CFA Level 1 - Quantitative Review
Calculate Mean and Standard Deviation of Expected Returns.
The mean is a calculation of central tendency.
Mean = A + B + C
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Expected Return E(r) = {(A)*(P)A} + {(B)*(P)B} + {(C)*(P)C}
E(r) = S ri * p(ri)
Variance σ2 ={(A – E(r)A,B,C )2 * PA}+{(B – E(r)A,B,C)2 * PB}+{(C – E (r)A,B,C )2 * PC}
s2i = S p(ri) * [ri – E(r)] 2
The variance indicates the adequacy of the mean as representative of the population by measuring the deviation from expectation. Basically it and the standard deviation are measures or the average deviation from the mean.
Standard Deviation = σ = Square Root of the Variance = (σ2)1/2
If the data is normally distributed:
68% or all observations will fall within … 1σ of the mean.
95% of all observations will fall within … 2σ of the mean.
99% or all observations will fall within … 2.5σ of the mean.
99.99% of all observations will fall within… 4σ of the mean.
Construct a Hypothesis Test Consisting of a Null Hypothesis and Test Statistic.
Hypothesis testing is a procedure for determining the likelihood that an inferential error has been made. First, state a claim that is to be tested. This is the null hypotheses, and is assumed to hold unless there is sufficient evidence to reject it. The goal of hypothesis testing is to decide whether to reject the null hypothesis, while identifying the likelihood of errors in the determination (JKE).
Below are the steps used in creating a hypothesis test:
1. Set up the null hypothesis
~Example of Null and Alternative
Null Hypothesis(Ho) .78 =.98
Alternative Hypothesis (H1) .78 ≠ .98
Remember that x ≠ y is a two tail test and x < y and x > y are one tests.
2. Chose a confidence level, e.g., 95%
~If a two tail test is used, then choose a 95% level of confidence; you are testing 2.50% in each tail.
~If a one tail test is used, then choose a 95% level of confidence; you are testing 5% in one tail.
3. Compute the T-Statistic: Value(2)-Value(1)
Standard Error
Assume standard error = (.98-.78)/.07 = .07
4. Look up the needed value in the T-Table.
5. Compare the T-statistic from the table to the calculated statistic. If the T-statistic that you calculated is less than the one in the table then you fail to reject the null hypothesis. If it is greater than the one in the table then you reject the null.
Interpret the significance of regression coefficients and results
The regression equation will calculate a constant alpha (α) intercept, and a beta (β) that is multiplied by the independent variable (X):
Y=α + βX
Know the correlation coefficient (ρ or r) measures the degree of correlation between two variables. Know that it ranges from –1 to 1. Perfect correlation =1 and perfect negative correlation =-1.
Coefficient of determination (R2) is the percentage of variation in the dependent variable (Y) caused by the independent variable. It is a measure of the goodness of fit of the regression line. The higher the R2 the better. R squared values range from 0 to 1.
The Coefficient of determination is the square of the correlation coefficient, i.e., R2=ρ2
Understanding Regression Analysis: An Introductory Guide
Review the assumptions and inferences underlying linear regression.
Know that Least Squares Regression, also known as Sum of the Squared Errors
[i.e., S(ci - ^ci)2] is the sum of the squared distances of all the points in the observation away from the mean.
When using linear regression to form a hypothesis, look for a functional relationship between two variables, such as how much is X changed by Y. The variable that causes the action is the independent variable and the variable that is acted upon is the dependent variable.
Y= A + B*X or Y = Alpha + Beta(x) or Y=α + βX
Alpha & Beta are the unknown parameters that define the intercept and slope of the line. After you calculate your Alpha and Beta, test their significance. Also note that the regression analysis does not prove anything; there could be other unknown variables that affect both X and Y.
Standard Error is the standard deviation or the dispersion about the regression line. It is also called unsystematic variation.
Discuss the results of regression and correlation analysis
After a regression analysis has been performed an equation will be produced.
Y=A + Bx
Y= Independent Variable
X= The independent Variable
B= The Slope
The regression line is produce by minimizing the sum of the least squared errors of the data points along the line.
The correlation coefficient measures a degree of linear association between two variables. The coefficient always falls between –1 and 1: +1 equals perfect correlation; –1 equals a perfect negative correlation; 0 means that there is no correlation between the two variables.
The square of the correlation coefficient (ρ) is the coefficient of determination (R2): R2=ρ2.
The coefficient of determination measures the goodness of fit of the regression line.
Discuss the relationship between the dependent and independent variables
Regression analysis is trying to measure a relationship between two variables. It try’s to measure how much the dependent variable is effected by the independent variable. The regression equation as a whole can be used to determine how related the independent and dependent variables are related. It should be noted that the relationship between the two variables can never be measured with certainty. There always exists other variables that are unknown that have may have an effect on the dependent variable.
Identify which coefficients are statistically significant
One uses the calculated t-statistic to test significance. Compare the calculated t-statistic that is given for the slope and the y intercept to the critical t-statistic. Using a 5% confidence level, or at the 0.05 level, the critical t-statistic equals 1.96; the critical statistic may also be given in the problem.
If the calculated t-statistic for the slope (β) is higher than the critical t-statistic then it proves that it is statistically significant. If the calculated t-statistic for the slope is lower than the critical t-statistic, the variable is not statistically significant.
If the calculated t-statistic for the Y intercept (α) is higher than the critical t-statistic then it proves that it is statistically significant. If the calculated t-statistic for the y intercept is lower than the critical t-statistic, the variable is not statistically significant.
The process involves comparing a calculated t-statistic that is given to the critical t-statistic that may also be given. If the calculated t-statistic is smaller than the critical t-statistic, it is not statically significant. If it is larger than the critical than it is statistically significant.
First Principles of Valuation: The Time Value of Money
Calculate net present and future values of lump sums and annuities.
Future Value = CF*(1+R)t
Future Value of an annuity = CF * ((1+R)t-1)/R
Present Value = CF/ (1+R)t
Present Value of an annuity = CF*(1-{1/(1+R)t))/R
As can be seen the present value calculation involves discounting the future cash flows to a present value. To save space here and to save time on the test you should be able to calculate PVs and FVs using your calculator.
N = Number of periods
I/Year = Yield in market place or the Required Rate of Return
PV = Present value
PMT = Payment amount per period
FV = The future value of the investment
One can solve for any of the above variables. Just input the other variables and solve for the unknown. Using the calculator on the test will prove to be a very time efficient manner of calculating present values and future values.
The Investment Setting
Explain the difference between arithmetic and geometric rates of return.
The arithmetic mean is sum of all the given variables divided by the number of variables.
Arithmetic average = (A + B + C)/3
Example - Three year investment returns: HPRs = Year 1 = .15; year 2 = .20; year 3 = -.20
Arithmetic average = (.15 + .20 + -.20)/3 = 0.05 or 5.0%
The geometric average is calculated by adding one to all holding period returns and then multiplying them, and taking this result to the 1/nth root.
Example - Three year investment returns: HPRs = Year 1 = .15; year 2 = .20; year 3 = -.20
Geometric average = {(1.15)*(1.20)*(.80)}1/3- 1 = 3.353%
1. One would chose to use the geometric average as a measure of long term investment results. 2. The arithmetic mean is biased upwards.
These are the two main points to remember about the averages.
Explain such measures of risk as variance, standard deviation, and coefficient of variation.
E(r) Expected Return = {(A)*(P)A} + {(B)*(P)B} + {(C)*(P)C}
Variance or σ2 = {(A – E(r)A,B,C )2 * PA} + {(B – E(r) A,B,C)2 * PB} + {(C – E (r) A,B,C )2 * PC}
A, B & C = Individual investment returns
E(r) =Expected Return
(P) = Probability
The variance indicates the adequacy of the mean as representative of the population by measuring the deviation from expectation. Basically, it and the standard deviation are measures or the average deviation from the mean. The standard deviation is the square root of the variance.
The coefficient of variation (CV) gives a measure of risk per unit of return. The lower the CV the better because the investment would offer less risk per unit of return.
Coefficient of variation equals the standard deviation divided by the expected rate of return:
CV = σ / E(r)
What Practitioners Need to Know…About Time Diversification
Explain the argument for time diversification.
Time diversification refers to the fact that over a long period of time horizons, above average returns tend to offset below average returns - the per annum average rate of return has a smaller standard deviation for a longer period of time. Using historic returns, and assuming a normal distribution, the graph and time shows that the risk of not losing money in investment is reduced. (JKE)
Discuss why that argument is fallacious when presented in terms of ending wealth rather than a rate of return.
While the per annum average rate of return has a smaller standard deviation for a longer time horizon, it is true that the uncertainty compounds over a greater number of years. The magnitude of a potential loss in ending wealth is much greater than suggested by the rate of return analysis. (JKE)
Outline the conditions under which time diversification may hold true.
1. The belief that investment return is not random.
2. The belief that the risk of bad outcomes would equally impact risk less assets.
3. Willing to accept greater risk because one could adjust consumption and work habits.
4. Discontinuous utility functions (JKE).
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5 CFA L1 – Quantitative Review