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Engineering Mathematics in Finance: BENG0019

创建日期：2024-06-11 17:34:05

所属分类：数学mathematics

标签： BENG0019

Engineering Mathematics in Finance

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Engineering Mathematics in Finance:
BENG0019
There are 3 questions on this assignment.
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) A farmer plans to fence off a rectangular area in a field to make a chicken run. He has 100 metres of
fence. What is the largest rectangular area A(x,y) that the fencing can enclose? 5 marks
b) By determining the bordered Hessian show that a maximum has been determined.
5 marks
c) 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 should refer to quadratic forms to derive the
sufficient conditions for the nature of these extrema.
10 marks
d) 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.
i. Find the extreme values of f subject to both constraints f(, , ) = ;
2 + 2 − 2 = 0 + + = 24
15 marks
ii. Find the maximum value of (, , ) = + 2 + 3
0 ℎ ℎ − + = 1 ℎ 2 + 2 = 1
15 marks
Hint: You do not need to determine the Bordered Hessian.
2. Difference equations and Laplace transforms.
a) Suppose we have the following 1st order difference equation = −1.2−1 + 176, find the particular
solution using any method of your choice. Take 0 = 60. 5 marks
b) Plot your solution to 2(a) on a graphical software/package of your choice and discuss the stability of the
solution.
5 marks
c) Using only Laplace transforms solve the following Samuelson model given below i.e. the second order
difference equation (where is national income):
i. +2 − 8+1 + 12 = 0, = 0 < 0, ℎ 0 = 1, 1 = 1
10 marks
ii. Verify your solution to 3c (i) above by assuming a solution of the type = ()
,
where symbols have their usual meanings. 5 marks
BENG0019 Page | 3
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: £42m, £73m, £124m,
£91m, £54m 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 7% 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). 10 marks
c) 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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