One thing to note is that volatility / standard deviation can be measured over many different time horizons and assumptions.
Is it annualized standard deviation that matters? Semi-annualized? Do you take daily measurements and scale up? Weekly? Annual only? How many time periods do you need to really know an accurate number?
Another thing you need to be careful of, is all of this implies frequent rebalancing!
Let's say I am investing for 10 years and I want to put half my money into a 10 year bond, and half into a stock market index. It doesn't matter what their correlations are, I didn't actually achieve any real benefit if I didn't rebalance between them and then withdrew on the same day at the end of the 10 years. My compound annual growth rate was the same whether their correlation was +1.0 or 0.0 or -1.0. This is what people intuitively understand. In the long run, carry and the actual value of their investment is more important than year-to-year fluctuations. My 10 year bond returned exactly what it was supposed to, regardless of what the fluctuations of its price were.
I prefer to think about my investments in terms of a sustainable withdrawal rate. If my investments pay me 15% dividends per year, I don't need to particularly worry about the price of the instruments. I would rather have $100K of investments paying a sustainable profit of $15K annually, over $500K investments paying only a sustainable profit of $10K.
To me, the holy grail in investing isn't being diversified. It's finding things where it is a good idea to be UNDIVERSIFIED. I'd take 5 businesses which are undervalued, over the index any day.
The volatility and correlations of instruments are simply whatever happens to be in fancy at the time. Whatever two people (the buyer and the seller) agree to. Thus the worst case scenario is a pretty wide range. Correlations probably go to 1.0, and drawdowns can be anywhere, up to 80% (see the great depression), and this isn't necessarily the worse case scenario either. Thus safe leverage over long time horizons is probably in the region of zero. This is known as "stochastic market efficiency". https://breakingthemarket.com/stochastic-efficiency-is-real-and-its-spectacular/
Imagine that [optimal leverage] > 1 in our model market. This would mean that the simple strategy of borrowing money to buy stock will achieve faster long-run growth than buying stock only with our own money. If we associate putting all our money in stock, [optimal leverage]= 1, with an investment in the market, then it would be a trivial matter for us to beat the market (by doing nothing more sophisticated than investing borrowed money). Similarly, imagine that [optimal leverage] < 1. In this scenario, the market could again be beaten very easily by leaving some money in the bank (and, if [optimal leverage] < 0, by short selling).
It would strain language to consider our market efficient if consistent out-performance were so straightforward to achieve. This suggests a different, fluctuations-based notion of market efficiency, which we call stochastic market efficiency: it is impossible for a market participant without privileged information to beat a stochastically efficient market simply by choosing the amount he invests in stock, i.e. by choosing his leverage.
Overall leveraging up won't actually improve your long run returns (unless you actually have an edge). As people often misunderstand.
One thing to note is that volatility / standard deviation can be measured over many different time horizons and assumptions.
Is it annualized standard deviation that matters? Semi-annualized? Do you take daily measurements and scale up? Weekly? Annual only? How many time periods do you need to really know an accurate number?
Here is a paper from Andrew Lo on the statistics relating to this. https://www.researchgate.net/publication/228139699_The_Statistics_of_Sharpe_Ratios. "I find that the annual Sharpe ratio for a hedge fund can be overstated by as much as 65 percent because of the presence of serial correlation in monthly returns"
Another thing you need to be careful of, is all of this implies frequent rebalancing!
Let's say I am investing for 10 years and I want to put half my money into a 10 year bond, and half into a stock market index. It doesn't matter what their correlations are, I didn't actually achieve any real benefit if I didn't rebalance between them and then withdrew on the same day at the end of the 10 years. My compound annual growth rate was the same whether their correlation was +1.0 or 0.0 or -1.0. This is what people intuitively understand. In the long run, carry and the actual value of their investment is more important than year-to-year fluctuations. My 10 year bond returned exactly what it was supposed to, regardless of what the fluctuations of its price were.
I prefer to think about my investments in terms of a sustainable withdrawal rate. If my investments pay me 15% dividends per year, I don't need to particularly worry about the price of the instruments. I would rather have $100K of investments paying a sustainable profit of $15K annually, over $500K investments paying only a sustainable profit of $10K.
To me, the holy grail in investing isn't being diversified. It's finding things where it is a good idea to be UNDIVERSIFIED. I'd take 5 businesses which are undervalued, over the index any day.
The volatility and correlations of instruments are simply whatever happens to be in fancy at the time. Whatever two people (the buyer and the seller) agree to. Thus the worst case scenario is a pretty wide range. Correlations probably go to 1.0, and drawdowns can be anywhere, up to 80% (see the great depression), and this isn't necessarily the worse case scenario either. Thus safe leverage over long time horizons is probably in the region of zero. This is known as "stochastic market efficiency". https://breakingthemarket.com/stochastic-efficiency-is-real-and-its-spectacular/
Overall leveraging up won't actually improve your long run returns (unless you actually have an edge). As people often misunderstand.