# Portfolio risk: three exercises

Portfolio Lab | 9 September 2026 | About 25 minutes
Interactive version: https://www.portfoliolab.app/examples/portfolio-diversification-exercise
You may reproduce this original worksheet and its answers with attribution to Portfolio Lab.
All figures are hypothetical teaching inputs, not current forecasts or investment advice.

## 1. Diversification (5 minutes)
Two assets, each with 15% annual volatility. Invest 50% in each.
Predict portfolio volatility at correlations +1, 0 and -0.5. Which input changes? Which inputs stay fixed?
Formula: variance = (0.5*15)^2 + (0.5*15)^2 + 2*0.5*0.5*15*15*correlation.
Volatility is the square root of variance. Volatility is not maximum possible loss.

## 2. Allocation and risk (5 minutes)
80% equities (15% volatility), 20% Bitcoin (45% volatility), correlation 0.2.
Predict whether Bitcoin contributes more or less than 20% of variance risk.
Calculate portfolio variance and Bitcoin's contribution.
Portfolio variance = (0.8*15)^2 + (0.2*45)^2 + 2*0.8*0.2*15*45*0.2.
Bitcoin contribution = (0.2*45)^2 + 0.8*0.2*15*45*0.2.
Share = contribution / total variance. Repeat with zero Bitcoin.

## 3. Withdrawals (5 minutes)
Start with 100,000. Withdraw 5,000 at the end of each year.
Path A earns +20%, then -20%. Path B earns -20%, then +20%.
Calculate both ending balances. Repeat without withdrawals.
Why is the arithmetic average of 0% not the compound return?

## Instructor answers
1. Volatility: 15%, approximately 10.6066%, and 7.5%. At correlation -1 the idealised equal-risk pair has zero variance, a special mathematical case, not an available risk-free investment.
2. Variance 268.2; volatility approximately 16.3768%. Bitcoin contribution 102.6; share approximately 38.255%. At zero Bitcoin its contribution is zero. This variance allocation is not a probability or predicted loss.
3. Path A: 120,000 - 5,000 = 115,000; then 92,000 - 5,000 = 87,000. Path B: 80,000 - 5,000 = 75,000; then 90,000 - 5,000 = 85,000. With no withdrawals both end at 96,000. Two-year wealth factor: 1.2*0.8=0.96; annual compound return is sqrt(0.96)-1, approximately -2.02%.

## Discussion (5 minutes)
Ask learners to identify what these examples omit: changing correlations, fees, taxes, uncertain forecasts, liquidity and implementation. They isolate concepts; they do not recommend an allocation.
Next: https://www.portfoliolab.app/for/students