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contributes less to the standard deviation of the portfolio, and the covariance terms contribute more. If
the portfolio contains 50 stocks, it has 50 variance terms, but the variance/covariance matrix is 50 by
50, 2,500 terms in all. Since there are 50 variance terms, there are also 2,450 covariance terms; now
the variance terms contribute very little to the standard deviation of the portfolio, and the covariance
terms dominate. In the limit, the variance of the individual stocks has zero effect on the standard
deviation of the portfolio, as there are 1/N variance terms, and (1 - 1/N) covariance terms, with N
being the number of stocks in the portfolio. This allows us to conclude that in a portfolio of many
different stocks, the variance of the individual stocks does not matter. The covariance between the
stocks in the portfolio is what determines the standard deviation of returns and risk of the portfolio,
and this covariance is based on the correlation between the stocks in the portfolio.
The appropriate measure of risk for any stock in a portfolio is its contribution to the risk of the
overall portfolio, which we have seen is the covariance of returns between the stock and the other
stocks in the portfolio. A stock with high covariance of returns with the other stocks in the portfolio
has returns that are highly correlated with the returns of the other stocks and has a high standard
deviation as well.
In nonstatistical language, this means a stock with a high covariance with the other stocks in the
portfolio adds to the risk of the portfolio, because its returns move in the same direction as the returns
of the other stocks, and these movements are large or volatile. If the portfolio increases in value
because the majority of the component stocks are earning positive returns, then a high-covariance
stock usually earns larger positive returns, further increasing the return of the overall portfolio. If the
portfolio instead decreases in value because the majority of the component stocks are earning negative
returns, then a high-covariance stock usually earns larger negative returns, further decreasing the
return of the overall portfolio. This is how a high-covariance stock adds to the risk of the portfolio. In
up years this type of stock is up more than the other stocks in the portfolio, pushing the portfolio up
even more. In down years this same stock is down more than the other stocks in the portfolio, pushing
the portfolio down even more.
A stock with low covariance of returns with the other stocks in the portfolio tends to move in a
different direction than the other stocks in the portfolio, or these movements are not large or volatile.
This type of stock does not increase in value as much as the other stocks in the portfolio in good years,
and it does not decrease in value as much as the other stocks in the portfolio in poor years. A stock
with a low covariance of returns, therefore, reduces the variability and standard deviation of returns
and the risk of the portfolio.
Capital Asset Pricing Model
The capital asset pricing model (CAPM) developed the concept that the contribution of an individual
stock to the risk of a portfolio is best measured by comparing the returns of the individual stock to the
returns of the overall stock market. We have previously discussed how common factors, such as
interest rates, foreign exchange rates, inflation, and consumer confidence, impact economic conditions
and affect stock prices and returns. These common factors are the cause of market or systematic risk,
which cannot be eliminated by diversification and is therefore relevant and priced.