Question

Question 4

Suppose 100,000 kilograms of gold can be obtained from a gold mine during its first year in operation. However, its subsequent yield is expected to decrease by 10% over the previous year’s yield. The gold mine has a proven reserve of 1,000,000 kilograms.

1. a) Suppose that the price of gold is expected to be $60 per gram for the next several years. What would be the present worth of the revenue earned at an interest rate of 12% per annum compounded annually over the next seven years?
2. b) Suppose that the price of gold is expected to start at $60 per gram during the first year, but to increase at the rate of 5% over the previous year’s price. What would be the present worth of the revenue earned at an interest rate of 12% per annum compounded annually over the next seven years?
3. c) Consider part b again, find the net worth of the revenues to be earned in the next 4 years at the end of year 3.

Answer 1

The extraction of gold is equal to 100000 KG in the first year which is decreasing at the rate of 10% next year.
The price is $60 per gram and interest rate is given as 12%

We need to calculate the PW

PW = PMT / (1+Interest Rate) ^ Duration

a) The duration is 7 years
The value of gold is $60000 per KG.

1st year PW

(100000 * 60000) / (1.12) = 5357.14 million

2nd year PW
(100000 * 0.9 * 60000) / (1.12 ^ 2) = 4304.85 million

We can create a table here

Year	Extraction (KG)	Value in Million	PW @ 12%
1	100000	6000.00	5357.14
2	90000.00	5400.00	4304.85
3	81000.00	4860.00	3459.25
4	72900.00	4374.00	2779.76
5	65610.00	3936.60	2233.73
6	59049.00	3542.94	1794.96
7	53144.10	3188.65	1442.38
NPW			21372.08 Million

b) The price of the gold is rising 5% each year and extraction data is the same as in the part a.

1st year PW
(100000 * 60000) / (1.12) = 5357.14 million

2nd year PW
((100000 * 0.9) * (60000 * 1.05)) / (1.12 ^ 2) = 4520.09 million

We can create a table here

Year	Extraction (KG)	Value in Million	PW @ 12%
1	100000	6000.00	5357.14
2	90000.00	5670.00	4520.09
3	81000.00	5358.15	3813.83
4	72900.00	5063.45	3217.92
5	65610.00	4784.96	2715.12
6	59049.00	4521.79	2290.88
7	53144.10	4273.09	1932.93
NPW			23847.90

c) We will have to find out NPW at the end of year 3.
The pricing and extraction schedule won't change but discounted value will change here.

PW of 4th year
5063.45 / (1.12 ^ 1) = 4520.94

PW of 5th year
4784.96 / (1.12 ^ 2) = 3814.54

PW of 6th year
4521.79 / (1.12 ^ 3) = 3218.52

PW of 7th year
4273.09 / (1.12 ^ 4) = 2715.63

Year	Extraction	Value in Million	PW @ 12%
4	72900.00	5063.45	4520.94
5	65610.00	4784.96	3814.54
6	59049.00	4521.79	3218.52
7	53144.10	4273.09	2715.63
NPW			14269.63