Use of Beta to Partition Risk

The total risk or variability in earnings can be attributed to two classes of factors:

1. Marketwide factors which create variability in all securities.

2. Company specific factors which cause variability in the returns of that company's shares only.

The investor would like to know the exact proportions of the systematic and unsystematic risks in the total risk. To determine the systematic risk of a share, the beta (or b) of the share must be ascertained.

Systematic Risk of a Security = ^b2 x Variance of the Market Returns

The variance of the market returns is the variance of the returns from the market index.

Unsystematic Risk of a Security = Variance of Returns of Security - Systematic Risk

It must be noted that though the standard deviation is commonly used to denote the risk. However, while partitioning the risk, it is the variance of returns that is partitioned into systematic and unsystematic components.

The coefficient of determination may also be employed to partition the risk.

PERIOD RETURN ON MARKET INDEX (%) (X) RETURN ON SECURITY INDEX (%) (Y)     1                         24                20 4 16 8.00 22.48 6.1504 2.5 2                         10                12 256 144 192.00 11.84 0.0256 148 3                         36                26 100 4 20.00 31.6 31.36 58 4             -16                -8 1764 1024 1344.00 -7.92 0.0064 1019 5                         20                26 36 4 -12.00 19.44 43.0336 21 6                         32                28 36 16 24.00 28.56 0.3136 21 7                         14                  8 144 256 192.00 14.88 47.3344 83 8                         30                36 16 144 48.00 27.04 80.2816 9.20 9                         60                48 1156 576 816.00 49.84 3.3856 670 10                         50                44 576 400 480.00 42.24 3.0976 335 Total 260            240 4088 2584 3112.00 240 214.9888 2362.70 Mean   26                24             Covariance         345.78       S.D 21.3125 16.9443             Variance 454.22 287.11

The coefficient of determination gives us the percentage of the variation of the security's return that is explained by the characteristic line. This time the variance of the security's return gives us the systematic risk.

In our example, let us compute the systematic risk by employing both R² and b

Total return of security = V(Y) = 287.11

R²     =   = 0.914

Partition of risk of security by employing β

Systematic risk=b² x Variance of the market returns

                           =(0.761)² x 454.22 = 263.05

Unsystematic risk    =Variance of returns of security - Systematic risk

                            =287.11 - 263.05 = 24.06

Partition of risk of security by employing R²

Systematic risk       =R² x Variance of the security's return

                            =0.914 x 287.11

                             =262.42 ≈ 263.00

Unsystematic risk    =Variance of returns of security - Systematic risk

                            =287.11 - 263.00 = 24.11