So far our analysis of risk-return was confined to single assets held in isolation. In real world, we rarely find investors putting their entire wealth into single asset or investment. Instead they build portfolio of investments and hence risk-return analysis is extended in context of portfolio.

A portfolio is composed of two or more securities. Each portfolio has risk-return characteristics of its own. A portfolio comprising securities that yield a maximum return for given level of risk or minimum risk for given level of return is termed as ‘efficient portfolio’. In their Endeavour to strike a golden mean between risk and return the traditional portfolio managers diversified funds over securities of large number of companies of different industry groups.

However, this was done on intuitive basis with no knowledge of the magnitude of risk reduction gained. Since the 1950s, however, a systematic body of knowledge has been built up which quantifies the expected return and riskiness of the portfolio. These studies have collectively come to be known as ‘portfolio theory’.

A portfolio theory provides a normative approach to investors to make decisions to invest their wealth in assets or securities under risk. The theory is based on the assumption that investors are risk averse. Portfolio theory originally developed by Harry Markowitz states that portfolio risk, unlike portfolio return, is more than a simple aggregation of the risk, unlike portfolio return, is more than a simple aggregation of the risks of individual assets.

This is dependent upon the interplay between the returns on assets comprising the portfolio. Another assumption of the portfolio theory is that the returns of assets are normally distributed which means that the mean (expected value) and variance analysis is the foundation of the portfolio.

**I. Portfolio Return:**

The expected return of a portfolio represents weighted average of the expected returns on the securities comprising that portfolio with weights being the proportion of total funds invested in each security (the total of weights must be 100).

**The following formula can be used to determine expected return of a portfolio:**

**Applying formula (5.5) to possible returns for two securities with funds equally invested in a portfolio, we can find the expected return of the portfolio as below:**

**II. Portfolio Risk:**

Unlike the expected return on a portfolio which is simply the weighted average of the expected returns on the individual assets in the portfolio, the portfolio risk, σp is not the simple, weighted average of the standard deviations of the individual assets in the portfolios.

It is for this fact that consideration of a weighted average of individual security deviations amounts to ignoring the relationship, or covariance that exists between the returns on securities. In fact, the overall risk of the portfolio includes the interactive risk of asset in relation to the others, measured by the covariance of returns. Covariance is a statistical measure of the degree to which two variables (securities’ returns) move together. Thus, covariance depends on the correlation between returns on the securities in the portfolio.

**Covariance between two securities is calculated as below:**

- Find the expected returns on securities.
- Find the deviation of possible returns from the expected return for each security
- Find the sum of the product of each deviation of returns of two securities and respective probability.

**The formula for determining the covariance of returns of two securities is:**

**Let us explain the computation of covariance of returns on two securities with the help of the following illustration:**

So far as the nature of relationship between the returns of securities A and B is concerned, there may be three possibilities, viz., positive covariance, negative covariance and zero covariance. Positive covariance shows that on an average the two variables move together.

A’s and B’s returns could be above their average returns at the same time or they could be below their average returns at the same time. This signifies that as the proportion of high return and high risk assets is increased, higher returns on portfolio come with higher risk.

Negative covariance suggests that, on an average, the two variables move in opposite direction. It means A’s returns could be above its average returns while B’s return could be below its average returns and vice-versa. This implies that it is possible to combine the two securities A and B in a manner that will eliminate all risk.

Zero covariance means that the two variables do not move together either in positive or negative direction. In other words, returns on the two securities are not related at all. Such situation does not exist in real world. Covariance may be non-zero due to randomness and negative and positive terms may not cancel each other.

In the above example, covariance between returns on A and B is negative i.e., -38.6. This suggests that the two returns are negatively related.

The above discussion leads us to conclude that the riskiness of a portfolio depends much more on the paired security covariance than on the riskiness (standard deviations) of the separate security holdings. This means that a combination of individually risky securities could still comprise a moderate-to-low-risk portfolio as long as securities do not move in lock step with each other. In brief, low covariance’s lead to low portfolio risk.

**III. Diversification****:**

Diversification is venerable rule of investment which suggests “Don’t put all your eggs in one basket”, spreading risk across a number of securities.

Diversification may take the form of unit, industry, maturity, geography, type of security and management. Through diversification of investments, an investor can reduce investment risks.

Investment of funds, say, Rs. 1 lakh evenly among as many as 20 different securities is more diversified than if the same amount is deployed evenly across 7 securities. This sort of security diversification is naive in the sense that it does not factor in the covariance between security returns.

The portfolio comprising 20 securities could represent stocks of one industry only and have returns which are positively correlated and high portfolio returns variability. On the other hand, the 7-stock portfolio might represent a number of different industries where returns might show low correlation and, hence, low portfolio returns variability.

Meaningful diversification is one which involves holding of stocks of more than one industry so that risks of losses occurring in one industry are counterbalanced by gains from the other industry. Investing in global financial markets can achieve greater diversification than investing in securities from a single country. This is for the fact that the economic cycles of different countries hardly synchronize and as such a weak economy in one country may be offset by a strong economy in another.

Fig. 5.2 portrays meaningful diversification. It may be noted from the figure that the returns overtime for Security X are cyclical in that they move in tandem with the economic fluctuations. In case of Security Y returns are moderately counter cyclical. Thus, the returns for these two securities are negatively correlated.

If equal amounts are invested in both securities, the dispersion of returns, up, on the portfolio of investments will be less because some of each individual security’s variability is offsetting. Thus, the gains of diversification of investment portfolio, in the form of risk minimization, can be derived if the securities are not perfectly and positively correlated.

**IV. Systematic and Unsystematic Risk:**

Thus, the variance of returns on a portfolio moving in inverse direction can minimize portfolio risk. However, it is not possible to reduce portfolio risk to zero by increasing the number of securities in the portfolio. According to the research studies, when we begin with a single stock, the risk of the portfolio is the standard deviation of that one stock.

As the number of securities selected randomly held in the portfolio increase, the total risk of the portfolio is reduced, though at a decreasing rate. Thus, degree of portfolio risk can be reduced to a large extent with a relatively moderate amount of diversification, say 15-20 randomly selected securities in equal-rupee amounts.

Portfolio risk comprises systematic risk and unsystematic risk. Systematic risk is also known as non- diversifiable risk which arises because of the forces that affect the overall market, such, as changes in the nation’s economy, fiscal policy of the Government, monetary policy of the Central bank, change in the world energy situation etc.

Such types of risks affect securities overall and hence, cannot be diversified away. Even if an investor holds well diversified portfolio, he is exposed to this type of risk which is affecting the overall market. This is why, non-diversifiable or unsystematic risk is also termed as market risk which remains after diversification.

Another risk component is unsystematic risk. It is also known as diversifiable risk caused by such random events as law suits, strikes, successful and unsuccessful marketing programmes, winning or losing a major contract and other events that are unique to a particular firm.

Unsystematic risk can be eliminated through diversification because these events are random, their effects on individual securities in a portfolio cancel out each other. Thus, not all of the risks involved in holding a security are relevant because part of the risk can be diversified away. What is relevant for investors is systematic risk which is unavoidable and they would like to be compensated for bearing it. However, they should not expect the market to provide any extra compensation for bearing the avoidable risk, as is contended in the Capital Asset Pricing Model.

Figure 5.3 displays two components of portfolio risk and their relationship to portfolio size.

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