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

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" »

Matrices and Determinants

• Inverse Matrix: |A|⁻¹ = 1/|A|; A⁻¹ = (Adj A)ᵀ / |A|; A⁻¹ · A = I.
• Multiplication: Row by column (fila por columna).
• Determinants: |A₃,₃| = 4, det(2A) = 2³ · 4.
• Resolution (Cramer's Rule): x = |B c₂ c₃| / |A|, y = |c₁ B c₃| / |A|...

Geometry and Spatial Relationships

Areas and Volumes

• Area of a Parallelogram: |a × b|
• Area of a Triangle: |a × b| / 2
• Volume of a Parallelepiped: [u, v, w]
• Volume of a Tetrahedron: [u, v, w] / 6

Relative Positions

• Position of 3 Planes:
  • rg(A) = rg(A*) = 3: System Consistent Determined (1 point).
  • rg(A) = rg(A*) = 2: Line (recta).
  • rg(A) = rg(A*) = 1: Coincident.
  • rg(A) = 1, rg(A*) = 2: Parallel.
  • rg(A) = 2, rg(A*) = 3: Two cases: if two rows are proportional, two planes are parallel

Consider an European call option with strike equal to 10, T = 1, r = 0.05 and σ = 0.2. Using the following time-series:

1. For every time moment t = 0, 1/360, 2/360:

• Calculate the Black-Scholes price.
• Calculate the corresponding delta.
• Calculate the price of the corresponding replicating portfolio and its composition, if we consider a dynamic hedging.

2. Calculate a static hedging. What is the final value of the portfolio? Compare with the dynamic hedging results.

Solution

• At time 0, time to maturity is 1, and then
  • The Black-Scholes price is given by S0N(d1)−Ke−rT N(d2) = 10N(0.35)−10e−0.05N(0.15) = 1.0450
  • The Delta is given by N(d1) = N(0.35) = 0.6368
  • At time t = 0, the value of the replicating portfolio coincides with the Black-Scholes price

Mathematical Aptitude - Average

Question 1

The average age of 30 students is 9 years. If the age of their teacher is included, it becomes 10 years. The age of the teacher (in years) is

_______

(A) 31

(B) 35

(C) 40

(D) 43

Question 2

The average weight of 10 men is decreased by 3 kg when one of them whose weight is 80 kg is replaced by a new person. The weight of the new person is

_______

(A) 50

(B) 60

(C) 70

(D) 73

Question 3

The average of 17 numbers is 45. The average of the first 9 of these numbers is 51 and the last 9 of these numbers is 36. What is the ninth number?

(A) 14

(B) 16

(C) 18

(D) 20

Question 4

The average weight of 9 mangoes increases by 20 gm if one of them weighing 120 gm is replaced by another. The weight of the new mango is

________

(A) 180 gm

(B)... Continue reading "Mathematical Aptitude Test: Average Practice Questions" »

Hypothesis Tests for Population Means

Test 1: Sample of 10 — Mean < 550

(a) In a random sample of 10 people who take a test, the average score is 532.4, and the standard deviation is 28.3. Do the data provide significant evidence, at the 5% level, that the population mean score is less than 550?

1. Step 1: Hypotheses.

   H₀: μ = 550 vs. H_a: μ < 550 (1 point) Where μ is the population mean score.

2. Step 2: Significance level.

   α = 0.05 (1 point)

3. Step 3: Test statistic.

   Test Statistic: t_obs = (X̄ − μ₀)/(s/√n) = (X̄ − 550)/(s/√n) (1 point)

4. Step 4: Rejection region.

   The test rejects H₀ at level α in favor of H_a if t_obs < −t_α,df=9, where t is based on 9 degrees of freedom. (1 point)

5. Step 5: Calculation and conclusion.

   Test Statistic: