The efficient frontier of stock investing

The original concept

The original concept of the efficient frontier was introduced by Markowitz in 1952. He modelled buy and hold investments of stocks and bonds as the weighted sum of expected individual returns. According to the central limit theorem in statistics, expected returns converge to their true historical values when the average or mean is taken over an arbitrarily large number of separate samples. Therefore, Markowitz’s investment model can be considered as a weighted sum of true historical individual returns. This weighted sum of historical returns represents, in turn, a weighted average or mean. In statistics, such a mean represents an expected portfolio return over the holding period. The sum of the weightings of the long positions is normalized to one, of the short positions to minus one. One may introduce a hedging ratio as the ratio of the two sums. The ingenuity of this model lies in modelling annual expected portfolio returns, R, as a weighted sum of n annual stock returns R/h with h[yrs] denoting the holding period in years

 R = ∑i={1,n} wRi /h ,

and providing ways to compute the weightings by optimizing one’s personal investment objective in terms of rewards and risks:

Investment Objective = Maxθ={w}[Reward/Risk = Maxθ={w}[R/Variance(R) ]

where θ represents the set of unknown weightings

Investment objectives can be of any kind. Markowitz chose to maximize the reward/risk ratios. He took the Compounded Annual Growth Rate (CAGR) as a measure of the annual expected portfolio returns and chose the Variance of the portfolio price fluctuations over time as his measure of risk. The Variance(R) is calculated from the fluctuations in the individual stock returns at sampled time periods and properly combined using the estimated weightings. The sampling frequency may be set to the inverse of a correlation time of a year, quarter, month, week, day, …, down to one tick. Markowitz calculated the portfolio weightings that maximized the mean (CAGR) with minimized Variance (Risks). His methodology is often referred to as Mean-Variance, or MV. He introduced the efficient frontier, which graphically represents a curve of maximized  portfolio rewards with minimized risks as a function of portfolio diversification (Sizing). Portfolio Sizing is directly proportional to Investment Sizing. As portfolio diversification also relates to risks, the curves reside in a Reward – Risk plane. These curves enable an investor to link portfolio diversification shown on the horizontal axis to maximized rewards with minimized risks on the vertical axis. Markowitz assumed a Normal probability density function (pdf) of the portfolio price fluctuations over time. By assuming any pdf, you simplify the numerical optimization process to a linear multivariate regression, the CPU of which increases quadratically with Sizing. This quadratic dependence may produce unreliable results for larger portfolios or when large market swings are present.

Limitations of MV methods

  • Markowitz noted already that he applied the optimization to a single holding period in a continuous time setting. From a mathematical point of view, that limitation is overcome by combining the weighted sum of n*N historical returns Ri, j of each individual stock i over all N holding periods hj in an advancing time scale:

    Rj = ∑i={1, n} wi, j Ri, j /hj         j = {1, 2, ..., N }.

    This combining is for compounding investments expressed by

    Cj+1 = Cj (1+Rj )     with C1 = Initial investment,

    and for fixed or constant investments by

    CN = (C1/N) ∑j={1, N} Rj .

    For fixed holding periods hj = h, and Rj can be considered as a back test (past performance) as well as a compounded or summed time series of Fourier components with exponential factors exp(i2πt/h) with time t = jh. These Fourier series are the power spectrum of compounding and constant investments. Each holding period has its own set θ of optimized weighting coefficients. Each back test or past performance has its best predictor of success: 

    Past performance is your best predictor of success (Jim Simons, 2005).

  • The mathematical proof of this statement is given by the Wiener-Khinchin-Einstein theorem, which generally holds for randomly fluctuating signals. According to this theorem, this best predictor corresponds to the peak in the power spectrum where the the holding period gives the strongest autocorrelations. This provides the investor with an opportunity to optimize the holding period and the hedging ratio in line with the chosen investment objective AFTER the optimal weighting coefficients have been computed. The CPU to perform the underlying computations for these portfolios increases linearly with increasing number of holding periods in the time series as well as with portfolio size. It is not self-evident that when each portfolio is optimized in terms of rewards and risks in the time series, the total sum of returns over all holding periods is also optimized accordingly. Our software always checks for that evidence. It is also not self-evident that the individual stock-return fluctuations in each holding period have the same pdf. Our software does not use any pdf. Nor does our software use any equation of motion for the price fluctuations like the Fokker-Planck equation or equations of the same form (Black-Scholes, Navier-Stokes, or any other diffusion-type of equation). The best fit (best predictor) to such equations is the so-called Cramer-Rao lower bound, which is a standard mathematical procedure to find the holding period that best fits the equations. We do not see any reason to add these equations as additional information to the price fluctuations and make the computations more cumbersome. In addition, we do not see any reason to use AI, ML, or NLP, because the best predictor is mathematically determined by the Wiener-Khinchin-Einstein theorem and can be programmed and computed in a relatively simple way.
  • As a second item of critique, it is stated that Variance and the square root thereof (standard deviation) may not be good measures of risks. From a mathematical point of view, you may choose any quantifiable risk measure that you would like to minimize. We use the maximum drawdown since the IPO as a measure of risk and the annual expected return as a measure of reward. The maximum drawdown is a measure that is usually determined over many holding periods including periods of crises. This implies that the validation period should span 15+ years. 
  • As a third item of critique stands the assumption of a Normal pdf fitting the portfolio return fluctuations. Like any other pdf, these will not fit the tails of real price fluctuations in the equity markets. As past performance is the best predictor of success, we systematically apply the screening and ranking conditions of our game plans to the historical data in our Dbase of the stocks in our WatchLists. Our ranking system is time-invariant, so that it depends on correlation times. Only eod prices and volumes are used. This systematic screening and ranking gives a time series of portfolios at preset holding periods. The weightings and timing of each portfolio are optimized to the risks and rewards that are in line with the investor’s personal investment objective. Hence, no pdf is needed for the optimization process, just a gradient descent or ascent method. The significant advantage of time-invariant ranking is that the CPU to optimize the investment objective increases linearly with Sizing and with the number of holding periods. During large swings of the market, it has proven its reliability and effectiveness. As many investors want to evaluate the volatility, skew and excess kurtosis of return distributions, we expand the Value at Risk (VaR) of our summed time series of optimal portfolios in a Cornish-Fisher expansion

    VaRReturn Spaceσ√N [-1.96 + 0.474μ1/√N - 0.0687μ2/N + 0.146μ12/N ] - 0.5σ2N ,

    where N is the number of holding periods in the recommended validation period, and σ, μ1, and μ2 are respectively the volatility, skew, and excess kurtosis measured from the return distribution. These quantities are calculated from the measured moments of the distribution of returns in accordance with the following:

    - the zero moment, M0, is the number of observations in the validation period

    - the first moment, M1, is the mean of all observed returns during validation

    - the second M2, third M3, and fourth moment M4, are defined in the standard manner:

    Mk = ∑j={1,N(Rj - M1 )k / M0      for k = 2, 3, 4 .

    The volatility, σ, is given by M2, the skew, μ1, by M33, and the excess kurtosis, μ2, by M44 - 3.

    The relative volatility, β, of annualized portfolio returns, Cj, is defined as

    β = COVCj, Ind/VARInd ,

    where relative means relative to the volatility of an Index, Ind. The relative annual return, α, is defined as

    α = (C - rrf ) - β (rInd - rrf ) ,

    where relative implies relative to the risk-free-rate, rrf, and the annual return of an Index, rInd, and where C denotes the annual rate of portfolio returns, either compounded or as generated free cash on a fixed investment. The Sharpe ratio is a reward/risk ratio defined by:

    Sharpe ratio = (C - rrf ) / σ .

    The Risk Indicator introduced for all asset markets by MiFid2 rules of the European Union is calculated from the Var-equivalent-Volatility (VeV), which takes on the form:

    VeV = {√(3.842 - 2*VaRReturn Space - 1.96)}/√(Nh) .

    Our software links the calculated VeV to the prescribed Risk Indicator of the MiFid2 rules. In conclusion, we broadly see three different systems of defining the {Reward, Risk} space:

    1. Rewards are defined as annual expected returns (AER), either compounded or on fixed investments. Risks are defined as the maximum drawdowns (MDD) on those returns.
    2. Rewards are expected annual returns relative to risk-free-rates and some Index. They are called “alpha”. Risks are defined as the spreads on those returns, relative to the corresponding spread on the same Index. They are called “beta”.
    3. Rewards are defined as the Value at Risk (VaR) in return space. Risks are defined as the VaR-equivalent-Volatility (VeV). These definitions are given in the MiFid2 rules of the European Union.

    The mathematical representations of these three different {Risk, Reward} spaces can be summarized as follows:

    {Reward, Risk} ↔ {AER, MDD} ↔ {α, β} ↔ {VaR, VeV} .

    It is our understanding that retail investors have their best grip on the first definition, professional investors manage their returns and risks in terms of alpha and beta, and European legislation uses the third set of definitions, which is based on the Cornish-Fisher expansion as the distribution function of the return fluctuations. The first two sets of definitions do not necessarily have to use a probability density function, as they can fully resort to actual historical data.
  • As a fourth opportunity for extension stands the stipulation of multiple return factors. The multiple-factor method was introduced by Fama and French in 1992. They expanded the portfolio returns into a weighted sum of market premiums or factors. Their original proposition added two new coefficients to the Capital Asset Pricing Model (CAPM) and changed the definition of the first coefficient β. These factor weightings should be computed in linear regression AFTER the weighting coefficients of the Markowitz expansion have been computed. You do not need these factors to compute optimally weighted portfolios. Such factors may be instrumental in doing your due diligence in ranking your assets in terms of their margins of safety. They are also instrumental in ETF design that track certain market premiums. That is presently a $900 Billion business. 

Past performance is the best predictor of success but no guarantee
When we scan the past for the time series of portfolios, we use the Annual Expected Result (AER) as reward and the Maximum DrawDown (MDD) as risk and do not make any assumption about the pdf of the reward fluctuations. The resulting portfolios are called optimal portfolios as they are maximized in rewards and/or minimized in risks. By varying the asset allocations and computing the resulting AER and MDD, you search for optima of combinations of these two quantities. Hence, you need a search algorithm to find portfolio weightings that maximize the MAR ratio (= AER/MDD) or just maximize the AER or minimize the MDD as objective functions. You could also take other ratios like the Sharpe ratio or the Sortino ratio as objective functions, but we prefer the MAR ratio. Maximizing this ratio is usually close to the investment objective of a retail investor. As stated above, we maximize the MAR ratio in a gradient ascent.

Enabling investors to link an investment to a maximum annual expected result
Portfolio managers usually select their stocks from a WatchList. We made a WatchList of some 1300 liquid stocks of Wall Street with daily-dollar volumes in excess of $1 million since 2005. We compute the time series of optimal portfolios from it with holding periods of 13 weeks. We compute these time series by using the historical eod prices from the data providers CSI and Yahoo. The chart of the efficient frontier gives the maximum annual returns with minimum drawdowns as a function of the # stocks in these optimal portfolios (solid lines):

Each investment has its own optimal Annual Expected Result (AER). Portfolios are optimized, so that the weightings maximize the MAR ratio = AER/Max Drawdown (solid lines: MAR>1.2), or maximize the AER (dotted lines: MAR>0.8). A variation of what mathematicians call a gradient-ascent method was used to find the optima. The CPU used to compute all these curves is less than 10 seconds. The validation period ran from 14-July-2009 to 14-July-2019   

The chart enables an investor to evaluate the maximum annual expected returns with minimum market risks given the size of his investments or portfolio diversification. This efficient frontier for our WatchList of liquid stocks can be summarized in the following table:

For example, a retail investor who wants to start investing with an amount between $1000 and $10,000 could let DigiFundManager select only one stock every 13 weeks from this WatchList of liquid stocks using some given screening and ranking conditions. His Annual Expected Return validated over the past ten years is 42% after fees and tax, with a maximum drawdown of -8.5%. During the ten years before these past ten years, the risks and rewards are significantly larger but still balanced. Using different WatchLists gives different efficient frontiers, often allowing for significant improvements by proper hedging conditions.

DigiFundManager and predicting the future, machine learning, artificial intelligence and NLP
DigiFundManager uses none of these concepts. All what it does is that it validates the past of screened, ranked, optimally weighted and timed portfolios. Validation or out-of-Sample testing is accomplished by computing the optimal portfolios on Fridays at closing and calculating the risks and rewards on the following Mondays at closing when actual rebalancing is assumed to take place. In digitizing the validation of these four activities of portfolio management, we only use the historical eod prices, volumes, dividends, and splits. According to the Efficient Market hypothesis, these historical prices fully reflect the available information. As we have seen, the past is the best predictor of success, and statistics does not distinguish between the various kinds of risks. We do not estimate or quantify a prediction interval in which a certain future observation will fall with a certain probability. We do not do such things, because the price fluctuations of the stock market cannot be fitted with probability distributions when there is blood in the Street. When we rank the 1300+ stocks out of our WatchList with liquid stocks, we do that on the basis of relative probabilities to increase in price only based on preceding price movements. For instance, on 10-Oct-2014, we computed on the basis of preceding price movements that FRO received the highest rank in the WatchList of 1300+ stocks. We did not know and had not quantified that the following quarter FRO’s share price would increase by a factor of four. Within our programmed rationale, FRO got the highest rank in the same way that other stocks got the highest ranks at other quarters. The proof is always in the pudding. Our quantitative investment system of maximum annual expected rewards with minimized risks with holding periods of 13 weeks as shown in the chart and table is a competitive system, even with High Frequency Trading (HFT).

Annual Expected Results and overdiversified portfolios
For investments between $1000 and $10,000, an investor could also decide to set up a portfolio of six long positions and pay relatively more fees. His Annual Expected Result would decrease from 42% to 22% with a maximum drawdown of -11%. An important finding is that holding periods of 13 weeks produce larger Annual Expected Results than holding periods of 1 week for portfolios selected from this WatchList. The efficient frontier of our WatchList of liquid stocks does not underperform the results of HFT. Scaling up investments to hedge-fund levels is a different expertise. If you were to scale up the investments in our largest portfolios of 384 liquid stocks, we could foresee an increase from $4 million to $500 million. However, the mission of our software service is to bring low-frequency quantitative investing to the retail investor.

Jan G. Dil and Nico C. J. A. van Hijningen ≅η
15 Feb 2021

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