Load Data

data <- read_csv("myetf4.csv")

## Rows: 751 Columns: 5
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): Index
## dbl (4): tw0050, tw0056, tw006205, tw00646
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

head(data)

## # A tibble: 6 × 5
##   Index      tw0050 tw0056 tw006205 tw00646
##   <chr>       <dbl>  <dbl>    <dbl>   <dbl>
## 1 2015/12/14   53.3   18.2     31.1    19.6
## 2 2015/12/15   53.3   18.4     31.6    19.6
## 3 2015/12/16   54.1   18.6     31.6    19.9
## 4 2015/12/17   54.8   18.8     32.2    20.0
## 5 2015/12/18   54.5   19.0     32.2    19.8
## 6 2015/12/21   54.4   19.0     33      19.6

Assume first column = Date, others = ETF prices.

data <- data %>%
  rename(Date = 1) %>%
  mutate(Date = as.Date(Date))

Compute Daily Returns

returns_daily <- data %>%
  arrange(Date) %>%
  mutate(across(-Date, ~ (. / lag(.) - 1))) %>%
  drop_na()

head(returns_daily)

## # A tibble: 6 × 5
##   Date          tw0050   tw0056  tw006205  tw00646
##   <date>         <dbl>    <dbl>     <dbl>    <dbl>
## 1 2015-12-15  0.000751  0.00712  0.0171    0.00102
## 2 2015-12-16  0.0152    0.00979  0.000317  0.0132 
## 3 2015-12-17  0.0116    0.0135   0.0199    0.00804
## 4 2015-12-18 -0.00493   0.00744 -0.00155  -0.00998
## 5 2015-12-21 -0.00165   0.00369  0.0255   -0.0106 
## 6 2015-12-22  0.00239  -0.00368  0.00303   0.00407

Q1: GMVP using Daily Returns

Mean and Covariance

R <- as.matrix(returns_daily[, -1])

mu <- colMeans(R)
Sigma <- cov(R)

Compute GMVP Weights

Formula:
w = Σ⁻¹ 1 / (1’ Σ⁻¹ 1)

one <- rep(1, ncol(R))

Sigma_inv <- solve(Sigma)

w_gmvp <- Sigma_inv %*% one / as.numeric(t(one) %*% Sigma_inv %*% one)

w_gmvp

##                [,1]
## tw0050   -0.2193579
## tw0056    0.7283718
## tw006205  0.1076234
## tw00646   0.3833627

Portfolio Return and Risk

ret_gmvp <- sum(w_gmvp * mu)
sd_gmvp  <- sqrt(t(w_gmvp) %*% Sigma %*% w_gmvp)

ret_gmvp

## [1] 0.0002536645

sd_gmvp

##             [,1]
## [1,] 0.005904941

Q2: GMVP using Monthly Returns

Convert to Monthly Prices

data_monthly <- data %>%
  mutate(YearMonth = floor_date(Date, "month")) %>%
  group_by(YearMonth) %>%
  summarise(across(-Date, last)) %>%
  ungroup() %>%
  rename(Date = YearMonth)

Monthly Returns

returns_monthly <- data_monthly %>%
  arrange(Date) %>%
  mutate(across(-Date, ~ (. / lag(.) - 1))) %>%
  drop_na()

head(returns_monthly)

## # A tibble: 6 × 5
##   Date        tw0050   tw0056 tw006205  tw00646
##   <date>       <dbl>    <dbl>    <dbl>    <dbl>
## 1 2016-01-01 -0.0198 -0.0138  -0.173   -0.0389 
## 2 2016-02-01  0.0286  0.0435  -0.0276  -0.00363
## 3 2016-03-01  0.0555 -0.00258  0.0828   0.0260 
## 4 2016-04-01 -0.0472 -0.0372  -0.0248   0.00964
## 5 2016-05-01  0.0252  0.0166   0.00442  0.0221 
## 6 2016-06-01  0.0364  0.0296  -0.0256  -0.0261

GMVP (Monthly)

R_m <- as.matrix(returns_monthly[, -1])

mu_m <- colMeans(R_m)
Sigma_m <- cov(R_m)

one <- rep(1, ncol(R_m))

Sigma_inv_m <- solve(Sigma_m)

w_gmvp_m <- Sigma_inv_m %*% one / as.numeric(t(one) %*% Sigma_inv_m %*% one)

w_gmvp_m

##                 [,1]
## tw0050   0.003183861
## tw0056   0.474049044
## tw006205 0.001203750
## tw00646  0.521563345

Monthly GMVP Return and Risk

ret_gmvp_m <- sum(w_gmvp_m * mu_m)
sd_gmvp_m  <- sqrt(t(w_gmvp_m) %*% Sigma_m %*% w_gmvp_m)

ret_gmvp_m

## [1] 0.005733667

sd_gmvp_m

##            [,1]
## [1,] 0.02490441

Q3: Tangency Portfolio (rf = 0)

Formula:
w = Σ⁻¹ μ / (1’ Σ⁻¹ μ)

one <- rep(1, ncol(R_m))

w_tan <- solve(Sigma_m) %*% mu_m / as.numeric(t(one) %*% solve(Sigma_m) %*% mu_m)

w_tan

##                [,1]
## tw0050    1.3050535
## tw0056   -0.1576807
## tw006205 -0.8475320
## tw00646   0.7001591

Tangency Portfolio Return and Risk

ret_tan <- sum(w_tan * mu_m)
sd_tan  <- sqrt(t(w_tan) %*% Sigma_m %*% w_tan)

ret_tan

## [1] 0.01809002

sd_tan

##            [,1]
## [1,] 0.04423638

Final Results Summary

results <- tibble(
  Portfolio = c("GMVP Daily", "GMVP Monthly", "Tangency"),
  Return = c(ret_gmvp, ret_gmvp_m, ret_tan),
  Risk_SD = c(sd_gmvp, sd_gmvp_m, sd_tan)
)

results

## # A tibble: 3 × 3
##   Portfolio      Return Risk_SD
##   <chr>           <dbl>   <dbl>
## 1 GMVP Daily   0.000254 0.00590
## 2 GMVP Monthly 0.00573  0.0249 
## 3 Tangency     0.0181   0.0442

Interpretation

GMVP minimizes portfolio risk regardless of return
Monthly GMVP differs due to lower frequency data
Tangency portfolio maximizes return per unit of risk (Sharpe ratio since rf = 0)

Higher return portfolios come with higher volatility, showing the classic risk-return tradeoff in financial markets.

Homework 04 - Portfolio Optimization (Taiwan ETFs)

Bat-Erdene Zorigtbaatar

2026-03-25