Mean-Variance Analysis: Traditional Approach Flashcards

Question

Characterising the MV frontier

Answer 1

the relationship between μ and σp2 is again quadratic → geometry of the mean-variance frontier is as in two-asset case means relate linearly to portfolio weights → any portfolio formed with returns on the frontier is again on the frontier

Answer 2

Optimal w depends on μ only through scaling factor λ → all mean-variance efficient portfolios contain the same composition of risky assets → this corresponds to a tangency portfolio as in special case II Can now substitute w into constraint to solve for w and frontier → you will do this in the next problem set

Answer 3

Traditional approach to mean-variance analysis: characterize frontier in (σ, μ)-space Special cases: two assets: (σ, μ)-combinations lie on (possibly degenerate) (σ, μ)-combinations lie on hyperbola if σgmv = 0, hyperbola is degenerate with risk-free asset, slope of degenerate hyperbola = Sharpe ratio of risky asset three assets (one risk-free): two step construction: first choose risky portfolio, then combine with risk-free asset all frontier portfolios invest in same tangency portfolio of risky assets General cases: solve for mean-variance frontier using constrained minimization no risk-free asset: analogous to two-asset case with σgmv > 0 with risk-free asset: analogous to three-asset case

Answer 4

Equation (1) says that the portfolio mean is simply the weighted average of the means (expected returns) of the two assets in the portfolio. The weights on each asset coincide with the portfolio weights wi, i.e. the fraction of the total amount invested in each of the two assets. Equation (2), in turn, says that the portfolio variance is a quadratic function of the portfolio weights w1 and w2. The portfolio variance may be smaller than what would be implied by simply taking weighted averages of standard deviations in analogy to how portfolio means are related to individual asset means in equation (1). We also see from the derivation that the inequality is strict, the portfolio variance is strictly smaller, if the asset returns are not perfectly positively correlated, i.e. ρ < 1. This illustrates the benefits from diversification: splitting wealth and investing into multiple risky assets that are not perfectly correlated reduces the riskiness of the overall portfolio (relative to a naive weighted average of standard deviations). This variance is minimized by buying both assets in equal proportions, w1 = w2 = 1/2. In this case, we obtain σp2 = (1 + ρ) /2σ2. If the asset returns are perfectly negatively correlated, ρ = −1, this portfolio is risk-free because the payoffs are perfectly hedged. If the asset returns are uncorrelated, the portfolio is still risky, but its variance is only half the variance of each of the two assets. As ρ approaches 1, the benefits of diversification disappear, σp2 approaches σ2.

Answer 5

The sharpe ratio relates the reward of investing in a risky asset, the excess return over the risk-free rate, to the risk of the asset, measured by its return standard deviation. If, for example, E1 > Rf, then investing more into the risky asset (by reducing the risk-free investment or even borrowing at the risk-free rate) increases the expected return of the portfolio, but it also increases the portfolio standard deviation. In fact, the equation |S1| = |Sp| tells us that the portfolio standard deviation increases proportionally such that the Sharpe ratio remains constant. An investor can achieve a higher expected return but not a higher Sharpe ratio by leveraging a risky investment with risk-free borrowing.

Answer 6

μp = S1σp. This equation tells us that all portfolios that invest a long position into the risky asset are located on a straight line in the (μ, σ)-space. Given our previous discussion of the case σgmv = 0, this result is not surprising: the line just describes one of the two lines that form the “degenerate” hyperbola of all mean-standard deviation combinations achievable when the global minimum variance portfolio is risk-free. If E1 > Rf , then S1 > 0 and the line is upward-sloping. It describes the “efficient” portion of the hyperbola, i.e. the portfolios that offer the highest mean return for a given variance. This line is sometimes called the capital allocation line because any investor that cares only about mean and variance (and dislikes the latter) would want to choose a portfolio on that line

Answer 7

In this case, both asses are risky, σ1, σ2 > 0, and we can write R2 = a + bR1 with b > 0 due to perfect positive correlation. We have assumed that R1 ̸=R2,so it cannot be that both a=0 and b=1 are satisfied. If a=0 is satisfied and b ̸= 1, then we have a violation of the law of one price.6 Consequently, it must be the case that a ̸= 0. In this case, R2 is effectively a portfolio consisting of a risk-free asset and the asset with return R1.7 We can therefore simply apply the previous analysis using the returns R1′ := R1 and R2′ := Rf (which have correlation ρ = 0) instead of R1 and R2. In other words, either the configuration of perfect positive correlation violates the law of one price or it can be handled by the previous analysis after describing the same asset market in terms of two different, but equivalent, returns.

Answer 8

If both assets have the same expected return E := E1 = E2, then also all portfolios formed with these assets have the expected return E as can be readily observed from equation (1). Consequently, the combinations of all (σ,μ)-pairs achievable by forming portfolios of the two assets lie on a horizontal line at the level μ = E. Because the two returns are by assumption not identical, they cannot have the same variance and be perfectly positively correlated at the same time.8 Hence, the portfolio variance σp2 is not constant across all portfolios. We can still achieve arbitrarily large portfolio variance by combining the assets. The minimum variance achievable σg2mv can be obtained by minimizing σp2 with respect to the portfolio weight w1

Answer 9

Two Funds Separation. The previous construction reveals a remarkable fact: there is a unique “best” risky asset portfolio, the tangency portfolio, that lies on the mean variance frontier. All other portfolios on the mean variance frontier combine this risky asset portfolio with the risk-free rate. Consequently, any investor who seeks to invest into a portfolio on the mean-variance frontier will chose the same combination of risky assets, the tangency portfolio. Different investors only differ with regard to how they combine that tangency portfolio with the risk-free rate. A very risk-averse investor might hold a large fraction of wealth in the risk-free asset and invests only little into the tangency portfolio, whereas a less risk-averse investor may follow the opposite strategy.

Mean-Variance Analysis: Traditional Approach Flashcards

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