Answer: -1.89%
Step-by-step explanation: This part of the current page talks about a portfolio that consists of two stocks, DW and Woodpecker, and asks for the smallest expected loss for the portfolio in the next month with a 16% probability. To answer this question, we need to use some concepts from portfolio theory and statistics.
- **Portfolio theory**: This is a branch of finance that studies how investors can construct optimal portfolios of risky assets to maximize their expected return for a given level of risk, or minimize their risk for a given level of expected return¹.
- **Expected return**: This is the average return that an investor can expect to earn from an investment over a long period of time. It is calculated by multiplying the possible returns by their probabilities and adding them up. For a portfolio of two stocks, the expected return is the weighted average of the expected returns of each stock, where the weights are the proportions of the portfolio invested in each stock².
- **Standard deviation**: This is a measure of how much the actual returns of an investment deviate from its expected return. It is also known as the volatility or risk of the investment. For a portfolio of two stocks, the standard deviation is not simply the weighted average of the standard deviations of each stock, but also depends on the correlation between the returns of the two stocks. The correlation measures how closely the returns of the two stocks move together. It ranges from -1 to 1, where -1 means perfect negative correlation (the returns move in opposite directions), 0 means no correlation (the returns are independent), and 1 means perfect positive correlation (the returns move in the same direction). The formula for the standard deviation of a portfolio of two stocks is given by³:
$$\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}}$$
where $\sigma_p$ is the standard deviation of the portfolio, $w_1$ and $w_2$ are the weights of the two stocks, $\sigma_1$ and $\sigma_2$ are the standard deviations of the two stocks, and $\rho_{12}$ is the correlation between the returns of the two stocks.
- **Z-score**: This is a measure of how many standard deviations a given value is away from the mean of a distribution. It is calculated by subtracting the mean from the value and dividing by the standard deviation. For a normal distribution, which is a bell-shaped curve that is symmetric around the mean, the z-score can be used to find the probability of observing a value within a certain range. For example, the probability of observing a value within one standard deviation of the mean is about 68%, within two standard deviations is about 95%, and within three standard deviations is about 99.7%. The z-score can also be used to find the value that corresponds to a given probability. For example, the value that has a 16% probability of being below it is about -0.994 standard deviations away from the mean⁴.
To find the smallest expected loss for the portfolio in the next month with a 16% probability, we need to do the following steps:
- Convert the annual return mean and standard deviation of each stock to monthly values by dividing by 12. This is because the question asks for the loss in the next month, not the next year.
- Calculate the expected return and standard deviation of the portfolio using the formulas above. Since the portfolio allocates equal funds to each stock, the weights are both 0.5. Since the correlation between the returns of the two stocks is zero, the last term in the standard deviation formula is zero.
- Find the z-score that corresponds to a 16% probability of being below it using a normal distribution table or a calculator. This is -0.994.
- Multiply the z-score by the standard deviation of the portfolio and subtract it from the expected return of the portfolio. This gives the smallest expected loss for the portfolio in the next month with a 16% probability.
Using the data given in the question, we can calculate the answer as follows:
- The monthly return mean and standard deviation of DW are 15%/12 = 1.25% and 44%/12 = 3.67%, respectively.
- The monthly return mean and standard deviation of Woodpecker are 11.4%/12 = 0.95% and 58%/12 = 4.83%, respectively.
- The expected return of the portfolio is 0.5 x 1.25% + 0.5 x 0.95% = 1.1%.
- The standard deviation of the portfolio is $\sqrt{0.5^2 x 3.67^2 + 0.5^2 x 4.83^2 + 2 x 0.5 x 0.5 x 3.67 x 4.83 x 0}$ = 3.01%.
- The smallest expected loss for the portfolio in the next month with a 16% probability is 1.1% - 0.994 x 3.01% = -1.89%.
Therefore, the answer is -1.89%. This means that there is a 16% chance that the portfolio will lose more than 1.89% in the next month. Note that this is an expected value, not a guaranteed outcome. The actual loss could be higher or lower depending on the actual returns of the two stocks.