cong thuc chuong 3 va 5

Author

LuciusVU (Vucius)

Module 3

xác định điểm trung bình của data (lãi suất trung bình, …. )

Arithmetic Mean (Average):
\[ \bar{X} = \frac{1}{N} \sum_{i=1}^{N} X_i \]
- Used to calculate the simple average of returns or values.
Geometric Mean: \[ \text{Geometric Mean} = \left( \prod_{i=1}^{N} (1 + X_i) \right)^{\frac{1}{N}} - 1 \]
- Best for compounded or multi-period returns to show growth rates over time.
Median/Mode

đo độ biến động của các điểm data

Mean Absolute Deviation (MAD): \[ \text{MAD} = \frac{1}{N} \sum_{i=1}^{N} |X_i - \bar{X}| \]
- MAD is the average of the absolute deviations from the mean

Sample Variance (\(s^2\)): \[ s^2 = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \bar{X})^2 \]
- Similar to population variance but adjusted for sample data.
Standard Deviation: \[ \quad s = \sqrt{s^2} \quad \text{(for sample)} \]
- The square root of variance, providing a more interpretable measure of spread.
Coefficient of Variation (CV): \[ \text{CV} = \frac{\sigma}{\bar{X}} \]
- A relative measure of risk (standard deviation) to return (mean), useful for comparing the risk per unit of return across different investments.
Target Downside Deviation (also known as Target Semideviation): \[ \text{Downside Deviation} = \sqrt{\frac{1}{N - 1} \sum_{i=1}^{N} \max(0, \text{Target} - X_i)^2} \]

định hình phân phối chuẩn

Skewness: \[ \text{Skewness} = \frac{\frac{1}{N} \sum_{i=1}^{N} (X_i - \bar{X})^3}{\sigma^3} \]
- Measures the asymmetry of data; a positive skew indicates a right tail, while a negative skew shows a left tail.

Kurtosis: \[ \text{Kurtosis} = \frac{\frac{1}{N} \sum_{i=1}^{N} (X_i - \bar{X})^4}{\sigma^4} \]
- Indicates ‘tailedness’ or the frequency of extreme values. High kurtosis implies frequent extreme deviations.

đo độ tương quan của data

Covariance: \[ \text{Cov}(X, Y) = \frac{1}{N} \sum_{i=1}^{N} (X_i - \bar{X})(Y_i - \bar{Y}) \]
- Measures the degree to which two variables move together. A positive value indicates that variables tend to increase or decrease together.
Correlation Coefficient (\(\rho\)): \[ \rho_{X,Y} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \]
- Standardizes covariance by dividing by the product of the standard deviations of each variable. Ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).
correlation [-1,1] correlation ~ 0: X và Y không tương quan (vd giá cổ phiếu/lợi suất 2 cổ phiếu) correlation ~ -1 tương quan âm correlation ~ 1 tương quan dương

Expected Portfolio Return: \[ E(R_p) = \sum_{i=1}^{n} w_i \times E(R_i) \]
- Where \(w_i\) is the weight of each asset \(i\) in the portfolio, and \(E(R_i)\) is the expected return of asset \(i\).

Variance of a Two-Asset Portfolio: \[ \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(A, B) \]
- Where \(w_A\) and \(w_B\) are weights of assets A and B, \(\sigma_A^2\) and \(\sigma_B^2\) are their variances, and \(\text{Cov}(A, B)\) is the covariance between A and B.
Alternative Portfolio Variance Formula Using Correlation: \[ \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{A,B} \sigma_A \sigma_B \]
- Where \(\rho_{A,B}\) is the correlation coefficient between assets A and B, showing how much the returns move together.

Covariance measures the degree to which two assets’ returns move together, which is essential for diversification analysis.

Covariance: \[ \text{Cov}(A, B) = E[(R_A - E(R_A))(R_B - E(R_B))] \]
- Where \(R_A\) and \(R_B\) are the returns of assets A and B, and \(E(R_A)\), \(E(R_B)\) are their expected returns.

Roy’s Safety-First Criterion is used to choose portfolios that minimize the probability of returns falling below a specified threshold.

Safety-First Ratio: \[ \text{SFRatio} = \frac{E(R_p) - R_L}{\sigma_p} \]
- Where \(E(R_p)\) is the expected portfolio return, \(R_L\) is the minimum acceptable return, and \(\sigma_p\) is the portfolio standard deviation.
- Higher values of the Safety-First Ratio indicate portfolios with a lower probability of falling below the target return threshold.