Question

Good news! Gold has just been discovered in your backyard. Mining engineers tell you that you can extract five ounces of gold per year forever. Gold is currently selling for \(\$ 400\) per ounce, and that price is not expected to change. If the discount rate is 5 percent per year, estimate the total value of your gold mine.

Short answer

The total value of the gold mine is \$ 40000

Step-by-step solution

Identify C and r

Here \(C\) is the cash flow each year from selling the gold. This is the number of ounces mined each year (5) times the selling price per ounce (\$400). So \(C = 5 * 400 = \$ 2000\). The discount rate \(r\) is given as 5 percent, or 0.05.

Apply the formula

We apply the perpetuity valuation formula, which gives \(V = C / r = 2000 / 0.05 = \$ 40000\).

Key concepts

Time Value of Money

The concept of the time value of money is foundational in finance and investment theory, affirming that a dollar today is worth more than a dollar tomorrow. This principle fits into our natural preference for immediate gratification over future benefit. But there's more to it than just impatience; it's about potential earnings. If you have money in hand today, you could invest it and earn interest, making it worth more in the future.

For instance, if you receive \(100 today and deposit it in a savings account with a 5% annual interest rate, in one year, you would have \)105. This illustrates the essence of the time value of money - money's potential to grow over time given a certain rate of return. Understanding this concept helps one to make better financial decisions, like evaluating whether to take a lump sum of money today or receive it in increments over time.

Discount Rate

The discount rate is a critical component in time value of money calculations. It is the interest rate used to 'discount' future cash flows back to their present value. This rate reflects the risk and time preference of the valuation. It's essentially the rate of return that could be earned on an alternative investment of similar risk.

So, why does the discount rate matter? It allows us to compare the value of money received in the future with money in hand today. A higher discount rate indicates a higher risk or opportunity cost and will reduce the present value of future cash flows. In the context of valuing a gold mine, the discount rate provides a way to measure the mine's worth in today's dollars, considering the risk and opportunity cost of mining and selling the gold over time.

Cash Flow

Cash flow, in the most basic sense, is the money that goes in and out of a business or an investment over a period. In the context of the gold mine exercise, cash flow represents the income generated from selling the gold extracted from the mine each year.

Cash flow is important because it determines the potential return on an investment. Positive cash flow means more funds are coming into the business than going out, which is a good indicator of financial health. When we calculate the present value of an investment, we need an estimate of these cash flows. The stability and predictability of the cash flow also play a crucial role. In the gold mine example, despite the simplicity of the perpetual cash flow, it provides a consistent stream of income.

Present Value

The concept of present value is intertwined with the time value of money. It is the current worth of a future sum of money or stream of cash flows given a specified rate of return, known as the discount rate. Calculating the present value allows us to make apples-to-apples comparisons between amounts of money available at different times.

In our gold mine scenario, the present value gives us the worth of all the future gold earnings today. It's a way of saying, 'If I were to sell this gold mine now, how much would it be worth?' This calculation helps investors make informed decisions when faced with investment opportunities that will pay off in the future. By bringing all future cash flows back to the present and summing them up, we get a clear picture of the investment's value right now.