# modern portfolio theory

• This efficient half-line is called the capital allocation line (CAL), and its formula can be shown to be In this formula P is the sub-portfolio of risky assets at the tangency
with the Markowitz bullet, F is the risk-free asset, and C is a combination of portfolios P and F. By the diagram, the introduction of the risk-free asset as a possible component of the portfolio has improved the range of risk-expected return
combinations available, because everywhere except at the tangency portfolio the half-line gives a higher expected return than the hyperbola does at every possible risk level.

• For example, while we can compute that the optimal portfolio position for 3 stocks is, say, 44%, 35%, 21%, the optimal position for a project portfolio may not allow us to
simply change the amount spent on a project.

• Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level.

• The CAPM is a model that derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and
the risk of the market as a whole.

• Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a
given level of risk.

• The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk vs expected return profile — i.e., if for
that level of risk an alternative portfolio exists that has better expected returns.

• In matrix form, for a given “risk tolerance” , the efficient frontier is found by minimizing the following expression: where • is a vector of portfolio weights and (The weights
can be negative); • is the covariance matrix for the returns on the assets in the portfolio; • is a “risk tolerance” factor, where 0 results in the portfolio with minimal risk and results in the portfolio infinitely far out on the frontier
with both expected return and risk unbounded; and • is a vector of expected returns.

• When risk is put in terms of uncertainty about forecasts and possible losses then the concept is transferable to various types of investment.

• A derivation [12] is as follows: (1) The incremental impact on risk and expected return when an additional risky asset, a, is added to the market portfolio, m, follows from
the formulae for a two-asset portfolio.

• The concept of capturing the risk tolerance of an investor by documenting how much risk is acceptable for a given return may be applied to a variety of decision analysis problems.

• portfolio risk or market risk) refers to the risk common to all securities—except for selling short as noted below, systematic risk cannot be diversified away (within one
market).

• Systematic risks within one market can be managed through a strategy of using both long and short positions within one portfolio, creating a “market neutral” portfolio.

• Mathematical model Risk and expected return MPT assumes that investors are risk averse, meaning that given two portfolios that offer the same expected return, investors
will prefer the less risky one.

• Once an asset’s expected return, , is calculated using CAPM, the future cash flows of the asset can be discounted to their present value using this rate to establish the correct
price for the asset.

• In this equilibrium, relative supplies will equal relative demands: given the relationship of price with supply and demand, since risk-to-reward is “identical” across all
securities, proportions of each security in any fully-diversified portfolio would correspondingly be the same.

• Every possible combination of risky assets, can be plotted in this risk-expected return space, and the collection of all such possible portfolios defines a region in this
space.

• As a result, when it is combined with any other asset or portfolio of assets, the change in return is linearly related to the change in risk as the proportions in the combination
vary.

• Asset pricing theory builds on this analysis, allowing MPT to derive the required expected return for a correctly priced asset in this context.

• The price paid must ensure that the market portfolio’s risk / return characteristics improve when the asset is added to it.

• additional risk Updated expected return Hence additional expected return (2) If an asset, a, is correctly priced, the improvement for an investor in her risk-to-expected return
ratio achieved by adding it to the market portfolio, m, will at least (in equilibrium, exactly) match the gains of spending that money on an increased stake in the market portfolio.

• Since a security will be purchased only if it improves the risk-expected return characteristics of the market portfolio, the relevant measure of the risk of a security is
the risk it adds to the market portfolio, and not its risk in isolation.

• [15] More fundamentally, investors are stuck with estimating key parameters from past market data because MPT attempts to model risk in terms of the likelihood of losses,
but says nothing about why those losses might occur.

• Within the market portfolio, asset specific risk will be diversified away to the extent possible.

• For given portfolio weights and given standard deviations of asset returns, the case of all correlations being 1 gives the highest possible standard deviation of portfolio
return.

• [14] In practice, investors must substitute predictions based on historical measurements of asset return and volatility for these values in the equations.

• In contrast, modern portfolio theory is based on a different axiom, called variance aversion,[24] and may recommend to invest into Y on the basis that it has lower variance.

• The variance of return (or its transformation, the standard deviation) is used as a measure of risk, because it is tractable when assets are combined into portfolios.

• More formally, then, since everyone holds the risky assets in identical proportions to each other — namely in the proportions given by the tangency portfolio — in market equilibrium
the risky assets’ prices, and therefore their expected returns, will adjust so that the ratios in the tangency portfolio are the same as the ratios in which the risky assets are supplied to the market.

• Updated portfolio risk Hence, risk added to portfolio but since the weight of the asset will be low re.

• If nuclear engineers ran risk management this way, they would never be able to compute the odds of a meltdown at a particular plant until several similar events occurred in
the same reactor design.— Mathematical risk measurements are also useful only to the degree that they reflect investors’ true concerns—there is no point minimizing a variable that nobody cares about in practice.

• Combinations along this upper edge represent portfolios (including no holdings of the risk-free asset) for which there is lowest risk for a given level of expected return.

• MPT uses historical variance as a measure of risk, but portfolios of assets like major projects do not have a well-defined “historical variance”.

• (There are several approaches to asset pricing that attempt to price assets by modelling the stochastic properties of the moments of assets’ returns – these are broadly referred
to as conditional asset pricing models.)

• Such measures often cannot capture the true statistical features of the risk and return which often follow highly skewed distributions (e.g.

• Very often such expected values fail to take account of new circumstances that did not exist when the historical data were generated.

• If the desired portfolio is outside the range spanned by the two mutual funds, then one of the mutual funds must be sold short (held in negative quantity) while the size of
the investment in the other mutual fund must be greater than the amount available for investment (the excess being funded by the borrowing from the other fund).

• [13][2] The risk, return, and correlation measures used by MPT are based on expected values, which means that they are statistical statements about the future (the expected
value of returns is explicit in the above equations, and implicit in the definitions of variance and covariance).

• They simply indicate the need to run the optimization with an additional set of mathematically expressed constraints that would not normally apply to financial portfolios.

• [10][11] This theorem states that any portfolio on the efficient frontier can be generated by holding a combination of any two given portfolios on the frontier; the latter
two given portfolios are the “mutual funds” in the theorem’s name.

• The risk-free asset has zero variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition, since its variance is zero).

• So in the absence of a risk-free asset, an investor can achieve any desired efficient portfolio even if all that is accessible is a pair of efficient mutual funds.

• In general: Expected return: where is the return on the portfolio, is the return on asset i and is the weighting of component asset (that is, the proportion of asset “i” in
the portfolio, so that ).Portfolio return variance: where is the (sample) standard deviation of the periodic returns on an asset i, and is the correlation coefficient between the returns on assets i and j. Alternatively the expression can
be written as: where for , or where is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as , or Portfolio return volatility (standard deviation): For a two-asset portfolio:Portfolio expected return:
Portfolio variance: For a three-asset portfolio: Portfolio expected return: Portfolio variance: The algebra can be much simplified by expressing the quantities involved in matrix notation.

• the log-normal distribution) and can give rise to, besides reduced volatility, also inflated growth of return.

• There many other risk measures (like coherent risk measures) might better reflect investors’ true preferences.

• When a risk-free asset is introduced, the half-line shown in the figure is the new efficient frontier.

• The fact that all points on the linear efficient locus can be achieved by a combination of holdings of the risk-free asset and the tangency portfolio is known as the one mutual
fund theorem,[10] where the mutual fund referred to is the tangency portfolio.

• In practice, short-term government securities (such as US treasury bills) are used as a risk-free asset, because they pay a fixed rate of interest and have exceptionally low
default risk.

• An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return This version of the problem requires that we minimize
subject to for parameter .

• A portfolio optimization method would have to take the discrete nature of projects into account.

• In this case, the MPT investment boundary can be expressed in more general terms like “chance of an ROI less than cost of capital” or “chance of losing more than half of the
investment”.

• Capital asset pricing model Main article: Capital asset pricing model The asset return depends on the amount paid for the asset today.

• Risk-free asset and the capital allocation line Main article: Capital allocation line The risk-free asset is the (hypothetical) asset that pays a risk-free rate.

• the sensitivity of the asset price to movement in the market portfolio’s value (see also Beta (finance) § Adding an asset to the market portfolio).

• Its vertical intercept represents a portfolio with 100% of holdings in the risk-free asset; the tangency with the hyperbola represents a portfolio with no risk-free holdings
and 100% of assets held in the portfolio occurring at the tangency point; points between those points are portfolios containing positive amounts of both the risky tangency portfolio and the risk-free asset; and points on the half-line beyond
the tangency point are portfolios involving negative holdings of the risk-free asset and an amount invested in the tangency portfolio equal to more than 100% of the investor’s initial capital.

• Post-modern portfolio theory extends MPT by adopting non-normally distributed, asymmetric, and fat-tailed measures of risk.

• Criticisms Despite its theoretical importance, critics of MPT question whether it is an ideal investment tool, because its model of financial markets does not match the real
world in many ways.

• In this context, the volatility of the asset, and its correlation with the market portfolio, are historically observed and are therefore given.

• “[20] A few studies have argued that “naive diversification”, splitting capital equally among available investment options, might have advantages over MPT in some situations.

• [21] When applied to certain universes of assets, the Markowitz model has been identified by academics to be inadequate due to its susceptibility to model instability which
may arise, for example, among a universe of highly correlated assets.

Works Cited

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Photo credit: https://www.flickr.com/photos/alexbrn/5357144886/’]