Derivation of Loan/Mortgage Monthly Payment Formula

Physics Ninja looks at the derivation of the Amortization Formula used to calculate the monthly payment on a loan or mortgage

[@watch7749]

You sir, are awesome! This is the BEST explanation I have come across! I don't know if its because loan sellers don't want to be super clear or if they just lack the math background, but I hadn't found any of their explanations to make sense. This made it crystal clear, and I feel so much safer knowing what I'm getting myself into!

[@boredomgotmehere]

What a wonderful and thoughtful explanation. You not only gave the superficial meaning but the contextual meaning by doing the heavy lifting of proving the geometry series. Super thankful.

[@PhysicsNinja]

Thanks for the comment. I appreciate it.

[@TusharMajumdar]

Thank you very much. Was looking for how the loan was calculated and only getting the formula everywhere. But your explanation clears it out. Thank you!

[@EverApprentice]

I've been looking for how this formula is determined all over the internet, and finally here it is!! Thanks for the amazing explanation.

[@tomera79]

Thank you. Very clear and useful! I wish more tutorials on the Internet were like this one!

[@avankm774]

I cannot give the amount of likes that I want. This video made my night after a whole day looking for explanations on how to calculate this and also how to make the table for each month. Awesome video and keep up the good work, thank you!

[@PhysicsNinja]

Thanks for the support!

[@fozzydare7987]

I’m determined to understand this fully - it’s going to take more than one watch 😅 Thank you for your work

[@PhysicsNinja]

It probably took me 4-5 times

[@otis299]

Love your videos! Just FYI, the dollar sign goes in front of a dollar amount e.g., $100.45 or $1M for one million dollars. The lesser used cent sign “¢” would go after the number and would be omitted in the presence of a dollar sign e.g., 99¢. Thanks for all you do!

[@arielfuxman8868]

This makes so much sense. Excellent for us Mathematically oriented people.

[@PhysicsNinja]

Thanks for watching.

[@wanghonghui1553]

Sir, you are a treasure for our society!

[@fozzydare7987]

I’m determined to understand this fully - it’s going to take more than one watch 😅 Thank you for your workI

[@EverApprentice]

Great video. Clear explanation and Python demonstration on this prolonged question I have in mind. Thanks for that!

[@makk3480]

just one line for the video: AMAZING EXPLANATION!

[@PhysicsNinja]

Thank you

[@woojin8369]

Thank you for exploring your curioisty and your due dilligence! This was inisghtful :))

[@tomaszandraka8985]

Take it one step closer to make the formula a little cleaner: take out the (1+r)^N term from the top and bottom - this simplifies to: M = (P*r)/(1-(1+r)^-N). Great explanation!

[@patomatma]

awesome job

[@JAMYLIVES]

Wow! wonderfully explained. Thank you!

[@PhysicsNinja]

Glad it was helpful!

[@arkleston3299]

Thank you for explaining this so well!

[@PhysicsNinja]

You're very welcome!

[@09blueumbrella]

This was very helpful. Thank you!

[@ramineslami6053]

thank you so much. it was really clear and useful.

[@bansheedearg]

Thanks for sharing the python code, for python3, print() is a function so you'll have to wrap the strings in parens. Great video, thank you!

[@prashantdahiya711]

Thank You for the same, brilliant explanation.

[@blackblather]

wonderful explanation. Thank you

[@majid1885]

great video, thank you

[@Kappy0001]

Very nice work! Thanks!

[@juant4821]

Excellent derivation!

[@funnelboss5721]

Awesome Thanks for sharing, may i ask why simplify geometry part (2) - (1)?

[@williamrhopkins]

Very nice -- a couple of nuances. The compounding period can vary. In your example interest is compounded monthly. In Canada mortgagee rates are compounded semi annually so the monthly interest rate is (1+r/2)^(2/12). Often banks actually calculate a daily interest rate. For semi annual compounding the daily rate for non leap years is (1+r/2)^(2/365) and for leap years (1+r/2)^(2/366). This allows for calculating weekly and biweekly repayments. For monthly payments the amount of interest that is applied to the principle will vary depending on the length of the month. Lastly the financial literacy of sales people and loan managers is abysmal -- just ask any of them to explain how the loan is amortized or challenge them to produce a spreadsheet that provides an amortization schedule. Come on this is high school math -- no calculus involved.

[@Epistemer]

Thanks man, great explanation :)

[@gtrecc5454]

Great video

[@orionfollett7056]

Isn't dividing the annual interest rate by 12 to get the monthly rate an approximation? Going up 12% a year is not the same as going up 1% a month. Anyone can easily see this by just inputting it into a calculator if they don't believe me. (1+.12)^1 = 1.12 =/= (1+.01)^12 = 1.1268... I know this is how most calculators work still but I wonder why this approximation is always used?

[@ScienceDiscoverer]

I noticed this too... I guess its just because of the en.wikipedia.org/wiki/Time_value_of_money If you pay your loan in the end of the year in one payment, you will have to pay more, as all the payment will be in the future, thus it cost will be more than if you paid small sum every day or every month. You are still paying bulk of the loan in the future, but few payments in first month worth much more than those in the end, so the end interest sum is lower in this case, although the "annual" interest stays the same!

[@ecolibertarian5622]

The assumption r=R/12 is never used in the actual derivation so the formula still stands. And you are right, r would be the monthly interest rate that translates to the R annual rate. i.e (1+r)^12 = R.

[@susovandey4531]

Awesome brother

[@adamidrisnuh4328]

Thank you... Very much appreciate. If my lecturer teach this formula, i think my financial management will get A.

[@CliffSedge-nu5fv]

I've actually been looking for a derivation of present value given future value, but I think I can use this as a guide for how to set it up.

[@mukamafrancois2776]

Awesome!

[@PhysicsNinja]

Thanks!

[@matteogriseri6513]

THANK YOU FINALLY

[@b.c.2836]

Taking python class, already built a calculator for amortization total interest paid, but no wanted to bulid one that took into account extra principal payments and I was looking for that formula.

[@NeonCurveZz]

You got rid of the porsche? I thought you were planning on getting an R8 next?!

[@youpattube1]

Thanks for the derivation. But...... And this may sound like something a flat earther would ask, but it seems like interest gets figured in twice. To get the Current Payment, you take the Prevous Balance, subtract the previous payment, and add the interest paid on the previous balance. But the interest was already figured in the monthly payment. I'm just a little confused.

[@martincstee5335]

This video explains it better: kzbin.info/www/bejne/qKWsc3qel7iYeLs&ab_channel=DrBobMaths%28OnlineMathsTuition%29 Essentially the first month when you obtain the loan, you do not make any payment. The interest for the first month accrues at the end of the first month, regardless if you make the payment either at the end of the first month or the beginning of the second month. So, at the end of the first month interest was only factored in for the first month. At the end of the second month, interest was only factored in for the balance minus the first constant payment. One can argue that the constant monthly payment M already contains the interest Pnr. But at least for the first month the interest of P0r is correct. Maybe some math genius can derive a more logical formula that can save us all some interest payment.

[@shivankagrawal9764]

16:21 But why did you multiply X by (1+r)

[@bansheedearg]

So he could eliminate all the internal terms of (1+r)^n. If I have 1 + A + A*A + A*A*A + A^4 + A^5, I can multiply and divide by A - 1 to get (A^6 - 1) / (A - 1), or why he subtracted equation (2) from equation (1)

[@cradleofrelaxation6473]

Thanks. I was like how the heck did my back come up with the money payment?

[@user-fy4iq6if4z]

짱입니당

[@Scott-jk5zk]

18:26 how does (1+r)X-X=rX ?

[@Scott-jk5zk]

one sec X((1+r)-1) X(r+1-1) X(r) rX I see now

[@NgocTruong-zt9dp]

Bro can you explain why p1 = p0 -M +p0*r ?

[@near5128]

5:40 and 20:14

[@SubhashChandraDey-h2m]

Well i can explain this a little more simply. Just explain what annuities are and how it's formula is derived. Done!

[@PhysicsNinja]

Clear as mud!

[@truthfirst9488]

Is this for USA or Canada?

[@fozzydare7987]

I’m determined to understand this fully - it’s going to take more than one watch 😅 Thank you for your work

[@swimxian]

Thank you very much. Was looking for how the loan was calculated and only getting the formula everywhere. But your explanation clears it out. Thank you!