Introduction
A business or investment choice rarely offers a certain reward in exchange for a certain cost. One alternative may have a higher average payoff but a greater chance of a loss; another may be less likely to miss a target but have a more severe worst-case outcome. Comparing choices therefore requires more than ranking them by one number.
The useful first step is to describe the possible outcomes on a consistent basis. Then compare both the potential reward and the uncertainty that matters to the decision. The appropriate comparison depends on the objective: a choice that suits an organization willing to accept substantial downside for higher expected returns may be unsuitable when preserving a minimum amount of capital is the priority.
Put Choices on a Common Basis
Before calculating, specify what counts as an outcome. It might be an investment return over one year or the net profit from a business initiative over a defined period. Compare like with like: a one-year percentage return should not be directly compared with a five-year dollar profit unless the difference in time horizon and units is addressed.
For a discrete model, suppose an outcome \(R\) can take values \(r_1,\ldots,r_n\), with probabilities \(p_1,\ldots,p_n\). Assume the outcomes are mutually exclusive and collectively exhaustive, and that \(p_j\) is the probability of outcome \(r_j\), with \(p_j\geq 0\) and \(\sum_{j=1}^n p_j=1\). The expected outcome is the probability-weighted average:
Expected return is a long-run average across repetitions of the modeled situation, not a prediction that any single trial will produce that value. If outcomes are dollar profits, the expected outcome is in dollars; if they are rates, it is a rate.
One measure of dispersion around the expected outcome is variance, the probability-weighted average squared deviation from the mean. Standard deviation is the square root of variance and is expressed in the same units as the outcomes.
These formulas require a specified probability model. If the probabilities are estimates rather than known values, the resulting expected return and dispersion inherit that uncertainty. Neither measure alone identifies the probability of a loss, the size of a severe loss, or the chance of meeting a particular target.
Worked Comparisons
A sound comparison combines a summary of central tendency with measures tied to the decision. Depending on the objective, those may include the probability of a negative outcome, the size of the loss if one occurs, or the probability of falling below a required threshold. The examples below use hypothetical choices and explicitly state their modeled outcomes and probabilities.
Worked Example: Comparing Two One-Year Investments
A hypothetical investor has $100,000 and is considering two one-year investments. Investment A returns 8% with probability 0.75 or loses 8% with probability 0.25. Investment B returns 3% with probability 0.90 or loses 6% with probability 0.10. Compare expected return, standard deviation, and the chance and size of a loss.
Method. Calculate the probability-weighted return for each investment, then examine dispersion and downside separately. Assume the stated outcomes are exhaustive for the one-year period, and ignore taxes, fees, and any interim cash flows.
Investment A: expected return. The weighted average is
The expected ending value is \(100{,}000(1.04)=\$104{,}000\). This is an average across the modeled outcomes, not a guaranteed ending value.
Investment A: dispersion and downside. Deviations from the 4% mean are 4 percentage points and -12 percentage points. Thus,
The probability of a loss is 0.25, and the loss in the loss outcome is 8% of $100,000, or $8,000.
Investment B. Its expected return is \(0.90(0.03)+0.10(-0.06)=0.027-0.006=0.021\), or 2.1%. The expected ending value is \(100{,}000(1.021)=\$102{,}100\). Deviations from the mean are 0.9 and -8.1 percentage points, so the variance is \(0.90(0.009)^2+0.10(-0.081)^2=0.0000729+0.0006561=0.000729\), and the standard deviation is \(\sqrt{0.000729}=2.7\%\). The probability of a loss is 0.10, and the loss in that outcome is 6%, or $6,000.
Investment A has the higher expected return, while B has lower standard deviation, a lower probability of loss, and a smaller loss in its loss outcome. Which is preferable depends on the investor’s objective and capacity to bear loss. Common mistake avoided: treating the 4% expected return as a promise that A will earn 4%. Its modeled outcomes are instead 8% and -8%.
Worked Example: Comparing Business Initiatives Against a Target
A hypothetical insurer is considering one of two initiatives. Initiative A has a 0.70 probability of producing a net profit of $600,000 and a 0.30 probability of a net loss of $500,000. Initiative B has a 0.90 probability of producing a net profit of $210,000 and a 0.10 probability of a net loss of $100,000. The decision objective requires at least $150,000 of profit. Compare expected profit, the chance of meeting the objective, and downside.
Method. Compute the probability-weighted profit first. Then classify each outcome against both zero profit and the $150,000 target. Assume the amounts are net outcomes over the same planning period and the probabilities sum to one for each initiative.
For A, expected profit is \(0.70(\$600{,}000)+0.30(-\$500{,}000)=\$420{,}000-\$150{,}000=\$270{,}000\). Its probability of a loss is 0.30. If the loss outcome occurs, the loss is $500,000; the probability-weighted loss amount, counting zero for the profitable outcome, is \(0.30(\$500{,}000)=\$150{,}000\). A meets the $150,000 target only in its profit outcome, so its probability of meeting the target is 0.70.
For B, expected profit is \(0.90(\$210{,}000)+0.10(-\$100{,}000)=\$189{,}000-\$10{,}000=\$179{,}000\). Its probability of a loss is 0.10. The loss in that outcome is $100,000, and the probability-weighted loss amount is \(0.10(\$100{,}000)=\$10{,}000\). B meets the target in its $210,000 outcome, so its probability of meeting the target is 0.90.
A has the higher expected profit, but B has a greater probability of meeting the stated target and a smaller loss if its loss outcome occurs. The target matters: comparing only expected profit would not answer whether the decision objective is likely to be achieved. Common mistake avoided: comparing the profit outcomes while ignoring the probabilities. A’s $600,000 profit is not its expected profit; the weighted expected profit is $270,000.
Worked Example: Equal Expected Outcomes, Different Downside Profiles
Two hypothetical projects have the same planning horizon and report net profit in thousands of dollars. Project C produces $200,000 with probability 0.50 or $0 with probability 0.50. Project D produces $130,000 with probability 0.90 or a loss of $170,000 with probability 0.10. Compare their expected profits, standard deviations, and probabilities of falling below a $50,000 minimum.
Method. Work in thousands of dollars throughout. Calculate each mean and standard deviation, then apply the target directly to each possible outcome. Assume the two listed outcomes for each project are mutually exclusive and exhaustive.
For C, the expected profit is \(0.50(200)+0.50(0)=100\), or $100,000. The deviations from the mean are 100 and -100, so its variance is \(0.50(100)^2+0.50(-100)^2=10{,}000\), in squared thousands of dollars. Its standard deviation is \(\sqrt{10{,}000}=100\), or $100,000. The only outcome below $50,000 is zero, so the probability of missing the minimum is 0.50.
For D, the expected profit is \(0.90(130)+0.10(-170)=117-17=100\), also $100,000. Its deviations from the mean are 30 and -270. Its variance is \(0.90(30)^2+0.10(-270)^2=810+7{,}290=8{,}100\), and its standard deviation is \(\sqrt{8{,}100}=90\), or $90,000. Only the -$170,000 outcome falls below $50,000, so the probability of missing the minimum is 0.10.
D has a lower standard deviation and a lower probability of falling below the stated minimum, but its adverse outcome is a $170,000 loss; C never produces a loss. Thus, a lower standard deviation does not mean that every aspect of downside is better. Common mistake and correction: averaging D’s two profits equally gives \((130-170)/2=-20\) thousand, but the outcomes are not equally likely. Weighting them by their stated probabilities gives \(0.90(130)+0.10(-170)=100\) thousand. The calculation must use the model’s probabilities, not an unweighted average.
A Practical Comparison Process
A structured comparison helps prevent a calculation from answering a different question than the decision-maker intended. Use the following sequence before selecting an alternative.
Identify what counts as success or failure, the relevant time period, and whether the decision concerns a dollar outcome, a rate, or another quantity.
Check that each alternative uses a consistent basis and that its modeled outcomes cover the possibilities being considered. State assumptions and limitations.
Find the probability-weighted expected outcome. If useful, calculate standard deviation as a summary of dispersion around that expected value.
Calculate the probability of a loss or of missing a target, and inspect the size of the adverse outcomes. Do not infer these details from the mean alone.
State which alternative performs better on each relevant dimension, then identify what additional preference or constraint is needed to choose between them.
Why No Single Summary Settles the Choice
Expected return gives equal mathematical weight to all outcomes according to their probabilities, but it does not say which outcomes matter most to a particular decision-maker. Standard deviation measures spread around the mean, including favorable outcomes as well as unfavorable ones. A chance of loss focuses on whether an outcome crosses zero, but it does not distinguish a small loss from a very large one. A target-based probability answers whether a specified threshold is met, but it does not describe the outcomes on either side of that threshold.
These measures are useful descriptions, not universal decision rules. A choice should be judged against a stated objective and relevant constraints. For example, a manager responsible for meeting a minimum funding need may prioritize the chance of falling short, while another decision-maker may accept a higher chance of loss in pursuit of a larger expected profit. The numbers clarify the trade-off; they do not establish a preference unless the objective or decision criteria do so.
Check Your Understanding
For each hypothetical situation, compare the stated outcomes using the requested measures. Keep the time horizon and units consistent.
- A one-year investment of $50,000 returns 12% with probability 0.60 and loses 5% with probability 0.40. Find its expected return, expected ending value, and probability of loss. (Answer: expected return 5.2%; expected ending value $52,600; probability of loss 0.4000.)
- Choice X produces a net profit of $90,000 with probability 0.80 or a loss of $60,000 with probability 0.20. Choice Y produces $55,000 with probability 0.90 or a loss of $20,000 with probability 0.10. Compare expected profit and loss probability. (Answer: X has expected profit $60,000 and loss probability 0.2000; Y has expected profit $47,500 and loss probability 0.1000.)
- Project R produces $40,000 or $10,000, each with probability 0.50. Project S produces $35,000 with probability 0.80 or a loss of $15,000 with probability 0.20. Find each expected profit and the probability of falling below $20,000. (Answer: both expected profits are $25,000; R has probability 0.5000 of falling below $20,000, while S has probability 0.2000.)
- A hypothetical business choice has a 0.75 probability of earning $240,000 and a 0.25 probability of losing $120,000. Find expected profit, loss probability, and probability of meeting a $200,000 target. (Answer: expected profit $150,000; loss probability 0.2500; probability of meeting the target 0.7500.)
- Two choices have expected profits of $80,000. One has a 0.50 probability of earning $160,000 and a 0.50 probability of earning $0; the other earns $100,000 with probability 0.90 or loses $100,000 with probability 0.10. For a minimum-profit target of $50,000, compare the probability of missing the target and the loss outcome. (Answer: the first choice misses the target with probability 0.5000 and has no loss outcome; the second misses it with probability 0.1000 and has a $100,000 loss outcome.)