**Portfolio Theory.**

**(1). Introduction.**

The analysis of risk and uncertainty concentrates in some way on altering future returns to allow for uncertainty of outcome (e.g. using probability distributions of returns). An alternative approach is to allow for uncertainty by increasing our required rate of return on risky projects.

This latter approach is commonly taken by investments. For example, if we were comparing a building society investment with one in equities we would normally require higher return form equities to compensate us for their extra risk. In a similar way if we were appraising equity investment in a food retailing company against a similar investment in a computer electronics firm we would usually demand higher returns from the electronics investment to reflect its higher risk.

Clearly use of a risk adjusted discount rate can be employed in almost any situation involving risk. The practical problem is how much return we should demand for a given level of risk. To solve this problem we can turn to the stock exchange-a place where risk and return combinations (securities) are bought and sold every day. If for example, we can better the return earned by investors on the stock market by investing in a physical asset offering the some level of risk, we can increase investor wealth and the investment should be adopted.

Unfortunately the required approach is not as simple as this. Investors seldom hold securities in isolation. They usually attempt to reduce their risks by ‘not putting all their eggs into one basket” and therefore hold portfolios of securities. Before we can deduce a risk-adjusted discount rate from stock exchange returns we need to identify the risks investors in their diversified investment portfolios.

**(2). The portfolio effect.**

A portfolio is simply a combination of investments. If an investor puts half of this funds into an engineering company and half into a retail ships firm then it is possible that any misfortunes in the engineering company (e.g. A strike) may by to some extent offset by the performance of the retail investment. It would be unlikely that both would suffer a strike in the same period.

**(3). Correlation.**

Correlation is a statistical measure of how strong the connection is between two variables. In portfolio theory the two variables are the returns of two investments. High positive correlation means that both investments tend to show increase (or decrease) in return at the same time.

The degree of risk reduction possible by combining the investments depends on the carrel between them.

The more negative the correlation the greater the possibilities for risk reduction.

**(4). Portfolio theory- the two-security portfolio.**

A formal analysis of the combination of two investments is now presented. Because portfolio theory has its roots in the management of stock exchange investments, this is referred to as the two security portfolio.

The analysis is usually presented in terms of rates of return over a single time period is simply:

**(End of period value – start of period value) + dividend paid /Start of period value.**

**(5). Covariance and correlation.**

The risk reduction in the last example was made possible by low correlation between the investments. Jest looking at the possible reruns of A and B shows that there is no consistent positive or negative relationship between them. The correlation coefficient will probably be jest higher then zero.

The covariance will be positive for positive correlation and negative for negative correlation, but its size depends on the size of the figures in the original data and is difficult to interpret.

The covariance will be positive for positive correlation and negative for negative correlation, but its size depends on the size of the figures in the original data and is difficult to interpret.

**(6). Formulae for the two – security portfolio.**

In general, the risk of a two –security portfolio will depend on:

**a.**The risk of the constituent investments in isolation;**b.**The correlation between them ; and**c.**The proportion in which the investments are mixed.**(7). Two – security portfolios-Effect of the correlation coefficient.**

Maximum risk reduction is possible with a correlation of -1. In this case, risk can be (but is not always) reduced to zero. If correlation is +1, the portfolio risk is simple a weighted average of the investment risks.

**(8). Combining a risky security and a risk-free security.**

There is a special case of the two-security portfolio which is particularly important for our later studies. This is the case of combining a risk-free security with a risky security.

A risk-free security is one which shows no variability in its predicated returns. In other words its return is known with certainty. In practice it can be approximated by an investment in government stocks or bank deposit accounts at fixed interest (although varying rates of inflation would mean the real on these investments becomes uncertainty.

A risk free security has a zero variance and a zero covariance with any other security.

In the other words the portfolio standard deviation is simply the standard deviation of the riskily investment times the proportion of that investment in the portfolio.

In the other words the portfolio standard deviation is simply the standard deviation of the riskily investment times the proportion of that investment in the portfolio.

The expected returns of the portfolio will still be weighted average of the expected returns of the two investments.