# Effective interest rate for mortgage loan

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%.

• 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. 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%).. 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% Aug 19, 2017 at 17:14
• The image shows a footnote 1 after the 0.9804% figure: does that say anything relevant? Aug 21, 2017 at 6:52

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 %
`````` • yes. But in the book, it states that the monthly rate is 0.9804%. Wondering if I missed something the textbook assumed. 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. Aug 19, 2017 at 10:44
• hey Chris, up there, I uploaded the screenshot of the case. second paragraph from the bottom. Aug 19, 2017 at 11:53
• 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 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? Aug 19, 2017 at 14:21