Expected Loss

Concept

Theoretically, Expected Loss is the probability-weighted average of losses that may arise from uncertain future events. Within decision theory and health economics, it represents the expected adverse consequence associated with risk and uncertainty, enabling alternative interventions or policies to be evaluated on the basis of their anticipated losses rather than individual outcomes. The concept underpins risk-based decision making and resource allocation where uncertainty is explicitly recognised.

Mathematically, expected loss is represented as the expected value of a loss function over all possible outcomes. The mathematical framework combines the probability of each outcome with its associated loss, producing a single summary measure of anticipated loss. In decision analysis, expected loss is commonly minimised to identify the preferred strategy under uncertainty.

In practice, expected loss is estimated using probabilities derived from clinical studies, epidemiological data, expert elicitation, or probabilistic models, together with monetary, health, or utility losses assigned to each outcome. It is implemented in decision trees, Markov models, Bayesian decision analysis, and value of information analyses to compare healthcare interventions and assess the consequences of uncertainty.

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Purpose

Used to quantify the average anticipated loss associated with uncertain healthcare decisions, compare competing interventions under uncertainty, support risk-based resource allocation, and identify strategies that minimise expected adverse outcomes.

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Mathematical Formulae

Primary Formula

EL = ????� P?L?

where:

• EL = expected loss
• P? = probability of outcome i
• L? = loss associated with outcome i

Supporting Formulae

Continuous form:

EL = ? L(x)f(x) dx

where:

• L(x) = loss function
• f(x) = probability density function

Related Mathematical Methods

• Expected utility theory
• Bayesian decision analysis
• Decision tree analysis
• Markov decision modelling
• Probabilistic sensitivity analysis
• Value of information analysis

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Example

A screening programme has two possible outcomes:

• Serious adverse event: probability = 0.02; loss = �250,000
• No serious adverse event: probability = 0.98; loss = �0

Expected loss:

EL = (0.02 ? 250,000) + (0.98 ? 0) = �5,000

The programme therefore has an expected loss of �5,000 per patient from adverse events, which can be incorporated into an overall economic evaluation alongside expected benefits.

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Excel Implementation

Function	Example Formula	Health Economics Application
SUMPRODUCT	=SUMPRODUCT(B2:B6,C2:C6)	Calculates expected loss from outcome probabilities and associated losses.
SUM	=SUM(D2:D6)	Totals probability-weighted losses after calculating each outcome separately.
IF	=IF(A2=""Adverse Event"",B2*C2,0)	Applies losses only to specified health outcomes.

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VBA (Optional)

Automate the calculation of expected losses across multiple probabilistic scenarios and generate comparative summaries for alternative health interventions.

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Sources

• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
• Raiffa H, Schlaifer R. Applied Statistical Decision Theory. Harvard University Press.
• ISPOR-SMDM Modeling Good Research Practices Task Force reports.