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 32,000+ readers and get free money management solutions delivered to your inbox! Subscribe to get posts via email! (Link takes you to our email sign-up form)
🔥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?
The answer is
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) nper –emi[(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) nper – emi[(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 7000+ users!
Use our Robo-advisory Tool for a start-to-finish financial plan! ⇐ More than 2,500 investors and advisors use this!
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 momentum, low-volatility stock screeners.
- 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 32,000+ readers and get free money management solutions delivered to your inbox! Subscribe to get posts via email! (Link takes you to our email sign-up form)
About The Author
Dr M. Pattabiraman(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 Twitter(X), Linkedin, or YouTube. 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 “Fee-only India,” 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!
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.Buy the book: Chinchu gets a superpower for your child!
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? Use our market valuation tool (it will work with any index!), or get the Tactical Buy/Sell timing tool!
We publish monthly mutual fund screeners and momentum, low-volatility stock screeners.
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 Google News. 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
- Twitter @freefincal
- Subscribe to our YouTube Videos
- Posts feed via Feedburner.
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! Get it or gift it to a young earner.Your Ultimate Guide to Travel
This is an in-depth dive into vacation planning, finding cheap flights, budget accommodation, what to do when travelling, and how travelling slowly is better financially and psychologically, with links to the web pages and hand-holding at every step. Get the pdf for Rs 300 (instant download)