**Tree Harvesting Problem in Investment**

The optimum investment timing analysis is of direct relevance in the case of tree harvesting problems. Suppose that we own a piece of land and are considering growing a crop of trees, we would like to maximize the net present value (NPV) of investment. The maximization of the investment’s net present value (NPV) would depend on when we harvest trees. The future value of trees increases when harvesting is postponed: but the opportunity cost of capital is incurred by not realizing the value by harvesting the trees. The net present value (NPV) will be maximized when the trees are harvested at the point where the percentage increase in value equals the opportunity cost of capital.

Suppose the net future value obtained over the years from harvesting the trees is At, and if the opportunity cost of capital is K, then the present value (PV) of the net realizable value of trees is given by:

**PV = At / (1+K) t = At (1+K)-t**

To discussion so far can be put in the formal terms. To determine the optimum harvesting time, which maximizes the net present value (NPV), we set the derivative of the net present value (NPV) with respect to t in equation equal to zero:

**NPV = Ate –kt – C**

**dNPV / dt = -kAte –kt + dAte –kt/ dt = 0**

Solving for k, we obtain

**K = (dAt / dt) / At**

The expression (dAt / dt) At is the rate at which the obtainable net future values changes with time; it is the incremental (marginal) rate of change in value. Thus the NPV will be maximized, when the marginal rate of change in value equals the opportunity cost of capital.

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