Present Value of Future Gold Production

Suppose the current price of gold is $1,200 per ounce. Hans Consultants forecasts that:

For the next two years, gold prices will grow at an average rate of 12% per year.
After two years, the growth rate will decline to a long-run trend of 3% per year.
You expect to produce 1 million ounces of gold in eight years.

Assumptions:
Gold prices have a beta of 0, meaning there is no market risk premium.
The risk-free rate is 5.5% per year.
Discounting is done continuously or annually (specify the approach if needed).
Required:
Calculate the present value of the 1 million ounces of gold produced in eight years, assuming the given growth and discount rates.

The correct answer and explanation is :

To calculate the Present Value (PV) of Future Gold Production, we follow these steps:

Step 1: Forecast Future Gold Price in Year 8

The gold price grows in two phases:

First 2 Years: Growth at 12% per year.
After Year 2: Growth at 3% per year.

Step 1.1: Price at Year 2

Using the compound growth formula:
[
P_2 = P_0 \times (1 + g_1)^2
]
[
P_2 = 1200 \times (1.12)^2 = 1200 \times 1.2544 = 1505.28
]

Step 1.2: Price at Year 8

From Year 2 to Year 8, price grows at 3% per year:
[
P_8 = P_2 \times (1 + g_2)^6
]
[
P_8 = 1505.28 \times (1.03)^6 = 1505.28 \times 1.19405 = 1796.63
]

Step 2: Compute Future Revenue

[
\text{Revenue at Year 8} = \text{Gold Price in Year 8} \times \text{Production}
]
[
= 1796.63 \times 1,000,000 = 1,796,630,000
]

Step 3: Discount to Present Value

Using continuous discounting:
[
PV = FV \times e^{-rt}
]
[
PV = 1,796,630,000 \times e^{- (0.055 \times 8)}
]
[
PV = 1,796,630,000 \times e^{-0.44}
]
[
PV = 1,796,630,000 \times 0.643
]
[
PV = 1,155,254,090
]

Conclusion

The present value of producing 1 million ounces of gold in 8 years is $1.155 billion. This accounts for expected gold price growth and risk-free discounting.