The Magic Dollar

A goblin came to my home yesterday. Like many goblin, he was a banker. He owed a favor to my great grand-fathered. To repay his debt, he offered me betwen receiving a million dollar or a magic dollar. That magic dollar and his children would double every day for a month. What would be the best choice?

We could select according to our intuition, but there must be a trick. One million is too good to be true. We must always be aware of what is too big to be true. If the goblin proposed it, the magic dollar is most likely worth as much. Lets take of our pencil & paper!

The dollar become 2$ after 1 day. He grows to 64$ after 7 days. That's a bad start for this choice, lets continue. After the second week, the dollar accrue to 8,192$.

It does not seems possible that the magic dollar is the right choice after almost half the month completed. The almost entirety of the road is still ahead (991,808$). What do you think, should we continue the experience?

For the third week, we pass to 131,072$ mid week. Finally 2 days later, we arrive at the mid-point at 524,288$. It took us 20 days to arrive to this point. How much days would be needed to arrive at a million?

Of course if the magic dollar double, it will take one day. At the end of the third week we are well over the million. The fourth week is staggering. We do not know what to do with our dollars. We pass by the 16 millions mark mid-week and above the 100 millions at week end.

The last few days are nothing to sneeze at. At day 30, we are at 500 millions and at the 31^th, we arrive at over 1,000$ million. If the goblin strike at your door, my best advice would be to ask him to come back in December or January to reach that 1 billion!

Now, lets come back to reality. 1 dollar invested in accessible asset do not double each week, but after roughly a decade. If you are able to put away 100,000$, then you can expect to have another 100,000$ at the end of 10 years. To be more precise the type of investment and his return determine the exact time to double.

Types of investissement	Gold or cash	100% Bonds	60% Equity 40% Bonds	100% Equity
Interest rate	2%	3,4%	5,5%	6,9%
Double after	35.0 years	20.7 years	12.9 years	10.4 years
30 years later on 100$	181$	273$	498$	740$

We can conclude two truths on these numbers.

• It will not suffice to invest 100,000 for your retirement, even with 30 years of time horizon. We must also contribute each year to retirement

• The type of investment as a rather big impact, You would have 5 times more money if you invest in equity versus in cash at the bank.

Value of the Magic Dollar = 2^{(number of days - 1)}