Expected value

Learning outcome statements

The learning outcome statements relevant for this section are:

1. calculate the expected value of random variables
2. identify the benefits and shortcomings of the expected value technique
3. use probability values to estimate future cash flows

The concept of Expected Value (EV) is widely used in financial decision-making to help estimate the likely outcomes of uncertain future events.

Definitions

Expected value

This represents the weighted average of all possible values a random variable can take, where the weights are the probabilities of each outcome.

By calculating expected values, decision-makers can quantify risk and make more informed financial and strategic choices.

Before diving into the expected value calculation, it’s useful to understand some basic probability principles. Probability measures the likelihood of an event occurring and ranges from $0\text{ (impossible event)}$ to $1\text{ (certain event)}$. In expected value calculations, probabilities are used to assign weights to each potential outcome, reflecting the likelihood of each scenario. The total of all probabilities should equal $1.0$ (or $100\%$), as these represent all possible outcomes.

Calculating the expected value of random variables

The formula for calculating the expected value of a discrete random variable is:

$$\text{Expected Value (EV)}=\sum \text{(P(x)}\times \text{x)}$$

Where:

• $\text{x}$ represents each possible outcome,
• $\text{P(x)}$ is the probability of that outcome occurring, and
• ∑ indicates the sum of all weighted outcomes.

Example: FiveFriends Café is analyzing its expected daily sales revenue for budgeting purposes. The café manager has observed five possible sales scenarios on a typical day, each with an estimated probability based on past sales trends. The manager wants to calculate the expected daily sales revenue to aid in planning and resource allocation.

Sale scenario	Daily sales ($)	Probability
Very high	2,000	10%
High	1,500	20%
Average	1,000	40%
Low	700	20%
Very low	400	10%

Using the formula for expected value, we can calculate the expected daily sales. We need to multiple each scenario by its probability and then add the amounts for each scenario into one single total.

$$\text{Expected Value (EV)}=\sum \text{(P(x)}\times \text{x)}$$

Sale scenario	Daily sales ($)	Probability	Expected value
Very high	2,000	10%	$200
High	1,500	20%	$300
Average	1,000	40%	$400
Low	700	20%	$140
Very low	400	10%	$40
Total			$1,080

Thus, the expected daily sales revenue for FiveFriends Café is $\$1,080$. This value represents the average daily revenue GreenLine Café can expect over time, accounting for the variability of sales under different conditions.

Using probability values to estimate future cash flows

The expected value technique is particularly helpful for estimating future cash flows when businesses face uncertainty. By assigning probability weights to various cash flow scenarios, companies can project expected inflows or outflows, allowing for more precise financial planning. This approach is especially useful in budgeting and investment decisions, where cash flows might depend on variables like market demand, cost fluctuations, or economic conditions.

The calculation of the estimated future cash flows will follow the same formula presented above where candidates will need to multiply the expected cash flow to its probability of occurrence and then summing up all the products. The probabilities will normally be provided because the detailed computation of the probability of occurrences are not covered in the learning outcome statements.

Benefits and shortcomings of the expected value technique

Benefits

• Improved decision-making: Expected value enables management to incorporate potential risks into decision-making by weighing each possible outcome. This helps in comparing different projects or investments on a risk-adjusted basis.
• Financial planning precision: By estimating future cash flows based on probable scenarios, the expected value method aids in budgeting, capital allocation, and long-term planning, aligning projected resources with strategic goals.
• Quantifiable risk assessment: Expected value calculations allow for a quantitative approach to risk assessment, helping managers decide how much risk is acceptable relative to the expected gains.

Shortcomings

• Dependency on accurate probability estimates: The reliability of expected value calculations heavily depends on accurate probability estimates, which can be challenging to determine. If probabilities are misjudged, the expected value may not reflect true potential outcomes.
• Simplification of complex scenarios: Expected value calculations often assume a linear relationship between probabilities and outcomes, which can oversimplify complex, non-linear real-world scenarios.
• Ignores variability: Expected value provides an average or “central” estimate but doesn’t account for the spread of potential outcomes (variance), which can be crucial in high-risk decisions.

In summary, expected value serves as a valuable tool for financial forecasting and planning, offering insights into potential outcomes based on probability-weighted averages. However, it’s essential to supplement expected value with other risk analysis tools, such as variance and sensitivity analysis, to capture the full picture of potential risks and returns.

Conclusion

The expected value (EV) technique is a useful tool for estimating average outcomes by considering all possible scenarios and their probabilities.

In business, EV helps managers predict future cash flows, sales, and other metrics, supporting budgeting, forecasting, and risk management. However, as an average measure (and long-term), EV doesn’t guarantee outcomes, so actual results may vary. While powerful, EV works best in conjunction with other planning tools for comprehensive decision-making.