# Payments and Interest Rates

Suppose you have two options for making a payment:

(A) Pay $90 \%$ of the purchase price two months after the date of the sale.

(B) Deduct $X \%$ off the purchase price and pay cash on the date of sale.

Determine $X$ so that a customer is indifferent between the two options when valuing them using an effective annual interest rate of $8 \%$.

Let $P$ be the purchase price. So option (B) is: $P \left(1- \frac{X}{100} \right)$. Now option (A) would be $0.9P(1.08)^{1/6}$ since we want to know what $0.9P$ would be worth in $2$ months? Then set $P \left(1- \frac{X}{100} \right) = 0.9P(1.08)^{1/6}$ to find $X$?

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## 1 Answer

Edited.

Not quite.

Here's how it works: let $P$ be the price. Under option (A), you take the money $P$ you would have paid, put it in the bank generating interest, and at the end of the two months you pay $0.9P$, keeping the difference between what you have at the end of the two months and the price you pay. So under option (A), at the end of two months you have both the item plus $P(1.08)^{1/6}-0.9P$ in cash.

Under option (B), you pay $(1- X/100)P$ now, keeping $(\frac{X}{100})P$ of the price, which you can then put in the bank for two months generating interest. So under option (B), at the end of the two months, you have both the item and $\left(\frac{X}{100}P\right)(1.08)^{1/6}$ in cash.

You are trying to find the value of $X$ that will make both options equal. (Note that the value $P$ does not really matter, because it will cancel out.) So the quantities you need to set equal to one another are not exactly what you have, but something slightly different.

Added. Equivalently, as you note in the comments, we may instead want to compare the present value of option (A) with option (B) (rather than comparing how much money you will end up with if you set aside $P$ under options (A) and (B), two months down the line). The present value of option (A) is the amount of money $Y$ you would need to put in the bank at $8\%$ interest so that at the end of two months you have $0.9P$ to pay for the item; that is, $Y(1.08)^{1/6} = 0.9P$, or $Y = \frac{0.9P}{(1.08)^{1/6}}$. So you want to find $x$ for which the current expenditure $(1-x)P$ equals the current value of option (A), $(1-x)P = \frac{0.9P}{(1.08)^{1/6}}$. Again, $P$ drops out, and we get the formula you have in the comments, $$1 - x = \frac{0.9}{(1.08)^{1/6}}.$$ This gives $x = \frac{X}{100} = 1 - \frac{0.9}{(1.08)^{1/6}}$, or a little over $11$% discount.

If you proceed as I described originally, and fix the mistake I originally made, you get exactly the same formula and the same answer.

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I see. I get $\left(1-\frac{X}{100} \right)(1.08)^{1/6} = 0.9$ as the equation to solve. – PEV Nov 22 '10 at 20:22
@Trevor: Hmmm... I don't get that. Let $x=\frac{X}{100}$. Then you have $[(1.08)^{1/6}-0.9]P = (1-x)(1.08)^{1/6}P$. Cancelling $P$, we have $(1.08)^{1/6}-0.9 = (1-x)(1.08)^{1/6}$. How do you go from there to your equation? – Arturo Magidin Nov 22 '10 at 20:28
For option (B) the amount paid is $P\left(1-\frac{X}{100} \right)$ immediately. For option (A) the amount paid in two months is $0.9P$. The present value on the date of the sale is $\frac{0.9P}{1.08^{1/6}}$. So then I got: $P \left(1-\frac{X}{100} \right) = \frac{0.9P}{1.08^{1/6}}$ which simplifies to the above. – PEV Nov 22 '10 at 20:32
@Trevor: Sorry! I had a mistake in my derivation of case (B), saying you were paying $\frac{X}{100}$ now instead of $1-\frac{X}{100}$; so I was computing the percentage of the price, you were computing the discount. That's why we were getting complementary answers. Yes, your answer is correct. I've fixed the discussion, and added the "present value" version. – Arturo Magidin Nov 22 '10 at 21:14