Theory of portfolio

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Theory of portfolio
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Theory of portfolio (portfolio theory, modern portfolio theory, MPT) - is the body of ideas, started by Harry Markowitz in his 1952 paper "Portfolio Selection", that explains how an investor should combine assets so that the expected return of the whole portfolio is as high as possible for a given level of risk, or the risk is as low as possible for a given expected return[1]. Risk is measured by the variance (or standard deviation) of portfolio returns, and the central lesson is that an asset should not be judged on its own but by what it does to the portfolio it joins.

Before 1952 investment advice mostly meant picking good stocks. Markowitz changed the question. He showed that two risky shares held together can be less risky than either of them alone, provided their returns don't move in lockstep. That one observation turned diversification from folk wisdom ("don't put all your eggs in one basket") into something you could calculate. The Royal Swedish Academy of Sciences later described his contribution as the theory of portfolio choice, which "analyzes how wealth can be optimally invested in assets which differ in regard to their expected return and risk, and thereby also how risks can be reduced"[2]. Markowitz shared the 1990 Nobel Memorial Prize in Economic Sciences with Merton Miller and William Sharpe.

Origins and development

Markowitz was a graduate student at the University of Chicago when he wrote the article. It appeared in The Journal of Finance (Vol. 7, No. 1, pp. 77-91) and rejected the then-common rule that an investor should simply maximise discounted expected returns. Under that rule, he argued, the investor would put everything into the single security with the highest expected return, which nobody sensible does. Investors diversify. So a theory of investing must explain why, and the explanation is that they care about variance as well as about the mean[1]. In 1959 he expanded the work into the book Portfolio Selection: Efficient Diversification of Investments, which also discussed semivariance as an alternative risk measure.

The theory then grew in steps:

  • 1952 - Markowitz introduces the mean-variance criterion and the set of efficient portfolios. In the same year A.D. Roy publishes a closely related "safety first" approach.
  • 1958 - James Tobin adds a riskless asset and shows that the choice of risky assets can be separated from the decision of how much risk to take[3].
  • 1964 - William Sharpe asks what happens if all investors behave the Markowitz way, and derives market equilibrium prices: the Capital Asset Pricing Model[4]. John Lintner and Jan Mossin reach similar results independently.
  • 1966 - Sharpe proposes the reward-to-variability ratio for judging mutual funds, now known as the Sharpe ratio[5].
  • 1990s - critics of the variance measure group their ideas under the label "post-modern portfolio theory"[6].

Tobin received the Nobel prize in 1981; Sharpe in 1990 together with Markowitz.

Assumptions of the mean-variance model

The classic model rests on a short list of assumptions. They're worth stating openly, because most of the later criticism attacks one of them:

  • investors are risk-averse and judge portfolios only by expected return and variance of return over one period,
  • the expected returns, variances and covariances of all assets are known (or can be estimated well enough),
  • investors prefer more return to less and less risk to more,
  • assets are infinitely divisible, there are no taxes and no transaction costs,
  • in the equilibrium extensions (Tobin, Sharpe) investors can lend and borrow at the same risk-free rate.

Mean and variance are sufficient to describe choice if returns follow a normal distribution or if investors have quadratic utility. Neither is strictly true. Markowitz knew this and presented mean-variance as a practical approximation, consistent with expected utility theory in many realistic cases, not as an exact law.

Formulas for a two-asset portfolio

Take two assets with expected returns \( E(R_1) \), \( E(R_2) \), standard deviations \( \sigma_1 \), \( \sigma_2 \) and correlation coefficient \( \rho_{12} \). If a share \( w \) of wealth goes into asset 1 and \( 1-w \) into asset 2, the expected return of the portfolio is a plain weighted average:

\[ E(R_p) = w \, E(R_1) + (1-w) \, E(R_2) \]

Risk is a different story. The portfolio variance is

\[ \sigma_p^2 = w^2 \sigma_1^2 + (1-w)^2 \sigma_2^2 + 2 w (1-w) \rho_{12} \sigma_1 \sigma_2 \]

The last term is where diversification lives. When \( \rho_{12} = 1 \), portfolio standard deviation is just the weighted average of the two standard deviations, and nothing is gained. When \( \rho_{12} < 1 \), the portfolio standard deviation is lower than that average. At \( \rho_{12} = -1 \) risk can, in theory, be eliminated entirely.

The weight of asset 1 that gives the smallest possible variance (the minimum-variance portfolio) is

\[ w^{*} = \frac{\sigma_2^2 - \rho_{12} \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho_{12} \sigma_1 \sigma_2} \]

For many assets the same logic is written in matrix form, \( \sigma_p^2 = w^{T} \Sigma w \), where \( \Sigma \) is the covariance matrix. With 100 assets that matrix holds 100 variances and 4,950 distinct covariances. That count alone explains why estimation became the practical headache of the whole approach.

Efficient frontier, capital market line and the Sharpe ratio

Plot every possible portfolio on a chart with standard deviation on the horizontal axis and expected return on the vertical axis. The feasible combinations form a region whose left edge curves like a bullet. The upper half of that edge, starting at the minimum-variance portfolio, is the efficient frontier: for every point on it no other portfolio offers more return for the same risk, or less risk for the same return. Portfolios below it are inefficient. Rational mean-variance investors pick somewhere on the frontier, and where exactly depends on their appetite for risk.

Tobin's step was to add a risk-free asset, such as a short-term government bill. Now an investor can mix any risky portfolio with lending or borrowing at the risk-free rate \( R_f \). Those mixes lie on straight lines. The steepest possible line touches the frontier at a single point, the tangency portfolio, and every investor is best off holding that same risky portfolio and simply adjusting the split between it and the riskless asset[3]. This is the separation theorem: the question "which risky assets?" is separate from the question "how much risk?". Cautious investors lend, bold ones borrow. Both hold the identical risky mix.

In Sharpe's equilibrium all investors hold the tangency portfolio, so it must be the market portfolio M. The line from \( R_f \) through M is the capital market line (CML)[4]:

\[ E(R_p) = R_f + \frac{E(R_M) - R_f}{\sigma_M} \, \sigma_p \]

Its slope is the market price of risk. The same kind of slope, applied to any portfolio, gives the Sharpe ratio[5]:

\[ S_p = \frac{E(R_p) - R_f}{\sigma_p} \]

A higher ratio means more excess return per unit of total risk. The tangency portfolio has the highest Sharpe ratio of all risky portfolios.

Systematic and unsystematic risk

Diversification has a limit. As more shares are added to a portfolio, the part of risk that is unique to each company (a failed product launch, a lawsuit, a strike) averages out. This is specific or unsystematic risk. What remains is systematic risk, also called market risk: recessions, interest rate changes, inflation shocks. They hit almost everything at once and can't be diversified away.

The CAPM draws the logical conclusion. If unsystematic risk can be removed at no cost, the market will not pay anyone for bearing it. Expected return should depend only on an asset's exposure to market movements, measured by beta:

\[ E(R_i) = R_f + \beta_i \left[ E(R_M) - R_f \right] \]

Plotted against beta, this relation is the security market line, and the bracketed term is the market risk premium. The CAPM is a separate theory with its own controversies; for portfolio theory it matters mainly as the equilibrium consequence of everyone following Markowitz.

Example of the theory of portfolio

The numbers below are hypothetical, chosen to illustrate the formulas.

An investor can split money between two funds:

  • Fund A (equities): expected return 10%, standard deviation 20%,
  • Fund B (bonds): expected return 6%, standard deviation 10%,
  • correlation between them: \( \rho = 0.2 \), so the covariance is \( 0.2 \times 0.20 \times 0.10 = 0.004 \),
  • risk-free rate: 3%.

Applying the two-asset formulas for several weights of Fund A:

Weight in A Weight in B Expected return Variance Standard deviation Sharpe ratio
0% 100% 6.0% 0.010000 10.00% 0.300
25% 75% 7.0% 0.009625 9.81% 0.408
50% 50% 8.0% 0.014500 12.04% 0.415
75% 25% 9.0% 0.024625 15.69% 0.382
100% 0% 10.0% 0.040000 20.00% 0.350

Look at the second row. Moving a quarter of the money from bonds into the riskier equity fund raises expected return from 6% to 7% and at the same time lowers standard deviation from 10.00% to 9.81%. Sounds wrong. It isn't. Low correlation does the work.

The minimum-variance weight follows from the formula:

\[ w^{*} = \frac{0.01 - 0.004}{0.04 + 0.01 - 2 \times 0.004} = \frac{0.006}{0.042} \approx 0.143 \]

So about 14.3% in Fund A and 85.7% in Fund B. This portfolio has expected return of about 6.57% and standard deviation of about 9.56%, lower than either fund held alone. Every portfolio with less than 14.3% in A is inefficient: the investor could get more return for less risk by moving towards w*.

Correlation changes the picture a lot. For the 50/50 mix, standard deviation equals 15.0% if \( \rho = 1 \), 11.2% if \( \rho = 0 \), and 5.0% if \( \rho = -1 \). Same funds, same weights, three very different risk levels.

Adding the risk-free asset, the tangency portfolio (the one with the highest Sharpe ratio) has about 38.7% in Fund A, expected return of about 7.55% and standard deviation of about 10.79%, giving a Sharpe ratio of about 0.421. Higher than any row in the table. A cautious investor might hold half in Treasury bills and half in this mix; a bold one might borrow to hold more than 100% of it. Both stay on the same capital market line. All figures were checked with a short Python script.

Post-modern portfolio theory and downside risk

Variance punishes upside surprises as much as downside ones. Investors, on the other hand, rarely complain when a fund beats expectations. That asymmetry is the starting point for what Brian Rom and Kathleen Ferguson called post-modern portfolio theory (PMPT)[6]. The main ideas:

  • risk is measured relative to a minimum acceptable return (MAR) chosen by the investor, not relative to the mean,
  • only returns below the MAR count, via downside deviation or semivariance,
  • return distributions may be skewed or fat-tailed and need not be normal,
  • performance is judged by measures such as the Sortino ratio, which divides excess return over the MAR by downside deviation instead of total standard deviation.

Markowitz had himself proposed semivariance in 1959, but it was computationally heavy at the time. Cheaper computers removed that obstacle. Other risk measures used in the same spirit include value at risk and conditional value at risk (expected shortfall).

Behavioural critique

Mean-variance theory describes how a rational investor should behave. Real investors often do something else. Hersh Shefrin and Meir Statman proposed behavioural portfolio theory, in which people build portfolios as layered pyramids: a safe bottom layer meant to protect against poverty and an upper layer of lottery-like holdings meant to give a shot at riches[7]. Each layer is managed as a separate mental account, and covariances between layers are largely ignored, which Markowitz's model would never allow. Such portfolios look poorly diversified by mean-variance standards, yet they match how many households actually behave.

Other well-documented behaviours sit uneasily with the theory: home bias (holding far too much domestic equity), loss aversion, over-trading and the tendency to judge each holding in isolation rather than by its contribution to total portfolio risk.

Advantages of the theory of portfolio

  • It gives a precise, measurable meaning to diversification and shows why correlation, not just individual risk, matters.
  • The efficient frontier turns a vague trade-off into a menu of concrete choices (see risk-return tradeoff).
  • The separation theorem justifies holding a broad market portfolio plus cash or bonds, which is the intellectual basis of index investing.
  • It supplies standard tools for performance evaluation (Sharpe ratio) and for strategic asset allocation in portfolio construction.
  • The framework is flexible: constraints such as no short selling, sector limits or liability matching can be added to the optimisation.
  • It led directly to the CAPM and to much of modern capital market theory.

Limitations of the theory of portfolio

  • Estimation error. The optimiser treats estimated means, variances and correlations as if they were exact. Small errors in expected returns produce large, unstable shifts in the weights. Richard Michaud argued that unconstrained mean-variance optimisers tend to overweight exactly those assets whose expected returns have been overestimated, so the "optimal" portfolio can be worse in practice than a simple rule of thumb[8].
  • Naive diversification often does as well. DeMiguel, Garlappi and Uppal compared 14 optimisation models with an equally weighted 1/N portfolio across several data sets and found that none of them consistently beat the naive strategy out of sample; the estimation windows needed for the optimised versions to win were far longer than available histories[9].
  • Normality. Real returns have fat tails and skewness, especially in crises. Variance then understates the probability of large losses.
  • Stationarity. The model assumes return distributions stay fixed. In fact volatility clusters and changes over time (see stochastic volatility), and correlations tend to rise exactly when markets fall, just when diversification is needed most.
  • Single period. The basic model looks at one period and ignores the investment horizon, consumption needs and rebalancing.
  • Unrealistic market assumptions. Taxes, transaction costs, borrowing limits and illiquid assets are absent from the classic version.
  • Variance as risk. It treats gains and losses symmetrically, which most investors don't.

Practitioners have answered with shrinkage estimators for the covariance matrix, the Black-Litterman model (which blends market-implied returns with the investor's own views), resampled efficient frontiers and worst-case optimisation over ranges of inputs. None of these throws out Markowitz. They patch the inputs.

Related approaches

  • Capital Asset Pricing Model - equilibrium extension of mean-variance theory based on beta.
  • Arbitrage pricing theory (Ross, 1976) - explains returns by several systematic factors instead of one market factor.
  • Factor models (e.g. Fama-French) - used in practice to estimate covariances and expected returns more reliably.
  • Risk parity - allocates so that each asset class contributes equally to total risk, avoiding expected-return estimates.
  • Liability-driven investing - optimises assets relative to a future stream of obligations, typical for pension funds and insurers.

References

Footnotes

  1. ↑ Markowitz H., Portfolio Selection, The Journal of Finance, 1952, pp. 77-91
  2. ↑ The Royal Swedish Academy of Sciences, Press release: The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1990, 1990
  3. ↑ Tobin J., Liquidity preference as behavior towards risk, The Review of Economic Studies, 1958, pp. 65-86
  4. ↑ Sharpe W.F., Capital asset prices: A theory of market equilibrium under conditions of risk, The Journal of Finance, 1964, pp. 425-442
  5. ↑ Sharpe W.F., Mutual fund performance, The Journal of Business, 1966, pp. 119-138
  6. ↑ Rom B.M., Ferguson K.W., Post-modern portfolio theory comes of age, The Journal of Investing, 1994, pp. 11-17
  7. ↑ Shefrin H., Statman M., Behavioral portfolio theory, Journal of Financial and Quantitative Analysis, 2000, pp. 127-151
  8. ↑ Michaud R.O., The Markowitz optimization enigma: Is 'optimized' optimal?, Financial Analysts Journal, 1989, pp. 31-42
  9. ↑ DeMiguel V., Garlappi L., Uppal R., Optimal versus naive diversification, The Review of Financial Studies, 2009, pp. 1915-1953
Theory of portfolio — recommended articles
Minimum-variance portfolio — Risk-return tradeoff — Portfolio risk — Portfolio construction — Systematic risk — Security market line — Capital market theories
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