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数学统计

Mean-Variance efficiency

时间:2015-09-25 22:26来源:未知 作者:admin 点击:

The seminal paper Markowitz (1952) develops the “mean-variance efficiency” theory based on the utility model of von Neumann and Morgenstern (1953). Based on the “mean-variance” framework, the Capital Asset Pricing Model (CAPM) is developed by Sharpe (1964) and Treynor (1961), and extended by Lintner (1965), Mossin (1966) and Black (1972). The “mean-variance” preference is the key assumption behind the portfolio theory and the CAPM. This essays is a discussion on the mean-variance portfolio theory. I start with the terminologies expected return and risk in section 1. I then move on to the definition of “mean-variance efficiency” in section 2, efficient frontiers in section 3 and 4, and capital market line in section 5. Lastly, I compare the well-diversified portfolio and individual stocks in terms of “reward-to-risk” in section 6.  

The first discussion point in the assignment is addressed in section 1 and 5. The second point is discussed in section 6 as a separate part. The third point is analyzed in section 4, efficient frontier with a risk-free asset. And the last point is illustrated in section 3, 4 and 5.

1.      Expected return and risk

Portfolio theory starts with two important terms, expected return and risk. Assuming risk aversion agents, an optimized portfolio choice is actually a trade-off between the expected return and risk. Intuitively, we expect higher returns to be associated with higher risks. But how much higher in return should we demand for a higher level of risk is the question addressed in portfolio optimization theory. So this points out a “high risk-high return” portfolio is not necessarily better than a “low risk-low return” portfolio. This point will be discussed further in the capital market line section where risk-return relationship is discussed.

Furthermore, a total risk can be decomposed into a systematic risk part, which is non-diversifiable, and an idiosyncratic, or firm-specific risk part, which is diversifiable. When we hold a portfolio with more than one asset and the correlation between these assets are between -1 and 1, the portfolio would benefit from diversification. This is illustrated in figure 1, where the diversifiable risk decreases as the number of securities in the portfolio increases, while the systemic risk does not change.

 
Figure 1. Portfolio Risk Decomposition

 
2.      Definition of “efficient portfolio”
An efficient portfolio is, given a certain level of expected return, the portfolio that has the least risk. The other way around, an efficient portfolio is, giving certain level of risk, the one with the highest expected return.
The next question is how to quantify expected return and risk. Although we know expected return should be a forward-looking measure, in practice it can be calculated as the average of historical time-series realized returns. How far back we should look into the history is a question of practice. Risk measure is even more complicated. Under the framework of “mean-variance” analysis, where investors only care about the mean and variance of a portfolio, risk is measured by the total volatility, the variance of a portfolio or an individual asset. Two assumptions can lead to the “mean-variance” preference. First, investors have a quadratic utility function. Second, asset returns follow a normal distribution, which the mean and variance parameters describe the whole distribution.
Thus the mathematical expression for the portfolio optimization problem is as the following:

in which  is the expected return rate of the portfolio and  the variance, with  being the weights. If there is no pairwise correlation among the assets, namely , it leads to the obvious fact that . Therefore equally weighted portfolio could eliminate the risk completely, and the optimal portfolio only selects the best single asset while portfolio diversification is meaningless. If the assets are correlated pairwise,  . That is to say diversification could eliminate the individual risk while the systematical risk could not be avoided. Therefore the diversified portfolio strategy is efficient to achieve expected return by minimalizing the risk.
3.      Efficient Frontier with only Risky Assets
With only risky assets, the efficient frontier is the upper part of the parabola (the green curve) in the following figure. The lower part of the parabola is dominated by the upper part because given a certain level of standard deviation, the expected return on the upper part is higher than the expected return on the low part. By the same argument, the portfolios located on the efficient frontier are preferable to those in the shaded area.
Figure 2. Efficient Frontier with only Risky assets
 
 
4.      Efficient Frontier with both Risky Assets and a Risk-free Asset
With one more risk-free asset, where the agents can borrow or lend on the capital market, the efficient frontier becomes the line from the risk-free asset and the tangency portfolio. Graphically, it is the blue line in the following figure which connects the risk-free asset and the tangency portfolio.

 
Figure 3. Efficient Frontier with both Risky Assets and a Risk-free Asset

Any point to the left of the line is not achievable by available assets. And any point to the right or under this line is inferior to the points on the line from the mean-variance efficient perspective. This can be seen clearly from figure 4. The slope of the line drawn to connect risk-free asset and any point on the efficient frontier is the reward to risk. Higher slope means a higher reward, or price, for per unit risk. Thus the steepest possible line contains the most efficient portfolios.

Figure 4. Reward to Risk Comparison

A rational investor, who has a preference for “efficient” portfolios will invest on this steepest possible line because this line provides the highest compensation for per unit risk.  The proportion of wealth investing in risky asset depends on the risk aversion level of the investor. Any point from risk-free asset and the tangency portfolio can be achieved by investing part in the risky portfolio (the tangent portfolio) and lending at the risk-free rate. And any point from tangency portfolio and to the northeast can be achieved by borrowing at the risk-free rate and investing in the risky portfolio. In the equilibrium, the demand for assets equals to the supply of all the assets. Thus, the tangency portfolio is the market portfolio. Thus, a preference for “efficient” portfolios would lead all investors to only invest on this line, which can be achieved by investing in a combination of the risk-free asset and the market portfolio, which is also the tangent portfolio in equilibrium.
 
5.      Capital Market Line
The capital market line (CML) describes the risk-return relationship of combinations of the risk-free asset and the market portfolio. The slope of the line is the price of risk. Any point above this line is underpriced relative to the CML and any point under this line is overpriced.  
Figure 5. The Capital Market Line (CML)

The price of risk is also widely known as Sharpe ratio, introduced in Sharpe (1966). Sharpe proposed the term “reward-to-volatility” ratio to describe it. For a portfolio p, it is defined as
where  is the excess return of the portfolio p. A higher Sharpe ratio implies a better risk-return trade-off. And the highest Sharpe ratio can be achieved is the slope of the CML.

Using the risk-return relationship, I will further discuss the point comparing a “high risk-high return” portfolio to a “low risk-low return” portfolio here. So it should depend on the reward to risk. A “high risk-high return” portfolio may give a lower reward to risk than a “low risk-low return” portfolio. For example, given a portfolio with 10% of expected return and 4% of volatility and a portfolio with 8% of expected return and 2% of volatility, and risk-free rate as 2%, we have the reward to risk for the “high risk-high return” portfolio as 2 and that of the “low risk-low return” portfolio as 3. Obviously, from the view of reward to risk measured as the total volatility, the “low risk-low return” portfolio is better than the “high risk-high return” portfolio.
 
6.      Well Diversified Portfolios versus Individual Stocks

A well-diversified portfolio is different from an individual stock in the sense that their risk component is different. As discussed in the first section that there are two types of risks, systemic and idiosyncratic, where the systemic risk is not diversifiable and the idiosyncratic risk is. By definition, a well-diversified portfolio, say market portfolio, is a portfolio with only systemic risk, or almost only. However, an individual stock has a larger component of idiosyncratic risk comparing to a well-diversified portfolio.

Recall we discussed in section 4 that all rational investors would hold a combination of the market portfolio and the risk-free asset, thus all investors would be well-diversified and no one would care about idiosyncratic risk. So in equilibrium, idiosyncratic risk is not priced and only systemic risk is priced. This is also the basic idea in CAPM that only beta risk is priced, where beta is a measure of sensitivity to market risk, or systemic risk. So if we compare the reward to risk between a well-diversified portfolio and an individual stock, it is more likely that a well-diversified portfolio has a higher reward to risk than the individual stock because all the risk in the well-diversified portfolio is priced while only systemic part of the risk of the individual stock is priced.

References
Brealey, R., Myers, S., and Allen, F. (2008) “Principles of Corporate Finance”

Fama, E., French. (2004) “The Capital Asset Pricing Model: Theory and Evidence”, Journal of Economic Perspectives, Vol. 18, No. 3, pp. 25-46

Pennacchi, G. (2007) “Theory of Asset Pricing”

Goetzmann, W., Kumar, A. (2008) “Equity Portfolio Diversification”, Review of Finance, Vol. 12, No. 3, pp. 433-463

Markowitz, H. (1952) “Portfolio Selection”, Journal of Finance, Vol. 7, No. 1, pp. 77-91

Sharpe, W. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”, Journal of Finance, Vol. 19, No. 3, pp. 425-442

Sharpe W. F., 1966, Mutual Fund Performance, The Journal of Business 39(1): 119-138

Statman, M. (1987) “How Many Stocks Make a Diversified Portfolio?”, Journal of Financial and Quantitative Analysis, Vol. 22, No. 3, pp. 353-363



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