SHORT TERM VOLATILITY CHANGES AND ACTIVE PORTFOLIO MANAGEMENT

Seppo Pynnönen
Department of Mathematics and Statistics, University of Vaasa,
P.O.Box 700, FIN-65100 Vaasa, Finland, e-mail sjp@uwasa.fi

In this paper the more traditional active portfolio management discussed in Treynor and Black (1973), Rosenberg (1979), Rudd and Clasing (1982), Grinold and Kahn (1995), and Pynnönen (1995) is extended to utilize the changing volatility in addition to the predicted short term residual returns. It is shown that taking this new aspect into account enables investors to perform a more detailed analysis of the sources of required return that should be gained by active portfolio management. Furthermore, the approach enables investors to predict future end period volatility in the investment horizon on the basis of current information. In this paper the necessary formulas are derived for practical portfolio analysis. In addition, in contrast to the above papers, the possible change in the market position is taken explicitly into account, and its impact on active management is included in the analysis. This situation becomes relevant when short selling is not allowed.

Main Results

We assume that the excess return series (excess over/under the risk free return), r_it, of the ith share can be modeled by the market model

(1)

where r_mt is the market excess return, is the residual return with zero mean and conditional (with respect to past information) variance, following a generalized ARCH-process, GARCH. The parameter measures a temporal mispricing or expected residual return of the security, and the Beta of the stock, indicates the sensitivity of the security with respect to changes of the market portfolio.

It may be noted that adoption of a GARCH-type model for residuals implies that the ordinary least squares (OLS) is no more efficient in estimating Betas. The method of maximum likelihood (ML) must be used instead. Furthermore, one could also model time varying Beta in the GARCH framework, but here we, however, treat Beta as a constant during short term portfolio revision periods. For estimating varying Beta in the GARCH framework, the interested reader is referred to Bollerslev et al. (1988).

In addition to the constant Beta, we assume that the residual returns are uncorrelated with the market portfolio and with other securities. This implies that, if (1) is a portfolio consisting of q shares, and Q denotes the total number of shares on the markets, then the residual variance can be written as

(2)

,

where is the active holding with market weight w_i and h_i the actual proportion invested on the i^th share. denotes the portfolio Beta. Note that , if share i is not in the portfolio.

Modeling the conitional heteroscedasticity by a GARCH(1,1) process has proven to be many times an adequate model to capture the volatility clusterings in the residual process. In that case we get

(3)

,

with the standard stationary assumptions, where denotes the information set available at time point t. Because ARCH-processes are not linear they do not aggregate as such. As a consequence the portfolio conditional variance depends on the individual stock conditional variances such that

(4)

,

where , , and with the unconditional variance of the residual term.

It is assumed that a quadratic utility function is a sufficient approximation for the investor's preferences, such that the indifference curves can be presented as

(5)

,

where C denotes the indifference curve, the expected (excess) return of the portfolio, the investor's risk aversion parameter, and the variance of the portfolio. Adopting this preference structure implies that long run optimality requires that in the normal situation when there is no special information available on the markets, the investor allocates

(6)

,

of his/her funds to the risky market and the rest, , to the riskless alternative. [see e.g. Rosenberg (1979) for details]. Here is the market's expected excess return and is the variance of the market return. Hence, knowing the long term policy one knows instantaneously also the investor's risk aversion parameter, . It is worth noting that the expected long term return, return the risky normal portfolio is then with variance

Under these assumptions it will be shown in the paper that taking an active position by deviating from the normal portfolio requires an extra return that should be gained by the policy is

(7)

,

where the first term on the right hand site denotes the required alpha (additional return) due to changed long term residual risk position, the second term the required alpha due to short term risk, and the last term the required alpha due to the change on the market risk position. Similar analysis and formula can be derived also for stockwise and market timing steps in portfolio revision.