# Risk and diversification

Diversification is the cornerstone of modern portfolio theory, yet despite being familiar with the term, few investment professionals fully understand the mechanics underlying this phenomenon.

The principle of diversification is best explained by drawing on the conventional wisdom of the old adage of not placing all one’s eggs in a single basket, but over and above the mere message of spreading one’s risk there is also a real statistical benefit derived from portfolio diversification.

To explain the benefits, let’s assume that we have a portfolio of different stocks.
These stocks are from different sectors, different industries, have different management teams, some are defensive, some are cyclical, and they have differing sensitivities to interest rates, consumer demand, and other market influences.

If something in the greater economy (or market) changes, the effects on the various stocks will be different—some stocks might increase in value as a result of the change and others will decrease in value. Moreover, those that increase (or decrease) in value will do so by differing degrees.

The point is that by holding different stocks, the reaction to an exogenous influence will cause some stocks to go up and some to go down, and assuming we hold sufficient stocks, and more importantly, *sufficiently different stocks*, the increases in some stock prices will cancel out the decreases in other stock prices.

The risk that the unique characteristics of a particular stock will cause it to decline in value in response to an exogenous influence is referred to as its unique or unsystematic risk. An example of an unsystematic risk might be a strike by motor vehicle manufacturing workers, which would presumably negatively affect only those industries associated with the motor vehicle industry. The same principle might be applied to any type of portfolio of financial assets such as a unit trust portfolio.

Sometimes an exogenous influence might cause a market-wide increase or decrease in stock prices irrespective of the unique characteristics of individual stocks. The risk that an exogenous influence will cause a market-wide decline in value is referred to as systematic risk or market risk.

Irrespective of how many stocks we hold in our portfolio or how different they are, there will be no cancellation of the market movement as a result of this type of influence, since all the stocks (or unit trusts) are influenced to the same degree.

An example of such an influence might be a decline in GDP, which causes all equities (or unit trusts) to be adversely affected. Another example might be a catastrophic political event, such as the September 11 terrorist attacks, which causes a market-wide decline in equities.

Before we are able to explain the mechanics of diversification, we need to first of all understand what “risk” is in the context of investment theory.

Risk is essentially defined as the unpredictability of returns from an investment. This definition of risk is usually quantified by a measure referred to as standard deviation. The use of standard deviation as a measure of dispersion implicitly assumes that the returns derived from an investment are normally distributed with most of the returns occurring close to the mean and with the frequency of returns falling off exponentially as one moves away from the mean.

That said, standard deviation is thus a measure of the dispersion of returns from the mean or expected return. This risk, as we saw previously, may be split up into unsystematic risk and systematic risk. More importantly, the unsystematic risk can be diversified away (by holding sufficient stocks and sufficiently different stocks), whereas the systematic risk or market risk cannot be diversified away.

Many investors know and understand that there is an intrinsic relationship between risk and expected return. The more risk an investor is willing to bear, the greater the potential reward for bearing this risk. What few investors realise, however, is that the relationship is not between total risk (systematic plus unsystematic risk) and the expected return, but rather between systematic risk and expected return.

What this boils down to is that investors are only rewarded for the risk which cannot be diversified away, that is, investors are only rewarded for bearing systematic (or market) risk. Since unsystematic risk may be diversified away at no cost (theoretically at least) there is no reward earned for bearing it.

Diversification should thus not be seen as an enhancement to a portfolio, but rather fundamental to constructing portfolios which maximise return for a particular level of risk or conversely minimise risk for a particular level of return. The level of diversification in a portfolio is dependent on how differently the assets (stocks or funds) in that portfolio move relative to each other.

The significant impact of diversification can be seen in the rather daunting equation below. This equation is used to calculate the portfolio risk of a two asset portfolio.

**σp2=Wa2σa2 + Wb2σb2 + 2WaWbσaσbρab**

where portfolio risk = standard deviation = σp

σa and σb are the standard deviations (risk) of fund a and b respectively

wa and wb are the weightings of funds a and b in the portfolio respectively

σab is the correlation of fund a with fund b

There is no need to commit this equation to memory. Rather, there are two important points to take into consideration here. Firstly, portfolio risk σp is not merely an arithmetic average of the risks of the individual funds (σa and σb). Secondly, the portfolio risk σp is a function of the correlation between the two funds ?ab in this case fund a and b.

Note that this theory applies equally to funds as they do to stocks or any financial assets for that matter which is why I have used funds and stocks interchangeably.

Correlation refers to the degree to which one asset’s returns move relative to another’s. It’s important to understand that this correlation is a statistical measure. Frequently, individuals associate correlation with the asset composition of the funds in question. Yet despite the fact that asset composition plays an important role in the behavior of funds and ultimately in the performance which they deliver, asset composition is but only one factor which influences the returns delivered by a fund.

Ultimately, the correlation (in the strictest statistical sense) is a measure of how fund *returns* move relative to each other. The importance of calculating correlation thus becomes apparent. However since correlation is a standard statistical measure, it may be quite easily calculated using a standard spreadsheet application such MS Excel.

The example below illustrates using Excel’s ‘correl’ formula on three arrays of data corresponding to the AG Equity Fund, SIM General Equity Fund and the SIM Active Income Fund for the period from the beginning of 2010 till the beginning of October 2010:

The correlation between the two equity funds is 98%, whilst the correlation between AG Equity and SIM Active Income is 62%. As expected the two equity funds have a higher correlation due to their similar asset compositions, whereas the AG Equity fund and the SIM Active Income fund have a lower correlation.

The point is that even though asset composition has a major effect on the correlation of two funds, the correlation itself is measured on the returns of the funds relative to each other. This is because it is ultimately the returns of the funds which contain the effects of the asset compositions, as well as the numerous other factors which influence the changes in the fund returns.

The previous demonstration made use of monthly data in order to illustrate the methodology for calculating correlation. In practice, one would use the frequency of data with the highest resolution, which in this case is daily data. This would lead to a much more accurate correlation measure, since monthly data aggregates away a lot of the intra-monthly movement in the returns.

Using daily data the correlation between the two equity funds over the same period of time is 90% whilst the correlation between the AG Equity Fund and the SIM Active Income fund is 32%.

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