Market risk can be measured by beta, which measures how sensitive the return is to market movements. Thus, beta measures the risk of an asset relative to the average asset. By definition the average asset has a beta of one relative to itself. Thus, stocks with betas below 1 have lower than average market risk; whereas a beta above 1 means higher market risk than the average asset.

Estimating beta

Beta is measuring the individual asset's exposure to market risk. Technically the beta on a stock is defined as the covariance with the market portfolio divided by the variance of the market:

In practice the beta on a stock can be estimated by fitting a line to a plot of the return to the stock against the market return. The standard approach is to plot monthly returns for the stock against the market over a 60-month period.

Intuitively, beta measures the average change to the stock price when the market rises with an extra percent. Thus, beta is the slope on the fitted line, which takes the value 1.14 in the example above. Abeta ofl.14means that the stock amplifies the movements in the stock market since the stock price will increase with 1.14% when the market rise an extra 1%. In addition it is worth noticing that r-square is equal to 8.4%, which means that only 8.4% of the variation in the stock price is related to market risk.

Portfolio risk and return

The expected return on a portfolio of stocks is a weighted average of the expected returns on the individual stocks. Thus, the expected return on a portfolio consisting of n stocks is:

Where wi denotes the fraction of the portfolio invested in stock: i and r i is the expected return on stock i.

Example:

- Suppose you invest 50% of your portfolio in Nokia and 50% in Nestlé. The expected return on your Nokia stock is 15°% while Nestlé offers 10%. What is the expected return on your portfolio? n

- A portfolio with 50°%invested in Nokia and 50°% in Nestlé has an expected return of 12.5%.

Portfolio variance

Calculating the variance on a portfolio is more involved. To understand how the portfolio variance is calculated consider the simple case where the portfolio only consists of two stocks, stock 1 and 2. In this case the calculation of variance can be illustrated by filling out four boxes in the table below.

Table 2: Calculation of portfolio variance

In the top) left corner of Table 2, you weight the variance on stock 1 by the square of the fraction of the portfolio invested in stock 1. Simikrly, the bottom left corner is the variance of stock 2 times the square of the fraction of the portfolio invested in stock 2. The two entries in the diagonal boxes depend on the covariance between stock 1 and 2. The covariance is equal to the correlation coefficient times the product °f the two standard deviations on stock 1 and 2.The portfolio variance is obtained by adding the content of the four boxes together:

The benefit of diversification follows directly from the formula of the portfolio variance, since the portfolio variance is increasing in the covariance between stock 1 and 2. Combining stocks with a low correlation coefficient will therefore reduce the variance on the portfolio.

Example:

- Suppose you invest 50% of your portfolio in Nokia and 50% in Nestle. The standard deviation on Nokia's and Nestle's return is 30% and 20%, respectively. The correlation coefficient between the two stocks is 0.4. What v the portfolio variance?

- A portfolio with 5t°/oinvestedin Nokia and 50°% in Nestle has a variance of445, which is equivalent standard deviatio0 of2S.1%>.

For a portfolio of n stocks the portfolio variance is equal to:

Note that when i=j, c„ is the variance of stock i, a2. Similarly, when i^j, 0. is the covariance between stock i and j as 0.. = 0,0.0,.

Portfolio's market risk

The market risk of a portfolio of assets is a simple weighted average of the betas on the individual assets.

Where wi denotes the fraction of the portfolio invested in stock i and Pi is market risk of stock i.

Example:

- Consider the portfolio consisting of three stocks A, B and C.

Amount invested

Expected return

Beta

Stock A

1000

10%

0.8

Stock B

1500

12%

1.0

Stock C

2500

14%

1.2

- What is the beta on this portfolio?

- As the portfolio beta is a weighted average of the betas on each stock, the portfolio weight on each stock should be calculated. The investment in stock A is $1000 out of the total investment of $5000, thus the portfolio weight on stock A is 20%, whereas 30% and 50% are invested in stock B and C, respectively.

- The expected return on the portfolio is:

- Similarly, the portfolio beta is:

- The portfolio investing 20% in stock A, 30% in stock B, and 50% in stock C has an expected return of 12.6% and a beta of 1.06. Note that a beta above 1 implies that the portfolio has greater market risk than the average asset.

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