Sunday, December 31, 2017

Published 3:24 PM by with 0 comment

How Much Should You Invest To Create Your Own Pension?

The concept of a pension fascinates me since it would be great to have your investments pay your living expenses while you work wherever you want without worrying about a salary. I've created a calculator here that allows you to figure out the numbers you need to create your own pension plan.

Calculator


Initial Amount ($) Investment Yield (%) Retirement Yield (%) Defer Years Monthly Payout ($) Payment Years Inflation (%)

Required Monthly Deposit ($)


Derivation

I haven't actually seen this solved anywhere else online so I'm posting the full math behind this since it took me a couple of hours to get right and was a fun problem. A number of places have very simplified versions of it, but none that I found included the investment, the payout, and the inflation. Note that no one reviewed this so I might have typos, and feel free to let me know in the comments if I do.

The basic problem being solved here is finding the monthly input that makes the future value of an investment at the end of the defer period equal to the present value of the payout at the end of the defer period:

PVout = present value of the payouts at the end of the defer period
FVin = future value of the total investments at the end of the defer period


First, we need to figure out how to calculate those. Start with the future value of the investment. 

We have an initial investment that's the 'Initial Amount' in the calculator. That will grow at the 'Investment Yield' for the entire defer period. Its value at the end of the defer period is simply:

PVo = initial investment
yin = investment yield per month
t = months in defer period


Next, we have the future value of the monthly investments. This is the hardest one to calculate and I was unable to come up with a simple equation for it. To start with, you want the value at the end of the defer period of the first two months. That's:

FV = future value of the term
Pin = amount invested each month in today's dollars
i = monthly inflation rate
yin = monthly investment yield
t = months in defer period



This continues and thinking about it all the way through the defer period, you end up with:

Pin = amount invested each month in today's dollars
i = monthly inflation rate
r = monthly investment yield
t = months in defer period


I don't know how to simplify this summation any further so we're left with this:

FVin = future value of investments at end of defer period
PVo = initial investment
Pin = amount invested each month in today's dollars
i = monthly inflation rate
r = monthly investment yield
t = months in defer period


Finally, we move to the present value of the payout at the end of the defer period. Do like before and take the first couple of months then write the general sum:

Pout = monthly payment
t = months in defer period
n = months in payout period
yout = monthly retirement yield
i = monthly inflation rate



Noting that t and i are constants, we can pull those out and get:

PVout = present value of payout at end of defer
Pout = monthly payment
t = months in defer period
n = months in payout period
yout = monthly retirement yield
i = monthly inflation rate


This is a Geometric Series and it has a known sum grouping the terms like the following:

see the link for definitions of a and r here

PVout = present value of payout at end of defer
Pout = monthly payment
t = months in defer period
n = months in payout period
yout = monthly retirement yield
i = monthly inflation rate



Now...plugging all of this into the very first equation and solving for Pin, you end up with:

Pin = amount invested each month in today's dollars
Pout = monthly payment
t = months in defer period
n = months in payout period
yout = monthly retirement yield
i = monthly inflation rate
PVo = initial investment
yin = monthly investment yield


This is the math behind this problem. It's worth noting that minor changes to this can give you things like:
  • how to calculate the yield of an annuity, including with inflation
  • how to calculate the yield required to get a given retirement amount
  • how much you end up with if you invest a given amount of money each month


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