# Theory of Portfolio Selection by Terence M. Ryan M.A. (auth.)

By Terence M. Ryan M.A. (auth.)

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Second, not all of the T roots will necessarily be real numbers. Some may be imaginary numbers (such roots always occur in pairs) and these may safely be ignored. Last, since the assetselection criteria involve a comparison between the IRR and the market rate of interest, we will generally be concerned with those roots in the vicinity of the latter. 2 illustrates the above situations. Here there are four roots: D 1 , being negative, corresponds to a negative value ofr1 ; D 2 and D 3 represent a pair of complex roots; while D 4 , ifless than unity, represents a positive value ofr1• Descartes's Rule of Signs An important theorem, due to Descartes, states that the number of positive real roots of a polynomial is equal to the number of changes in sign of the coefficients, minus an even non-negative number.

1 + ... +Sr. 49) Sr + r)T = "'Pv. 04. 04 is in fact the net present value of the investment, calculated in the conventional manner of page 29. It is also worth noting that since 1/(1 r)T is a constant for any discount rate, r, there is a one-to-one relationship between the NPV criterion and the NTV criterion in that any project which is accepted on the basis of its NPV would also have been accepted on the basis of its NTV. Similarly, the ranking of projects ~nder the NPV criterion will be exactly the same as their ranking underNTV.

1. The IRR, like the NPV, is thus a measure of the rate of profit on an investment project, and consequently the implied criteria for asset selection based on the IRR measure are similar to those described above based on the NPV measure. Accept/reject criterion. Since the market rate ofinterest, r, represents the opportunity cost of the funds used to purchase the investment project, and since the rate of profit generated by that project is its internal rate of return, r1, it follows that if the latter exceeds the former the funds used in acquiring the project are earning a higher rate of return than their epportunity cost and so the project should be accepted.