Question

2. [Theory] For the following questions, the present is Year 0, and the relevant interest rate is 10% per year. Round your answers to the nearest cent. Show your work, or you will not receive marks. There’s a formula sheet at the end of the assignment.

a. (2 marks) Use (P/F,i,N) to find the present value (Year 0 value) of $1,000 in Year 10.

Present Value: $ _______________

b. (4 marks) Maintenance costs for a machine are $100 in Year 1, after which they go up by $50 a year. Thus, costs are $150 in year 2, $200 in Year 3, etc.  The last maintenance costs are incurred in Year 30.  Use (P/A,i,N) and (A/G,i,N) to find the present value (Year 0 value) of these maintenance costs.

Present Value: $ _______________

c. (4 marks) An after-school sports program for disadvantaged children provides benefits worth $500 in Year 1. Each year after that, the benefits from the program go up by 10%, so they are $550 in Year 2, $605 in Year 3, etc. The last year in which benefits are provided is Year 20. Use (P/A,g,i,N) (or equivalently, (P/A,i^o,N)/(1+g) ) to calculate the present value of these benefits.

Present Value: $_____________

d. (2 marks) You want to save up $1,000,000 by the time you retire in Year 45.  Use (A/F,i,N) to calculate how much money you must put in a bank account every year, starting in Year 1 and with the last payment being in Year 45, in order to have $1,000,000 in Year 45.

Annual Payment: $______________

(F/P,i,N) = (1+i)N	(P/F,i,N) = (1+i)−N	(A/F,i,N) = i(1+i)N−1
(F/A,i,N) = [(1+i)N−1]i	(A/P,i,N) = i(1+i)N(1+i)N−1	(P/A,i,N) = (1+i)N−1i(1+i)N
(A/G,i,N) = 1i−N(1+i)N−1	(P/A,g,i,N) = (PA,io,N)⁄1+g	io=1+i1+g−1

Answer 1

A. PV = $ 1000(P/F, 10%,10)

$\large PV = \$ 1,000*\frac{1}{(1+0.1)^{10}}$

$\large PV = \$ 1,000*0.3855$

$\large PV = \$ 385.50$

B. The present value of cash flow can be calculated using the following formula

PV = $ 100(P/A,10%,30) + $50(A/G,10%,30)

PV = $ 100* 9.427 + $ 50* 8.176

PV = $ 942.70 + $ 408.80

PV = $ 1351.50

C. The present value of cash flow can be written as follows

$\large PV = \frac{A}{1+i} +\frac{A(1+g)}{(1+i)^2}+..+\frac{A(1+g)^{n-1}}{(1+i)^n}$

$\large PV = \frac{A}{i-g}[1-(\frac{1+g}{1+i})^n]$

Here i=g

Therefore,

$\large PV = A*\frac{n}{1+i}$

$\large PV = 500*\frac{20}{1+0.1}$

$\large PV = \$ 9090.91$

D. The cash flow can be written as follows

FV = A+ A(1+i)+A(1+i)²+...+A(1+i)⁴⁴

$\large FV =\frac{ A(1+i)^{45} -A}{i}$

$\large FV =A*\frac{ (1+i)^{45} -1}{i}$

$\large 1,000,000 =A*\frac{ (1+0.1)^{45} -1}{0.1}$

$\large 1,000,000 =A*718.9048$

A = $ 1391.005

Every year he must deposit $ 1391.

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