It is important to put down on paper clear goals about what we expect from our investments and the different outcomes that are possible. This will help to stay the course during rough times.

Frankly this is very hard to do and always just a general estimate. It´s impossible to know what the future will bring. I try to corner the problem from different angles to get a feel for what might happen to my portfolio and from that I try to generate practical rules to achieve a high rate of capital growth while keeping the downside risks in check.

Rather than trying to calculate expected risk and return from all the individual portfolio elements and their correlation, I like to take the opposite approach first and

**look outside at the examples of other investors and market studies to find a realistic benchmark.**From that I estimate what results my approach might bring. I try to analyze my own risk tolerance visualizing my past investing experiences, scale that down to be more realistic and decide on my portfolio exposure from there.

I concentrate on the

**Sharpe ratio**as a measure of risk-adjusted return even though it has some limitations. It is the most widely used measure, which allows comparison of a variety of different studies to each other. Return numbers alone are quite useless, as they don´t give any indication of the risks taken to achieve them.The Sharpe ratio is the excess return (above the risk free rate) divided by the volatility (as a proxy for risk).

As a rule of thumb the higher the ratio, the better. For example a Sharpe ratio of 1 implies excess return and volatility of the same magnitude; e.g. 10% excess return with a standard deviation of 10%. You should expect maximum drawdowns between 2 and 3 times your annual return, depending on the skew of your return distribution, with a Sharpe ratio of 1 at some point in the future.

Long term US equity returns have been about 10% (excess returns about 6%) with a standard deviation of 19% and a Sharpe ratio of 0,32. Drawdowns of -40% to -50% have happened quite frequently, but have reached more than -80% at a maximum.

The Sharpe ratio has two major drawbacks to be aware of: It treats upside and downside deviation the same way and it uses the return´s standard deviation, implying an independent, normal distribution of returns, which ignores the reality of negatively skewed, correlated returns with abnormaly fat tails. This is the reason why long term equity returns have had larger drawdowns than the standard deviation might lead us to expect. Negative skew causes the Sharpe ratio to overestimate the risk-adjusted returns, which, for example, is visible in short options strategies, which have highly overstated Sharpe ratios.

Sharpe ratios fluctuate a lot over time. US large cap stocks have had Sharpe ratios per decade ranging from -0,08 (the 1970s) to 1,4 (the 1950s). That means the historical Sharpe ratio of strategies will show great variation depending on the backtested time frame.

For realistic estimates the historical time frame should be across whole market cycles including several good and bad times, for example 1990-2010. Including data from the Great Depression will give a good worst case scenario of the possible magnitude of losses.

**The upper ceiling of possible returns**

Looking at the all time best investors, one has to be aware that they are absolute outliers and simply define an upper ceiling of returns that are possible at all. This gives a reality check when encountering return claims that are way above that. If that were consistently possible, compounding would lead to the investor owning the entire world´s assets after some decades. As it is super investors already own a fair chunk of it.*

- Star investors like Warren Buffet, George Soros, Paul Icahn and others have managed to earn 20% – 30% yearly returns over a span of decades boasting Sharpe ratios between 0,7 and 1. This implies a very high risk tolerance, as drawdowns frequently top -50% for such return distributions.
- The best long term hedge fund returns I came across are James Simons´s Renaissance Technology´s Medallion fund at 35% annual return over decades and Edward Thorp´s: Princeton Newport Partners averaging 20% annually with a minute 6% standard deviation and an incredible Sharpe Ratio of 2,33 (the highest I ever came across spanning decades) – this was achieved using leverage around 8 times in a completely hedged statistical arbitrage portfolio.
- A very interesting example of concentrated value investing during very bad times is John Maynard Keynes investment management of the Chest Fund, King’s College, Cambridge, during the great depression, 1927-1945. Beating the market considerably, he made 9% per year with a Sharpe ratio of 0,4.

**The realistic average of the market with timing and strategy tweaks**

- The classic portfolio mix of 60/40 stocks and bonds historically had 8,5% annual returns and a Sharpe ratio of 0,4-0,5.
- Different buy and hold Global Asset Allocations including smart beta factors had 9,5% to 12% annual returns and historical Sharpe ratios around 0,8. The return dispersion of different allocations over long time frames has been astonishingly small.
- Global Asset Allocations including market timing (moving average trend following) boosts Sharpe ratios to 1-1,1
- Trend following Managed Futures have achieved long term Sharpe ratios between 0,6 and 1 and are able to target specific volatility levels very well.
- Volatility selling historical Sharpe ratios vary more widely because strategies differ. Examples are PUT index 0,7; S&P systematic strangle portfolio 1,53 or Vix strategies 0,85 – 1,3 (biased towards overestimation because of negative skew).
- A reality check back with the actual historical returns of professional money managers shows that a Sharpe ratio of 1 has been a long term ceiling for virtually all of them.

If the future rhymes with the past, judging from these outside sources,

**combining different strategies could land a portfolio´s Sharpe ratio at 0,7 – 1 at a maximum. That is the long term goal for my portfolio.**

Adding up the historical returns of different strategies, I use and accounting for correlation benefits, I actually arrive at a 1-1,5 Sharpe ratio. There seems to be a reduction in portfolio performance, when moving from theory to practice. Maybe in general, Sharpe ratios in backtests are better than Sharpe ratios measured on actual returns, because we are seeing the effect of costs, fees and taxes? I will definitely keep a close eye on the distribution of real returns my portfolio generates and adjust my strategies accordingly.

An intriguing opposing position is that, during a shareholders meeting in 1999, Warren Buffett lamented that he could generate 50% returns if only he had less money to invest. Maybe a smaller portfolio can more easily generate relatively high returns?

So, what does all that imply?

Averaging statistics from a wide range of studies,

**historically a fully invested diversified portfolio (mixing the strategies above) would have returned between 11% to 15% annually with a standard deviation between 8% and 12%**depending on weighting and exact rules. The Sharpe ratio historically had a fairly stable value around 1, as higher returns coincide with higher volatility. The maximum drawdown since 1973 was approximately -20% to -25%. The reality check above leads me to pull these estimates down a bit and anticipate higher volatility and drawdowns.

As option selling strategies allow the use of no-cost leverage, I look at optimizing these benchmark values to my personal goals and risk tolerance. This is a very personal process and it would not be a good idea to simply copy it.

To me the portfolio statistics feel very benign, I have seen a -25% to -30% drawdown in my portfolio many times and I wouldn’t mind it to be a little bit higher, provided I can get improved overall returns. To someone else that may feel quite different – it is easy to simply invest a fraction of the available capital in such a portfolio to target a fraction of the risk and average returns.

**My goal is to build my investment portfolio to be a primary source of my current income as well as a growing retirement account.**

As I am self-employed and running my own business, I don´t depend on the investment income and – in contrast to someone in a full-time job – am very flexible in scaling my business activities up or down to fit my financial needs and the time demands of doing investment research. Although I have a long time until retirement, I would be quite interested in more financial independence in the not too distant future. As I am in a phase where I aim primarily at growing my wealth, I think a high level of risk is justified. Later on in my investing career, when wealth preservation becomes my focus, I will dial down the risk.

Being comfortable with a higher level of risk than the portfolio statistics suggest (always adding a margin of safety in case the future turns out to be quite different), my next question is:

**What would my optimal exposure be, to reach my financial goals as soon as possible while minimizing risk of ruin?**

There is a very neat way to calculate the portfolio exposure, that will lead to the highest equity over time while avoiding ruin (assuming the portfolio statistics accurately describe the future):

**The Kelly Criterion**.** With your annual excess return and its standard deviation (careful – a normal distribution of returns is assumed), Kelly´s formula calculates the optimal amount of leverage to use for the highest compound growth of your portfolio over time. The full Kelly leverage will result in an extremely geared up and volatile portfolio. Drawdowns of more than -90% can easily happen (a drawdown has a 10% chance to reach -90% and a 50% chance to reach -50%). That is not suitable for real life investing with all its uncertainties.But using a fraction of Kelly leverage reduces volatility dramatically, returns less so and it becomes a powerful tool.

**My rule of thumb is to use conservative estimates of my portfolio statistics and then employ one quarter Kelly leverage.**

In practice, running the average numbers from above leads to

**a quarter Kelly leverage of 1,6x to 2,5x times available investment capital.**To implement that, it would be possible to allocate all our capital to the different portfolio strategies except for option selling. We could then sell systematically diversified options, until they add up to the portfolio´s short volatility allocation, capping exposure at the desired overall leverage.

An ace in the hole is the time-varying level of leverage applied through adaptive portfolio allocation. In good times (when the actual Sharpe ratios are above average) I apply up to 2,5 times leverage, reducing exposure to one, when the market heats up (volatility is on the rise and fundamentals point to strong overvaluation) and finally go to a large proportion of cash in bad times. On average the portfolio should be only slightly levered and mostly at the right time, avoiding overbetting and large drawdowns.

The main problem could be a flash crash or 1987 type event, when higher drawdowns are to be expected. The leverage should still prove to be low enough to avoid ruin – the margin of error until reaching full Kelly leverage (about 5-6 times leverage) is high enough.

**The levered portfolio could yield 17% to 22% annually with 15% to 25% volatility and I expect to experience drawdowns around -30% to -50%.**Much better prospects than an equity portfolio, which experiences similar drawdowns with much lower return expectations. Compounded over time this would definitely make you financially independent no matter where you start. Your money would double roughly every four years.

I would consider this a very aggressive risk level – it will definitely feel highly uncomfortable at times.

Using optimal Kelly betting, it still has room to be pushed, if you don´t mind an extreme rollercoaster ride. It could be done on a small speculative portion of one´s wealth – a barbell approach of a very conservative and a very risky portfolio – combined. Aggressive risk taking could also be justified, when the invested capital base is quite small and the savings rate to grow that capital is comparatively high. This would often be the case in the beginning of an investment career, but unfortunately the level of knowledge required to successfully run a combination of strategies is very easily underestimated. One tends to feel, that one has everything necessary figured out at each stage one is at – even, if that is just the beginning of a long journey.

I don´t think we are straying into fantasy land with these numbers, as there are many real life examples that boast the same Sharpe ratio. Most of the time they yield returns between 10% and 15% like our unlevered version, which is a much more tolerable approach.

On the other hand higher numbers than above, imply very high risk (likely drawdowns far beyond -60%), or the presence of some serious alpha which is exceedingly rare and could not be systematized for very long as the superior returns would soon be arbitraged away by the competition.

**Analyzing the robustness of your investment approach – will it likely hold up in the future?**

All strategies in the portfolio should have a positive expectation over a very wide range of variations (the wider, the better) for parameters, time frames and asset classes.

For example momentum and trend strategies work over virtually all asset classes with lookback periods ranging between 2 and 15+ months.

For a value strategy you can employ any of dozens of value metrics successfully.

Volatility strategies, even though they target a narrower field, work with all kinds of different instruments and different signals to produce positive returns.

**I use strategy parameters that fall somewhere in the middle of the range that has worked historically and do not optimize for the perfect parameters.**Doing that would lead to curve fitting and future underperformance as your parameter set will most likely regress back to the mean performance of all parameters. It is also quite possible to diversify across different parameters – e.g. strategies implemented across different time frames often behave quite differently.

Simplicity (few inputs and parameters) equals robustness and enhances consistency and our trust in our portfolio. Future performance is more likely to be similar to the past than with complex strategies with lots of moving parts that are all subject to fluctuation.

Focusing on building a solid investment process is more important than looking at historical returns.

*a good piece on logical return limitations is “MISSION IMPOSSIBLE: BEATING THE MARKET OVER LONG PERIODS OF TIME“ by Alpha Architects

**„Fortune´s formula“ by William Poundstone goes deep into the subject of bet sizing.

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