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I'm currently deciding between taking loan with down payment and no down payment. To get behind the idea, I read a case study from a finance textbook.

I get the basic math and understand concept of present value, but couldn't understand how some numbers came up while analysing it. Here is the case:

Mortgage loan is $100,000, with 30-year duration. You can choose either to pay down payment or not. If you don't, annual interest rate is 12%, if you do take the offer of paying $2,000 (2% discount point off initial $100,000), you get 11.5% annual interest rate.

Case 1. No down payment, annual interest rate is 12%, therefore, monthly is 12%/12 = 1%. Compounding monthly: 

Effective annual rate =  (1.01)^12 - 1 = 0.1268, which is = 12.68%

Case 2. Down payment = $2,000. (so, now we owe $100,000-$2,000 = $98,000) Interest rate at 11.5%, therefore, monthly should be 11.5%/12 = 0.9583%.

In this case, using finance calculator, monthly payment would be $990.29

Now, this is where the confusion begins. My monthly rate as what I manually calculated is 0.9583%, BUT the book states it should be 0.9804%.

Hence, Question: How and why does the monthly rate turn out to be 0.9804%??

If we reverse the calculation with that rate, it turns out we actually get higher interest rate; 0.9804% * 12 = 11.76%, higher than initial 11.5%.

Quote/screenshot from the book: enter image description here enter image description here

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    The 2 'points' for which you pay 2000$ will buy you the lower interest rate. They are not a down payment, you still owe the full amount.
    – Aganju
    Aug 19, 2017 at 13:56
  • @Aganju thanks for the comment. Agreed. I misunderstood the concept. Feel free to edit my question. That being said, how would the calculation using 'points' affect the interest rate? As you can see, the book suggest 0.9804%. That's 11.76% annual. Higher than what I would be paying at 11.5% (0.9583%)..
    – Mr. Slow
    Aug 19, 2017 at 14:25
  • The 2K is interest. If you payoff the loan before the breakeven point the interest rate is very high. But it will always be larger then 11.5%
    – mhoran_psprep
    Aug 19, 2017 at 17:14
  • The image shows a footnote 1 after the 0.9804% figure: does that say anything relevant?
    – TripeHound
    Aug 21, 2017 at 6:52

1 Answer 1

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With the $2000 downpayment and interest rate of 11.5% nominal compounded monthly the monthly payments would be $970.49

As you state, that is a monthly rate of 0.9583%

Edit

With the new information, taking the standard loan equation

s = (d - d (1 + r)^-n)/r

where

s is the loan principal
d is the periodic payment
r is the periodic interest rate
n is the number of periods

Let

s = 100000
r = 0.115/12 = 0.00958333
n = 30*12 = 360

d = (r (1 + r)^n s)/((1 + r)^n - 1) = 990.291

Now setting s = 98000, with d = 990.291 solve for r

r = 0.980354 %

enter image description here

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  • yes. But in the book, it states that the monthly rate is 0.9804%. Wondering if I missed something the textbook assumed.
    – Mr. Slow
    Aug 19, 2017 at 9:01
  • @Mr.Slow Can you quote more of the book, or include an image, because that rate seems unrelated to anything else you're mentioned.
    – Chris Degnen
    Aug 19, 2017 at 10:44
  • hey Chris, up there, I uploaded the screenshot of the case. second paragraph from the bottom.
    – Mr. Slow
    Aug 19, 2017 at 11:53
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    Hey Chris, just a thought for you. There is no down payment. The $2000 is "points". To the OP - points or money paid to the bank or lender in exchange for a bit of a lower interest rate on the loan. I will let Chris take it from here
    – JTP - Apologise to Monica ♦
    Aug 19, 2017 at 12:25
  • @JoeTaxpayer, My bad, you're right, just realised they are points which means in both cases actually there are no money down. However, wouldn't the rates be unaffected? I believe that monthly rate should be 0.9583% (11.5% annual)? If you look at the screenshot, the book suggest higher interest rate, 0.9804% (11.76%). How does it get to that number?
    – Mr. Slow
    Aug 19, 2017 at 14:21

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