# Did you know: A single equation governs investing and borrowing?

Published: May 22, 2024 at 6:00 am

We discuss how pretty much all of investing (lump sum or SIP) and borrowing (debt., EMI) can be described by a single equation! In finance workshops, people are often taught to use spreadsheet commands like PV, FV, PMT, NPER, etc., without a deeper understanding.

Consider a lump sum investment we shall label as pv (for present value). What is the future value (fv) of this investment? The well-known compounding formula gives this.

fv=pv(1+rate) nper

Here, the rate is the interest rate or the rate of return, and nper refers to the number of periods corresponding to the rate of return. We shall keep things simple here and assume the rate is the annual return and nper is in years. Other variations like monthly rates or quarterly rates are also possible.

What if I wanted to invest each year? Then, the formula is

fv= pmt[(1+rate) nper-1]/rate  if the payments are made at the end of the period or

fv= (1+rate)pmt[(1+rate) nper-1]/rate if the payments are made at the start of the period

This is also known as the SIP formula. Here, pmt is the periodic payment. This can be each year, each quarter, or each month with a corresponding rate. We shall keep things simple and assume a yearly SIP. Over the long term, it matters little whether you use the monthly SIP or year SIP variants. The markets and not this formula determine the return you get!

Join over 32,000 readers and get free money management solutions delivered to your inbox! Subscribe to get posts via email!
🔥Enjoy massive discounts on our robo-advisory tool & courses! 🔥

So what if you have a lump sum and periodic investments?

fv= pv(1+rate) nper + pmt[(1+rate) nper-1]/rate –> [1]

This is the combined formula (we have assumed payments are made at the end of the period).

This equation can compute fv, pv, nper, rate and pmt if the other quantities are known. Those familiar with spreadsheet formulae would immediately recognise these quantities.

The above equation represents investing. What about borrowing? I will introduce the equation first and then explain it.

balance= loanamt(1+rate) nper – emi[(1+rate) nper-1]/rate –> [2]

Let us consider the example of a home loan. Given a loan rate, how is the emi calculated? Suppose you want a loan to buy a home. Let us call the loan amount =loanamt. The bank would ask itself, suppose instead of giving this loan to you, if it invests the amount = loanamt at the home loan rate, what would be the future value fv at the end of the home loan tenure nper?

fv=loanamt(1+rate) nper

Therfore, for the loan to make financial sense to the bank, it asks what monthly payments (emi) should be made by you at the same rate so that at the end of the loan tenure (nper), the corpus from these EMIs is equal to the fv?

In other words

fv = emi[(1+rate) nper-1]/rate

So, at the end of the loan tenure

loanamt(1+rate) nper = emi[(1+rate) nper-1]/rate

Since both of them are equal. Or we can write

0 = loanamt(1+rate) nperemi[(1+rate) nper-1]/rate

Let us consider an example.

• loanamt = 50,00,000
• nper = 20 years = 240 months
• rate = 10%

So if the bank invests the loanamt for 20 years at 10%, it would get

loanamt(1+rate) nper =5000000*(1+(10%/12))^(20*12) = 3,66,40,368

If the bank gives it to you, the emi is 48,251. Why?

emi[(1+rate) nper-1]/rate =48251*((1+(10%/12))^(20*12)-1)/(10%/12) =3,66,40,368

So, after 20 years,

loanamt(1+rate) nperemi[(1+rate) nper-1]/rate= zero

That is, the future values of a lump sum and SIP (= EMI) are the same at the end of the loan tenure.

What is the situation after one year?

loanamt(1+rate) nper = 5000000*(1+(10%/12))^(12*1) = 55,23,565

emi[(1+rate) nper-1]/rate = 48251*((1+(10%/12))^(12*1)-1)/(10%/12) = 6,06,302

These two numbers do not ring a bell, but

55,23,565 – 6,06,302 = 49,17,263 = home loan balance after one year of paying EMIs

Similarly

loanamt(1+rate) nper – emi[(1+rate) nper-1]/rate = home loan balance after nper years of paying EMIs

So, the full equation is

balance= loanamt(1+rate) nper – emi[(1+rate) nper-1]/rate –> [2]

This is our 2nd equation, hence the [2]. Now, compare this with the first equation.

fv= pv(1+rate) nper + pmt[(1+rate) nper-1]/rate –> [1]

We can now combine the two into one “master equation” to represent all of investing and borrowing!

fv= pv(1+rate) nper ± pmt[(1+rate) nper-1]/rate

If it is investing, use the + sign and

• fv = corpus value
• pv = lump sum investment
• rate = rate of return
• nper = duration of the investment
• pmt = periodic investment

If it is borrowing, use the – sign and

• fv = loan balance
• pv = amount borrowed
• rate = rate of borrowing
• nper = duration of the loan
• pmt = periodic payment to close the loan

All the spreadsheet formulae like PV, FV, PMT, RATE, and NPER use this master equation. I would strongly recommend students of finance and financial advisors base their results on the master equation without blindly using spreadsheet formulas.

🔥Enjoy massive discounts on our courses, robo-advisory tool and exclusive investor circle! 🔥& join our community of 5000+ users!
Use our Robo-advisory Tool for a start-to-finish financial plan! More than 1,000 investors and advisors use this!
New Tool! => Track your mutual funds and stock investments with this Google Sheet!
We also publish monthly equity mutual funds, debt and hybrid mutual funds, index funds and ETF screeners and
Podcast: Let's Get RICH With PATTU! Every single Indian CAN grow their wealth!  You can watch podcast episodes on the OfSpin Media Friends YouTube Channel. 🔥Now Watch Let's Get Rich With Pattu தமிழில் (in Tamil)! 🔥
• Do you have a comment about the above article? Reach out to us on Twitter: @freefincal or @pattufreefincal
• Have a question? Subscribe to our newsletter using the form below.
• Hit 'reply' to any email from us! We do not offer personalized investment advice. We can write a detailed article without mentioning your name if you have a generic question.

Join over 32,000 readers and get free money management solutions delivered to your inbox! Subscribe to get posts via email!

(PhD) is the founder, managing editor and primary author of freefincal. He is an associate professor at the Indian Institute of Technology, Madras. He has over ten years of experience publishing news analysis, research and financial product development. Connect with him via or Pattabiraman has co-authored three print books: (1) You can be rich too with goal-based investing (CNBC TV18) for DIY investors. (2) Gamechanger for young earners. (3) Chinchu Gets a Superpower! for kids. He has also written seven other free e-books on various money management topics. He is a patron and co-founder of “” an organisation promoting unbiased, commission-free investment advice.
Our flagship course! Learn to manage your portfolio like a pro to achieve your goals regardless of market conditions! More than 3,000 investors and advisors are part of our exclusive community! Get clarity on how to plan for your goals and achieve the necessary corpus no matter the market condition is!! Watch the first lecture for free!  One-time payment! No recurring fees! Life-long access to videos! Reduce fear, uncertainty and doubt while investing! Learn how to plan for your goals before and after retirement with confidence.
Our new course!  Increase your income by getting people to pay for your skills! More than 700 salaried employees, entrepreneurs and financial advisors are part of our exclusive community! Learn how to get people to pay for your skills! Whether you are a professional or small business owner who wants more clients via online visibility or a salaried person wanting a side income or passive income, we will show you how to achieve this by showcasing your skills and building a community that trusts and pays you! (watch 1st lecture for free). One-time payment! No recurring fees! Life-long access to videos!
Our new book for kids: “Chinchu Gets a Superpower!” is now available! Most investor problems can be traced to a lack of informed decision-making. We made bad decisions and money mistakes when we started earning and spent years undoing these mistakes. Why should our children go through the same pain? What is this book about? As parents, what would it be if we had to groom one ability in our children that is key not only to money management and investing but to any aspect of life? My answer: Sound Decision Making. So, in this book, we meet Chinchu, who is about to turn 10. What he wants for his birthday and how his parents plan for it, as well as teaching him several key ideas of decision-making and money management, is the narrative. What readers say!
Must-read book even for adults! This is something that every parent should teach their kids right from their young age. The importance of money management and decision making based on their wants and needs. Very nicely written in simple terms. - Arun.
How to profit from content writing: Our new ebook is for those interested in getting side income via content writing. It is available at a 50% discount for Rs. 500 only!
Do you want to check if the market is overvalued or undervalued? (it will work with any index!), or get the Tactical Buy/Sell timing tool!
We publish monthly mutual fund screeners and
About freefincal & its content policy. Freefincal is a News Media Organization dedicated to providing original analysis, reports, reviews and insights on mutual funds, stocks, investing, retirement and personal finance developments. We do so without conflict of interest and bias. Follow us on Freefincal serves more than three million readers a year (5 million page views) with articles based only on factual information and detailed analysis by its authors. All statements made will be verified with credible and knowledgeable sources before publication. Freefincal does not publish paid articles, promotions, PR, satire or opinions without data. All opinions will be inferences backed by verifiable, reproducible evidence/data. Contact information: letters {at} freefincal {dot} com (sponsored posts or paid collaborations will not be entertained)
Connect with us on social media
Our publications

You Can Be Rich Too with Goal-Based Investing

Published by CNBC TV18, this book is meant to help you ask the right questions and seek the correct answers, and since it comes with nine online calculators, you can also create custom solutions for your lifestyle! Get it now.
Gamechanger: Forget Startups, Join Corporate & Still Live the Rich Life You Want This book is meant for young earners to get their basics right from day one! It will also help you travel to exotic places at a low cost!