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Accessible, useful and human finance · sagora.eu
Portfolio management course · SAGORA Workshop
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Portfolio management: advanced pathway
Eight steps, one per course objective, followed by a review. Each step asks a question, has you work with a single chart and checks your understanding with a few multiple-choice questions. The calculations are those of the full simulator, on the same data.
1. Readthe question for the step, at the top of the page.
2. Adjustthe settings and observe the chart.
3. Answerthe questions: the answer and its explanation appear immediately.
4. Rememberthe sentence that closes the step.
Five US assets in dollars, from 31/07/2002 to 27/08/2026 (Yahoo Finance). A trading month has 21 days. Risk-free rate: 3-month US Treasury bills, 1.70% a year on average over the period. To go further: the full simulator.
Step 1 · Diversify
Why not put everything into the asset that has returned the most?
Each point is an asset, placed according to its volatility σ (the risk, on the horizontal axis) and its expected return μ (on the vertical axis). The curve is the Markowitz mean-variance efficient frontier: for each level of volatility, the highest expected return you can obtain by mixing the selected assets. The star is the Max. Sharpe portfolio, the one with the highest Sharpe ratio. The line from the risk-free rate that touches it is the capital market line (CML): by adjusting the exposure, you move along this line.
Portfolio assets
Share invested in the Max. Sharpe portfolio
Cost of borrowing above 100%
Composition of the Max. Sharpe portfolio
Efficient frontierFrontier with all five assetsCapital market lineYour positionRandomly drawn portfolios
Key takeawayDiversification pays off as soon as the correlation between the assets is below 1: the volatility of the mix falls below the weighted average of the volatilities. You first choose the Max. Sharpe portfolio, then adjust the risk with the risk-free asset, by lending or borrowing: this is the separation theorem. In practice, borrowing costs more than the risk-free rate and puts a kink in the line.
Step 2 · Measure
An accurate average can still tell the wrong story.
The histogram shows the 6,023 monthly returns of the chosen portfolio, one per rolling trading month. The expected return μ is 12 times their average: it is an arithmetic mean. The CAGR tells you how fast 100 invested at the start of the period and rebalanced every trading month actually grew: it is a geometric mean. The orange bars are the monthly returns below the monthly risk-free rate: they are the only ones counted by the semi-deviation, which is calculated relative to the risk-free rate (not to 0 or to the mean) and is the denominator of the Sortino ratio. μ and the CAGR come from the same history: μ serves as the forecast for the optimisation, the CAGR records what actually happened.
In large type, the value over one year; below it, over one month, on the histogram’s scale. Annualisation is linear, not compounded: μ is 12 times the monthly average; σ, the semi-deviation, Sharpe and Sortino are √12 times their monthly value. Only the CAGR is compounded: 1 + CAGR = (1 + compounded monthly growth)12. The course case study applies the same rule to daily returns (× 252 and × √252): equity volatility comes out higher there.
Portfolio
+50% then −50%The arithmetic mean of the two returns is 0%. Yet 100 becomes 150 then 75: the capital has fallen. Volatility pulls the CAGR below the arithmetic mean, by about half the variance.
Key takeawayμ is 12 times an arithmetic mean, the CAGR is an annualised geometric mean: compounding adds a little to the CAGR, the variance takes away about σ²/2. For a low-volatility portfolio, the CAGR can exceed μ; the higher the variance, the more it tends to fall below. The Sharpe ratio divides the excess expected return by the volatility; the Sortino ratio divides it by the semi-deviation alone, which counts only the monthly returns below the risk-free rate.
Step 3 · Choose a criterion
Seven ways to define risk, seven portfolios.
Each row is the best portfolio for one criterion, on the five assets and the 6,023 monthly returns. The 95% VaR is the 5th percentile of monthly returns; the CVaR is the average of the worst 5% of monthly returns. A negative value is a loss. Click a row to see its distribution and what the criterion is looking for.
What each criterion seeks
For all seven criteria, the weights of the five assets are positive or zero and add up to 100%: no short selling, no borrowing. All calculations use the same 6,023 monthly returns.
95% VaR95% CVaRWorst monthly return
Key takeawayChanging the criterion changes the allocation. VaR ignores the size of losses beyond the threshold. CVaR and Maxi-min maximise a ratio of excess return to loss rather than seeking the smallest possible loss: a very cautious portfolio can lose less and still obtain a worse ratio.
Step 4 · Question the optimum
An optimum shown to a tenth of a point is still an estimate.
Four experiments, one per tab. Each changes just one thing: the risk-free rate, the data period, the history itself (resampled) or the moment at which the allocation is known.
Risk-free rate
The minimum variance portfolio looks only at volatilities and correlations. The Max. Sharpe portfolio maximises (μ − Rf) / σ.
Data period
Max. Sharpe portfolio for the period
Frontier for the periodBar: μ ± 1 standard error
Number of randomly drawn histories
Each fictitious history has 287 trading months drawn with replacement from the 6,023 actual rolling months. The tool recalculates the frontier on each one, averages the frontiers rank by rank, then selects the maximum Sharpe point of this average frontier (Michaud’s method). With 2,000 histories, the calculation takes about ten seconds.
Historical weightAverage resampled weight10th-90th percentile range
To choose the Max. Sharpe portfolio, you need past data. An investor can only use the data that exist when they invest their money. The portfolio the tool shows elsewhere is calculated over the whole period, including the years after the investment: it already knows what comes next. The timeline shows which data each one uses.
Investment date
Key takeawayAn optimum is an estimate: it depends on the period observed and the uncertainty about the expected return (its standard error) is roughly σ divided by the square root of the number of years. Resampling shows how large this dependence is. The minimum variance portfolio, which does without expected returns, is the most stable without being the best performer.
Step 5 · Time
What might happen to this portfolio and what difference does the horizon make?
Each optimisation model is re-optimised for the chosen horizon: it chooses its weights by looking at capital growth over 1, 5, 10, 15 or 20 years, not over one month. The projection then replays this portfolio over 2,000 possible futures, chaining together trading months drawn at random from the history. Each future starts at 1. Below the chart, a card explains how the chosen model calculates its weights. The second tab shows what happens to Omega and Sortino when their threshold changes.
Portfolio
Horizon
Rebalancing
Contribution
With rebalancing, 0.15% of the amount traded is deducted each time the portfolio returns to its target weights. Money waiting to be contributed earns the risk-free rate.
Between the 5th and 95th percentiles, at each dateBetween the 30th and 70th percentiles, at each dateMedianFinal mean
Threshold for the downside measures
Horizon
Omega and Sortino treat annualised growth below the threshold as disappointing. At each horizon, the calculation looks for each criterion’s best portfolio over 1,000 paths, as the Horizons tab of the full simulator does: a few seconds per threshold.
Key takeawayThe mean multiple is not the typical case: read the median. Rebalancing is not there to earn more; it keeps the risk you chose. Spreading contributions protects in the short term and costs a little in the long term. With a low threshold and a long horizon, almost no path ends with annualised growth below the threshold: Omega and Sortino no longer see any risk.
Step 6 · Reading a distribution
Compare two portfolios on all their outcomes, not on a single figure.
The cumulative distribution function gives, for each level x, the share of outcomes less than or equal to x. Where the curve crosses 50%, you read the median; at 5%, the 5th percentile. The further right a curve lies, the better the outcomes. Compare the optimisation models on their monthly returns (the portfolios from step 3), then on their simulated annualised growth at 1, 5, 10, 15 or 20 years. At each horizon, each model is re-optimised on the simulated growth at that horizon, as in the Horizons tab of the full simulator: its weights change with the horizon (1,000 paths, the same for all portfolios).
What you are comparing
Portfolio A
Portfolio B
Which of the seven models dominates which?
Key takeawayA dominates B at first order when its curve is never above B’s and lies below it at least once: any investor who prefers to earn more then chooses A. When the curves cross, neither dominates the other at first order and the choice depends on the investor’s preferences. Over one month, the optimisation models do not dominate one another; over a long horizon, dominance relationships appear.
Step 7 · Crises
Average risk is not crisis risk.
Volatility summarises every month, calm or turbulent. Four tabs look at the moments when markets fall: extreme values, far more frequent than a normal distribution predicts; what bonds and gold do during a crisis; the assets that hold up on the worst days and those that fall together; extreme days, which come in clusters and weigh heavily on performance.
Asset
US equities (S&P 500), daily returns
One question: are very large falls as rare as a normal distribution with the same standard deviation predicts? The observed days are compared with a normal distribution with the same mean and the same standard deviation.
US equities, day by day
Every day, from -6% to +6%
Zoom: falls of more than 4%
Observed daysFalls of more than 4%Normal distribution with the same daily mean and standard deviation
Key takeawayDaily returns do not follow a normal distribution with the same mean and standard deviation: they include more very calm days, fewer intermediate days and, above all, far more extreme days, both down and up. These excess extreme days form what are called fat tails. Volatility alone therefore underestimates crisis risk: to measure it, you need to look beyond volatility, as CVaR and Maxi-min do in step 3.
Episode
Correlation over the whole periodDuring the episode
Reference asset
The tool isolates the reference’s worst 5% of trading days and looks at what the other assets did on those days.
When everything falls at once
Portfolio
A removed day is replaced by what Treasury bills earned that day. Logarithmic scale.
The 60 worst days, year by yearValue of 100 invested, with and without the extreme days
Key takeawayExtreme losses are far more frequent than a normal distribution predicts and come in clusters: volatility alone underestimates crisis risk. A negative correlation over the whole period does not guarantee protection: bonds cushioned the 2008 crisis and Covid, but not 2022, when equities and bonds fell together. What protects depends on what causes the crisis. A few days weigh heavily on performance: the best often come just after the worst.
Step 8 · CAPM and beta
What return does an asset’s risk justify?
The characteristic line is the regression line of the asset’s monthly returns in excess of the risk-free rate on those of the market portfolio. Its slope is the beta, its intercept is Jensen’s alpha; the R² tells you what share of the asset’s variance the market portfolio explains. The CAPM rewards only systematic risk: E(Ri) = Rf + βi × (E(Rm) − Rf).
Asset
Market portfolio used
The asset’s variance, broken down
Security market line (SML)
An asset above the line has an observed μ higher than the one its beta justifies: its alpha is positive.
Betas and alphas against the market portfolio used
Key takeawayThe CAPM rewards only systematic risk, measured by beta. Specific risk, which diversification can eliminate, is not rewarded. A beta and an alpha only make sense relative to the market portfolio used; a historical alpha is an observation, not a forecast.
Review · What the tool is not
The tool reorganises and replays the past. It does not forecast what comes next.
Bitcoin’s Yahoo Finance series only starts on 18/09/2014. Adding it brings all calculations back to the common period: the figures then change for all assets, not just for bitcoin.
Bitcoin
What to remember
Diversification pays off as soon as the correlation between assets is below 1: the volatility of the mix falls below the weighted average of the volatilities.
Choose, then scale: the Max. Sharpe portfolio first, the risk-free asset next.
Each criterion measures risk differently; changing criterion changes the allocation.
An optimum is an estimate: it depends on the period observed.
Time changes how risk should be read: median and mean drift apart as the horizon lengthens.
A distribution is read as a whole; if the curves cross, neither dominates the other at first order.
Extreme days, both down and up, are far more frequent than a normal distribution predicts: measuring risk by volatility alone underestimates them.
The CAPM rewards only systematic risk, measured by beta.