Minimum days until expected values reach $1,000,000?

[nigahiga]

Given the left page scenario, what is the minimum number of days until the expected value rebalances to $1,000,000?
The diagram on the right demonstrates pictorially how expected values are calculated per day iteration

 

[tkhunny]

Awesome problem. There is much to consider, here. Why don't you share with us some of what you have considered?

Thought Question: with a .5 + .25 * 2 + .25 * (1/2) = .5 + .5 + .125 = 1.125 daily expected portfolio growth, are we SURE to get anywhere?

 

[nigahiga]

Awesome problem. There is much to consider, here. Why don't you share with us some of what you have considered?

Thought Question: with a .5 + .25 * 2 + .25 * (1/2) = .5 + .5 + .125 = 1.125 daily expected portfolio growth, are we SURE to get anywhere?

Hey! I believe I have solved it. I calculated the expected values at the first three consecutive days, and derived a sequence formula for the nth expected value. Then I set it equal to 1,000,000 and voila! I got around n≈117.29 which rounds up to 118 days.

Confirmation or flat-out rejection of my proposed solution would be appreciated

 

[tkhunny]

Well, okay, but why did we do that? It's a consistent, well-defined, recursive process.

\(\displaystyle 1.125^{n} = 1,000,000 \implies n = 117.2962683\)


Did we need to draw the tree?

 

[nigahiga]

Well, okay, but why did we do that? It's a consistent, well-defined, recursive process.

\(\displaystyle 1.125^{n} = 1,000,000\implies n = 117.2962683\)

Did we need to draw the tree?

Wow, stumped. That's amazing... I need to study more.

 

[tkhunny]

Wow, stumped. That's amazing... I need to study more.

Keep up the good work. Remember my signature.