Question

Explain how the use of a two-factor model can facilitate the development of MVO models with n assets.

Answer 1

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. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance of asset prices as a proxy for risk

An investor can reduce portfolio risk simply by holding combinations of instruments that are not perfectly positively correlated (correlation coefficient − 1 ≤ ρ i j < 1 {\displaystyle -1\leq \rho _{ij}<1} ). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk. The mean-variance framework for constructing optimal investment portfolios was first posited by Markowitz and has since been reinforced and improved by other economists and mathematicians who went on to account for the limitations of the framework.

If all the asset pairs have correlations of 0—they are perfectly uncorrelated—the portfolio's return variance is the sum over all assets of the square of the fraction held in the asset times the asset's return variance (and the portfolio standard deviation is the square root of this sum).

If all the asset pairs have correlations of 1—they are perfectly positively correlated—then the portfolio return’s standard deviation is the sum of the asset returns’ standard deviations weighted by the fractions held in the portfolio. 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.

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. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact trade-off will not be the same for all investors. Different investors will evaluate the trade-off differently 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-expected return profile – i.e., if for that level of risk an alternative portfolio exists that has better expected returns.

Under the model:

• Portfolio return is the proportion-weighted combination of the constituent assets' returns.
• Portfolio volatility is a function of the correlations ρij of the component assets, for all asset pairs (i, j).

In general:

• Expected return:

E ⁡ ( R p ) = ∑ i w i E ⁡ ( R i ) {\displaystyle \operatorname {E} (R_{p})=\sum _{i}w_{i}\operatorname {E} (R_{i})\quad }

where R p {\displaystyle R_{p}} is the return on the portfolio, R i {\displaystyle R_{i}} is the return on asset i and w i {\displaystyle w_{i}} is the weighting of component asset i {\displaystyle i} (that is, the proportion of asset "i" in the portfolio).

• Portfolio return variance:

σ p 2 = ∑ i w i 2 σ i 2 + ∑ i ∑ j ≠ i w i w j σ i σ j ρ i j {\displaystyle \sigma _{p}^{2}=\sum _{i}w_{i}^{2}\sigma _{i}^{2}+\sum _{i}\sum _{j\neq i}w_{i}w_{j}\sigma _{i}\sigma _{j}\rho _{ij}} ,

where σ {\displaystyle \sigma } is the (sample) standard deviation of the periodic returns on an asset, and ρ i j {\displaystyle \rho _{ij}} is the correlation coefficient between the returns on assets i and j. Alternatively the expression can be written as:

σ p 2 = ∑ i ∑ j w i w j σ i σ j ρ i j {\displaystyle \sigma _{p}^{2}=\sum _{i}\sum _{j}w_{i}w_{j}\sigma _{i}\sigma _{j}\rho _{ij}} ,

where ρ i j = 1 {\displaystyle \rho _{ij}=1} for i = j {\displaystyle i=j} , or

σ p 2 = ∑ i ∑ j w i w j σ i j {\displaystyle \sigma _{p}^{2}=\sum _{i}\sum _{j}w_{i}w_{j}\sigma _{ij}} ,

where σ i j = σ i σ j ρ i j {\displaystyle \sigma _{ij}=\sigma _{i}\sigma _{j}\rho _{ij}} is

the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as σ ( i , j ) {\displaystyle \sigma (i,j)} , c o v i j {\displaystyle cov_{ij}} or c o v ( i , j ) {\displaystyle cov(i,j)} .

• Portfolio return volatility (standard deviation):

σ p = σ p 2 {\displaystyle \sigma _{p}={\sqrt {\sigma _{p}^{2}}}}

For a two asset portfolio:

• Portfolio return: E ⁡ ( R p ) = w A E ⁡ ( R A ) + w B E ⁡ ( R B ) = w A E ⁡ ( R A ) + ( 1 − w A ) E ⁡ ( R B ) . {\displaystyle \operatorname {E} (R_{p})=w_{A}\operatorname {E} (R_{A})+w_{B}\operatorname {E} (R_{B})=w_{A}\operatorname {E} (R_{A})+(1-w_{A})\operatorname {E} (R_{B}).}
• Portfolio variance: σ p 2 = w A 2 σ A 2 + w B 2 σ B 2 + 2 w A w B σ A σ B ρ A B {\displaystyle \sigma _{p}^{2}=w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}}

For a three asset portfolio:

• Portfolio return: E ⁡ ( R p ) = w A E ⁡ ( R A ) + w B E ⁡ ( R B ) + w C E ⁡ ( R C ) {\displaystyle \operatorname {E} (R_{p})=w_{A}\operatorname {E} (R_{A})+w_{B}\operatorname {E} (R_{B})+w_{C}\operatorname {E} (R_{C})}
• Portfolio variance: σ p 2 = w A 2 σ A 2 + w B 2 σ B 2 + w C 2 σ C 2 + 2 w A w B σ A σ B ρ A B + 2 w A w C σ A σ C ρ A C + 2 w B w C σ B σ C ρ B C {\displaystyle \sigma _{p}^{2}=w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+w_{C}^{2}\sigma _{C}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}+2w_{A}w_{C}\sigma _{A}\sigma _{C}\rho _{AC}+2w_{B}w_{C}\sigma _{B}\sigma _{C}\rho _{BC}}

Diversification

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. 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. 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). 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.

When a risk-free asset is introduced, the half-line shown in the figure is the new efficient frontier. It is tangent to the hyperbola at the pure risky portfolio with the highest Sharpe ratio. 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 leveraged portfolios involving negative holdings of the risk-free asset (the latter has been sold short—in other words, the investor has borrowed at the risk-free rate) and an amount invested in the tangency portfolio equal to more than 100% of the investor's initial capital. This efficient half-line is called the capital allocation line (CAL), and its formula can be shown to be

E ( R C ) = R F + σ C E ( R P ) − R F σ P . {\displaystyle E(R_{C})=R_{F}+\sigma _{C}{\frac {E(R_{P})-R_{F}}{\sigma _{P}}}.}

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. 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,[7] where the mutual fund referred to is the tangency portfolio.