For this assignment, we will be considering goal-seeking with respect to loans and working
For this assignment, we will be considering goal-seeking with respect to loans and working with the formula connecting present value, monthly payment, and interest, so we reproduce the formula here:
Although this formula technically has five variables, we will take n to be 12, so we will not particularly worry ourselves about that variable and instead focus on the remaining ones. Also, we will not concern ourselves very much with t, since in many situations it is not possible to have a completely free choice of t -- there may be two or three choices (e.g., a mortgage can often be set up for 20, 25, or 30 years), but not a completely arbitrary choice. (It is more common to want to solve for t when looking into the future: how long until a target value is reached, for example.) For our purposes below, we'll consider t to be known.
That leaves us with three variables: PV, PMT, and r. Certainly if we know PMT and r, we can find PV (i.e., "how much house can I afford"), because the above formula is already solved for PV. Since PMT appears in the formula above in a quite straightforward away (multiplying the rest of the right-hand side), we can easily solve for PMT in terms of PV and r (i.e., "if I buy this, what will my monthly payment be"). However, isolating r is difficult (and "difficult", in this case, means "impossible except for very small values of nt"). We can convince ourselves graphically that is at least possible to find r; the below graph uses $400 for PMT and 4 for t, and graphs the value of PV against r: