Central Limit Theorem (CLT)

Check the sample size n. If n is 30 or more, the sampling distribution is normal. The mean of the sample mean equals the population mean. The standard error is calculated as σ divided by the square root of n.

Z Distribution for Population Mean

• Identify values: Sample mean, population mean, σ, and n.
• Compute standard error: SE = σ / √n.
• Compute z: z = (sample mean - population mean) / standard error.
• Use the z-table to find the probability or critical value.
• Confidence interval: Sample mean ± (z-critical × standard error).

T Distribution for Population Mean

• Find the sample mean and sample standard deviation (s).
• Standard error: SE = s / √n.
• Determine the degrees of freedom (df).
• Compute t: t = (sample mean - population

Data Types and Vector Creation

• Nominal → names
• Ordinal → order
• Interval → equal spacing
• Ratio → real math possible

Creating Vectors

c(1, 2, 3) # numeric
c("a", "b") # text
c(TRUE, FALSE) # logical

Vector Functions

numeric(5) # 0 0 0 0 0
rep(2, 5) # 2 2 2 2 2
seq(1, 10) # 1 to 10

Matrix Operations

A matrix is a table of numbers: matrix(1:12, nrow=3)

• + → add
• %*% → matrix multiply
• t() → transpose
• solve() → inverse
• det() → determinant

Note: Standard deviation is the square root of the variance.

Understanding trim=0.10 in R

It means to trim 10% from each end of the dataset.

Reading Data

return <- fund_return[,1]

Descriptive Statistics Functions

length(return)

Gives the number of observations (e.g., 76).

mean(return)

Calculates the ordinary average.

median(

Residual claim: shareholders get only what is left-over after everyone has been paid. Discount rate: the rate at which present and future values are traded off. Future value of cash flow: C x (1+ r)^n. R= (FV/PV)^1/n - 1. Present value= C/(1+ r)^n. Compound interest: exponential growth in wealth, initial amount of money + the interest that has been added the previous years. Perpetuity, Growing Perpetuity, Annuity: multiple payments overtime, same amount, regular interest. NPV: the difference between and investment's value and its cost. Positive NPV=investment yes; NPV=0 project earns the required rate. . IRR measures the average return on the investment: accept project if IRR is greater than the discount rate(minimum return you want). Payback

Question 6: Vector Spaces and Subspaces

Finding the Zero Vector

1. Take the general scalar multiplication formula r ◦ (a, b, c, d).
2. Plug in r = 0.
3. Read off the resulting matrix; that is the zero vector. (Note: 0 ◦ anything = the zero vector in any vector space.)

Proving a Subset U is a Subspace

1. Identify which entries of U are locked (fixed numbers) and which are free (variables).
2. Check if the zero vector (from part a) fits the locked entries of U.
3. Take two general elements of U (using the same locked numbers but different free letters) and combine them with addition. Check that the locked entries stay locked.
4. Take one general element of U and scale it with ◦. Check that the locked entries stay locked.
5. If all three subspace conditions hold, then U

Time Series Analysis

Identifying Trends

• Upward trend: Data increases over time.
• Downward trend: Data decreases over time.
• Stationary trend: No clear increase or decrease.

Smoothing Data (Reducing Fluctuations)

• 3-point moving mean: Average of 3 consecutive values.
• 5-point moving mean: Average of 5 consecutive values.

Seasonal Indices (SI)

Formula:

Seasonal Index = Actual Value ÷ Trend Value

• SI > 1 → Above-average season.
• SI < 1 → Below-average season.

Deseasonalising Data

Formula:

Deseasonalised Value = Actual Value ÷ Seasonal Index

Regression Line (Trend Line)

Equation of the line:

y = mx + c

• m = gradient (rate of change).
• c = y-intercept (starting value).

Bivariate Data Analysis

Types of Correlation

• Positive → Both variables increase together.
• Negative

Statistical Goals

• Describe: Explain what's happening in the data (e.g., mean, mode, average, minimum, variation).
• Explore: Understand how different variables relate to each other.
• Draw Inference: Test hypotheses or theories to make generalizations. Important: Correlation doesn't equal causation.
• Predict: Forecast future outcomes (e.g., weather networks).
• Draw Causal Inference: Determine cause-and-effect relationships, which requires experiments.

Variables in Statistics

Variable Types

1. Categorical (e.g., color, name, religion) vs. Numerical (Discrete: whole numbers OR Continuous: decimals, e.g., movie ranking).
2. • Nominal: Categories with no inherent order.
   • Ordinal: Categories with a meaningful order.
   • Interval: Ordered, equal intervals, but zero is arbitrary

Probability Basics

• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Complement Rule: P(A') = 1 - P(A)
• Conditional Probability: P(B | A) = P(A ∩ B) / P(A) (if P(A) > 0)
• Law of Total Probability: P(B) = Σ P(B | A_i)P(A_i)
• Bayes' Theorem: P(A_k | B) = P(B | A_k)P(A_k) / Σ P(B | A_i)P(A_i)
• Independence: P(A ∩ B) = P(A)P(B) → then P(A | B) = P(A)

Counting Principles

• Multiplication Rule: n₁ × n₂ × ...
• Permutation (Order Matters): nPr = n! / (n - r)!
• Combination (No Order): C(n, r) = n! / (r!(n - r)!)
• Distinguishable Permutations: n! / (n₁! n₂! ...)
• Probability (Equal Outcomes): P(A) = (# favorable) / (# total)

Discrete Distributions

Mean: E[X] = Σ x f(x)   Variance: Var(X) = E[X²] - (E[X])²

• Binomial(n, p): P(X = x) = C(n, x)p^x(1 - p)^{n - x}, E = np, Var

Chapter 1: Margin of Error and Sample Size

• Margin of Error (ME) = (Upper bound - Lower bound) / 2
• Sample size for a proportion type: n = z² × p̂ × (1 - p̂) / ME²
• Standard Deviation = √(pq/n)
• Width = (√n / √new n) × Old width
• Confidence Interval: Helps make correct business decisions by obtaining more information about the population parameter.

Chapter 2: Mean Intervals and T-Tests

• True mean interval for one sample: Use 8:TInterval.
• Critical value of t^*: VARS → InvT(area, df), where df = n - 1 and area = ((1 - confidence level) / 2) + confidence level.
• Sample size for T-test: n = (z × σ / ME)²
• Standard Error of the Mean: SE = s / √n. Alternatively, use 1:T-test to find Sx / √n.
• A t-interval has a larger Margin of Error than a z-interval

Financial Valuation Metrics

Market Capitalization = Market value of equity = Shares × Price of share

Enterprise Value = Market Value of Equity + Debt − Cash

EBITDA and EBIT Formulas

• EBITDA = Net Income + DA, Taxes, and Interest
• EBITDA = Revenue − Expenses (excluding DA, Taxes, and Interest)
• EBITDA = EBIT + Depreciation + Amortization
• EBIT = Revenues − Costs − Depreciation

Income and Cash Flow Calculations

Income Tax = EBIT × τ

Unlevered Net Income = (Revenues − Costs − Depreciation) × (1 − τ)

Free Cash Flow = (EBIT) × (1 − τ) + Depreciation − CapEx +/− ΔNWC

Investment Decision Rules

Sunk costs are costs that have been or will be paid regardless of the decision to undertake an investment.

NPV = PV(Benefits) − PV(Costs). Invest... Continue reading "Corporate Finance Formulas and Valuation Metrics" »