Question

Explain, with the aid of an equation, the differences between the determinants of i) the risk of a portfolio with large no. of assets (e.g. 200 stocks in a portfolio) and ii) the risk of a portfolio with small no. of assets (e.g. 5 stocks in a portfolio).

Answer 1

The risk of an individual asset can be measured by the variance on the returns. The risk of individual assets can be reduced through diversification. Diversification reduces the variability when the prices of individual assets are not perfectly correlated. In other words, investors can reduce their exposure to individual assets by holding a diversified portfolio of assets. As a result, diversification will allow for the same portfolio return with reduced risk.

Example:

- A classical example of the benefit of diversification is to consider the effect of combining the investment in an ice-cream producer with the investment in a manufacturer of umbrellas. For simplicity, assume that the return to the ice-cream producer is +15% if the weather is sunny and -10% if it rains. Similarly the manufacturer of umbrellas benefits when it rains (+15%) and looses when the sun shines (-10%). Further, assume that each of the two weather states occur with probability 50%.

	Expected return		Variance
Ice-cream producer Umbrella manufacturer	0.515% + 0.5-10% = 2.5% 0.5-10% + 0.515% = 2.5%	0.5- [15-2.5]2 +0.5- [-10-2.5]2 = 12.52% 0.5- [-10-2.5]2 +0.5- [15-2.5]2 = 12.52%

- Both investments offer an expected return of +2.5% with a standard deviation of 12.5 percent

- Compare this to the portfolio that invests 50% in each of the two stocks. In this case, the expected return is +2.5% both when the weather is sunny and rainy (0.5*15% + 0.5*-10% = 2.5%). However, the standard deviation drops to 0% as there is no variation in the return across the two states. Thus, by diversifying the risk related to the weather could be hedged. This happens because the returns to the ice-cream producer and umbrella manufacturer are perfectly negatively correlated.

Obviously the prior example is extreme as in the real world it is difficult to find investments that are perfectly negatively correlated and thereby diversify away all risk. More generally the standard deviation of a portfolio is reduced as the number of securities in the portfolio is increased. The reduction in risk will occur if the stock returns within our portfolio are not perfectly positively correlated. The benefit of diversification can be illustrated graphically:

Figure 2: How portfolio diversification reduces risk

As the number of stocks in the portfolio increases the exposure to risk decreases. However, portfolio diversification cannot eliminate all risk from the portfolio. Thus, total risk can be divided into two types of risk: (1) Unique risk and (2) Market risk. It follows from the graphically illustration that unique risk can be diversified way, whereas market risk is non-diversifiable. Total risk declines until the portfolio consists of around 15-20 securities, then for each additional security in the portfolio the decline becomes very slight.

Portfolio risk

Total risk = Unique risk + Market risk Unique risk

- Risk factors affecting only a single assets or a small group of assets

- Also called

• Idiosyncratic risk

• Unsystematic risk

• Company-unique risk

• Diversifiable risk

• Firm specific risk

- Examples:

• A strike among the workers of a company, an increase in the interest rate a company pays on its short-term debt by its bank, a product liability suit.

Market risk

- Economy-wide sources of risk that affects the overall stock market. Thus, market risk influences a large number of assets, each to a greater or lesser extent.

- Also called

• Systematic risk

• Non-diversifiable risk

- Examples:

• Changes in the general economy or major political events such as changes in general interest rates, changes in corporate taxation, etc.

As diversification allows investors to essentially eliminate the unique risk, a well-diversified investor will only require compensation for bearing the market risk of the individual security. Thus, the expected return on an asset depends only on the market risk.