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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