Question

Arbor Systems and Gencore stocks both have a volatility of 39 %. Compute the volatility of a portfolio with 50 % invested in each stock if the correlation between the stocks is ​(a​) plus 1.00​, ​(b​) 0.50​, ​(c​) 0.00​, ​(d​) negative 0.50​, and ​(e​) negative 1.00. In which of the cases is the volatility lower than that of the original​ stocks?

Answer 1

Standard deviation is a measure of the volatility of stocks.

Weight of Arbor Systems in the portfolio = w_A = 0.5, Volatility or standard deviation of Arbor systems = σ_A = 39%

Weight of Gencore in the portfolio = w_G = 0.5, Volatility or standard deviation of Gencore = σ_G = 39%

ρ = Correlation between Arbor systems and Gencore

The variance of a portfolio is calculated using the below formula:

σ_P² = w_A²*σ_A² + w_G²*σ_G² + 2*ρ*w_A*w_G*σ_A*σ_G

Part a

Correlation between Arbor systems and Gencore = ρ = +1

w_A = 0.5, w_G = 0.5, σ_A = 0.39, σ_G = 0.39, ρ = 1

σ_P² = (0.5)²*(0.39)² + (0.5)²*(0.39)² + 2*1*0.5*0.5*0.39*0.39 = 0.038025 + 0.038025 + 0.07605 = 0.1521

Standard deviation or Volatility of the portfolio is the square root of the variance of the portfolio

Volatility of the portfolio when correlation is +1 = 0.1521^1/2 = 0.39 = 39%

We see that the volatility of the portfolio is same as the individual stocks

Part b

Correlation between Arbor systems and Gencore = ρ = 0.5

w_A = 0.5, w_G = 0.5, σ_A = 0.39, σ_G = 0.39, ρ = 0.5

σ_P² = (0.5)²*(0.39)² + (0.5)²*(0.39)² + 2*0.5*0.5*0.5*0.39*0.39 = 0.038025 + 0.038025 + 0.038025 = 0.114075

Standard deviation or Volatility of the portfolio is the square root of the variance of the portfolio

Volatility of the portfolio when correlation is 0.5 = 0.114075^1/2 = 0.337749907475931 = 33.77% (Rounded to two decimals)

We see that the volatility of the portfolio (33.77%) is lower than the individual stocks (39%)

Part c

Correlation between Arbor systems and Gencore = ρ = 0

w_A = 0.5, w_G = 0.5, σ_A = 0.39, σ_G = 0.39, ρ = 0

σ_P² = (0.5)²*(0.39)² + (0.5)²*(0.39)² + 2*0*0.5*0.5*0.39*0.39 = 0.038025 + 0.038025 + 0 = 0.07605

Standard deviation or Volatility of the portfolio is the square root of the variance of the portfolio

Volatility of the portfolio when correlation is 0 = 0.07605^1/2 = 0.275771644662754 = 27.58% (Rounded to two decimals)

We see that the volatility of the portfolio (27.58%) is lower than the individual stocks (39%)

Part d

Correlation between Arbor systems and Gencore = ρ = -0.5

w_A = 0.5, w_G = 0.5, σ_A = 0.39, σ_G = 0.39, ρ = -0.5

σ_P² = (0.5)²*(0.39)² + (0.5)²*(0.39)² + 2*(-0.5)*0.5*0.5*0.39*0.39 = 0.038025 + 0.038025 + (-0.038025) = 0.038025

Standard deviation or Volatility of the portfolio is the square root of the variance of the portfolio

Volatility of the portfolio when correlation is -0.5 = 0.038025^1/2 = 0.195 = 19.5%

We see that the volatility of the portfolio (19.5%) is lower than the individual stocks (39%)

Part e

Correlation between Arbor systems and Gencore = ρ = -1

w_A = 0.5, w_G = 0.5, σ_A = 0.39, σ_G = 0.39, ρ = -1

σ_P² = (0.5)²*(0.39)² + (0.5)²*(0.39)² + 2*(-1)*0.5*0.5*0.39*0.39 = 0.038025 + 0.038025 + (-0.07605) = 0

Standard deviation or Volatility of the portfolio is the square root of the variance of the portfolio

Volatility of the portfolio when correlation is -1 = 0^1/2 = 0%

We see that the volatility of the portfolio (0%) is lower than the individual stocks (39%)

We see that in parts b, c, d & e the volatility of the portfolio is lower than the individual stocks