**DIVERSIFICATION/CORRELATION**

Don't put all your eggs in one basket ….. or the virtue of diversification.

Learning objectives:

Understand how risk reduction can be achieved through asset diversification.

Correlation coefficient

Assets' weights

AssetA Return: %

AssetA StdDev: %

AssetB Return: %

AssetB StdDev: %

Correlation: 0.1

AssetA Weight: 0

AssetB Weight: 0

We saw in previous pages that during the past 83 years there was no strong case for diversification for investors willing to stay fully invested for more than 30 years. But for those who may not be able to stay fully invested because of some future possible financial constraints or because they have shorter time horizons, we will illustrate how diversification can increase return for a given risk or reduce risk for a given return. Also note that diversification might also be useful for investors who have the capacity to be fully invested for a long period of time but don’t have the appropriate level of risk tolerance. Ability and willingness are the two aspects that must be taken into account when evaluating risk tolerance. For instance when the stock market reached its lows during the recent financial crisis in 2008/2009 you might have been able to restrain yourself from selling your stock positions but did you have the stomach to buy stocks? Reflecting on your past behaviours is a good way to assess your own attitude towards risk; it is a lot more valuable than filling out a survey intended to profile your risk level.

Now before describing why we can benefit from diversification, we need to explain the concept of covariance and correlation. Covariance and correlation are two statistical terms that quantify the degree of similarity between two variables. Correlation is dimensionless and is equal to the covariance divided by the standard deviations of the two variables. Correlation is one of the most common statistics used in investment industry and it allows you with a single number to describe the degree of relationship between two assets’ returns. Correlation is normalized into a number comprised between -1 and +1 and called correlation coefficient. Two assets that move always in the same direction will have a correlation coefficient equal to 1. Two assets that move always in the opposite direction will have a correlation coefficient equal to -1. And two assets that move completely independently of one another have a correlation coefficient equal to 0.

As saw in previous sections, investments can be described in statistical terms with their expected return rate and their expected volatility (standard deviation). The graph illustrates the return in function of the risk of a portfolio composed of two assets.

Let’s take our two main assets stocks and bonds with their expected returns and standard deviations which we assume are (6%, 20%) and (2%, 9%) respectively. Using the same data set of historical returns over the last 83 years, we get a correlation coefficient equal to 0.06.

One sliding bar allows you to modify the correlation coefficient and the other one the assets’ weights in the portfolio.

Let the correlation coefficient unchanged and slide the ‘weight’ bar and observe how the portfolio risk and return move along the curve. The curve represents the risk return relationship with all the possible combinations of asset A and asset B.

First important observation to make is that contrary to what most people believe, a portfolio composed of the safest asset (asset B in this example) only does not offer you the lowest risk since the curve goes to the left of point B. Try to determine the composition of the lowest risk portfolio by sliding the ‘weight’ bar. You should reach the point of minimum risk with about 20% of asset A and 80% of asset B. The corresponding portfolio shows an expected return and expected standard deviation of 2.7% and 8.4% respectively compare to 2% and 8% for asset B only.

Another important observation is that for the same level of risk that asset B offers you, in this example 9% you can get a higher return than the one of asset B. Try by yourself and you will see that a portfolio composed on 30%of asset A and 70% of asset B will give you risk of 9% but the return is about 3.2%.

Default values have been set based on the historical data of stock and bond returns over the last 83 years and have been used to illustrate some key observations but of course you can adjust the numbers and use this interactive graph to simulate the effect of diversification using different assets with their own return, risk, and correlation characteristics.

Note that by changing the ‘correlation coefficient’ you can easily see that the less the assets are correlated the more you can benefit from diversification by reducing the risk. At the extremes, when correlation coefficient is set to 1 there is a straight line between the two assets, thus there is no possible benefit. When correlation coefficient is set to -1 you can find a combination that offers risk level of 0. Note that this case cannot exist in real life, otherwise you would have a ‘free lunch’ situation where one can borrow money with a risk free rate and buy the portfolio. You may have some investments that have a strong negative correlation, such as the VIX and the SP500 but they cannot show positive expected return at the same time.

Important to note that you can benefit from diversification as long as the correlation coefficient is low you don’t absolutely need to have a strong negative value.

It is also important to mention that correlation coefficient between two assets does not tell you anything about the respective volatility of the assets. For example if you take two assets that are very much correlated (coefficient correlation close to 1) then you can say that they experience ups and downs at almost the same time, but one asset can experience stronger moves than the other one, and thus be a lot more volatile. That is why for instance many financial advisors will recommend including commodities-linked assets in a portfolio because they are in general negatively correlated but due to their high volatility their portion in the total portfolio should stay relatively small.

If X and Y are two random variables, their covariance is the expected value of the product between their respective deviations from their means.

Correlation is defined as:

The volatility or the standard deviation of a two-asset portfolio is:

The volatility or standard deviation of a N-asset portfolio is:

When correlation of rate or return of two assets is calculated from historical data then the following formula is used:

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