**The Diversified Shareholder and The Capital market.**

**(1). Mixing many risky securities.**

Portfolio theory has its roots in the management of investors’ portfolios of stock exchange investments and fixed interest stocks. The following sections show how the attempt to identify an optimal portfolio for investors has led to a comprehensive but simple theory of how the capital market relates risk and return. This, in turn, will assist us in our attempt to adjust discount rates to allow for risk.

The previous section considered portfolios of two securities. It is easy to expend this theory to cover portfolios of many securities, noting that where returns are assessed in percentage terms:

**(a)**The expected return of a portfolio is equal to the weighted average of the returns of the individual securities in the portfolio.

**(b)**The risk of the portfolio depends on:

i. The risk of each security in isolation:

ii. The proportions in which the securities are mixed:

iii. The correlations between every pair of securities in the portfolio.

**(2). Efficient portfolios.**

It is possible to identify which of these portfolios are really worth holding.

A rational risk –adverse investor would define an efficient portfolio as one that has.

This simple approach is known as the mean-variance efficiency rule (return = mean or expected return; risk = variance or standard deviation).

So, out of all the possible portfolios which an investor could make out of his chosen securities, which are mean variance efficient? (Put another way, which portfolios would the investor select from, given logical assessment of the mean returns and variances of all those available).

A rational risk –adverse investor would define an efficient portfolio as one that has.

**a).**A higher return than any other portfolio with the same risk; and**b).**A lower risk than any other with the same return.This simple approach is known as the mean-variance efficiency rule (return = mean or expected return; risk = variance or standard deviation).

So, out of all the possible portfolios which an investor could make out of his chosen securities, which are mean variance efficient? (Put another way, which portfolios would the investor select from, given logical assessment of the mean returns and variances of all those available).

You need to be familiar with the following terminology.

**(a).**Domination an investment dominates another if it provides a better return for the same risk or less risk for the same return.

**(b).**An efficient investment is one which is not dominated by any other investment, where as an inefficient investment is one which is dominated.

**(c).**Investor utility curves (or indifference curves) are curves alone which the investor is indifferent between the combinations of risk and return.

**(d).**Optimal portfolio- the efficient portfolio that has the highest utility for a given investor. This idea was discussed at the start of the chapter & identifying this portfolio requires knowledge of the investor’s indifference curves.

**(3). The market portfolio.**

**The assumptions**A few assumptions are now made, in order to build a simple model:

**(a).**Investors base their portfolio investment decisions on expected returns, standard deviation and correlations between all pairs of investments.

**(b).**All investor have the same expectations about future outcomes over a one-period time horizon.

**(c).**Investors may lend and borrow without limit at the risk-free rate of interest.

**(d).**There are no market imperfections: investments are infinitely divisible, information is costless, there are no taxes, transaction costs or interest rate charges, and no inflation.

Some of these assumptions are obviously unrealistic, but they greatly simplify the model- building process. Furthermore, even if the assumptions are relaxed, the theory will still hold approximately.

**Building the model.**

**(a).**Firstly , consider all the portfolio which could be constructed out of risky securities quoted on the stock market.

**(b).**Then identify the efficient portfolios from these.

**(c).**The consider mixing any one of these efficient portfolios, with a risk-free investment, Rf.

**Conclusion.**

**Out of all the possible portfolios that could be constructed from risky investment, only one portfolio is worth considering-portfolio M.**

A combination of Rf and M produces portfolios which are better than any others in terms of the return which is offered for any given level of risk.

However, given the existence of risk-free investment, investors would choose form those on the revised efficient frontier represented by line RfM.

Portfolios on the line RfM are achieved by mixing portfolio M with risk-free investments. Portfolios on the line M N are achieved by borrowing at the risk-free rate (remember we have assumed that the risk-free rate applies to borrowing as well as lending) and investing our own funds plus borrowed funds in portfolio M.

**What is portfolio M.**

**Because we have assumed that all investors have the same expatiation's about the future outcomes of investments, it follows that:**

**All investors will come to the conclusion that portfolio M is the best portfolio consisting solely of risky investments to hold.**Now, if any quoted share was not in portfolio M, then nobody would with to hold it. It would therefore have not value. We must therefore conclude that:

**Portfolio M includes every risky security which is quoted on the market.**

Portfolio M is in fact simply a slice of the whole stock market; the proportions of shares held in it are the same as the total market capitalization of the shares on the stock market:

**Portfolio M is called the market portfolio.**

**All rational risk- averse investors will hold the market portfolio, according to the model we have jest constructed. Note that it is not necessary for very investor to hold very share on the stock market. Replicas of portfolio M may be generated by holding as few as fifteen shares. Investment in unit trusts will also achieve the same result.**

However, all investors do not have the same attitude to risk. By using the market portfolio, and by either lending or borrowing suitably at the risk-free rate, the investor can choose any level of risk he likes and can predict the return which the market will give him. This return will be the best that he could possibly get for the risk taken.

**(4). constructing the capital market line.**

We have already seen that combinations of risk-free and investments give a straight line trade-off between risk and return.

To draw the capital market line we therefore need only two observations,

**1.**Rf- The risk-free rate of interest, which can be approximated by the return on government stock.

**2.**Rm and sigma m- The risk and return of the market portfolio. As the market portfolio should contain all risk investments, this can be estimated by using the risk and return on a stock market index such as the Financial times all share index.

**Value of the Capital market line.**

The capital market line tells us for a given level 0 risk the return an investor should expect on the stock exchange. It is often referred to as giving the market price of risk. That is if we choose to take a given level of risk on the stock exchange then we can expect a given level of return.

If this is less than that offered by the project, it is tempting to say that the project should be accepted. However, there is a flaw in this logic.

The problem with this analysis is not in determining the capital market line but in determining the risk of an individual investment.

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