Notes, abstracts, papers, exams and problems of Mathematics

Management of Interest Rate Risk

Major Risks in Bond Market

A) Interest Rate Risk (change in market prices of bonds due to varying interest rates)

• Increase in rates, decrease in market prices, increase in reinvestment rate risk (coupons reinvested at lower return)

B) Reinvestment Rate Risk (uncertainty of rate at which interim cash flows can be reinvested)

• High coupon rate, high reinvestment rate risk
• Greater for longer holding periods (high interim cash flows)

C) Default Risk (credit risk) (issuer unable/unwilling to pay interest and principal of bond)

• High credit rating, lower yield
• Short-term T-bills (almost risk-free, no default risk, and low return and reinvestment risk because of short duration)

D) Call Risk (risk bond issuer will redeem bond before... Continue reading "Interest Rate Risk Management: A Comprehensive Guide for Investors" »

Product Information and Constraints

Fran Farnsworth sells three health care products: Flex, Feelgood, and Firmup. Her costs, sales time, delivery costs, and commission for each product are as follows:

• Flex: $3 cost, 10 minutes to sell, $0.50 delivery cost, $2 commission
• Feelgood: $5 cost, 20 minutes to sell, no delivery cost, $4 commission
• Firmup: $4 cost, 12 minutes to sell, $1 delivery cost, $3 commission

Fran's weekly constraints are:

• Maximum $1500 worth of products
• Maximum $85 delivery expense
• Maximum 40 hours (2400 minutes) for sales

Linear Programming Model

Let x, y, and z represent the number of boxes of Flex, Feelgood, and Firmup sold per week, respectively. Let C be Fran's weekly commission. The LP model is:

Maximize:

C = 2x + 4y + 3z

Subject

What is a Shapefile?

A shapefile is an Esri vector data storage format for storing the location, shape, and attributes of geographic features. It is stored as a set of related files and contains one feature class. Google Earth, an Online Satellite Imagery (OSI) Technique, is a tool for generating shapefiles.

Geometric Analysis Tools in Projects

Example 1: Population Maps of To Kwa Wan using ArcGIS

We used ArcGIS to create population maps of To Kwa Wan.

1. Selected population and working population data from the C&SD department.
2. Opened TPUs with the population data in ArcGIS.
3. Calculated the area of the TPUs based on Hong Kong 1980 Grid, determining population densities.
4. Created maps with different population densities, using graduated colors to represent

Formal letter

We are writing in reference to your new products advised in March 23 edition of “Info Globe”. We are a software company based in Berlin which specializes in Games Development.

Our company is currently developing a new software to improve old games quality. We are interested in designing a new video game console which would works with old games with a better graphics quality.

 We would like to request a catalogue of your lasted collection that was advised in Info Globs. Please let us know if you would be interested in scheduling a meeting.

 I look forward to hearing from you.

Yours sincerely, M Walton

Faculty and Student Responses

A university president proposed that all students must take a course in ethics as a requirement for graduation. The table below gives the responses of a sample of faculty and students at the university when asked their opinion on this issue.

Y: FAVOR O: Oppose N: Neutral

Totals

F: Faculty         45             15             10                    70

S: Student        90             110             30                  ₂₃₀ 

 Totals   135               125                40

Probabilities

1. Find the following probabilities; each time ONE person is being randomly selected.

P(S or Y) = (230 + 135 – 90)/... Continue reading "Calculating Probabilities from a Table: Ethics Course Requirement" »

Systematic Sampling: Type of Probability Sampling Method

In systematic sampling, sample members from a larger population are selected according to a random starting point but with a fixed, periodic interval. The sampling interval, calculated by dividing the population size by the desired sample size, determines the selection.

Oversampling: Techniques to Adjust Class Distribution

Oversampling is a data analysis technique used to adjust the class distribution of a data set, ensuring a balanced representation of different classes/categories.

P Value: Probability of Obtaining Extreme Test Results

The p value represents the probability of obtaining test results at least as extreme as the observed results during the test, assuming the null hypothesis... Continue reading "Understanding Systematic Sampling and Statistical Significance" »

Worksheet: Revisiting Calculating Probabilities from a Table

1. Goals

P(F) = 251/478 = 0.525

The probability that a randomly selected student is a girl is 0.525.

P(P^C) = (247 + 90)/478 = (478 – 141)/478 = 337/478 = 0.705

The probability that a randomly selected student did not indicate being popular as a primary goal is 0.705.

P(B or S) = (227 + 90 – 60)/478 = 257/478 = 0.538

The probability that a randomly selected student is a boy or indicated a primary goal is to excel at sports or both is 0.538.

P(B and S) = 60/478 = 0.126

The probability that a randomly selected student is a boy and indicated a primary goal is to excel at sports is 0.126.

P(S|B) = 60/227 =... Continue reading "Calculating Probabilities from a Table" »

In the examples C = {1,2,3,4} and D = {3,4,5}

SymbolMeaningExample{ }Set: a collection of elements{1,2,3,4}A ∪ BUnion: in A or B (or both)C ∪ D = {1,2,3,4,5}A ∩ BIntersection: in both A and BC ∩ D = {3,4}A ⊆ BSubset: A has some (or all) elements of B{3,4,5} ⊆ DA ⊂ BProper Subset: A has some elements of B{3,5} ⊂ DA ⊄ BNot a Subset: A is not a subset of B{1,6} ⊄ CA ⊇ BSuperset: A has same elements as B, or more{1,2,3} ⊇ {1,2,3}A ⊃ BProper Superset: A has B's elements and more{1,2,3,4} ⊃ {1,2,3}A ⊅ BNot a Superset: A is not a superset of B{1,2,6} ⊅ {1,9}A^cComplement: elements not in AD^c = {1,2,6,7}
When  = {1,2,3,4,5,6,7}A − BDifference: in A but not in B{1,2,3,4} − {3,4} =

By the Vertical Angles Congruence Theorem (Theorem 2.6), m∠4 = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines.

m∠4 + (x + 5)° = 180° Consecutive Interior Angles Theorem

115° + (x + 5)° = 180° Substitute 115° for m∠4.
x + 120 = 180 Combine like terms.

x = 60 Subtract 120 from each side.7

By the Alternate Exterior Angles Theorem, m∠8 = 120°.
∠5 and ∠8 are vertical angles. Using the Vertical Angles Congruence Theorem
(Theorem 2.6), m∠5 = 120°.
∠5 and ∠4 are alternate interior angles. By the Alternate Interior Angles Theorem,
∠4 = 120°. So, the three angles that each have a measure of 120° are ∠4, ∠5, and ∠8.

By the Linear Pair Postulate (Postulate 2.8), m∠1 = 180° − 136° = 44°.