The Stock Market

Quantifying risk and performance.

Generally speaking, when the market goes up, individual stocks (and mutual funds) go up. (They go down together as well.) Therefore plots that show the price of a stock over time tend to look similar to plots that show general market performance over that same time. While there's no way to measure "general market performance," most followers would agree that market indices serve as effective proxies. In this example I use the Standard & Poor 500 Index (S&P 500) to stand for the market. The index is computed by taking a weighted average of the share prices of 500 of the largest companies in the United States.

I've chosen Procter & Gamble (P&G) for this demonstration. You may view the data on share prices of Procter and Gamble and values of the S&P 500 index. The data is drawn from the one year period ending April 19, 1997.

The relationship between the price of a share of P&G and the value of the S&P 500 are not compared directly. Instead, we compare the percent gain/loss in P&G (over a given period of time) to the percent gain/loss in the S&P 500 (over the same period of time). Taking a look at a little bit of the data. . .



S&P 500





















For the week of April 23 P&G closed at $84-7/8 (Wall Street uses divisions of 1/2, 1/4, 1/8, 1/16, etc, in pricing stocks. . .the figures in my table are, to the nearest 0.01, decimal equivalents). The following week (April 30) saw a price drop; P&G closed at 83, for a loss of $1.88 per share. This is a 2.21% loss.

For the week of April 23 the S&P 500 Index closed at 766.34. The following week saw a it drop to 737.65--a 3.74% loss.

For the week of May 6, P&G saw a gain of 3.92% while the S&P 500 gained 2.75%.

And so on. . .

People who study the financial markets are interested in questions like "How closely does P&G follow the market?" That is, given the market's performance, what happens to P&G? Since we're using the S&P 500 to measure "the market," we'll take % change in S&P 500 to be the explanatory variable, with % change in P&G the response variable. Both are quantitative variables. For the year's worth of data (approximately 50 cases then), we have the scatterplot shown below. (The pink points are taken from the table above.)

I would characterize this as a weak, positive, linear association.

I've fit a least squares regression line through the data--basically a "best fit line." The equation of this line is: Y = 0.93% + 0.63 X. You can see that there is considerable variability about this line.

People who analyze financial information use the slope of such a line as a measure of risk. In this case slope is interpreted as the expected change in P&G's % gain when the S&P 500 % gain increases by 1%. For Procter & Gamble the slope is 0.63. You may think this is bad--that P&G doesn't go up as much as the market does. In fact, every additional 1% gain in the market is, on average, accompanied by only an additional 0.63% gain in P&G. You might then ask: Why invest in Procter and Gamble? Well, think about it. . .an additional 1% decline in the market is, on average, accompanied by only an additional 0.63% loss in P&G. Basically then, the slope does not measure performance, it measures risk. Procter and Gamble is less risky than is the market!

The slope of this line measures risk. A slope below 1 indicates below average risk; a slope above 1 indicates above average risk. In the financial world this slope is called a stock's BETA. BETA is measured relative to some index of "overall market price." For U.S. markets, the S&P 500 is often used. Other indices are used to price foreign markets. (Other indices, such as the Russell 2000, are used to measure the overall performance of smaller U.S. companies. The index that is used depends on what market is being tracked.)

Note that a horizontal line has 0 slope (0 BETA). If a companyhas 0 BETA we conclude that there's no risk involved relative to the market as a whole. Many foreign stocks have BETA = 0 (relative to the U.S. markets). A negative slope (BETA < 0) would accompany a stock that systematically improves when the market worsens. Unfortunately, such a stock would decline when the market improves. Because the market improves more often than not, such a stock would, in the long run, lose money!

The y-intercept of this line is a measure of performance. (The intercept is where the line crosses the y-axis--the value of y when x = 0.) The intercept measures the percent change in the stock when the market change is 0. For Procter & Gamble the intercept is 0.93%. This may seem like a very small number; in fact it is huge. That's because we're dealing with weekly changes. This number says In a week when the market held steady, on average Procter & Gamble gained 0.93%. Think about that--that's almost 1% per week! This number is always published in an annualized form. It turns out that 0.93% per week is equivalent to about 62% annually.

The (annualized) intercept measures performance relative to the market. A value above 0 indicates that on average the stock outperforms a stand-still market. A value below 0 indicates that on average the stock underperforms a stand-still market. In the financial world this intercept is called a stock's ALPHA.

In Procter & Gamble we see a stock that outperforms a stagnant market, and does so while being well below average in the risk department.

Remember--the information provided on this page relects only one year's past performance. Procter & Gamble did very very well last year. Historically, P&G's ALPHA is not nearly so high. Its BETA is greater than 0.63 (it's not generally as safe as it was for the year studied here) but still significantly less than 1. Due to the nature of its products, the company tends to do relatively well in even the worst of markets. Procter & Gamble has traditionally been one of the least risky investments on the market. Many financial counselors recommend Procter & Gamble as a good low-risk investment; one that should outperform a bear market.


Forecasting is folly--in no way should this information be considered a guarantee of similar future performance. Equities markets are inherently risky. Consult a prospectus before investing.

For Procter & Gamble. . .Are weekly volume and closing share price associated? How about weekly voume and spread (the difference between the week's high and low)? You'll need price and volume information to get started.