Notes, summaries, assignments, exams, and problems for Mathematics

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" »

Investment Analysis II: Key Cash Flow Components

Financial leverage is debt. "Unleveraged" Free Cash Flow (FCF) means we calculate FCF without regard to how the firm is financed. Ultimately, we are valuing the assets of the project. We achieve this by estimating how much cash is generated by the asset side of the balance sheet.

Capital Expenditure (CAPEX)

Capital Expenditure (CAPEX) is defined as the original cost of investment in property, plant, or equipment and other long-term assets.

Formula for CAPEX

NET PPE (current) - NET PPE (prior) + Depreciation

Depreciation and Asset Valuation

Depreciation: Depreciation charges are intended to represent the cost of wear and tear over the asset's life. The Straight-Line Depreciation method divides the asset’s... Continue reading "Capital Budgeting: Calculating Cash Flows and Valuation Metrics" »

Gross Income and Business Deductions

Gross Income: Includes revenue from sales, services, rents, dividends, interest, etc. Exclusions include life insurance proceeds and municipal bond interest. Expenses associated with the production of tax-exempt income are not deductible.

Deductions Above the Line (for AGI)

Deductions for Adjusted Gross Income (AGI) include business expenses, rental expenses, self-employed health insurance, and IRA contributions.

• Ordinary & Necessary Test: An expense must be customary and appropriate for the business.
• Meals Deduction: Can deduct 50% of meals if the expense is ordinary, necessary, and reasonable, the employee is present, the person eating is a current or potential client, and the meal is purchased separately

This table provides a structured reference for key settings and functional components within SAP Treasury and Risk Management.

Sample Size Calculation

To find what n must be:

n ≥ (Z_1-α/2 / MOE)² * p(1-p)

Wald Confidence Interval

SE^ = sqrt((p^ * (1 – p)) / n)

Odds Ratio Analysis

Calculated by criss-cross of 2x2 table.

Example: The odds for someone to smoke ≥ 5 cigarettes and have lung cancer is 3.40 times that of someone who smokes < 5 cigarettes daily to have lung cancer.

Fisher's Exact Test

Used when cell frequencies are < 5 in a 2x2 table. Tests for independence.

• H₀: Odds ratio = 1
• H_A: Odds ratio ≠ 1

Chi-Square Test of Independence

• H₀: The two variables are independent.
• H_A: The two variables are not independent.

Logit Model

To find the number of odds used in the model, find all combinations of levels (e.g., a 2x3 model has 6 combinations).

Exp(estimate) = odds for... Continue reading "Statistical Analysis Cheat Sheet: Formulas and Methods" »

SAP S/4HANA Financial and Treasury Functions

The following table details various aspects, components, and features related to SAP S/4HANA, particularly focusing on Cash Management and Treasury functions.

Bond Valuation and Interest Rate Premiums

Calculating Bond Rates and Premiums

1. Long-Term Treasury Bond Rate Calculation

You read in The Wall Street Journal that 30-day T-bills are currently yielding 2%. Your brother-in-law, a broker at Kyoto Securities, has provided the following estimates of current interest rate premiums on a 1-year bond:

• Liquidity Premium: 3%
• Maturity Risk Premium (MRP): 1.5%
• Default Risk Premium (DRP): 1.2%

Based on these data, what is the long-term Treasury bond rate?

For a Treasury bond, the Liquidity Premium and Default Risk Premium are typically not applicable. The 30-day T-bill yield (2%) can be considered the short-term risk-free rate plus inflation premium (r* + IP). To find the long-term Treasury bond rate, we add the... Continue reading "Financial Calculations: Bond Valuation & Stock Risk Analysis" »