Abstract
In this chapter we introduce the theory and the application of the computer program of modern portfolio theory. The notion of diversification is age-old: “don’t put your eggs in one basket,” obviously predates economic theory. However, a formal model showing how to make the most of the power of diversification was not devised until 1952, a feat for which Harry Markowitz eventually won the Nobel Prize in economics. Markowitz portfolio shows that as you add assets to an investment portfolio the total risk of that portfolio – as measured by the variance (or standard deviation) of total return – declines continuously, but the expected return of the portfolio is a weighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return. In the second part we introduce the mean–variance spanning test that follows directly from the portfolio optimization problem.
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Notes
- 1.
Markowitz model assumes that investors are risk averse. This means that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, 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 tradeoff will differ by investor based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-return profile (i.e., if for that level of risk an alternative portfolio exists that has better expected returns). Using risk tolerance, we can simply classify investors into three types: risk-neutral, risk-averse, and risk-lover. Risk-neutral investors do not require the risk premium for risk investments; they judge risky prospects solely by their expected rates of return. Risk-averse investors are willing to consider only risk-free or speculative prospects with positive premium; they make investment according the risk-return tradeoff. A risk-lover is willing to engage in fair games and gambles; this investor adjusts the expected return upward to take into account the “fun” of confronting the prospect’s risk.
- 2.
High covariance indicates that an increase in one stock’s return is likely to correspond to an increase in the other. A low covariance means the return rates are relatively independent and a negative covariance means that an increase in one stock’s return is likely to correspond to a decrease in the other.
- 3.
The efficient frontier will be convex – 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 efficient frontier is a parabola (hyperbola) when expected return is plotted against variance (standard deviation).
- 4.
Rational investors will not short sell a high-return asset and buy a low-return asset. This case is just for extreme assumption.
- 5.
Whether an investor engages in any of this short-selling activity depends on the investor’s own unique set of indifference curves.
- 6.
The solution uses the minimization techniques of calculus. Write out the expression for portfolio variance from Equation (10.6), substitute 1 − w A for w B , differentiate the result with respect to w A , set the derivative equal to zero, and solve for w A to obtain \({w}_{A} = \frac{{\sigma}_{B}^{2}-{\rho}_{\mathit{AB}}{\sigma}_{A}{\sigma}_{B}} {{\sigma}_{A}^{2}+{\sigma}_{B}^{2}-2{\rho}_{\mathit{AB}}{\sigma}_{A}{\sigma}_{B}}\) and \({w}_{B} = 1 - {w}_{A}\).
- 7.
The solution procedure for two risky assets is as follows. Substitute for expected return from Equation (10.5) and for standard deviation from Equation (10.13). Substitute 1 − w A for w B . Differentiate the resulting expression for CAL s with respect to w A , set the derivative equal to zero, and solve for w B .
- 8.
The complete portfolio means that the entire portfolio including risky and risk-free assets.
- 9.
The expected return and standard deviation of each index needs to be annualized.
- 10.
The correlation matrix is calculated in Excel using the data analysis function that is found under the Tool Menu. Note that if Data Analysis does not appear on the Tool Menu you will need to select Add-in and add to the Menu.
- 11.
Because the expected returns and the portfolio weights are represented by column vectors (denoted e and w, respectively, with row vector transposes e T and w T), and the variance-covariance terms by matrix V, then the expressions can be written as simple matrix formulas. So, the calculation of expected return and variance of portfolio can use the Excel array functions.Matrix notation Excel formula:Portfolio return: w T e = SUMPRODUCT(w, e)Portfolio variance: w T Vw = MMULT(TRANSPOSE(w), MMULT(V, w))
- 12.
If Solver does not show up under the Tools menu, you should select Add-Ins and then select Analysis. This should add Solver to the list of options in the Tools menu.
- 13.
We use 3-month Treasury Bill interest rate as the risk-free rate; the average interest rate of a 3-month T-Bill is 4.09% from 1990/01 to 2006/12.
- 14.
The total of all weights in a portfolio is 1.
- 15.
When entering a target rate of return (PortReturn), enter NumPorts as an empty matrix [].
- 16.
The Default of lower asset bound is all 0s (no short-selling); default upper asset bound is all 1s (any asset may constitute the entire portfolio).
- 17.
Groups(i, j) = 1 (jth asset belongs in the ith group). Groups(i, j) = 0 (jth asset not a member of the ith group).
- 18.
In reality, portcons is an entry point to a set of functions that generate matrices for specific types of constraints; portcons allows you to specify all the constraints data at once, while the specific portfolio constraint functions allow you to build the constraints incrementally. These constraint functions are pcpval, pcalims, pcglims, and pcgcomp.
- 19.
varargin is variable length input argument list. The varargin statement is used only inside a function M-file to contain optional input arguments passed to the function. The varargin argument must be declared as the last input argument to a function, collecting all the inputs from that point onwards. In the declaration, varargin must be lowercase.
- 20.
If borrowing is not desired, or not an option, set to NaN (default).
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Chen, WP., Chung, H., Ho, KY., Hsu, TL. (2010). Portfolio Optimization Models and Mean–Variance Spanning Tests. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_10
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