Question 559 variance, standard deviation, covariance, correlation
Which of the following statements about standard statistical mathematics notation is NOT correct?
The covariance between variables X and Y is represented by ##\sigma_{X,Y}##.
Unfortunately, statisticians commonly use lower case sigma ##(\sigma)## to represent covariance and standard deviation. The only way to distinguish between them is to look at the subscript. The covariance will have a comma between the variables listed in the subscript but the standard deviation won't. Standard deviation is of a single variable such as X ##(\sigma_X)##, while covariance is between two variables such as X and Y ##(\sigma_{X,Y})##.
To avoid the confusion caused by the clashing symbol notation, some people prefer to use the function notation ##cov(X,Y)## for ##\sigma_{X,Y}## and ##sd(X)## for ##\sigma_{X}##.
Diversification in a portfolio of two assets works best when the correlation between their returns is:
When the correlation of returns between assets A and B are perfectly negative (-1), every time asset A's return rises, asset B's return falls, and vice versa. They will always move in opposite directions in the same ratio. They always tend to cancel each other out.
Therefore a risk-free portfolio of A and B is possible with a certain weighting in each. A risk free portfolio has all risk diversified away which is the ideal situation.
In common sense terms, if the correlation is negative one then when stock A falls, stock B will rise, which means you've overall lost nothing, the risks have offset and this is ideal. What you don't want is a correlation of positive one since that means that when stock A falls, stock B also falls, causing losses in your portfolio.
Mathematically, when the correlation between two stocks is negative, the portfolio variance (##\sigma_p^2## and standard deviation ##\sigma_p##) will be lower since the last term ##(2.x_A.x_B.\rho_{A,B}.\sigma_{A}.\sigma_{B})## will be negative when the correlation (##\rho_{A,B}##) is negative and the weights (##x_A## and ##x_B##) are positive:###\begin{aligned} \sigma_p^2 =& x_A^2.\sigma_A^2 + x_B^2.\sigma_B^2 + 2.x_A.x_B.\rho_{A,B}.\sigma_{A}.\sigma_{B} \\ \end{aligned} ###
All things remaining equal, the variance of a portfolio of two positively-weighted stocks rises as:
As correlation rises, there's less diversification, so variance (risk) rises, so long as both stocks have positive weights so neither are sold short.
Mathematically, this can be seen in the two-stock portfolio variance formula:
###\sigma_p^2 = x_1^2.\sigma_1^2 + x_2^2.\sigma_2^2 + 2.x_1.x_2.\rho_{1,2}.\sigma_1.\sigma_2###| Portfolio Details | ||||||
| Stock | Expected return |
Standard deviation |
Correlation ##(\rho_{A,B})## | Dollars invested |
||
| A | 0.1 | 0.4 | 0.5 | 60 | ||
| B | 0.2 | 0.6 | 140 | |||
What is the standard deviation (not variance) of returns of the above portfolio?
Using the two-asset portfolio variance formula,
###\begin{aligned} \sigma_p^2 =& x_1^2.\sigma_1^2 + x_2^2.\sigma_2^2 + 2.x_1.x_2.\sigma_{1,2} \\ =& x_1^2.\sigma_1^2 + x_2^2.\sigma_2^2 + 2.x_1.x_2.\rho_{1,2}.\sigma_{1}.\sigma_{2} \\ =& \left( \frac{60}{60+140} \right) ^2 \times 0.4^2 + \left( \frac{140}{60+140} \right) ^2 \times 0.6^2 + \\ & 2 \times \left( \frac{60}{60+140} \right) \times \left( \frac{140}{60+140} \right) \times 0.5 \times 0.4 \times 0.6 \\ =& 0.2412 \\ \sigma_p =& (0.2412)^{1/2} \\ =& 0.491121166 \\ \end{aligned} ###
Two risky stocks A and B comprise an equal-weighted portfolio. The correlation between the stocks' returns is 70%.
If the variance of stock A's returns increases but the:
- Prices and expected returns of each stock stays the same,
- Variance of stock B's returns stays the same,
- Correlation of returns between the stocks stays the same.
Which of the following statements is NOT correct?
The covariance of returns will actually increase since it is the product of stock A's standard deviation which increases, stock B's standard deviation which stays the same and the two stock's correlation of returns which also stays the same.
In mathematical notation using Greek symbols: ### \sigma_{A,B} = \sigma_A.\sigma_B.\rho_{A,B} ### Which is the same as: ### \text{cov}(r_A, r_B) = \text{std}(r_A).\text{std}(r_B).\text{corr}(r_A,r_B) ###All things remaining equal, the higher the correlation of returns between two stocks:
When the correlation between two assets' returns increases, if one stock has a negative return, it is more likely that the other stock will also have a negative return. This means that the asset returns will not cancel each other out as often, so there will be less diversification and greater portfolio variance and volatility (standard deviation).
The higher the correlation, the higher the covariance. This is because the covariance between two stocks' returns is a function of correlation:
### cov(r_A, r_B) = corr(r_A, r_B).std(r_A).std(r_B)###Or in mathematical notation with Greek symbols: ### \sigma_{A,B} = \rho_{A,B}.\sigma_A.\sigma_B###
An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 6% pa.
- Stock A has an expected return of 5% pa.
- Stock B has an expected return of 10% pa.
What portfolio weights should the investor have in stocks A and B respectively?
Use the portfolio return equation.
###\mu_p = \mu_A.x_A + \mu_B.x_B### ###0.06 = 0.05 x_A + 0.1 x_B###But we have two unknowns and one equation. To solve for the two weights, we need another equation, which is that the sum of the weights must equal one. ###x_A + x_B = 1### ###x_B = 1 - x_A ###
Substitute this into the above portfolio return equation.
###0.06 = 0.05 x_A + 0.1 (1-x_A)### ###0.06 = 0.05 x_A + 0.1 -0.1 x_A### ###0.05 x_A = 0.04### ###\begin{aligned} x_A &= 0.04 / 0.05 \\ &= 0.8 = 80\% \\ \end{aligned}### ###\begin{aligned} x_B &= 1 - x_A \\ &= 1 - 0.8 \\ &= 0.2 = 20\% \\ \end{aligned}###Therefore the investor should long both stocks in the ratio 80:20, so for every $100 he intends to invest he should buy $80 of stock A and $20 of stock B.
Question 556 portfolio risk, portfolio return, standard deviation
An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 12% pa.
- Stock A has an expected return of 10% pa and a standard deviation of 20% pa.
- Stock B has an expected return of 15% pa and a standard deviation of 30% pa.
The correlation coefficient between stock A and B's expected returns is 70%.
What will be the annual standard deviation of the portfolio with this 12% pa target return?
Use the portfolio return equation and the 'sum of the weights equals one' equation to solve simultaneously for the weights ##x_A## and ##x_B##.
###\mu_p = \mu_A.x_A + \mu_B.x_B### ###0.12 = 0.1 x_A + 0.15 x_B### ###0.12= 0.1 x_A + 0.15 (1-x_A)### ###\begin{aligned} x_A &= 0.03 / 0.05 \\ &= 0.6 = 60\% \\ \end{aligned}### ###\begin{aligned} x_B &= 1 - x_A \\ &= 1 - 0.6 \\ &= 0.4 = 40\% \\ \end{aligned}###Since both weights are positive, the investor should long (buy) both stocks. He should buy the stocks in the ratio 60:40, so for every $100 he intends to invest he should buy $60 of stock A and $40 of stock B.
To find the standard deviation of the portfolio, use the two-asset portfolio variance formula.
###\begin{aligned} \sigma_p^2 =& x_A^2.\sigma_A^2 + x_B^2.\sigma_B^2 + 2.x_A.x_B.\sigma_{A,B} \\ =& x_A^2.\sigma_A^2 + x_B^2.\sigma_B^2 + 2.x_A.x_B.\rho_{A,B}.\sigma_{A}.\sigma_{B} \\ =& 0.6^2 \times 0.2^2 + 0.4^2 \times 0.3^2 + \\ &2 \times 0.6 \times 0.4 \times 0.7 \times 0.2 \times 0.3 \\ =& 0.04896 \\ \end{aligned} ### ###\begin{aligned} \sigma_p =& (0.04896)^{1/2} \\ =& 0.221269067 \\ =& 22.1269067 \text{ percentage points} \\ \end{aligned} ###What is the correlation of a variable X with itself?
The corr(X, X) or ##\rho_{X,X}## equals:
The correlation of a variable with itself is positive one, which is perfect correlation.
Since the covariance of a variable with itself is its variance, and the correlation of two variables is the covariance divided by the product of each variables' standard deviation, and standard deviation squared is variance, then:
###\begin{aligned} corr(X,Y) &= \dfrac{cov(X,Y)}{sd(X).sd(Y)} \\ corr(X,X) &= \dfrac{cov(X,X)}{sd(X).sd(X)} \\ &= \dfrac{var(X)}{(sd(X))^2} \\ &= \dfrac{var(X)}{var(X)} \\ &= 1 \\ \end{aligned}###What is the correlation of a variable X with a constant C?
The corr(X, C) or ##\rho_{X,C}## equals:
The correlation of a variable with a constant is undefined due to a divide-by-zero problem.
Since the covariance of a variable with a constant is zero, and the correlation of two variables is the covariance divided by the product of each variables' standard deviation, and the standard deviation of a constant is zero, then:
###\begin{aligned} corr(X,C) &= \dfrac{cov(X,C)}{sd(X).sd(C)} \\ &= \dfrac{0}{sd(X) \times 0} \\ &= \dfrac{0}{0} \\ &= \text{mathematically undefined} \\ \end{aligned}###The covariance and correlation of two stocks X and Y's annual returns are calculated over a number of years. The units of the returns are in percent per annum ##(\% pa)##.
What are the units of the covariance ##(\sigma_{X,Y})## and correlation ##(\rho_{X,Y})## of returns respectively?
Hint: Visit Wikipedia to understand the difference between percentage points ##(\text{pp})## and percent ##(\%)##.
Similarly to the variance of returns, covariance is measured in percentage points per annum all squared ##\left( (\text{pp pa})^2 \right)##. But correlation is a pure number, it has no units.
Many people prefer to be shown correlations rather than covariances, because the units of correlations are pure numbers that are always between positive one and negative one. Therefore the extent of correlation can be more easily judged since it always has the same maximum (1) and minimum (-1), unlike the covariance whose bounds are harder to calculate.
Let the standard deviation of returns for a share per month be ##\sigma_\text{monthly}##.
What is the formula for the standard deviation of the share's returns per year ##(\sigma_\text{yearly})##?
Assume that returns are independently and identically distributed (iid) so they have zero auto correlation, meaning that if the return was higher than average today, it does not indicate that the return tomorrow will be higher or lower than average.
Annual standard deviation is equal to monthly standard deviation multiplied by the square root of twelve. This will occur so long as there is no auto-correlation of monthly returns.
To see why annual variance is equal to twelve times the monthly variance, see the answer to Question 307. Since standard deviation ##(sd(r)## or ##\sigma_r)## is the square root of variance ##(var(r)## or ##\sigma_r^2)##:
###\begin{aligned} sd(r_\text{yearly}) &= \sqrt{var(r_\text{yearly})} \\ &= \sqrt{12.var(r_\text{monthly})} \\ &= \left( 12.var(r_\text{monthly}) \right)^{1/2} \\ &= 12^{1/2}.\left( var(r_\text{monthly}) \right)^{1/2} \\ &= \sqrt{12}.sd(r_\text{monthly}) \\ \end{aligned}###