# 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 _{i }/h with h[yrs] denoting the holding period in years*

* R = ∑ _{i={1,n}} w_{i }R_{i} /h ,*

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

*Investment Objective = Min _{θ=}*

_{{}

_{w}_{}}[

*Risk/Reward*]

*= Min*

_{θ=}_{{}

_{w}_{}}[

*Variance(R)/R*]

*where* *θ* *represents the set of unknown weightings*.

Investment objectives can be of any kind. Markowitz chose to minimize the risk/reward 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 Risk – Reward 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 doing so, he simplified 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 historical returns
*R*_{i, j}*i**N**h*_{j}*R*_{j}= ∑_{i=}_{{}_{1, n}_{}}*w*{_{i, j }R_{i, j}/h_{j}j =*1, 2, ..., N*}*.*This combining is for compounding investments expressed by

and for fixed or constant investments by*C*_{j+1}= C_{j}(1+R_{j}) with C_{1}= Initial investment,

Each holding period has its own set of optimized weighting coefficients. One ends up with a time series of optimized portfolios. The CPU increases with increasing number of elements in the time series. Hence, the CPU increases with the number of holding periods. This provides the investor with the opportunity to optimize the holding period and the hedging ratio in line with the chosen investment objective AFTER the weighting coefficients have been computed. It is not self-evident that when each portfolio is optimized in terms of risks and rewards in the time series, the total sum of returns over all holding periods is also optimized accordingly. Our software always checks that evidence. It is also not self-evident that in each holding period, the individual stock-return fluctuations have the same pdf. Our software does not use any pdf.*C*_{N}= (C_{1}/N) ∑_{j=}_{{}_{1, N}_{}}*R*_{j}. - 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.
*Based on the assumption that 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

where*VaR*=_{Return Space}*σ√**N*[*-1.96 + 0.474μ*]_{1}/√N - 0.0687μ_{2}/N + 0.146μ_{1}^{2}/N*- 0.5σ*,^{2}N*N*is the number of holding periods in the recommended validation period, and*σ*,*μ*, and_{1}*μ*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:_{2}- the zero moment,

*M*, is the number of observations in the validation period_{0}- the first moment,

*M*, is the mean of all observed returns during validation_{1}- the second

*M*, third_{2}*M*, and fourth moment_{3}, are defined in the standard manner:*M*_{4}

The volatility,*M*_{k}= ∑_{j=}_{{1,N} }(*R*)_{j}- M_{1}for^{k}/ M_{0}*k = 2, 3, 4*.*σ*, is given by √*M*, the skew,_{2}*μ*, by_{1}*M*, and the excess kurtosis,_{3}/σ^{3}*μ*, by_{2}*M*. The Risk Indicator introduced for all asset markets by MiFid2 rules of the European Union is calculated from the Var-equivalent-Volatility (_{4}/σ^{4}- 3*VeV*), which takes on the form:

Our software links the calculated*VeV =*{*√(3.842 - 2*VaR*_{Return Space}*- 1.96)*}*/√(Nh)*.*VeV*to the prescribed Risk Indicator of the MiFid2 rules. - 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. Their original proposition added two new coefficients to the Capital Asset Pricing Model (CAPM). These should be computed in linear regression AFTER the weighting coefficients 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.

*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 indicator (= 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 indicator. Maximizing this indicator is usually close to the investment objective of a retail investor. As stated above, we maximize the MAR indicator 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 indicator = 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 chart enables an investor to evaluate the maximum annual expected results 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 and volumes. According to the Efficient Market hypothesis, these historical prices fully reflect the available information. Further, we assume that the past is the best predictor of success and that 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

14 July 2019