Corporate Finance

Risk and Return 
 67

An Illustration

Exhibit 3.8 presents the average portfolio (weekly) return and standard deviation for various compositions
of the stocks of HLL and Colgate Palmolive. The correlation between the returns from the two stocks is 0.5.
The portfolio variance is the least when 60.40 percent of the money is invested in HLL stock. The portfolio
variance decreases at a faster rate than the portfolio return in line with our expectation.

HLL Colgate

Average weekly return (percent) 0.974 0.073
Standard deviation (percent) 4.012 4.463
Covariance 8.743
Correlation coefficient 0.0493
Three-asset and N asset portfolios

The variance of a three-asset portfolio can be written as:

σp^2 = X 12 × σ 12 + X 22 × σ 22 + X 32 σ 32 + 2X 1 × X 2 × Cov 12 + 2X 2 × X 3 × Cov 23 + 2X 1 × X 3 × Cov 13

where Xi are weights; σi^2 are variances.

Similarly, for an N asset portfolio:

σp^2 = Σ Σ Xi Xj σI σj σij

There are gains from diversification. But how large should the portfolio be? In other words, how many
stocks should an investor hold before marginal costs exceed marginal benefits of diversification? As the
number of assets in the portfolio increases, the variance of the portfolio approaches the covariance or the
average systematic risk of the stocks, i.e., unsystematic risk is diversified away. The marginal benefits from

Exhibit 3.8 Portfolio return and risk

Portfolio return and risk

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

HLL (%)

Standard dev.& avg. return