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PART IX — MATHEMATICAL FINANCE🔵 Intermediate

How Do You Measure the Risk of Losing Money on an Investment?

Probability and Statistics in Finance

Expected value, variance, the normal distribution, and correlation — the statistical toolkit for quantifying risk and building portfolios.

math financeprobabilitystatisticsexpected returnvariancenormal distributioncorrelation

Live Result

$7.50

ƒ: p1 * r1 + p2 * r2 + p3 * r3raw: 7.50000

ƒ(x)

Use variable names from the panel above (e.g. FV, r, n) — or type numbers directly: 10000 / (1 + 0.08)^10

p1% as decimal

Probability of scenario 1 (decimal)

p1 = 0.25

Return in scenario 1 (%)

r1 = 30

p2% as decimal

Probability of scenario 2 (decimal)

p2 = 0.50

Return in scenario 2 (%)

r2 = 10

p3% as decimal

Probability of scenario 3 (decimal)

p3 = 0.25

Return in scenario 3 (%)

r3 = -20

Observed value for z-score calculation

x = 12

Mean / expected value

mu = 7.5

sigma

Standard deviation

sigma = 17.85

💡 You can also enter values directly in the formula: 10000 / (1 + 0.08)^10

⬇ Export Calculation

Exports a plain .txt file with your expression, formula, all variable values, result, and educational notes — ready to paste into any report, Word doc, Notion, or Google Docs.

The exported file includes the formula in standard mathematical notation — you can paste it directly into Excel, Google Sheets, or back into FinanceSheep.

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Learn: Probability and Statistics in Finance

PART IX — MATHEMATICAL FINANCE · Educational Guide

The Core Idea

Finance is fundamentally about making decisions under uncertainty. Probability and statistics let us quantify that uncertainty — measure expected outcomes, model the spread of possible results, and combine assets in ways that reduce risk without sacrificing return. These tools underpin CAPM, Black-Scholes, VaR, and every portfolio optimization model.

How It Works

Expected Value E[X] = Σ p_i × x_i — the probability-weighted average outcome. Variance = Σ p_i × (x_i − E[X])² — measures dispersion around the mean. Standard deviation σ = √Variance — the most common risk measure in finance (same units as returns). Normal distribution: 68% of outcomes within ±1σ, 95% within ±2σ, 99.7% within ±3σ. Z-score = (x − μ) / σ — how many standard deviations an observation is from the mean. Correlation ρ (−1 to +1) — when ρ < 1, combining assets in a portfolio reduces total risk below the weighted average of individual risks.

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Real-World Example: Stock with three scenarios: Bull +30% (25%), Base +10% (50%), Bear −20% (25%). E[R] = 0.25×30 + 0.50×10 + 0.25×(−20) = 7.5 + 5 − 5 = 7.5%. Variance = 0.25×(30−7.5)² + 0.50×(10−7.5)² + 0.25×(−20−7.5)² = 0.25×506.25 + 0.50×6.25 + 0.25×756.25 = 318.75. σ = √318.75 = 17.85%. 95% confidence interval: 7.5% ± 2×17.85% = −28.2% to +43.2%. Z-score of +30% return: (30−7.5)/17.85 = +1.26σ above mean.

How Do You Measure the Risk of Losing Money on an Investment?

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