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AN ANALYSIS OF PORTFOLIO REVISION STRATEGIES UTILIZING VARIABLE REVISION INTERVALS

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Page 1: AN ANALYSIS OF PORTFOLIO REVISION STRATEGIES UTILIZING VARIABLE REVISION INTERVALS

APPLICATIONS AND IMPLEMENTATION

AN ANALYSIS OF PORTFOLIO REVISION STRATEGIES UTILIZING VARIABLE REVISION INTERVALS*

Donald A. Nast, Florida State University George C. Philippatos, The Pennsylvania State University

ABSTRACT

This paper investigates portfolio revision with an emphasis on the decision of when to revise. A statistical technique based on the sequential analysis of the time series of portfolio return relatives determines when revision is to occur. The tech- nique detects changes in the time series which are an indication that the underlying generating process of the time series has changed and that the portfolio should be, if necessary, revised. Thus, the length of the revision interval is variable and a function of the data. The statistical technique is utilized in conjunction with three portfolio revision strategies. These three revision strategies are compared to a buy and hold policy over three nonoverlapping, 12-year investment horizons. The basis of com- parison is the net terminal values which include adjustments for transaction costs and taxes. The sensitivity of the statistical technique to its parameters is also analyzed.

INTRODUCTION

Portfolio revision has received increasing attention in recent years. Theoretical research has attempted to extend portfolio revision into a dynamic, rnultiperiod frame- work [8] [3] . Concurrently, empirical research has tested the feasibility of revising common stock portfolios over time.' Smith [9] concluded that investors can profitably revise portfolios by taking advantage of new information as it is revealed in the changing prices of the securities. Evans [ 6 ] , however, determined that a buy and hold policy yields a superior rate of return over a rebalancing strategy. The rebalancing strategy assumes constant expectations over time and, thus, adjusts the security pro- portions in the portfolio to their initial levels. Cheng and Deets [4], when excluding transaction costs, found that rebalancing a portfolio is superior to buy and hold. They did not, however, extend their empirical work to include transaction costs.

*The authors wish to thank J. R. Ezzell, J. Hayya, and J. Herendeen, The Pennsylvania State University, for valuable comments on an earlier draft of this paper. Special thanks are also due to two referees of the Journal for very constructive criticism.

'Considerable empirical research has also been directed at the buying, holding, and selling of individual securities. See Fama [7].

71

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7 2 DECISION SCIENCES [Vol. 7

The empirical studies have assumed that investors adjust their porttolios according to their constant or revised expectations. However, these portfolio adjust- ments are assumed to occur only at fixed, equal length intervals. For instance, Smith [9] and Evans [6] revised portfolios at semiannual intervals, while Cheng and Deets [4] tested various fixed length intervals of one week to 90 weeks.' In this study, we relax the fixed interval restriction and allow revision of a portfolio when a change has occurred in the process underlying the observed returns of the portfolio. In particular, we employ a sequential analysis technique presented by Brown 121 to detect such a change. When this change is detected, the portfolios are revised according to three revision strategies. The net terminal values (after transaction costs and taxes) of the portfolios are then compared to a buy and hold portfolio value with the result that even with the more realistic assumption of variable revision intervals, the buy and hold policy was superior during the time periods tested.

METHODOLOGY

Sequential Analysis

An extension of sequential analysis known as the parabolic mask is a statistical technique that permits the detection of turning points in a time series. Wald [ lo ] noted that the probability of obtaining a sample o f x l , x 2 , . . . ,x, consecutive obser- vations on x when x is considered a normally distributed random variable with an unknown mean p and a known standard deviation u is

Wald suggested a one-sided sequential analysis test based on the sequential probability ratio, pl/po, with control limits specified in terms of the Type I (a) and Type I1 @) errors. As each observation on x is obtained, the cumulative sum, Exi, is determined for comparison to the specified control limits. At each observation, the decision is to either accept the null hypothesis that p = po, accept the alternative hypothesis that p = pl, or to continue testing when the control limits are not violated. Figure l(a) describes the one-sided sequential analysis test.

Two-sided Sequential Test

The one-sided sequential analysis test can be extended to a two-sided test by considering the observationsxi as errors from a forecasting model. The null hypothesis states that the mean of the errors is zero or p = po = 0. The alternative hypothesis states that the errors are either negatively or positively biased, thus indicating that the

'Smith employed a 20-year investment horizon of 1946-1965. Evans's investment horizon was the 10-year period, 1958-1967. A 31-year period, 1937-1969, was utilized by Cheng and Deets in their empirical testing.

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19761

t

xxi 0

-

PORTFOLIO REVISION STRATEGIES

Con tinue Testing

e m .e .. .*.

Observation Number, n

0

(b) Two-sided Sequential Analysis Test V-Mask Reversed

73

FIGURE 1

. Continue Testing

. Observation Number, n

(a) Sequential Analysis

t

EXi 0

- Test

zxi -k

0 Observation

-

(c) Parabolic Mask

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74 DECISION SCIENCES [Vol. 7

observations are from another population. The critical level of bias is represented by cu; therefore, the alternative hypothesis is p = p 1 = +_ cu. The control limits of the V-shaped sequential analysis test for the cumulative sums of the forecast errors are derived as

1- cu n i= 2 I X i = + l n J a 2 t -n .

With such a forecasting model, the rejection of the null hypothesis, Ho, is the item of interest. If the cumulative sum of the forecast errors, Exi, lies in either the continue testing or accept Ho range, the model is considered adequate and the testing continues. The purpose is to detect when the observations are being generated from some other population as indicated by a violation of a control limit and the rejection of Ho. Barnard [ l ] altered the application of the sequential test by reversing the direction of the V-mask such that it is always centered on the most recent cumulative sum. Figure I(b) diagrams the reversed V-mask sequential test.

Parabolic Mask

Brown’s 121 interest in sequential analysis and V-masks was not in detecting a particular critical size change of the forecast errors, but instead detecting any size change. Consequently, Brown obtained the envelope curve over the parameter c of all V-masks. This envelope curve is a parabola congruent to

(.gl xi) 2 - - 2 0 2 h((l - (3)

which is reversed and centered on the most recent cumulative sum. The parabolic mask detects any size change in the forecast errors and the direction of the change with large changes detected quickly. The inclusion of the a and 0 errors allows the detections to be made within specified risks.

Figure I(c) indicates the nature and use of the parabolic mask. Point A is the current cumulative sum of the forecast errors of the time series model. When the parabolic mask is centered on point A, the cumulative sum at point B falls outside the parabola boundary, thus indicating that a change in the underlying process occurred at point B. The null hypothesis that the mean of the forecast errors is zero is therefore rejected, and the alternative hypothesis that the observations are from another popu- lation is accepted.

Use of the Parabolic Mask

The parabolic mask technique is employed in our study to detect when a change occurs in the underlying process generating the time series of portfolio returns. It does not indicate what economic factors, if any, caused the change, but only that a change has occurred and that the portfolio should be revised according to altered expectations. The return relatives of the portfolio being tracked are the input

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19761 PORTFOLIO REVISION STRATEGIES 75

observations for the parabolic mask.3 A base period is established at the beginning of each revision interval to determine the necessary parameters: the standard deviation, u, and the mean, fi. The cumulative sums are derived from the difference between the consecutive portfolio return relatives and the mean of the return relatives in the associated base period.

Figure l(c) presents a revision interval with its associated base period. As previously discussed, the cumulative sum outside the parabolic mask at point B indicates that a change in the time series of portfolio returns has occurred. The portfolio is thus revised utilizing three revision strategies. A new base period is established to include point A; new parameters are determined from the base period; and, the next revision interval is begun.

The standard of comparison over the entire investment horizon is a buy and hold (BH) policy. The BH policy is based on holding the initial market portfolio during the entire time period tested. The initial market portfolio is r n e r a t e d using the mean-variance efficiency criterion and the risk-free rate of return.

Revision Strategies

The first revision strategy (LC) permits only long positions in the portfolio or a strictly cash position. When the parabolic mask indicates a change has occurred and that the portfolio return relatives have turned downward, then, the LC revision strategy dictates a cash position. When an upward change is signalled by the parabolic mask, a long position is established in a new market portfolio.

The second revision strategy (LS) is an extension of the previous LC strategy by allowing a short position in the portfolio rather than an al l cash position when the parabolic mask indicates a persistent negative bias in the portfolio return relatives. The short position consists of those securities and their proportions that were in the immediately previous long position portfolio. Thus, the LS strategy is a series of long and short positions, while the LC strategy is a series of long and cash positions. For both strategies, a new mean-variance, market portfolio is generated whenever a new long position is required. The market portfolios in all the revision strategies are derived using the historical means, variances, and covariances of the securities as proxies for the expectational inputs in the mean-variance portfolio selection model.

A transitional rebalancing strategy (TR) is the third revision strategy in our study. When the parabolic mask signals that a long position is in order, the available funds are invested in equal proportions in the worst performing securities during the revision interval just ended. On the other hand, when a short position is indicated, the

3The return relative of the portfolio for period t is the value of the portfolio at the end of period t divided by the value of the portfolio at the end of period t-1. The value of the portfolio includes the reinvestment of dividends paid during the period.

4The investor is assumed able to alter the risk-return level of his total position by lending or borrowing at the risk-free rate. Thus, we concentrate on the market portfolio. This assumption is consistent with the approach in Smith's study. Evans and Cheng and Deets rebalancing strategies did not require the use of a risk-free rate.

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best performing securities of the previous revision period are sold short in equal proportions.' The number of securities in the resultant portfolios is equivalent to the number of securities in the initial market portfolio for the LC strategy.

Data

To test the three revision strategies and the parabolic mask technique, a random sample of 50 securities was chosen from those trading on the New York Stock Exchange during the entire 1932-67 period6 The 36-year period was divided into three equal, non-overlapping 12-year subperiods as follows: Period 1, January 1932 to December 1943; Period 2, January 1944 to December 1955; and Period 3, January 1956 to December 1967. Each revision strategy and the buy and hold policy were tested separately by beginning each 12-year period with an initial cash balance of $200,000 and determining the net terminal cash value at the end of each period. The strategies were then compared on the basis of their resultant net terminal value^.^ The net terminal values include the appropriate adjustments for transaction costs and

8 taxes. For each security, the monthly return relatives were obtained. The security

return relatives are then weighted proportionately to determine the monthly return relatives of the portfolio which become the input observations for the parabolic mask technique. A six-month base period is used to determine the necessary parameters for the parabolic mask. The cumulative sum at each observation point is derived from the difference between the current and prior consecutive portfolio return relatives and the mean of the return relatives in the base period. Each time a new investment position is assumed, the base period is updated to include the observation at which detection of a change occurred. New parameters for the parabolic mask are obtained from this updated base period. Since the critical risk for the investor is the possibility of rejecting a true change, the Type I1 @) error, this risk was set at 0.01. The Type I (a) error risk was set a t 0.20. The net terminal values were also tested for their sensitivity to the length of the base period and the level of the a risk.

'The rationale for the transitional rebalancing strategy derives from the martingale concept. Those securities performing poorer than the historical mean return in one time period are, on the average, expected to perform better than the historical mean return in the following period. Also, those performing better in the first time period are, on the average, expected to do poorer in the next. The TR strategy combines these considerations with the parabolic mask technique. See 141 16) [7] .

6A listing of the securities chosen is available form the authors upon request.

7The net terminal value (geometric total) is used as the basis of comparison rather than the geometric mean return since as Cheng and Deets [ 5 ] point out, the geometric mean is a biased statistic when the length and number of revision periods differ between strategies.

'Transaction costs of 1% of the value and taxes at a 59% marginal rate and 25% capital gains rate were included. Cash dividends were assumed reinvested immediately in the security that paid them.

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19761 PORTFOLIO REVISION STRATEGIES 77

EMPIRICAL RESULTS

Revision Strategies Results

The testing of the three revision strategies in the three time periods yielded the net terminal values in Table 1 . As noted, the BH policy was definitely superior to the three revision strategies in all three periods. The LC strategy produced gains over the initial investments of $200,000, but was clearly inferior to the gains of the BH policy. The performance of the LS strategy, on the other hand, did not even yield a gain on the initial investment. Permitting short positions instead of holding cash when the parabolic mask indicated a downward change in the series of portfolio return relatives resulted in losses from those short positions. The TR strategy produced the poorest performance of the three revision strategies. In the first period, the TR strategy yielded a negative terminal value. This negative value resulted from a loss on the final short position which was greater than the positive intermediate value at the beginning of the short position.

TABLE 1

Net Terminal Values for Portfolio Revision Strategies and Buy and Hold Policy

Period 1 Period 2 Period 3 Strategy 1/32- 12/43 1/44-12/55 1 /56-12/67

Buy and Hold $566,478 $725,8 19 $5 14,245 (BH)

Long and Cash (LC)

Long and Short (LS)

275,330

47,589

247,252

101,675

286,720

140,669

Transitional - 10,738 17,635 196,277 Rebalancing

(TR)

'The three strategies and the BH policy were also compared over the entire 36-year period with no change in the pattern of results.

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Net Terminal

Value

Parabolic Mask Operation

Further analysis of the results reveals the operation of the parabolic mask. For the LC and LS revision strategies during the three time periods, there were 17, 17, and 19 revision intervals, respectively. For the LS strategy in Period 1, the length of the revision interval ranged from a 1-month position to a 37-month position; in Period 2, from a 1-month position to a 24-month position; and in period 3, from a I-month position to a 15-month position. In each time period, the TR strategy experienced a longer maximum revision interval. For all three strategies the parabolic mask technique resulted in a wide variety of revision interval lengths.

There were also detections of changes in the time series of the portfolio return relatives that did not necessitate a revision of the investor’s position. This situation occurred when the detected change did not represent a change in direction of the portfolio return relatives, but only a change in the rate of increase or decrease of the portfolio’s value. Under these conditions, the investor retained the current portfolio. On average, the parabolic mask registered a detection of a change every 2.3 months. Many of the detections occurred only one month following the indicated change, thus suggesting that weekly or daily data may be more appropriate for use with the parabolic mask.

Net Type I Terminal Error Value

Sensitivity of Strategies to Parameter Changes

Due to the surprisingly poor performance of the revision strategies in conjunction with the parabolic mask, the sensitivity of the strategies to the length of the base period and to the level of the a risk was investigated. The analysis, however, was restricted to the LC strategy and Period 3. Table 2 presents the sensitivity

6 months $286,720

8 months 256,201

12 months 213,295

16 months 305,331

20 months 3 10,470

TABLE 2

~ ~~~

0.20 $286,720 0.30 319,551

0.40 309,115

0.50 287,944

Net Terminal Values for Portfolio Strategy LC in Period by Length of Base Period and by Type I (a) Error

I Base Period

Length

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19761 PORTFOLIO REVISION STRATEGIES 79

results.'' As the base period lengthened, there was no appreciable change in the net terminal values, and all the results remained substantially lower than the BH policy. The longer base period lengths had no effect on the number of revision intervals during the 12-year period; however, the months in which they occurred were different as the base period lengths changed.

Similarly, increasing the level of the Type 1 risk did not result in any appreciable improvement in the performance of the LC revision 3trategy. The larger the a risk, the narrower is the acceptance region of the parabolic mask. Thus, as expected, the number of detections and consequent changes in investment position increased. At a 0.50 Type I risk, the number of revision intervals was 28, with a maximum revision interval of 14 months. As this additional analysis indicates, altering the level of these two parameters, the Type I risk and the length of base period, did not endanger the superiority of the BH policy.

CONCLUSIONS

This study has examined the use of a sequential analysis technique, the parabolic mask, to determine the appropriate time to revise common stock portfolios. Three different portfolio revision strategies were employed. The results of the empirical testing indicated that the buy and hold policy is superior to the three revision strategies during the three 12-year time periods analyzed. Indeed, the BH superiority was substantial. The additional sensitivity analysis of increasing the base period length and Type I utilized in the parabolic mask technique did not reveal any beneficial effect on the performance of the revision strategy. Thus, permitting the data itself to dictate the length of the revision interval rather than employing a predetermined, fixed revision did not result in the alternative revision strategies outperforming the BH policy. Our results are consistent with those of Evans [6], but contrary to those of Cheng and Deets [4] and Smith [9]. It should be noted, however, that Cheng and Deets's study did not include the effects of transaction costs.

A closer investigation of the operation of the parabolic mask suggested that with monthly observations, the technique was inadequate in signalling the appropriate time to revise the portfolios. The majority of detections of changes in the time series were separated by one month. Thus, although it appeared that changes were detected quickly, some of the resultant portfolio revisions were inappropriate. Further research employing weekly or daily observations to permit a more adequate functioning of the parabolic mask would appear appropriate. Another area of future study is to examine the underlying determinants of the Type 1 and Type I1 risks, and their relationship to the investor's risk-return tradeoff. Further sophistication of the revision strategies to allow individual securities to enter and leave the portfolios prior to complete revision is also necessary.

In conclusion, while our study demonstrated the superiority of the BH policy, it also demonstrated the use of the parabolic mask sequential analysis technique with the problem of portfolio revision.

"During the testing for the various base period lengths, the a and p risks were held constant at 0.20 and 0.01, respectively. For the different levels of a error, the base period was 6 months, and the p risk was 0.01.

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DECISION SCIENCES [Vol. 7

REFERENCES

Barnard, G. A. “Control Charts and Stochastic Processes.” Journal of Royal Statistical Sociery, 21 (1959), pp. 239-257.

Brown, Robert G. “Detection of Turning Points in a Time Series.” Decision Sciences, 2 (October, 1971), pp. 383403.

Chen, Andrew H. Y., Frank C. Jen, and Stanely Zionts. “The Optimal Portfolio Revision Policy.”Journal ofBusiness,44 (January, 1971),pp. 51-61.

Cheng, Pao L., and M. King Deets. “Portfolio Returns and the Random Walk Theory.” Journal of Finance, 26 (March, 1971), pp. 11-30.

Cheng, Pao L., and M. King Deets. “Statistical Biases and Security Rates of Return.” Journal oj’Financial and Quantitative Analysis, 6 (June, 197 l) , pp. 977-994.

Evans, John L. “An Analysis of Portfolio Maintenance Strategies.” Journal of Finance, 25 (June, 1970), pp. 561-571.

Fama, Eugene F. “Efficient Capital Markets: A Revision of Theory and Empirical Work.” Journal of Finance, 25 (May, 1970), pp. 383417.

Mossin, Jan. ‘‘Optimal Multiperiod Portfolio Policies.” Journal of Business, 41 (April, 1968),

Smith, Keith V. “Alternative Procedures for Revising Investment Portfolios.” Journal of Financial and Quantitative Analysis, 3 (December, 1968), pp. 371403.

pp. 215-229.

\ l o ] Wald, Abraham.Sequential Analysis. New York: John Wiley & Sons, Inc., 1947.