ñòð. 33 |

Portfolio Management

and the portfolio standard deviation is

s2 Â¼ g2 s1 2 Ã¾ (1 Ã€ g)2 s2 2 Ã¾ 2g(1 Ã€ g)s12 (10:1:6)

In (10.1.6), s12 is the covariance between the returns of asset 1 and

asset 2. For simplicity, it is assumed further that the asset returns are

uncorrelated, that is, s12 Â¼ 0. The value of g that yields minimal risk

for this portfolio equals

gm Â¼ s2 2 =(s1 2 Ã¾ s2 2 ), (10:1:7)

This value yields the minimal portfolio risk sm

sm 2 Â¼ s1 2 s2 2 =(s1 2 Ã¾ s2 2 ) (10:1:8)

Consider an example with E[R1 ] Â¼ 0:1, E[R2 ] Â¼ 0:2, s1 Â¼ 0:15,

s2 Â¼ 0:3. If g Â¼ 0:8, then s % 0:134 < s1 and E[R] Â¼ 0:12 > E[R1 ].

Hence, adding the more risky asset 2 to asset 1 decreases the portfolio

risk and increases the portfolio return. This somewhat surprising

outcome demonstrates the advantage of portfolio diversification.

Finally, let us combine the risk-free asset with a portfolio that

contains two risky assets. The optimal combination of the risky

asset portfolio and the risk-free asset can be found at the tangency

point between the straight risk-return trade-off line with the intercept

E[R] Â¼ Rf and the risk-return trade-off curve for the risky asset

portfolio (see Figure 10.1). For the portfolio with two risky uncorrel-

ated assets, the proportion g at the tangency point T equals

gT Â¼ (E[R1 ] Ã€ Rf )s2 2 ={(E[R1 ] Ã€ Rf )s2 2 Ã¾ (E[R2 ] Ã€ Rf )s1 2 }

(10:1:9)

Substituting gT from (10.1.9) into (10.1.5) and (10.1.6) yields the

coordinates of the tangency point (i.e., E[RT ] and sT ). A similar

approach can be used in the general case with an arbitrary number

of risky assets. The return E[RT ] for a given portfolio with risk sT is

â€˜â€˜as good as it gets.â€™â€™ Is it possible to have returns higher than E[RT ]

while investing in the same portfolio? In other words, is it possible to

reach say point P on the risk-return trade-off line depicted in Figure.

10.1? Yes, if you borrow money at rate Rf and invest it in the portfolio

with g Â¼ gT . Obviously, the investment risk is then higher than that of

sT .

114 Portfolio Management

0.16

E[R]

P

T

0.12

0.08

RF

0.04

sigma

0

0 0.04 0.08 0.12 0.16

Figure 10.1 The return-risk trade-off lines: portfolio with the risk-free

asset and a risky asset (dashed line); portfolio with two risky assets (solid

line); Rf Â¼ 0:05, s1 Â¼ 0:12, s2 Â¼ 0:15, E[R1 ] Â¼ 0:08, E[R2 ] Â¼ 0:14.

10.2 CAPITAL ASSET PRICING MODEL (CAPM)

The Capital Asset Pricing Model (CAPM) is based on the portfolio

selection approach outlined in the previous section. Let us consider

the entire universe of risky assets with all possible returns and risks.

The set of optimal portfolios in this universe (i.e., portfolios with

maximal returns for given risks) forms what is called a efficient

frontier. The straight line that is tangent to the efficient frontier and

has intercept Rf is called the capital market line.2 The tangency point

between the capital market line and the efficient frontier corresponds

to the so-called super-efficient portfolio.

In CAPM, it is assumed that all investors have homogenous expect-

ations of returns, risks, and correlations among the risky assets. It is

also assumed that investors behave rationally, meaning they all hold

optimal mean-variance efficient portfolios. This implies that all invest-

ors have risky assets in their portfolio in the same proportions as the

entire market. Hence, CAPM promotes passive investing in the index

115

Portfolio Management

mutual funds. Within CAPM, the optimal investing strategy is simply

choosing a portfolio on the capital market line with acceptable risk

level. Therefore, the difference among rational investors is determined

only by their risk aversion, which is characterized with the proportion

of their wealth allocated to the risk-free assets. Within the CAPM

assumptions, it can be shown that the super-efficient portfolio consists

of all risky assets weighed with their market values. Such a portfolio is

called a market portfolio.3

CAPM defines the return of a risky asset i with the security market

line

E[Ri ] Â¼ Rf Ã¾ bi (E[RM ] Ã€ Rf ) (10:2:1)

where RM is the market portfolio return and parameter beta bi equals

bi Â¼ Cov[Ri , RM ]=Var[RM ] (10:2:2)

Beta defines sensitivity of the risky asset i to the market dynamics.

Namely, bi > 1 means that the asset is more volatile than the entire

market while bi < 1 implies that the asset has a lower sensitivity to the

market movements. The excess return of asset i per unit of risk (so-

called Sharpe ratio) is another criterion widely used for estimation of

investment performance

Si Â¼ (E[Ri ] Ã€ Rf )=si (10:2:3)

CAPM, being the equilibrium model, has no time dependence. How-

ever, econometric analysis based on this model can be conducted

providing that the statistical nature of returns is known [2]. It is

often assumed that returns are independently and identically distrib-

uted. Then the OLS method can be used for estimating bi in the

regression equation for the excess return Zi Â¼ Ri Ã€ Rf

Zi (t) Â¼ ai Ã¾ bi ZM (t) Ã¾ ei (t) (10:2:4)

It is usually assumed that ei (t) is a normal process and the S&P 500

Index is the benchmark for the market portfolio return RM (t). More

details on the CAPM validation and the general results for the mean-

variance efficient portfolios can be found in [2, 3].

As indicated above, CAPM is based on the belief that investing in

risky assets yields average returns higher than the risk-free return.

Hence, the rationale for investing in risky assets becomes question-

able in bear markets. Another problem is that the asset diversification

116 Portfolio Management

advocated by CAPM is helpful if returns of different assets are

uncorrelated. Unfortunately, correlations between asset returns may

grow in bear markets [4]. Besides the failure to describe prolonged

bear markets, another disadvantage of CAPM is its high sensitivity to

proxy for the market portfolio. The latter drawback implies that

CAPM is accurate only conditionally, within a given time period,

where the state variables that determine economy are fixed [2]. Then

it seems natural to extend CAPM to a multifactor model.

10.3 ARBITRAGE PRICING THEORY (APT)

The CAPM equation (10.2.1) implies that return on risky assets is

determined only by a single non-diversifiable risk, namely by the risk

associated with the entire market. The Arbitrage Pricing Theory

(APT) offers a generic extension of CAPM into the multifactor

ñòð. 33 |