FINM7008 Chap.7 The Markowitz Frontier and the Power of Diversification
The Markowitz Frontier and the Power of Diversification
What diversification removes
Adding stocks to an equally weighted portfolio shrinks the average-variance term towards zero, leaving the average covariance. Firm-specific risk disappears; systematic, market-wide risk stays.
The Markowitz process
With many assets, Markowitz optimisation traces the minimum-variance frontier from expected returns and a covariance matrix.
Its upper half is efficient, and the separation property splits the choice of risky portfolio from the choice of how much to hold.
Evidence and practice
The lecture backs the theory with numbers: single stocks have typically shown annual standard deviations of 35% to 50% against 14% to 16% for the market, and many households still hold only one to three shares.
The tutorials ask which events are market risks and which stock to add for the best diversification.
What this chapter covers
- 01
Household under-diversification
- 02
Single stocks against the market
- 03
Equal-weight portfolio variance
- 04
Systematic and idiosyncratic risk
- 05
Why cherry-picking fails
- 06
The minimum-variance frontier
- 07
Efficient frontier and separation
Worked example · free
Portfolio risk with ten and thirty stocks
- 1Ten stocks: 0.16/10 + (9/10)(0.024) = 0.016 + 0.0216 = 0.0376, so σ = 19.39%.
- 1Thirty stocks: 0.16/30 + (29/30)(0.024) = 0.00533 + 0.0232 = 0.02853, so σ = 16.89%.
- 1Floor: √0.024 = 15.49%, the systematic risk that remains with very many stocks.
Key terms
- Idiosyncratic Risk
- Risk specific to one firm or industry, such as a recall or a strike, that diversification removes.
- Efficient Frontier
- The upper half of the minimum-variance frontier, where no portfolio is dominated.
- Separation Property
- The result that the choice of risky portfolio is separate from the choice of how much to hold in it.
- Global Minimum-Variance Portfolio
- The single portfolio with the lowest variance of all risky combinations.
- Covariance Matrix
- The table of every asset's variance and every pair's covariance used as the optimiser's risk input.
The Markowitz Frontier and the Power of Diversification FAQ
How many stocks make a well-diversified portfolio?
The lecture cites Statman's 1987 study, which found that a random selection of stocks needs at least thirty to forty names to count as well diversified. Most of the risk reduction comes from the first few additions.
Why does picking the stocks with the best Sharpe ratios not give the best portfolio?
An optimal portfolio is defined by how assets move together, not by their stand-alone merits. Stocks chosen only for high returns or Sharpe ratios can overlap heavily, concentrating risk, while weaker assets with low correlations can combine into a better mix.
What inputs does Markowitz optimisation need?
It needs the expected return of every asset and the full covariance matrix, whose entries are the variances and each pair's standard deviations times their correlation. Solvers such as Excel's Solver or Python's PyPortfolioOpt then trace the frontier.
What is the difference between the minimum-variance frontier and the efficient frontier?
The minimum-variance frontier is the whole curve of lowest-variance portfolios for each target mean. The efficient frontier is only its upper half, from the global minimum-variance point upwards, because every portfolio on the lower half is beaten by one with the same risk and a higher mean.
Why is the separation property important for fund managers?
It says the optimal risky portfolio is the same for everyone with the same inputs, and only the split between that portfolio and bills depends on risk aversion. A manager could therefore offer one risky fund to all clients, which the lecture calls the theoretical basis of the mutual fund industry.
Can a portfolio lie to the left of the minimum-variance frontier?
No. The frontier is, by construction, the lowest variance attainable for each expected return using the available assets, so no combination of those assets can have less risk at the same mean. Adding new assets can, however, move the whole frontier further left.
Exam move
Practise the equal-weight variance formula for three portfolio sizes and plot the results. Then explain the Markowitz process aloud, from inputs to complete portfolio, without notes. Then classify a list of business events as systematic or firm-specific and justify each label in one clause.
If you build the Assignment 2 model in Excel, use it to trace a frontier for a few assets and see the leftward shift as each asset is added.
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