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!

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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) 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


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.

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