Section I.A. Statistics Flashcards
variance
Variance Defined:
Measures how far a set of numbers is spread out.
Defined as the average squared difference between the mean and each item in the population or in the sample.
Variance is always non-negative.
Drawback: variance gives added weight to outliers since squaring the numbers can skew interpretations.
Standard deviation is easier to use and understand.
Application of Variance:
A high variance means that data points are very spread out.
Variance value of zero means that all values within the set are identical.
Calculate the variance of the following returns for Clean Energy fund in its only years:
Year 1 = 8%
Year 2 = -7%
Year 3 = 11%
Year 4 = 22%
Solve by calculating the mean first: .08 + -.07 + .11 + .22 = .34; .34/4 = .0850
Plug into the variance equation from the formula sheet: [(.08-.085) 2+(-.07 -.085) 2+(.11 -.085) 2+(.22 -.085)2] = .0429
.0429 divided by (n) 4 = .0107
______ is the square root of ______
Standard deviation is the square root of variance
The mid-cap blend fund you’re tracking has an expected return of 9.7% and a standard deviation of 14.1%. The S&P 400 mid-cap index has an expected return of 8.9% and a standard deviation of 15.0%. The correlation between the two is .88. What is the covariance between the two?
Covariance = correlation of .88 multiplied by the standard deviation of the fund (.141) multiplied by the standard deviation of the index (.15) = 0.0186.
Semi-variance
Defined:
• Measures data that is below the mean or target
value of a data set.
• Considered a better measurement of downside risk.
• Semi-variance is the average of the squared
deviations of all values less than the average or
mean
Coefficient of Correlation
= Covariance divided by the product of the standard deviations of the two assets
The frontier markets fund you’re tracking has an expected return of 16.8% and a standard deviation of 21.7%. The S&P 500 index has an expected return of 9.7% and a standard deviation of 12.5%. The covariance between the two is .0123. What is the correlation between the two?
The Coefficient of Correlation = Covariance divided by the product of the standard deviations of the two assets = 0.0123 / (0.217 * 0.125) = 0.4535.
Tech Fund has a standard deviation of 14.96 and Energy Fund has a standard deviation of 12.43. What is the correlation coefficient between the two investments below?
Year Tech Fund Energy Fund
1 36.45 14.10
2 -5.17 15.17
3 12.46 -15.70
4 9.10 2.20
First, calculate the average returns of both funds. Tech Fund = 13.21. Energy Fund = 3.94.
Now take the differences between the tech fund’s single and average returns and multiply by the differences between the energy fund’s single and average returns. [(36.45 – 13.21) x (14.10 – 3.94)] + [(-5.17 – 13.21) x (15.17 – 3.94)] + [(12.46 – 13.21) x (-15.70 – 3.94)] + [(9.10 – 13.21) x (2.20 – 3.94)] = (236.12) + (-206.410) + (14.73) + (7.15) = 51.59/4 = 12.90.
Second, calculate the correlation coefficient by plugging in the covariance you calculated into the formula:
correlation coefficient = [(covarianceAB)/(standard deviationA x standard deviationB)] = 12.90/(14.96 x 12.43) = 12.90/185.95 = .0694.
Which of the following statements regarding correlation coefficients are accurate?
I. When two stocks have a correlation coefficient that is greater than zero but less than 1.00, it means that they have a positive correlation and there may be some benefit through diversification.
II. Mutual fund A and mutual fund B have a correlation coefficient of -0.89. This indicates beneficial diversification given the low level of positive correlation or association between these funds.
III. In a situation where two investments have a perfectly negative correlation coefficient of -1.00, the variance or standard deviation would be zero in a perfectly balanced portfolio allocation.
IV. A portfolio of investments that exhibits a vast range of correlation coefficients from 0.20 to 0.60 would be considered highly diversified and ensure a very low risk level as measured by standard deviation.
I. TRUE: When two stocks have a correlation coefficient that is greater than zero but less than 1.00, it means that they have a positive correlation and there may be some benefit through diversification.
II. FALSE: Mutual fund A and mutual fund B have a correlation coefficient of -0.89. This indicates beneficial diversification given the low level of positive correlation or association between these funds. THESE FUNDS DO NOT HAVE A POSITIVE CORRELATION TO EACH OTHER.
III. TRUE: In a situation where two investments have a perfectly negative correlation coefficient of -1.00, the variance or standard deviation would be zero in a perfectly balanced portfolio allocation. The negative correlation relates to positive and negative movements from a mean, and thus can equal positive or negative returns. But the variance or standard deviation of those returns would be zero.
IV. FALSE: A portfolio of investments that exhibits a vast range of correlation coefficients from 0.20 to 0.60 would be considered highly diversified and ensure a very low risk level as measured by standard deviation. THE RISK LEVELS OF THESE INVESTMENTS COULD ALL BE VERY HIGH INDIVIDUALLY; AND WHILE DIVERSIFICATION MAY BRING THE PORTFOLIO’S STANDARD DEVIATION DOWN, IT MAY STILL BE VERY HIGH AND CONSIDERED RISKY.
Consider the following probability distribution for stocks A and B: The coefficient of correlation between A and B is:
State Probability Return on Stock A Return on Stock B
1 0.10 10% 8%
2 0.20 13% 7%
3 0.20 12% 6%
4 0.30 14% 9%
5 0.20 15% 8%
The coefficient of correlation between A and B is:
covA,B = 0.1(10% - 13.2%)(8% - 7.7%) + 0.2(13% - 13.2%)(7% - 7.7%) + 0.2(12% - 13.2%)(6% - 7.7%) + 0.3(14% - 13.2%)(9% - 7.7%) + 0.2(15% - 13.2%)(8% - 7.7%) = 0.76;
rA,B = 0.76/[(1.1)(1.5)] = 0.46.
Kurtosis
Kurtosis defined: measures the peakedness of a probability distribution or normal distribution curve;
if kurtosis is positive (leptokurtic), the chart will show a more slender distribution, i.e., higher peak, that is concentrated more around the mean and may have fatter tails;
if kurtosis is low (platykurtic), a chart will show thinner tails and distribution that is less concentrated around the mean, thus a flatter peak.
With a ______negative/positive skew______, we have fewer, but more extreme outcomes to the __left/right____ of the mean. Those outcomes pull the distribution and mean to the ____left/right____. Thus, the standard deviation may be ____over/underestimating_______ the risk because the possibility of that extreme left tail event is not captured by the statistic.
negative skew, left, underestimating
Skewness applied:
For distributions that are non-normal (e.g., exhibit skewness, which is a measure of the distribution’s asymmetry around the mean), the standard mean-variance analysis is limited, which means standard deviation is simply less meaningful.
With a negative skew, we have fewer, but more extreme outcomes to the left of the mean. Those outcomes pull the distribution and mean to the left. For a negative skew, the standard deviation may be underestimating the risk because the possibility of that extreme left tail event is not captured by the statistic.
With a positive skew, we have fewer, but more extreme outcomes to the right of the mean. Those outcomes pull the distribution and mean to the right. For a positive skew, the standard deviation may be overestimating the risk.
Expected Return =
Expected Return = Sum [probability of state x the return if state occurs]
Calculate the expected return of the following scenario.
Probability Risk Return
.20 .20 .10
.45 .30 .08
.25 .25 .18
.10 .15 .06
Expected Return = Sum [probability of state x the return if state occurs]
Calculation:
20% x 10% = 2.0%
45% x 8% = 3.6%
25% x 18% = 4.5%
10% x 6% = 0.6%
Then 2% + 3.6% + 4.5% = 0.6% = 10.70%
