Category “Prime Brokerage Basics”

Cash Dividends

October 12th, 2010

Regular cash dividends are those paid out of a company’s profits to the owners of the business (i.e., the shareholders). A company that has issued preferred stock must pay dividends on those shares before paying dividends on the common stock. The preferred stock dividend is usually set whereas the common stock dividend is determined at the discretion of the board of directors.  Cash dividends are normally paid quarterly.

There are three important dates associated with dividends:

  • Declaration date: The declaration date is the day the board of directors announces their intention to pay a dividend. On this day, a company creates a liability on its books; it now owes the money to the stockholders. On the declaration date, the board will also announce a date of record and a payment date.
  • Record date: This date is also known as the “ex-dividend” date. It is the day on which the stockholders of record are entitled to the upcoming dividend payment. A stock will usually begin trading ex-dividend or ex-rights the fourth business day before the payment date. In other words, only the owners of the shares on or before that date will receive the dividend. If you purchased shares of Coca-Cola after the ex-dividend date, you would not receive its upcoming dividend payment; the investor from whom you purchased your shares would receive it.
  • Payment date: This is the date the dividend will actually be given to the shareholders of company

The dividend yield is calculated by dividing the actual or indicated annual dividend by the current price per share.

A dividend accrual recognizes dividend income or expense that has become payable or receivable during a reporting period, but for which cash has not been paid or received in the period. Companies record a dividend accrual at the end of the month in which the ex-date occurs for those dividend payments that will be made in a subsequent month. During the dividend accrual period, a foreign currency cash dividend is subject to translational P&L.

Dividends from preferred shares take precedence over those from common shares. That is, if a preferred dividend is missed, a common dividend cannot be paid.  If the preferred dividends are cumulative, then missed dividends remain in arrears until paid. Most preferred stock pays a fixed dividend, stated in a dollar amount or as a percentage of par. Outside of the U.S., payments made to owners of preferred stock are considered interest, though all other characteristics of this interest/dividend paying class of capital stock are the same. Payments made at a specified rate have preference over common stock dividend payments.

Convertible preferred is preferred stock that may be converted into a fixed number of shares of common stock, paying dividends at a specified rate and period.

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Financial Statistics (8) – Prediction Intervals

October 7th, 2010

- Eric Bank

Prediction interval

We continue our review of elementary statistical concepts that are commonly used in the financial industry (i.e. by prime brokerages, hedge funds, financial analysts, etc.). Recall from a recent article that the formula for a linear regression is:

Yi = b0 + b1Xi + εi for i = 1, …, n

where:

Yi is the ith value of the dependent variable

b0 is the y-intercept

b1 is the slope coefficient

Xi is the ith value of the independent variable

εi is the ith value of an error term

i is the index of a particular variable

n is the maximum value of i

Unfortunately, we do not have access to the population values of b0 and b1, so we are forced to estimate these values with b0estimated and b1estimated.  This is one cause of uncertainty in the predicted value of Yi. The second cause of uncertainty is the error term εi , which is the difference between the estimated and true value of Yi.  These two uncertainties beg the question: How confident are we about the forecast results? To answer this question, we calculate a prediction interval which is an estimated interval into which future observations will fall, with a given probability, in light of past observations.

For example, if we forecasted that sales for ABC Corporation would grow by 8 percent this year, our prediction would be more meaningful if we were 95 percent confident that sales growth would fall in the interval from 7 percent to 9 percent.  A value outside the 7% to 9% range would not instill confidence in the value.

We can compute confidence intervals using our old friend, the standard error of the estimate s. The variance of the prediction error is equal to the square of the standard error of the estimate, namely sf2.   This estimated variance can be calculated using this formula:

Note that sx2 is the variance of the independent variable X.

After you calculate the variance of the prediction error, you choose a significance level α, say 0.05.  We apply another old friend, the t-statistic, which is the critical value for the forecast interval and can be looked up in the back of any statistics textbook..  By using (1 – α) = 0.95, we can compute the percent prediction interval Y as

Y = ± tc sf

Let’s take a numerical example[i] as follows:

1)     Assume a linear regression equation Y = 1.3478 + 30.0169(0.10) = 4.3495; the standard error of the estimate s = 0.7422; the mean value of X = 0.0647; the variance of the mean sx2 = 0.004641; the number of observations n = 9, the number of coefficients (the y-intercept and the slope) = 2.

2)     Assume we are interested in the 95% confidence interval.

3)     Compute the variance of the prediction error:

sf2 = 0.74222 [1 + 1/9 +  (0.10 – 0.0647)2 /  (9 – 1)0.004641] = 0.630556

4)     Take the square root of the variance of the prediction error sf2, giving the standard deviation of the forecast error sf = (0.630556)1/2 =  0.7941.

5)     The degrees of freedom = (observations n – number of coefficients) = (9 – 2) = 7. From the back of a statistics book, the critical t-statistic for 7 degrees of freedom at the 95% confidence interval, tc = 2.365.

6)     We compute the prediction interval for the 95% level of confidence. It is equal to the following:

Y = ± tc sf = 4.3495 – 2.365(0.7941) to 4.3495 + 2.365(0.7941) = 2.4715 to 6.2275.

From this example, we are 95% confident that a value of the dependent variable will have a value between 2.4715 and 6.2275, the prediction interval.

Well, we have now reviewed the basic concepts pertaining to single-variable linear regressions. We’ll pick up our voyage through financial statistics next time by examining multiple regressions.

[i] DeFusco, McLeavey, Pinto and Runkle, “Quantitative Methods for Investment Analysis, Second Edition”, pages 323-324.

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Financial Statistics (7) – Analysis of Variance and the F-Test

October 4th, 2010

-Eric Bank

Analysis of variance (ANOVA) is used to determine how useful an independent variable X is at explaining variation of the dependent variable Y. For this article, we’ll confine our discussion to linear regressions with a single independent variable, although ANOVA is also appropriate for multi-variable regressions. Recall that we have examined linear regressions and the meaning of the slope coefficient. The F-test is used within ANOVA to see whether the slope coefficient (b1) is equal to zero. That is, we test the null hypothesis H0 of b1 = 0 against the alternative hypothesis H1 that b1 ≠ 0.  If H0 turns out to be true, then X is not a good predictor of Y.

To calculate the F-statistic, we need the following items of data:

  • the number of observations n
  • the number of parameters (intercept and slope coefficient) = 2
  • the sum of squared errors SSE =

  • the total variation in Y that is explained by the regression, known as the regression sum of squares RSS =

  • total variation  TSS = SSE + RSS

The F-statistic is the ratio of the RSS to the average SSE. This average is calculated by dividing the non-averaged SSE by n-2, the degrees of freedom (observations less parameters). Thus, the F-statistic equals RSS / (SSE / (n -2)).

To clarify, suppose H0 was true, and that X did nothing to predict the value of Y. In that case, the predicted value of Y for Xi is equal to the mean value of the dependent variable.  But if this were true, then RSS = zero, and the F-statistic = zero.  The higher the value of F, the more predictive X is.

In a previous blog, we examined the t-statistic. Note that F is equal to the square of t for a single-variable linear regression. In the next blog, we will look at prediction intervals.

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Financial Statistics (6) – The Coefficient of Determination

September 27th, 2010

- Eric Bank

Determination

As we pointed out in our discussion of the standard error of estimate, it would be nice to know how well the independent variable X explains variation in the dependent variable Y. To calculate the fraction of the total variation in the dependent variable that is explained by the independent variable, one uses the coefficient of determination (R2).

There are two ways to calculate R2. The easier method involves squaring the correlation coefficient for a linear regression with a single independent variable. Recall from a previous blog that the correlation coefficient, r, is equal to the covariance of the two variables divided by the product of their standard deviations (sxsy).  (We pointed out that covariance measures the extent to which two variables (X, Y) change together).   The formula for the correlation coefficient is:

r  = Cov(X, Y) / sxsy.

We square it, giving us R2 as the coefficient of determination. However, this doesn’t work when we are dealing with more than one independent variable (X).

The alternate calculation of R2 for multiple independent variables is to use the following definition:

Total variation = Unexplained variation + Explained variation

Since R2 stands for the fraction of the total variation that is explained by a linear regression, we get this solution:

R2 = Explained Variation/Total Variation = 1 – (Unexplained Variation / Total Variation)

There is one more alternative for calculating R2 . Linear regression packages typically report a statistic called multiple R, which is the correlation between actual Y values and predicted Y values.  R2 is the square of multiple R.

As an example, let’s take the results from a hypothetical multiple regression which regresses inflation rate on money supply growth rate for several different countries over a particular period of time. We calculate the following results:

Given that

  • total variation is the sum of the squared deviations (Yi – Yavg)2 = 0.001598
  • the unexplained variation is 0.000386

the value for R2 is (0.001598 – 0.000386) / 0.001598 = 0.7586.

Now when you inspect the generated results from a linear regression, you’ll have an understanding of the reported R2 statistic, and can judge the meaningfulness of the predicted Y values.

We are making great progress with our review of elementary financial statistics. Next time, we’ll look at analysis of variance (ANOVA) and the F-test.

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Mortgage-Backed Securities Calculations

September 24th, 2010

There are several calculations specific to amortizing fixed-income securities such as MBS. We assume a fixed mortgage rate in the following examples.

Mortgage Interest Income Component

The interest component of a mortgage payment is P&L income. A monthly mortgage payment’s interest component can be calculated as follows:

Calculation:

For most mortgage-related securities, interest accrues according to the following standard calculation:

Accrued Interest = Original Face * Accrued Interest Factor * (Coupon/100) * (N/360)

Where:

Coupon = Annual coupon rate of the security, in percent

Accrued Interest Factor = Days in accrual cycle / Total days in coupon interval

N = number of days from the first day of the accrual period (the “as-of” date for accrued interest factor) to the settlement date itself.  The day count is computed according to the 30/360 calendar.

Example:

Assume a 6% $200,000 mortgage with monthly payments of $1,198, with an accrued interest factor of .98, and N = 30.

Accrued Interest = 200,000 * .98 * (6/100) * (30/360) = $980

Mortgage Principal Component

A monthly mortgage payment’s principal component is the remainder of the payment after subtracting the interest component.

Calculation:

Principal = X – B(n) * y/12 where X is the monthly mortgage payment and the second term is the interest component.

Example:

From the previous example, principal = $1,198 – $980 = $218

This payment reduces the mortgage principal to $200,000 – $218 = $199,782.

Unamortized Principal

The outstanding (unamortized) principal can be computed with the following formula.

Calculation:

X ∑ 1/(1 + y/12)n where X is the monthly mortgage payment, y is the annual mortgage rate, and n is the remaining number of payments.

The limits of the summation are from 1 to n, yielding a working calculation of X * (1 – 1/(1 + y/12)n ) / y/12.

Example:

With a monthly payment of $1198, and interest rate of 6%, and 300 remaining payments,

Outstanding principal = $1198 * (1 – 1/1+ .06/12)300) / .06/12 = $185,937.80

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Financial Statistics (5) – Standard Error of Estimate

September 22nd, 2010

Plot A has a smaller standard error of estimate than does Plot B

- Eric Bank

We left off last time having concluded a discussion of the t-test for evaluating correlations. Next, using the standard error of estimate, we’ll examine how to assess the strength of a relationship between an independent and a dependent variable as determined by a linear regression.

Recall that the equation for a linear regression is:

Yi = b0 + b1Xi + εi for i = 1, …, n

where the residual error term, ε, gives an indication of how certain we are about a particular predicted Y value via a linear regression.   The standard error of estimate tells us how spread out actual values of Y are with respect to their predicted values. The bigger the standard deviation of the error term, the less precise is the relationship between the two variables.

The standard error of estimate (SEE) measures the variability of the error term:

Don’t panic: the equation just adds up the squares of the error terms, divides the sum by number of degrees of freedom, and takes the square root of the whole thing. Another way of saying this is that the SEE is the difference between the dependent variable’s actual value for each observation and its predicted value for each observation.

SEE and standard deviation are almost identical, except that SEE has n-2 degrees of freedom (to account for the two parameters and and standard deviation has n-1 degrees of freedom.  This little difference in the denominator ensures that SEE is unbiased. Whereas the standard deviation is the square root of the average squared deviation from the mean, the standard error of the estimate is the square root of the average squared deviation from the regression line.

To get a general feel of the meaning of a particular SEE value, know that if the error residuals (ε = Yactual – Ypredicted) are normally distributed around the prediction line, about 68% of actual scores will fall between ±1 SEE of their predicted values.

While we can say that smaller SEE values result in better predictions, it would be nice to know how well the independent variable X explains variation in the dependent variable Y. To calculate the fraction of the total variation in the dependent variable that is explained by the independent variable, one uses the coefficient of determination, which will be our next topic.

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Financial Statistics (4) – Testing Correlations for Significance: the t-Test

September 20th, 2010

Tea test

- Eric Bank

Now that we have examined correlation and linear regression, we now need to understand whether a correlation describes a real relationship or is just the result of chance.  Only real relationships are predictive.  Another way of saying this is that we want to test the null hypothesis (H0) that a correlation coefficient ϱ in the population is equal to zero (ϱ = 0), versus the alternative hypothesis (H1) that it is significantly different from zero (ϱ ≠ 0).

Since we are testing whether the correlation is not zero (i.e. significantly bigger or smaller than zero), we need to perform a two-tailed test. We assume that the variables (X and Y) are normally distributed – this permits us to perform a t-test:

where the sample correlation r is an estimate of the population correlation ϱ, and n is the sample size. We use (n -2) degrees of freedom to see if the test statistic has a t-distribution; if it does, then H0 is true. By using n – 2 instead of n for the degrees of freedom, we avoid introducing a bias into the calculation.  If the calculated t-value exceeds the critical t-value for the degrees of freedom, then H0 can be rejected. By the way, you can look up the critical t-value in a table at the back of any statistics book. Note that as n increases, the absolute value of the critical t-value decreases: it’s easier to reject the null hypothesis with a larger sample size. Also note that the numerator of the t-test increases with increasing n, meaning you get larger values of t for larger samples. The bottom line is that the likelihood of failing to reject a false H0 decreases with sample size.

When we perform a t-test, we need to specify a level of statistical significance.  For example, if we choose the 0.05 level of significance, we are confident in the results of test 95 times out of 100. All things being equal, a lower level of significance produces a higher critical t-value: it becomes harder to reject H0, but you have more confidence in the predictive value of the correlation.

Let’s work a numerical example[1].  We determine that the sample correlation r between monthly returns on long-term U. S. government bonds and 30-day T-bills was 0.1119 over 924 months of observations. Is this value of r high enough to reject the hypothesis that returns on the bonds were uncorrelated to returns on the T-bills?  For the 0.05 level of significance, the critical t-value is 1.96, and we can plug in the values into the t-test:

tactual > tcritical =  0.1119 (924 – 2).5 / (1 – 0.11192).5 = 3.4193 > 1.96

Thus, in this example we are able to reject the null hypothesis, and say that there is correlation between government bonds and T-bills.

We want next to assess the strength of a relationship between an independent and a dependent variable as determined by a linear regression. We will examine this test in our next blog using a statistic called the standard error of estimate.

[1] Quantitative Methods for Investment Analysis, Second Edition, by Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, 294-295.

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Financial Statistics (3) – Linear Regression: Assumptions, Limitations, and Uses

September 13th, 2010

Lineal regression

Last time, we defined linear regression and explained the relevant equations.  We’ll continue today with a look at the assumptions underlying the proper use of linear regression, limitations on the interpretation of linear regression results, and uses of correlation analysis for financial and economic forecasting.

Assumptions

1)     There must be more data points than there are variables.  For the two-variable examples we have been discussing, this just trivially requires at least 3 data points.  For multi-variable regressions, the number of data points must always exceed the number of variables; otherwise you encounter the dreaded multi-collinearity, which results in coefficient estimates that may change haphazardly in response to small changes in the model or the data.

2)     The regressors (the independent variables on the X-axis that predict the value of the dependent variable on the Y-axis) must be free from measurement error.

3)     Some estimation methods prefer that observations not be strongly correlated to each other, although there are techniques to handle this occurrence.

4)     It is preferred that the error terms (ε) all have the same mean and standard deviation. This leads to the situation where each probability distribution for different Y–values all have the same standard deviation, independent of associated X-values – a condition called homoscedasticity. Unequal standard deviations in the error terms, heteroscedasticity, are allowed but decrease the accuracy of certain parameter-estimation methods.

Limitations

1)     There may be a strong nonlinear relation among the variables that is not detected by a linear regression. An example would be a quadratic relationship.

2)     Outliers (a few observations with values far away from all others) can compromise the accuracy of a linear regression. Judgment is required to know whether to include or exclude outliers.

3)     Correlation does not imply causality.  Furthermore, spurious correlations can imply a relationship between variables when in fact none exists. There are three causes of spurious correlations:

  • correlation between two variables that exhibits chance relationships in a particular set of observations
  • correlation created by a calculation that mingles each of two variables with a third
  • correlation between two variables created not from a direct relation between them but from their relation to a third variable

Examples of Uses for Correlation Analysis

1)     Evaluating the accuracy of economic forecasts that are based on linear regression of forecast and actual economic results.  For example, the outlook for inflation may be forecast by changes in the consumer price index – how accurate would such a forecast be?  One could do a linear regression between forecast and actual inflation rates.  The higher the correlation, the more useful the forecast.

2)     It is important to measure portfolio manager performance as compared to a specific benchmark, such as the S&P 500. Style analysis is used to choose a benchmark appropriate to the portfolio choices of a specific portfolio manager. If two styles show a very high correlation to each other, there may be no justification for differentiating the two styles. For example, if small-cap growth and small-cap value had a correlation near 1, then it would be just as relevant to use just small-cap as the relevant benchmark.

3)     Currency traders attempt to optimize the amounts allocated to each currency. By using a multiple regression matrix, one can see the cross-correlations between any pair of currencies.  This information can help a currency trader decide how to hedge currency risks by selecting currencies with low correlation coefficients relative to currencies that dominate a portfolio.

4)     Portfolio managers who seek to diversify risks across different asset classes need to know how the returns of each asset class correlate to the returns of other classes.  In this way, the manager can determine if an investment in a particular asset class actually provides a sufficient increment of diversification.

It is important to know whether apparent relationships among variables are caused by chance or are a reflection of the real world.  Therefore, it is important to know how to test the significance of a correlation coefficient.  We’ll tackle this subject next time.

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Financial Statistics (2) – Linear Regression: Definition

September 9th, 2010

A linear regression is a statistical method that helps one understand the relationship between two (or more) variables.  It does this in three ways:

  1. It uses one variable to predict the value of another variable
  2. It tests hypotheses concerning the relationship between two variables
  3. It quantifies the strength of the relationship between two variables

As we did in our discussion of linear correlation, we will denote two variables as X and Y; X is the independent variable, Y the dependent one.  A linear regression assumes that there is a linear relationship between X and Y, and is given by the following formula:

Yi = b0 + b1Xi + εi for i = 1, …, n

where:

Yi is the ith value of the dependent variable

b0 is the y-intercept

b1 is the slope coefficient

Xi is the ith value of the independent variable

εi is the ith value of an error term

i is the index of a particular variable

n is the maximum value of i

In English, the value of the dependent variable Yi is equal to {the value of dependent variable when the independent variable’s value is zero (b0)} plus {the product of the slope b1 and the independent variable b1} plus {some error term εi}. The error term is that part of Yi that is not explained by Xi . We call b0 and b1 the regression coefficients.

When we speak about the relationship between two variables, we think in terms of many contemporaneous observations (a cross-sectional series) or observations over a period of time (a time-series). Observations are indexed by values 1 to n.  For example, you may be interested in the effect in various countries of money supply (Xi where i refers to a particular country) on the country’s inflation rate (Yi) – that would be a cross-sectional analysis.  Conversely, you would use a time-series analysis to test the money supply/inflation rate relationship in one country over a period of time.

A perfect linear regression would be one where all of the error terms equaled zero.  This would indicate that all changes to Y were accounted for by changes to X.  For instance, if I eat every cookie handed to me, then there would be no error values when I plot cookies offered versus cookie consumed. In this case, the regression line’s y-intercept would be zero and the slope would equal 1; all actual data values would be points directly on the regression line. Thus, if you offered me 3 cookies, I’d eat 3 cookies. Obviously this example is unrealistic when the number of cookies offered rises above some critical value, say 3-dozen in my case.

A more realistic case is one that plots a straight regression line through the data in which the errors are minimized – the best fit.  In real life, we are interested in imperfect correlations, so we need a method to achieve the best fit, which we define as the regression line that minimizes the sum of the squared vertical distances (deviations) between observations and the regression line.  This method is called the linear least-square method.  Nifty, but how do we calculate the best fit?

To achieve the best fitting regression line, we need to find the slope b1 and y-intercept b0 that produces the minimum sum of the squared errors. (We square the errors, which are simply the vertical deviations from the regression line, because we don’t want positive and negative values to cancel each other out).  How do we find these magic regression coefficients? We need to make estimates, which we call the fitted parameters, according the following formula:

The funny little hat (^) above b0 and b1 designates that the regression coefficients are estimated. We are summing, for all index values of i, the squares of the following difference: the actual value of the dependent variable minus the predicted value of the dependent variable. When this sum (the sum of the squared error terms) is minimized, we have a best-fit regression line. The actual method of calculating this minimum is complicated, and we leave it to a computer spreadsheet or math package to do the nitty-gritty work.

A note about the slope coefficient b1: when a linear regression contains a single independent variable, the slope coefficient is equal to the following:

b1 = Cov(Y, X) / Var(X) = Cov(Y, X) / sxsx where s = standard deviation

which is the covariance of Y and X divided by the variance of X.  Alert readers will recall from the previous blog that this formula is very similar to that for the correlation coefficient (r). The difference here is that the denominator, the variance of X, is the equivalent to the square of the standard deviation of X (sx). For the correlation coefficient, the denominator is the product of the standard deviations for X and Y:

r = Cov(Y, X) / sxsy

Conceptually, one can see that the coefficients are very similar – they both give a scale to the covariance of the two variables.

Next time, we will address the assumptions one makes in order to calculate a proper linear regression.

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Financial Statistics (1) – Correlation

September 7th, 2010

Scatter plot

Many people who work at financial institutions, such as prime brokerages and hedge funds, have had formal financial training, including the use of statistics and other quantitative methods.  Today we are launching a series of blogs that cover these important topics at a straightforward, accessible level. We’ll assume you have had some exposure to the subject matter (for instance, you are familiar with terms like population and sample) and that you can handle simple algebra.

Statistics play a key role in financial modeling, so we’ll begin by looking at linear correlations and linear regressions.

Data analysis and prediction are the reasons for employing statistical method.  Data can be organized and presented in many ways.  One of the most popular presentations is a scatter plot, in which two series of observations are plotted on an x-y coordinate graph.  For each data pair (that is, two simultaneous observations), the appropriate point is shown on the graph as the intersection of the x and y values.  For instance, if we place money-supply growth on the x-axis and inflation rate on the y-axis, we can plot a series of unconnected points that indicate some kind of relationship between the two data series.

To indicate how closely two data series are related, we use a measure of their linear association, the correlation coefficient (r). The values that r can have range from -1 (perfect negative correlation) through zero (no linear correlation) to +1 (perfect positive correlation).  To calculate the r of a data sample, we must first understand another statistic: sample covariance.

Covariance measures the extent to which two variables (X, Y) change together. It is given by the following equation:

where

n is the number of data pairs

i is a particular value from 1 to n

is the ith X variable,  is the ith Y variable

and are the mean X and Y values, respectively

In English, this states that the sample covariance is the average value of the product of the deviations of observations on two random variables from their sample means. The use of (n – 1) instead of n to calculate the mean is used to ensure that sample covariance is an unbiased estimate of population variance.

To show the relationship between covariance and r, we note that if we take the covariance of X with itself, we have calculated the variance of X. Variance (denoted by the symbol s2) is a measure of how far values deviate from their mean, and is given by the following equation:

This is the variance of X, a measure of X’s dispersion around its mean   Standard deviation (sx) is the positive square root of variance:

Now we have all of the elements in place to calculate the sample correlation coefficient:

Thus, the correlation coefficient, r, is equal to the covariance of the two variables divided by the product of their standard deviations.  Think of it as the covariance normalized for the dispersion of each variable.

It is assumed that for the correlation coefficient the means and covariances of X, Y, and Cov(X,Y) are finite and constant. Note that r refers solely to linear associations between X and Y, that is, no exponents greater than 1.

A value of r equal to, say, 0.9, would indicate a strong linear relationship between X and Y, but not necessarily any causal relationships between the two variables.  A classic example of spurious correlation is one between vocabulary and height.  One may infer that the real relationship has something to do with age.

Forecasters use correlations to analyze trends and changes in trends. For instance, a change in the consumer price index (CPI) is correlated with a change to the inflation rate. So whenever a new CPI figure is released, economists revise their forecasts for inflation, which in turn affect interest rates and bond prices. When dealing with more than two variables, a correlation matrix is used to sort out the various linear relationships among the variables.

Next time out, we’ll tackle linear regressions.

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