Financial Math Answers

Find the present value of $10,000 received at the start of every year for 20 years if the interest rate is J1 = 12% p.a. and if the first payment of $10,000 is received at the end of 10 years.

4. A consumer has the utility function U = 4X0.75Y 0.25 and can buy good X at £12
a unit and good Y at £2 a unit. Find the combination of X and Y that they should
purchase to minimize the cost of achieving a utility level of 580 and check that
second-order conditions are met using the bordered Hessian matrix.

1. A firm has the production function Q = K0.5L0.5 and buys input K at £12 a unit
and input L at £3 a unit and has a budget of £600. Use the Lagrange method to
find the input combination that will maximize output, checking that second-order
conditions are satisfied by using the bordered Hessian.
2. A firm operates with the production function Q = 25K0.5L0.4 and buys input K
at £20 a unit and input L at £8 a unit. Use the Lagrange method to find the input
combination that will minimize the cost of producing 400 units of Q, using the
bordered Hessian to check that second-order conditions are satisfied.
3. A consumer has the utility function U = 20X0.5Y 0.4 and buys good X at £10 a
unit and good Y at £2 a unit . If their budget constraint is £450, what combination
of X and Y will maximize utility? Check that second-order conditions are satisfied
by using the bordered Hessian.

7)Find and compare the future value after two years of a deposit of $100 attracting interest at a rate a) annually and b) semiannually.
8)Which will deliver a higher future value after one year, a deposit of $1, 000 attracting interest at 15% compounded daily, or at 15.5% compounded semi-annually?
9)What initial investment subject to annual compounding at 12% is needed to produce $1, 000 after two years?
10)Which will deliver a higher future value after one year, a deposit of $1, 000 attracting interest at 15% compounded daily, or at 15.5% compounded semi-annually?
11)What initial investment subject to annual compounding at 12% is needed to produce $1, 000 after two years?
12)Find the present value of $100, 000 to be received after 100 years if the interest rate is assumed to be 5% throughout the whole period and a) daily or b) annual compounding applies.

1)A sum of $9, 000 paid into a bank account for two months (61 days) to attract simple interest will produce $9, 020 at the end of the term. Find the interest rate r and the return on this investment.
2)How much would you pay today to receive $1, 000 at a certain future date if you require a return of 2%?
3)How long will it take for a sum of $800 attracting simple interest to become $830 if the rate is 9%? Compute the return on this investment.
4)Find the principal to be deposited initially in an account attracting simple interest at a rate of 8% if $1, 000 is needed after three months (91 days).
5)How long will it take to double a capital attracting interest at 6% compounded daily?
6)What is the interest rate if a deposit subject to annual compounding is doubled after 10 years?

QUESTION 4: BUSINESS DECISIONS [20 MARKS]
Suppose that ABC Ltd is considering purchasing one of three new processing machines. Either machine would make it possible for the company to produce its products more efficiently.
Estimates regarding each machine are provided below:

Machine A Machine B Machine C
Original cost $79,000 $110,000 $244,000
Estimated life 7 years 8 years 10 years
Salvage value Nil Nil $30,000
Estimated annual cash inflows $30,000 $ 60,000 $58,500
Estimated annual cash outflows $ 7,000 $ 35,000 $18,500



A. If the projects cannot be repeated, which machine should ABC Ltd choose based on the NPV criteria at an 8% cost of capital? (9 marks)

B. If the projects can be repeated, which machine should ABC Ltd choose based on the NPV criteria at an 8% cost of capital? (6 marks)

Suppose that ABC Ltd is considering purchasing one of three new processing machines. Either machine would make it possible for the company to produce its products more efficiently.
Estimates regarding each machine are provided below:

Machine A Machine B Machine C
Original cost $79,000 $110,000 $244,000
Estimated life 7 years 8 years 10 years
Salvage value Nil Nil $30,000
Estimated annual cash inflows $30,000 $ 60,000 $58,500
Estimated annual cash outflows $ 7,000 $ 35,000 $18,500



A. If the projects cannot be repeated, which machine should ABC Ltd choose based on the NPV criteria at an 8% cost of capital? (9 marks)

A loan of R1 000 is granted at an interest rate of 16% p.a. compounded monthly. The loan is to be amortised by means of ten consecutive, equal monthly payments starting one and a half years after the granting of the loan. The balance outstanding on the loan (to the nearest cent) immediately after the seventh monthly payment has been made is equal to R

: LOAN AMORTIZATION

I. A family buys a house worth $326,000. They pay $75,000 deposit and take a mortgage for the balance at J12=9% p.a. to be amortized over 30 years with monthly payments.

A. Find the value of the mortgage on their house?
B. Find the value of the monthly payment?
C. Find the loan outstanding after making 20 payments?
D. Find the principal repaid in the 21st payment?
II. Fill out the loan amortization schedule provided in the solution template for the first 5 loan payments. What do you notice about the composition of the payment amount?

I. A family buys a house worth $326,000. They pay $75,000 deposit and take a mortgage for the balance at J12=9% p.a. to be amortized over 30 years with monthly payments.

A. Find the value of the mortgage on their house? (1 mark)
B. Find the value of the monthly payment? (3 marks)
C. Find the loan outstanding after making 20 payments? (4 marks)
D. Find the principal repaid in the 21st payment? (5 marks)