###### Dependent and Independent Events

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A portfolio is a collection of investments a company, mutual fund, or individual investor holds. A portfolio consists of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.

To calculate the portfolio’s expected return, you take the expected returns of each security in the portfolio. Then, you multiply each security’s expected return by its proportion in the portfolio and add them up. The formula below helps you find the portfolio’s expected return:

$$ E(R_p)= w_1 E(R_1)+w_2 E(R_2)+\cdots w_n E(R_n) $$

Where:

\(w_1,w_2,\dots,w_n\) = Weights (market value of asset/market value of the portfolio) attached to assets \(1,2,\dots,n\).

\(R_1,R_2,\dots,R_n\) = Expected returns for assets \(1,2,\dots,n\).

**Example: Portfolio Expected Return**

Assume we have a simple portfolio of two mutual funds, one invested in bonds and the other invested in stocks. Let us further assume that we expect a stock return of 8% and a bond return of 6%, and our allocation is equal in both funds. The expected return would be calculated as follows:

$$ E(Rp)=(0.5 \times 0.08)+(0.5 \times 0.06)=0.07 \text{ or } 7\% $$

The variance of a portfolio’s return is a function of the individual asset covariances as well as the covariance between each of them.

Consider a portfolio with three assets: A, B, and C. The portfolio variance is given by:

$$ \begin{align*}

\text{Portfolio} & \text{ Variance} \\ & = W_A^2 \sigma^2 (R_A)+W_B^2 \sigma^2 (R_B)+W_C^2 \sigma^2 (R_C)+2(W_A) (W_B)Cov(R_A,R_B ) \\ & +2 (W_A)(W_C)Cov(R_A,R_C )+2 (W_B)(W_C)Cov(R_B,R_C) \end{align*} $$

If we have two assets, \(A\) and \(B\), then:

$$ \text{Portfolio Variance}= W_A^2 \sigma^2 (R_A)+W_B^2 \sigma^2 (R_B)+2 (W_A)(W_B)Cov(R_A,R_B) $$

Where:

\(W_A\) = Weight of assets \(A\) in the portfolio.

\(W_B\) = Weight of assets \(B\) in the portfolio.

\(\sigma^2 (R_A)\) = Variance of the returns on assets \(A\).

\(\sigma^2 (R_B)\) = Variance of the returns on assets \(B\).

Portfolio variance is a measure of risk. The higher the variance, the higher the risk. Investors usually reduce the portfolio variance by choosing assets with low or negative covariance, e.g., stocks and bonds.

Portfolio standard deviation is simply the square root of the portfolio variance. It is a measure of the riskiness of a portfolio.

Considering a portfolio with two assets, A and B, the portfolio standard deviation is given by:

$$ \text{Standard deviation}= \sqrt{ (W_A^2 \sigma^2 (R_A)+W_B^2 \sigma^2 (R_B)+2(W_A)(W_B)Cov(R_A,R_B) } $$

Covariance is a measure of the degree of co-movement between two random variables. The general formula used to calculate the covariance between two random variables, \(X\) and \(Y\) is:

$$ Cov(X,Y)=\sigma(X,Y)=E[(X-E[X])(Y-E[Y]) $$

Where:

\(Cov(X, Y)\) = Covariance of \(X\) and \(Y\).

\(E[X]\) = Expected value of the random variable X.

\(E[Y]\) = Expected values of the random variable Y.

This formula calculates the population covariance. It does this by taking the probability-weighted average of the cross-products of the random variables’ deviations from their expected values for every possible outcome.

The sample covariance between two variables, \(X\) and \(Y\), based on a sample data of size \(n\) is:

$$ Cov(X,Y)=\sum_{i=1}^n {\frac { (X_i-\bar X)(Y_i-\bar Y)}{n-1}} $$

Where:

\(\bar X\)= Sample mean of \(X\).

\(\bar Y\)= Sample mean of \(Y\).

\(X_i\) and \(Y_i\) = i-th data points of \(X\) and \(Y\), respectively.

The covariance between two random variables can be positive, negative, or zero.

- A positive number indicates co-movement. The variables tend to move in the same direction.
- A value of zero indicates no relationship.
- A negative value shows that the variables move in opposite directions.

A covariance matrix displays a complete list of covariances between assets needed to calculate the portfolio variance. Consider a portfolio with three assets A, B, and C. The covariance matrix is as follows:

$$ \begin{array}{c|c|c|c}

\textbf{Asset} & \bf A & \bf B & \bf C \\ \hline

A & \bf{Cov(R_A,R_A)} & Cov(R_A,R_B) & Cov(R_A,R_C) \\ \hline

B & Cov(R_B,R_A) & \bf{Cov(R_B,R_B)} & Cov(R_B,R_C) \\ \hline

C & Cov(R_C,R_A) & Cov(R_C,R_B) & \bf{Cov(R_C,R_C)}

\end{array} $$

The off-diagonal (bolded) terms represent variances since, for example:

$$ Cov(R_A,R_A )=\rho(A,A) \sigma_A \sigma_A =1⋅\sigma_A^2=\sigma_A^2 $$

As such, the table above transforms:

$$ \begin{array}{c|c|c|c}

\textbf{Asset} & \bf A & \bf B & \bf C \\ \hline

A & \sigma_A^2 & Cov(R_A,R_B) & Cov(R_A,R_C) \\ \hline

B & Cov(R_B,R_A) & \sigma_B^2 & Cov(R_B,R_C) \\ \hline

C & Cov(R_C,R_A) & Cov(R_C,R_B) & \sigma_C^2

\end{array} $$

Intuitively, a three-asset portfolio would have \(3 \times 3 = 9\) entries of covariances. However, we do not count the off-diagonal terms since they contain the individual variances of the assets. As such, we have \(6 (= 9 – 3)\) covariances.

Note that:

$$ \begin{align*}

Cov(R_B,R_A)&=Cov(R_A,R_B) \\

Cov(R_A,R_C) &=Cov(R_A,R_C) \\

Cov(R_C,R_B) & =Cov(R_B,R_C)

\end{align*} $$

Therefore, there are \(\frac {6}{2}=3\) distinct covariance terms in the above covariance matrix.

In general, if we have \(n\) securities in a portfolio, there are \(\frac {n(n-1)}{2}\) distinct covariances and \(n\) variances to estimate.

Correlation is the covariance ratio between two random variables and the product of their two standard deviations. The correlation formula for random variables \(X\) and \(Y\) is:

$$ \begin{align*}

\text{Correlation (X,Y)}& =Corr(X,Y)=\rho(X,Y)\\ &=\frac {Cov(X,Y)}{\text{Standard deviation(X)} \times \text{Standard deviation(Y)} } \\

& =\frac { Cov(X,Y)}{\sigma_X \sigma_Y} \end{align*} $$

Correlation measures the strength of the linear relationship between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is always between -1 and +1:

- +1 indicates a perfect linear relationship (i.e., the two variables move in the same direction with equal unit changes).
- Zero indicates no linear relationship at all.
- -1 indicates a perfect inverse relationship, i.e., a unit change in one means that the other will have a unit change in the opposite direction.

**Example: Calculating Correlation Coefficient from the Covariance Matrix \(\#1\)**

Harrison is a portfolio manager who oversees three assets: A, B, and C. The covariance matrix of these assets is shown below:

$$ \begin{array}{c|c|c|c}

\textbf{Asset} & \bf A & \bf B & \bf C \\ \hline

A & 0.04 & 0.02 & 0.01 \\ \hline

B & 0.02 & 0.05 & 0.015 \\ \hline

C & 0.01 & 0.015 & 0.09

\end{array} $$

Using this information, what is the correlation coefficient between assets \(B\) and \(C\)?

**Solution**

Note:

$$ \begin{align*}

\text{Correlation } (B,C)&=\frac {Cov(B,C)}{\sigma_B \sigma_C } \\

&=\frac {0.015}{\sqrt{0.05 \times 0.09}}=0.224 \end{align*} $$

**Example: Calculating the Correlation Coefficient \(\#2\)**

We expect a 15% chance that ABC Corp’s stock returns for the next year will be 6%. There’s a 60% probability that they will be 8% and a 25% probability of a 10% return. The expected return is 8.2%, and the standard deviation is 1.249%.

We also anticipate that the same probabilities and states are associated with a 4%, 5%, and 5.5% return for XYZ Corp. The expected value of returns is then 4.975%, and the standard deviation is 0.46%.

To calculate the covariance and the correlation between ABC and XYZ returns, then:

$$ \begin{align*} Cov(R_{ABC},R_{XYZ} )& =0.15(0.06-0.082)(0.04-0.04975) \\ & +0.6(0.08-0.082)(0.05-0.04975)\\ & +0.25(0.10-0.082)(0.055-0.04975)\\ &=0.0000561 \end{align*} $$

$$ \begin{align*}

& \text{Correlation}(Ri,Rj) \\ & =\frac {\text{Covariance}(R_{ABC},R_{XYZ})}{\text{Standard deviation(RABC)} \times \text{Standard deviation(RXYZ)}} \end{align*} $$

Therefore:

$$ \text{Correlation}=\frac {0.0000561}{(0.01249 \times 0.0046)}=0.976 $$

The correlation between the returns of the two companies is very strong (almost +1), and the returns move linearly in the same direction.

**Example: Calculating Correlation Coefficient \(\#3\)**

An analyst studied five years of historical data to examine how changes in Central Bank interest rates affect the country’s inflation rate. The covariance between the interest rate and inflation rate is -0.00075. The standard deviation of the interest rate is 5.5%, and the inflation rate is 12%. Now, let’s calculate and interpret the correlation between these two variables.

**Solution**

$$ \begin{align*}

&\text{Correlation}_{\text{Interest rate, Inflation}}\\&=\frac { \text{Covariance}_{\text{Interest Rate, Inflation}}}{\text{Standard deviation}_{\text{Interest rate}} \times {\text{Standard deviation}}_{\text{Inflation}}} \\

& \text{Correlation}_{\text{Interest rate, Inflation}} = \frac {-0.00075}{(0.055 \times 0.12)}=-0.11364 \end{align*} $$

A correlation of -0.11364 indicates a negative correlation between the interest rate and the inflation rate.

Note that if we consider, say, assets \(A\) and \(B\), then:

$$ \begin{align*} Corr(A,B) & =\rho(A,B)=\frac {Cov(A,B)}{\sigma_A \sigma_B } \\

\Rightarrow Cov(A,B) & =\sigma_A \sigma_B \rho(A,B) \end{align*} $$

Consequently, in the formula for calculating portfolio variance, consisting of two assets, A and B, we substitute for \(Cov(A, B)\) so that:

$$ \text{Portfolio Variance}= W_A^2 \sigma^2 (R_A)+W_B^2 \sigma^2 (R_B)+2(W_A)(W_B) \sigma_A \sigma_B \rho(A,B) $$

## Question

Assume that we have investments in two companies, ABC and XYZ. For ABC, there’s a 15% chance of a 6% return, a 60% chance of an 8% return, and a 25% chance of a 10% return. The expected return for ABC is 8.2%, and the standard deviation is 1.249%. For XYZ, there are similar probabilities of 4%, 5%, and 5.5% returns. The expected return for XYZ is 4.975%, and the standard deviation is 0.46%.

The portfolio standard deviation is

closest to:

- 0.0000561.
- 0.00007234.
- 0.00851.

The correct answer is C.$$ \text{Portfolio Variance}= W_A^2 \sigma^2 (R_A)+W_B^2 \sigma^2 (R_B)+2(W_A)(W_B)Cov(R_A,R_B) $$

First, we must calculate the covariance between the two stocks:

$$ \begin{align*} Cov(R_{ABC},R_{XYZ}) & =0.15(0.06-0.082)(0.04-0.04975) \\ & +0.6(0.08-0.082)(0.05-0.04975) \\ & +0.25(0.10-0.082)(0.055-0.04975) \\ & =0.0000561 \end{align*} $$

Since we already have the weight and the standard deviation of each asset, we can proceed and calculate the portfolio variance:

$$ \begin{align*} \text{Portfolio variance} & =0.5^2 \times 0.01249^2 + 0.5^2 \times 0.0046^2 \\ & +2 \times 0.5 \times 0.5 \times 0.0000561 \\ & =0.00007234 \end{align*} $$

Therefore, the standard deviation is:

$$ \sqrt{0.00007234}=0.00851 $$