Last Study Topics

• Measuring Portfolio Risk• Measuring Risk Variability• Unique Risk vs Market Risk

Today’s Study Topics

• Portfolio Risk• Market Risk Is Measured by Beta• Beta as a Portfolio Risk Measurement

Case: Lambeth Walk• Example: Lambeth Walk invests 60% of his

funds in stock I and the balance in stock J. The standard deviation of returns on I is 10%, and on J it is 20%. Calculate the variance of portfolio returns and Standard deviations, assuming;– a. The correlation between the returns is 1.0.– b. The correlation is .5.– c. The correlation is 0.

Continue

• a. The correlation between the returns is 1.0.

% 14 0.0196 Deviation Standard

0196.00)1x0.10x0.22(.60x.40x

]x(0.20)[(.40)

]x(.10)[(.60) Valriance Portfolio22

22

Continue

• b. The correlation is .5.

% 12.16 0.0148 Deviation Standard

0196.00.20)0.50x0.10x2(.60x.40x

]x(0.20)[(.40)

]x(.10)[(.60) Valriance Portfolio22

22

Continue

• c. The correlation is 0.

% 10 0.0100 Deviation Standard

0100.00)0x0.10x0.22(.60x.40x

]x(0.20)[(.40)

]x(.10)[(.60) Valriance Portfolio22

22

General Formula for Computing Portfolio Risk

• The method for calculating portfolio risk can easily be extended to portfolios of three or more securities.

• We just have to fill in a larger number of boxes. Each of those down the diagonal—the shaded boxes - contains the variance weighted by the square of the proportion invested, on the next slide.

Portfolio RiskThe shaded boxes contain variance terms; the remainder contain covariance terms.

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

STOCK

To calculate portfolio variance add up the boxes

Understanding

• Notice that as N increases, the portfolio variance steadily approaches the average covariance. If the average covariance were zero, it would be possible to eliminate all risk by holding a sufficient number of securities.– Unfortunately common stocks move together, not

independently. Thus most of the stocks that the investor can actually buy are tied together in a web of positive covariance which set the limit to the benefits of diversification.

HOW INDIVIDUAL SECURITIES AFFECT PORTFOLIO RISK

• The risk of a well diversified portfolio depends on the market risk of the securities included in the portfolio.

• Wise investors don’t put all their eggs into just one basket:

• They reduce their risk by diversification. • They are therefore interested in the effect

that each stock will have on the risk of their portfolio.

Betas for selected U.S common stock

STOCK Beta (B) STOCK Beta (B)

Amazon 3.25 Boeing .56

Coca-Cola .74 Dell Computer 2.21

Exxon Mobile .40 General Electric 1.18

General Motors .91 McDonald’s .68

Pfizer .71 Reebok .69

Market Risk Is Measured by Beta

• If you want to know the contribution of an individual security to the risk of a well diversified portfolio, it is no good thinking about how risky that security is if held in isolation—you need to measure its market risk, and that boils down to measuring how sensitive it is to market movements.

• This sensitivity is called beta (B).

Continue

• Stocks with betas greater than 1.0 tend to amplify the overall movements of the market.

• Stocks with betas between 0 and 1.0 tend to move in the same direction as the market, but not as far.

• Of course, the market is the portfolio of all stocks, so the “average” stock has a beta of 1.0.

Dell Computer• Dell Computer had a beta of 2.21. If the future

resembles the past, this means that on average when the market rises an extra 1%, Dell’s stock price will rise by an extra 2.21 percent. – When the market falls an extra 2%, Dell’s stock

prices will fall an extra 2 x 2.21 %= 4.42 %.– Thus a line fitted to a plot of Dell’s returns versus

market returns has a slope of 2.21.

Return on Dell Computer %

Expected

return

Expectedmarketreturn

1.0%+

stock

2.21%

Beta and Unique Risk

Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P Composite, is used to represent the market.

Beta - Sensitivity of a stock’s return to the return on the market portfolio.

Measuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Uniquerisk

Unique Risk

• Of course Dell’s stock returns are not perfectly correlated with market returns.

• The company is also subject to unique risk, so the actual returns will be scattered about the line in Figure.

• Sometimes Dell will head south while the market goes north, and vice versa.

Why Security Betas Determine Portfolio Risk

• Let’s review the two crucial points about security risk and portfolio risk;

– 1) Market risk accounts for most of the risk of a well-diversified portfolio.

– 2) The beta of an individual security measures its sensitivity to market movements.

Explanation 1: Where’s Bedrock?

• Where’s bedrock? It depends on the average beta of the securities selected.– The portfolio beta would be 1.0, and the

correlation with the market would be 1.0. – If the standard deviation of the market were 20 %

(roughly its average for 1926–2000), then the portfolio standard deviation would also be 20%.

• What if the portfolio beta would be 1.5?– What would be the portfolio’s S.D?

Continue

• The general point is this:

– The risk of a well-diversified portfolio is proportional to the portfolio beta, which equals the average beta of the securities included in the portfolio.

• This shows how portfolio risk is driven by security betas.

Beta and Covariances

• A statistician would define the beta of stock i as;

– It turns out that this ratio of covariance to variance measures a stock’s contribution to portfolio risk.

2m

imiB

Beta and Unique Risk

2m

imiB

Covariance with the market

Variance of the market

Coca-Cola & Reebok

• Remember that the risk of this portfolio was the sum of the following cells:

• If we add each row of cells, we can see how much of the portfolio’s risk comes from Coca-Cola and how much comes from Reebok.

2222

22

211221

2112212221

21

)5.58()35(.σx5.585.312.

35.65.σσρxxReebok

5.585.312.

35.65.σσρxx)5.31()65(.σxCola-Coca

ReebokCola-Coca

Continue• Coca-Cola’s contribution to portfolio risk

depends on its relative importance in the portfolio (.65) and its average covariance with the stocks in the portfolio (774.0).

Continue• The proportion of the risk that comes from the

Coca-Cola holding is;

• Similarly, Reebok’s contribution to portfolio risk depends on its relative importance in the portfolio (.35) and its average covariance with the stocks in the portfolio (1,437.3).

• =.35 x 1,437.3 = .35 x 1.43 = .5

• 1,006.1

Understanding• In each case the proportion depends on two

numbers: – The relative size of the holding (.65 or .35) and a

measure of the effect of that holding on portfolio risk (.77 or 1.43).

• The latter values are the betas of Coca-Cola and Reebok relative to that portfolio. – On average, an extra 1% change in the value of

the portfolio would be associated with an extra 0.77% change in the value of Coca-Cola and a 1.43% change in the value of Reebok.

Does Standard Deviation Relevant?

• Example: Lonesome Gulch Mines has a standard deviation of 42% per year and a beta of .10. Amalgamated Copper has a standard deviation of 31% a year and a beta of .66. Explain why Lonesome Gulch is the safer investment for a diversified investor?

Continue• Solution: In the context of a well-diversified

portfolio, the only risk characteristic of a single security that matters is the security’s contribution to the overall portfolio risk. This contribution is measured by beta. Lonesome Gulch is the safer investment for a diversified investor because its beta (+0.10) is lower than the beta of Amalgamated Copper (+0.66). For a diversified investor, the standard deviations are irrelevant.

Summary

• Portfolio Risk• Market Risk Is Measured by Beta• Beta as a Portfolio Risk Measurement