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Now, let's go to the next step - calculation of the expected return, the variance and the standard deviation of our portfolio, consisting of the two assets: 60% of the asset A and 40% of the asset B. Future returns of the assets A and B are considered as a random variables R _A и R_B.

Expected return of the asset A is 10% and the standard deviation is 8.66%. Expected return of the asset B is 15%, the standard deviation is 12%.

The calculation of the portfolio expected return is a fairly straightforward. But the calculation of the standard deviation and variance of the portfolio is more complicated, because portfolio variability (standard deviation) is not the weighted-average of the variabilities of the individual assets. Diversification reduces the variability of the portfolio, because the prices of different assets vary differently. In many cases, the decrease in price of one asset is compensated by the price growth for another.

Portfolio expected return is the weighted average of the expected returns of the individual assets held in our portfolio:

r = 0,6*r_A + 0,4*r_B
r = 0,6*10% + 0,4*15% = 12%

For calculation of the variance σ² and the standard deviation σ of the portfolio return, we need to know the covariance between assets A and B. Covariance is the measure of how much returns of two assets vary together. This is distinct from variance, which measures how much a single asset varies.

The formula for the covariance is:

σ₁₂ = E(R_A - r_A)(R_B - r_B)

It follows from the formula above, that the covariance of an asset with itself σ₁₁ is its variance σ₁².

The value of the covariance will be given in the exam problems. Or, instead of the covariance, it will be given the value of the coefficient of correlation. The coefficient of correlation is a dimensionless measure and can be expressed as the standardized covariance. The correlation coefficient varies between -1 and +1.

The correlation coefficient formula is:

ρ₁₂ = σ₁₂/σ₁σ₂

Say ρ₁₂ = 0.7.

The correlation coefficient ρ₁₂  and the covariance σ₁₂ are positive, if the returns of assets move in the same direction (in most cases). Correlation and covariance are negative, when returns move in the opposite directions. If the returns are independent then the correlation coefficient is 0.

For calculation of the portfolio variance we need to fill the following matrix. The (i; j) element is the (X_iX_j σ_ij ), where X_i - the weights of assets A and B in the portfolio.

 

This matrix is very like covariance matrix. The (i; j) element of covariance matrix is (σ_ij ).  

 

Next, we simply summarized the elements of out matrix and calculate the portfolio variance:
σ² = X₁² σ₁² + X₂² σ₂² + 2(X₁X₂ ρ₁₂ σ₁σ₂)
or for general case of N assets:
 
Calculation of portfolio variance:
σ² = 0,6² * 8,66² + 0,4² *12² + 2*( 0,6 * 0,4 * 0,7 * 8,66 * 12) = 84,96

Standard deviation is equal the square root of the variance:
σ = √84,96 = 9,22

it is easy to verify that if the correlation coefficient is +1 then portfolio standard deviation is equal the weighted average of the standard deviations of the individual portfolio assets σ = X₁ σ₁ + X₂ σ₂. If the correlation coefficient is -1 then portfolio standard deviation is equal σ = x₁ σ₁ - X₂ σ₂ and it is possible to achieve the zero portfolio standard deviation by varying the proportion of assets weights X₁ and ;X₂ in the portfolio. Unfortunately, it is not possible in practice.

 

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