Financial Mathematics and Business Statistics Project - Semester 1

This project delves into financial mathematics and business statistics, addressing real-world scenarios through various analytical techniques. It begins with a decision analysis for a game retailer, evaluating international and domestic expansion strategies under different market scenarios. The project then tackles mortgage calculations, comparing interest rates and payment options. Linear programming is applied to optimize production for a furniture manufacturer, considering resource constraints and marketing campaigns. Finally, investment appraisal methods are used to evaluate new product development strategies for Pierce Plc. The assignment provides detailed calculations, recommendations, and justifications for each problem, offering a comprehensive understanding of financial modeling and statistical analysis.

Financial Mathematics and Business
Statistics

Table of Contents
INTRODUCTION...........................................................................................................................1
QUESTION 1..................................................................................................................................1
A. Decision Analysis of three alternative actions........................................................................1
B. Recommendation to Dreamcatcher with arguments...............................................................2
QUESTION 2..................................................................................................................................3
A. Identify three best opportunities..............................................................................................3
B. Monthly payment....................................................................................................................3
C. Partial monthly payment for remaining 18 years....................................................................4
D. Recommendations...................................................................................................................4
QUESTION 3..................................................................................................................................5
A. Formulating the problem as linear program and using Excel’s solver for solution................5
B. Recalculate the solution in case of renegotiation with supplier..............................................7
C. Recalculation of the solution in case of marketing campaign.................................................8
D. Short report, description and justifying the formulation with recommendations...................8
QUESTION 4..................................................................................................................................9
A. Methods of investment appraisal............................................................................................9
B. Application of investment appraisal methods.......................................................................10
C. Recommendations to Pierce Plc............................................................................................12
QUESTION 5................................................................................................................................13
A. Summarizing the distribution of expected grades for both exams and key characteristics...13
B. Statistical Test.......................................................................................................................15
C. Results...................................................................................................................................16
CONCLUSION..............................................................................................................................16
REFERENCES..............................................................................................................................17

INTRODUCTION
Rising the uncertainty and complexities in the market requires thorough examination of
necessary data set and predict future applying various mathematical and statistical tools and
techniques. In the practical world, every company applies various analytical methods and tools to
look at the past trends and determines future opportunity to stay competitively strong. This
assignment is based on a number of case problems which will be resolved using distinctive
techniques like linear programming, decision-tree, investment appraisal techniques and statistical
methods. Sound knowledge and the application of such tools will helps to determine the most
preferable and justifiable option that is more profitable and contributes towards long-term
success.
QUESTION 1
A. Decision Analysis of three alternative actions
Dreamcatcher is a game retailer that was established five year ago and had showed strong
performance since establishment. Therefore, now owners are looking for expanding the presence
for the prospective growth. After undertaking industry analysis and their own position in relation
to key market players, company had found three possible courses of actions with three possible
scenarios. Managers want to know the best profitable alternative in which they must proceed. It
can be done with the analysis through decision model that presents all the alternatives with their
prospective probability to come to a final point that gives maximum profit (Anderson and
et.al.,2016).
Scena
rios
Probabil
ity
Project (Profit in GBP 000)
Internatio
nal
Expansio
n
Expected
profitabilit
y
Domestic
Expansion
Expected
profitabilit
y
No
change
Expected
profitabilit
y
Pessi
misti
c 25% 3750 937.5 4300 1075 5500 1375
Aver
age 55% 6900 3795 7000 3850 6150 3382.5
Opti
misti
20% 10950 2190 9750 1950 8350 1670

c
6922.5 6875 6427.5
Working note:
Expected profitability under different market scenarios
¿ Profitability∗Probability
Total expected profitability under different alternatives:
1. International Expansion: (£3,750*0.25) + (£6,900*0.55) + (£10,950*20%) = £6,922.5
2. Domestic Expansion: (£4,300*0.25) + (£7,000*0.55) + (£1,950*20%) = £6,875
3. No Change: (£5,500*0.25) + (£6,150*0.55) + (£8,350*20%) = £6,427.5
B. Recommendation to Dreamcatcher with arguments
There are three alternatives available to the Dreamcatcher that is either to expand
business globally or domestically or do not expand. In every alternative, there is a possibility of
three different market scenarios, pessimistic, optimistic and average with having different
probability of 25%, 20% and 55% respectively. Considering the outcome of the analysis, it is
seen that in the alternative of international expansion, total expected profitability is greater than
the other alternative options. In the global expansion project, Dreamcatcher is expected to get
total return of £6,922.5 whereas if it expands business domestically then the return is £6,875 and
if it does not make any change then the profit is just £6427.5.
The main reason behind this is by expanding the operations of a business globally means
on a large scale, Dreamcatcher would be able to generate a greater number of customers and can
serve more people. As a result, it will significantly contribute towards maximizing its total sales.
Moreover, through delivering its game services to maximum number of buyers, it can get scale-
based benefits that will helps to minimize cost each unit and attract high return. However,
domestic expansion attract less number of buyers hence, although sales will be maximize but to a
less extent than international expansion (Howell, 2016). In despite of this, if neither it expands
globally nor domestically then un-doubtedly, the total expected profitability will be
comparatively less than other two options. As per the results, although in pessimistic and average
market scenario, domestic expansion shows greater yield, still, high profit prediction in

optimistic market in the alternative of international expansion provided maximum net return of
£69,225,000 that suggest Dreamcatcher to go for internationalization strategy and expand their
business in the global market place. It will help building an international reputation and improve
its competitiveness. Hence, it can be said that international expansion is an optimal course of
action.
QUESTION 2
A. Identify three best opportunities
HSBC UK: Initial base rate: 1.24%
Subsequent rates: 3.94%
Overall interest rate: 3.7%
No need to pay application fee, arrangement fee, product fee and completion fee
Barclays: 1.33% fixed for 2 year
Subsequent years: 3.99% variable
Overall rate of interest: 3.6%
If the mortgage will be paid early than 3% of the amounts paid will be charged
Chelsea Building Society
Interest rate: 1.49% fixed
Subsequent years: 4.99% variable
Overall interest rate: 4.5%
Early payment of mortgage resulted in charge of 2.5%
B. Monthly payment
Most favorable rate: 3.6% by Barclays
Monthly Payment on the mortgage on one bedroom

¿ £ 350,000/( {[(1+ 3.66 %)20 ]−1 }/[3.66 %(1+3.66 %)20])
¿ 24,984.25
Mont hly payment =£ 24,984.25/12
¿ £ 2082
Monthly Payment on the mortgage on three-bedroom house
¿ £ 500,000/( {[(1+ 3.66 %)20 ]−1 }/[3.66 %(1+3.66 % )20])
¿ 35,691.79
Mont hly payment on mortgage=35,691.79/12 mont h s
¿ 2,974
C. Partial monthly payment for remaining 18 years
One-off partial payment worth 75000 on loan
One−bedroom house
Remaining period :18 years
Interest rate :3.6 %
Monthly mortgage payment :£ 1,731.73
Three−bedroom house
Monthly mortgage payment :£ 2,676.31
As per the results, on the second mortgage, it will need to pay greater payment while on
the one-bedroom house, mortgage payment would be £1,731.
D. Recommendations
As per the decisions, it is recommended to the Kyle to take mortgage from Barclays due
to lower rate of interest. Moreover, he must make partial payment of 75,000 on the loan and in
order to purchase the one-bedroom house, he would be requires to pay £1731 and to purchase

three-bedroom house, he would b required to pay £2,676 due to high amount of mortgage
(Fabozzi, 2016). It is totally based upon the financial conditions. if he is able to make high
payment and is in good financial conditions, then he must go for the second mortgage otherwise,
first seems more preferable.
QUESTION 3
A. Formulating the problem as linear program and using Excel’s solver for solution
Linear programming is a mathematical programming which is used by many companies
to maximize their profit or minimize their total cost. The model perform necessary calculations
to find out the point at which company will have maximum return or least cost subjected to
various constraints or scarce availability of resources (Dantzig, 2016). As per the scenario, a
furniture manufacturer recently noticed falling in their profit in the Tables product line.
Therefore, company wants to identify that which models are more profitable that must be
concentrated. It produces four different kinds of tables using different material but it is subjected
with key constraints because of limited availability of resources. The problem can be solved
using linear programming modeling as follows:
Linear programming equation:
Maximum function:
U 1C 1+ U 2 C 2+U 3 C 3+U 4 C 4
Constraints:
U 1W 1+U 2 W 2+ U 3 W 3+U 4 W 4 ≤ 52,000
U 1 M 1+U 2 M 2+ U 3 M 3+U 4 M 4 ≤ 36,000
U 1O 1+U 2O 2+U 3 O3+U 4 O 4 ≤ 48,500
U 1 L 1+U 2 L 2+U 3 L 3+U 4 L 4 ≤ 20,000
U 1≤ 3,500 units
U 2≤ 3,000 units

U 3 ≤1,500 units
U 4 ≤2,000 units
Here: C: Contribution
W: Wood
M: Metal
O: Other material
L: Labor
Calculation of contribution per unit
Square Round Rectangular Elliptic
Selling Price £600.00 £700.00 £900.00 £1,050.00
Variable cost
Wood £165.00 £180.00
Metal £140.00 £160.00
Other material
£60.00 £40.00 £90.00 £80.00
Labor £37.50 £68.75 £50.00 £87.50
Total variable cost £237.50 £273.75 £300.00 £347.50
Contribution per unit £362.50 £426.25 £600.00 £702.50
Maximum function:
3 , 500∗362.50+3 ,000∗426.25+1,500∗600+2,000∗702.50
Constraints:
3,500∗0+3,000∗11+ 1,500∗0+ 2,000∗12 ≤52,000
3,500∗7+3,000∗0+1,500∗8+2,000∗0 ≤36,000
3,500∗6+3,000∗4 +1,500∗9+2,000∗8≤ 48,500
3,500∗1.5+3,000∗2.75+ 1,500∗2+2,000∗3.50 ≤20,000

Square Round Rectangular Elliptic
Requirement
s Constraints
Demand 1,470 2,545 1,500 2,000
Wood 0 28,000 0 24,000 52,000 52,000
Metal 10,288 0 12,000 0
Other material 8,818 10,182 13,500 16,000 22,288 36,000
Labor 2,205 7,000 3,000 7,000 48,500 48,500
Contribution
per unit 362.50 426.25 600.00 702.50 19,205 20,000
Total
contribution 532,765
1,085,00
0 900,000
1,405,00
0 3,922,765
After solving the linear programming equation in the Excel solver, it is discovered that as
per current situation, furniture manufacturer must produce rectangular and Elliptic products in
full capacity. It means these items must be produce in 1,500 and 2,000 units because it drive
highest contribution of £600 and £702.50 each unit. However, remaining tables, Square and
Round should be produce in 1,470 and 2,545 units. At this level, firm will achieve maximum
contributed amounted to £3,922,765.
B. Recalculate the solution in case of renegotiation with supplier
If firm renegotiate with the wood supplier and able to maximize its supply from 52,000 to
55,000 than only first equation constraints value will be change as follows:
3,500∗0+3,000∗11+ 1,500∗0+ 2,000∗12 ≤55,000
Square Round Rectangular Elliptic Requirements Constraints
Demand 1
,288
2,
818
1,500 2,
000
Wood
-
31,
000
- 24,
000
55,000 55,000
Metal 9
,015 -
12,000
-
21,015 36,000
Other
material
7
,727
11,
273
13,500 16,
000
48,500 48,500
Labor 1
,932
7,
750
3,000 7,
000
19,682 20,000
Contributio
n per unit 363 426
600
703
Total
contribution
466
,856
1,201,
250
900,000 1,405,
000
3,973,106

By renegotiation with the supplier, availability of quantum of wood will be increase. As
Rectangular and Elliptic products already produced to the extent of full demand therefore, it does
not change. However, production volume of Round table will be increase to 2,818 units which
shows that after allocating 24,000 units of wood to Elliptic, rest 31,000 units will be used in the
production of Round table. As a result, total contribution will be increase from £3,922,765 to
£3,973,106 evidence greater profitability (Vanderbei, 2015).
C. Recalculation of the solution in case of marketing campaign
If a marketing campaign is likely to be conducted in future which will increase market
demand for Elliptic tables to 2,500 units, then constraints about maximum demand of the
mentioned table will be change as follows:
U 4 ≤2,000 units
Square Round Rectangular Elliptic
Requirem
ents
Constr
aints
Demand 1,167 2,000 1,500 2,500
Variable cost
Wood - 22,000 - 30,000 52,000 52,000
Metal 8,167 - 12,000 - 20,167 36,000
Other material 7,000 8,000 13,500 20,000 48,500 48,500
Labor 1,750 5,500 3,000 8,750 19,000 20,000
Contribution per
unit 362.50 426.50 600.00 702.50
Total
contribution 422,917 853,000 900,000 1,756,250 3,932,167
The outcome reveals that by raising the demand for Elliptic table from 2000 to 2500
units, company must take the benefits of huge demand and produce 500 more units of it which
will increase overall contribution by £9,402 and reached to £3,932,167 (Rardin and Rardin,
2016). Thus, no-doubt, firm must spend on marketing less than the increase in profit of £9402.
D. Short report, description and justifying the formulation with recommendations
The problem of linear programming solved in Excel putting maximum, changing sale
rule and constraints and the results are summarized below:
Situation Square Round Rectangular Elliptic TC
A
1,470 2,545 1,500 2,000
3,922,765
B 1,288 2,818 1,500 2,000 3,973,106

C
1,167 2,000 1,500 2,500
3,932,167
The results shows that option B is more suitable because it will gain maximum yield of
£3,973,106, thus, it is better to suggest the manufacturer to renegotiate with the supplier and
raise the available wood supply to 55,000 units.
QUESTION 4
A. Methods of investment appraisal
Pierce Plc is looking for applying new product development strategy by manufacturing
two products, X and Y, both requires an equal initial investment of £5 million. However, due to
financial constraints, company can only developed one of these. Out of these, product X has
longer life cycle of 10 years while product Y just need 5 years. The problem can be solved
through investment appraisal techniques which are of great importance to assess the
attractiveness of an investment proposal.
Payback period: It is the simplest method that is useful to determine that how long a
project is expected to take to recoup its initial investment (Brooks, 2015). Pierce can use the
techniques to find out that in which project, it will recover its cost of investment quicker before
other.
Advantages
 Determines recovery period of initial outlay
 Easy calculation
 Easy to understand
Demerits
 Although the technique is useful to determine recoup period, still, does not has too much
practical significance.
 The method is criticized because it does not look at the cash flows in the subsequent
years and also does not consider time value of currency (Siegel, 2016).
 Does not consider time value of money
 Does not takes into account the pattern of the cash flows

Net Present Value: Another discounted method of capital budgeting is determining net
present value that represent the amount of present value of cash flows left over after deducting
beginning outlay (Anderson, Sweeney and Williams, 2014).
Advantages:
 Time value of money is considered
 Helps to increase firm’s value
 It considers risk inherent while projecting future
Disadvantages:
 Difficult to identify the correct discounting factor
 Sudden volatility affects future cash flows projections and the NPV as well (Andor,
Mohanty and Toth, 2015).
Internal rate of return: It is measured by determining the rate of interest at which sum of
discounted cash flows became equivalent to the beginning outlay means zero NPV (Baum and
Crosby, 2014).
Advantages:
 Time value of money is considered
 Simple way of measurement
 Simplicity
Disadvantages:
 Do not require hurdle rate
 Ignores project side
B. Application of investment appraisal methods
Payback period
Year Product X Product Y Cumulative
cash flows
Cumulative
cash flows
1 500 1,000 500 1,000
2 750 2,000 1250 3,000