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

1) What is an Inverse Matrix and How Do You Calculate It?

A matrix A^-1 is called the inverse matrix of a matrix A (nxn) if AxA^-1= A^-1xA=E (where E is the identity unit matrix).

We calculate it by performing row operations on the augmented matrix (A | I) to transform it into (I | B). If this reduction is possible, then B=A^-1, which is the inverse matrix of A.

2) Define the Model of a Game

Games can be modeled in various forms:

• Tree Form Model (Game Tree): Represents the game as a sequence of decisions (moves) made by players.
• Normal Form Model: Represents the game using:
  • List of players
  • List of strategy spaces for each player
  • List of payoff functions (decision matrix) defining outcomes for each combination of strategies.
• Characteristic Function Form:

KEYS

1. You must be wondering why I haven’t phoned you.

2. The heavy rain prevented us from having the picnic we had planned.

3. Recently, there has been a rise in the number of people who study a degree.

4. It was a matter of minutes before the police came.

5. Can anyone come up with a solution to this problem?

6. Are you accusing me of not telling the truth about what happened?

7. I haven’t got round to phoning you, but I’ll do it soon.

8. Economists say that there is no doubt that the situation will get better.

9. It was my mother who/that got me interested in reading.

10. What is that’s causing you such a lot of confusion?

11. She made a very quick decision without giving enough thought to the matter.

12. It was a while until/ till/ before... Continue reading "Common English Phrases and Expressions for Language Learners" »

Introduce Research

• Study design/methods
• Appraisal
• Interpretation
• Application/utilization

Understand

• Evidence Informed Decision Making Model
• Sources of Evidence
• Types of research that inform practice (quantitative, qualitative, mixed methods)
• Why research studies should be critically appraised

1.2 Research, EBP & EIDM

• There are different ways of knowing; Empirical, Personal, Aesthetics, Ethical

1. Empirical (focus of class)

• Theories, models, facts
• Validation, confirmation
• Scientific competence

2. Personal knowledge

• Person stories, self
• Reflection, response
• Therapeutic use of self

3. Aesthetics knowledge

• Experience of nursing, health, illness
• Appreciation, grasp meaning
• Transformative act/acts

4. Ethical Knowledge

• Principles, codes
• Justification, dialogue

Evidence-Based Practice... Continue reading "HTHSCI 3CO4 Cheat Sheet" »

Chapter 6 Review: Bond Valuation Problems

1. Coupon Rate Calculation

   A $1,000 face value bond is currently quoted at 100.8. The bond pays semiannual payments of $22.50 each and matures in six years. What is the coupon rate?

   Coupon rate = ($22.50 × 2) / $1,000 = 0.0450, or 4.50 percent

2. Current Price of a Bond

   A $1,000 face value bond currently has a yield to maturity of 8.22 percent. The bond matures in five years and pays interest semiannually. The coupon rate is 7.5 percent. What is the current price of this bond?

   FV = 1000; PMT = 75/2 = 37.5; N = 5 x 2 = 10; I/Y = 8.22 / 2 = 4.11 => PV = 970.96

3. Determining Coupon Rate

   A six-year, semiannual coupon bond is selling for $991.38. The bond has a face value of $1,000 and a yield to maturity of 9.19

Distribution of Moody's Ratings

Historical Default Rates

• Rating Agencies
• Default rates from bonds
• Market Data

CAPM Model for Bond Returns

The usual CAPM has to be modified to include expected loss per year.

What must the β be to justify the excess spread for Baa Bonds?

Merton's Model

A Structural Model That Predicts Default Rates

How Do We Find V₀ and σ_V?

We know E₀ and σ_E, and

An application of Ito's lemma shows

These allow us to find V₀ and σ_V from E₀ and σ_E.

Default Correlation

Defaults: If Firm A Defaults, is Firm B More Likely to Default Too?

No Default Correlation

Consider two firms, X and Y:

Q_X = 0.1, Q_Y = 0.2

If X and Y are uncorrelated, then the probability that they both default is Q_X × Q_Y = 0.02

Building Default Correlation

If X and Y are correlated,... Continue reading "Understanding Bond Default Rates and Correlation" »

Final Exam Cheat Sheet

Machine Rate Problems

Chapter 1: Categorical (Qualitative) Data

What is Categorical Data?

Categorical data describes the qualities of individuals or objects, rather than quantities. It's about characteristics that can be grouped into categories.

Examples of Categorical Data:

• Hair Color
• Preferred Clothing Brand
• Nationality

Visualizing Categorical Data:

Categorical data is often displayed using:

• Bar Charts
• Pie Charts
• Tables

Chapter 2: Quantitative Data

What is Quantitative Data?

Quantitative data deals with numbers and measurements. It allows for comparisons and mathematical operations.

Examples of Quantitative Data:

• Height
• Weight
• Age

Visualizing Quantitative Data:

Quantitative data is often presented using:

• Dot Plots
• Stem Plots
• Histograms

Measures of Center

• Mode: The most frequent value