Investments, tenth edition

Asset

Standard 

Deviation (%)

Correlation Matrix

A

B

C

A

20

1.00

0.90

0.90

B

20

0.90

1.00

0.00

C

20

0.90

0.00

1.00

 Suppose that you construct a portfolio with weights  2 1.00; 1.00; 1.00, for assets A; B; C, 

respectively, and calculate the portfolio variance. You will find that the portfolio  variance 

appears to be negative ( 

2 

200). This of course is not possible because portfolio 

variances cannot be negative: We conclude that the inputs in the estimated correlation matrix 

must be mutually inconsistent. Of course,  true  correlation coefficients are always consistent.  

2

   

But we do not know these true correlations and can only estimate them with some impreci-

sion. Unfortunately, it is difficult to determine at a quick glance whether a correlation matrix 

is inconsistent, providing another motivation to seek a model that is easier to implement.

 

 Introducing a model that simplifies the way we describe the sources of security risk 

allows us to use a smaller, consistent set of estimates of risk parameters and risk pre-

miums. The simplification emerges because positive covariances among security returns 

arise from common economic forces that affect the fortunes of most firms. Some examples 

of common economic factors are business cycles, interest rates, and the cost of natural 

resources. The unexpected changes in these variables cause, simultaneously, unexpected 

     8.1 

A Single-Factor Security Market 

  

1

 

We are grateful to Andrew Kaplin and Ravi Jagannathan, Kellogg Graduate School of Management, 

 

Northwestern University, for this example. 

   

2

 The mathematical term for a correlation matrix that cannot generate negative portfolio variance is “positive 

definite.”  

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258 

P A R T   I I

  Portfolio Theory and Practice

changes in the rates of return on the entire stock market. By decomposing uncertainty into 

these  systemwide versus firm-specific sources, we vastly simplify the problem of estimat-

ing covariance and correlation.