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Comparing Two Discounts

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THE SCENARIO A while back, I covered a discount offered by a storage place in which I rent a unit. Some time has passed, and they now have two different discounts that I can choose between. They're as follows:
  1. Pre-pay for 6 months, and get the last month half off (i.e. pay for 5 months, but get 6)
  2. Pre-pay for 12 months, and get the last month for free (i.e. pay for 11 months, but get 12)
In looking at those deals, they seem like the same thing! All the numbers have the same proportion, so wouldn't the discount be the same? After all, at the end of the year of doing the first one twice or the second one once, I'd be in the same position - I'd have paid for 11 months of rent, but gotten 12. That's my intuition as well - or it would be if I hadn't practiced a bunch with my financial calculator. So let's figure out which of these discounts I should take advantage of. The question: If my rent is $200 per month, which of the options gives me a better return on my money?
THE SOLUTION This one is pretty simple, and has three parts, none of which is overly complicated.
  1. Determine the yield for the 6-month discount
  2. Determine the yield for the 12-month discount
  3. Compare the yields
First things first, make sure the calculator is using 12 Payments per Year. Part 1: Evaluating paying for 5 months but getting 6 First off, I have to figure out my prepay rent. $200 x 5.5 = $1,100 N: 6 (I get 6 months worth of rent by prepaying) I/YR: (This is what I'm trying to find) PV: -1,100 (When I prepay, I fork over $1,100) PMT: 200 (I'm saving myself $200 per month by prepaying) FV: 0 (At the end, I owe rent again)

My yield on 6-month prepayment is 30.53%.

Part 2: Evaluating paying for 11 months but getting 12 First off, I have to figure out my prepay rent. $200 x 11 = $2,200 N: 12 (I get 12 months of rent by prepaying) I/YR: (This is what I'm trying to find) PV: -2,200 (I have to give them $2,200 up front to pay my rent for a year) PMT: 200 (I'm saving myself $200 per month by prepaying) FV: 0 (At the end, I owe rent again)

My yield on 12-month prepayment is 16.38%.

Part 3: Comparing the two If I prepay for 6 months, my yield is 30.53, but if I prepay for 12 months, my yield drops to 16.38. And that's generally the way this works - doubling the time cuts the yield (roughly) in half. Halving the time (roughly) doubles the yield. But why? The short answer is: Time. In the 6 month scenario, I don't have to wait nearly as long to get my money back, to 'make a profit', or to 'be in the black' on the deal. I get half a month's rent after 5 months, rather than waiting twice as long to see my profit. If that's a hard concept to wrap your mind around (it was for me, at first), try playing around with your calculator, running different scenarios, and you'll see that it ends up being true - get your profit sooner, and your yield goes up. Wait for your profit, and your yield goes down. That's because the way we represent yield is Per Year - Time is baked into the equation.

What do you think? Did you expect this outcome? Are you surprised by it? Do you even believe it? Have you run across similar prepayment discount opportunities in your life? Let us know in the comments!