Why Correlation Changes a Portfolio Decision
Two development programs can have attractive individual probabilities of technical success and still provide less diversification than their separate probabilities suggest. They may rely on the same biological mechanism, platform, manufacturing process, or operational network. If a shared weakness causes both programs to fail together, the portfolio has a greater chance of producing no success than an independence calculation would imply.
For two programs, the probability of at least one success depends on both programs’ marginal success probabilities and their joint success probability. Expected monetary return has a different property: when portfolio return is the sum of program returns, its expectation is the sum of their expectations, whether or not outcomes are correlated. Dependence matters for the spread of returns and for probabilities of portfolio events such as no success.
The success indicators are binary outcomes, not measures of the underlying biology or operations. A correlation between indicators describes how the two success events co-occur. It does not, by itself, identify the mechanism that creates their dependence. A shared-risk scenario, such as a platform being either fit or impaired, can give a more interpretable model of that mechanism.
Joint Probabilities and Correlation
For two binary indicators, the covariance is \(q-p_Ap_B\). If both success probabilities are strictly between 0 and 1, their correlation is \(\rho=(q-p_Ap_B)/\sqrt{p_A(1-p_A)p_B(1-p_B)}\). Equivalently, a specified correlation implies a joint success probability through this equation. But not every numerical correlation is feasible: the joint probability must satisfy the bounds below.
These bounds follow because an intersection cannot be larger than either event, and the probability of the union cannot exceed 1. A dependence assumption outside these bounds cannot describe any joint distribution with the stated marginal probabilities. When either marginal probability is 0 or 1, the correlation formula has a zero denominator; use the joint-probability formulation instead.
Worked Example: Calculating the Chance of Any Success
Suppose two hypothetical programs have success probabilities \(p_A=0.60\) and \(p_B=0.50\). A joint-risk assessment assigns probability \(q=0.4225\) to both succeeding. First check that this joint probability is feasible: its lower bound is \(\max(0,0.60+0.50-1)=0.10\), and its upper bound is \(\min(0.60,0.50)=0.50\). Thus \(q=0.4225\) is feasible.
The probability of at least one success is
For comparison, if the programs were independent, the joint success probability would be \(0.60(0.50)=0.30\), and the probability of at least one success would be \(0.60+0.50-0.30=0.80\). Here the positive dependence raises the probability of joint success and lowers the chance of at least one success, relative to independence. The probability of neither succeeding is \(1-0.6775=0.3225\).
The joint probability can also be checked by listing the four possible outcomes: both succeed, A alone succeeds, B alone succeeds, and neither succeeds. Their probabilities are respectively \(0.4225\), \(0.60-0.4225=0.1775\), \(0.50-0.4225=0.0775\), and \(0.3225\). They are nonnegative, sum to 1, and reproduce both stated marginal probabilities.
Justification: Why the Union Formula Works
Adding \(p_A\) and \(p_B\) counts the outcome in which both programs succeed twice: once as an A success and once as a B success. Subtracting the joint probability removes that extra count. This inclusion-exclusion argument gives the union formula without requiring independence.
Independence is a special case, not a default consequence of having separate programs. Under independence, the probability that neither succeeds is \((1-p_A)(1-p_B)\), so the probability of at least one is \(1-(1-p_A)(1-p_B)\). Under dependence, the relevant no-success probability is \(P(S_A=0,S_B=0)\); it must come from the joint model or data, not from multiplying marginal failure probabilities.
Representing Shared Risks with Scenarios
One practical way to describe a common risk is to define a shared state \(Z\), such as whether a platform performs adequately. Specify the probability of each state and each program’s conditional success probability in that state. If programs are assumed conditionally independent given \(Z\), their conditional joint probability is the product of the conditional probabilities. Averaging those conditional results over the states gives the marginal and joint probabilities used for the portfolio calculation.
Conditional independence is an assumption, not an automatic property of a scenario model. It says that after the shared state has been specified, no additional dependence between program outcomes remains. If programs also share risks not captured by \(Z\), the conditional-independence assumption may be inadequate.
Worked Example: A Shared Platform Risk
Consider two hypothetical programs using the same platform. The platform is adequate with probability 0.30 and impaired with probability 0.70. Conditional on an adequate platform, each program succeeds with probability 0.80; conditional on an impaired platform, each succeeds with probability 0.20. Assume the programs are conditionally independent within either platform state.
For either program, the marginal success probability is the state-weighted average:
The probability that both succeed is the weighted average of the conditional joint probabilities:
Therefore, the probability of at least one success is \(0.38+0.38-0.22=0.54\). Treating the programs as independent based only on their marginal probabilities would instead give \(1-(1-0.38)^2=0.6156\). The difference arises because the shared platform state makes joint failure more likely than the product of the marginal failure probabilities. In fact, the probability that both fail is \(1-0.38-0.38+0.22=0.46\), whereas the independence calculation gives \(0.62^2=0.3844\).
This example isolates a shared-risk mechanism; it does not establish that the stated scenario probabilities are well supported. In an actual decision, the state definitions and probabilities need scientific and operational justification, and results should be checked under plausible alternatives.
Expected Return and Portfolio Risk
Dependence does not change the expected value of a sum when each program’s expected return is held fixed. Let a hypothetical program’s net return be \(V_i=R_iS_i-C_i\), where \(R_i\) is its fixed value if successful and \(C_i\) is its fixed investment cost. If both investments are made, total return is \(V_A+V_B\). By linearity of expectation, its expected value is \(R_Ap_A-C_A+R_Bp_B-C_B\). No independence assumption is needed for this result.
Variance does depend on the joint outcomes. For fixed costs, the covariance of net returns is \(R_AR_B(q-p_Ap_B)\). The portfolio variance equals the sum of the program variances plus twice that covariance. Thus positive dependence increases portfolio variance relative to independence, while negative dependence reduces it, given the same marginal success probabilities and fixed payoffs.
Worked Example: Same Expected Return, Different Portfolio Risk
Use the hypothetical programs from the first example, with \(p_A=0.60\), \(p_B=0.50\), and \(q=0.4225\). Suppose program A costs \$40 million and returns \$100 million if successful; program B costs \$40 million and returns \$80 million if successful. Both investments are committed, and costs and successful returns are fixed.
The expected net return for A is \(0.60(\$100\text{ million})-\$40\text{ million}=\$20\text{ million}\). For B it is \(0.50(\$80\text{ million})-\$40\text{ million}=\$0\). The expected portfolio net return is therefore \$20 million, whether the program outcomes are independent or dependent, as long as the marginal success probabilities and payoffs remain unchanged.
For the dependent case, \(q-p_Ap_B=0.4225-0.30=0.1225\). The variance of A’s return is \((\$100\text{ million})^2(0.60)(0.40)=2400\) million-squared dollars. The variance of B’s return is \((\$80\text{ million})^2(0.50)(0.50)=1600\) million-squared dollars. The covariance is \((\$100\text{ million})(\$80\text{ million})(0.1225)=980\) million-squared dollars. Hence:
The corresponding standard deviation is \(\sqrt{5960}=\$77.20\) million. Under independence, covariance would be zero, so the variance would be \(2400+1600=4000\) million-squared dollars and the standard deviation would be \(\$63.25\) million. The expected return is unchanged, but the dependent portfolio has greater return variability. These summaries do not alone decide whether the portfolio is attractive; the decision depends on the relevant objectives, constraints, and consequences of unfavorable outcomes.
Model Checks, Alternatives, and Limitations
A sound portfolio calculation starts with the actual decision quantity: at least one success, both successes, total net return, or another outcome. Then check that the marginal and joint probabilities are on the same definition of success and refer to the same time horizon. A probability of clinical success under one endpoint cannot be combined casually with a probability of commercial success under another definition.
Useful checks include verifying that joint probabilities satisfy their bounds, that all outcome probabilities are nonnegative and sum to 1, and that averaging conditional probabilities reproduces the intended marginals. Compare the dependent result with independence as a reference case, then test whether plausible changes in shared-risk assumptions alter the decision. A large difference is a signal to examine the dependence assumptions closely, not proof that one model is correct.
For two programs, a joint probability or a fully specified joint table is enough to calculate the probability of at least one success. For three or more programs, pairwise correlations generally do not determine the probability that all fail. The probability of at least one success is \(1-P(\text{all fail})\), so its calculation requires the joint distribution of the failure events, or a model that determines it. A scenario model with shared states is one option; an explicitly specified joint distribution is another. Monte Carlo evaluation can compute portfolio quantities from a specified joint model, but simulation cannot repair an unjustified dependence assumption.
The main alternative is to assume independence and multiply marginal probabilities. That approach is appropriate only when independence is substantively defensible for the outcomes and risks being modeled. A shared-state model makes a specified common risk explicit, but may omit other common causes. A direct joint-probability model can represent dependence without explaining its mechanism, but its inputs may be difficult to estimate. In all approaches, sparse data can make dependence estimates uncertain, and historical co-failure may not predict future programs if technologies, teams, or operating conditions change.
Dependence matters most when it changes the probability of a consequential portfolio event or the range of possible returns. It does not automatically make a portfolio worse: negative dependence can improve diversification, and a positive expected return can remain unchanged under different dependence assumptions. The decision analysis should therefore report the portfolio quantity being evaluated, the assumed dependence structure, and how the result compares with a defensible alternative.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two programs have success probabilities 0.40 and 0.40, and their joint success probability is 0.25. What is the probability of at least one success?
- A shared-state model has two equally likely states. Conditional probabilities that both programs succeed are 0.91 in the first state and 0.19 in the second. What is the marginal probability of joint success?
- Why can the expected total return be unchanged by dependence while its variance changes?
- For three programs, why is knowing every pairwise correlation not generally enough to calculate the probability of at least one success?
Solutions. (1) \(0.40+0.40-0.25=0.55\). (2) Averaging across the equally likely states gives \(0.50(0.91)+0.50(0.19)=0.55\). (3) Linearity of expectation gives the sum of the marginal expectations without an independence assumption, while the variance of a sum includes covariances between program returns. (4) Pairwise correlations do not generally determine the joint probability that all three programs fail; that joint failure probability is needed to calculate the chance of at least one success.