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Each step prepares one or two case questions. Follow them in order: you place the portfolios, find the best one, size it according to the risk you accept, then test how fragile the estimates are and how the portfolios behave in crises. The last step gives the solutions. Annualised data 2006-2026.
In Markowitz's mean-variance framework, each portfolio is described by two numbers: its expected return (the annual average of its past returns, used as an estimate) and its risk, measured by volatility (the standard deviation of returns, the square root of the variance). The risk of a mix, however, is not the average of the risks. Take two assets whose movements are almost independent, for example US equities (S&P 500) and gold: their correlation was only over 2006-2026. Once mixed, their risk becomes lower than the average of their risks. This is not magic, it is correlation: ranging from −1 to +1, it tells you whether two assets rise and fall together (+1: always together; 0: unrelated; −1: in opposite directions). The lower it is, the more diversification reduces risk.
“Diversification is the only free lunch in finance.” Quote attributed to Harry Markowitz, 1990 Nobel laureate in Economics
This is what the case is all about: achieving the best expected return for a given risk or the lowest risk for a given expected return.
Move the slider: as the correlation falls, the curve bends further to the left and some mixes become less risky than the less risky of the two assets.
Each dot is a portfolio from the case. Its position gives its risk (annual volatility, on the horizontal axis) and its expected return (on the vertical axis), measured over 2006-2026. The mauve dots contain only one asset. The green curve is the mean-variance efficient frontier: the best possible expected return for each level of risk, by freely mixing the six assets of the case (US, European and emerging-market equities, US bonds, US real estate and gold, all held through dollar-denominated listed index funds, known as ETFs) without short selling (no negative share: you do not bet on an asset falling). For question 1, several portfolios reach 8%: keep in mind what is at stake in step 1. Hover over a dot to read its figures, click to see its composition.
1. You are aiming for an expected return of at least 8%. Which portfolio do you choose among the 33?
2. You accept at most 8% risk. Which portfolio do you choose?
Each asset column gives the share of the portfolio invested in that asset: the darker the cell, the larger the share. Click a header to sort (for example “Gold” to see the portfolios most invested in gold), or a row to select it. All prices are in dollars.
Start from the risk-free rate ( a year, the average of 3-month US Treasury bills over 2006-2026, the green diamond): this is what an investment with no risk at all earns. Connect it to a portfolio: the slope of this line is its Sharpe ratio, the expected return earned above the risk-free rate for each point of risk. The steeper the line, the better the risk is rewarded. Click on dots or show all the lines.
3. Which portfolio has the best Sharpe ratio? According to Markowitz, why is it the only risky portfolio you should hold, whatever your profile?
Click a row to draw its line on the chart. Sort by “Sharpe ratio” to check your answer to question 3.
Choose a risky portfolio, then the share of your money you invest in it. The rest is invested at the risk-free rate. Above 100%, you borrow at this rate to invest more. All these mixes lie on the line connecting the risk-free rate to the portfolio: the larger the share, the higher the risk and the expected return.
4. Take the risk aversion test. What annual risk do you accept? What share do you invest in the portfolio from question 3 and what share at the risk-free rate (or borrowed)?
From your profile to the risk to enter. The test places you on a scale from 1 to 7. Risk measures the usual deviation around the average: with 10% risk, in two years out of three, the return stays within 10 points of its average. To go from your profile to a risk, here are some reference points:
| Test score | Profile | Indicative annual risk |
|---|---|---|
| 1 | Very defensive | 3% |
| 2 | Defensive | 6% |
| 3 | Cautious | 9% |
| 4 | Balanced | 12% |
| 5 | Moderately dynamic | 16% |
| 6 | Dynamic | 20% |
| 7 | Very dynamic | 25% |
Example: Cautious profile, 9% risk, i.e. 9 / 9.97 ≈ 90% in No. 8 and 10% at the risk-free rate.
These reference points are illustrative: they extend the test question on the acceptable annual variation (±3, ±8, ±15 or ±25%). They are not regulatory thresholds.
Risk and expected return are linked by the line: set either one, then click “Apply” or press Enter: the share and the other value follow.
Two separate situations: the same initial investment, which you would need either in 5 years or in 20 years. The expected returns and risks shown are those of the 2006-2026 period, measured in dollars: currency risk is not counted here (see step 8). The amounts obtained are estimates derived from the expected return and the risk, not actual history (replayed in step 8). For each horizon, start from the amount you will need, then choose the risky portfolio and the share you invest in it: the rest is invested at the risk-free rate.
5. You will need this money in 5 years, then in 20 years. Is your risk aversion the same? Which mix do you choose in each case? Does the risky portfolio change?
For each horizon, enter the amount you will absolutely need, then choose the share invested in the risky portfolio. On the slider, the green zone shows the shares that secure this amount with the chosen level of certainty.
“95 times out of 100”: if the future played out 100 times, you would have at least your amount in 95 of them. “99 times out of 100” is stricter: the green zone narrows.
We start from the expected return and risk of your mix. The share invested in the risky portfolio stays the same: you regularly bring it back to its starting level (rebalancing). No simulation: a statistical formula (log-normal distribution) directly gives the amount you will have at least 95 times out of 100 (or 99), and the amount reached half of the time. It is a model: real crashes can be more violent (step 8).
For example, an expense planned in 5 years.
For example, supplementary retirement income.
Compare the two cards, then answer the three case questions. The answers are in step 9.
To aim for a higher expected return, there are two options: take a riskier portfolio among the 33 or keep the optimal portfolio from step 3 and borrow at the risk-free rate to buy more of it. Enter the target expected return and compare the risk of the two options.
6. You are aiming for an expected return of . How do you reach it? Compare 100% S&P 500 with borrowing at the risk-free rate and investing in the portfolio from question 3.
Borrowing at the risk-free rate is a textbook assumption: a real loan costs more and the bank may require a top-up if the portfolio falls (step 8).
An expected return is not observed: it is estimated from the past. Change the measurement period (the last 5, 10 or 20 years) and see what moves: the expected returns of the assets, the position of the 33 portfolios and the portfolio with the best Sharpe ratio, the one whose line from the risk-free rate is the steepest. Note: this is not your investment horizon, it is the data window used for the estimates. The risk-free rate is also measured over each period: .
7. Measure the expected returns and risks over the last 5, 10 and 20 years. How do they vary? Does the best Sharpe ratio always go to the same portfolio? What does this difference tell you?
The green star is the true optimum for the period: the tangency portfolio you would get by freely mixing the six assets. The exercise is limited to the 33 portfolios of the case. With all mixes allowed, the star is the one to choose.
Case assumption: the correlations of the full 2006-2026 period are applied to all three measurement periods.
The figures in the previous steps are estimates. Now replay the real history of 2006-2026 and take on the challenges: each one reveals a key point for question 8. You can also try everything freely.
8. Beyond expected returns, which real-world factors could influence the performance of your portfolios? Identify at least three key points and justify them.
“Buy and hold” replay, without rebalancing: the starting weights are never reset and drift with the markets. The value is tracked trading day after trading day, on closing prices: drops during a trading session do not appear. Borrowing is granted with the portfolio held as collateral (Lombard loan), at the day’s Treasury bill rate. The bank lends only a share of the value of each asset (): if the debt exceeds this amount, it requires a top-up (margin call). Unlike step 5, the starting share is not maintained. Fees: assumption of 1% per year, deducted pro rata by day from the net value of the investment. In euros: conversion at the day’s EUR/USD rate, top-ups included. Inflation: assumption of 2% per year, the target of both the European Central Bank and the US Federal Reserve. Tax: simplified example of 10% deducted on exit, not the exact tax rule.
Choose a question, try to find your answer first, then reveal the solution. The answers you saved under the questions in steps 2 to 8 appear here, next to the solution. The button under each solution takes you back to the corresponding step.