## 832 Internal Rate of Return

Life cycle savings are the true savings if all the input is known correctly and without doubt. But future energy prices and system performance are uncertain, and the choice of the discount rate is not clear-cut. An investment in DG equipment is uncertain, and it must be compared with competing investments that have their own uncertainties. The limitation of the life cycle savings approach can be circumvented if one evaluates the profitability of an investment by itself, expressed as a dimensionless number. Then, one can rank different investments in terms of this index of profitability and in terms of risk. General business experience can serve as a guide for expected profitability as function of risk level. Among investments of comparable risk, the choice can then be based on profitability.

More precisely, the profitability is measured as so-called internal rate of return rr, defined as that value of the discount rate rd at which the life cycle savings S are zero:

For an illustration, take the case of Eq. 8.38 with energy escalation rate re = 0 (so that rd,e = rd), and suppose an extra investment DCcap is made to provide annual energy savings (-DQ). The initial investment DCcap provides an annual income from energy savings:

If DCcap were placed in a savings account instead, bearing interest at a rate rr, the annual income would be annual income = (A/P,rr,N) DCcap (8.41)

The investment behaves like a savings account whose interest rate rr is determined by the equation

Dividing by (A/P,rr,N), we see that the right and left sides correspond to the two terms in Eq. 8.38 for the life cycle savings c _ -DQpe DCcap (8.43)

and that rr is, indeed, the discount rate rd for which the life cycle savings are zero; it is the internal rate of return. Now the reason for the name is clear; the internal rate of return is the profitability of the project by itself, without reference to an externally imposed discount rate. When the explicit form of the capital recovery factor is inserted, one obtains an equation of the Nth degree, generally not solvable in closed form. Instead, one must resort to iterative or graphical solution. (There could be up to N different real solutions, and multiple solutions can indeed occur if there are more than two sign changes in the stream of annual cash flows, but the solution is unique for the case of interest here — an initial investment that brings a stream of annual savings.)

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