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Data That Informs Choices
Quantitative Analysis · Chapter 5: Decision Analysis with ProbabilitiesSection 5.2 · lesson 21 of 5512 min · free with a free account
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In this lesson
Using probability and economic metrics in decision-making.
- Assign probability four ways: objective, subjective, prior, conditional
- Compute NPV and EMV as core decision metrics
- Reason with opportunity cost and marginal analysis
- Apply EMV to choose between two investment alternatives
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Probability and Economic Metrics · Section 5.2
Data That Informs Choices
Every serious decision under uncertainty needs two things, and only two. It needs probabilities, which say how likely each possible future is. And it needs economic metrics, which say how much each future is worth. Miss the probabilities, and you cannot weight the futures you face. Miss the metrics, and you cannot compare outcomes at all. This section builds both ingredients from the ground up. First, the four kinds of probability a quantitative analyst has to work with. Then four economic metrics that recur across almost every business decision. And finally, a fully worked example that puts them together.
Two Ingredients
What Every Decision Under Uncertainty Needs
Start with why the two ingredients are genuinely separate. A probability with no payoff attached tells you nothing about what to do. A payoff with no probability attached is just a number on a page. Probability answers one question. How likely is this future? Economic metrics answer the other. How valuable is this future? The section runs in three stages. Stage one covers four kinds of probability: objective, subjective, prior, and conditional. Stage two covers four economic metrics, from net present value through to marginal analysis. Stage three works a full example, comparing two investments across three market scenarios. None of this is a recommendation yet. It is the raw material recommendations are built from.
Objective Probability
Probability Read Straight Off the Data
An objective probability is computed directly from observed historical data. Suppose sixty out of two hundred past projects of a similar type overran budget. To count, the overrun had to exceed five per cent. Then the objective probability that a future project overruns is sixty divided by two hundred. That is zero point three, or thirty per cent. This is the frequentist reading, treating probability as a long run relative frequency. Objective probabilities convince because they rest on evidence, but they are only as good as that evidence. Two pitfalls come up again and again. First, the historical sample may simply not represent the future. A new competitor, a regulatory shift, or a pandemic breaks the link. Analysts call that a structural break. Second, the sample may be far too small for the estimate to be stable. So quote the sample size whenever you quote the probability.
Subjective Probability
When There Is No Data, There Is Judgment
When data is sparse, or the situation is genuinely novel, judgment has to take over. A subjective probability is an expert's degree of belief, expressed as a number between zero and one. I would give it a seventy per cent chance. That is a subjective probability. Such estimates are unavoidable in mergers and acquisitions, in product launches, and in geopolitical risk. They are also vulnerable to a well documented set of biases. Overconfidence, anchoring, availability, and motivated reasoning. So what makes a subjective number trustworthy at all? Calibration. Testing the expert's past forecasts against what actually happened. A well calibrated forecaster, saying seventy per cent, should be right about seventy per cent of the time. That holds across many such forecasts, never on any single one. Calibration training, using observable benchmarks and feedback, improves accuracy, and is now standard practice.
Prior and Conditional
Where You Start, and What Changes It
A prior probability is your estimate before any new information is observed. It captures everything currently known: history, expert judgment, and base rates. Priors can be objective, subjective, or a blend of the two. The key idea is that a prior is provisional. Evidence arrives, and you update it. A conditional probability is the probability of one event, given that another has already occurred. We write it as P of B given A. Formally, it is the probability of A and B together, divided by the probability of A. This is the bedrock of Bayesian updating, developed fully in Chapter Six. A firm is deciding whether to launch a new product, with an unconditional success probability of sixty per cent. If the economy turns out strong, the conditional probability of success rises to eighty per cent. If it turns out weak, success drops to thirty per cent. Weight a strong economy at zero point six and a weak one at zero point four. Combine the branches, and you recover exactly the sixty per cent you started with.
Side By Side
The Four Types of Probability
Put the four types side by side, and the structure becomes clear. Objective probability comes from frequency in historical data. Thirty per cent of two hundred past projects overran budget. Subjective probability is a calibrated expert's degree of belief. An expert saying, twenty five per cent chance of a recession next year. A prior is the initial estimate, made before any new evidence arrives. A base rate probability of failure of zero point four, for instance. The probability of default, given a weak economy, might be zero point one five. Objective and subjective describe where a number came from. Prior and conditional describe when in the analysis it gets used. The two pairs are independent.
Net Present Value
Discounting Value Back to Today
Probabilities tell us how likely each future is. Economic metrics tell us how valuable. The first metric is net present value. It is the sum of every cash flow associated with a project. Each one discounted back to today's value, at an appropriate discount rate. That rate carries the time value of money, and the risk of those particular cash flows. Inflows count as positive, outflows as negative. A positive net present value means discounted inflows exceed discounted outflows. The project creates value. A negative one destroys value, even when the undiscounted sum looks perfectly healthy. Take a five year project with an initial outlay of one hundred, and uneven inflows. The nominal inflows total one hundred and ninety, but discounted at ten per cent, their present value is only about one hundred and forty three. Net present value is therefore roughly forty three. Still positive, so the project adds value.
Expected Monetary Value
One Number for a Whole Distribution
The second metric is expected monetary value, usually shortened to E M V. It is the probability weighted average payoff across all possible states of nature. Multiply each payoff by its probability, then add the products together. What that buys you is comparability. A whole distribution of outcomes collapses into one single number. And single numbers can be ranked. So the basic decision rule is simple. Pick the alternative with the highest E M V. But that rule rides on two conditions, and both of them matter. First, the decision maker must be risk neutral. Second, the outcomes must not be large enough to threaten solvency. A bet with the best average payoff is still a bad bet, if losing would end the company. Drop either condition, and expected monetary value can point you the wrong way.
Cost and Margin
Opportunity Cost and Marginal Analysis
Two more metrics, and both are really about what you are giving up. Opportunity cost is the value of the next best alternative that was forgone. Suppose a firm invests one million dollars in Project A, and drops Project B. Project B would have earned one hundred and fifty thousand. Then the opportunity cost of Project A is that one hundred and fifty thousand. Notice what it does not depend on. How much Project A itself earns. Accounting profits ignore opportunity cost. Economic profits do not. If you are only comparing do the project against do nothing, you have almost certainly overlooked an opportunity cost. Marginal analysis asks a narrower question. What if we change by one small unit? Produce one more, hire one more, hold one more day of stock? Expand while marginal benefit exceeds marginal cost. Contract when it falls below. The optimum sits exactly where marginal benefit equals marginal cost. Only the incremental revenues and costs of the change matter. Total and sunk costs are irrelevant here.
Worked Example
Two Investments, Three Scenarios
Now put all of it to work on a real comparison. A manufacturing firm is evaluating two competing investment alternatives. Both require a similar capital outlay, but their payoffs diverge sharply. What each one pays depends on which market scenario unfolds over the next two years. Optimistic, at thirty per cent, meaning strong growth and high demand. Base, at fifty per cent, meaning moderate growth and the demand everyone expects. Pessimistic, at twenty per cent, meaning a slowdown with weak demand. Check the arithmetic before going any further. Those three probabilities add to one. If a payoff table's probabilities do not sum to one, nothing computed from it means anything. Alternative A, the aggressive one, pays one hundred and eighty thousand, then ninety thousand, then minus forty thousand. Alternative B, the conservative one, pays one hundred and twenty thousand, then eighty thousand, then twenty thousand. So A pays well in the good scenarios, but loses money in a downturn. B never loses money.
Computing EMV
Multiply, Then Sum
Apply the formula. Multiply each payoff by its probability, then sum across the states. Start with Alternative A. Zero point three times one hundred and eighty thousand gives fifty four thousand. Zero point five times ninety thousand gives forty five thousand. Zero point two times minus forty thousand gives minus eight thousand. Add them up: fifty four thousand, plus forty five thousand, minus eight thousand. The expected monetary value of A is ninety one thousand. Now take Alternative B. Zero point three times one hundred and twenty thousand gives thirty six thousand. Zero point five times eighty thousand gives forty thousand. Zero point two times twenty thousand gives four thousand. Adding those three gives eighty thousand. So ninety one thousand for A, against eighty thousand for B. A wins by eleven thousand.
Decide, Then Stress-Test
Why EMV Is an Input, Not the Decision
Under the expected monetary value rule, Alternative A is preferred. But a responsible analyst would not stop there. Two issues need surfacing before that recommendation goes anywhere. First, the risk profiles are not the same at all. Alternative A can lose forty thousand in the pessimistic scenario. Alternative B never loses money. Someone who is risk averse may rationally prefer B, despite its lower expected value. Especially if a forty thousand loss would be catastrophic for the firm. Remember, expected monetary value implicitly assumes risk neutrality. Second, the probabilities themselves are only estimates. Suppose the pessimistic case is really thirty five per cent. Both expected values fall, but A falls far faster, because A holds the only losing payoff. The ranking can reverse entirely, and B becomes the better choice. So vary each probability up and down by ten percentage points, and see which recommendation survives. None of this invalidates the metric. It shows why it is one input, not the whole decision.
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