Report: Financial Mathematics and Business Statistics Analysis

This report delves into the application of financial mathematics and business statistics to solve real-world business problems. It begins with a decision analysis for market expansion, evaluating international, domestic, and no-change scenarios. The report then addresses mortgage alternatives, calculating monthly costs and recommending the best option for a property purchase. A linear programming problem is formulated to optimize production mix, followed by recalculations for supplier negotiations and marketing campaigns. Investment appraisal techniques are used to select a profitable project with a shorter payback period. Finally, the report presents a statistical analysis of expected grades and formulates confidence intervals. The report provides detailed calculations, recommendations, and justifications for each scenario, demonstrating the importance of statistical and mathematical tools in making sound business decisions.

Financial Mathematics and
Business Statistics

TABLE OF CONTENTS
INTRODUCTION...........................................................................................................................1
Question 1........................................................................................................................................1
A) Decision analysis...............................................................................................................1
B) Recommendation...............................................................................................................1
Question 2........................................................................................................................................2
A) Mortgage alternatives and interest rates............................................................................2
B) Computation of monthly cost of available alternatives of mortgages...............................3
C) Computation of remaining monthly payments for mortgage............................................3
D) Recommendation on choosing desired property on financial terms.................................3
Question 3........................................................................................................................................3
A) Design linear programming problem................................................................................3
B) Recalculation for further negotiation with wood suppliers...............................................5
C) Recalculation for increasing demand through marketing campaign.................................7
D) Justify and provide recommendation on the basis of production mix...............................7
Question 4........................................................................................................................................8
A) Advantages and disadvantages of various investment appraisal techniques.....................8
B) Choosing investment project with less payback period....................................................9
C) Recommendation to chose profitable project..................................................................12
Question 5......................................................................................................................................12
A) Provide summarised distribution of expected grades......................................................12
B) Formulate confidence interval for analysing difference..................................................14
C) Expectation of actual results to have similar pattern.......................................................15
CONCLUSION..............................................................................................................................15
REFERENCES..............................................................................................................................16

INTRODUCTION
Mathematics and statistics plays crucial role in making appropriate decision so that
business may be flourished in the best possible way. The present report discusses about various
cases related to business statistics and mathematics which are required so that adequate decision
may be taken with much ease. In this report, decision analysis is carried out for expansion of
organisation in different markets which best suits to it in terms of profitability. Moreover,
mortgage costs and interest payment is calculated for purchasing property. Linear programming
problem is formulated for adopting the best production mix. Investment appraisal techniques is
carried out for selecting profitable project for investment purpose and that too having less
payback period. Statistical data is provided to show whether there is significant difference
between variables or not. Thus, all these statistical techniques and mathematical methods are
discussed for taking better decisions with much ease.
Question 1
A) Decision analysis
Pessimistic Average
approach
Optimistic Total of courses of action
International
Expansion
937.5 3795 2190 6922.5
Domestic
Expansion
1075 3850 1950 6875
No Change 1375 3382.5 1670 6427.5
B) Recommendation
The above calculations are provided to analyse better investment decision so that firm
may be able to expand operations in the best possible way. Dreamcatcher which is engaged in
retail business in games is planning to expand its market reach in various alternative courses of
action which best suits for achieving desired goals with much ease. In this study, there are three
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market scenarios which are required to be attain expansion of the firm. The three alternatives are
International expansion, domestic expansion and no change. These expansion options are carried
out by taking average of profit arrived in these markets by the business. Probabilities of varied
market options are provided to show best of them.
The profit of each of the alternative market options are provided with their respective
probabilities. Moreover, the approaches such as pessimistic, optimistic and average views are
listed which gives clarity about the effectiveness of each of the options in the best possible way.
It can be interpreted that Dreamcatcher should invest in international market as total of
probability of this investment is higher than other alternatives. The total probability of
international expansion is 6922.5 considering the views of pessimistic, optimistic and average
approaches in the given scenario. The calculation of probability is made by taking 25 % of
several views of market options.
On the other hand, domestic expansion is calculated by taking probability of 55 %
considering views is carried out which amounts to 6875. While, other option for no change is
made having probability of 20 % amounting to 6427.5. Thus, it is clarified that Dreamcatcher
should invest in international expansion of related to markets and as such, it is recommended that
adequate profits will be generated and financial strength may be enhanced in a better way.
Question 2
A) Mortgage alternatives and interest rates
There are various alternatives of mortgages Kyle can easily avail from the money
available with him. He can easily take low operating costs of interest so that he may chose homes
with adequate mortgage interest rates. There are alternatives of mortgage such as seller
financing, hard money lenders and loan taken against insurance policy. This will help him to
avail adequate mortgage for making right decision and also cost-effective one for him. Various
alternatives with different interest rates are as follows-
Seller financing
This is useful alternative to Kyle to avail low operating interest rates. Seller financing is
means that seller of the property agrees to lend his home and in return buyer of the property signs
a promissory note such as mortgage deed which carries obligation to repay the loan amount in
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instalments. If buyer of the mortgage property defaults on repaying loan, then seller can take
possession of his property. The applicable interest rate on seller financing is 6 % on this type of
alternative available to Kyle.
Hard money lenders
This is another better option available to Kyle which can easily take for alternative
against mortgage. Hard money lenders provide loan to firm or individuals which is backed by
some real estate. The main lenders in this category are banks and credit unions imparting such
facility. There can be a mortgage broker which provides agreement between lender and
borrower. He brings all required documentation for making out the deal between them in the best
possible way. However, they are different from banks as various Basel norms are required to be
followed which leads to increment on service costs. Increment in expenditures leads to hefty
interest payment. Thus, hard money lenders provides loan at lower interest rates and that too at
monthly payment of loan along with the interest accrued on it. The interest rate applicable on this
alternative of mortgage is around 9 % and Kyle can avail this loan.
Loan taken against insurance policy
This is another useful interest rate policy which may be helpful for Kyle to avail loan
against any insurance policy. This option is available at various insurance policies such as
endowment plans, money back policies and ULIP (Unit Linked Insurance Policy). The main
essence of getting loans against insurance policy is that person need not to repay amount of loan
on its whole life insurance policy. The heirs of the policy will get amount of the policy reduced
up to the loan availed. Thus, it will be better for Kyle to not to repay loan amount as it will be
deducted afterwards on the insurance policy. The interest rate applicable on this policy is around
9 %.
B) Computation of monthly cost of available alternatives of mortgages
For seller financing
= 350000/{[(1+6%)^20]- 1}[1+6%)^20
= 109131 /12
= 9094.25
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For hard money lenders
= 155902/12
= 12991.83
C) Computation of remaining monthly payments for mortgage
D) Recommendation on choosing desired property on financial terms
It is recommended that seller financing option should be availed by Kyle as it has lest
interest
Question 3
A) Design linear programming problem
Particulars of
Tables
Square Round Rectangular Elliptic Maximum Available
Price £600.00 £700.00 £900.00 £1,050.
00
Wood table 11 12 52,000
Metal table 7 8 36,000
Other
Materials
6 4 9 8 48,500
Labour hours 1.5 2.75 2 3.5 20,000
Maximum
Demand
3,500 3,000 1,500 2,000
Square Round Rectangular Elliptic
Selling Price of tables £600.00 £700.00 £900.00 £1,050.00
Associated Variable
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cost
Wood table £165.00 £180.00
Metal table £140.00 £160.00
Other material £60.00 £40.00 £90.00 £80.00
Labour £37.50 £68.75 £50.00 £87.50
Total variable cost £237.50 £273.75 £300.00 £347.50
Contribution per unit
(CPU)
£362.50 £426.25 £600.00 £702.50
Square Round Rectang
ular
Elliptic Require
ments
Constrain
ts
Demand of the tables 1,470 2,545 1,500 2,000
Associated Variable
cost
Wood table 0 11 0 12
Metal table 7 0 8 0
Other material 6 4 9 8
Labour 2 3 2 4
Wood table 0 28,000 0 24,000 52,000 52,000
Metal table 10,288 0 12,000 0 22,288 36000
Other material 8,818 10,182 13,500 16,000 22,288 36,000
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Labour 2,205 7,000 3,000 7,000 48,500 48,500
Contribution per unit
(CPU)
362.50 426.25 600.00 702.50 19,205 20,000
Contribution 532,765 1,085,00
0
900,000 1,405,000 3,922,765
The linear programming problem is being provided which is associated with various
constraints provided in the case of furniture manufacturer. It can be said that manufacturer has
various production price of constraints and requirements of each of the varieties of tables.
Moreover, there are price, labour hours associated with the production. There are wood, metal
and other materials categories having the constraints and requirements of each of them which is
available to furniture manufacturer.
There are various shapes of tables such as Square, Round, Rectangular and Elliptic
categories. It can be analysed that demand of Round category is 2545 which is higher among the
other categories. This is evident from the fact that Square has 1470 demand, Rectangular has
1500 in terms of demand and last category Elliptic has 2000. These figures are arrived by the
requirements provided by the suppliers. In terms of wood table, the shape of Round and Elliptic
such as 28000 and 24000 are segregated which amounts to 52000 of the total requirements of the
manufacturer.
B) Recalculation for further negotiation with wood suppliers
Square Round Rectang
ular
Elliptic Requireme
nts
Constraints
Demand of
the tables
1,288 2,818 1,500 2,000
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Associated
Variable cost
Wood table - 11 - 12
Metal table 7 - 8 -
Other
material
6 4 9 8
Labour 2 3 2 4
Wood table - 31,000 - 24,000 55,000 55,000
Metal table 9,015 - 12,000 - 21,015 36,000
Other
material
7,727 11,273 13,500 16,000 48,500 48,500
Labour 1,932 7,750 3,000 7,000 19,682 20,000
Contribution
per unit
(CPU)
363 426 600 703
Contribution 466,856 1,201,250 900,000 1,405,000 3,973,106
It can be analysed that marketing campaign has been successfully done by the furniture
manufacturer and as such, it is able to negotiate wooden table to 55000 in the best possible way.
In relation to this, wooden table has increased and requirements was maximised by almost 3000
units. This is obtained by increasing Round and Elliptic by 31000 and 24000 respectively which
has provided the requirements of 55000 units quite easily. By calculating the total requirement of
the tables, contribution per unit is arrived at 3973106 after negotiating with the suppliers.
Moreover, constraints are also 55000 in the total contribution
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C) Recalculation for increasing demand through marketing campaign
Square Round Rectang
ular
Elliptic Requirements Constrain
ts
Demand of the
tables
1,167 2,000 1,500 2,500
Associated
Variable cost
Wood table 11 12
Metal table 7 8
Other material 6 4 9 8
Labour 2 3 2 4
Wood table - 22,000 - 30,000 52,000 52,000
Metal table 8,167 - 12,000 - 20,167 36,000
Other materials 7,000 8,000 13,500 20,000 48,500 48,500
Labour 1,750 5,500 3,000 8,750 19,000 20,000
Contribution per
unit (CPU)
362.50 426.50 600.00 702.50
Contribution 422,917 853,000 900,000 1,756,250 3,932,167
It can be interpreted that Elliptic category has been increased to 2500 units while earlier it
was 2000 units. Moreover, the requirements also changed in the wooden table as Round and
Elliptic consists of 22000 and 30000 which has arrived the total requirement of 52000 units.
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D) Justify and provide recommendation on the basis of production mix
It can be justified from the above formulated linear programming problem that
production mix should be adopted which has maximum contribution. It can be analysed that
maximum contribution is obtained in the second option of the production which is related to
increasing 55000 units of wooden by implementing marketing mix in the best possible way. This
production mix is quite good as total contribution obtained here is 3973106 which is the highest
among various different options. All the categories such as wood, metal having various sub-
categories are adequate in this production mix. In addition to this, labour hours are also adequate
in it. Thus, it is recommended that furniture manufacturer should use this production mix as it
has maximum contribution and higher production may be achieved following this mix.
Question 4
A) Advantages and disadvantages of various investment appraisal techniques
NPV
NPV (Net Present Value) is useful investment appraisal technique which evaluates
effectiveness and attractiveness of the project in terms of profitability. This provides how much
return will be generated in accordance with the present and future value of cash flows. Higher
the NPV, better for the company to invest in the project. It will be helpful for Pierce Plc to assess
effectiveness of Product X or Product Y.
Advantages
1. The main advantage of NPV is that it considers time value of money while assessing
effectiveness of the project quite easily
2. Another advantage is that cost of capital is being used which help to assess risk
associated with the project.
Disadvantages
1. It is not suitable as it is complex to calculate and interpret results. Moreover,
discounting rate is harder to calculate.
2. It is not useful when there are two mutually exclusive projects having different time
period.
IRR (Internal Rate of Return)
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IRR is quite effective way of evaluating project as it involves estimating profitability of
the new project. It is discounting rate which makes NPV of cash flows equal to zero. Higher the
IRR, better fort the company to invest in the project as it carries good rate of return on it.
Advantages
1. It is easy to calculate and obtain results and also it does not ignore the concept of time
value of money to assess project.
2. It assesses profitability of new project by considering whole life of the project's cash
flows. Thus, better evaluation is obtained.
Disadvantages
1. It is not useful as it considers only profitability aspect of the project and does not
consider when return will be produced to reoccupy investment money.
2. It provides conflicting results when two projects have different time and size of the
investment.
Payback Period
Payback period is quite useful technique to assess project's attractiveness in terms of
yielding return within less time frame. It is recommended by market analysts that less the
payback period, investment should be made by the company to attain results.
Advantages
1. It is valuable concept which evaluating how much time project will require to yield
results. Moreover, it provides more liquidity to assess effectiveness of project.
2. It provides risk associated with new project which is the main essence of payback
period.
Disadvantages
1. It is not suitable as it ignores time value of money.
2. It ignores profitability aspect of project while evaluating it.
B) Choosing investment project with less payback period
Year Product Y Present
value @
Discounted cash
flows (DCF) (Y)
10