Math CentralQuandaries & Queries


Question from Darrin, a parent:
In his finance class, my son is being asked to take the following equation and solve for "D":
1 / [1 - (1 + D/12)^T] = (1 / D) * [((12*B) / (P*T)) + Y + ((12N * Z) / (360T))] - [(12Z) / (360T)] * F
We started at the equation with the "B" variable isolated, but have come this far (and validated it), and are now at an impasse. (I think due to the exponent).


A lot of the complications here are in blocks that you can keep together. Thus we can think instead about

1 / [1 - (1 + D/12)^T] = (1 / D) * Q - R

Q depends on B,P,T,Y,N,Z; R on Z,T,F. It's not too hard to see that by selecting appropriate Y and F (say) we can get any value for Q,R. So this really is generic.

Now isolate the exponential term (1+D/12)T on one side. You will probably spot one rather simple solution easily. If you plot both sides as functions of D, you should be able to show (if all constants in the original are positive) that one side goes to infinity for D=Q/R; that there is no other solution with 0<D<Q/R ; and that there is always a second solution with D > Q/R , and no other. Details are left to you. It seems to me almost certain that the second solution will not be expressible in closed form. Is it possible that a numerical solution is expected?

Good hunting!


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