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This is the problem I have to solve:

Assume you are 30 years old right now. You believe you could save for the next 20 years, i.e., once you have 50 years old. For the next 10 years, and until your retirement when you are 10 years older, you can not save due to some expenses (taxes, university tuition, etc.) If you want to have incomes of $100.000.000 USD per year when you are 61 years old, how much you would save every year (the same amount during all the years), for the next 20 years, starting at the end of this year (when you are 31 years old)? Suppose your savings/investments will yield an effective rate of 8%.

My solution:

20 years * $100.000.000=C*(1+8%)^20

C ~=43M

Is it correct?

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    No, you are assuming only one pay out rather than multiple ones. – JB King Aug 16 '16 at 22:10
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    This question is much more complicated than a single formula. From age 61 to 80 you'll receive a total of $2B assuming the last $100mm payment will exhaust the account. From 51 to 60 you will not contribute to the account but it will still grow. From 31 to 50 you will contribute an annual amount that will grow by 8% each year. You assume an 8% growth rate throughout. There are a couple of numbers you need to back in to before you can determine the amount you will need to have in your account by age 51 to determine your annual deposits from age 31 to 50. – quid Aug 16 '16 at 22:14
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    These numbers seem really off. Are you sure you want $100m every year? – Superbest Aug 16 '16 at 22:24
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    Maybe they are Jamaican J$. 100.000.000 J$ would be about US$790K today. – shoover Aug 17 '16 at 17:40
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    Having clicked on the OP's profile, I see it's entirely reasonable for OP to be talking about Colombian pesos. 100M of those would be an entirely reasonable US$34K. – shoover Aug 17 '16 at 17:45
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As I understand it, you want to save from 30 to 50, invest, and start collecting a return of $100M on this investment after your 61st birthday.

I'm assuming that when you are "unable to save" from 50 to 60, at least the proceeds from your investment are automatically invested back in.

A single investment that yields r return annually, if held for y years (with reinvestment) will amount to (1+r)^y.

On your ith year, you will make an investment x, which will amount to x*1.08^(61-i). For example, in your first year i=31 and in your last i=50.

Now you can sum these to get the total value of your investment at 61: Sum_{i=31-50} (x*1.08^(61-i)) = x* Sum{i=31-50} (1.08^(61-i)). There's an analytic formula for taking this sum, but I did a simple calculation in Python and it comes out to x*106.7.

This is your capital, from which you will extract x*106.7*0.08 annual income. Since you wish to have $100M, the investment must equal x=100M/(106.7*0.08), which comes out to $11.7M, this is the amount you must invest every year between ages 30-50 to obtain your $100M/year indefinitely from 61 onward.

Alternatively, one could expect to live, say, 70 years, and then exhaust the investment over the course of this 10 year retirement. In year j, your investment p(j) will be worth p(j)=p(j+1)/1.08 +100M, with p(70)=100m (since your last withdrawal will empty the account). You can sum starting back from p(70) all the way to p(61). Again, there is a way to sum this but from Python I get 674.7M. This is the capital you must have at 60, so x*106.7=674.7M and x=674.7M/106.7 comes out to $6.32M invested every year between ages 30-50. As you can see it's quite a bit less than the other 11.7M, and this is how retirement calculations are usually done, since nobody lives very long after retiring (although if you have the misfortune of outliving your expectations, you'd have to find an alternative source of income). But since you didn't give an expected duration of retirement I'm guessing this isn't what you want.

Here's a plot of what happens to your net worth with either strategy: enter image description here

Note what happens if you overshoot your projected lifespan, and also the minute, but perceptible change in slope around age 50.

And since Xalorous is taking the CPA spot, I'll put in my application for financial advisor. :)

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    When you write : "x*1.08^(61^i)", shouldn't that be "61-i", not "61^i". – user41790 Aug 16 '16 at 23:05
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    I think that is basically correct. I totally misread the question when I posted my answer. You may also note that you still have one more "61^i" which needs to be changed to "61-i". – user41790 Aug 16 '16 at 23:07
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    And those numbers should be modified by the probability of you dying each year - see annuity calculations – mmmmmm Aug 16 '16 at 23:20
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    @Mark Yes, that would make it an even more accurate calculation (and more laborious sum) but I think a bit too much for this question. – Superbest Aug 16 '16 at 23:30
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    @Superbest, What is 1.25B in 1.25B=x/(106.7*0.08)? – InfZero Aug 21 '16 at 21:28
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To receive $100m p.a. perpetually from a balance of x with interest rate r

c = 100
r = 0.08

x r = c

∴ x = c / r = 100 / 0.08 = 1250

A balance of $1,250m would be required at the start of each year to yield $100m interest at the end of each year.

However, to receive $100m p.a. for, say, ten years, ending aged 70

n = 10

enter image description here

A balance of $671m would be required at age 60 to receive $100m at ages 61, 62 ... 69, 70.

(See annuity table below.)

For more information on this calculation see Calculating the Present Value of an Ordinary Annuity

To produce a value of x from age 51 to 60 requires 9 years of compounding, starting with balance y

y = x / (1 + r)^9

So for x = 671.00814 the balance y (aged 51) would be $335.67113m

To save y by saving d every year for n years

y = 335.67113
n = 20

enter image description here

∴ d = (r y) / ((1 + r) (-1 + (1 + r)^n)) = 6.79181

Twenty years of saving $6.79181m would be required to meet your goal.

Annuity Table (with some slight rounding differences)

enter image description here

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You need to map your cash flow. Use graph paper.

Divide this into three parts.

Contribution phase

  • starting value is zero
  • annual, monthly, or per payday contribution is CONTRIBUTION
  • final value of contributions is SAVINGS
  • estimated rate of return. Do some research to find a number that fits your risk profile (8% is a bit aggressive, but at your age it's fine)

Coasting phase

  • starting value is SAVINGS from above
  • estimated rate of return. Research from above should uncover that as you approach retirement age, you should become more risk averse. A lower estimate here is appropriate.
  • ending value is FINALSAVINGS

Spending phase

  • starting value is FINALSAVINGS from above
  • rate of return is even more risk averse than coasting phase
  • PAYOUT is periodic payout
  • FINAL value is what you want it to be at 81.

First go through and assign all rates. Start at the end and work your way back. Three separate problems using time value of money calculations.

  1. The third phase is a starting value problem. How much do you need (FINALSAVINGS) to have saved in order to withdraw PAYOUT per year and end up with FINAL at the end of 20 years. Given FINAL, PAYOUT, and yearly ROR, calculate FINALSAVINGS.

  2. Second phase, you know final value and rate of return and periodic cashflow (0). Calculate the starting value again.

  3. First phase, you know ROR, starting value and final values, but need the periodic contribution.

About rates of return. You're going to have to work really hard to maintain 8% ROR after taxes and with volatility. You should do your calculations based on a more conservative number. If you exceed your estimate, so much the better, you can trim the saving period, extend the coasting period, or extend the spending period, or increase the withdrawal amounts, or some combination. If you fall short of your ROR estimates, you will have to reduce somewhere.

Using somewhat conservative numbers for ROR, and a handy online calculator, I got a number of the same order of magnitude but a bit lower than the number you listed.

I will say that if this is remotely achievable for you, you should hire a CPA. If you can swing 43 million a year, I'll quit my job, go get my CPA certification and work for you full time for a small fraction of that.

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    It's not remotely achievable to him, you guys are doing his homework. – quid Aug 16 '16 at 22:57
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    @quid I did approach it as homework. My answer, + appropriate formulae, will get his answer. Mapping the cashflow is key. He has to pull the right formula for each calculation, adjust the rates to fit the periods, and make the right estimates. If his instructor is anything like my professors, he'll have to explain what he's done. If he doesn't graph it out, he won't get why there's three parts. If he doesn't develop some kind of reasonable estimate for ROR, he won't be able to defend the rates he's chosen. Finally, if he hasn't paid attention, he won't be able to find and use the equations. – Xalorous Aug 16 '16 at 23:00

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