University of Illinois at Chicago
College of Business Administration
Department of Information & Decision Sciences
IDS 371:     Business Statistics II
Instructor:   Prof. Stanley L. Sclove
Textbook:   McClave, Benson & Sincich, 7th ed. (MBS)

Introductory Statistical Aspects of Portfolio Analysis
Copyright © 1999 Stanley Louis Sclove

MBS Statistics in Action 4.1 / Portfolio Selection, pp. 174-5

We'll begin by watching Against All Odds: Inside Statistics Program #6: "Time Series Analysis" and Program #17, "Binomial Distribution" (6:38-16:22): T-bills and Stocks; A Finance Experiment (Diversification) The distributions of returns on T-bills (mean 6%, standard deviation 2%) and Stocks (mean 14%, standard deviation 8%) are compared and contrasted. A portfolio of half of each is analyzed. An econ class at Wheaton "invests"; some students construct diversified portfolios, others don't.

Concepts and Definitions

Rate of Return

Portfolio

Against All Odds: Inside Statistics -- Program #6, "Time Series Analysis" (second half)

As emphasized by Yale economics professor Burton Malkiel in Program 6 ("Time Series") of "Against All Odds: Inside Statistics", portfolio selection involves choosing a diversity of stocks, some of which are negatively correlated. Here we study the situation with a portfolio of just two stocks, the returns of which are negatively correlated.

Two criteria to be simultaneously optimized, to the extent possible, are the mean return, to be maximized, and the variance of the return, to be minimized. Let the variable X represent the percentage return on Stock A and Y the percentage return on Stock B. Suppose that a proportion "a" is invested in A and "b" (=1-a) in B. Then the return   P   on one dollar of the portfolio is aX+bY. the expected return is

E(P) = E(aX+bY) = aE(X) + bE(Y) .

The standard deviation, SD(P), that is, SD(aX+bY), is the square root of the variance Var(aX+bY) .   The variance is given by the formula below in the next section. It involves the covariance, Cov(X,Y).
Combination of the Two Criteria

The two criteria, mean and standard deviation, can be meaningfully combined into a single criterion to be maximized, such as the probability that the return exceeds some nominal value, say, 5% available from CDs. When the portfolio P has a normal distribution, probabilities can be computed from the "z-score," z = (P - E[P])/SD[P].

Covariance and Correlation

Covariance

For two random variables X and Y , denote the values in a finite population of N by (tj, uj), j =1, 2, ..., N. In the finite population model the covariance of X and Y is the ordinary arithmetic average of the products (tj - µx)(uj - µy).   That is, the covariance is

(t1 - µx)(u1 - µy) + (t2 - µx)(u2 - µy) + ... + (tN - mx)(uN - µy)

When the variance of a sum or difference of two variables is computed, the covariance is what emerges from the cross-product.

Var(X +/- Y) = Var(X) + Var(Y) +/- 2Cov(X,Y).

For a linear combination we have

Var(aX+bY) = a2Var(X) + b2Var(Y) + 2abCov(X,Y) .

Exercise. For the data of Problem 9.28, n = 6 (x,y)-pairs, verify that the variance of the difference is the variance of x plus the variance of y minus twice the covariance.

Correlation

The correlation coefficient or, more simply, the correlation of X and Y is their covariance, divided by the product of their standard deviations.

Corr(X,Y) = Cov(X,Y) /[SD(X)SD(Y)] .

The correlation is a dimensionless (unitless) quantity, ranging between -1 and +1.   The covariance can be computed as the product of the correlation and the standard deviations by reversing the above formula:
Cov(X,Y) = Corr(X,Y)SD(X)SD(Y).
The sample quantities SSxx, SSyy and SSxy are defined in MB Chapter 10 (Simple Linear Regression).   The sample variance of X is sx2 = SSxx/(n-1).   The sample variance of Y is sy2 = SSyy/(n-1).   The sample covariance is sxy = SSxy/(n-1).   The covariance is related to the regression coefficient and to the correlation coefficient. The coefficient of X in the regression of Y on X is SSxy/SSxx = sxy/sx2 .   The correlation coefficient is rxy = SSxy/[SSxxSSyy]1/2, or rxy = sxy/(sxsy) .

Spreadsheets

In the accompanying spreadsheet the probability Prob(P > c) is computed, assuming the returns have normal distributions.   (Note that the distribution of aX+bY may be well approximated by a normal distribution, even when the distributions of X and Y separately are not so well approximated by normal distributions.)   For any given value of   c, the optimal portfolio can be approximated.   That is, the spreadsheet allows for the user to specify different criteria   c, corresponding to different goals of investment clients, some of whom are conservative, others of whom are less averse to risk.
Created: 12 June 1998       Updated: 19 June 1999