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

Understanding Annuity Due

An annuity due is a series of equal payments made at the beginning of each period. Below are practical examples to help you master these calculations.

Example 1: Lease vs. Cash Purchase

You are buying a new vehicle and must decide between leasing or paying cash. The salesperson offers a cash price of $23,000 or a five-year lease with:

• Monthly payments of $300 (first payment due at signing).
• A buyout of $9,500 at the end of five years.

Since the first payment is made at the start, the lease requires 59 additional equal payments. Assuming a 6% annual interest rate compounded monthly (0.5% per month):

• BGN Mode: N = 60, I = 0.5, FV = $9,500, PMT = $300
• Result: CPT PV = –$22,638

Conclusion: The lease is the better deal because... Continue reading "Annuity Due Calculations: Financial Analysis and Examples" »

) What is the difference between PEST, PESTEL, STEEPLE, and SLEPT Analysis?

16.) What are a corporation's dynamic capabilities and why is it important to be aware of them as part of the strategic management process?

17.) Different internationalization strategies are normally compared based on two main dimensions. What are these two dimensions and why were they originally chosen by companies to compare their internationalization strategies?

18.) How can a company create competitive advantage in its most basic form?

19.) What are sources of competitive advantage?

20.) What is the next step after you have conducted a PEST Analysis? How do you use such an Analysis?

21.) In class we discussed four basic levels of internationalization. Name and compare... Continue reading "International Business and Global Competitiveness: Key Concepts and Strategies" »

Essential Math Formulas and Concepts

Volume of Shapes

(a) Cube:

• Formula: V = s³ (where s is the length of a side)
• Example: A cube with side length 4 cm

V = 4³ = 64 cm³

(b) Cylinder:

• Formula: V = πr²h (where r is the radius and h is the height)
• Example: A cylinder with radius 2 cm and height 6 cm

V = π(2²)(6) = 24π cm³ (approximate value)

Surface Area of Shapes

(a) Cube:

• Formula: SA = 6s² (where s is the length of a side)
• Example: A cube with side length 3 cm

SA = 6(3²) = 54 cm²

(b) Cone:

• Formula: SA = πr(r + √(r² + h²)) (where r is the radius and h is the height)
• Example: A cone with radius 5 cm and height 8 cm

SA = π(5)(5 + √(5² + 8²)) = 129.74 cm² (approximate value)

Stem and Leaf Plot

(a) Mean:

• Example: Data: 5, 7, 8, 9, 10, 12, 15

Mean = (5 + 7 + 8... Continue reading "Essential Math Formulas and Concepts: A Quick Reference" »

Discrete Math Concepts

Binary Relations

Cartesian Product

The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.

Example: If A = {0, 1}, the Cartesian product A × A is {(0,0), (0,1), (1,0), (1,1)}.

Properties of Binary Relations

A binary relation R on a set A can have several properties:

1. Reflexive: For all x ∈ A, (x, x) ∈ R.
2. Symmetric: For all x, y ∈ A, if (x, y) ∈ R, then (y, x) ∈ R.
3. Transitive: For all x, y, z ∈ A, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
4. Irreflexive: For all x ∈ A, (x, x) ∉ R.
5. Antisymmetric: For all x, y ∈ A, if (x, y) ∈ R and (y, x) ∈ R, then x = y.

Examples of Relation Properties

Consider a set A = {0, 1}.

• The relation R = A

Cost of the Building: Land+Construction

When was the building bought? YTD amort.construction 2020 / construction P&L --> it has been amortized x years until 2020

At which price did the company sell the building? ((Land + Construction) - YTD amort.constr) + Profit from selling

useful life of the building? construction / construction P&L

Occupancy: 33%. VERY LOW !! ===> 70 rooms x 600 ADR x occupancy x 365 days = 5.059.000 €

F&B Menu: 7 € ==> VERY LOW & doesn't match with an 600 ADR !! [(80% of 100 people for lunch) + (80% of 100 people per Dinner) ] * Menu Price * 365 days = 409.000 €
F&B Breackfast: 18€/person. Reasonable with 600 ADR 80% of 1,5 clients x 33% occupancy * 70 rooms * 365 days * Breackfast Price... Continue reading "Hotel Financial Analysis and Recommendations" »

Stock Valuation and WACC Examples

Problem 1: Constant Dividend Growth

Thomas Brothers is expected to pay a $0.35 per share dividend at the end of the year (D1 = $0.35). The dividend is expected to grow at a constant rate of 6% a year. The required rate of return is 13%. What is the stock’s current price?

P0 = D1 / (r – g) = 0.35 / (0.13 - 0.06) = $5.00

Problem 2: Current Stock Price with Constant Growth

A stock just paid a dividend of D0 = $1.50. The required rate of return is r = 10%, and the constant growth rate is g = 4.0%. What is the current stock price?

P0 = D0(1+g) / (r – g) = 1.50(1 + 0.04) / (0.10 - 0.04) = $26.00

Problem 3: Expected Stock Price in One Year

Using your answer for problem 2, what is the expected stock price in one year... Continue reading "Calculating Stock Prices and Weighted Average Cost of Capital (WACC)" »

Actuarial Science Practice Problems

Problem 11: Investment Option Comparison

11. $50,000 is invested in two options:

• Option A: Invested into a fund with an annual effective rate i.
• Option B: Purchase of an annuity-immediate with 24 level annual payments at a rate of 10%.

Option B is equivalent to Option A with an interest rate of 5%. The accumulated value for Option A is 50,000(1+i)²⁴. For Option B: * = Option A. Find i.

Problem 14: Monthly Deposits and Future Value

14. Smith deposits $1,000 at the end of each month with a nominal rate i⁽¹²⁾ = 0.12. The first deposit is on 1/31/2010, and the last is on 12/31/2034. From 2035 to 2059, the accumulated value Y is:

Problem 15: Interest Rate Changes and Balances

15. Since 6/30/2014, a deposit of $100... Continue reading "Actuarial Mathematics: Annuity and Loan Practice Problems" »

Linguistic Exercise: Defining Corners

3.4.3. Match the definitions to the words.

Definitions:

• 1) Convex area where two edges, sides, or surfaces of something join.
• 2) Concave area where two edges, sides, or surfaces of something join.
• 3) Area where two edges, sides, or surfaces of something join.

Words:

• a) Corner (English)
• b) Esquina (Spanish)
• c) Rincón (Spanish)

Exercise Solutions:

1. Convex area: Esquina
2. Concave area: Rincón
3. General area: Corner

Figure Matching:

• Fig A: Corner / Esquina
• Fig B: Corner / Rincón

Semantic Analysis: Distinctive Features

Discuss the following issue: Is "concave" a distinctive semantic feature (seme) in English? And in Spanish?

According to structuralism, distinctive features are those which help us differentiate the units within the... Continue reading "Semantic Analysis of Corner in English and Spanish" »

Core Probability Definitions and Formulas

• Independence: Events A and B are independent if P(A ∩ B) = P(A)P(B).
• Mutual Exclusivity: Events A and B are mutually exclusive if P(A ∩ B) = 0.

Key Relationships

• If P(A) + P(B) > 1, then A and B are not mutually exclusive. (True)
• If P(A) + P(B) = 1, A and B are not necessarily mutually exclusive. (False, unless A and B are complements)

Conditional Probability and Addition Rule

• Multiplication Rule: P(A ∩ B) = P(A) P(B|A).
• Conditional Probability Definition: P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0.
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Advanced Probability Relationships

Bayes' Theorem Setup

Given P(A|B) = p, P(B|A) = q, and P(A) = r. Using the definition of conditional probability:... Continue reading "Conditional Probability and Discrete Distributions" »

Limits of Sequences

We will often need to specify that a number x is close to another number a. However, this doesn’t mean anything unless we specify how close. If ε is a positive number, then the statement “x is within ε of a” does have meaning. It means that the distance between x and a is less than ε – that is, |x − a| ε.

Theorem 2.1.1

If x, y, a, and ε are real numbers with ε > 0, then (a) |y|.

Theorem 2.1.2 (Triangle Inequality)

If a and b are real numbers, then (a) |a + b| ≤ |a| + |b|; and (b) ||a| − |b| ≤ |a − b|. Proof: For part (a), we observe that −|a| ≤ a ≤ |a| and −|b| ≤ b ≤ |b|. If we add these inequalities, the result is
−(|a| + |b|) ≤ a + b ≤ |a| + |b|. By the preceding theorem (with “&... Continue reading "Understanding Limits of Sequences in Mathematics" »