Question 1:
Looking over a text, it asks to prove that a simple discount rate \(\displaystyle d\) is equivalent to a simple interest rate \(\displaystyle r\) where \(\displaystyle r = \frac{d}{1-dt}\). The solution in the book does this by setting total interest \(\displaystyle I = Prt\) (where \(\displaystyle P\) is the principal, and \(\displaystyle t\) time) equal to the discount \(\displaystyle D = Sdt\) (where \(\displaystyle S\) is the amount of the loan, and \(\displaystyle d\) the discount rate) and set the principal of the interest equal to the proceeds of the loan, both represented by \(\displaystyle P\). They then solve for \(\displaystyle r\).
With those assumptions in place, I totally get the math. But what I guess I don't understand about the problem is, why would we set these equal to each other? In what sense are we showing that the two are equivalent? Since the principal of the investment is what you start with before accruing any interest, I would think that we would equate this with \(\displaystyle S\), the total loan amount before interest discounts.
Question 2:
So looking at an explanation of ordinary annuities, it says that for a payment of \(\displaystyle R\) after \(\displaystyle n\) payment periods, with interest rate \(\displaystyle i\), is described by the table:
Payment...............Conversion Periods.............Amount
1.........................n-1.................................R(1+i)^(n-1)
2.........................n-2.................................R(1+i)^(n-2)
... And so on. What I find a little odd about this is that, at the last payment period, no interest is charged. It seems like the basic way this works is: Get a loan or whatever, and at the end of the first payment period, you pay back a portion of the total, and then pay interest on what's left. But the business which loaned you money seems like they're losing an opportunity to charge you interest on the money you've been using up until the end of the first payment period. Am I misunderstanding something, or is this just not something that businesses care about?
Looking over a text, it asks to prove that a simple discount rate \(\displaystyle d\) is equivalent to a simple interest rate \(\displaystyle r\) where \(\displaystyle r = \frac{d}{1-dt}\). The solution in the book does this by setting total interest \(\displaystyle I = Prt\) (where \(\displaystyle P\) is the principal, and \(\displaystyle t\) time) equal to the discount \(\displaystyle D = Sdt\) (where \(\displaystyle S\) is the amount of the loan, and \(\displaystyle d\) the discount rate) and set the principal of the interest equal to the proceeds of the loan, both represented by \(\displaystyle P\). They then solve for \(\displaystyle r\).
With those assumptions in place, I totally get the math. But what I guess I don't understand about the problem is, why would we set these equal to each other? In what sense are we showing that the two are equivalent? Since the principal of the investment is what you start with before accruing any interest, I would think that we would equate this with \(\displaystyle S\), the total loan amount before interest discounts.
Question 2:
So looking at an explanation of ordinary annuities, it says that for a payment of \(\displaystyle R\) after \(\displaystyle n\) payment periods, with interest rate \(\displaystyle i\), is described by the table:
Payment...............Conversion Periods.............Amount
1.........................n-1.................................R(1+i)^(n-1)
2.........................n-2.................................R(1+i)^(n-2)
... And so on. What I find a little odd about this is that, at the last payment period, no interest is charged. It seems like the basic way this works is: Get a loan or whatever, and at the end of the first payment period, you pay back a portion of the total, and then pay interest on what's left. But the business which loaned you money seems like they're losing an opportunity to charge you interest on the money you've been using up until the end of the first payment period. Am I misunderstanding something, or is this just not something that businesses care about?
