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

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" »

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" »

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" »