1. **State the problem:** We need to calculate the beta coefficient of Tausi Ltd.'s ordinary shares using the given share prices and stock exchange index values from 2019 to 2024. 2. **Formula and explanation:** Beta ($\beta$) measures the sensitivity of a stock's returns to the returns of the market. It is calculated as: $$\beta = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$$ where $R_i$ is the return of the stock and $R_m$ is the return of the market index. 3. **Calculate returns:** Calculate the annual returns for Tausi Ltd. and the stock exchange index using: $$R_t = \frac{P_t - P_{t-1}}{P_{t-1}}$$ where $P_t$ is the price at year $t$. - Tausi Ltd. returns: - 2020: $\frac{78 - 75}{75} = 0.04$ - 2021: $\frac{81 - 78}{78} \approx 0.0385$ - 2022: $\frac{79 - 81}{81} \approx -0.0247$ - 2023: $\frac{85 - 79}{79} \approx 0.0759$ - 2024: $\frac{76.5 - 85}{85} \approx -0.1$ - Stock Exchange Index returns: - 2020: $\frac{815.5 - 752}{752} \approx 0.0846$ - 2021: $\frac{875 - 815.5}{815.5} \approx 0.0723$ - 2022: $\frac{840 - 875}{875} \approx -0.04$ - 2023: $\frac{900 - 840}{840} \approx 0.0714$ - 2024: $\frac{795 - 900}{900} \approx -0.1167$ 4. **Calculate covariance and variance:** - Mean return of Tausi Ltd.: $\bar{R_i} = \frac{0.04 + 0.0385 - 0.0247 + 0.0759 - 0.1}{5} \approx 0.00594$ - Mean return of market: $\bar{R_m} = \frac{0.0846 + 0.0723 - 0.04 + 0.0714 - 0.1167}{5} \approx 0.01432$ - Covariance: $$\text{Cov}(R_i, R_m) = \frac{1}{n-1} \sum_{t=1}^n (R_{i,t} - \bar{R_i})(R_{m,t} - \bar{R_m})$$ Calculate each term: - $(0.04 - 0.00594)(0.0846 - 0.01432) = 0.03406 \times 0.07028 = 0.00239$ - $(0.0385 - 0.00594)(0.0723 - 0.01432) = 0.03256 \times 0.05798 = 0.00189$ - $(-0.0247 - 0.00594)(-0.04 - 0.01432) = -0.03064 \times -0.05432 = 0.00166$ - $(0.0759 - 0.00594)(0.0714 - 0.01432) = 0.06996 \times 0.05708 = 0.00399$ - $(-0.1 - 0.00594)(-0.1167 - 0.01432) = -0.10594 \times -0.13102 = 0.01388$ Sum: $0.00239 + 0.00189 + 0.00166 + 0.00399 + 0.01388 = 0.02381$ Divide by $n-1=4$: $$\text{Cov}(R_i, R_m) = \frac{0.02381}{4} = 0.00595$$ - Variance of market returns: Calculate each squared deviation: - $(0.0846 - 0.01432)^2 = 0.00494$ - $(0.0723 - 0.01432)^2 = 0.00336$ - $(-0.04 - 0.01432)^2 = 0.00295$ - $(0.0714 - 0.01432)^2 = 0.00326$ - $(-0.1167 - 0.01432)^2 = 0.01717$ Sum: $0.00494 + 0.00336 + 0.00295 + 0.00326 + 0.01717 = 0.03168$ Divide by 4: $$\text{Var}(R_m) = \frac{0.03168}{4} = 0.00792$$ 5. **Calculate beta:** $$\beta = \frac{0.00595}{0.00792} \approx 0.75$$ 6. **Interpretation:** A beta of approximately 0.75 means Tausi Ltd.'s shares are less volatile than the market. They tend to move in the same direction as the market but with lower magnitude. **Final answer:** $$\boxed{\beta \approx 0.75}$$