Mathematical Methods in Finance Solutions

Introduction

The aim of this project was learn how to use scientific and mathematical methods to provide an optimum solution to a realistic financial problem. In order to do this the information has to be put together in a structured way so that the desired outcome can be maximized by altering inputs subject to a series of constraints. To find the best possible combination of allocations without an optimizing tool like Solver would be almost impossible. The key steps were firstly understanding the problem, then formalizing it in a way that it could be solved using an optimizing tool. Having done this the requirement was to present the whole project in a consultancy type format.

Background

Vista Properties has purchased 140,000 square metres of land on which a shopping centre is to be built, and it has an option to buy an additional 20,000 square metres of adjacent land. It wants to know how to allocate the space on the land it has already purchased and whether this option should be exercised.

The problem regarding allocation is one of choosing the mix of shop types and floor areas will give the best financial results over a 7 year period. This has to be done within several constraints on minimum and maximum sizes of shops and financial targets.

Method

The Midvale Shopping Centre project has already been agreed, so this was tackled first. The option to purchase 20,000 square metres of adjacent land was then tackled.

Midvale Shopping Centre

The issue was how to allocate the floor-space available between the various types of shop in a way that would be most beneficial to Vista Properties. The method recommended is based on finding the highest level of net present value (NPV) less costs not already included in net present value. That is, less improvement and construction costs. This is referred to as profit in the remaining text. The net present value figures supplied include fixed charges. Mr Wasser had challenged whether fixed charges should be included. He was quite right to do so as the model assumes that all the component charges vary with floor space within each shop type. Fixed charges are however constant and should be modelled as such.

I order to calculate the fixed charges that have been included, the net present value of 7 years fixed cost was found to be £608,522. This was divided by floor area of 45,000 as it is assumed that the accountant would have done this. This gave a figure of £13.52 to add back for each shop type. In Excel a column was set up with the size of floor space available to allocate to each shop type. This is the column that Solver could change to achieve the best profit, subject to the various constraints. The profit was found by first multiplying all floor space allocations by the adjusted NPV’s and then subtracting construction and improvement costs. Construction costs were £6 times the total floor space in metres. Improvement costs were the sum of each shop floor space times the individual shop type cost of improvement.

Constraints were applied as follows:

• All floor spaces must be greater or equal to zero (otherwise Solver can allocate negative floor spaces).
• The guaranteed rent, sum of rent times floor space for 7 years must be greater than the cost of improvements plus interest at 10%.

The sum of floor-spaces within each group must be greater than the group minimum. For example in Group A the sum of Supermarket an Dept. Store floor space had to be greater than the group total of 20. Supermarket an Dept. Store floor space had to be individually more than 10 000 sq mtrs each.

The maximum floor space was constrained for each shop type according to the supplied table.

Option

In evaluating the option, the available floor space not allocated above was allocated optimally with Solver. It was assumed that the minimum requirements had been satisfied by the above phase.

Analysis

The first thing to set in Solver was the cell to be maximised, which was the one with NPV less improvement and construction costs. It also had to be told that this was to be maximised. The next thing was to tell it the cell range that it was allowed to alter in order to produce a maximum solution.

The constraints were then added to the Solver tool in Excel. This involve in each case telling it the cell that was to be above or below a certain value, which could be a number or another cell. For example the cell representing the sum of the floor-space was told that this must be less or equal to the cell containing the 45 for the maximum amount that could be allocated. Having set up the cell to maximize, the cells that could be changed and the constraints as described above, the Solver tool was told to produce a solution. The dialogue indicated that it could produce a solution meeting all the criteria and this was an optimal value. Like all solutions of this type it can only be as good as the input and relies on accurate data and assumptions such as that all space allocated would be taken up with no breaks in tenancy.

Also there was the possibility of extra revenue when sales exceeded targets, but there was no information to enable this to be used.

Conclusion

The allocations for the Meadvale Project are dominated by the ones with the highest profit potential which are the Supermarket and the Department Store.

The remaining space available under local planning rules would therefore be less profitable. In this case, although the optimum mix has been obtained it does not represent a great return on the Capital Outlay of £200,000. However, the company had experienced large capital gains on previous projects and would need to consider whether this would justify going ahead as the project would at least return a small profit.

Appendix 1: Midvale

Group	No	Type of Shop	Cost of interior improvements (a)	Present Value(a) plus fixed	Guarantee Rent (a)	Group min(b)	Shop max(b)	Size (a)
A	1	Supermarket	9	73.52	3.2	20	20	20.0
A	2	Dept Store	13	93.52	4.1	0	20	20.0
B	3	Shoes	12	58.52	3	0	0.9	0.0
B	4	Women's clothing	8	63.52	3.2	0	3	0.0
B	5	Men's Clothing	7	61.52	3.2	0	2	0.0
C	6	Opticians	6	63.52	3	2	4	3.0
C	7	Chemists	7	59.52	3.1	0	1.6	0.0
D	8	Gift	8	48.52	2.5	2	3	0.0
D	9	Mobile phone	9	63.52	2.4	0	1.3	1.3
D	10	Café	10	53.52	2.6	0	1.5	0.0
D	11	Music	7	59.52	2.3	0	1.5	0.7
D	12	Bakery	11	48.52	3	0	1	0.0
(a) £ per sq mtr
(b) 000's sq mtrs
SIZE CONSTRAINTS	Allocated	Min(a)	Max(a)
Total Square metres	45.0		45
Group A		40.0	2
Group B		0.0	0
Group C		3.0	2
Group D		2.0	2
Shop 1		20.0	10	20
Shop 2		20.0	10	20
INCOME (£000)		£ 000
Minimum Rent		£ 1,118.11	Must exceed Improvements + interest
Sales excess		£ -
Other				£ -
Total income		£ 1,118.11
Present value		£ 3,655.60
COSTS			£ 000
Fixed				£125
Construction		270
Improvements		475
Interest on improvements		332
Improvements + interest		807
				0
Total costs excluding fixed		1,077
COST CONSTRAINTS
Improvements	Les than	450
PROFIT (PRESENT VALUE LESS COSTS)	2,579
NPV of 7 year fixed		608.55
Nett profit			1,970

Appendix 2: Options

Group	No	Type of Shop	Cost of interior improvements (a)	Present Value(a) plus fixed	Guarantee Rent (a)	Group min(b)	Shop max(b)	Size (a)
A	1	Supermarket	9	73.52	3.2	0	0.0	0.0
A	2	Dept Store	13	93.52	4.1	0	0.0	0.0
B	3	Shoes	12	58.52	3	0	0.9	0.0
B	4	Women's clothing	8	63.52	3.2	0	3.0	3.0
B	5	Men's Clothing	7	61.52	3.2	0	2.0	2.0
C	6	Opticians	6	63.52	3	0	1.0	1.0
C	7	Chemists	7	59.52	3.1	0	1.6	0.0
D	8	Gift	8	48.52	2.5	0	3.0	0.0
D	9	Mobile phone	9	63.52	2.4	0	0.0	0.0
D	10	Café	10	53.52	2.6	0	1.5	0.0
D	11	Music	7	59.52	2.3	0	0.8	0.0
D	12	Bakery	11	48.52	3	0	1.0	0.0
(a) £ per sq mtr
(b) 000's sq mtrs
SIZE CONSTRAINTS	Allocated	Min(a)	Max(a)
Total Square metres	6.0		6
Group A		0.0	0
Group B		5.0	0
Group C		1.0	0
Group D		0.0	0
Shop 1			0.0	0	0.0
Shop 2			0.0	0	0.0
INCOME (£000)		£ 000
Minimum Rent		£ 133.00	Must exceed improvements + interest
Sales excess		£ -
Other				£ -
Total income		£ 133.00
Present value		£ 377.12
COSTS			£ 000
Fixed				£29
Construction		36
Improvements		44
Interest on improvements		31
Improvements + interest		75
				0
Total costs		215
COST CONSTRAINTS
Improvements	Les than	450
PROFIT (PRESENT VALUE LESS COSTS)	163
NPV of 7 year fixed		141
Nett profit			22

Appendux 3: Formulas Midvale Cols A to E

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Group	No	Type of Shop	Cost of interior improvements (a)	Present Value(a) plus fixed
A	1	Supermarket	9	73.52
A	2	Dept Store	13	93.52
B	3	Shoes	12	58.52
B	4	Women's clothing	8	63.52
B	5	Men's Clothing	7	61.52
C	6	Opticians	6	63.52
C	7	Chemists	7	59.52
D	8	Gift	8	48.52
D	9	Mobile phone	9	63.52
D	10	Café	10	53.52
D	11	Music	7	59.52
D	12	Bakery	11	48.52
(a) £ per sq mtr
(b) 000's sq mtrs
SIZE CONSTRAINTS			Allocated	Min(a)
Total Square metres			=SUM(I2:I13)
Group A			=SUM(I2:I3)	2
Group B			=SUM(I4:I6)	0
Group C			=SUM(I7:I8)	2
Group D			=SUM(I9:I13)	2
Shop 1			=+I2	10
Shop 2			=+I3	10
INCOME (£000)				£ 000
Minimum Rent				=SUMPRODUCT(F2:F13,I2:I13)*7
Sales excess				0
Other				0
Total income				=SUM(E32:E34)
Present value				=SUMPRODUCT(E2:E13,I2:I13)
COSTS				£ 000
Fixed				125
Construction				=+D20*6
Improvements				=SUMPRODUCT(D2:D13,I2:I13)
Interest on improvements				=+E43*0.1*7

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