Diversification: the only free lunch, and its limits

Almost every claim in investing involves a trade: you accept more risk to earn more expected return, you give up liquidity to earn a premium, you pay a fee to buy access. Diversification is the one construction choice that appears to break the pattern. Combining assets whose returns do not move in lockstep reduces the volatility of the combined portfolio without reducing its expected return. Nothing is given up. That is why the line "diversification is the only free lunch in finance," widely attributed to Harry Markowitz, has survived seventy years of contrary fashion.

The claim is true, and it is also more limited than most investors assume. The free lunch is finite in size, it runs out faster than people expect, it does not protect against the risk that everything is priced expensively at once, and it partially withdraws itself at exactly the moment you most want it. Understanding the mechanism — the actual algebra of why volatility falls — tells you where the free portion ends and where you have started paying.

Why combining assets reduces volatility

Portfolio volatility is not the average of the volatilities of the holdings. It is driven by covariance, the tendency of two assets to deviate from their own means at the same time and in the same direction. For two assets with weights w₁ and w₂, volatilities σ₁ and σ₂, and correlation ρ, the portfolio variance is:

σ²ₚ = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂

The first two terms are unavoidable — they are the assets' own risk, scaled down by the squares of their weights. The third term is where diversification lives. If ρ is less than 1, that cross term is smaller than it would be for a single asset, and the total comes out below the weighted average.

Take two assets, each with 20% annual volatility, held 50/50, with the same expected return. If ρ = 1 (they are effectively the same asset), portfolio volatility is 20%: nothing gained. If ρ = 0.5, variance is 0.01 + 0.01 + 0.01 = 0.03, so volatility is 17.3%. If ρ = 0, variance is 0.02 and volatility is 14.1%. If ρ = −1, the two cancel exactly and volatility is zero.

Expected return, meanwhile, is simply the weighted average — it does not care about correlation at all. Return adds linearly; risk adds sub-linearly. That asymmetry is the entire free lunch, and it is a mathematical fact rather than a market opinion.

The lunch is real, and you can measure it in francs

Lower volatility is not just a comfort. It raises the return you actually compound, because compounded (geometric) return sits below average (arithmetic) return by roughly half the variance:

geometric ≈ arithmetic − σ²/2

An asset with an 8% arithmetic mean return and 20% volatility compounds at roughly 8% − 0.02 = 6%. Two such assets with zero correlation, held 50/50, still have an 8% arithmetic mean but 14.1% volatility, so they compound at roughly 8% − 0.01 = 7%. One full percentage point of compounding, per year, purchased by rearranging holdings rather than by taking more risk or paying more fees. Over thirty years, a percentage point compounds into roughly a third more terminal wealth.

This is the honest core of the argument, and it is why the lesson on rebalancing later in this course matters so much: rebalancing is the mechanism that keeps you actually holding the diversified mix rather than drifting into whatever has recently won.

Where the free lunch runs out

Now the limit. Assume you hold N assets in equal weights, each with volatility σ, and every pair has the same correlation ρ. Portfolio variance simplifies to:

σ²ₚ = σ² [ 1/N + (1 − 1/N)ρ ]

The first term shrinks toward zero as you add holdings. The second does not — it converges to σ²ρ. So no matter how many positions you own, portfolio volatility never falls below σ√ρ. That floor is systematic risk: the shared exposure that all the holdings have to the same underlying economy.

With σ = 25% and ρ = 0.3, the floor is 25% × √0.3 = 13.7%. Now watch how quickly you approach it:

• 1 holding: 25.0%
• 10 holdings: 15.2%
• 30 holdings: 14.2%
• 100 holdings: 13.9%
• infinite holdings: 13.7%

Going from one holding to ten removes about 10 percentage points of volatility. Going from ten to thirty removes one. Going from thirty to one hundred removes about 0.3. The free lunch is served almost entirely in the first course. This is the quantitative version of a result that goes back to Evans and Archer (1968) and was refined by Elton and Gruber (1977) and Statman (1987): the marginal risk reduction from an additional name falls away very fast.

The practical consequence is not "hold ten stocks." It is that once you are broadly spread within an asset class, adding a twentieth similar fund buys you essentially nothing, while adding a genuinely different risk — a different economy, a different currency, a different driver of cash flow — still buys something. Breadth of count is cheap; breadth of kind is what remains scarce.

The limits that actually cost people money

Fake diversification. Holding six equity funds is not six exposures if all six own the same global large-cap companies. Many portfolios that look spread across a dozen products contain, on inspection, one dominant risk: global equity beta, with fee layers on top. The test is not how many lines are on your statement but how many independent things could go wrong.

Concentration hiding inside an index. A market-capitalisation index is diversified by construction only to the extent that market capitalisation is spread out. Finland around the turn of the millennium is the extreme illustration: at Nokia's peak in 2000 the company alone accounted for roughly two-thirds of the Helsinki exchange's market value, so an investor who owned "the Finnish market" owned, in substance, one telecoms company. Smaller national indices routinely behave this way, and the word "index" says nothing about how many independent bets sit inside it — a point the lesson on index funds takes up with the Swiss market specifically.

Correlations are not constants. The ρ in the formula above is estimated from history and is unstable, and it tends to rise when markets fall — precisely when the reduction was supposed to help. The next lesson in this course is devoted to that mechanism, so it is enough here to say that the diversification you measure in calm periods systematically overstates the diversification you will receive in a violent one.

Diversification does not price-protect. Spreading across many assets reduces the risk that any one of them disappoints. It does nothing about the risk that all of them were bought at valuations implying low future returns. A well-diversified portfolio of expensive assets is still a portfolio of expensive assets.

It cuts both tails. Diversification narrows the distribution of outcomes. That removes the disaster and it also removes the spectacular win. Anyone who wants to be very rich from one insight has to accept concentration and the ruin risk that comes with it — which is why this course has a separate lesson on position sizing for concentrated portfolios. What you cannot do is hold a concentrated portfolio and claim the risk properties of a diversified one.

What this means for a Swiss-resident investor

Two structural facts make the Swiss case distinctive.

First, home bias is unusually costly here. Switzerland is a small share of global market capitalisation, its listed market is dominated by a handful of very large pharmaceutical and food companies, and a Swiss investor's human capital, property, and pension claims are already Swiss-franc exposures. Loading a securities portfolio with more Swiss risk adds to a position that is already large. The course covers home bias and currency in its own lesson; the point for now is that "diversified across twenty Swiss names" is a much narrower statement than it sounds.

Second, the bond half of the standard diversification recipe is much thinner in francs than the textbook assumes. The Swiss franc bond market is small next to the dollar and euro markets, and domestic issuance is dominated by a short list of names: the Confederation, cantons and communes, the two mortgage bond institutions that fund Swiss mortgage lenders, the large banks, and a modest number of domestic corporates. Whole stretches of the credit spectrum that a dollar investor takes for granted — deep high-yield, a broad range of industrial issuers — barely exist in CHF. A "diversified CHF bond portfolio" of thirty lines can therefore contain far fewer independent things that could go wrong than thirty lines suggest: most of those issuers depend on the same domestic economy, several are exposed to the same domestic property market, and all of them move with the same franc yield curve.

That leaves a real dilemma rather than a clean answer. Going abroad for bond diversification buys genuinely different issuers and different rate cycles, but reintroduces currency risk, which for a franc-based investor is large relative to bond volatility and has to be hedged at a price. The lesson on home bias and currency exposure works through that trade explicitly. The point here is only that the number of holdings in a bond sleeve is an even weaker measure of diversification than it is in equities.

What to take away

• Portfolio risk is driven by covariance, not by the average risk of the holdings, which is why combining imperfectly correlated assets lowers volatility without lowering expected return.
• The benefit shows up as real money through compounding: cutting volatility from 20% to 14% at an unchanged 8% arithmetic mean adds roughly one percentage point to the annual compounded return.
• Diversification has a hard floor of σ√ρ — with 25% asset volatility and 0.3 average correlation, no number of holdings takes portfolio volatility below about 13.7%.
• The great majority of the achievable risk reduction arrives within the first ten to thirty genuinely different holdings; adding more similar positions buys almost nothing while adding fees and complexity.
• Counting positions is not a diversification test; counting independent things that could go wrong is, and by that test a Swiss investor holding Swiss equities, Swiss property and a Swiss pension holds far fewer than they think.

Sources

• Harry Markowitz, "Portfolio Selection," Journal of Finance, 1952 — the paper that established portfolio variance, not asset variance, as the object of interest.
• Harry Markowitz, Portfolio Selection: Efficient Diversification of Investments, 1959.
• John L. Evans and Stephen H. Archer, "Diversification and the Reduction of Dispersion: An Empirical Analysis," Journal of Finance, 1968.
• Edwin J. Elton and Martin J. Gruber, "Risk Reduction and Portfolio Size: An Analytical Solution," Journal of Business, 1977.
• Meir Statman, "How Many Stocks Make a Diversified Portfolio?", Journal of Financial and Quantitative Analysis, 1987.
• Andrew Ang, Asset Management: A Systematic Approach to Factor Investing, Oxford University Press, 2014.
• Nasdaq Helsinki (formerly HEX) index factsheets and historical constituent weights — the source for Nokia's share of the Finnish market around 2000.
• SIX Swiss Exchange, index rulebooks and factsheets for the SMI and SPI — the authoritative source for current index weights and concentration.
• Swiss National Bank, statistics on Swiss franc bond issuance and outstanding volumes by issuer category — the source for the composition of the domestic bond market.
• SIX Swiss Exchange, bond market segment rules and issuer listings.