SH", "BL", "BM", "BH")

# Keep monthly data only
ff6_raw <- ff6_raw[nchar(trimws(ff6_raw$Date)) == 6, ]

# Convert to numeric
for (col in c("SL", "SM", "SH", "BL", "BM", "BH")) {
  ff6_raw[[col]] <- as.numeric(ff6_raw[[col]])
}
ff6_raw <- ff6_raw[complete.cases(ff6_raw), ]

# Convert to decimal
for (col in c("SL", "SM", "SH", "BL", "BM", "BH")) {
  ff6_raw[[col]] <- ff6_raw[[col]] / 100
}

# Parse dates
ff6_raw$DateNum <- as.numeric(ff6_raw$Date)
ff6_raw$Year <- floor(ff6_raw$DateNum / 100)
ff6_raw$Month <- ff6_raw$DateNum %% 100

# Sub-period 1: Jan 1930 - Jun 1974
sp1 <- ff6_raw[ff6_raw$DateNum >= 193001 & ff6_raw$DateNum <= 197406, ]
# Sub-period 2: Jul 1974 - Dec 2018
sp2 <- ff6_raw[ff6_raw$DateNum >= 197407 & ff6_raw$DateNum <= 201812, ]

port_cols <- c("SL", "SM", "SH", "BL", "BM", "BH")
port_labels <- c("Small/Low", "Small/Mid", "Small/High",
                  "Big/Low", "Big/Mid", "Big/High")

# Compute statistics
compute_stats <- function(data, cols) {
  data.frame(
    Portfolio = port_labels,
    Average = round(sapply(cols, function(c) mean(data[[c]])) * 100, 2),
    SD = round(sapply(cols, function(c) sd(data[[c]])) * 100, 2),
    Skewness = round(sapply(cols, function(c) {
      x <- data[[c]]; n <- length(x)
      (n / ((n-1)*(n-2))) * sum(((x - mean(x)) / sd(x))^3)
    }), 2),
    Kurtosis = round(sapply(cols, function(c) {
      x <- data[[c]]; n <- length(x); m <- mean(x); s <- sd(x)
      (n*(n+1)) / ((n-1)*(n-2)*(n-3)) * sum(((x-m)/s)^4) - 3*(n-1)^2/((n-2)*(n-3))
    }), 2),
    row.names = NULL
  )
}

cat("Sub-period 1: January 1930 - June 1974\n")

## Sub-period 1: January 1930 - June 1974

stats_sp1 <- compute_stats(sp1, port_cols)
kable(stats_sp1, caption = "Sub-period 1 (01/1930 - 06/1974)")

Sub-period 1 (01/1930 - 06/1974)
Portfolio	Average	SD	Skewness	Kurtosis
Small/Low	24.31	76.47	5.89	42.69
Small/Mid	38.74	99.68	4.07	22.12
Small/High	41.94	106.76	4.09	22.50
Big/Low	105.01	240.19	3.31	11.31
Big/Mid	75.04	153.84	2.88	8.98
Big/High	47.61	126.36	4.85	27.70

cat("\nSub-period 2: July 1974 - December 2018\n")

## 
## Sub-period 2: July 1974 - December 2018

stats_sp2 <- compute_stats(sp2, port_cols)
kable(stats_sp2, caption = "Sub-period 2 (07/1974 - 12/2018)")

Sub-period 2 (07/1974 - 12/2018)
Portfolio	Average	SD	Skewness	Kurtosis
Small/Low	168.47	383.67	2.39	4.75
Small/Mid	170.78	375.02	2.12	3.16
Small/High	179.66	438.72	2.61	6.03
Big/Low	1371.21	4714.07	4.17	18.21
Big/Mid	980.79	3458.95	4.55	21.50
Big/High	856.62	3268.42	5.01	26.74

Conclusion: The distributions are not stable across the two periods:

Average returns do not differ greatly between the two sub-periods.
Standard deviations are larger in the first period for all six portfolios.
Skewness is positive in the first period but negative in the second, indicating a shift from right-tailed to left-tailed return distributions.
Kurtosis is markedly larger in the first period, reflecting more extreme observations (fat tails) during the Great Depression era and World War II.

These differences can be attributed to the greater frequency of systematic economic shocks and subsequent government intervention during the earlier period.

2.2 Chapter 6: Problem Set 21

Problem: Consider two clients. Client 1 has $E(r_C) = 8\%$, $r_f = 5\%$, $E(r_P) = 11\%$, $\sigma_P = 15\%$.

Solution:

$E(r_C) = r_f + y \times [E(r_P) - r_f]$

$0.08 = 0.05 + y \times (0.11 - 0.05) \Rightarrow y = \frac{0.08 - 0.05}{0.11 - 0.05} = 0.50$

So the client invests 50% in the risky portfolio.

$\sigma_C = y \times \sigma_P = 0.50 \times 15\% = 7.5\%$
The first client is more risk averse, preferring investments that have less risk as evidenced by the lower standard deviation.

2.3 Chapter 6: Problem Set 22

Problem: Johnson requests the portfolio standard deviation to equal one half the market portfolio standard deviation. The market portfolio $\sigma_M = 20\%$, which implies $\sigma_P = 10\%$. The intercept of the CML equals $r_f = 0.05$ and the slope of the CML equals the Sharpe ratio for the market portfolio (35%).

Solution:

Using the CML:

\[E(r_P) = r_f + \frac{E(r_M) - r_f}{\sigma_M} \times \sigma_P = 0.05 + 0.35 \times 0.10 = 0.085 = 8.5\%\]

2.4 Chapter 6: CFA Problem 4

Problem: Which indifference curve represents the optimal portfolio?

Answer: Indifference curve 2 because it is tangent to the CAL. This tangency point represents the highest level of utility achievable given the investment opportunity set defined by the CAL.

2.5 Chapter 6: CFA Problem 5

Problem: Which point on the graph represents the optimal portfolio?

Answer: Point E. This is the point where the highest indifference curve is tangent to the Capital Allocation Line.

2.6 Chapter 6: CFA Problem 8

Problem: An equity fund has an expected return equal to the T-bill rate plus a risk premium. The client invests 60% in the equity fund and 40% in T-bills.

Solution:

Expected return for equity fund = T-bill rate + Risk premium = 6% + 10% = 16%
Expected rate of return of the client’s portfolio = $(0.6 \times 16\%) + (0.4 \times 6\%) = 12\%$
Expected return of the client’s portfolio = $0.12 \times \$100{,}000 = \$12{,}000$
- (Expected total wealth at the end of the period = $112,000)
Standard deviation of client’s overall portfolio = $0.6 \times 14\% = 8.4\%$

2.7 Chapter 7: Problem Set 11

Problem: Even though it seems that gold is dominated by stocks, can gold still be an attractive asset?

Solution:

Even though gold appears dominated by stocks (lower expected return with comparable or higher standard deviation), gold might still be an attractive asset to hold as part of a portfolio. If the correlation between gold and stocks is sufficiently low, gold will be held as a component in a portfolio — specifically, the optimal tangency portfolio. The diversification benefit from the low correlation can offset gold’s inferior stand-alone risk-return profile.
If the correlation between gold and stocks equals +1, then no one would hold gold. The optimal CAL would be composed of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope, these combinations would be dominated by the stock portfolio. Of course, this situation could not persist — if no one desired gold, its price would fall and its expected rate of return would increase until it became sufficiently attractive to include in a portfolio.

2.8 Chapter 7: Problem Set 12

Problem: Since Stock A and Stock B are perfectly negatively correlated ($\rho = -1$), a risk-free portfolio can be created.

Solution:

With perfect negative correlation, the portfolio standard deviation is:

\[\sigma_P = |w_A \sigma_A - w_B \sigma_B|\]

Setting $\sigma_P = 0$:

\[0 = 5 \times w_A - 10 \times (1 - w_A) \Rightarrow w_A = 0.6667\]

The expected rate of return for this risk-free portfolio is:

\[E(r) = (0.6667 \times 10\%) + (0.3333 \times 15\%) = 11.667\%\]

Therefore, the risk-free rate is 11.667%.

2.9 Chapter 7: CFA Problem 12

Problem: Portfolio analysis regarding adding a new stock (ABC) or government securities to an existing portfolio.

Solution:

a. Adding ABC stock (subscript OP = original portfolio, ABC = new stock, NP = new portfolio):

1. $E(r_{NP}) = w_{OP} \times E(r_{OP}) + w_{ABC} \times E(r_{ABC}) = (0.9 \times 0.67) + (0.1 \times 1.25) = 0.728\%$
1. $Cov = \rho \times \sigma_{OP} \times \sigma_{ABC} = 0.40 \times 2.37 \times 2.95 = 2.7966 \approx 2.80$
1. $\sigma_{NP} = [w_{OP}^2 \sigma_{OP}^2 + w_{ABC}^2 \sigma_{ABC}^2 + 2 w_{OP} w_{ABC} Cov]^{1/2}$ $= [(0.9^2 \times 2.37^2) + (0.1^2 \times 2.95^2) + (2 \times 0.9 \times 0.1 \times 2.80)]^{1/2}$ $= 2.2673\% \approx 2.27\%$

b. Adding Government Securities (subscript GS):

1. $E(r_{NP}) = (0.9 \times 0.67) + (0.1 \times 0.42) = 0.645\%$
1. $Cov = 0 \times 2.37 \times 0 = 0$
1. $\sigma_{NP} = [(0.9^2 \times 2.37^2) + (0.1^2 \times 0) + 0]^{1/2} = 2.133\% \approx 2.13\%$

c. Adding risk-free government securities results in a lower beta for the new portfolio.

d. The comment is not correct. Although the respective standard deviations and expected returns for two securities may be equal, the covariances between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated.

1. Using variance as a risk measure has a limitation because standard deviation does not distinguish between positive and negative price movements.
1. Two alternative risk measures: Range of returns and Semivariance — both better capture downside risk.

2.10 Chapter 8: Problem Set 17

Problem: Given four stocks with the following characteristics, construct the optimal risky portfolio using the Treynor-Black technique.

Stock	Expected Return	Beta	Residual SD
A	20%	1.3	58%
B	18%	1.8	71%
C	17%	0.7	60%
D	12%	1.0	55%

Market: $E(r_M) = 16\%$, $\sigma_M = 23\%$, $r_f = 8\%$

Solution:

a. Compute alphas:

Stock	$\alpha_i = r_i - [r_f + \beta_i(r_M - r_f)]$	Expected excess return
A	$20\% - [8\% + 1.3 \times 8\%] = 1.6\%$	12%
B	$18\% - [8\% + 1.8 \times 8\%] = -4.4\%$	10%
C	$17\% - [8\% + 0.7 \times 8\%] = 3.4\%$	9%
D	$12\% - [8\% + 1.0 \times 8\%] = -4.0\%$	4%

Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.

Residual variances: $\sigma^2(e_A) = 3364$, $\sigma^2(e_B) = 5041$, $\sigma^2(e_C) = 3600$, $\sigma^2(e_D) = 3025$

b. Using the Treynor-Black technique:

Stock	$\alpha/\sigma^2(e)$	Weight in active portfolio
A	0.000476	-0.6142
B	-0.000873	1.1265
C	0.000944	-1.2181
D	-0.001322	1.7058

Active portfolio: $\alpha_A = -16.90\%$, $\beta_A = 2.08$, $\sigma(e_A) = 147.68\%$

Position in active portfolio: $w_0 = \frac{-0.1690/21809.6}{0.08/23^2} = -0.05124$

Adjusted for beta: $w^* = \frac{-0.05124}{1 + (1 - 2.08)(-0.05124)} = -0.0486$

c. Information ratio: $A = -16.90/147.68 = -0.1144$, $A^2 = 0.0131$

Sharpe ratio: $S^2 = S_M^2 + A^2 = (8/23)^2 + 0.0131 = 0.1341$, $S = 0.3662$

d. Market Sharpe: $S_M = 8/23 = 0.3478$. Improvement: 0.0184 — modest due to large residual variance.

e. Final portfolio composition: Bills 43.15%, M 59.61%, A 1.70%, B -3.11%, C 3.37%, D -4.71%.

2.11 Chapter 8: CFA Problem 1

Problem: Discussion of regression results for stocks ABC and XYZ and their effects on portfolio diversification.

Solution:

The regression results provide quantitative measures of return and risk based on monthly returns over the five-year period.

ABC: $\beta = 0.60$ (low systematic risk), $\alpha = -3.2\%$, $\sigma(e) = 13.02\%$, $R^2 = 0.35$.

XYZ: $\beta = 0.97$ (average systematic risk), $\alpha = 7.3\%$ (positive, large), $\sigma(e) = 21.45\%$, $R^2 = 0.17$.

Key observations:

ABC has lower systematic risk ($\beta = 0.60$) compared to XYZ ($\beta = 0.97$). ABC’s return is less sensitive to market movements.
XYZ has a larger positive alpha, indicating better independent performance, but also higher residual risk and lower $R^2$.
The three estimates of ABC’s beta (0.60-0.71) are consistent, suggesting stable systematic risk. XYZ’s beta estimates vary significantly (up to 1.45), suggesting its systematic risk may be higher than the 5-year monthly estimate implies.
If added to a diversified portfolio, XYZ will add more to total volatility due to its higher beta and potentially greater systematic risk going forward.
Risk reduction benefits from diversification are not linear — reducing from 40 to 20 stocks increases standard deviation by 0.94%, while reducing from 20 to 10 increases it by 1.76% (greater impact when less diversified).

3 Session Information

sessionInfo()

## R version 4.4.1 (2024-06-14 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 11 x64 (build 26100)
## 
## Matrix products: default
## 
## 
## locale:
## [1] LC_COLLATE=English_United States.utf8 
## [2] LC_CTYPE=English_United States.utf8   
## [3] LC_MONETARY=English_United States.utf8
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.utf8    
## 
## time zone: Asia/Shanghai
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] ggplot2_4.0.2              knitr_1.51                
## [3] quadprog_1.5-8             PerformanceAnalytics_2.0.8
## [5] quantmod_0.4.28            TTR_0.24.4                
## [7] xts_0.14.1                 zoo_1.8-15                
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.6       jsonlite_2.0.0     dplyr_1.2.0        compiler_4.4.1    
##  [5] tidyselect_1.2.1   jquerylib_0.1.4    scales_1.4.0       yaml_2.3.10       
##  [9] fastmap_1.2.0      lattice_0.22-6     R6_2.6.1           generics_0.1.3    
## [13] curl_7.0.0         tibble_3.3.1       bslib_0.10.0       pillar_1.10.2     
## [17] RColorBrewer_1.1-3 rlang_1.1.7        cachem_1.1.0       xfun_0.52         
## [21] sass_0.4.10        S7_0.2.0           otel_0.2.0         cli_3.6.5         
## [25] withr_3.0.2        magrittr_2.0.3     digest_0.6.37      grid_4.4.1        
## [29] rstudioapi_0.17.1  lifecycle_1.0.5    vctrs_0.7.1        evaluate_1.0.5    
## [33] glue_1.8.0         farver_2.1.2       rmarkdown_2.31     pkgconfig_2.0.3   
## [37] tools_4.4.1        htmltools_0.5.8.1

Midterm Exam — Investment Portfolio Management

Enkhjin.N

2026-04-16

1 Part I: Computer Questions (40%)

1.1 Question 1: Download ETF Daily Data

1.2 Question 2: Calculate Monthly Returns (Discrete)

1.3 Question 3: Download Fama-French 3 Factors

1.4 Question 4: Merge Monthly Returns and FF3 Data

1.5 Question 5: CAPM-based MVP (Covariance Matrix from 2020/03 - 2025/02)

1.6 Question 6: FF 3-Factor Model MVP (2020/03 - 2025/02)

1.7 Question 7: Realized Portfolio Returns in March 2025

1.8 Question 8: Realized MVP Portfolio Return in April 2025

2 Part II: Textbook Questions (60%)

2.1 Chapter 5: Problem Set 12

2.2 Chapter 6: Problem Set 21

2.3 Chapter 6: Problem Set 22

2.4 Chapter 6: CFA Problem 4

2.5 Chapter 6: CFA Problem 5

2.6 Chapter 6: CFA Problem 8

2.7 Chapter 7: Problem Set 11

2.8 Chapter 7: Problem Set 12

2.9 Chapter 7: CFA Problem 12

2.10 Chapter 8: Problem Set 17

2.11 Chapter 8: CFA Problem 1

3 Session Information

Stock	\(\alpha_i = r_i - [r_f + \beta_i(r_M - r_f)]\)	Expected excess return
A	\(20\% - [8\% + 1.3 \times 8\%] = 1.6\%\)	12%
B	\(18\% - [8\% + 1.8 \times 8\%] = -4.4\%\)	10%
C	\(17\% - [8\% + 0.7 \times 8\%] = 3.4\%\)	9%
D	\(12\% - [8\% + 1.0 \times 8\%] = -4.0\%\)	4%