Does $500 turn into $1 million after 3 years by increasing 1% every trading day?

As others have pointed out, this can be solved by first determining the number of days required to turn $500 into $1,000,000 at a 1% daily interest rate and then adjusting for the number of trading days per year. 

1. Solving for number of days requires solving for N where 500 * (1+.01)^N = 1,000,000
2. This is based on the formula to calculate a lump sum compounded at any rate
   1. P * (1 + R)^N
      1. Where P=initial investment, R=interest rate, and N=number of periods
3. (500*(1.01)^N)/500=1,000,000/500
4. 1.01^N=1,000,000/500
5. 1.01^N=2,000
6. N ln1.01=ln 2,000
   1. For those unfamiliar, ln is a mathematical tool known as a natural logarithm
7. N=ln 2,000 / ln1.01
8. N=763.88439457
9. Rounded, N is approximately equal to 764 days
10. On the NYSE and NASDAQ, which are open an average of 253 days per year, the number of years required is equal to 764/253or 3.019 years. Unfortunately you would be just shy of staying inside the 3 year mark via the historical averages of these two exchanges. 
11. Other markets which remain open on weekends will obviously accelerate the rate of compounding and bring the final result below three years

How fast could you turn $500 into $1 million if you added an extra $500 every trading day?

As a bonus, it may be interesting to determine how much faster we can reach the million dollar mark by adding another $500 to the pool every day while still reinvesting all of the profits. To do this, we can use the formula for the future value of an annuity due. An annuity due refers to a stream of payments which are invested at the beginning of each period (in this case each trading day) and are compounded at a given interest rate. 

1.  p  *  ( [ ((1+r)^n) – 1] / r )   * (1+r) = 1,000,000
   1. Where p=initial investment amount each period, r=interest rate, and n=number of periods
2. 500 * ([((1+.01)^n) -1] / .01) * (1+.01) =1,000,000
3. Let’s remove the .01 denominator on the right by multiplying both sides by .01
   1. .01*(500 * ([((1+.01)^n) -1] / .01) * (1+.01)) =1,000,000*.01
   2. This gives us: 500 * ([((1+.01)^n) -1]) * (1+.01) = 10,000
4. Lets remove the 500 and (1+.01) on the left side by dividing 10,000 by them 
   1. ([((1+.01)^n) -1]  = 10,000/ (500*1.01)
   2. ([((1+.01)^n) -1]  = 19.80198
5. Add 1 to both sides to further isolate n
   1. 1.01^n = 20.80198
6. Now we simply need to use a natural logarithm again to find the answer
   1. n ln 1.01 = ln 20.80198
   2. n = ln 20.80198 / ln 1.01
   3. n = approximately 305 days
   4. Using the same 253 trading days per year, it takes 1.2055 years to reach the million dollar mark. 
7. Now we can see that by investing $500 each day we reduced the time required to reach $1,000,000 by about two and a half times. 

For those interested in performing similar calculations, financial calculators can make the process much more efficient. For example, the TI-84’s TVM Solver feature will allow you to solve for any variable in an annuity by simply plugging in the numbers. Above you can see the math that underlies those calculators. 

The reality of making 1% every trading day

Although this is (hopefully) a fun mathematical exercise, it is worth adding the disclaimer that making a 1% daily return is effectively impossible. The best hedge fund and private equity managers in the world fail to come anywhere near this figure, regardless of how much risk they take. Numbers like 1% can be eerily deceptive to novices in the world of trading; after all, it sounds pretty easy to do until you’ve actually participated in the markets. Having said that, these calculations do illustrate the profound power of compound interest, which Albert Einstein once said is the strongest force in the universe.