Question

An investment has a 50 percent chance of generating a 10 percent return and a 50 percent chance of generating a 16 percent return. What is the investment's average expected rate of return? a. 10 percent. b. 11 percent. c. 12 percent. d. 13 percent. e. 14 percent. f. 15 percent. g. 16 percent.

Short answer

The investment's average expected rate of return is 13% (option d).

Step-by-step solution

Understand Expected Value

To find the average expected rate of return for this investment, we use the concept of expected value. The expected value is a weighted average of all possible values, where each value is weighted by its probability of occurrence.

Identify Probabilities and Returns

The investment has two possible outcomes: a 10% return with a 50% probability and a 16% return with a 50% probability. These probabilities and returns will be used to calculate the expected return.

Calculate Expected Return

To calculate the expected return, multiply each return by its probability and sum the results. This can be expressed with the formula: \[ E(R) = (P_1 \times R_1) + (P_2 \times R_2) \]Where \( P_1 = 0.5 \), \( R_1 = 10\% \), \( P_2 = 0.5 \), and \( R_2 = 16\% \).

Perform the Calculation

Substitute the values into the formula: \[ E(R) = (0.5 \times 10) + (0.5 \times 16) \]\[ E(R) = 5 + 8 = 13 \]So, the expected return is 13%.

Choose the Correct Answer

Compare the calculated expected return with the given options. The correct answer is 13%, which corresponds to option d.

Key concepts

Probability

Probability is a key concept in mathematics and statistics that measures how likely an event is to occur. In the context of investments, it helps in evaluating the chances of different returns. Probability is expressed as a number between 0 and 1. Here, the investment scenario gives us a 50% probability for two different returns. A 50% probability means there is an equal chance, or half the likelihood, of either outcome occurring.
To effectively use probabilities, follow these steps:

• Identify all possible outcomes of an event.
• Assign probabilities to these outcomes, ensuring they sum up to 1.
• Use these probabilities to make informed decisions, particularly in finance where potential profits and losses are concerned.

Understanding probabilities allows investors to better gauge the risks and expected returns of investments.

Weighted Average

The weighted average is an essential concept used whenever different values have different levels of importance. Unlike a simple average, which treats all values equally, a weighted average assigns different weights to different values. This is essential in financial calculations, like determining the expected return on investment.
The formula for a weighted average is:
\[ \text{Weighted Average} = \sum (w_i \times x_i) \]
where \( w_i \) is the weight of each value, and \( x_i \) is each value itself. For investment returns, the probability acts as the weight. This makes the weighted average a powerful tool to summarize the varying impacts of different returns based on their probabilities. By focusing on weighting, investors can assess expected returns accurately.

Return on Investment

Return on Investment (ROI) is a measure used to evaluate the performance of an investment. It is expressed as a percentage comparing the profit or loss from an investment relative to its cost. ROI helps investors understand how efficiently their money is being used.
To calculate ROI, use the formula:
\[ \text{ROI} = \left( \frac{{\text{Net Profit}}}{{\text{Cost of Investment}}} \right) \times 100 \]
This measurement is valuable because it provides a straightforward indication of profitability. In scenarios with multiple potential returns, like our given exercise, the expected rate of return becomes a critical factor. Calculating ROI based on expected returns ensures that investment choices consider all possible outcomes accurately.

Investment Analysis

Investment Analysis involves assessing financial assets to make decisions about where to allocate resources. Expected value, weighted averages, and ROI are crucial components of this analysis. By leveraging these tools, investors can comprehensively evaluate potential investments and their associated risks.
Steps in investment analysis include:

• Identifying potential investment vehicles.
• Calculating potential returns and risks using probabilities and weighted averages.
• Estimating long-term profitability through ROI.

Using these analyses improves the decision-making process by quantifying the trade-off between risk and return. This analytical approach accommodates various outcomes, equipping investors with insights to optimize their portfolios and achieve desired financial goals.