Portfolio analysis
1. Portfolio returns
Question 1.
Suppose you buy 500 shares of stock A, buy 300 shares of stock B, and short sell 100 shares of stock C. The corresponding prices of shares A, B, C: $70, $50, $90. Compute the weights of your portfolio.
# Define the vector values
values=c(500,300,100)
# Define the vector weights
weights=values/sum(values)
# Print the resulting weights
print(weights)## [1] 0.5555556 0.3333333 0.1111111Question 2.
Suppose on 10/12/2023, Anna buy 2000 shares of VFS and 3000 shares of TESLA at the close price. Compute the daily rate of return of her portfolio from 11/12/2023 to 20/12/2023.
## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: TTR## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## [1] "VFS"## [1] "TSLA"# Extract adjusted close prices
VFS_prices <- Cl(VFS)
Tesla_prices <- Cl(TSLA)
# Calculate daily returns
VFS_returns <- dailyReturn(VFS_prices)
Tesla_returns <- dailyReturn(Tesla_prices)
# Define the number of shares
VFS_shares <- 2000
Tesla_shares <- 3000
# Calculate the portfolio value for each day
portfolio_values <- VFS_shares * VFS_prices + Tesla_shares * Tesla_prices
# Calculate the daily portfolio returns
portfolio_returns <- dailyReturn(portfolio_values)
# Print the daily portfolio returns from 11/12/2023 to 20/12/2023
portfolio_returns['2023-12-11/2023-12-20']## daily.returns
## 2023-12-11 0.000000000
## 2023-12-12 -0.011779041
## 2023-12-13 0.010248189
## 2023-12-14 0.048643148
## 2023-12-15 0.012033416
## 2023-12-18 -0.006131959
## 2023-12-19 0.0209462062. Mean and variance of a two-asset portfolio
Question 1
A person invests a portfolio buy buying 100 stock A and 400 stock B. The price for a stock A and a stock B are $40 and $20, respectively. Assume that the expected rate of return of stock A and B are 17%/year and 13%/year, respectively. In addition, the correlation between the stock returns of A and B is 0.7. Note that the volatility of stock and A and B are 30%/year and 20%/year, respectively.
- Compute the expected rate of return r of the portfolio
# Define the parameters
investment_A <- 100 * 40
investment_B <- 400 * 20
total_investment <- investment_A + investment_B
weight_A <- investment_A / total_investment
weight_B <- investment_B / total_investment
weight_A## [1] 0.3333333## [1] 0.6666667expected_return_A <- 0.17
expected_return_B <- 0.13
# a) Calculate the expected return of the portfolio
expected_return_portfolio <- (weight_A * expected_return_A) + (weight_B * expected_return_B)
expected_return_portfolio## [1] 0.1433333- Compute the variance of the portfolio
# Define the parameters
volatility_A <- 0.3
volatility_B <- 0.2
correlation_AB <- 0.7
#Calculate the variance of the portfolio
variance_A <- volatility_A^2
variance_B <- volatility_B^2
covariance_AB <- correlation_AB * volatility_A * volatility_B
covariance_AB## [1] 0.042portfolio_variance <- (weight_A^2 * variance_A) + (weight_B^2 * variance_B) + (2 * weight_A * weight_B * covariance_AB)
portfolio_variance## [1] 0.046444443. Global minimum variance portfolios
Question 1
Consider two assets S1 and S2 with the annual mean return \(\overline r_1\) = 0.22,¯ r2 = 0.15, the volatilities σ1 = 0.2,σ2 = 0.3. The correlation coefficient of the two assets is ρ = 0.2. Assume the short sell is not permitted. Compute the weights of S1 and S2 that constitute a portfolio having the minimum variance.
Solution:
Since short selling is not permitted (meaning weights must be non-negative), we need to adjust the weights to the feasible region. If short selling were permitted, \(w_{1}\) would be greater than 1 and \(w_{2}\) would be negative. To respect the no short-selling constraint, we must set \(w_{1}= 1\) and \(w_{2}=0\) because \(w_{2}\) must be non-negative.
Therefore, the minimum variance portfolio with the constraint that short selling is not allowed has the weights: \(w_{1}=1\) and \(w_{2}=0\)
Problem 2:
Consider two assets \(S_{1}\) and \(S_{2}\) with the annual mean return \(\hat{r_{1}}=0.22\),\(\hat{r_{2}}=0.15\), the volatilities \(\sigma_{1}= 0.2\),\(\sigma_{2}= 0.3\). The correlation coefficient of the two assets is \(\rho = 0.7\). Assume the short sell is permitted. Compute the weights of \(S_{1}\) and \(S_{2}\) that constitute a portfolio having the minimum variance.
Solution:
Denote that:
- \(w_{1}\) is the weight of Assets \(S_{1}\)
- \(w_{2}= 1-w_{1}\) is the weight of Assets \(S_{2}\)
The covariance of portfolio is: \[Cov(S_{1}, S_{2})=\sigma_{12}= \rho\times\sigma_{1}\times\sigma_{2}=(0.7)(0.2)(0.3)= 0.042\]
The variance of portfolio is: \[ \begin{eqnarray} \sigma^{2}&=& w_{1}^{2}(\overline{r}_{1})^{2} + w_{2}^{2}(\overline{r}_{2})^{2} + 2w_{1}w_{2}\sigma_{12} \\&=& w_{1}^{2}(0.2)^{2} + w_{2}^{2}(0.3)^{2}+2w_{1}w_{2}(0.042) \end{eqnarray} \]
The problem is optimal weight of the asset A and B: \[ \begin{array}{cl} Minimize & \sigma^{2}\\ s.t & w_{1}+w_{2}=1\\ \end{array} \] Since short selling is permitted, negative weights are allowed.
By using method of Larange multipliers, we obtain solution: \[w_{1}= \frac{\sigma_{2}^{2}-\sigma_{12}}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{12}}= \frac{(0.3)^{2}-0.042}{(0.2)^{2}+(0.3)^{2}-2(0.042)}= 1.04348,\\ w_{2}=1-w_{1}=1-1.04348= -0.04348\]
4. Efficient portfolios
Consider two assets \(S_{1}\) and \(S_{2}\) with the annual mean return \(\hat{r_{1}}=0.22\),\(\hat{r_{2}}=0.15\), the volatilities \(\sigma_{1}= 0.2\),\(\sigma_{2}= 0.3\). The correlation coefficient of the two assets is \(\rho = 0.2\). Assume the short sell is not permitted. Compute the weights of S1 and S2 that constitute an efficient portfolio having the expected return \(r = 20\%\) and the lowest variance.
Solution:
Denote that:
- \(w_{1}\) is the weight of Assets \(S_{1}\)
- \(w_{2}= 1-w_{1}\) is the weight of Assets \(S_{2}\)
The variance between \(S_{1}\) and \(S_{2}\) is: \[Cov(S_{1}, S_{2})= \rho \times \sigma_{1} \times \sigma_{2}= (0.2)(0.2)(0.3)= 0.012\] Now the covariance matrix is: \[\sum = \begin{bmatrix} \sigma_1^2 & \text{Cov}(S_1, S_2) \\ \text{Cov}(S_1, S_2) & \sigma_2^2 \end{bmatrix} = \begin{bmatrix} 0.04 & 0.012 \\ 0.012 & 0.09 \end{bmatrix}\]
The problem is to maxminimize the variance \((\sigma_{1}, \sigma_{2})\)for a given expected return \(r=0.2\). The formula for the variance of a portfolio \(\sigma_{p}^{2}\) with weight 1 (\(w_{1}\)) and (\(w_{2}\)) is \[ \begin{array}{cl} Minimize & \sigma^{2}=w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}Cov(S_{1}, S_{2}) \\ s.t & r_{p}= w_{1}\hat{r_{1}}+\hat{w_{2}}r_{2}=0.2\\ & w_{1}+w_{2}=1 \end{array} \] Then the expectation of portfolio is calculated: \[\begin{eqnarray*} r_{p}= w_{1}\hat{r_{1}}+\hat{w_{2}}r_{2}=0.2 &\iff& w_{1}0.22+ w_{2}0.15= 0.2\\&\iff&w_{1}0.22+(1-w_{1})0.15=0.2\\&\iff& w_{1}= 0.7143 \end{eqnarray*}\] The weight of \(S_{1} w_{1}\) is: 0.7143 and the weight of \(S_{2} w_{2}\) is: \(w_{2}=1-w_{1}=0.2857\)
Therefore, the minimum variance of portforlio is given as: \[\begin{eqnarray*} \sigma^{2}&=&w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}Cov(S_{1}, S_{2})\\&=& (0.7143)^{2}(0.2)+(0.2857)^{2}(0.3)+2(0.7134)(0.2857)(0.012)\\ &=& 0.1314 \end{eqnarray*}\]
5. Portfolio of multiple stocks
Question 1:
Considering a portfolio including three stocks A, B, C. Compute the exected rate of return r of the portfolio.
| Asset_name | Share_number | price expected_return_rate |
|---|---|---|
| A | 100 | $30 |
| B | 400 | $20 |
| C | 200 | $50 |
- Calculate the total market value of the portfolio: \[Asset\ A= 100\times 30= 3000\\ Asset \ B = 400\times 20= 8000\\ Asset \ C = 200\times 50= 10000\]
Then, \[Total\ Asset = 3000+8000+10000= 21000\]
- Calculate the weight of each asset:
- Weight of Asset A:
\[w_{A}=\frac{3000}{21000}=0.1429\]
- Weight of Asset B:
\[w_{B}=\frac{8000}{21000}=0.3801\]
- Weight of Asset C:
\[w_{C}=\frac{10000}{21000}=0.4762\]
- The expected rate of return \(r_{p}\) of the portfolio is: \[r_{p}= w_{A}\times r_{A}+ w_{B}\times r_{B} +w_{C}\times r_{C}= 0.1832 \]
Question 2:
Consider three assets A, B, C. The annual mean returns of A, B, C are 17%, 5%, 8%, respectively. The volatilities (the standard deviation of mean returns) of A, B, C are 22%, 7%, 8%, respectively. The correlation matrix of A, B, C are given as:
\[\begin{matrix} &A & B &C \\ A & 1.00 & -0.25 & -0.05 \\ B & -0.25 & 1.00 & 0.45 \\ C & -0.05 & 0.45 & 1.00 \\ \end{matrix}\]
We build a portfolio consisting of A, B, C with weights \(w_{A}=0.6\), \(w_{B}=0.15\), \(w_{B}=0.25\).
- Compute the covariance matrix of A, B, C. Using this formula, we compute the covariances:
\[ Cov(A,B)= \rho_{AB}\sigma_{A}\sigma_{B}=(-0.25)(0.22)(0.07)= -0.00385\\ Cov(B,C)= \rho_{BC}\sigma_{B}\sigma_{C}=(0.45)(0.07)(0.08)= 0.00252\\ Cov(A,C)= \rho_{AC}\sigma_{A}\sigma_{C}=(-0.05)(0.22)(0.08)= 0.00088 \]
\[\sum = \begin{bmatrix} 0.0484 & -0.00385 & 0.00088 \\ -0.00385& 0.0049 & 0.00252\\ 0.00088 & 0.00252 & 0.0064\\ \end{bmatrix}\] b) Compute the mean return of the portfolio.
The mean return of the portfolio \(\hat{r}_{p}\) is given by:
\[\begin{eqnarray*} \hat{r}_{p} &=& w_{A}\hat{r}_{A}+ w_{B}\hat{r}_{B} +w_{C}\hat{r}_{C}\\ &=& (0.6)(0.17)+ (0.15)(0.05)+ (0.25)(0.08)= 0.1295 \end{eqnarray*}\]
- Compute the variance return of the portfolio.
The variance of the portfolio \(\sigma_{p}^{2}\) is given by: \[\begin{eqnarray*} \hat{\sigma}^{2}_{p} &=& w_{A}^{2}\hat{\sigma}^{2}_{A}+ w_{B}^{2}\hat{\sigma}^{2}_{B} +w_{C}^{2}\hat{\sigma}^{2}_{C}\\ &=& (0.6^{2}\times0.0484)+(0.15^{2}\times0.0049)+(0.25^{2}\times0.0064)\\ &+&2(0.6)(0.15)(−0.00385)+2(0.6)(0.25)(−0.00088)+2(0.15)(0.25)(0.00252)\\ &=& 0.0172 \end{eqnarray*}\]
The standard deviation (volatility) of the portfolio is: \[\sigma_{p}= \sqrt{0.0172}= 0.1311\]