Chapter 1

Modern Portfolio Theory

The concept of the market portfolio has a long history and dates back to the

seminal work of Markowitz (1952). In his paper, Markowitz deﬁned precisely

what portfolio selection means: “the investor does (or should) consider ex-

pected return a desirable thing and variance of return an undesirable thing”.

Indeed, Markowitz showed that an eﬃcient portfolio is the portfolio that max-

imizes the expected return for a given level of risk (corresponding to the vari-

ance of portfolio return). Markowitz concluded that there is not only one

optimal portfolio, but a set of optimal portfolios which is called the eﬃcient

frontier.

By studying the liquidity preference, Tobin (1958) showed that the eﬃcient

frontier becomes a straight line in the presence of a risk-free asset. In this

case, optimal portfolios correspond to a combination of the risk-free asset and

one particular eﬃcient portfolio named the tangency portfolio. Sharpe (1964)

summarized the results of Markowitz and Tobin as follows: “the process of

investment choice can be broken down into two phases: ﬁrst, the choice of a

unique optimum combination of risky assets

1

; and second, a separate choice

concerning the allocation of funds between such a combination and a single

riskless asset”. This two-step procedure is today known as the Separation

Theorem (Lintner, 1965).

One diﬃculty when computing the tangency portfolio is to precisely deﬁne

the vector of expected returns of the risky assets and the corresponding co-

variance matrix of asset returns. In 1964, Sharpe developed the CAPM theory

and highlighted the relationship between the risk premium of the asset (the

diﬀerence between the expected return and the risk-free rate) and its beta (the

systematic risk with respect to the tangency portfolio). By assuming that the

market is at equilibrium, he showed that the prices of assets are such that the

tangency portfolio is the market portfolio, which is composed of all risky assets

in proportion to their market capitalization. This implies that we do not need

assumptions about the expected returns, volatilities and correlations of assets

to characterize the tangency portfolio. This major contribution of Sharpe led

to the emergence of index funds and to the increasing development of passive

management.

In the active management domain, fund managers use the Markowitz

framework to optimize portfolios in order to take into account their views

1

It is precisely the tangency portfolio.

3

4 Introduction to Risk Parity and Budgeting

and to play their bets. However, the implementation of portfolio theory is not

simple. It requires the estimation of the covariance matrix and the forecasting

of asset returns. One problem is that optimized portfolios are very sensitive to

these inputs. Some stability issues make the practice of portfolio optimization

less attractive than the theory (Michaud, 1989). In this case, regularization

techniques may be employed to attenuate these problems. This approach is

largely supported by Ledoit and Wolf (2003), who propose to combine dif-

ferent covariance matrix estimators to stabilize the solution. Today, the most

promising approach consists in interpreting optimized portfolios as the so-

lution of a linear regression problem and to use lasso or ridge penalization.

However, regularization is not suﬃcient to obtain satisfactory solutions,

which is why practitioners introduce some constraints in the optimization

problem. These constraints may be interpreted as a shrinkage method (Jagan-

nathan and Ma, 2003). By imposing weight constraints, the portfolio manager

implicitly changes the covariance matrix. This approach is then equivalent to

having some views and is therefore related to the model of Black and Litter-

man (1992).

1.1 From optimized portfolios to the market portfolio

In this section, we review the seminal framework of Markowitz and the

CAPM theory of Sharpe.

1.1.1 The eﬃcient frontier

Sixty years ago, Markowitz introduced the concept of the eﬃcient frontier.

It was the ﬁrst mathematical formulation of optimized portfolios. For him,

“the investor does (or should) consider expected return a desirable thing and

variance of return an undesirable thing”. By translating these principles into a

problem of mean-variance optimization, Markowitz (1952) showed that there

is no one optimal portfolio, but a set of optimized portfolios.

We consider a universe of n assets. Let x = (x

1

, . . . , x

n

) be the vector of

weights in the portfolio. We assume that the portfolio is fully invested meaning

that

P

n

i=1

x

i

= 1

>

x = 1. We denote R = (R

1

, . . . , R

n

) the vector of asset

returns where R

i

is the return of asset i. The return of the portfolio is then

equal to R (x) =

P

n

i=1

x

i

R

i

. In a matrix form, we also obtain R (x) = x

>

R.

Let µ = E [R] and Σ = E

h

(R − µ) (R − µ)

>

i

be the vector of expected returns

and the covariance matrix of asset returns. The expected return of the portfolio

is:

µ (x) = E [R (x)] = E

x

>

R

= x

>

E [R] = x

>

µ

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