Mean-Variance Analysis: Traditional Approach

Question 1

Motivation for MV analysis

Answer 1

The traditional approach to asset pricing starts with portfolio choice, not SDFs
Classic approach to portfolio choice: mean-variance portfolio theory (Markowitz 1952) decision problem if investors only care about mean and variance of returns
e.g. because they have mean-variance preferences (e.g. quadratic utility)
or because means and variances fully describe the statistical properties of all portfolios (e.g. normally distributed returns)

Mean-variance portfolio choice is typically presented as a trade-off between risk (variance) and return (mean/expected return)

To analyze this trade-off it makes sense to first understand the mean and variance characteristics of available asset returns
in particular: characterize the mean-variance frontier

these are the portfolios with lowest variance for given mean return
Here, we do not care about the mean-variance portfolio problem per se
but: it is nevertheless insightful to characterize the mean-variance frontier of returns

Question 2

Setup for two asset case

Answer 2

Question 3

Benefits of diversification

Answer 3

Portfolio mean is simply a weighted average of asset means μp = w1E1 + w2E2

But portfolio variance is quadratic function of weights w1, w2 which implies for w1, w2 ≥ 0 σp ≤ w1σ1 + w2σ2

Question 4

Benefits of diversification example

Answer 4

Suppose σ1 = σ2 = σ and we invest into both assets in equal proportion (w1 = w2 = 1/2)

Question 5

Diversification benefits for different p

Answer 5

Question 6

Relationship between portfolio mean and variance

Answer 6

Question 7

assumptions for this

Answer 7

E1 ̸= E2 (the two assets have different expected returns)

σ12 < σ1σ2 (no perfect positive correlation, ρ < 1)

Question 8

Solving MV equation

Answer 8

Question 9

Calculating global MV portfolio

Answer 9

We can characterize the vertex of the parabola by finding the portfolio that minimizes σp2 this portfolio is called the global minimum variance (gmv) portfolio

Question 10

GMV example

Answer 10

Question 11

The MV diagram

Answer 11

Question 12

Illustration of MV diagram

Answer 12

Question 13

The degenerate case where sigma GMV =0

Answer 13

In the special case σgmv = 0, the gmv portfolio is risk-free hence, let’s write Rf instead of Rgmv

then Rf =E[Rf]=μ gmv

Question 14

Illustration of degenerate case

Answer 14

Question 15

Special case with risk-free assets showing Sharpe ratios

Answer 15

Question 16

Interpretation of the Sharpe Ratio

Answer 16

The Sharpe ratio relates two characteristics of a risky investment

1 reward of investing in risky asset (excess mean return over Rf )

2 risk of the investment (standard deviation)

It is a more interesting characteristic of an asset than mean return alone

leveraging an investment with risk-free borrowing increases mean return (if E1 > Rf )

but is also increases risk proportionally,

Sharpe ratio remains unaffected

this is ultimately what |Sp| = |S1| tells us

The higher is the Sharpe ratio, the higher the compensation for taking on extra risk

Question 17

The Capital Allocation Line

Answer 17

Consider the case S1 > 0 (E1 > Rf ) and w1 ≥ 0

(long position in risky asset)

Then previous equation becomes
μp =Rf +S1σp

This is an upward-sloping line whose slope is the sharpe ratio S1 of the risky asset

This line is called the capital allocation line
(because any investor with mean-variance preferences would want to invest in a portfolio on the line)

Remark: the capital allocation line is simply the “efficient” portion of the degenerate hyperbola in the σgmv = 0 case

Question 18

Setup with two risky and one risk-free asset

Answer 18

Question 19

Two-step portfolio construction

Answer 19

We can split the portfolio construction in two logical steps

1 form portfolio of risky assets

return
Rpr = w1R1+ w2R2 (with w1+w2 =1)
→ can analyze mean-variance structure using two-asset special case

2 form portfolio of risk-free asset and risky return Rpr
total portfolio return

Rp = wf Rf + wr Rpr (with wf + wr = 1)
→ can again analyze mean-variance structure using two-asset special case

Question 20

The MV frontier

Answer 20

With more than two assets, any mean return μp can be achieved with many portfolios

Natural question: given μ, which portfolio minimizes σp2 among portfolios with μp = μ?

we call such a portfolio mean-variance efficient

The set of all mean-variance efficient portfolios is called the mean-variance frontier

How can we determine the mean-variance frontier?

1 geometric construction → discussed below
2 optimization-based construction → problem set

Question 21

Geometric construction of the MV frontier

Answer 21

Question 22

Illustration of MV frontier

Answer 22

Question 23

The Tangency Portfolio and Two funds separation

Answer 23

There is a unique risky asset portfolio that maximizes |Spr |

the line connecting this portfolio with the risk-free asset is tangent to the risky asset frontier

for any other risky asset portfolio, the line is not tangent

This portfolio is called the tangency portfolio

Remarkable result (two funds separation; Tobin 1958):

any mean-variance efficient portfolio combines two “funds” (with different weights)

1 the risk-free asset
2 the tangency portfolio of risky assets

In particular: all investors that hold mean-variance efficient portfolios hold exactly the same risky asset portfolio

Question 24

Setup with N-risky assets and no risk free assets

Vector Notation

Answer 24

Question 25

Characterising the MV frontier

Answer 25

Question 26

Solving the problem part 1

Answer 26

Question 27

Solution part 2

Answer 27

Question 28

MV frontier solution

Answer 28

Question 29

Some conclusions

Answer 29

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

Question 30

minimizing σp2 over μ yields the weights on the gmv portfolio

Answer 30

Question 31

Setup for many risky assets and 1 risk free

Answer 31

Question 32

The Minimization problem with this setup

Answer 32

Question 33

Conclusions from solving this

Answer 33

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

Question 34

Summary

Answer 34

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

Question 35

Implicit benefits of diversification

Answer 35

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.

Question 36

Vector notation

Answer 36

Question 37

Constraints

Answer 37

Question 38

Parabola of combinations of mean (μ) and variance (σ2) achievable by forming port- folios of two assets.

Answer 38

Question 39

Hyperbola of combinations of standard deviation (σ) and mean (μ) achievable by forming portfolios of two assets. The figure depicts the exact same situation as Figure 1, but represents it in a more conventional plot. The black dashed lines depict the asymptotes of the hyperbola.

Answer 39

Question 40

The Sharpe ratio

Answer 40

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.

Question 41

The capital allocation line

Answer 41

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

Question 42

The Case of Perfect Positive Correlation, ρ = 1.

Answer 42

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.

Question 43

The Case of Identical Mean Returns.

Answer 43

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

Question 44

Two Funds Separation.

Answer 44

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.

Question 45

Full depiction

Answer 45

Question 46

The general case

Answer 46