Modern portfolio theory Guide, Meaning , Facts, Information and Description
Modern portfolio theory (MPT) proposes how rational investors will use diversification to optimize their portfolios, and how an asset should be priced given its risk relative to the market as a whole. The basic concepts of the theory are the efficient frontier, Capital Asset Pricing Model and beta coefficient, the Capital Market Line and the Securities Market Line.MPT models the return of an asset as a random variable and a portfolio as a weighted combination of assets; the return of a portfolio is thus also a random variable and consequently has an expected value and a variance. Risk in this model is identified with the standard deviation of portfolio return. Rationality is modeled by supposing that an investor choosing between several portfolios with identical expected returns, will prefer that portfolio which minimizes risk.
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2 The risk free asset 3 Asset pricing 4 References 5 See also 6 External links |
The model assumes that investors are risk averse. This means that an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. The exact trade-off will differ by investor. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favourable risk-return profile - i.e. if for that level of risk an alternative portfolio exists which has better expected returns.
It is further assumed that investor's risk / reward preference can be described via a quadratic utility function. The effect of this assumption is that only the expected return, i.e. mean return, and the volatility, i.e. the standard deviation, matter to the investor. The investor is indifferent to other characteristics of the distribution of returns, such as its skew. Note that the theory uses an historical parameter, volatility, as a proxy for risk while return is an expectation on the future.
Under the model:
In general:
For a two asset portfolio:
Risk and reward
Mean and variance
Mathematically:
The variance of the portfolio will be the sum of the product of every asset pair's weights and covariance, - this sum includes the squared weight and variance (or ) for each individual asset. Covariance is often expressed in terms of the correlation in returns between two assets where
For a three asset portfolio, the variance is:
(As can be seen, as the number of assets (n) in the portfolio increases, the calculation becomes “computationally intensive” - the number of covariance terms = n (n-1) /2. For this reason, portfolio computations usually require specialized software. These values can also be modeled using matrices; for a manageable number of assets, these statistics can be calculated using a spreadsheet.)
An investor can reduce portfolio risk simply by holding unrelated instruments. In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk. For diversification to work the component assets must have unrelated risks.
Mathematically:
From the formulae above: if any two assets in the portfolio have a correlation of less than 1 (i.e. are not perfectly correlated) the portfolio variance and hence volatility will be less than the weighted average of the individual instruments' volatilities.
Every possible asset combination can be plotted in risk-return space. For every level of return there exists one portfolio with the lowest risk; conversely, for every level of risk there is one portfolio with the highest return. The combination of all such portfolios is called the efficient frontier (sometimes “the Markowitz frontier”.)
The efficient frontier is illustrated above, with return on the y axis, and risk on the x axis.
The efficient frontier will be concave – this is because the risk-return characteristics of a portfolio change in a non-linear fashion as its component weightings are changed. (As described above, portfolio risk is a function of the correlation of the component assets, and thus changes in a non-linear fashion as the weighting of component assets changes.)
The region above the frontier is unachievable by holding risky assets alone. No portfolios can be constructed corresponding to the points in this region. Points below the frontier are suboptimal. A rational investor will hold a portfolio only on the frontier.
Because both risk and return change linearly as the risk free asset is introduced into a portfolio, this combination will plot a straight line in risk return space. The line starts at 100% in cash and weight of the risky portfolio = 0 (i.e. intercepting the return axis at the risk free rate) and goes through the portfolio in question where cash holding = 0 and portfolio weight = 1.
Mathematically:
Return is the weighted average of the risk free asset, rf, and the risky portfolio, p, and is therefore linear:
All rational investors will hold some combination of the market portfolio and the risk free asset.
The CML is illustrated above, with return on the y axis, and risk on the x axis.
Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Systematic risk, or market risk, refers to the risk common to all securities - systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio.
Since a security will be purchased only if it improves the risk / return characteristics of the market portfolio, the risk of a security will be the risk it adds to the market portfolio.
The volatility of the asset, and its correlation with the market portfolio, is historically observed and is therefore a given. The (maximum) price paid for any particular asset (and hence the return it will generate) should also be determined based on its relationship with the market portfolio.
The CAPM is usually expressed:
A more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. In theory, an asset is correctly priced when its observed price is the same as its value calculated using the CAPM derived discount rate. If the observed price is higher than the valuation, then the asset is overvalued; it is undervalued for a too low price.
Mathematically:
(1) The incremental impact on risk and return when an additional risky asset, a, is added to the market portfolio, m, follows from the formulae for a two asset portfolio. These results are used to derive the asset appropriate discount rate.
Risk =
Diversification
The efficient frontier
The risk free asset
The risk free asset is the (hypothetical) asset which pays a risk free rate - it is usually proxied by an investment in short-dated Government bonds. The risk free asset has zero variance in returns (hence risk free); it is also uncorrelated with any other asset. As a result, when it is combined with any other asset, or portfolio of assets, the change in return and also in risk is linear.
Since the asset is risk free, portfolio standard deviation is simply a function of the weight of the risky portfolio in the position. This relationship is linear.
Portfolio leverage
An investor can add leverage to the portfolio by holding the risk free asset. The addition of the risk free asset allows for a position in the region above the efficient frontier. Thus, by combining a risk-free asset with risky assets, it is possible to construct portfolios whose risk-return profiles are superior to those on the efficient frontier. The market portfolio
To ensure that the combination held is always above the efficient frontier, the line plotted must be tangential to the efficient frontier (as opposed to going through it). For a given risk free rate, there is thus only one portfolio on the efficient frontier which can be combined with cash efficiently; i.e. the tangent portfolio. This is the market portfolio. See the illustration at top.Capital Market Line
When the market portfolio is combined with the risk free asset, the result is the Capital Market Line. All points along the CML have superior risk-return profiles to any portfolio on the efficient frontier. (A position with zero cash weighting is on the efficient frontier - the market portfolio.)Asset pricing
A rational investor would not invest in an asset which does not improve the risk-return characteristics of his existing portfolio. Since a rational investor would hold the market portfolio, the asset in question will be added to the market portfolio. MPT derives the required return for a correctly priced asset in this context.Systematic risk and specific risk
Capital Asset Pricing Model
The asset return depends on the amount paid for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model which derives the theoretical required 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.
Once the 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. (Here again, the theory accepts in its assumptions that a parameter based on past data can be combined with a future expectation.)
Return =
(2) If an asset, a, is correctly priced, the improvement in risk to return achieved by adding it to the market portfolio, m, will at least equal the gains of spending that money on an increased stake in the market portfolio. The assumption is that the investor will purchase the asset with funds borrowed at the risk free rate, rf; this is rational if .Securities Market Line
The relationship between Beta and required return is plotted on the Securities Market Line (SML) which shows expected return as a function of . The intercept is the risk free rate available for the market, the slope is unit of return per unit of risk , .
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