Portfolio theory
Modern Portfolio Theory
Portfolio Theory (Harry Markowitz introduced Modern Portfolio Theory (MPT) in a 1952 article) proposes to allocate assets in the portfolio as a function of their volatilities, returns and correlations. The pictorial way is to think about the two-dimensional plane of the returns (first moment) and volatilities (second moment) of different assets. See [1]
As any simple model, MPT has plenty of oversimplifying assumptions. See the review and the criticisms [1].
MPT is a basic framework, while the application of MPT in practice raises more questions than answers.
Since the historical returns never predict but only suggest possible future returns, therefore the future expected returns require further analysis and assumptions. For instance, see [2].
The risk, in general, is the risk of significant to investor losses, and therefore beyond the standard deviation, the loss can include the downside risk and skew(asymmetric) effects. The generalization of MPT for more general risk/loss is addressed in some versions of PMPT.[3], see below. Yet there is a recent contribution suggesting the risk is fully related to tail-risk skewness[4] (after all, profits and losses are symmetrical under normal distribution and on average zero).
Correlations, in general, are not stable or well defined in practice (stable and not-noisy correlations are rather exception), and the portfolio construction should be seen more generally through the analysis of historical series and future stress tests.
Harry Markowitz (Markowitz, Levy) claims that for essentially all utility functions, the optimization of utility is equivalent to mean-variance analysis. Perhaps, Markowitz is right in the analysis of averages but not regarding the outcomes of possible temporary drawdowns. Bill Sharpe also expressed the limitations of Markowitz's claim due to tail-risk and "satiation" of utility.[5]
Post-Modern Portfolio Theory
There seem to be a couple of major development directions of MPT which can be called Post-Modern Portfolio Theory (PMPT).
A first direction deals with the leveraging of lower return parts of the portfolio. Ray Dalio of Bridgewater[6] and others proposed to leverage bonds in a typical equity and bond portfolio so that to match the equity return. The main idea of risk parity[7] is to match the size of assets by risk (risk parity) which is inversely proportionally to the volatility[8]. Bridgewater financial results are very impressive, although they can be a consequence of multiple other inputs of success beyond general principles of risk-parity.
A second direction deals with the generalization of the risk to the drawdown (stop-loss) [3]. The illustrative basic math results are nearly impossible in this case, and there seem to be only numerical analysis based results. Perhaps, this is why originally formulated by Markowitz under standard MPT's assumptions. Tail Risk Parity[9] is one of the most important developments.
Kelly Criterion
Kelly criterion proposes assets leverage to maximize the growth without risking bankruptcy. Practical and successful applications have been made and reviewed by Ed Thorp[10] (a paper is attached as well).
There is important connection between Markowitz and Kelly approaches. Kelly suggests the optimal size allocation for the long run, even for a single asset. Markowitz discusses relative allocation between the assets without any overall leverage. In the case of quadratic utility function, a general result for asset allocation is
$F=C^{-1} * M$
where $F$ is a $N$ vector indicating the fraction of the equity to be allocated to each asset, $C$ is the covariance matrix of the assets, and $M$ is the vector for the expected excess returns of these assets. This result covers both the leverage and relative allocation.[11]
Even for normal distribution, Kelly system has large (larger than ?) volatility which can destroy the expected long-run performance. See the illustrative discussion and the calculator[12] showing negative returns for not too large number of bets for 5th and 10th percentiles. Any results in the literature for tail distributions? It seems Kelly solution is mean-field type (as in physics) with large effects of fluctuations, extreme ruin-run paths. But this problem of ruin-run is a general problem and a nice illustration for investing in the unpredictable probabilistic world. Notice also a remark by M. Moubassin how small number of outcomes with large loss will violate an individual's utility function.
Sharpe ratio
The Sharpe ratio $\frac{\mu}{\sigma}$, where $\mu$ and $\sigma$ are the risk-premium return and volatility correspondingly, is a portfolio performance measure (among others). The investment portfolios are frequently characterized by Sharpe Ratio assuming the risk is defined as a normal volatility (historical or expected). If we decide to maximize/optimize the portfolio in the return-volatility $\mu$ - $\sigma$ plane, then, indeed, the Capital Line of MPT has the highest Sharpe ratio. Kelly criterion, perhaps, suggest than even in the Gaussian world (without skewness and tail risk), it makes more sense to discuss portfolios, which can be leveraged, in the return-variance $\mu$- $\sigma^2$ plane.
Utility theory
A good review of utility functions and efficient markets by Andrew Lo. Accessible summary about deficiencies of the dominant economics theory in a talk[13] and "Adaptive efficient hypothesis".