4QQMN503 Mathematics for Accounting & Finance
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4QQMN503 Mathematics for Accounting & Finance
1. Question 1
A principal of £8,000 is invested at an annual rate of 6% interest.
(i) Determine the future value after 12 years if the interest is compounded monthly.
(ii) Determine the future value after 12 years if the interest is compounded continuously.
(iii) Derive the annual equivalent rate for each method of compounding in (i) and (ii).
(iv) If the interest is compounded continuously, how long would it take for the future value to reach £20,000?
[20 Marks]
2. Question 2
An individual is looking to secure a £300,000 mortgage. The bank offers an interest rate of 4% compounded an- nually.
(i) Show that the monthly repayment (m) for borrow- ing P at interest rate r compounded annually over n years is given by:
(ii) Calculate the monthly repayment if they take the mortgage over 25 years.
(iii) They do not want the monthly repayment to exceed £1,800 per month. Derive what is the minimum loan duration that satisfies this criterion.
[20 Marks]
3. Question 3
A firm is a monopolistic producer of two goods G1 and G2. The prices are related to quantities Q1 and Q2 according to the demand functions:
And the total cost function is
(i) Show that the firm’s profit function is
(ii) Find the values of Q1 and Q2 which maximise ⇡ and deduce the corresponding prices.
(iii) Show that the profit function has a maximum. Which conditions have to be met?
[20 Marks]
4. Question 4
An individual’s utility function is given by:
where x1 and x2 denote the number of items of two goods, G1 and G2. The prices of the goods are £2 and £10, respectively. Assume that the individual has £400 available to spend on these two goods.
(i) Write down the constraint and briefly describe the individual’s maximisation problem.
(ii) Find the utility-maximising values of x1 and x2 through the method of substitution.
(iii) Verify that the ratio of marginal utility to price is the same for both goods at the optimum.
(iv) Set up the Lagrangian and show that we find the same values for x1 and x2, as derived under (ii).
(v) What is the interpretation of the Lagrange multi- plier? What does it tell us in this particular exam- ple?
[20 Marks]
5. Question 5
A six equation macroeconomic model of two trading na- tions, labelled by 1 and 2, can be reduced to two equa- tions given by:
where Y1 and Y2 denote the national incomes of country
1 and country 2, respectively. Similarly, I
and I
stand for the respective levels of investment in country 1 and 2.
(i) Express this system in matrix form and find the values of Y1 and Y2 as a function of I
and I
.
(ii) Let I
= 100 and I
= 200, Compute the values of of Y1 and Y2 .
(iii) Assume that in country 1 imports M1 are given by: M1 = 0.3Y1
And for country 2 they are described by: M2 = 0.1Y2
Recall, that M1 = X2 and M2 = X1, where X1 and X2 denote the exports of both countries. Based on this information, calculate the balance of payments between these countries.
[20 Marks]
2021-12-26