How do you calculate the constant payment by hand ?

Question

It is an armotized loan problem: "A homeowner borrows $100,000 on a mortgage loan, and the loan is to be repaid in five equal payments at the end of each of the next 5 years. The lender charges 6 percent on the balance at the beginning of each year." I know you can use the PMt function on Excel, but how does it do it?

There are some hints in the book but I didn't get: "You could also factor out the PMT term, find the value of the remaining summation term (it’s 4.212364), and then divide it into the $100,000 to find the payment, $23,739.64.

Where does 4.212364 come from?

Answer 1

Let annual payment = x
Convert everything to the present value:

Present value of the loan = P
Present value of first payment = x / (1 + r)
ie x/(1+r) is the amount NOW that would be worth x in one year from now.
Present value of 2nd payment = x / (1+r)^2 via the same reasoning

Present value of nth payment = x / (1+r)^n

So we require that
x/(1+r) + x/(1+r)^2 + x/(1+r)^3 + x/(1+r)^4 + x/(1+r)^5 = P

Now substituting the numerical values, we get
x * (4.212) = 100,000
x = $23,739.64

**EDIT**
The 4.212 comes from adding up all the powers of 1/(1+r)
ie 1/1.06 + 1/1.06^2 + ... + 1/1.06^5 = 4.212
Before this I factored out the x, which was common to every term on the LHS.

Answer 2

Set up the appropriate equation. You start with 100,000, multiply by 1.06, and subtract x at the end of the year. Repeat five times, and this should equal zero, since the 100,000 is supposed to be paid off. So we get
((((106000-x)*1.06-x)*1.06-x)*1.06-x)*1.06-x = 0

Work this out, you won't get any square terms or anything hard, but it is kind of tedious.