Natural Resource Example

Problem Description

This example is adpated from two problems in David Luenberger's book Investment Science (see p. 337).

A company has the option to acquire a ten-year lease to extract gold from a gold mine. The lease should be acquired if its cost is less than the value of the lease.

Here are relevant data for this example:

Price of gold will be assumed constant over the life of the lease. It is currently $400 per ounce.
The mine's capacity is 50,000 ounces.
Operating costs reflect inherent nonlinearities in mining: decreasing returns-to-scale, and increasing costs as reserves are depleted. Specifically, when the begin-of-year mine inventory is I, the cost to mine Q ounces for the upcoming year is 500•Q²/I.
The yearly cost of capital is 10%.

The following assumptions are made:

Cash flows will be estimated on a yearly basis.
Taxes will be ignored.
Production occurs instantaneously.
Production quantities will be in multiples of the lot size, in this case 1000 ounces.
Profits each year accrue at the beginning of the year.

A production policy specifies the quantity and timing of production. Each production policy yields a corresponding profit cash flow stream, that is, yearly profits over time. The present value of yearly profits (at an appropriate cost of capital) represents the value of the lease given this production policy. A reasonable objective is to find the production policy that maximizes the present value of yearly profits. The resulting present value is the value of this lease.

Hava Preliminaries

The first step is to record the problem’s parameters. In Hava, they take the following form:

P = 400;     // per ounce price of gold
I0 = 50;     // initial inventory of gold (in 000's)
T = 10;      // lease duration (in years)
d = 1/1.1;   // per-year discount factor

The next step is to define key economic and state variables, namely, profit, inventory and production. For this purpose, let I(t) denote the inventory at the beginning of year t, and let Q(t) denote the production quantity mined for year t+1, t = 0, 1, … , T.

From what we have learned so far:

leaseValue = sum(t = 0 to T-1) {d^t*profit(t)};
profit(t) = 400*Q(t) - 500*Q(t)^2/I(t);

Material balance implies that end-of-year inventory equals the begin-of-year inventory less the quantity produced in the year:

I(t) = if (t == 0) {I0} else {I(t-1) - Q(t-1)};

Constant Production Plan

It's a good idea to evaluate simple production policies. A natural candidate for evaluation is the constant production plan:

Q(t) = I0/T;

Click here to see the constant production solution.

Optimal Myopic Production Plan

Analysis of the constant production plan shows that it is not a good idea to mine all the mine's inventory over the life of the lease.

Another idea worth investigating is the optimal myopic production plan. Here, producton quantities each year are chosen to maximize the profits for that year only.

The inventory, I(t), is known at the beginning of year t. For a fixed value of inventory, the profit function is a (concave) quadratic function of the production quantity, Q(t). Setting its derivative with respect to Q(t) to zero yields the optimal myopic production plan as

Q(t) = 0.40*I(t);

Click here to see the myopic production solution.

Optimal Lease Value

The state of a system is all the information an analyst would need to know to determine the best course of action from this state forward. For this problem, an analyst needs to know two pieces of information:

how many years, t, the mine has been operated, and
how much inventory, I, is left in the mine.

Let optValue(t, I) denote the optimal value of the lease if the system state is (t, I). The optimal lease value is simply optValue(T, I0). Let Value(t, I, Q) denote the value of the lease if the state is (t, I) and the production quantity Q is chosen. The value of the lease is zero when t = T, since the mine is returned to its owner. Otherwise:

optValue(t, I) 
= if (t < T) {max(Q = 0 to I) {Value(t, I, Q)}}
  else {0};

The function V(t, I, Q) is the sum of two components:

the profit generated now, namely,
P*Q - 500*Q^2/I;
the present value of all future profits determined by optimally operating the mine hereafter.

As for the second component, given a state (t, I) and a production quantity choice of Q, the next state will be (t+1, I-Q). (There will be one less year in the lease, and the new inventory will be the old inventory less the production quantity.) The optimal value of the lease in state (t+1, I-Q) is optValue(t+1, I-Q)! Since this value accrues next year, its present value today is d*optValue(t+1, I-Q).

Putting it all together, we can express V(t, I, Q) in terms of optValue(t, I), as follows:

private Value(t, I, Q) 
= if (I > 0) {(P*Q - 500*Q^2/I) + d*optValue(t+1, I-Q)}
  else {0};

We have defined the variables and their relationships so that Hava can take it from here.

Click here to see the code created so far.

Optimal Production Policy

The optimal lease value has been determined (good), but we do not know the optimal production policy! Without it, the company cannot realize the optimal lease value, since it will not know what course of action to take over the life of the lease.

Let optI(t) and optQ(t) denote, respectively, the optimal begin-of-year inventory and corresponding optimal production quantity for year t = 0, 1, … , T-1. In terms of optI(t), optQ(t) satisfies the following equation:

optQ(t) = argmax(Q = 0 to optI(t)) {Value(t, optI(t), Q)};

Given the path of optimal inventory and production up to time t-1, the optimal inventory at time t is given by

optI(t) 
= if (t == 0) {I0}
  else {optI(t-1) - optQ(t-1)};

Click here to see the optimal production solution.

Sensitivity and Economic Analysis

This Hava program can be used to answer such questions as: How do the lease values for the constant, myopic optimal and optimal production plans compare in percentage terms? What is the sensitivity of the lease value to changes in the cost of capital? lease duration? initial inventory? cost structure?