Portfolio Size and Risk What we have seen for a portfolio with two assets can be extended to

Portfolio Size and Risk What we have seen for a portfolio with two assets can be extended to

include any number of assets. The calculations become very compli- cated, because we have to consider the covariance between every possi- ble pair of assets! But the basic idea is the same. The risk of a portfolio declines as it includes more assets whose returns are not perfectly corre- lated with the returns of the assets already in the portfolio.

The idea of diversification is based on beliefs about what will hap- pen in the future: expected returns, standard deviation of all possible returns, and expected covariance between returns. How valid are our beliefs about anything in the future? We can get an idea by looking at the past. So we look at historical returns on assets—returns over time— to get an idea of how some asset’s returns increase while at the same time others do not or decline.

Let’s look at the effects of diversification with common stocks. As we add common stocks to a portfolio, the standard deviation of returns on the portfolio declines—to a point. We can see this in Exhibit 10.6, where the portfolio standard deviation is plotted against the number of different stocks in the portfolio. After around twenty different stocks, the portfolio’s standard deviation is about as low as it is going to get.

Why does the risk seem to reach some point and not decline any far- ther? Because common stocks’ returns, in general, are positively corre- lated with one another. There just aren’t enough negatively correlated stocks’ returns to reduce portfolio risk beyond a certain point.

We refer to the risk that goes away as we add assets as diversifiable risk. We refer to the risk that cannot be reduced by adding more assets as nondiversifiable risk. Diversifiable and nondiversifiable components of a portfolio’s risk are shown in Exhibit 10.6.

The idea that we can reduce the risk of a portfolio by introducing assets whose returns are not highly correlated with one another is the basis of modern portfolio theory (MPT). MPT tells us that by combin- ing assets whose returns are not correlated with one another, we can determine combinations of assets that provide the least risk for each possible expected portfolio return.

Though the mathematics involved in determining the optimal com- binations of assets are beyond this text, the basic idea is provided in

Risk and Expected Return

Exhibit 10.7. In panel (a), the expected return and standard deviation for all possible portfolios is shown. Each point in the graph represents a possible portfolio that can be put together comprising different assets and different weights. The points in this graph represent every possible portfolio. As you can see in this diagram:

■ Some portfolios have a higher expected return than other portfolios with the same level of risk. ■ Some portfolios have a lower standard deviation than other portfolios with the same expected return.

Because investors like high returns and low risk, some portfolios are preferable to others. Portfolio that deliver the highest return for the level of risk make up what is called the efficient frontier. If investors are rational, they will go for the portfolios that fall on this efficient frontier. All the possible portfolios, as well as the efficient frontier, are dia- grammed in Exhibit 10.7, panel (b).

So what is the relevance of MPT to financial managers? MPT tells us that:

■ We can manage risk by judicious combinations of assets in our portfo- lios. ■ There are some combinations of assets that are preferred over others.

In the next section, we will see what MPT can teach us about valuation. EXHIBIT 10.6 The Average Standard Deviation of a Portfolio for Different Portfolio

Sizes

THE FUNDAMENTALS OF VALUATION

EXHIBIT 10.7 Possible Portfolios and the Efficient Frontier

Panel A: The Expected Return and Risk for Different Portfolios

Panel B: The Efficient Frontier