The Only Free Lunch in Modern Finance

“Diversification is the only free lunch in finance” is a well-known quote from the economist and Nobel Prize winner Harry Markowitz, the father of modern portfolio optimization. This quote refers to the fact that improving the risk-return profile of a portfolio by optimizing for returns (picking investments that will perform well in the future) is in general a difficult and costly endeavor. Comparatively, optimizing for risk is rather easy; risk can be reduced by simply spreading the portfolio (and the risk) out into different assets. The result is that the risk-return profile of the portfolio can be improved at virtually no cost. Is this too good to be true? How exactly does this work?

Risk in finance is typically measured in terms of volatility — an investment’s tendency to change in value. Naturally, it follows that if diversification reduces risk, it should reduce the volatility of the resulting portfolio. Let’s review the mathematics for the case of a portfolio with two investments and confirm this is indeed this is the case. Perhaps we’ll discover some subtleties along the way.

Suppose that a portfolio is evenly split between two assets X and Y. The returns of these two assets will be modeled as two random variables with a correlation (p) of zero and equal standard deviation. We can then use the following general formula to calculate the resulting portfolio standard deviation:

The correlation p was previously defined to be zero. The weights a,b can be set to 1/2 since the portfolio is evenly split between them. Let’s plug in these assumptions:

Simplify:

Refactor:

Refactor:

Recall that Var(X) = Var(Y):

Refactor:

Standard deviation is the square root of variance:

The result is that the standard deviation of the portfolio is lower than either of the individual portfolio components by a factor of sqrt(2). By spreading the investment over two assets, we have in essence caused a significant portion of the risk to cancel out while retaining the returns of the portfolio components. This is exactly where the diversification free lunch comes from. We have, however, glossed over one important mathematical detail:

Let’s loop back to the assumption that p = 0. What is this term and what is its significance? This term is the correlation of assets X and Y; it’s their degree of co-movement. If p is close to one, the assets are highly correlated and there is only a small benefit to the diversification, that is, only a small volatility reduction. If p = 0 there is a sqrt(2) volatility reduction. Interestingly, if p < 0 the volatility reduction is even greater than sqrt(2). In fact, if p = -1 it is theoretically possible to diversify away the portfolio volatility entirely.

Pearson R correlation

Now that we have built an intuition for the volatility damping effect of diversification, let’s look at a real-world example: a classic portfolio comprised of 60% treasuries and 40% equities (known as the 60/40 portfolio). The treasuries component will be represented using the IEF ETF and the equities component will be represented by the SPY ETF.

SPY vs IEF vs 60/40

SPY Return vs IEF Returns

In the above scatter plot, it’s clear that treasuries and equities have historically been negatively correlated (this has been true for about the past 25 years) with a correlation coefficient of p=-0.4. It follows from the previous mathematical exercise that the negative correlation of the 60/40 portfolio components should lead to a significant volatility reduction at the portfolio level. Let’s examine the real-world data and confirm it matches our mathematical model.

As suspected, we find a significant risk reduction. In fact, the 60/40 portfolio has about as much risk as a portfolio that is 100% treasuries, even though 40% of the portfolio is held in assets far riskier than treasuries! This is also reflected in the Sharpe Ratio, which is up approximately 50% — a substantial improvement in risk-adjusted returns. Many hedge funds would exploit this reduction in vol (volatility) by using leverage, enabling the portfolio to reach higher absolute returns. This line of thinking falls under the umbrella of risk-parity strategies that Bridgewater in particular is famous for.

Practically speaking, it’s incredibly hard to predict future returns — yet conveniently correlations are relatively stable over time. The result is that it is far easier to boost risk-adjusted returns by diversifying your portfolio than to predict outperformance of any individual security or asset class. This is virtually the only free lunch in modern finance. Don’t leave it on the table.

Additional Considerations:

• In general, diversification reduces the volatility of the portfolio by the sqrt(n), where n is the number of among uncorrelated assets in the portfolio.
• There are very few uncorrelated assets to diversify into. Pretty much all equities are significantly correlated. One notable exception is Virtu (VIRT).
• Correlations increase during periods of extreme market panic; this is an unfortunate result because periods of market panic are precisely when investors need the benefits of diversification most.