Chapter 5 - Portfolio Theory Flashcards
Mean-variance portfolio theory (or modern portfolio theory MPT)
It specifies a method for an investor to construct a portfolio that gives the maximum return for a specified risk, or the minimum risk for a specified return.
However, the theory relies on some strong and limiting assumptions about the properties of portfolios that are important to investors. In the form described here the theory ignores the investor’s liabilities, although it is possible to extend the analysis to include them.
Two parts of the application of the mean-variance framework to portfolio selection
The application of the mean-variance framework to portfolio selection fall conceptually into two parts:
- The definition of the properties of the portfolios available to the investor (opportunity set). -looking at risk and return
- The determination of how the investor chooses one out of all the feasible portfolios in the opportunity set. -determination of the investor’s optimal portfolio from those available
Assumptions underlying mean-variance portfolio theory (4+2)
In specifying the opportunity set it is necessary to make some assumptions about how investors make decisions. Then the properties of portfolios can be specified in terms of relevant characteristics.
Main:
- All expected returns, variances and covariances of pairs of assets are known.
- Investors make their decisions purely on the basis of expected return and variance over a single time period.
- Investors are non-satiated. (At a given level of risk, they will always prefer a portfolio with a higher return to one with a lower return)
- Investors are risk-averse. (For a given level of return they will always prefer a portfolio with lower variance to one with higher variance)
Not main:
- There are no taxes or transaction costs
- Assets may be held in any amounts, ie short-selling, infinitely divisible holdings, no maximum investment limits.
Risk of the portfolio
Variance or standard deviation
Inefficient portfolio
A portfolio is inefficient if the investor can find another portfolio with the same or higher expected return and lower variance, or the same or lower variance and higher expected return.
Efficient portfolio
A portfolio is efficient if the investor cannot find a better one in the sense that it has either a higher expected return and the same or lower variance, or a lower variance and the same or higher expected return.
Ignored key factors that might influence investment decision in practice
- The suitability of the asset(s) for an investor’s liabilities.
- The marketability if the asset(s)
- Higher moments of the distribution of returns such as skewness and kurtosis.
- Taxes and investment expenses.
- Restrictions imposed by legislation.
- Restrictions imposed by the fund’s trustees.
Investor invests in N securities (i=1,…N) and a proportion if xi is invested in security Si. Define
- Return of portfolio
- Expected return on the portfolio
- Variance
- Return on the portfolio Rp is:
Rp = Σ (i=1 to N) of xi*Ri
where Ri is the return on security Si - Expected return on the portfolio is:
E = E[Rp] = Σ (i=1 to N) of xi*Ei
where Ei is the expected return on security Si - The variance is V = var[Rp] = Σ(j=1 to N) Σ(i=1 to N) of xi * xj * Cij
where Cij is the covariance of the returns on securities Si and Sj and we write Cii = Vi
State an expression for the value of xA at which the variance is minimised
Xa = (Va - Cab) / (Va + Vb - 2Cab)
What types of curves are in the diagram tha shows how the portfolio expected return varies with the standard deviation or variance?
In the (E-σ) space the curves representing possible portfolios of two securities are hyperbolas.
It is possible to plot the same results in (E-V) space, where the lines would be parabolas.
What can we say about the diagram of E-σ when the correlation coefficient is -1 or +1?
It is possible to obtain risk-free portfolios with 0 standard deviation of return.
How can the variance minimisation problem be solved if N>2
One method of solving such a minimisation is the method of Lagrangian multipliers.
The Lagrangian function is:
W = V - λ (E - Ep) - μ ((Σi xi) - 1)
This can be used to find the efficient frontier in the case of N > 2 securities.
How can the Langrangian functions be used to determine a set of simultaneous equations for finding the efficient frontier in the case of N>2
To find the minimum, we set the partial derivatives of W wrt all the xi and λ and μ equal to 0. The result is a set of linear equations that can be solved.
How are the portfolio variance V and the portfolio proportions xi related to the portfolio expected return E? (N>2)
The solution to the problem shows that the minimum variance V is a quadratic in E and and xi is linear in E.
What is the usual way of representing the results of the calculations to derive the minimum variance curve
The usual way of representing the results is by plotting the minimum standard deviation for each value of Ep as a curve in E-σ space.
In this space, with expected return on the vertical axis, the efficient frontier is the part of the curve lying above the point of the global minimum of the standard deviation.
