Last year, when we published the Unified Pension Scheme vs National Pension Scheme Calculator, many readers were surprised that we used a single formula to determine future NPS corpus by treating it as a step-up SIP. Since then, several users of that tool have asked us to explain the formula. So here are the details.
A step-up SIP refers to periodic investments where the invested amount increases at some rate. This is technically called a growing annuity.
SIP formula
- SIP = monthly SIP amount. We shall multiply this by 12 and consider the corpus from a yearly SIP. Since the returns are variable, the difference between a monthly SIP and a yearly SIP is insignificant over time,
- n = no of years the investment is made
- r = expected rate of return (you can account for taxes by suitably lowering this value)
Corpus = 12*SIP*[(1+r)^n-1]/r
investment made at the end of each period (year in this case)
Corpus = 12*SIP*(1+r)*[(1+r)^n-1]/r
investment made at the start of each period (year in this case)
Here ^ refers to, to the power of. x^2 = x*x. 2^3 = 2*2*2 = 8
Step-up SIP formula
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Now, we include another variable
- g – the rate at which investments increase each year. For example, if g = 10%, and the initial monthly SIP amount is Rs. 1000. After one year, the SIP amount is Rs. 1100 and so on.
Corpus = 12*SIP*[(1+r)^n- (1+g)^n]/(r-g)
investment made at the end of each period (year in this case)
Corpus = 12*SIP*(1+r)*[(1+r)^n- (1+g)^n]/(r-g)
investment made at the start of each period (year in this case)
What if g > r?
Many readers wonder if this formula will work if g >r. The corpus will not become negative. This is because when g>r, the factor [(1+r)^n- (1+g)^n] is negative. But (r-g) in the denominator is also negative and the signs cancel off.
There is, however, one issue in this formula.
What if r =g? Then, the denominator becomes zero, and we will get a 1/0 zero error, also known as a singularity in math.
When r=g, the following formula applies for a step-up SIP
Corpus = 12*SIP*n*(1+r)^(n-1)
investment made at the end of each period (year in this case)
Corpus = 12*SIP*n*(1+r)^n
investment made at the start of each period (year in this case)
Play around with these formulae and have fun. If you are interested in investment analysis, financial planning, financial advisory, learning, and creating custom versions of calculators, you can consider joining the freefincal investor circle! This is an exclusive space for investors, advisors, fintech employees and students to access financial planning and insurance tools, mutual fund and stock analysis tools, coding strategies and Excel macros for data extraction.