Personal Capital was included last week on a short list of apps that help you save money. The author of the article, Katie Roof, noted that a great use of Personal Capital is to “look at your assets to see where there are opportunities for investment diversification.”

It’s a great insight – while diversification is a well-used investing tool, it can be overlooked *as a way to save money*. In our quest to bring transparency to investing, we present in this post the theory and math of portfolio diversification.

**The Theory: “The Only Free Lunch in Investing”**

Diversification is a relatively intuitive concept. We’ve all heard the axiom “don’t put all your eggs in one basket.” Its simplistic application to investing is as follows: if all your eggs (stocks) are in one basket and you drop it, you lose everything. However, if you have 10 eggs (stocks), each with its own basket and you drop one, you’ll still have nine eggs (stocks) left.

While diversification is not a new concept in investing, it was not until the 1950s that investors began to quantify its benefits. In 1952, Harry Markowitz – then a 25-year old economist at the University of Chicago – wrote the seminal paper called *Portfolio Selection*. The remarkable insights in this paper formed the basis for Modern Portfolio Theory, and ended up earning him the Nobel Prize in 1990.

A key insight of Modern Portfolio Theory is the classification of risk into two types: *unsystemic risk* and *systemic risk*. *Unsystemic risk*, or diversifiable risk, is the risk that comes with owning a single stock that may decline on its own. For example, if there’s a management scandal at Company X, that is considered unsystemic risk. *Systemic risk*, or un-diversifiable risk, is the risk that the economy as a whole will experience a downturn. A recent example of systemic risk is the subprime mortgage crisis, which sparked a worldwide decline in financial markets.

In his work, Markowitz showed that the unsystematic risk of a portfolio can be decreased through diversification. In other words, while the return of a diversified portfolio will be equal to the average rates of return of the individual holdings, the risk (as measured by the standard deviation) of the portfolio will be *less than the risk of the individual holdings*. That’s why diversification has been called the only free lunch in investing.

**Your Portfolio: the Benefits of Diversification**

Before we get to the math, a last note on the practical application to your portfolio. Our example focuses on a two-stock portfolio to keep the math simple – we show that because there’s some diversifiable risk, even blending two identical stocks in a portfolio lowers the risk. As you can imagine, by investing a number of different companies across asset classes and industries, you can substantially reduce risk and maximize your risk-adjusted returns. The math gets more complicated, but it turns out you can also solve for maximizing mean for a given level of variance – which is called an “efficient” portfolio. The “efficient frontier” is then the set of investment portfolios that provide the greatest return for the risk you take.

How Efficient is Your Portfolio? Check in with our Investment Checkup

So we can now return to our initial question: how does diversification help you to save money? By reducing the risk of potential loss. If you invest in a portfolio on the efficient frontier, you can minimize risk while maximizing returns, and save money in the process. Read on for the math.

**The Math: Measuring Risk and Return**

If you’ve gotten to this point in the blog, congrats! You’re on your way to proving for yourself that diversification lowers the risk in your portfolio. In this section, we walk through the calculations for the return and risk of two identical companies, Company X and Company Y.

As you’ll see, these companies have identical risk and return profiles in various market scenarios (systemic risk) and in the event of a management scandal (unsystemic risk). Because the stocks have unsystemic risk the standard deviation of the portfolio (2.5%) is less than that of the individual stocks (3.5%).

*Expected Return*

To calculate the expected return for a stock or a portfolio, you need either historical returns or the expected returns in various “states of the word.” The math is simple: expected return is equal to the probability-weighted average of the returns of each case (scandal or no scandal in under three market scenarios: downside, base and upside). In the chart below, we show the return expectations for Company X and Company Y.

For each stock individually, we calculate the expected returns as follows.

*Risk – Standard Deviation*

The way we express risk is through standard deviation. Mathematically, standard deviation represents the volatility, or variability of a stock’s return relative to its average.

You may remember from statistics that the standard deviation is equal to the square root of the variance of the stock. In the same simplified world, you calculate the variance by summing the square of the probability of each scenario times the difference between the return in that scenario by the expected return.

As you’ll notice, variance is expressed as a percentage squared. To compare apples to apples, we take the square root of the variance to get standard deviation.

Unfortunately, calculating the portfolio standard deviation is more complicated than is the case for an individual stock (fortunately, that’s where the beauty of diversification kicks in!). Before we can get started on standard deviation, we need to calculate the covariance and correlation.

*Covariance*

Covariance is the measurement of how the stocks move together. If the stocks move in the same direction, the covariance is positive; if different, it’s negative. If it’s zero, they have no relationship.

To determine the covariance between stocks of Company X and Company Y, we measure the difference between the return of each stock and the average return over the whole time period, and multiply the two together. Add them up and divide them by the number of periods to get the covariance. This is what the formula looks like in shorthand.

In the below table, we show how you get from historical returns to the calculation of the covariance based on the above formula.

The covariance between Stock X and Stock Y is negative, which reflects the fact that the stocks tend to move in different directions.

*Correlation*

Our second step is to calculate the correlation between Stock X and Stock Y. Correlation ranges from -1 to 1; +1 indicates that the two assets move completely together and a -1 indicates that the two assets move in opposite directions from each other.

We calculate the correlation by dividing the covariance by the product of the standard deviation of each stock. Using the formula in Example 1, we calculate that σ_{x }= 3.6%, σ_{y }= 3.6%.

Stock X and stock Y have a correlation of negative 0.07, so the two assets are moving in different directions.

**Portfolio Standard Deviation**

Our final step is to calculate the standard deviation of the portfolio.

And voila! By blending two stocks, we’ve lowered the risk of the portfolio.

As a last note, while we’ve shown an example of two stocks that are negatively correlated, you can also lower the risk if the correlation coefficient is positive. It’s only when the correlation coefficient between stocks x and y is 1 – when they move perfectly in sync – that the standard deviation is equal to the average standard deviation of the two stocks.

And finally, congrats! You’ve proven that diversification lowers the risk in your portfolio.