Question

you are a well-paid engineer with a well-established international corporation. in planning for your retirement, you are optimistic and expect to make an investment of $10,000 in year 1 and increase this amount by 12% each year. how long will it take for your account to have a future worth of $1,000,000 at a rate of return of 10% per year?

Answer 1

Our target is to get $1000000 from initial cashflow of $10000 (in first year) which grows by 12% each year. There are 2 ways to calculate this. First method is by putting values in the formula given below (it requires a bit complex mathematical calculations). second method is a bit lengthy but quite easy which i have explained in detail.

FV = Initial amount * ((1 + i)n - (1 + g)n) / (i - g)
Where FV is future value (here 1000000), i is discount rate (here, 10%), g is growth rate of initial amount each year (here, 12%) and n is time period.

Another easier way to calculate future value of a growing annuity (annuity is payment each year here) is to work out each cashflow by growing initial cashflow ($10000) by the rate G (12%). Then find future value of each cashflow by compounding each cashflow at 10% for remaining years. And finally, summing each value we get. This process becomes very easy especially when you have access to Ms Excel.
For initial amount of $10000 growing 12% each year at interest rate of 10%, it will take 17 years to get final amount (future value) as $1000000.
To illustrate, first calculate future value of $10000 for 17 years at 10% compound interest rate.
FV = $10000 * (1.1^17) = 50545.
Then, increase $10000 by 12% for next year; $10000 + 12% of $10000 = 11200.
fv = $11200 * (1.1^16) = 51464.
Note that we compounded it here for 16 years and not 17 years because $11200 is now have 16 years to grow and not 17 years which was there for $10000.
again increase $11200 by 12% for third years which becomes $12544.
fv = $12544 * (1.1^15) = 52399.
Here, we compounded $12544 for 15 years because now it we are at begining of third year and this amount has 15 years remaining to grow.
If we continue this process, we will get 17 future values (1 for each cashflow). when we sum all those values, we get $996364 at end of 17 years which is very close to $1000000.