Question

Compute the risk \(\operatorname{Var}\left(K_{1}\right), \operatorname{Var}\left(K_{2}\right)\) and \(\operatorname{Var}\left(K_{3}\right)\) in each of the following three investment projects, where the returns \(K_{1}, K_{2}\) and \(K_{3}\) depend on the market scenario: \(\begin{array}{llllc}\text { Scenario } & \text { Probability } & \text { Return } K_{1} & \text { Return } K_{2} & \text { Return } K_{3} \\\ \omega_{1} & 0.25 & 12 \% & 11 \% & 2 \% \\ \omega_{2} & 0.75 & 12 \% & 13 \% & 22 \%\end{array}\) Which of these is the most risky and the least risky project?

Short answer

K3 is the most risky project with variance 62.5×10^-3. K1 is the least risky with variance 0.

Step-by-step solution

- Compute the Expected Return for Each Project

The expected return (\text{E}) of each project is a weighted average of returns based on their probabilities. Use the formula: \[ \text{E}(K_i) = \text{P}(\text{Scenario 1}) \times \text{Return in Scenario 1} + \text{P}(\text{Scenario 2}) \times \text{Return in Scenario 2} \] For each project, calculate the expected return.

- Calculate Expected Return for Project K1

\[ \text{E}(K_1) = 0.25 \times 12\text{\text%} + 0.75 \times 12\text{\text%} = 12\text{\text%} \]

- Calculate Expected Return for Project K2

\[ \text{E}(K_2) = 0.25 \times 11\text{\text%} + 0.75 \times 13\text{\text%} = 12.5\text{\text%} \]

- Calculate Expected Return for Project K3

\[ \text{E}(K_3) = 0.25 \times 2\text{\text%} + 0.75 \times 22\text{\text%} = 17\text{\text%} \]

- Compute Variance for Each Project

Variance (\text{Var}) measures the dispersion of returns from the expected return. Use the formula: \[ \text{Var}(K_i) = \text{P}(\text{Scenario 1}) \times (\text{Return in Scenario 1} - \text{E}(K_i))^2 + \text{P}(\text{Scenario 2}) \times (\text{Return in Scenario 2} - \text{E}(K_i))^2 \] Calculate the variance for each project.

- Calculate Variance for Project K1

\[ \text{Var}(K_1) = 0.25 \times (12\text{\text%} - 12\text{\text%})^2 + 0.75 \times (12\text{\text%} - 12\text{\text%})^2 = 0 \]

- Calculate Variance for Project K2

\[ \text{Var}(K_2) = 0.25 \times (11\text{\text%} - 12.5\text{\text%})^2 + 0.75 \times (13\text{\text%} - 12.5\text{\text%})^2 = 0.625 \times 10^{-3} \]

- Calculate Variance for Project K3

\[ \text{Var}(K_3) = 0.25 \times (2\text{\text%} - 17\text{\text%})^2 + 0.75 \times (22\text{\text%} - 17\text{\text%})^2 = 62.5 \times 10^{-3} \]

- Compare the Risks

Compare the variances calculated: \[ \text{Var}(K_1) = 0 \] \[ \text{Var}(K_2) = 0.625 \times 10^{-3} \] \[ \text{Var}(K_3) = 62.5 \times 10^{-3} \] \(K_3\) is the most risky and \(K_1\) is the least risky project.

Key concepts

expected return

The expected return is a crucial concept in investment and finance. It represents the average return that an investor anticipates to earn from an investment project based on different possible market scenarios and their probabilities. To compute the expected return (\text{E}) of a project, you can use the formula:

\[ \text{E}(K_i) = \text{P}(\text{Scenario 1}) \times \text{Return in Scenario 1} + \text{P}(\text{Scenario 2}) \times \text{Return in Scenario 2} \]

This formula is straightforward to use. You simply multiply the return from each market scenario by the probability of that scenario occurring, and then sum these values. It's like finding a weighted average. For instance, let's calculate the expected returns (\text{E}(K_1)) and \text{E}(K_2). Using the probabilities and returns given, we get: \[ \text{E}(K_1) = 0.25 \times 12\% + 0.75 \times 12\% = 12\% \] \[ \text{E}(K_2) = 0.25 \times 11\% + 0.75 \times 13\% = 12.5\% \] For project K3, you use the same formula but with its respective returns and probabilities: \[ \text{E}(K_3) = 0.25 \times 2\% + 0.75 \times 22\% = 17\% \] Notice how the expected return helps investors understand the average potential outcome.

variance calculation

Variance is a measure of risk in investment. It shows how much the returns of a project deviate from the expected return. A higher variance indicates more risk because the returns vary widely, whereas a lower variance shows that the returns are more stable. Think of it as a measure of uncertainty. To calculate the variance (\text{Var}) of a project, use the formula:

\[ \text{Var}(K_i) = \text{P}(\text{Scenario 1}) \times (\text{Return in Scenario 1} - \text{E}(K_i))^2 + \text{P}(\text{Scenario 2}) \times (\text{Return in Scenario 2} - \text{E}(K_i))^2 \]

Let’s look at project K1 first. Given that the returns are the same for both scenarios, the variance is zero, as shown: \[ \text{Var}(K_1) = 0.25 \times (12\% - 12\%)^2 + 0.75 \times (12\% - 12\%)^2 = 0 \]

For project K2, the variance calculation will be:
\[ \text{Var}(K_2) = 0.25 \times (11\% - 12.5\%)^2 + 0.75 \times (13\% - 12.5\%)^2 = 0.625 \times 10^{-3} \]

Lastly, project K3:
\[ \text{Var}(K_3) = 0.25 \times (2\% - 17\%)^2 + 0.75 \times (22\% - 17\%)^2 = 62.5 \times 10^{-3} \]

By comparing variances, you see that project K3 has the highest variance, suggesting it is the most risky, whereas project K1, with zero variance, is the least risky.

market scenarios

Understanding market scenarios is vital in evaluating investment projects. These scenarios represent different possible states of the market which could affect the returns of an investment. Each scenario is associated with a probability that indicates how likely it is to occur.

In our exercise, we consider two market scenarios with associated probabilities and returns. For example:

• Scenario \(\text{w}_1\): Probability = 0.25, Return on K1 = 12%, Return on K2 = 11%, Return on K3 = 2%
  
• Scenario \(\text{w}_2\): Probability = 0.75, Return on K1 = 12%, Return on K2 = 13%, Return on K3 = 22%
  

These scenarios illustrate the uncertainty of returns due to different market conditions. It's the combined impact of these probabilities and returns that provides an investor with a comprehensive view of potential outcomes. When analyzing investment projects, always consider various market scenarios to get a well-rounded perspective of potential risks and returns.