Go to the companion website for more details.

Let's suppose it is December 30, 2001, and we are contemplating a 10-year lease on this mine. The price of gold on that date was $278 per ounce and the trailing annual volatility of gold prices from 1970 through 2001 was 20.5 percent. The yield on the 10-year Treasury was 5.07 percent. We shall assume that the extraction cost of gold is $200 per ounce and that 10,000 ounces are produced per year. Our parameters are therefore , with an annual discount rate equal to 1/1.0507. In the absence of price volatility, the annual profit would be equal to ($278 – $200)/1.0507^t for . In this case, the value of the lease is $6 million (see the spreadsheet for details). But, in this case, gold would be a risk-free asset, which it is not. The price dynamic, therefore, over the 10-year period consistent with these parameters is given in the top half of Figure 16.20; the lease computations follow in the bottom half of the figure. The defining assumptions here are that the price of gold is the price that holds at the beginning of each year and that all cash flows occur at the end of the year. Thus, because the gold mine has to be returned to the owner at the end of year 10, then the mine has no lease value in the final year.

It is worth noting that in June of 2011, the true price of gold exceeded $1,500 per ounce and that this price is captured in the range specified on our spreadsheet. The valuations at the beginning of year nine (bottom half) are tied directly to the year nine prices of gold (see the spreadsheet for details on the formulas). Letting (i,j) index the row and column in the lattice, we therefore have:

The lease values for years 1 through 8 at each node are a function of the risk-neutral values of the two subsequent years plus the value of the gold mine in the current year, all discounted using the annual Treasury yield of 5.07 percent. That is:

Clearly, gold prices below the cost of extraction ($200 per ounce) will result in zero valuations. Working backward through the lattice, we find that the lease would have had expected value equal to $11.29 million at the close of 2001. The realized value over the subsequent 10 years would have depended on the actual path that gold prices took.

Now suppose that, in addition to this information, you are told that a one-time $1 million expenditure will purchase an enhancement to the production process, increasing productivity to 12,500 ounces per year (a 25 percent enhancement), which is partly offset by raising per unit costs from $200 per ounce to $240 per ounce. You have the option to purchase this enhancement at any time. As we shall see, the value of this option depends entirely on the path that gold prices take.

Figure 16.21 shows the value of the lease with this enhancement in place. The valuation functions are now given by:

The value of the lease with the new technology is now $10.23 million, and after netting out the fixed cost of the technology, $9.23 million. Clearly, it does not pay to adopt the technology in the first year of the lease. The question remains as to whether it is ever optimal to adopt the new technology, and the answer is that it depends on how high the price of gold goes. In Figure 16.22 therefore, we compare the periodic value of the enhanced mine to the original lease, denoting in each year the cases in which the net cost of the enhanced mine exceeds the original lease values. That is, the enhancements are attractive as long as . These cases are highlighted in the third lattice, given in Figure 16.22. Beginning with year nine and substituting these enhancements back into the original lattice and working backward, continuing to substitute as long as the highlighted values exceed those in the original lattice, we find that only when gold prices reach $514.20 by year three (and continues to rise) will it pay to exercise the option during that year. In any case, it neither pays to enhance the mine nor to wait to exercise the option on the lease. The decision hinges on the cost of the enhancement, the increased variable costs of extraction and the price of gold. In this case, the price of gold was not sufficiently high to make the option viable.

The exercise of the option in this case is not optimal until year three, and that is because of the possibility that gold prices had reached a level sufficient to sustain the viability of the expense of the new technology. That decision was not so obvious in year one, when prices were only $278. It is important to understand the intuition here. The basic valuation logic is still intact—in general, the lease value is the discounted present value of the future payoffs modeled by the lattice. The option itself has value that is a function of the profitability of the lease, which, in turn, depends on the underlying movement in gold prices. The factors affecting the value of the option are the underlying price volatility, the discount rate, and the duration of the lease. It would be an interesting exercise to find out how sensitive this exercise would be to changes in each of these factors.