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 uncertain outcomes 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, ranging from $0$ (impossible) to $1$ (certain), and the probabilities of all possible outcomes must sum to $1.0$ (or $100\%$).

Calculating the expected value of random variables

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

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

Where:

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

Example: expected daily sales revenue

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.

Multiply each sales level by its probability and sum the products:

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

Answer: The expected daily sales revenue for FiveFriends Café is $\$1,080$. This value represents the average daily revenue FiveFriends 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 estimated future cash flows follows the same formula: multiply each possible cash flow by its probability of occurrence, then sum the products. The probabilities will normally be given, since computing probabilities of occurrence isn’t covered in the learning outcome statements.

Example: expected value and choosing between projects

Achievable Corp is evaluating a new product launch and must choose between two possible approaches. Management estimates the first-year cash flow outcomes for each:

Product A (uncertain outcomes):

• $\$50,000$ with a $30\%$ probability
• $\$30,000$ with a $50\%$ probability
• $\$10,000$ with a $20\%$ probability

Product B (contracted outcome):

• $\$32,000$ with a $100\%$ probability

Multiply each possible cash flow by its probability of occurrence, then sum the products:

$${\text{EV}}_A=(0.30\times \$50,000)+(0.50\times \$30,000)+(0.20\times \$10,000)$$

$${\text{EV}}_A=\$15,000+\$15,000+\$2,000=\$32,000$$

$${\text{EV}}_B=1.00\times \$32,000=\$32,000$$

Answer: Both products have the same expected value of $\$32,000$, but they are not equally risky. Product A’s actual outcome could range from $\$10,000$ to $\$50,000$, while Product B’s outcome is certain. Expected value treats the two as identical because it averages away that spread - this is what it means to say the technique assumes risk neutrality and ignores variability. A risk-averse manager would likely prefer Product B even though the expected values match.

Benefits and shortcomings of the expected value technique

Benefits

• Improved decision-making: Expected value gives one probability-weighted figure for comparing alternatives under uncertainty, making it easier to line up different projects or investments side by side.
• 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 outcome estimate: Expected value turns a range of uncertain outcomes into a single quantitative estimate; it does not by itself measure or adjust for risk, since that requires looking at the variability behind the average (see shortcomings below).

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.
• Assumes risk neutrality: Expected value collapses an entire distribution of outcomes into a single average, which implicitly treats the decision-maker as risk-neutral. It doesn’t capture how averse a manager or investor might be to downside risk, even when that risk matters more than the average outcome.
• 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.

Conclusion

Expected value estimates an average outcome by weighting each possible scenario by its probability. It’s most reliable for decisions that repeat many times, since the weighted average reflects a long-run result - for a one-off decision, the actual outcome can differ substantially from the expected value. For that reason, it works best alongside other planning tools like variance and sensitivity analysis.