Expected Value and Risk

The bid looked profitable until the downside arrived

A industrial services firm won a competitive RFP (request for proposal, a formal bidding process) by pricing below full cost, assuming a 70% win probability on follow-on work worth $2M. They lost the follow-on. The original contract lost $400K. Finance had computed a positive expected value on paper by treating unlikely upside as half-certain. The arithmetic of expectation was not wrong. The application was: they ignored who bore the loss, treated a one-shot existential hit like a repeatable casino bet, and never reported variance (spread of outcomes around the expected value) to the board.

Expected value (EV) is the probability-weighted average of outcomes. It answers: "If we faced this same decision many times under the same probabilities, what would the average payoff be?" EV is essential for portfolios of small bets, insurance pricing, and inventory policies with many periods. It is dangerous alone for single large gambles, career-risk decisions, and situations where probabilities themselves are unknown.

This lesson connects Lesson 1 scenario tables to Lesson 3 distribution parameters. You will compute EV, variance, and standard deviation for discrete scenarios, interpret downside metrics, and separate risk (known probabilities) from uncertainty (unknown probabilities). Lesson 5 adds simulation when closed-form EV is not enough.

Expected value for discrete scenarios

For outcomes x₁, x₂, … with probabilities p₁, p₂, … that sum to 1:

E[X] = Σ pᵢ × xᵢ

Example from Lesson 1 Harbor delay table: delays 0, 4, 12 weeks with probabilities 0.35, 0.45, 0.20.

E[delay] = 0.35(0) + 0.45(4) + 0.20(12) = 4.2 weeks.

EV is not the "most likely" outcome. It is the balance point. A skewed distribution can have EV far from the median. Managers presenting only EV without the distribution shape invite surprise.

Spreadsheet:

Always include =SUM(B) check cell labeled "prob sum."

Variance, standard deviation, and downside metrics

Variance measures squared deviation from EV:

Var(X) = Σ pᵢ × (xᵢ − E[X])²

Standard deviation SD(X) = √Var(X) returns to original units (dollars, weeks).

Two projects with EV = $100K:

• Project A: $100K certain → Var = 0
• Project B: 50% $250K, 50% −$50K → EV = 0.5(250) + 0.5(−50) = 100K; Var = 0.5(250−100)² + 0.5(−50−100)² = 0.5(22,500) + 0.5(22,500) = 22,500; SD = $150K

Risk-averse stakeholders prefer A; a venture fund with diversified bets might accept B if capital reserves absorb −$50K.

Downside metrics leaders request:

VaR requires a distribution or scenario set. For discrete scenarios, sort payoffs ascending and accumulate probabilities until you reach α.

From Unit 2, percentiles connect directly: p5 profit is the value where 5% of simulated or enumerated outcomes fall below.

Risk versus uncertainty (Knightian distinction)

Risk: you can assign meaningful probabilities (insurance pools, casino games, mature conversion funnels with thousands of trials).

Uncertainty: the probability model is not stable (new market, regulatory regime change, one-off M&A (mergers and acquisitions, combining companies)).

Frank Knight's distinction matters for governance. Teams sometimes assign precise probabilities to feel rigorous when the real issue is model ambiguity. Label scenarios as "risk-modeled" or "uncertainty-judgment." For uncertainty, use scenario ranges, pre-mortems, and flexible options rather than a single EV point.

Risk attitude, incentives, and who bears downside

Organizations are not expected-value maximizers in practice. Risk aversion means preferring a sure moderate outcome to a fair gamble with the same EV. Reasons include:

• Capital constraints: missing payroll is not offset by symmetric upside
• Regulatory covenants: banks face capital triggers on losses
• Career convexity: managers face asymmetric personal downside from visible failures
• Reputation and trust: customers remember service failures more than quiet wins

Separate company EV from stakeholder tolerance. A EV-positive project may be unacceptable if failure triggers covenant breach. Document the loss bearer: equity, lenders, customers, regulators.

Expected utility (advanced concept, managerial intuition): diminishing marginal value of dollars means a $1M loss hurts more than a $1M gain helps. A 50% chance to win $25M and 50% chance to lose $10M can be EV-positive ($7.5M) but unacceptable to a manager who loses their job at −$10M. Certainty effect: people overweight guaranteed outcomes; $900 sure versus 90% chance of $1,000 feels different though similar EV.

Decision rules used in practice:

• Proceed if EV > 0 and P(loss) below threshold and worst case survivable
• Use risk-adjusted value = EV − λ × Var for a rough penalty (λ chosen by policy)
• Require real options (pilot, staged investment) when uncertainty dominates

EV with continuous distributions (link to Lesson 3)

If X ~ Normal(μ, σ), E[X] = μ, Var(X) = σ².

If X ~ Poisson(λ), E[X] = λ, Var(X) = λ.

If X ~ Binomial(n,p), E[X] = np, Var(X) = np(1−p).

For profit = Revenue − Cost with independent normal approximations, E[Profit] = E[R] − E[C], Var(Profit) = Var(R) + Var(C) when independent. If correlated, Var(R−C) = Var(R) + Var(C) − 2ρσ_Rσ_C (Lesson 5).

Managers use these shortcuts for quick ranges: mean profit ± 2σ covers roughly 95% under normal approximation.

Covariance and portfolio thinking (preview)

When combining random variables, variances add only under independence. For profit₁ and profit₂ with correlation ρ:

Var(Profit₁ + Profit₂) = Var₁ + Var₂ + 2ρσ₁σ₂

Positive ρ increases total variance; negative ρ hedges. A conglomerate evaluating two projects should not assume diversification unless ρ is credibly low or negative. Lesson 5 simulates joint draws; Lesson 4 warns that EV adds linearly while risk may not shrink.

Real options and staging when EV is thin

When EV is positive but variance is large, staging (sequential investment that preserves abandon options) changes the decision. Spend $200K on pilot instead of $5M on full rollout. The pilot is a real option: you gain information (Bayes update) before committing. Probability language supports stage gates: "Proceed to full build only if pilot conversion ≥ 4% with 90% one-sided confidence" (Unit 4 formalizes inference).

Building decision tables in spreadsheets

Template columns:

1. Scenario name
2. Probability (sum to 1)
3. Cash payoff
4. Weighted payoff =B*C
5. Deviation =C−EV (EV cell absolute reference)
6. Squared deviation =E^2
7. Weighted squared =B*F

Cells:

• EV =SUM(D:D)
• Var =SUM(G:G)
• SD =SQRT(Var)
• P(loss) =SUMIF(C:C,"<0",B:B)

Add a managerial flag cell: =IF(AND(EV>0, SUMIF(C:C,"<0",B:B)<0.1), "Review", "Decline") illustrative only.

Apex Industrial considers bidding a maintenance contract.

Part A: Setup

Prob sum: 0.25 + 0.35 + 0.40 = 1.00 ✓.

Part B: Expected value

EV = 0.25(800) + 0.35(150) + 0.40(−120) = 200 + 52.5 − 48 = $204.5K

Positive EV, but not a slam dunk.

Part C: Variance and risk metrics

Deviations from 204.5:

Var ≈ 88,618 + 1,040 + 42,120 = 131,778 (thousand²)

SD ≈ √131,778 ≈ $363K

P(loss) = P(lose bid) = 0.40 (only negative scenario)

40% chance of −$120K from bid loss alone.

Part D: Managerial read

CFO should not approve solely on $204.5K EV. With 40% loss probability and SD larger than EV, Apex needs cash reserves or a smaller bid team to reduce sunk cost. Board question: "Is follow-on work independent or correlated with winning this bid?" If correlated, scenarios need restructuring (Lesson 1 dependence). Operator read: engineering should price overrun mitigation to shift mass from 150K to 800K scenario.

Lumen SaaS debates a 20% price promotion for annual plans.

Part A: Scenarios (incremental quarterly profit vs status quo)

Part B: EV and spread

EV = 0.30(600) + 0.45(200) + 0.25(−400) = 180 + 90 − 100 = $170K

P(loss) = 0.25.

Part C: VaR-style read at 10% worst outcomes

Sorted losses: only −400K scenario has negative mass 25%. The 10% worst tail average loss is −400K (only one negative scenario). Expected shortfall at 25% tail = −400K.

Median scenario by cumulative probability: cum 0.30 strong, cum 0.75 modest → median outcome is modest $200K, below EV $170K? Actually EV 170 < 200; wait EV is 170, modest is 200 - the distribution is pulled by strong upside and churn tail.

Check probabilities sum 1 ✓.

Part D: Managerial read

Marketing cannot cite EV $170K without noting 25% chance of −$400K and training customers to wait for discounts (second-order effect). Product should run a holdout geography. Finance should model LTV (lifetime value) beyond one quarter. If Lumen is preparing for fundraising, variance may matter more than EV to investors.

Jensen's inequality and why mean inputs lie

For nonlinear payoffs, E[f(X)] ≠ f(E[X]) in general. If profit = max(0, Revenue − Cost) with random revenue, plugging mean revenue into the formula overstates mean profit when the max truncates losses. Simulation (Lesson 5) exists partly to handle Jensen gaps. EV spreadsheets with scenario tables remain linear in payoffs per scenario; keep scenarios mutually exclusive and capture nonlinearity inside scenario definitions (e.g., "covenant triggered" scenario with discrete penalty).

Value at Risk worked mini-example

Using Apex RFP outcomes with profits 800, 150, −120 and probabilities 0.25, 0.35, 0.40. Sort outcomes: −120 (0.40), 150 (0.35), 800 (0.25). 95% VaR: the loss at the 5% left tail. Here worst outcome −120 has mass 0.40 > 0.05, so 95% VaR profit is −120K (floor scenario). For 90% VaR, still −120K until cumulative passes 0.10. Expected shortfall at 40% tail average of −120K here. These metrics speak to lenders better than EV $204.5K.

Organizational risk policy template

Document explicit thresholds:

Risk policy does not replace judgment; it prevents one optimistic scenario from bypassing review.

Connecting to Lesson 1 scenario coherence

EV calculations inherit garbage from bad probabilities. Before EV, enforce Lesson 1 checks: probabilities sum to 1, scenarios mutually exclusive, calibration documented. A beautifully computed Var is meaningless if "70% commit" means 40% historically.

Review drill: decision memo footer

Every investment memo footer lists: EV, SD, P(loss), worst case, probability sum check, and named loss bearer. Committees reject memos missing any line. The footer takes five minutes in Excel and prevents one-sided storytelling.

Harbor Insurance sells property policies. Loss L is $0 with prob 0.92, $50K with prob 0.05, $500K with prob 0.03. Deductible $25K: payout = max(0, L - 25K).

Scenarios after deductible: $0 (0.92 + part of 0.05 if L=50K pays 25K net to customer on 50K loss... actually L=50K → payout 25K; L=500K → payout 475K).

EV payout = 0.05(25) + 0.03(475) = 1.25 + 14.25 = $15.5K. Premium must cover EV plus expenses plus capital cost. P(any payout) = 0.08. Risk team tracks P(catastrophic) = 0.03 separately from EV. Regulators care about tail, not mean alone.

Spreadsheet: column loss, prob, payout =MAX(0, loss-deductible), weighted sum.

Atlas Ventures places five seed bets, each: 10% chance $10M return, 90% chance $0, $200K cost per bet. Per bet EV = 0.10(10M) - 200K = $800K positive. Founder loves EV. LP (limited partner, passive investor in the fund) cares about P(fund zero): if independent, P(all fail) = 0.9^5 ≈ 0.59. Fund likely returns little despite positive per-deal EV. Diversification across uncorrelated bets helps only if enough bets and capital reserves exist.

Compute P(at least one win) = 1 - 0.9^5 ≈ 0.41. Fund economics need carry, fees, and follow-on reserves; EV per deal is not fund EV. Lesson 1 complement rule on independent failures.

Hedging commodity cost adds premium but compresses left tail. EV may fall slightly while P(covenant breach) drops. CFOs should present variance reduction to boards, not only EV after hedge premium. Decision table with columns: unhedged EV, hedged EV, unhedged P(loss), hedged P(loss).

Part E: Apex bid team sizing

Sunk bid cost −$120K with probability 0.40. If lean bid costs $40K with higher lose prob 0.55, recompute EV and P(loss). Option: spend $40K on proposal, EV = 0.25(800)+0.35(150)+0.55(-40) = 200+52.5-22 = $230.5K similar EV with different loss profile. Risk committee may prefer lower absolute loss exposure.

Part E: Lumen holdout design

Run promotion in 60% of regions; holdout 40%. Measure incremental profit distribution, not only pooled EV. Lesson 1 comparison group meets Lesson 4 EV. Without holdout, second-order churn risk is invisible.

Capital budgeting with strategic abandonment

Nimbus Energy can invest $10M now or wait one year. Wait costs $1M option fee. If wait: 50% market opens (NPV +$25M), 50% closes (NPV −$3M on sunk fee only). Invest now: 70% +$12M, 30% −$8M. Enumerate:

Invest now EV = 0.7(12)+0.3(-8)=8.4-2.4=$6.0M.

Wait EV = -1 + 0.5(25)+0.5(-3) = -1+12.5-1.5 = $10.0M.

Wait dominates on EV but requires carrying option fee risk. If Nimbus cannot survive −$3M path, P(loss) on wait is 0.50 on incremental −3 after fee; board may still prefer wait with capital buffer. EV and tail together decide.

Spreadsheet: two tables, compare EV and P(loss) lines side by side.

Extended scenario: R&D portfolio with abandonment

Project: 60% fail at phase 1 (loss $2M), 40% reach phase 2; of those, 50% fail (loss $5M total cumulative), 50% succeed (gain $40M). Exhaustive paths:

• Fail phase 1: prob 0.60, payoff -2M
• Fail phase 2: prob 0.40*0.50=0.20, payoff -5M
• Success: prob 0.40*0.50=0.20, payoff +40M

Check sum probs: 0.60+0.20+0.20=1. EV = 0.6(-2)+0.2(-5)+0.2(40) = -1.2-1+8 = $5.8M. P(loss)=0.80. Pharma CFO may kill despite positive EV if P(loss) unbearable; biotech VC may fund portfolio of ten similar bets. Same math, different appetite.

Tree in Excel with joint probabilities prevents missing paths.

Extended variance comparison with dollars at risk

Project Y from practice: SD $122.5K on EV $100K. Coefficient of variation CV = SD/EV ≈ 1.23. Project X CV = 0. Rule of thumb thresholds vary by industry; banks want CV low, venture tolerates CV >> 1. Report CV alongside EV for cross-project ranking.

Extended spreadsheet instructions for Northgate practice

Cells A2:A4 scenarios, B2:B4 probs, C2:C4 NPV. D2=D2=B2C2. EV=SUM(D2:D4). E2=C2-$D$5 deviation. F2=E2^2. G2=B2F2. Var=SUM(G2:G4). SD=SQRT(var). P(loss)=SUMIF(C2:C4,"<0",B2:B4). Label each cell in plain language for audit.

Northgate Logistics route investment. Northgate can open a dedicated route.

1. Verify probabilities sum to 1.
2. Compute EV, variance, and SD.
3. Compute P(NPV < 0).
4. Northgate requires EV > 0 and P(loss) < 0.30. Should they proceed?
5. Explain in a paragraph why a lender might still decline despite positive EV.

Solution

1. 0.35 + 0.40 + 0.25 = 1 ✓.

2. EV = 0.35(12) + 0.40(4) + 0.25(−6) = 4.2 + 1.6 − 1.5 = $4.3M

Deviations: 12−4.3=7.7; 4−4.3=−0.3; −6−4.3=−10.3

Var = 0.35(59.29) + 0.40(0.09) + 0.25(106.09) = 20.75 + 0.04 + 26.52 = 47.31

SD = √47.31 ≈ $6.88M

3. P(NPV<0) = P(low) = 0.25 ✓.

4. EV > 0 and P(loss)=0.25 < 0.30 → passes Northgate rule provisionally (subject to strategic fit).

5. Lender perspective: SD nearly 1.6× EV; −$6M scenario may breach debt covenants or consume collateral. Expected payoff does not protect against default in the left tail. Lender wants minimum NPV floor or guarantee.

Two-project choice. Project X: certain $90K. Project Y: 60% $200K, 40% −$50K.

1. Compute EV for both.
2. Which has higher EV?
3. Which would a risk-averse CEO likely pick? Use variance to justify.
4. If the CEO is risk-neutral and repeats similar bets 20 times independently, what is the approximate distribution logic (link to Lesson 3 CLT)?

Solution

1. EV(X) = $90K. EV(Y) = 0.6(200) + 0.4(−50) = 120 − 20 = $100K.

2. Project Y higher EV by $10K.

3. Var(X)=0. Var(Y)=0.6(200−100)²+0.4(−50−100)²=0.6(10,000)+0.4(22,500)=6,000+9,000=15,000; SD≈$122.5K. Risk-averse CEO likely picks X for certainty despite lower EV.

4. With 20 independent Y-like bets, average payoff tends toward EV $100K by CLT, diversification reduces per-portfolio variance; risk-neutral CEO may prefer many Y bets over one X.

Stakeholder conflict over the same EV number

The board, CFO, and business unit head can agree on EV arithmetic and still disagree on proceed or stop. The board worries about tail loss and reputation. The CFO worries about covenant and cash runway. The BU head sees career upside from bold bets. Probability and EV are shared inputs; risk appetite is the weighting function over those inputs. Good governance separates "what we believe" (scenario table) from "what we tolerate" (policy thresholds). When appetite is unstated, teams default to personal career risk aversion or reckless promotion of pet projects.

Document risk appetite in the same workbook as the scenario table. If appetite is unstated, the loudest speaker wins, not the best analysis.

Discrete vs continuous EV (link to Lesson 3)

When outcomes are continuous (profit ~ simulation output), EV is still integral of x × density, estimated by simulation average. Lesson 4 discrete tables are the finite case. Managers should not mix discrete scenario EV with continuous tail metrics without labeling. A three-scenario EV for strategy plus a simulated p5 for covenants is fine if labeled two layers: strategic scenarios and financial stress simulation.

Median versus EV in skewed promotions

Lumen incremental profits skew left because cannibalization scenario is −400K while upside is +600K. Median scenario is modest +200K (cumulative prob 0.75). EV +170K. EV = 0.3(600)+0.45(200)+0.25(-400)=180+90-100=170. Median outcome is +200K at 0.75 cumulative. EV pulled down by tail loss. Marketing headline "expected +170K" is honest; "typical +200K" is median language; using both prevents spin.

Utility sketch without heavy math

Utility U(x) increasing concave: U(0)=0, U(100)=10, U(200)=15. EV dollars 100 but utility EV 7.5 from 50-50 on 0 and 200: 0.5(0)+0.5(15)=7.5 < U(100)=10. Manager prefers sure 100K EV equivalent in utility terms. Document when committee uses EV vs utility verbally.

Complete Northgate lender paragraph for practice alignment

Northgate's EV $4.3M sounds attractive, but SD $6.88M implies wide dispersion relative to mean. A lender funding 60% of capex cares that P(NPV<0)=25% is a one-in-four chance of value destruction on the enumerated downside scenario, not a smooth normal tail. Covenants on interest coverage should use simulated p5 cash flow (Lesson 5) even when strategic EV is positive. The lender paragraph is why EV alone fails capital committees.

Risk-adjusted score for project ranking

Score = EV − 0.5×SD (illustrative λ=0.5). Project X certain 90K scores 90. Project Y EV 100 SD 122.5 scores 100−61.25=38.75. Rank changes under risk penalty. Committees document λ implicitly through verbal risk appetite; make λ explicit to reduce politics.

When EV and NPV diverge in timing

Discounted cash flow timing can make NPV unlike simple scenario EV if payoffs arrive over years. This lesson uses single-period payoffs for clarity. Multi-period projects should build scenario trees by year or simulate cash paths (Lesson 5). Managers should ask whether quoted EV is undiscounted operating profit or NPV; mixing definitions across projects invalidates ranking.

Expected value of perfect information (preview)

Before spending on market research, compare EV with research to EV without. If research cost exceeds the expected improvement in decision value, skip it. Formally, expected value of perfect information is the weighted average of best actions under each future signal minus current EV. You do not need the full formula in daily work; you do need the habit of asking whether a study will change a decision enough to pay for itself.

Practice reminder: show checks in every solution

When you solve EV problems by hand or in Excel, include three checks on the page: probabilities sum to 1, weighted payoffs reproduce EV when summed, and P(loss) equals the sum of probabilities on negative scenarios. Reviewers trust solutions that show reconciliation lines such as "0.35+0.40+0.25=1 ✓" and "EV=4.3M ✓". Habitual checks separate finance-grade work from back-of-napkin guesses.

Stakeholder table for risk conversations

Use the table to order metrics in the deck: lead with the metric your audience owns, but never hide the others in an appendix they will not read.

One-line EV discipline

Before any approval meeting, write: "We proceed if EV = ___, P(loss) = ___, worst case = ___, and ___ bears the loss." If any blank is empty, the meeting is not ready.

Key takeaways

• Expected value is the probability-weighted average, not the single planned outcome.
• Variance and P(loss) determine whether EV-positive projects are survivable.
• Separate risk (modeled probabilities) from uncertainty (unstable models).
• Incentives and covenants often make stakeholders effectively risk-averse even when EV is positive.
• Spreadsheet decision tables need probability sum checks, EV, variance, and downside metrics together.

1. Model a real decision with three scenarios. Compute EV and P(loss). Would you proceed under your firm's capital constraints?
2. Where do managers in your org confuse EV with "what will happen"?
3. Continue to Lesson 5: Simulation for Managerial Decisions.