By Philippe Alezard
With Wiener, modern finance had found a way to give uncertainty a mathematical form. Brownian motion made it possible to conceive of price fluctuations as the cumulative effect of an infinite number of elementary shocks – random, unpredictable, yet nevertheless capable of being placed within a rigorous probabilistic framework. In other words, uncertainty ceased to be a mere market intuition and became a mathematical object. However, this initial breakthrough did not exhaust the issue. For knowing that prices fluctuate according to a stochastic dynamic does not yet tell us how an investor should behave in the face of this instability. Once uncertainty had been described, the next step was to structure how it was used.
It is precisely at this point that Harry Markowitz enters the picture. Whereas Wiener provided finance with the probabilistic grammar of uncertainty, Markowitz developed its decision-making logic. His primary concern was no longer the evolution of a price over time, but rather the selection of a portfolio of assets in a world where the future remains irreducibly uncertain. With him, finance crossed a new threshold of formalisation: it no longer contented itself with modelling market movements, but sought to determine how a rational agent should allocate their capital within these movements.
The only son of a Chicago grocery store owner couple, Harry Markowitz was born in Chicago on 24 August 1927. At secondary school, his first interests were physics and philosophy [1]. One of the arguments that particularly impressed him was David Hume’s claim that, even if we drop a ball a thousand times and it falls to the ground each time, we have no necessary proof that it will fall the thousand and first time. In 1947, he enrolled at the University of Chicago to study for a Bachelor of Philosophy degree on the OII (‘Observation, Interpretation and Integration’) programme, but by 1950, he was drawn to the economics of uncertainty. At that time, he was interested in the work of von Neumann [2] and Morgenstern on expected utility, as well as that of Milton Friedman and Leonard Savage on the utility function.
Having never taken a single finance course and never bought a single share, he was invited to join the renowned Cowles Commission for Research in Economics [3], an institution that has produced twelve Nobel Prize winners in Economics to date, including Markowitz himself. At the time, its director was Tjalling Koopmans [4], who would go on to win the Nobel Prize in 1975 for his work on the theory of the optimal allocation of resources. Among the lecturers who would teach the young Markowitz were, notably, Milton Friedman and Leonard J. Savage [5]. For the record, it is to the latter that we owe the rediscovery of Louis Bachelier’s thesis, which he had Paul Samuelson read. It was also while reading John Burr Williams’ The Theory of Investment Value [6] that Markowitz came up with the idea for his doctoral thesis topic. As he himself recounts in his autobiography, published on the occasion of his Nobel Prize award, ‘Williams proposed that the value of a share should be equal to the present value of its future dividends. Since future dividends are uncertain, I interpreted Williams’ proposition as meaning that a share should be valued on the basis of its expected future dividends. But if the investor were concerned only with the expected values of securities, he or she would be concerned only with the expected value of the portfolio; and, to maximise the expected value of a portfolio, it would be sufficient to invest in a single security. I knew that this was neither how investors behaved nor how they should behave. Investors diversify because they are as concerned about risk as they are about return.”
Markowitz’s insight [7], though seemingly simple, is in fact of considerable significance. Prior to him, financial analysis tended to assess investments primarily on an individual basis, based on their promised returns, their soundness or their reputation. Markowitz radically shifted the focus of the problem by demonstrating that, from the investor’s perspective, the relevant unit is not the individual asset, but the portfolio as a whole. In his seminal article, ‘Portfolio Selection’, published in the March 1952 issue of the Journal of Finance, he proposed representing a portfolio not as the simple sum of several securities, but as an overall structure resulting from their combination. This is because what matters is not only the intrinsic quality of each asset, but the combination of their behaviours, since the risk of a portfolio depends not only on the risk inherent in each security, but also on how these securities vary in relation to one another. The financial decision thus ceases to be a simple matter of selecting good securities and becomes a question of portfolio composition. This shift is decisive, as it reveals that risk itself cannot be considered in a purely individual manner. An asset may be highly volatile when considered in isolation, but it can become fully acceptable within a portfolio if, through its relationships with other assets, it helps to stabilise the whole. Conversely, apparently safe securities, if they all react in the same way to the same events, can collectively generate greater fragility than one might imagine. In this way, Markowitz introduces the idea that financial rationality is not a property of individual instruments, but of the way they are combined.
This idea is formulated through the distinction between expected return and risk. According to Markowitz, the investor optimises this return–risk pair at the portfolio level, either by seeking the highest return for a given risk or by seeking the lowest risk for a given return. It thus becomes possible to represent the portfolio using utility functions that depend solely on the expected return, i.e., the mathematical expectation, and on a measure of risk, i.e., variance. In Markowitz’s work, variance derives its legitimacy not from its common use in statistics, but from its economic relevance. It is justified by the fact that the prudent investor does not assess gains and losses symmetrically: an additional loss is more than proportionally burdensome, whereas an additional gain is not proportionally more favourable. Therefore, the dispersion of returns around their expected value constitutes a relevant measure of risk.
Therefore, if we denote wi as the proportion of wealth invested in asset i, and E(Ri) as its expected return, then the expected return of the portfolio can be expressed as:
This relationship expresses the fact that, in a basic sense, the overall return is merely a weighted average of the individual returns. At this stage, nothing yet appears to depart from common intuition. However, it is precisely when we move on to measuring risk that the theoretical novelty emerges. For while returns combine in an additive manner, risk, on the other hand, does not aggregate through a simple sum. The variance of the portfolio is expressed as:
In other words, the total risk depends not only on the variances specific to each asset, but also on all the cross-covariances between their returns. In the basic case of two assets, the formula becomes:
This expression lies at the very heart of the Markowitz revolution. It means that the risk of a portfolio is never the simple juxtaposition of individual risks, but the product of a relational structure. What matters is not only the volatility specific to each security, but also the way in which these volatilities interact, reinforce each other or offset one another.
This is where covariance comes into play, a concept that is crucial because it measures the direction and magnitude of the joint variation of two assets. When the covariance is strongly positive, the assets tend to rise and fall in unison; combining them therefore offers little protection, as any losses incurred by one asset are likely to be mirrored by the other. When the covariance is low, the overlap between the behaviours of the assets decreases, and the portfolio becomes relatively more stable. Finally, when the covariance is negative, the movements of one asset tend to partially offset those of the other, so that part of the individual risk is absorbed by the overall structure. Thus, in Markowitz’s work, diversification is given its first strictly mathematical foundation. It is no longer a prudent maxim based on simple common sense – don’t put all your eggs in one basket – but rather a calculable property of the combination of assets. Therefore, diversifying does not mean mechanically accumulating a large number of securities; it means combining positions whose behaviours are not superimposable. In essence, a large number of highly correlated assets are merely a disguised repetition of the same risk. Conversely, a truly diversified portfolio is one whose internal heterogeneity transforms the dispersion of exposures into an effective reduction in overall variance.
From this logic derives the concept of the efficient frontier, which encapsulates the theoretical objective of the model. If we plot all possible portfolios in a space where the horizontal axis measures risk sp and the vertical axis measures expected return E(Rp), we can see that certain portfolios outperform the others (see below).
This graph illustrates the logic of portfolio optimisation within the Markowitz mean–variance framework.
The black curve represents the frontier of risky portfolios. Its upper part is the efficient frontier: it comprises the portfolios that offer the highest possible return for a given level of risk. The lower part, below the minimum variance portfolio, is not efficient, as it is possible to find better options in terms of either return or risk.
The point labelled ‘Minimum Variance Portfolio’ represents the minimum variance portfolio. No other risky portfolio offers lower volatility. It therefore represents the starting point of the efficient frontier.
The Tangency Portfolio point is of particular importance because it lies at the point where the red dotted line is tangent to the frontier. This line is the capital market line, or capital allocation line: it starts at the risk-free rate and intersects the frontier at the portfolio that maximises the best excess return per unit of risk. In other words, if a risk-free asset is introduced, the tangent portfolio becomes the best benchmark risky portfolio.
The Maximum Return Portfolio point (Volatility = 6.0%) shows that, at a given level of volatility – in this case 6% – it is possible to identify, on the frontier, the portfolio that delivers the maximum return consistent with this constraint. This clearly illustrates the Markowitzian idea that a good portfolio depends on the trade-off chosen between risk and return.
The Equally-Weighted Portfolio point represents an equally weighted portfolio. As can be seen, it lies below the efficient frontier: it is therefore sub-optimal in the Markowitz sense. For a similar level of risk, there is a portfolio on the frontier that offers a higher return; or, for a similar return, there is a less risky portfolio.
The points 100% SPY (SP500 equities), 100% GLD (Gold), 100% EFA (iShare ETF on developed countries) and 100% IEF (ETF on 7–10-year Treasury bonds) correspond to portfolios fully invested in a single asset. They are all located to the right of or below the frontier, which demonstrates the value of diversification. Taken in isolation, these assets either offer too much risk for their return or an insufficient return compared to what a diversified portfolio would deliver.
The overall message of the graph is unambiguous. Diversification leads investors to more efficient combinations than holding a single asset or adopting a naively equal-weight allocation. This confirms Markowitz’s key insight: the value of a portfolio lies not merely in the sum of the individual qualities of its components, but in the structure of their interdependencies. In other words, financial rationality is no longer limited to the pursuit of maximum return in isolation; it is now defined within the joint space of expected return and overall risk, as determined by the efficient frontier.
The significance of the model lies precisely in this shift. With Markowitz, prudence ceases to be a vague disposition and becomes a formalisable principle. The portfolio is no longer conceived as an empirical juxtaposition of securities, but as an object of optimisation. Correspondingly, the investor no longer appears as a mere selector of individual assets, but as an agent who trades off between risk and return configurations. Financial decision-making is thus embedded within a rigorous framework, where calculation replaces intuition and where choices are formulated in terms of dominance, efficiency and trade-off.
However, such formalisation is only possible at the cost of a set of assumptions, the scope of which must be assessed. Building an efficient portfolio requires the availability of sufficiently robust estimates of expected returns E(Ri), variances and covariances Cov(Ri,Rj). However, these quantities are never provided immediately. They are inferred from past data series, extrapolated into the future, adjusted, stabilised, and ultimately treated as if they could provide a reliable basis for the present decision. Therefore, the optimal portfolio is not based on certain knowledge, but on inevitably imperfect statistical information.
This fragility is all the more significant given that markets are not stationary environments. Regimes change, correlations shift, crises abruptly reconfigure the dependencies between assets, and discontinuities can undermine the patterns observed in historical data. In this sense, the Markowitzian approach by no means eliminates uncertainty; rather, it reformulates it in a probabilistic language that is itself subject to estimation error. The calculated risk is never more than the risk as it can be grasped from a statistical picture of reality that is always partial.
More fundamentally still, reducing risk to variance entails a specific definition of financial uncertainty. By treating positive and negative deviations from the mean symmetrically, it equates fluctuations that are, however, unequally desirable from the investor’s perspective to a single phenomenon. However, investors do not fear variability as such; they primarily fear loss. This point does not negate the model’s usefulness, but it does indicate its conceptual limitation: variance is not risk per se; it is a specific mathematical representation of risk, based on certain assumptions regarding the preferences of the agents.
In addition, there is an equally crucial anthropological assumption. The model assumes an investor capable of ordering their risk preferences in a consistent and stable manner. It thus overlooks what real-world market experience continually brings to light: mimetic behaviours, self-referential expectations, cognitive biases, regulatory constraints, heterogeneous horizons, information asymmetries, and a plurality of strategies. The rationality portrayed by the theory is therefore less an empirical description of actual behaviour than a standard of consistency applied to decision-making in an uncertain world.
However, these reservations do not diminish the work’s historical significance; on the contrary, they enable us to precisely determine its status. Markowitz’s theory is not an exhaustive reproduction of the actual functioning of markets. It belongs to those major normative constructs through which modern finance has sought to make choices under uncertainty intellectually calculable. Its significance extends far beyond the mere technique of portfolio construction. By establishing the idea that a rational financial decision must be considered within the combined framework of expected return and measured risk, it has permanently transformed the very categories of the discipline.
A significant part of modern finance stems directly from this overhaul. The distinction between diversifiable and non-diversifiable risk, asset equilibrium models, asset allocation theory, index management, and subsequently a large part of contemporary quantitative management all represent, in various forms, an extension of Markowitz’s seminal work. In this sense, Markowitz not only added a new instrument to the financial toolkit; he also changed the intellectual framework within which financial problems can be formulated. Following Wiener, who had made the random movement of prices mathematically intelligible, Markowitz gave investment a genuinely decision-making form. The former made it possible to conceive of the dynamics of chance; the latter made it possible to conceive of the rational organisation of capital exposed to these dynamics. Between the two, one of the most profound developments in financial modernity took shape: not to abolish uncertainty, but to transform it into a calculable structure.
Markowitz’s theory already contained, in embryo, a decisive implication. If the risk of a portfolio depends on how the assets are combined, then part of that risk can be eliminated through diversification. However, this consequence called for a conceptual shift. Once an investor holds a sufficiently large portfolio, the risks associated with the specific characteristics of a company – managerial failure, industrial accident, loss of competitiveness, failed innovation – tend to dissipate within the overall picture. On the other hand, more general risks remain, linked to overall market movements and to general fluctuations in asset prices. It is on the basis of this new understanding of risk that another decisive stage in modern finance can unfold: no longer merely considering the optimal composition of a portfolio, but determining the value of contingent assets derived from this structure of uncertainty. Whereas Markowitz had shown how to rationally organise capital exposed to randomness, the Black–Scholes model would demonstrate that, under certain assumptions, it is possible to neutralise this randomness through dynamic hedging strategies and to derive a theoretical price for options. Portfolio theory thus paved the way for a theory of valuation: following the formalisation of choice in a risky environment came the formalisation of the price of the rights relating to that very risk.
[1] [1] https://www.nobelprize.org/prizes/economic-sciences/1990/markowitz/biographical/
[2] John von Neumann, 1903–1957, Hungarian, a genius in mathematics and physics, was, among other things, one of the key contributors to the Manhattan Project alongside Oppenheimer.
[4] Tjalling Koopmans, 1910–1985, mathematician, psychiatrist and economist, Nobel Prize in Economics in 1975
[5] Leonard Jimmie Savage, 1917–1971, American mathematician and statistician, Von Neumann’s assistant on the Manhattan Project
[6] John Burr Williams, 1900–1983, American economist responsible for the well-known Discounted Cash Flow (DCF) theory
[7] https://cowles.yale.edu/sites/default/files/2