How diversification lowers risk

Diversification is not owning a lot of things. It is owning things whose bad years do not all arrive together. The mechanism rests on one asymmetry that almost nothing else in finance offers:

• The expected return of a portfolio over any one period is the weighted average of the expected returns of its parts. Put half your money in something you expect to earn 8 percent and half in something you expect to earn 6 percent, and you expect 7 percent. How the two move relative to each other changes nothing about that.
• The risk of the portfolio is not the weighted average of the risks of its parts. It is almost always lower, and how much lower depends entirely on how the parts move relative to each other.

Return adds up. Risk does not. Everything else on this page follows from those two sentences.

Take two holdings with the same expected return and the same volatility, and split your money evenly between them. Write $\rho$ for the correlation between their returns and $\sigma$ for the volatility of either one. The blend's volatility is:

$$\sigma_p = \sigma\sqrt{\frac{1 + \rho}{2}}$$

If they move in perfect lockstep, $\rho = 1$ and the blend is exactly as volatile as either holding: you have gained nothing. If they move independently, $\rho = 0$ and the blend's volatility falls to about 71 percent of a single holding's, a reduction of roughly 29 percent, with no change whatever in what you expect to earn. If they move exactly opposite, $\rho = -1$ and the volatility goes to zero.

Harry Markowitz, whose work on portfolio selection founded modern portfolio theory, is widely quoted describing diversification as the only free lunch in investing. The phrase is fair. It also needs the careful statement in the last section, because what is free is the removal of one specific kind of risk, not the removal of risk.

Correlation is one number between -1 and +1 that says how closely two holdings' returns move together.

• +1 means perfect step. When one is up on the day, so is the other, in a fixed proportion. Blending them removes nothing.
• 0 means no straight-line relationship between the two. Read it carefully: zero correlation is not the same as independence, because two holdings can score zero and still be tied together in a way a straight line cannot capture. This is where most of the benefit available in practice lives.
• -1 means exactly opposite. A correctly weighted blend of two such holdings has no volatility at all, which also means it can earn no more than the risk-free rate, or someone would borrow to hold it forever.

Run the equal-weight, equal-volatility formula from the last section across a range of correlations and the shape of the benefit is easy to see:

Three things about correlation get missed. It describes direction and timing, not size: two holdings can be perfectly correlated while one moves five times as far as the other. It is a description of a past window, not a property of an asset, and it moves. And you do not need uncorrelated holdings to benefit, because anything below +1 lowers portfolio volatility relative to the weighted average of the parts. Partial help is still help.

Risk in a portfolio comes in two kinds, and diversification treats them completely differently.

Specific risk, also called unsystematic risk, belongs to one holding. A factory burns down, a drug trial fails, a chief executive is arrested, an accounting fraud surfaces, a patent is refused. The model treats these events as unrelated to each other, which is what lets them partly offset in a portfolio of many holdings: some years a few of your holdings get the bad news and a few get the good. Real events are less obliging than the model, and an accounting fraud at one company has a way of drawing auditors toward its peers.

Market risk, or systematic risk, belongs to everything at once. A recession, an interest-rate shock, a war, a change in how much investors want to be paid for taking risk at all. No amount of spreading within the affected market makes this go away, because the thing that moves is the market.

The arithmetic for an equally weighted portfolio of $n$ holdings makes the split explicit:

$$\sigma_p^2 = \frac{\bar{\sigma}^2}{n} + \frac{n-1}{n}\,\overline{\text{cov}}$$

Here $\bar{\sigma}^2$ is the average variance of a single holding and $\overline{\text{cov}}$ is the average covariance between pairs of them. The first term is specific risk, and it is divided by $n$, so it collapses quickly: the first ten holdings do most of the work, the next twenty do far less, and after that the curve is close to flat. The second term has no $n$ in a place that helps. As you add holdings it converges on the average covariance and sits there.

That average covariance is the floor. You cannot diversify below the amount by which the things you own move together. Which is why, past a modest number of holdings, the character of what you own matters far more than the count.

Both portfolios hold thirty names, so both look diversified by the only measure most people apply. They are not remotely the same portfolio.

Thirty mid-sized banks in one country share a balance-sheet shape, a funding model, a regulator and an exposure to the same interest-rate move. The risk specific to any one of them is diversified away properly: if one of the thirty has a bad loan book, it costs you a thirtieth of a portfolio. What is left is the risk that the whole sector reprices, and that is left at close to full strength, because the average covariance between two of those banks is high. In the formula above, the first term is small and the second term is large. Thirty names, one bet.

Thirty names spread across consumer staples, energy, software, healthcare and industrials, alongside government bonds, alongside companies domiciled outside your own country, have a much lower average covariance. Same count of holdings, lower floor.

The arithmetic of a sector fall makes the point without any theory at all. If a sector drops 45 percent and it is your entire portfolio, you are down 45 percent. If the same sector is a fifth of your portfolio and the rest of it happens to be flat, you are down 9 percent. The second is a bad year. The first is a year some people never recover from, because after a 45 percent fall you need an 82 percent gain to get back to where you were.

Roughly in order of how much they tend to add, the dimensions a portfolio can spread across are: asset type, then country and currency, then sector and industry, then company size and style, and only then the number of individual names. Investors in most countries have been measured holding far more of their own domestic market than its share of world market value, a pattern researchers call home bias.

This is the part that surprises people, and it is why the CAGR calculator sits at the top of this page rather than at the bottom.

Gains and losses are not symmetric. A 25 percent fall needs a 33 percent gain to undo. A 50 percent fall needs a 100 percent gain. So two investments with the same average yearly return do not end up with the same money: the one whose returns scatter more ends up with less. The gap even has an approximate size:

$$g \approx \mu - \frac{\sigma^2}{2}$$

where $g$ is the compound rate the money actually grew at, $\mu$ is the plain average of the yearly returns and $\sigma$ is their standard deviation. It is an approximation, closest for modest swings, but its direction is exact. The more the yearly returns scatter, the further compound growth falls below the average.

So diversification does something past making the ride more comfortable. Cutting $\sigma$ while leaving $\mu$ untouched raises the rate the money grows at. One condition is easy to miss: a portfolio only keeps the lower $\sigma$ if its weights stay near their targets, which means rebalancing. Left to drift, a blend quietly turns into whatever grew fastest inside it.

The worked examples below put figures on it: an investment averaging 7.5 percent a year through a 40 up, 25 down cycle reaches $12,762.82 after ten years, while the same average held as a blend rebalanced each year, with no swing at all, reaches $20,610.32. About 1.6 times the money, from the same average return over the same decade.

Hold that identical pair for the same decade and never rebalance, though, and it finishes at $12,762.82 as well, because each half runs the same ten yearly factors in a different order and order does not change a product. In this example the entire gap is the rebalancing. Lower correlation is what creates the opportunity; rebalancing is what collects it.

The examples use a pair of holdings that move exactly opposite, which is an idealisation chosen to show the mechanism cleanly. Taken literally it collides with the arbitrage argument in the correlation section: a blend with no volatility at all cannot go on paying 7.5 percent a year, because anyone could borrow at the risk-free rate and hold it forever. Nothing in the real world is that obliging. What survives is the direction of the effect, at every correlation below +1. Put your own start and end values into the CAGR calculator above to see the steady rate that connects them, and compare it to the average of the individual years.

Diversification is the rare idea in investing that asks for no forecast, no skill and no information advantage. It is also routinely oversold, so here is what it does not do.

• It does not protect against a market-wide fall. Correlations measured in calm markets tend to rise toward +1 in severe declines, exactly when a low number would be worth most. A portfolio of thirty stocks across ten sectors and four countries still falls in a global bear market. Diversification changes how far, not whether.
• It does not raise the expected return of anything. It does not make a poor holding good, and adding a holding you expect to lose money on lowers the portfolio's expected return in exact proportion to its weight.
• It is not free of cost. Funds charge fees, trades cost money, and in many countries selling to rebalance inside a taxable account creates a taxable gain. The free lunch is free of a cost in *expected return*, which is not the same as free.
• It does not remove the risks you control directly. A cash buffer for emergencies, a time horizon that matches the money, and not being forced to sell at the bottom all do work that no amount of spreading can do.
• It does not protect purchasing power. A well diversified portfolio can still lose ground to inflation, which is a separate calculation entirely: see real return after inflation.
• It stops paying past a point. Once the average covariance dominates, more holdings add complexity and nothing else. Twelve overlapping funds are not twelve times the diversification of one.

The careful version, then. Diversification is free in one precise sense: shedding the risk specific to individual holdings costs nothing in expected return, which stands out in a field where nearly every other reduction in risk is paid for out of expected return. That is the exception the free lunch line is pointing at, and it needs no forecast to claim. This page is education rather than advice about your own money. See risk and return for what the kind you pay for looks like.

The statistic doing the work here has a name. Whether two holdings offset each other is measured by the correlation between their returns, and diversification stops helping as that correlation approaches 1: correlation.