Goal

What are we trying to do with our money?

Basically money serves two purposes:

• To keep score and provide psychological benefits
• To fund consumption

The first use of money can be seen in for example Warren Buffett, the frugal billionaire. He could lose 99.9% of his money and his actual life would not change at all. Others like the WallStreetBets crowd gamble with money as a video game. But also many high earning professionals just keep accumulating until they die, ever increasing their safety buffer.

The second use of money is what is the more traditional view in economics. The general idea is that you can trade money for goods and services that provide you value (utility). The more money you have, the more value you can get. In particlar, a standard utility funcion from money to value is the logarithm. This is the utility function that gives the optimal growth in the long run, and also more or less matches with real human behavior.

The goal for the first use can be anything. Usually some growth is desired, but many other factors are also influental, like:

• Low volatility
• high volatility
• Difference from some benchmark
• Alignment with values or beliefs
• Expression of personality As it is not possible to take all of these into account, I will focus on the growth objective only.

The goal for the second use is to maximize the total value over the lifetime. This is a classic question addressed definitively by Merton (1070). The solution is quite simple: invest for the highest possible growth and spend 1/N each year where N is the number of years left.

Either way we want to have maximum growth.

To invest for the highest possible growth we need two things:

• The portolio with the highest Sharpe ratio, the average excess return \( \mu \) divided by the volatility \( \sigma \).
• To invest the optimal fraction \( f^* = \mu / \sigma^2 \) of our money into our portfolio.

(Note that these are second-order approximations that ignore skew, kurtosis etc.)