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BENG0019 Page | 1

Engineering Mathematics in Finance:

BENG0019

UNIVERSITY COLLEGE LONDON

There are 3 questions on this assignment.

Handout Date: 3

rd February 2020.

Submission Date: 2

nd March 2020, 23:59.

BENG0019 Page | 2

Answer all questions, and provide explanations or suitable references for any results or theorems used.

1. Constrained Optimization.

By setting up a Lagrangian function only solve the following problems:

a) Maximize 𝑓(𝑥, 𝑦, 𝑧) = 5𝑥𝑦 + 8𝑥𝑧 + 3𝑦𝑧 subject to 2𝑥𝑦𝑧 = 1920. 10 marks

b) Derive an expression for the Bordered Hessian when the objective function z(x,y) is a function of two

dependent variables x and y, you may assume that the constrained functional relation can be

written as g(x,y) = c where c is a real constant. You can refer to quadratic forms to derive the

sufficient conditions for the nature of these extrema.

10 marks

c) This question must be attempted using vector calculus (and Lagrange multipliers) i.e. to be awarded

marks for this question you must use the gradient of a scalar field.

𝑓(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥2 + 𝑦2 = 1 𝑎𝑛𝑑 𝑥 + 𝑧 = 1 15 marks d)

i. Minimize costs for a firm with the cost function c(𝑥, 𝑦) = 5𝑥

2 + 2𝑥𝑦 + 3𝑦

2 + 800 subject to the

production quota 𝑥 + 𝑦 = 39 8 marks

ii. Estimate additional costs if the production quota is increased to 40. 2 marks

Hint: You do not need to determine the Bordered Hessian.

e) A student has three exams to prepare for. She speculates that the following functions are valid for

her grade scores, each of the 𝑡𝑖 𝑤ℎ𝑒𝑟𝑒 𝑖 = 1,2,3 represent the time she can allocate to a particular

module and the 𝑔𝑖 𝑤ℎ𝑒𝑟𝑒 𝑖 = 1,2,3 represent the grades obtained, such that:

𝑔(𝑡1) = 30 + 15√𝑡1𝑔(𝑡2) = −60 + 2𝑡2𝑔(𝑡3) = 21 + 𝑡3

There are however some limitations on the times that she can allocate to each module for her revision

and her overall time these are:

By using the Lagrange multiplier technique (only) find the values of 𝑡1, 𝑡2 𝑎𝑛𝑑 𝑡3 that optimize her

grade average for the three modules. You do not need to show that you have obtained a maximum.

15 marks

2. Difference equations and Laplace transforms.

a) Suppose we have the following 1st order difference equation 𝑦𝑡 = −1.2𝑦𝑡−1+ 198, find the particular

solution using any method of your choice. Take 𝑦0 = 50. 3 marks

b) Plot your solution to 2(a) on a graphical software/package of your choice and discuss the stability of the

solution. 2 marks

c) Using only Laplace transforms solve the following Samuelson model given below i.e. the second order

difference equation (where 𝑦𝑡

is national income):

𝑦𝑡+2 − 5𝑦𝑡+1 + 6𝑦𝑡 = 4𝑡, 𝑖𝑓 𝑦𝑡 = 0 𝑓𝑜𝑟 𝑡 < 0, 𝑎𝑛𝑑 𝑦0 = 0, 𝑦1 = 1 10 marks

3. Net Present Value and Internal Rate of Return.

a) You buy a mining site, including exploration rights and there are set up costs of £285m. You expect to

extract the following value of gold over the next 6 years, net of running costs: £40m, £73.5m, £123.5m,

£90.5m, £54.5m and 21m. At the end of year 6 you pay £30m clean-up costs. The site will then be

handed back to authorities (as worthless). Should you go ahead with the project? The cash flows are

discounted at 6.8% p.a. 5 marks

b) By using only linear interpolation or the Newton-Raphson method or the secant method (which you

must code) determine the IRR of the project in 3(a). 5 marks

c) Find the IRR for an investment that costs £96,000 today and pays £1028.61 at the end of the month for

the next 60 months and then pays an additional £97,662.97 at the end of the 60th month if the investor

discounts expected future cash flows monthly. You will once again have to do this iteratively.

5 marks

d) Explain what is meant by the internal rate of return (IRR) in the context of project appraisal. What are

the drawbacks of the IRR method? 10 marks

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