Risk and Action Value Answer Different Questions
A prediction can identify people who are likely to experience an outcome. That is useful, but it does not by itself say whom an organization should contact, assist, or otherwise target. An intervention is worthwhile only when its expected benefit, compared with taking no action, justifies its cost and any other consequences.
Consider a hypothetical insurer deciding whom to contact before policy renewal. A model might estimate each policyholder’s probability of lapsing. A high lapse probability describes risk. But a call may have little effect on some people who are very likely to lapse, while people with lower lapse probabilities might be much more likely to renew if contacted. The most at-risk person need not be the person for whom a call creates the most value.
The distinction is aligned with the SOA Exam PA (Predictive Analytics) syllabus: a prediction describes an outcome under a specified context, while an action decision compares the expected consequences of alternative actions. Predictive accuracy alone does not establish that an intervention is effective or worthwhile.
A Simple Model of Action Value
For a clear comparison, suppose the outcome is an adverse event, and an intervention may reduce its probability. For a person or group with known characteristics, let \(p_0\) be the probability of the event without the intervention and \(p_1\) the probability with it. Let \(V\) be the monetary value assigned to avoiding one event, and let \(c\) be the cost of the intervention. Assume these values apply over the same time period and that there are no other consequences to include.
The difference \(p_0-p_1\) is the intervention’s absolute reduction in event probability. If it is positive, the intervention reduces risk under these assumptions. But a positive reduction is not sufficient for the intervention to be worthwhile: its expected benefit must also exceed its cost. A negative net value means that, on the specified assumptions, taking the action has lower expected value than not taking it.
The probabilities must represent a valid comparison of the two alternatives. In causal notation, imagine that \(Y(0)\) denotes the outcome that would occur without action and \(Y(1)\) the outcome that would occur with action. For an adverse binary outcome, \(p_0=P(Y(0)=1\mid X)\) and \(p_1=P(Y(1)=1\mid X)\) are the corresponding probabilities for a person or group with characteristics \(X\). Only one of these outcomes is observed for a given person. Estimating the difference therefore requires evidence about how outcomes change under the intervention; ordinary risk predictions alone do not provide that evidence.
Why the Net-Value Formula Holds
The formula follows by comparing expected outcomes under the two choices. Assume an adverse event creates a loss of \(V\), action costs \(c\), and there are no other costs or benefits. Without action, the event occurs with probability \(p_0\), so expected loss is \(Vp_0\). With action, expected loss from the event is \(Vp_1\), and the action also costs \(c\). Thus:
This derivation also shows the assumptions behind the result. It does not apply unchanged if the intervention has additional effects, if the value of an event varies across people, or if costs and benefits occur at different times and must be discounted. Those consequences would need to be included in the comparison. In the examples below, all amounts are hypothetical, apply over one period, and are undiscounted (an interest rate of 0% is used). No mortality assumption or separate probability distribution is required; the stated event probabilities are treated as Bernoulli probabilities with parameters \(p_0\) and \(p_1\).
Worked Example: Renewal Outreach and Unequal Responsiveness
An insurer considers a hypothetical renewal call costing $20 per policyholder. If a lapse is prevented, the insurer assigns that retained policy a value of $600. For a high-risk group, the estimated lapse probability is 0.40 without a call and 0.35 with a call. For a second group, it is 0.25 without a call and 0.10 with a call. Calculate each group’s net expected action value per call and identify which group has higher risk and which has higher action value.
Method. Use \(V(p_0-p_1)-c\) for each group. The given probabilities are assumed to represent the lapse rates under the two alternatives, and the $600 value applies to each lapse avoided. All amounts are undiscounted for one period; no mortality assumption or additional distribution is needed.
High-risk group. Its lapse probability falls from 0.40 to 0.35, so the absolute reduction is \(0.40-0.35=0.05\). The expected value of prevented lapses is \(600(0.05)=\$30\). Subtracting the $20 call cost gives \(\$30-\$20=\$10\) net expected value per call.
Second group. Its lapse probability falls from 0.25 to 0.10, so the absolute reduction is \(0.25-0.10=0.15\). The expected value of prevented lapses is \(600(0.15)=\$90\). Subtracting the $20 call cost gives \(\$90-\$20=\$70\) net expected value per call.
Check and interpretation. The high-risk group has the greater no-call lapse probability, 0.40 versus 0.25. The second group has the greater net action value, $70 versus $10, because the call is assumed to reduce its lapse probability by more. The calculation avoids the common mistake of using the highest predicted risk as a substitute for expected benefit from outreach.
Risk Does Not Measure Responsiveness or Consequence
Two distinct reasons can make risk a poor stand-in for action value. First, people with similar risk can respond differently to an intervention. The difference \(p_0-p_1\) captures this response in the simple model. Second, the consequences of the same event can have different values across people or cases. If the value of avoiding an event varies, action value depends on both the risk reduction and the event’s value.
The following comparison holds the intervention cost fixed but allows event values and risk reductions to differ. It shows why neither risk alone nor responsiveness alone always determines the best action.
Worked Example: Considering the Value of an Avoided Event
A hypothetical organization can offer the same one-period intervention to three groups. It costs $15 per person. For Group A, the adverse-event probability is 0.50 without action and 0.45 with action; an avoided event is valued at $1,000. For Group B, the probabilities are 0.30 and 0.20, and an avoided event is valued at $300. For Group C, the probabilities are 0.15 and 0.05, and an avoided event is valued at $1,000. Calculate each group’s net expected action value. All amounts are undiscounted (interest rate 0%); no mortality assumption or distribution beyond the stated Bernoulli probabilities is used.
Method. For each group, calculate the absolute probability reduction, multiply it by that group’s stated event value, and subtract the $15 intervention cost.
Group A. The probability reduction is \(0.50-0.45=0.05\). The expected benefit is \(0.05(\$1{,}000)=\$50\). The net value is \(\$50-\$15=\$35\).
Group B. The probability reduction is \(0.30-0.20=0.10\). The expected benefit is \(0.10(\$300)=\$30\). The net value is \(\$30-\$15=\$15\).
Group C. The probability reduction is \(0.15-0.05=0.10\). The expected benefit is \(0.10(\$1{,}000)=\$100\). The net value is \(\$100-\$15=\$85\).
Check and interpretation. Group A has the highest no-action event probability, 0.50, but Group C has the highest net action value, $85. Group B and Group C have the same probability reduction, but their event values differ, so their net values differ. The calculation avoids treating either baseline risk or probability reduction as sufficient when the consequences of an event are not equal.
Prioritizing When Outreach Is Limited
When the number of available interventions is limited, a risk ranking may produce a different priority list from a value-based ranking. Under the simple assumptions above, and when each action uses one unit of capacity, one approach is to calculate net expected action value for every eligible person or group and prioritize positive values from highest to lowest. If values are negative, the action is not justified by this calculation. Real decisions may also require other constraints or considerations, but they should be stated rather than hidden inside a risk score.
Worked Example: Choosing Two Targets Under a Call Limit
A hypothetical outreach team can call two policyholders. Each call costs $10. The value of avoiding a lapse is $800 for A, $300 for B, $500 for C, and $1,000 for D. Their estimated lapse probabilities without and with a call are, respectively: A, 0.50 and 0.45; B, 0.35 and 0.20; C, 0.25 and 0.10; D, 0.15 and 0.05. Calculate net expected value per call, then choose two targets. All amounts are undiscounted for one period (interest rate 0%); no mortality assumption or distribution beyond the given Bernoulli probabilities is used.
Method. Calculate the net value \(V(p_0-p_1)-c\) for each policyholder. With capacity for two calls, compare the two largest positive net values. This assumes the stated values are additive across calls and that calling one person does not change another person’s outcome.
A. The reduction is \(0.50-0.45=0.05\). Expected benefit is \(0.05(\$800)=\$40\), so net value is \(\$40-\$10=\$30\).
B. The reduction is \(0.35-0.20=0.15\). Expected benefit is \(0.15(\$300)=\$45\), so net value is \(\$45-\$10=\$35\).
C. The reduction is \(0.25-0.10=0.15\). Expected benefit is \(0.15(\$500)=\$75\), so net value is \(\$75-\$10=\$65\).
D. The reduction is \(0.15-0.05=0.10\). Expected benefit is \(0.10(\$1{,}000)=\$100\), so net value is \(\$100-\$10=\$90\).
Decision and check. The two highest net values are D at $90 and C at $65, so the team should call D and C under the stated assumptions. The two highest no-call lapse probabilities belong to A and B, so selecting targets by risk alone would choose a different pair. This calculation avoids that ranking mistake and checks that both selected actions have positive net value.
Common Traps in Framing the Decision
- Equating high risk with high benefit. A high \(p_0\) does not establish that an intervention substantially reduces risk. The comparison also needs \(p_1\), or other credible evidence about the action’s effect.
- Ignoring the event’s value. A probability reduction of 0.10 is not automatically worth the same amount in every case. The consequences of the event and the action’s other effects matter.
- Using the wrong comparison. The intervention’s value is measured against the no-action alternative, not by comparing treated people with a different group whose characteristics may differ.
- Assuming an association is an intervention effect. A model that predicts outcomes accurately does not, by that fact alone, show what would happen if a person were contacted. The treated-versus-untreated comparison must have a defensible basis.
- Forgetting costs or time period. A gross expected benefit is not net action value. Benefits and costs must refer to a consistent population and period; if timing matters, the comparison must account for it.
Check Your Understanding
For each question, use the stated assumptions and distinguish event risk from the value of acting.
- A hypothetical service call costs $18. An avoided adverse event is valued at $400. Its probability is 0.32 without the call and 0.24 with it. Calculate net expected action value and state whether the call has positive value under this calculation. All amounts are undiscounted for one period (interest rate 0%); no mortality assumption or distribution beyond the stated Bernoulli probabilities is used. (Answer: \(400(0.32-0.24)-18=\$14\); positive net value.)
- Two groups are eligible for a $10 intervention. For Group A, \(p_0=0.55\), \(p_1=0.53\), and \(V=\$1{,}000\). For Group B, \(p_0=0.30\), \(p_1=0.18\), and \(V=\$200\). Calculate both net values and identify the higher-value action. All amounts are undiscounted for one period (interest rate 0%); no mortality assumption or distribution beyond the stated Bernoulli probabilities is used. (Answer: A: \(1{,}000(0.02)-10=\$10\); B: \(200(0.12)-10=\$14\); Group B has higher action value.)
- A hypothetical prevention program costs $12 per person. The event probability is 0.20 without it and 0.12 with it; an avoided event is valued at $250. Calculate net action value and check whether the action is worthwhile under the stated assumptions. All amounts are undiscounted for one period (interest rate 0%); no mortality assumption or distribution beyond the stated Bernoulli probabilities is used. (Answer: \(250(0.20-0.12)-12=\$8\); positive net value.)
- A team can take one action. For A, the event probability changes from 0.50 to 0.48, an avoided event is valued at $1,000, and the action costs $10. For B, the probability changes from 0.25 to 0.10, an avoided event is valued at $300, and the action also costs $10. Which target has greater net action value? All amounts are undiscounted for one period (interest rate 0%); no mortality assumption or distribution beyond the stated Bernoulli probabilities is used. (Answer: A has net value \(1{,}000(0.02)-10=\$10\); B has net value \(300(0.15)-10=\$35\); choose B.)
- A model estimates a person’s probability of an adverse event without outreach as 0.40. The intervention cost and event value are known, but there is no estimate of the event probability with outreach. Can net action value be calculated from these facts alone? (Answer: No. The action’s effect, represented here by the with-action probability or a credible estimate of its change, is also needed.)