CFA Level I - Quantitative Methods
3 parts of the Binomial Probability of Success Formula
1. Total # of Possible Successful Outcomes (nCr) 2. Probability of success in x attempts (p^x) 3. Probability of failures in x attempts [(1-p)^n-x]
Expected Return of a 2-asset portfolio
E(R) = [w₁ * E(R₁)] + [w₂ * E(R₂)]
Expected value of sum of random variables
E(X + Y) = E(X) + E(Y)
Expected value of difference between random variables
E(X - Y) = E(X) - E(Y)
How many trading days are there each year?
250 trading days
Suppose you are the manager of a mutual fund indexed to the Bloomberg Barclays US Government/Credit Index. You are exploring several approaches to indexing, including a stratified sampling approach. You first distinguish among agency bonds, US Treasury bonds, and investment grade corporate bonds. For each of these three groups, you define 10 maturity intervals—1 to 2 years, 2 to 3 years, 3 to 4 years, 4 to 6 years, 6 to 8 years, 8 to 10 years, 10 to 12 years, 12 to 15 years, 15 to 20 years, and 20 to 30 years—and also separate the bonds with coupons (annual interest rates) of 6 percent or less from the bonds with coupons of more than 6 percent. Q1: How many cells or strata does this sampling plan entail? Q2: If you use this sampling plan, what is the minimum number of issues the indexed portfolio can have?
Stratified Sampling Answer 1: Multiplication Rule of Counting 3 issuer classifications 10 maturity classifications 2 coupon classifications Total Cells = (3)(10)(2) = 60 Answer 2: You cannot have fewer than one issue for each cell, so the portfolio must include at least 60 issues.
Required Rate of Return (RRR)
The minimum annual percentage earned by an investment that will induce individuals or companies to put money into a particular security or project.
Chebyshev's Inequality
The percentage of the observations that lie within k standard deviations of the mean is at least 1 - (1/k^2) when k > 1
Correlation Formula
correlation measures the strength of the relationship between variables. Correlation is the scaled measure of covariance. It is dimensionless. In other words, the correlation coefficient is always a pure value and not measured in any units. ρ(X,Y) = COVx,y / σxσy Where: ρ(X,Y) = correlation between the variables X and Y Cov(X,Y) = covariance between the variables X and Y σX = the standard deviation of the X-variable σY = the standard deviation of the Y-variable
How do you calculate the degrees of freedom?
n - 1
Proportion (simple random sampling)
n = [ ( z² * p * q ) + ME² ] / [ ME² + z² * p * q / N ]
Mean (simple random sampling)
n = { z² * σ² * [ N / (N - 1) ] } / { ME² + [ z² * σ² / (N - 1) ] }
n factorial
n! = n * (n-1) * (n - 2) * . . . * 3 * 2 * 1. By convention, 0! = 1.
Combinations of n things, taken r at a time
nCr = n! / r!(n - r)! = nPr / r!
Permutations of n things, taken r at a time
nPr = n! / (n - r)!
Proportionate stratified sampling
nh = ( Nh / N ) * n
Neyman allocation (stratified sampling)
nh = n * ( Nh * σh ) / [ Σ ( Ni * σi ) ]
Optimum allocation (stratified sampling)
nh = n * [ ( Nh * σh ) / sqrt( ch ) ] / [ Σ ( Ni * σi ) / sqrt( ci ) ]
Pooled sample proportion
p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)
Key difference between population/sample variance and continuous random variable / probability distribution variance?
population/sample variance = finite number = /n continuous distribution = infinite number = not /n
If the Expected Earnings Per Share E(EPS) for Bank Corp is $2.34, given the probability distribution of EPS for the current fiscal year below, What are the variance and standard deviation of BankCorp's EPS for the current fiscal year? Probability EPS 0.15 2.600 .45 2.450 .24 2.200 .16 2.00
σ² = Σ P(xi)( Xi - E(x))² + P(xii)( Xii - E(x))²........ σ² = Σ(0.15)( 2.6 - 2.34)² + (.45)( 2.45 - 2.34)² + (.24)( 2.20 - 2.34)² + (.16)( 2.00 - 2.34)² σ² = 0.038785 σ(EPS) = √0.038785 = 0.196939, or approximately 0.20.
Assume the 5-year annualized total returns for the five investment managers used in the earlier example represent all of the managers at a small investment firm. What is the population variance of returns?
σ² = Σ ( Xi - μ )² / N
Population variance
σ² = Σ ( Xi - μ )² / N
unbiased estimator
The mean of its sampling distribution is equal to the true value of the parameter being estimated
Variance of a linear transformation
Var(Y) = a² * Var(X).
State the Decision Rule Form of Hypothesis Testing
"if the test statistic is (greater, less than) the value X, reject the null."
Two-sample z-test for proportions
z-score = z = z = [ (p₁ - p₂) - d ] / SE
Margin of error (sample)
(Critical value) * (Standard error of statistic)
Margin of error (population)
(Critical value) * (Standard deviation of statistic)
Standardized test statistic
(Statistic - Parameter) / (Standard deviation of statistic)
3 Key questions for probability formulas
1. Are the events dependent or independent? 2. Are the events mutually exclusive? 3. Are the events exhaustive?
Name the 4 means
1. Arithmetic Mean 2. Weighted Mean 3. Geometric Mean 4. Harmonic Mean
Name the 3 critical z-values for a 1-tailed test
1. 90% confidence = +/- 1.28 2. 95% confidence = +/- 1.65 3. 99% confidence = +/- 2.33
Name the 3 critical z-values for a 2-tailed test
1. 90% confidence = +/- 1.65 2. 95% confidence = +/- 1.96 3. 99% confidence = +/- 2.575
State the 2 key characteristics of the correlation between confidence intervals and changing degrees of freedom in t-distributions?
1. As degrees of freedom increase, confidence intervals become more narrow 2. As degrees of freedom decrease, confidence intervals become wider
What are the 5 steps to use a t-table ?
1. Calculate the degrees of freedom 2. Identify your confidence interval 3. Calculate the tail value 4. Refer to the t-table row with the correct df 5. Find the column matching the tail probability value, and locate the "critical value"
When is the Geometric Mean Used? (hint: 2 reasons)
1. Calculating investment returns over multiple periods 2. Measuring compound growth rates
When is the Harmonic Mean Used? (hint: 2 reasons)
1. Calculating the average of ratios 2. Calculating the average of rates. Explanation: It is the most appropriate measure for ratios and rates because it equalizes the weights of each data point.
State the scale of measurement for each of the following: - Credit ratings for bond issues - Cash dividends per share - Hedge fund classification types - Bond maturity in years
1. Credit ratings = ordinal scale 2. Cash dividends per share = ratio scale 3. Hedge fund classification types = nominal scale 4. Bond maturity = ratio scale.
What are the 5 major bias types in sampling?
1. Data mining 2. Sample selection 3. Survivorship 4. Look ahead 5. Time period
What are the 2 key elements needed for sampling?
1. Data type 2. Method
3 key elements of Time Value of Money
1. Payment Structure (begin of period, end, annuity) 2. Number of Payments 3. Interest Rates (EAY, CAGR, stated rate)
What is the 1 key factor for selecting the appropriate p-values for the t-distribution?
1. Is the test two-tailed or single tailed? a. if two-tailed, a/2, because p-value is for both sides of the symmetrical distribution. b. if one-tailed, a = p-value because p-value is only for one side of the distribution
What are elements result from a sampling?
1. Sample Distribution 2. Sample Variance Measures (standard error, variance, deviation) 3. Possible Sample bias
Name the 2 key factors for applying difference of means formula
1. Samples are dependent 2. Variances are assumed to be unequal
Name the 2 key factors for applying mean differences formula
1. Samples are independent 2. Variances are assumed to be equal
What are the 3 Sampling Methods?
1. Simple Random Sampling 2. Systematic sampling (select every nth item) 2. Stratified Random Sampling
What are the 7 steps of hypothesis testing?
1. State the hypothesis 2. Identify the test statistic and it's distribution 3. Specify the significance level 4. State the decision rule 5. Collect the data and perform the calculations 6. Make the statistical decision 7. Make the economic or investment decision
What are the 4 characteristics of a t-distribution?
1. Symmetrical 2. Defined by degrees of freedom 3. Has fatter tails (more probability in the tails) than a normal distribution 4. As the degrees of freedom get larger, the t-distribution approaches the shape of a normal distribution.
5 Guidelines to determine which counting method (!) to employ for counting problems
1. The multiplication rule of counting is used when there are two or more groups. 2. Factorial is used by itself when there are no groups—we are only arranging a given set of n items. 3. The labeling formula applies to three or more subgroups of predetermined size. 4. The combination formula applies to only two groups of predetermined size. Look for the word "choose" or "combination." 5. The permutation formula applies to only two groups of predetermined size. Look for a specific reference to "order" being important.
What are the 4 types of data used for sampling?
1. Time Series Data 2. Cross Sectional Data 3. Longitudinal Data 4. Panel Data
What are the 3 desirable properties of an estimator?
1. Unbiased 2. Efficient 3. Consistent
3 key factorial equations (hint: choice & labeling)
1. nCr - choosing, no order 2. nPr - choosing, specific order 3. Labeling - # of different labels = n! / [n₁! * n₂!*.....]
What are the 2 key rules of sample selection?
1. samples must be randomly selected, even when categorized 2. equal number of items selected for each sample
What is the minimum percentage of any distribution that will lie within ±2 standard deviations of the mean?
1−1/k2 Chebyshev's inequality: the percentage of the observations that lie within k standard deviations of the mean is at least 1−1/k2 for all k>1
Chebyshev's Key Levels
36% lie within ±1.25 std. dev. 56% lie within ±1.50 std. dev. 75% lie within ±2 std. dev. 89% lie within ±3 std. dev. 94% lie within ±4 std. dev.
Effective Holding Period Return (HPRt)
= [e^(rcc * t)] - 1 where: rcc = stated rate t = time period
Expected Value (mean) of Binomial Distribution
= n * p where: n = number of successful outcomes p = probability of success
Variance of Binomial Distribution
= np(1-p) where: n = number of successful outcomes p = probability of success 1 - p = probability of failure
consistent estimator
A consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as the sample size increases
For 1 sided hypothesis tests, the +/- sign does what?
A: it matches the direction of the Ha sign < or >. If the Ha sign is <, then the value should be -. If the Ha sign is >, then the value should be +
The probability of the monetary authority increasing interest rates is 40%. The probability of a recession given an increase in interest rates is 70%. If the unconditional probability of a recession is 34%, determine the probability that either interest rates will increase or a recession will occur.
Addition Rule = P(A or B) = P(A) + P(B) − P(AB) P(R or I) = P(R) + P(I) − P(RI) P(R) = 0.34 P(I) = 0.40 P(RI) = 0.28 P(R or I) = 0.46
What is longitudinal data?
Are observations over time of multiple characteristics of the same entity, such as unemployment, inflation, and GDP growth rates for a country over 10 years.
There is a 60% probability the economy will outperform, and if it does, there is a 70% chance a stock will go up and a 30% chance the stock will go down. There is a 40% chance the economy will underperform, and if it does, there is a 20% chance the stock in question will increase in value (have gains) and an 80% chance it will not. Given that the stock increased in value, calculate the probability that the economy outperformed.
Bayes Formula P(Event | Information) = [P(Information | Event) * P(Event)] / P(Information) 4 Factors involved: P(Event) P(Information) P(Event | Information) P(Information | Event) A: We sum the probability of stock gains in both states (outperform and underperform) to get 42% + 8% = 50%. Given that the stock has gains, the probability that the economy has outperformed is 42% / 50% = 84%
Given a discount rate of 10%, what is the present value of an annuity that makes $200 payments at the beginning of each of the next three years, starting today?
BGN mode N=3; I/Y=10; PMT=-200; CPT→PVAD =$547.11
Calculate the number of different groups of three stocks from a list of eight stocks
Binomial Combination Formula nCr = n! / (n-r)!r! enter 8 [2nd] [nCr] 3 [=], which yields 56 n = 8 r = 3
Calculate the number of differently ordered groups of three stocks that can be selected from a list of eight stocks
Binomial Permutation Formula nPr = n! / (n-r)! enter 8 [2nd] [nPr] 3 [=] to get 336 n = 8 r = 3
Assuming a binomial distribution, compute the probability of drawing three black beans from a bowl of black and white beans if the probability of selecting a black bean in any given attempt is 0.6. You will draw five beans from the bowl.
Binomial Probability Formula p(x) = [n!(n−x)!x!] * px(1−p)^n−x = nCr * px(1-p)^n-x P(x=3) = p(3) = 5!/(2!3!) * (0.6)^3 (0.4)^2 = (120/12)(0.216)(0.160) = 0.3456 or 34.56%
Consider a stock with current price S that will, over the next period, either increase in value or decrease in value (the only two possible outcomes). The probability of an up-move (the up transition probability, u) is p and the probability of a down-move (the down transition probability, d) is (1 − p). With - initial stock price S = 50, - factor of up move = 1.01, down move = 1/1.01 - Prob(u) = 0.6, Calculate the possible stock prices after two periods.
Binomial Tree Formula uuS = 1.01² * 50 = 51.01, p(0.6)² = 0.36 udS = 1.01(1/1.01) * 50 = 50, p(0.6*0.4)=0.24 duS = (1/1.01)1.01 * 50 = 50, p(0.6*0.4)=0.24 ddS = (1/1.01)² * 50 = 49.01, p(0.4)² = 0.16 = 0.24 + 0.24 = 0.48 o 48% Since a stock price of 50 can result from either ud or du moves, the probability of a stock price of 50 after two periods (the middle value) is 2 × (0.6)(0.4) = 48%.
You have just been presented with a report that indicates that the mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%, and the mean monthly return for the S&P 500 is 1.09% with a standard deviation of 7.30%. Your unit manager has asked you to compute the CV for these two investments and to interpret your results.
CV = σ / μ CVt-bills = 0.36 / 0.25 = 1.44 CVsp500 = 7.30 / 1.09 = 6.70 These results indicate that there is less dispersion (risk) per unit of monthly return for T-bills than there is for the S&P 500 (1.44 versus 6.70).
Coefficient of Variation (CV)
CV = σ / μ the standardized measure of the risk per unit of return; calculated as the standard deviation divided by the expected return
A stock was purchased for $100 and sold one year later for $120. Calculate the investor's annual rate of return on a continuously compounded basis.
Calculating Continuously Compounded Returns ln (S1 / S0) = ln(1 + HPR) = Rcc Answer: ln(120/100) = 18.232% IF we had been given the HPR of 20% instead of the $100, $120, the calculation is: ln(1+0.20) = 18.232%
Assume that the annual earnings per share (EPS) for a population of firms are normally distributed with a mean of $6 and a standard deviation of $2. What are the z-values for EPS of $2 and $8?
Calculating Z Value (standardizing random variables) z = (x - μ) / σ Answer: If EPS = $8, z = ($8 − $6) / $2 = +1 If EPS = $2, z = ($2 − $6) / $2 = -2 Meaning: z = +1 indicates that an EPS of $8 is one standard deviation above the mean z = -2 means that an EPS of $2 is two standard deviations below the mean.
a priori probability
Comes from a formal reasoning and inspection process; an objective probability
Total Probability Rule
Conditional Probabilities: P(A) = P(A l B₁)(P(B₁) + P(A l B₂)P(B₂) +....P(A l Bn)P(Bn) Unconditional probability of event, given conditional probabilities. Independent Probabilities: P(AB) = P(AB₁) + P(AB₂) .... P(ABn) Where all B₁, B₂ .... Bn are mutually exclusive and exhaustive events
Consider a practice exam that was administered to 36 Level I candidates. The mean score on this practice exam was 80. Assuming a population standard deviation equal to 15, construct and interpret a 99% confidence interval for the mean score on the practice exam for 36 candidates.
Confidence Interval Formula Confidence Interval = x̄ +- z(σ/√n) Answer: = 80 +- 2.58(15/√36) = 80 +- 6.45 99% confidence interval = 73.55 to 86.45
The average return of a mutual fund is 10.5% per year and the standard deviation of annual returns is 18%. If returns are approximately normal, what is the 95% confidence interval for the mutual fund return next year?
Confidence Interval Formula The 90% conf. interval of X = x̄ +- 1.65s The 95% conf. interval of X = x̄ +- 1.96s The 99% conf. interval of X = x̄ +- 2.58s. Answer: μ = 10.5% σ = 18% 95% confidence interval for the return, R = 10.5 ± 1.96(18) = -24.78% to 45.78%
A researcher has gathered data on the daily returns on a portfolio of call options over a recent 250-day period. The mean daily return has been 0.1%, and the sample standard deviation of daily portfolio returns is 0.25%. The researcher believes that the mean daily portfolio return is not equal to zero. 1. Construct a 95% confidence interval for the population mean daily return over the 250-day sample period. 2. Construct a hypothesis test of the researcher's belief.
Confidence Intervals & Two Tailed Hypothesis Tests -critical value ≤ test statistic ≤ +critical value Answer 1: Step 1: sx̄ = 25% / √250 = 0.0158% Step 2: critical z-values for the confidence interval are z0.025 = 1.96 and -z0.025 = -1.96 Step 3: 0.1 − 1.96(0.0158) ≤ μ ≤ 0.1 + 1.96(0.0158), or 0.069% ≤ μ ≤ 0.131% Answer 2: H0:μ0 =0 ; Ha:μ0 ≠0 Reject H0 if test statistic < -1.96 or test statistic > +1.96 0.001 / (0.0025/√250) = 6.33 Since 6.33 > 1.96, we reject the null hypothesis
Annual returns on energy stocks are approximately normally distributed with a mean of 9% and standard deviation of 6%. Construct a 90% confidence interval for the annual returns of a randomly selected energy stock and a 90% confidence interval for the mean of the annual returns for a sample of 12 energy stocks.
Confidence Intervals for a Population Mean Confidence Interval = x̄ +- z(σ/√n) A 90% confidence interval for a single observation is 1.645 standard deviations from the sample mean = 9% ± 1.645(6%) = -0.87% to 18.87% A 90% confidence interval for the population mean is 1.645 standard errors from the sample mean. = 9% ± 1.645 (6%/√12) = 6.15% to 11.85%
What is "the critical value" ?
Critical values are essentially cut-off values that define regions where the test statistic is unlikely to lie
X is uniformly distributed between 2 and 12. Calculate the probability that X will be between 4 and 8.
Continuous Uniform Distribution For all a≤x1<x2 ≤b (i.e.,for all x1 and x2 between the boundaries a and b) P(X < a or X > b) = 0 (i.e., the probability of X outside the boundaries is zero). P(x1 ≤ X ≤ x2) = (x2 − x1)/(b − a). This defines the probability of outcomes between x1 and x2. Answer: 8-4 / 12-2 = 4/10 = 40%
You have a portfolio of two mutual funds, A and B. 75% of the portfolio is invested in A. Calculate the correlation matrix for this problem. Carry out the answer to two decimal places. Fund A Fund B E(RA) = 20%B E(RB) = 12% Wa = 75% Wb = 25% Covariance Matrix Fund A B A 625 120 B 120 196
Correlation Formula ρ(Ra,Rb) = Cov(Ra,Rb) / [σ(Ra)σ(Rb)] Step 1: Calculate σRa and σRb COV(Ra, Ra) = Var(Ra) = σ²(Ra) σ(Ra) = √625 = 25% COV(Rb, Rb) = Var(Rb) = σ²(Rb) σ(Rb) = √196 = 14% Step 2: Calculate the Correlation of (Ra,Rb) ρ(Ra,Rb) = 120/[25(14)] = 0.342857, or 0.34
Covariance Formula
Covariance measures the total variation of two random variables from their expected values. Using covariance, we can only gauge the direction of the relationship (whether the variables tend to move in tandem or show an inverse relationship). However, it does not indicate the strength of the relationship, nor the dependency between the variables. COVx,y = [Σ(xᵢ - x̄ ) ( yᵢ - ˉy)] / n - 1 where: COVx,y = covariance between variable x and y xᵢ = data value of x yᵢ = data value of y x̄ = mean of x ˉy = mean of y n = number of data values
The joint probabilities of the returns of Asset A and Asset B are given in the following figure. Calculate the covariance of returns for Asset A and Asset B. Joint Probabilities RB = 0.40 RB = 0.20 RB = 0.00 RA = 0.20 0.15 0 0 RA = 0.15 0 0.60 0 RA = 0.04 0 0 0.25
Covariance of Joint Probability COV(a,b) = P(a,b)[(Ra - RA)(Rb - RB)] Step 1: calculate E(Ra) & E(Rb) E(R) = P(Ra)(Ra) + P(Rb)(Rb) E(Ra) = (0.15)(0.20) + (0.60)(0.15) + (0.25)(0.04) = 0.13 E(Rb) = (0.15)(0.40) + (0.60)(0.20) + (0.25)(0.00) = 0.18 Step 2: Calculate the variance of the asset returns = COV(a,b)1 = P(a,b) [(Ra - RA)(Rb - RB)] + COV(a,b)2 = P(a,b) [(Ra - RA)(Rb - RB)] + COV(a,b)3 = P(a,b) [(Ra - RA)(Rb - RB)] Cov(RA, RB) = 0.15(0.20 − 0.13)(0.40 − 0.18) + 0.60(0.15 − 0.13)(0.20 − 0.18) + 0.25(0.04 − 0.13)(0.00 − 0.18) = 0.0066
Assume that the economy can be in three possible states (S) next year: boom, normal, or slow economic growth. An expert source has calculated that P(boom) = 0.30, P(normal) = 0.50, and P(slow) = 0.20. The returns for Stock A, RA, and Stock B, RB, under each of the economic states are provided in the probability model as follows. What is the covariance of the returns for Stock A and Stock B? Event P(S) RA RB Boom 0.3 0.20 0.30 Normal 0.5 0.12 0.10 Slow 0.2 0.05 0.00
Covariance of Returns via Probability Model COV(Ri,Rj) = P(event)[( Ri - μi )² * ( Rj - μj )²] Step 1: Calculate E(Ra) Step 2: Calculate E(Rb) Step 3: Calculate covariance of each event = P(boom)*COV(Ra,Rb) = P(Normal)*COV(Ra,Rb) = P(slow)*COV(Ra,Rb) Step 4: Calculate ΣCOV = COV(Ra,Rb boom) + COV(Ra,Rb normal) + COV(Ra,Rb slow)
Return on equity for a firm is defined as a continuous distribution over the range from -20% to +30% and has a cumulative distribution function of F(x) = (x + 20) / 50. Calculate the probability that ROE will be between 0% and 15%.
Cumulative Distribution Function for a Random Variable F(x) = P(X ≤ x). P(0 ≤ x ≤ 15) = F(15) - F(0) F(15) = (15 + 20) / 50 = 0.70 F(0) = (0 + 20) / 50 = 0.40 F(15) - F(0) = 0.70 - 0.40 = 0.30 = 30%
Chi-square test for homogeneity
DF = (r - 1) * (c - 1)
Chi-square test for independence
DF = (r - 1) * (c - 1)
Two-sample t-test
DF = (s₁²/n₁ + s₂²/n₂)² / { [ (s₁² / n₁)² / (n₁ - 1) ] + [ (s₂² / n₂)² / (n₂ - 1) ] }
Chi-square goodness of fit test
DF = k - 1
One-sample t-test
DF = n - 1
Simple linear regression, test slope
DF = n - 2
Two-sample t-test, pooled standard error
DF = n₁ + n₂ - 2
Determine p(6), F(6), and P(2 ≤ X ≤ 8) for the discrete uniform distribution function defined as: X = {2, 4, 6, 8, 10}, p(x) = 0.2
Discrete Uniform Distribution Function F(xn) = np(x) p(x)k where k=possible outcomes Answer 1: p(6) = 0.2, since p(x) = 0.2 for all x Answer 2: F(6) = P(X ≤ 6) = np(x) = 3(0.2) = 0.6. Note that n = 3 since 6 is the third outcome in the range of possible outcomes Answer 3: P(2 ≤ X ≤ 8) = 4(0.2) = 0.8. Note that k = 4, since there are four outcomes in the range 2 ≤ X ≤ 8.
The probability distribution of EPS for Ron's Stores is given in the figures below. Calculate the expected earnings per share. 10% Probability of EPS £1.80 20% Probability of EPS £1.60 40% Probability of EPS £1.20 30% Probability of EPS £1.00
E(X) = ΣP(xi)xi = P(x1)x1 + P(x2)x2 + ... P(xn)xn E[EPS] = 0.10(1.80)+0.20(1.60)+0.40(1.20)+0.30(1.00) = £1.28
Expected value of X
E(X) = μx = Σ [ xi * P(xi) ]
Mean of a linear transformation
E(Y) = ȳbar = ax̄ + b.
Effective Annual Rate (EAR)
EAR = (1 + I/Y)^n - 1 the actual rate paid (or received) after accounting for compounding that occurs during the year
Using a stated rate of 6%, compute EARs for semiannual, quarterly, monthly, and daily compounding.
EAR = [(1 + I/Y)^m] − 1
Effective Annual Rate based on Continuous Compounding (Rcc)
EAR = e^rcc - 1
e^x (the function) on calculator
Effective annual rate with continuous compounding for a stated annual rate (1 + I/Y)^n = effective annual rate with continuous compounding
Based on empirical data, the probability that the Dow Jones Industrial Average (DJIA) will increase on any given day has been determined to equal 0.67. Assuming that the only other outcome is that it decreases, we can state p(UP) = 0.67 and p(DOWN) = 0.33. Further, assume that movements in the DJIA are independent (i.e., an increase in one day is independent of what happened on another day). Using the information provided, compute the expected value of the number of up days in a 5-day period.
Expected Value of a binomial random variable E(X) = np E(X | n = 5, p = 0.67) = (5)(0.67) = 3.35 The statement is read as: the expected value of X given that n = 5, and the probability of success = 67% is 3.35.
Future Value (FV)
FV = PV (1 + I/Y)^n
A pension fund manager estimates that his corporate sponsor will make a $10 million contribution five years from now. The rate of return on plan assets has been estimated at 9 percent per year. The pension fund manager wants to calculate the future value of this contribution 15 years from now, which is the date at which the funds will be distributed to retirees. What is that future value?
FV = PV(1 + I/Y)^N PV = $10 million I/Y = 9% = 0.09 N = 10 we have followed the convention of indexing today as t = 0 and indexing subsequent times by adding 1 for each period. The additional contribution of $10 million is to be received in five years, so it is indexed as t = 5 and appears as such in the figure. The future value of the investment in 10 years is then indexed at t = 15; that is, 10 years following the receipt of the $10 million contribution at t = 5.
You are the lucky winner of your state's lottery of $5 million after taxes. You invest your winnings in a five-year certificate of deposit (CD) at a local financial institution. The CD promises to pay 7 percent per year compounded annually. This institution also lets you reinvest the interest at that rate for the duration of the CD. How much will you have at the end of five years if your money remains invested at 7 percent for five years with no withdrawals?
FV = PV(1 + I/Y)^N PV = $5,000,000 I/Y = 7% = 0.07 N = 5
An institution offers you the following terms for a contract: For an investment of ¥2,500,000, the institution promises to pay you a lump sum six years from now at an 8 percent annual interest rate. What future amount can you expect?
FV = PV(1 + I/Y)^N PV = ¥2,500,000 I/Y = 8% = 0.08 N = 6
Suppose your company's defined contribution retirement plan allows you to invest up to €20,000 per year. You plan to invest €20,000 per year in a stock index fund for the next 30 years. Historically, this fund has earned 9 percent per year on average. Assuming that you actually earn 9 percent a year, how much money will you have available for retirement after making the last payment?
FV annuity factor = A * [(1+I/Y)^N−1 / (I/Y)] A = €20,000 I/Y = 9% = 0.09 N = 30 FV = €20,000(136.307539)
Using a rate of return of 10%, compute the future value of the three-year uneven cash flow stream at the end of the third year. T0 = 0 T1 = 300 T2 = 600 T3 = 200
FV of cash flow stream = ΣFVindividual FV = PV * (1+I/Y)^n FV1:PV=-300;I/Y=10;N=2;CPT→FV=FV1 =363 FV2:PV=-600;I/Y=10;N=1;CPT→FV=FV2 =660 FV3:PV=-200;I/Y=10;N=0;CPT→FV=FV3 =200 FV = ΣFVindividual = 1,223
What is the future value of an annuity that pays $200 per year at the beginning of each of the next three years, commencing today, if the cash flows can be invested at an annual rate of 10%?
FVAd = FVAo × (1 + I/Y) BGN mode N = 3; I/Y = 10; PMT = -200; CPT → FV = $728.20
Joint Probability Formula's (2 - dependent, independent evens)
For Dependent Events 1. P(AB) = P(B) x P(A | B) For Independent Events: 2. P(AB) = P(A) x P(B)
For the last three years, the returns for Acme Corporation common stock have been -9.34%, 23.45%, and 8.92%. Compute the compound annual rate of return over the 3-year period.
Geometric Mean = ^n√[(1+Xi)(1 + Xii)...(1 + Xn)] ^3√1.21903 = 1.21903^(1/3) - 1 = 1.06825 - 1 On the TI, enter 1.21903 [yx] 3 [1/x] [=] = 1.06825 - 1 = 6.825%
An investor purchases $1,000 of mutual fund shares each month, and over the last three months the prices paid per share were $8, $9, and $10. What is the average cost per share?
Harmonic Mean = x̄ h = N / Σ[(1 / xi) + (1 / xii) + (1 / xi)] x̄ h = 3 / [1/8 + 1/9 + 1/10] = $8.926 per share
The Effect of Compounding Frequency on Future Value
I/Y = Annual Rate / compounding periods per year
Compute the FV one year from now of $1,000 today and the PV of $1,000 to be received one year from now using a stated annual interest rate of 6% with a range of compounding periods.
I/Y = the annual interest rate / m N = the number of years × m PVmonthly: FV=-1,000; I/Y=6/12=0.5; N=1×12=12: CPT → PV = PVM = 941.905 FVmonthly: PV=-1,000; I/Y=6/12=0.5; N=1×12=12: CPT → FV = FVM = 1,061.68
Bayes' Formula
P(Event | Information) = [P(Information | Event) * P(Event)] / P(Information) 4 Factors involved: P(Event) P(Information) P(Event | Information) P(Information | Event)
Quantile Position Formula - What is the formula for finding the Position of an observation at a given Percentile?
L = (n + 1) * y/100
When to use the Natural Log function (Ln)
Lognormal Distributions describe asset price distributions. (Normal distributions are for %returns) When you are given prices or $ amounts and asked to find the Rcc, or when you are given the Effective Holding Period Return and asked to find the stated rate. Example: EHPR = (10.52%), Stated Rate = Ln (1.1052) = 10% Rcc = Ln ( $1300 / $1000) = Ln (30% HPR) = EHPR 26.24%
What is the third quartile for the following distribution of returns? 8%, 10%, 12%, 13%, 15%, 17%, 17%, 18%, 19%, 23%
Ly = (n + 1) * (y/100) Ly = (10 + 1) * (75/100) = 8.25
Mean Absolute Deviation (MAD)
MAD = Σ|x − μ| / N the average distance between each data value and the mean
Binomial formula
P(X = x) = b(x; n, P) = nCx * P^x * (1 - P)^[n-x] = nCx * P^x * Q^[n - x]
Consider a portfolio consisting of eight stocks. Your goal is to designate four of the stocks as "long-term holds," three of the stocks as "short-term holds," and one stock as "sell." How many ways can these eight stocks be labeled?
Multinomial Labeling = total # of ways n! / [(na!) * (nb!) * (nc!)] the number of different ways to label the eight stocks is: 8! / [4! * 3! * 1*] = 280
The probability of the monetary authority increasing interest rates is 40%. The probability of a recession given an increase in interest rates is 70%. What is the probability of a recession and an increase in interest rates?
Multiplication Rule = Joint Probability = P(AB) = P(A | B) × P(B) P(RI) = P(R | I) × P(I) P(R | I) = 0.7 P(I) = 0.4 P(RI) = 0.28
What is the probability of rolling three 4s in one simultaneous toss of three dice?
Multiplication Rule = Joint Probability of more than two independent events P(ABC) = P(A) * P(B) * P(C) P(4) = 1/6 P(three 4s on the roll of three dice) = 1/6 * 1/6 * 1/6 = 1/216 = 0.00463
What is the PV of an annuity that pays $200 per year at the end of each of the next three years, given a 10% discount rate?
N = 3; I/Y = 10; PMT = -200; FV = 0; CPT → PV = $497.37
A bond will make coupon interest payments of 70 euros (7% of its face value) at the end of each year and will also pay its face value of 1,000 euros at maturity in six years. If the appropriate discount rate is 8%, what is the present value of the bond's promised cash flows?
N = 6; PMT = 70; I/Y = 8; FV = 1,000; CPT PV = -953.77 With a yield to maturity of 8%, the value of the bond is 953.77 euros. Note that the PMT and FV must have the same sign, since both are cash flows paid to the investor (paid by the bond issuer). The calculated PV will have the opposite sign from PMT and FV.
Negative Binomial formula
P(X = x) = b*(x; r, P) = x-1Cr-1 * Pr * (1 - P)^[x - r]
Geometric formula
P(X = x) = g(x; P) = P * Q^[x - 1]
Hypergeometric formula
P(X = x) = h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
Poisson formula
P(x; μ) = (e^-μ) (μ^x) / x!
Ln function on calculator
Natural Log, gives us the continuously compounded rate (stated annual rate) Ln(1 + HPR) = Rcc
Considering again EPS distributed with μ = $6 and σ = $2, what is the probability that EPS will be $9.70 or more?
Negative Z-Values using +z value table F(-Z) = 1 − F(Z) Here we want to know P(EPS > $9.70), which is the area under the curve to the right of the z-value corresponding to EPS = $9.70 z-value = (9.70 - 6) / 2 = 1.85 Z-table value for F(1.85) = 0.9678 F(-Z) = 1 - 0.9678 = 0.0322 or 3.2% P(EPS > $9.70) = 3.2%
Considering again EPS distributed with μ = $6 and σ = $2, what percent of the observed EPS values are likely to be less than $4.10?
Negative Z-Values using -z value table F(-Z) = -z value from -z value table z = (4.10 - 6) / 2 = -0.95 F(-0.95) = 0.1711 or 17.11% P(EPS < $4.10) = 17.11%
An insurance company has issued a Guaranteed Investment Contract (GIC) that promises to pay $100,000 in six years with an 8 percent return rate. What amount of money must the insurer invest today at 8 percent for six years to make the promised payment?
PV = FV / [1 / (1+I/Y)^n] FV=$100,000 I/Y = 8% = 0.08 N=6
Suppose you own a liquid financial asset that will pay you $100,000 in 10 years from today. Your daughter plans to attend college four years from today, and you want to know what the asset's present value will be at that time. Given an 8 percent discount rate, what will the asset be worth four years from today?
PV = FV / [1 / (1+I/Y)^n] FV^N=$100,000 I/Y=8%=0.08 N=6 the future payment of $100,000 that is to be received at t = 10. The time line also shows the values at t = 4 and at t = 0. Relative to the payment at t = 10, the amount at t = 4 is a projected present value, while the amount at t = 0 is the present value (as of today).
Perpetuity Formula
PV = PMT / (I/Y) This is present value. You would need to manipulate for Future Value.
Odds Formula
Odds(E) = P(E) / [1 - P(E)]
Annuity Due Formula
Ordinary Annuity * (1+I/Y) Since Annuity Due payments are made at the beginning of the period, we must add the (1+I/Y) to provide enough value over the last payment period.
Multinomial formula
P = [ n! / ( n₁! * n₂! * ... nk! ) ] * ( p₁^n₁ * p₂^n₂ * . . . * pk^nk )
Addition Rule of Probability
P(A or B) = P(A) + P(B) - P(AB) If A + B are mutually exclusive, then P(AB) = 0
Rule of multiplication
P(A ∩ B) = P(A) P(B|A)
Rule of addition
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Rule of subtraction
P(A') = 1 - P(A)
Probability Formula
P(E) = X / X + Y is the ratio of number of favorable outcomes to the total number of possible outcomes.
Present Value (PV)
PV=FV/(1+I/Y)^n
Alice would like to have $5,000 saved in an account at the end of three years. If the return on the account is 9% per year with monthly compounding, how much must Alice deposit today in order to reach her savings goal in three years?
PV=FV/(1+I/Y)^n EAR = [(1 + I/Y)^n] − 1 FV = 5000 I/Y = 9% compounded monthly = 9/12 = 0.75% = 0.0075 N = 36 (3 years of monthly compounding) - we can calculate the present value of $5,000 three years (36 months) from now as 5,000 / (1.0075)36 = $3,820.74
Kodon Corporation issues preferred stock that will pay $4.50 per year in annual dividends beginning next year and plans to follow this dividend policy forever. Given an 8% rate of return, what is the value of Kodon's preferred stock today?
PVperpetuity = PMT / (I/Y) PMT = 4.50 I/Y = 0.08 PV = $56.25
Kodon Corporation issues preferred stock that will pay $4.50 per year in four years, and is non-cumulative (i.e., does not pay any dividends for the first three years). Given an 8% required rate of return, what is the value of Kodon's preferred stock today?
PVperpetuity = PMT / (I/Y) + PVperpetuity / (1 + I/Y)^n PMT = 4.50 I/Y = 0.08 PV = $56.25 n = 3 = 56.25 / (1+0.08)^3 = $44.65
What is panel data?
Panel data contain observations over time of the same characteristic for multiple entities, such as debt/equity ratios for 20 companies over the most recent 24 quarters.
Single Observation Confidence Interval Formula
Population Mean +- critical value x standard deviation [this formula works when the confidence interval for a single value is constructed (i.e. n=1)]
You have a portfolio of two mutual funds, A and B. 75% of the portfolio is invested in A. Compute portfolio standard deviation of return Fund A Fund B E(RA) = 20%B E(RB) = 12% Wa = 75% Wb = 25% Covariance Matrix Fund A B A 625 120 B 120 196
Portfolio Standard Deviation Formula σ(Rp) = √[(wa)²(σ²Ra) + (wb)²(σ²Rb) + 2(wa)(wb)Cov(Ra,Rb)] or, if correlation is given instead of covariance matrix σ(Rp) = √[(wa)²(σ²Ra) + (wb)²(σ²Rb) + 2(wa)(wb)(σRa)(σRb)(ρ)]
Holding Period Return (HPR)
Rate of return over a given investment period. Found by calculating-- (Ending Price - Beginning price + Cash dividend) / Beginning Price.
relative frequency
Relative Frequency = Actual occurrences / population the fraction or percent of the time that an event occurs in an experiment
For the next year, the managers of a $120 million college endowment plan have set a minimum acceptable end-of-year portfolio value of $123.6 million. Three portfolios are being considered which have the expected returns and standard deviation shown in the first two rows of the table below. Determine which of these portfolios is the most desirable and the probability that the portfolio value will fall short of the target amount. Portfolio A B C E(Rp) 9% 11% 6.6% σp 12% 20% 8.2%
Roy's Safety First Criterion (ShortFall Ratio) SFratio = [E(Rp) - Rl)] / σp Step 1: Calculate the threshold return Rl Rl = (123.6-120)/120 = 0.03 = 3% Step 2: Calculate the SFratio's A = (9-3) / 12 = 0.5 B = (11-3) / 20 = 0.4 C = (6.6 - 3) / 8.2 = 0.44 Step 3: Choose the best portfolio Portfolio A has highest SFratio = 0.5 Step 4: Calculate probability of shortfall using z-table for F(-0.5) F(-0.5) = 0.3085 = 30.85%
Standard Error of Sample Means
SE = σ / sqrt(n)
Standard error of difference of sample means
SEd = sd = sqrt[ (s₁² / n₁) + (s₁² / n₂) ]
Standard error of difference of paired sample means
SEd = sd = { sqrt [ (Σ(di - d)² / (n - 1) ] } / sqrt(n)
Standard error of proportion
SEp = sp = sqrt[ p * (1 - p)/n ] = sqrt( pq / n )
Standard error of difference for proportions
SEp = sp = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
Standard error of the mean
SEx̄ = sx̄ = s/sqrt(n)
Portfolio Short Fall Risk
SFRatio = [E(R) - Target Return] / σ Choose the portfolio with the highest short fall risk value because it means the Expected Return of the portfolio lies within a highly probable interval within the distribution of returns. A larger shortfall risk means that there is a small chance of the portfolio return falling short of the mean (expected return). A smaller shortfall risk means that there is a large chance of the portfolio return falling short of the mean (expected return). Think about this in terms of standard deviation. The higher the standard deviation (short fall risk), the lower the probability of short fall.
Confidence interval
Sample statistic + Critical value * Standard error of statistic
efficient estimator
Sampling distribution that is less than that of any other unbiased estimator
What rate of return will you earn on an ordinary annuity that requires a $700 deposit today and promises to pay $100 per year at the end of each of the next 10 years?
Solve for I/Y N = 10; PV = -700; PMT = 100; CPT → I/Y = 7.07%
Suppose you have the opportunity to invest $100 at the end of each of the next five years in exchange for $600 at the end of the fifth year. What is the annual rate of return on this investment?
Solve for I/Y N = 5; FV = $600; PMT = -100; CPT → I/Y = 9.13%
Suppose you have a $1,000 ordinary annuity earning an 8% return. How many annual end-of-year $150 withdrawals can be made?
Solve for N I/Y = 8; PMT = 150; PV = -1,000; CPT → N = 9.9 years
How many $100 end-of-year payments are required to accumulate $920 if the discount rate is 9%?
Solve for N I/Y = 9%; FV = $920; PMT = -$100; CPT → N = 7 years
Suppose you are considering applying for a $2,000 loan that will be repaid with equal end-of-year payments over the next 13 years. If the annual interest rate for the loan is 6%, how much will your payments be?
Solve for PMT N = 13; I/Y = 6; PV = -2,000; CPT → PMT = $225.92
At an expected rate of return of 7%, how much must be deposited at the end of each year for the next 15 years to accumulate $3,000?
Solve for PMT N = 15; I/Y = 7; FV = +$3,000; CPT → PMT = -$119.38 (ignore sign)
Suppose a sample contains the past 30 monthly returns for McCreary, Inc. The mean return is 2% and the sample standard deviation is 20%. Calculate and interpret the standard error of the sample mean.
Standard Error of a Sample Mean (unknown population variance) sx̄ = s / √n sx̄ = 20% / √30 = 3.6%
The mean hourly wage for Iowa farm workers is $13.50 with a population standard deviation of $2.90. Calculate and interpret the standard error of the sample mean for a sample size of 30.
Standard Error of a Sample Mean (known population variance) σx̄ = σ / √n σx̄ = $2.90 / √30 = $0.53
A security will make the following payments at the end of the next four years: $100, $100, $400, and $100. Calculate the present value of these cash flows using the concept of the present value of an annuity when the appropriate discount rate is 10%.
Step 1: Calculate the PV of the annuity N = 4; PMT = 100; FV = 0, I/Y = 10; CPT → PV = -$316.99 Step 2: Calculate the PV of the single pmt N = 3; PMT = 0; FV = 300; I/Y = 10; CPT → PV = -$225.39 Step 3: Add both PV's 316.99 + 225.39 = $542.38.
What is the present value of four $100 end-of-year payments if the first payment is to be received three years from today and the appropriate rate of return is 9%?
Step 1: Find PV of the annuity as of the end of year 2 (PV2). (END mode) N=4; I/Y=9; PMT=-100; FV=0; CPT→PV=PV2 =$323.97 Step 2: Find the present value of PV2. Input the relevant data and solve for PV0 N=2; I/Y=9; PMT=0; FV=-323.97; CPT→PV=PV0 =$272.68 We need to stress this important point. The PV annuity function on your calculator set in "END" mode gives you the value one period before the annuity begins. Although the annuity begins at t = 3, we discounted the result for only two periods to get the present (t = 0) value.
Suppose you must make five annual $1,000 payments, the first one starting at the beginning of Year 4 (end of Year 3). To accumulate the money to make these payments, you want to make three equal payments into an investment account, the first to be made one year from today. Assuming a 10% rate of return, what is the amount of these three payments?
Step 1: Solve for PV3 in BGN Mode N=5; I/Y=10; PMT=-1,000; CPT→PV=PV3 =$4,169.87 Step 2: Solve for PMT in END Mode N = 3; I/Y = 10; FV = -4,169.87; CPT → PMT = $1,259.78
What is the difference between the observed value of a statistic and the quantity it is intended to estimate?
The Sampling Error sampling error of the mean = sample mean − population mean = x − µ
When your company's gizmo machine is working properly, the mean length of gizmos is 2.5 inches. However, from time to time the machine gets out of alignment and produces gizmos that are either too long or too short. When this happens, production is stopped and the machine is adjusted. To check the machine, the quality control department takes a gizmo sample each day. Today, a random sample of 49 gizmos showed a mean length of 2.49 inches. The population standard deviation is known to be 0.021 inches. Using a 5% significance level, determine if the machine should be shut down and adjusted.
The Z-Test Setup: H0: μ = 2.5 ; H0: μ = 2.5 z = (x̄ -μ) / (σ/√n) Reject H0 if: -1.96 > z-statistic > + 1.96 Answer: z = (2.49 - 2.5) / (0.021 / √49) = -3.33 -3.33 < -1.96 = reject H0 Meaning: The machine is out of adjustment and should be shut down for repair.
Discount Rate
The interest rate used in discounted cash flow (DCF) analysis to determine the present value of future cash flows.
What is the future value of an ordinary annuity that pays $200 per year at the end of each of the next three years, given the investment is expected to earn a 10% rate of return?
This problem can be solved by entering the relevant data and computing FV. N = 3; I/Y = 10; PMT = -200; CPT → FV = $662.00
You have a portfolio of two mutual funds, A and B. 75% of the portfolio is invested in A. Calculate the expected return of the portfolio. Fund A Fund B E(RA) = 20%B E(RB) = 12% Wa = 75% Wb = 25% Covariance Matrix Fund A B A 625 120 B 120 196
Total Expected (Weighted) Return E(Rp) = (wa)E(Ra) + (wb)E(Rb) = 0.75(20%) + 0.25(12%) = 18%.
The probability of the monetary authority increasing interest rates is 40%. The probability of a recession given an increase in interest rates is 70%. The probability of recession if interest rates do not rise, is 10%. What is the probability of a recession?
Total Probability Rule P(A) = P(A|B) * P(B) + P(A|1-B) * P(1-B) P(R) = P(R | I) × P(I) + P(R | 1-I) × P(1-I) P(I) = 0.4 P(1-I) = 0.6 P(R | I) = 0.7 P(R | 1-I) = 0.10
subjective probability
Uses a probability value based on an educated guess or estimate, employing opinions and inexact information.
Variance of the sum of independent random variables
Var(X + Y) = Var(X) + Var(Y)
Variance of the difference between independent random variables
Var(X - Y) = E(X) + E(Y)
Variance of X
Var(X) = σ² = Σ [ xi - E(x) ]² * P(xi) = Σ [ xi - μx ]² * P(xi)
Suppose as a bond analyst you are asked to estimate the number of bond issues expected to default over the next year in an unmanaged high-yield bond portfolio with 25 US issues from distinct issuers. The credit ratings of the bonds in the portfolio are tightly clustered around Moody's B2/Standard & Poor's B, meaning that the bonds are speculative with respect to the capacity to pay interest and repay principal. The estimated annual default rate for B2/B rated bonds is 10.7 percent. Estimate the standard deviation of the number of defaults over the coming year.
Variance of a Binomial Random Variable variance of X = np(1 − p) The variance is np(1 − p) = 25(0.107)(0.893) = 2.388775. The standard deviation is (2.388775)1/2 = 1.55. Thus, a two standard deviation confidence interval about the expected number of defaults would run from approximately 0 to approximately 6, for example.
When to use the e^x function
When you are given stated rate and asked to find the Effective Annual Rate (with continuous compounding) Example: Stated Rate = 10% Effective Annual Rate = e^.10 = 10.52%
geometric mean
X = [(1 + Xi)(1 + Xii)...(1 + Xn)]^1/n - 1 The mean of n numbers expressed as the n-th root of their product. Only used for positive numbers Used when calculating investment returns over multiple periods or to measure compound Growth Rates
Harmonic Mean
X = n / [(1/Xi) + (1/Xii) + ... (1/Xn)] The mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers
Standardized score
Z = (X - μ) / σ
Regression slope intercept
b0 = ȳ - b₁ * x̄
Regression coefficient
b1 = Σ [ (xi - x̄) (yi - ȳ) ] / Σ [ (xi - x̄)²]
Regression coefficient
b₁ = r * (sy / sx)
f statistic
f = [ s₁²/σ₁² ] / [ s₂²/σ₂² ]
what does the t-statistic indicate?
indicates the distance of a sample mean from a population mean in terms of the estimated standard error
Linear correlation (sample data)
r = [ 1 / (n - 1) ] * Σ { [ (xi - x̄) / sx ] * [ (yi - ȳ) / sy ] }
Sample correlation coefficient
r = [ 1 / (n - 1) ] * Σ { [ (xi - x̄) / sx ] * [ (yi - ȳ) / sy ] }
Pearson product-moment correlation
r = Σ (xy) / sqrt [ ( Σ x² ) * ( Σ y² ) ]
Sample standard deviation
s = sqrt [ Σ ( xi - x̄ )² / ( n - 1 ) ]
Pooled sample standard error
s pooled = sqrt [ (n₁ - 1) * s₁² + (n₂ - 1) * s₂² ] / (n₁ + n₂ - 2) ]
Population Mean Confidence Interval Formula
sample mean +- critical value x standard error [this formula is used when a sample is provided (i.e. n>1) and the resulting sample mean is used to infer the value of the population parameter
Standard error of regression slope
sb1 = sqrt [ Σ(yi - ŷi)² / (n - 2) ] / sqrt [ Σ(xi - x̄)² ]
Standard error of difference of sample proportions
sd = sqrt{ [p₁(1 - p₁) / n₁] + [p₂(1 - p₂) / n₂] }
Pooled sample standard deviation
sp = sqrt [ (n₁ - 1) * s₁² + (n₂ - 1) * s₂² ] / (n₁ + n₂ - 2) ]
Variance of sample proportion
sp² = pq / (n - 1)
Sample variance
s² = Σ ( xi - x̄ )² / ( n - 1 )
t-score
t = (x - μx) / [ s/sqrt(n) ].
tdf
t- distribution, is defined by degrees of freedom (df)
One-sample t-test for means
t-score = t = (x̄ - μ) / SE
Two-sample t-test for means
t-score = t = [ (x̄₁ - x̄₂) - d ] / SE
Matched-sample t-test for means
t-score = t = [ x̄₁ - x̄₂) - D ] / SE = (d - D) / SE
What is alpha?
the level of significance & the probability of making a type 1 error
empirical probability
when the probability comes from the long-run relative frequency of the event's occurrence
Sample mean
x̄ = ( Σ xi ) / n
You have calculated the stock returns for AXZ Corporation over the last five years as 25%, 34%, 19%, 54%, and 17%. Given this information, estimate the mean of the distribution of returns.
x̄ = ( Σ xi ) / n (25+34+19+54+17) / 5 = 29.8%
A portfolio consists of 50% common stocks, 40% bonds, and 10% cash. If the return on common stocks is 12%, the return on bonds is 7%, and the return on cash is 3%, what is the portfolio return?
x̄ w = (wiXi) +(wiiXii) +...+(wnXn)
Normal random variable (z-score)
z = (X - μ)/σ
Standardized score
z = (x - μx) / σx.
One-sample z-test for proportions
z-score = z = (p - P₀) / sqrt( p * q / n )
Simple linear regression line:
ŷ = b₀ + b₁x
Chi-square statistic
Χ² = [ ( n - 1 ) * s² ] / σ²
Chi-square test statistic
Χ² = Σ[ (Observed - Expected)² / Expected ]
probabilistic variance
δ²(X) = ∑P(x)x = P(x₁)[x₁ - E(X)]² + P(x₂)[x₂ - E(X)]² +... Probabilistic Standard Deviation would be the square root of the above.
Population mean
μ = ( Σ Xi ) / N
Mean of sampling distribution of the proportion
μp = P
Mean of geometric distribution
μx = Q / P
Mean of binomial distribution
μx = n * P
Mean of hypergeometric distribution
μx = n * k / N
Mean of negative binomial distribution
μx = rQ / P
Mean of Poisson distribution
μx = μ
Mean of sampling distribution of the mean
μx̄ = μ
Linear correlation (population data)
ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }
Population correlation coefficient
ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }
Population standard deviation
σ = sqrt [ Σ ( Xi - μ )² / N ]
Standard Deviation (σRp) of a 2-asset portfolio
σRp = sqrt[(w₁²σ²) + (w₂²σ²) + (2*w₁*w₂*COV₁,₂)]
Variance of the distribution of sample means formula
σ^2 / n
Standard deviation of difference of sample means
σd = sqrt[ (σ₁² / n₁) + (σ₂² / n₂) ]
Standard deviation of difference of sample proportions
σd = sqrt{ [P₁(1 - P₁) / n₁] + [P₂(1 - P₂) / n₂] }
Standard deviation of proportion
σp = sqrt[ P * (1 - P)/n ] = sqrt( PQ / n )
Variance of population proportion
σp² = PQ / n
Variance of geometric distribution
σx² = Q / P²
Variance of binomial distribution
σx² = n * P * ( 1 - P )
Variance of hypergeometric distribution
σx² = n * k * ( N - k ) * ( N - n ) / [ N² * ( N - 1 ) ]
Variance of negative binomial distribution
σx² = r * Q / P²
Variance of Poisson distribution
σx² = μ
Standard deviation of the mean
σx̄ = σ/sqrt(n)