Advances in Risk Management

THADAVILLIL JITHENDRANATHAN 267

14.2 EMPIRICAL METHODOLOGY AND DATA

In portfolio optimization models, the objective is to maximize the return and
minimize the risk of the portfolio. The expected return of a portfolio is the
weighted average of the returns of individual securities in the portfolio and
the weights are the proportion of each of the securities in the portfolio and
can be expressed as follows:

−

RP=

∑N

i= 1

Xi

−

Ri (14.1)

whereXiistheweightoftheithsecurityintheportfolioandRiistheexpected
return of that asset.
The standard deviation of a portfolio can be expressed as:

σP^2 =

∑N

i= 1

X^2 iσ^2 i+

∑N

i= 1

∑N

k= 1

k=i

XiXkσi,k (14.2)

whereσ^2 s are the variances andσi,kis the covariance between the two
securitiesiandk.
The standard method of optimization is to find a set of portfolios, which
will give the maximum return for a given level of risk. This set of portfolios
are called the efficient set of portfolios and based on their individual risk
preferences investors can choose a specific portfolio from this set of optimal
portfolios.
Mathematically the optimization problem can be stated as follows:

Minσ^2 P=

∑N

i= 1

Xi^2 σi^2 +

∑N

i= 1

∑N

k= 1

k=i

XiXkσi,k (14.3)

Subject to the following constraint:

∑N

i= 1

Xi= 1 (14.4)

Portfolios can be created with or without short-selling constraints. In this
chapter the portfolios are constructed with short selling constraints, which
require the following additional constraint:

0 ≤Xi< 1 (14.5)

In this chapter I use two different approaches to estimate the expected
returns, variances and co-variances using the historic data. The first method