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Abstract

In this paper we present a simple approach of incorporating a Value at Risk (VAR)

constraint to tactical asset allocation (TAA).We outline a dynamic VAR TAA strategy

which is useful in controlling the risk and expected losses of any balanced product. From

our results it is evident that controlling losses can improve returns and at the same time

reduce risk. The attractive feature of the strategy is that it is easy to implement and does

not require assumptions about the distribution of returns or estimating investors utility

function. In summary, the strategy provides pension fund managers with prescribed

tactical tilts in asset allocation which is consistent with their level of risk aversion. The

VAR TAA strategy significantly outperforms the buy hold strategy. This approach can be

used as a stand alone strategy or can also be used in conjunction with the views of a TAA

manager. We show that when the VAR TAA strategy is combined with more traditional

TAA strategies this produces a more robust investment process with improved

information ratios.

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Introduction

In this paper we present a simple approach of incorporating a Value at Risk (VAR)

constraint to tactical asset allocation (TAA).1 This is quite different to the traditional

approaches where TAA is based upon forecasting equity and bond returns. Numerous

authors have proposed a number of factors in forecasting the relative performance

between equities, bonds and cash. See for example, Arnott and Sorenson (1988), Perold

and Sharpe (1988), Arnott and Hendriksson (1989), Nam and Branch (1994). 2

Depending upon the relative attractiveness of equities, bonds and cash tilts are made

towards the most attractive asset class, and tilts are made away from the least attractive

asset class. The extent of the tilts are determined by the strength of view from the

modeling.

Typically, pension fund managers will determine their long term strategic asset allocation

and will periodically rebalance their portfolios back to this benchmark. Rebalancing can

be passive or tactical. By passive we mean, when deviations from the strategic asset

allocation reach a certain limit the portfolio is rebalanced back to neutral. This approach

is commonly used in the industry because the manager may not have the expertise to add

value consistently through tactical shifts in asset classes or on the other hand the portfolio

manager does not believe that TAA can add value consistently in the longer term.3

TAA

on the other hand can be performed by in-house expertise or by employing an outside

manager to implement a TAA overlay on the portfolio

Previous studies have applied a VAR constraint in determining optimal asset allocation in

maximizing returns. See for example, Alexander and Baptista (1999), Huisman, Koefijk

and Pownall (1999) in relation to optimal asset allocation and Basak and Shapiro (2001),

1 For an excellent overview of VAR, see Duffie and Pan (1997), Jorion (1997).2

Typical factors include, earnings yield minus the yield on long term government bonds, dividend yield

minus the yield on long term government bonds, earnings yield minus yield on long term government

bonds plus long term growth of earnings and normalized versions of the above.3 Weigel (1991) has shown that TAA managers do have market timing skills. However Watson Wyatt

(2005) report that evidence to support TAA has been mixed when it was used as a total fund overlay.

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Wang, Shyu, Liao and Chen (2004) for a discussion of dynamic asset allocation in a

VAR framework.

Typically, the asset allocation in relation to the equity/bond split favors bonds to equities.

The relative split being a function of the VAR limit. For example, Huisman, Koefijk and

Pownall (1999) found that the optimal asset allocation was 64% bonds and 36 % equities

at a 95% confidence interval.4 This asset allocation being calculated for the period 1980

to 1998. 5 The optimal asset allocation could be quite different when estimated for

different sample periods. Rather than model the optimal VAR asset allocation we propose

a dynamic strategy which tilts the portfolio towards the most attractive asset class based

upon movements in the VAR. In the following discussion we make no assumptions about

the distribution of returns or investors utility function.

The VAR TAA model we are proposing is easy to implement and provides pension fund

managers with prescribed tactical tilts in asset allocation which is consistent with their

level of risk aversion. This approach can be used as a stand alone strategy or can be used

in conjunction with the views of the TAA manager.

The Model

Initially, we need to determine the extent of losses the investment manager is willing to

accept. For example, lets assume the investment manager is prepared to accept a loss of

5% of the value portfolio in any month, but this loss can only occur 1% of the time. This

means that one in every hundred months we would expect to experience a loss of 5%.

This is a VAR of 5% at the 99% level of confidence. Once we have specified the VAR

of the portfolio we can then determine what combination of equities and bonds match the

VAR.

To illustrate the variability of the VAR of a portfolio consider a portfolio of 60% SP500

and 40% 10 year government bonds. This is a typical asset allocation for a balanced fund.

4 With no borrowing permitted.5 See also Gaivoronski and Plug (2004) for a description of generating Mean VAR efficient portfolios

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Figure 1 shows a time series plot of the VAR at the 99% level of confidence from 1950 to

2005.6 It is apparent from Figure 1 that the VAR for a constant mix portfolio can vary

from -2.3% to -8.2% on a monthly basis. Even with a constant mix portfolio the VAR can

vary considerably. Pension fund managers may not be aware that the VAR of the passive

portfolio is changing so dramatically and may wish to change the asset allocation to be

more conservative when markets are more volatile. The average VAR was -5.2% over the

55 year period.

Typically what seems to happen is that as equity prices fall this increases the risk of a

potential loss, and correspondingly increases the VAR. To maintain the specified VAR

limit the asset allocation will automatically shift the portfolio more to bonds such that the

VAR will always be same. In other words the asset allocation strategy will move up or

down the efficient frontier such that the VAR limit is maintained. When equity prices rise

the VAR model will move more into equities as the risk of a potential loss is decreasing.

Figure 1

Value at Risk of 60/40 Benchmark - 1/1950 to 8/2005

Average = -5.2% , Standard Deviation = 1.35%

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4

Date

VAR%

6 In order to obtain timely movements in expected losses, VAR was calculated on a 5 year rolling window.

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To illustrate the process consider the case of a balanced fund which invests in the SP500

and 10 year government bonds. The strategic benchmark asset allocation is 60% equities

and 40% bonds. In the following discussion no short sales are permitted this is in keeping

with a typical balanced fund. Table 1 presents the results of various scenarios from

1/1950 to 08/2005. The second column presents the returns/standard deviation of the

benchmark portfolio. The next five columns present the results of 99%, 98%, 97%, 96%

and 95% confidence intervals. The VAR limit was set at the average of the benchmark

which was -5.2%.

Table 1 - Returns

Benchmark is 60% SP500 and 40% 10 year Government Bonds

1/1950 8/2005, VAR limit set at -5.2%

Benchmark 99% CI 98% CI 97% CI 96% CI 95% CI

Av

Return

9.7% 11.4% 11.5% 11.9% 12.1% 12.2%

Stdev 9.4% 10.2% 11.1% 11.7% 12.3% 12.6%IR 1.02 1.12 1.04 1.01 0.99 0.96

Max Eq 60% 100% 100% 100% 100% 100%

Min Eq 60% 0% 40% 40% 45% 50%

Av Eq 60% 64.2% 76% 82% 86% 89%

Min Ret -10.6% -7.2% -11.7% -14.5% -14.5% -18.8%

The average return of the benchmark over this period was 9.7% with a standard deviation

of 9.4%. The various scenarios indicate out performance of the benchmark ranging from

1.7% to 2.5% pa with increased risk over the benchmark. The reason for the out

performance is that when equities are rising in value the VAR decreases and to maintain

the prespecified VAR the weighting to equities is increased. This is evident by the

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average equity exposure across all confidence intervals being greater than the equity

exposure of the benchmark. Using the information ratio (IR) as a measure risk/reward it

appears that the 99% level of confidence strategy offers the highest value. The average

equity exposure for the 99% level of confidence was 64% which represents a small tilt to

equities over the 55 year period. The average equity exposure for the remaining scenarios

range from 76% to 89%. This is likely to be too aggressive for the majority of pension

fund managers. The results from Table 1 suggest that if one is willing to accept an

expected loss of 5.2%, then the 99% confidence interval combination is probably the

preferred strategy. Table 2 provides similar information to Table 1 but presents the results

in terms of excess returns. In terms of information ratios all confidence intervals produce

similar information ratios. 7 The percentage of positive excess returns was 56.1%.

Table 2 Excess Returns


1/1950 8/2005 VAR limit set at -5.2%

99% CI 98% CI 97% CI 96% CI 95% CI

Av Excess

Return

1.8% 1.8% 2.2% 2.4% 2.5%

Stdev 2.8% 3.2% 3.7% 4.0% 4.4%

IR* 0.64 0.57 0.60 0.60 0.57

* Significant at the 1% level

Tables 1 and 2 presented the results of the various strategies if an investor was willing to

accept a loss of 5.2%. In Tables 3 and 4 we present alternative empirical results if we

consider the case of a more risk averse investor who is only willing to accept a loss of

4%. Note in this case the 99% level of confidence strategy is a far more conservative

portfolio, the average equity exposure is 46%. The 99% level of confidence strategy

outperforms the benchmark by 1.1%. The overall risk of the 99% confidence interval

strategy is 8.8% compared to the benchmark risk of 9.4%. The 98% and 97%

7 Although not shown here partitioning the data set into different sub samples produced similar results.

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confidence intervals have similar characteristics to the benchmark with similar standard

deviations and average equity exposure. Table 4 presents the results in terms of excess

returns and suggest that the 97% confidence interval would be preferred to the 98%

confidence intervals owing to its higher information ratio.

Table 3 - Returns


1/1950 8/2005 VAR limit set at -4.0%

Benchmark 99% CI 98% CI 97% CI 96% CI 95% CI

Av Return 9.7% 10.8% 10.8% 11.3% 11.4% 11.7%

Stdev 9.4% 8.8% 9.5% 10.0% 10.6% 11.1%

IR 1.02 1.23 1.14 1.12 1.07 1.05

Max Eq 60% 100% 100% 100% 100% 100%

Min Eq 60% 0% 0% 0% 35% 40%

Av Eq 60% 46% 57% 64% 72% 76%

Min Ret -10.6% -7.0% -7.9% -10.8% -12.6% -14.4%

Table 4 Excess Returns


1/1950 8/2005 VAR limit set at -4.0%

99% CI 98% CI 97% CI 96% CI 95% CI

Av Excess

Return

1.1% 1.2% 1.7% 1.7% 2.1%

Stdev 3.1% 3.0% 2.9% 3.1% 3.3%

IR* 0.36 0.40 0.59 0.55 0.64


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We next consider the question what is a reasonable confidence interval and VAR limit.

To address this issue we assume that the pension fund manager is willing to accept the

same level of risk (standard deviation) as the benchmark and the same average equity

exposure as the benchmark. Table 5 presents the results of different combinations of

confidence intervals and VAR limits which closely match the risk and average equity

exposure of the benchmark.

Table 5 Choosing CI and VAR


1/1950 8/2005

Benchmark 99% CI

VAR -4.7%

98% CI

VAR -4.2%

97% CI

VAR -3.8%

96% CI

VAR -3.6%

95% CI

VAR -3.2%

Av

Return

9.7% 10.7% 10.7% 10.7% 10.8% 10.8%

Stdev 9.4% 9.5% 9.5% 9.5% 9.5% 9.4%

IR 1.02 1.11 1.11 1.11 1.14 1.16

Max Eq 60% 80% 80% 80% 80% 80%

Min Eq 60% 40% 40% 40% 0% 0%

Av Eq 60% 60% 60% 60% 60% 60%

Min Ret -10.6% -7.2% -8.0% -9.8% -10.8% -10.8%

It is evident from Table 5 that all of the combinations produce similar results. Any of the

above combinations would be a suitable choice. However, the 99% confidence interval

with VAR limit of -4.7% is probably the preferred strategy, as it has the maximum

monthly loss of 7.2% compared to the maximum loss of 10.6% of the benchmark.

It is likely that many pension fund managers would feel uncomfortable allowing TAA

managers the flexibility to vary the equity positions from 0% to 100%. To consider more

reasonable scenarios we present results of constraining deviations from benchmark by +/-

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10%, +/- 20%, +/- 30% and +/-40%. In this case we set the confidence interval at 99%

and the VAR limit at -4.7%

Table 6 Excess Returns Constrained bets


1/1950 8/2005, 99% CI and VAR = -4.7%

+/- 10% +/- 20% +/- 30% +/- 40% +40% -

60%

Av Excess

Return

0.65% 1.0% 1.30% 1.60% 1.8%

Stdev 1.34% 2.0% 2.36% 2.70% 3.0%

IR* 0.49 0.50 0.56 0.56 0.6


The results from Table 6 indicate that all strategies have similar information ratios. It is

apparent that by restricting deviations from benchmark to 10% results in an excess return

of 0.65%. This is probably too little return when considering the cost/benefits of

employing a TAA manager. However, it may be an appropriate strategy if it is employed

as a rebalancing tool by the asset allocation committee, thereby avoiding the costs ofemploying a TAA manager. Watson Wyatt (2005) state that the most skilled TAA

managers have attained information ratios in the range 0.5 to 1.0, and one might expect to

obtain a gross information ratio of 0.5 from a TAA manager. All of the above strategies

attain this information ratio. If on the other hand one employs a TAA manager they

should be given the flexibility of varying the equity exposure by +/- 20%. In this case one

would expect to generate an excess return of 1%.

A major criticism of domestic TAA is the lack of diversification, or multiple sources of

alpha. In our example of two assets, if the bond/equity bet was incorrect then the strategy

underperformed. To overcome this lack of diversification one could apply the same

strategy to a multiple number of bets. For example, the asset base could be further

partitioned into large cap, mid cap and small cap value, large cap, mid cap and small cap

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growth for equities. The bond universe could be extended to 2 year, 5 year and 10 year

government bonds, investment grade corporate bonds and treasury bills. This is a more

complicated process but is feasible in the VAR TAA framework. Even adopting this

approach has its limitations, ideally one should consider global TAA which has a far

larger universe of investment opportunities.

Previously we mentioned that the VAR TAA strategy could be used in conjunction with

the views of other TAA managers. It is not unusual to have two or three TAA managers

mandated for a total portfolio overlay. Ideally the correlations between the managers

should be low thus providing diversification and hopefully more consistent performance

with increased information ratios. It is difficult to obtain benchmark returns for TAA

managers owing to the variability of benchmarks and differing constraints placed on

mandates, we therefore illustrate this point by including the results of two commonly

used TAA strategies. Arnott and Sorenson (1998) suggested a risk premium approach. A

proxy for the risk premium was the earnings yield minus the yield on the 10 year bond

(EY-LB). This variable was found to be a significant predictor of future excess returns.

Using a rolling regression, forecasts were made of next months excess return. If the

forecast was positive invest 100% in equities, and if the forecast was negative invest

100% in bonds. The second strategy is the constant proportion portfolio insurance (CPPI)

approach as outlined in Perold and Sharpe(1988). This approach has the effect of

providing a floor on the value of the portfolio. CPPI can be categorized as a momentum

type strategy as the strategy buys equities when the price rises and sells equities when the

price falls. Table 7 provides the results of the three strategies and the equally weighted

aggregation of the three strategies.

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Table 7 Excess Returns, EY-LB, CPPI and VAR strategies

Benchmark is 60% SP500 and 40% 10 year Government Bonds 1/1950-8/2005

EY-LB CPPI VAR Aggregate

Equal Wgt

Av Excess

Return

4.2% 1.2% 1.8% 2.4%

Stdev 6.8% 2.4% 2.8% 3.0%

IR* 0.62 0.50 0.64 0.80


Table 8 presents the correlation matrix. The correlations between the strategies is low and

provides reasonable diversification. It is apparent that combining strategies with low

correlations produces a more robust investment process with an improved information

ratio.

Table 8 Correlations between EY - LB, CPPI and VAR strategies

Benchmark is 60% SP500 and 40% 10 year Government Bonds 1/1950-8/2005

EY-LB CPPI VAR

EY-LB 1.0 0.39 0.20

CPPI 1.0 0.29

VAR 1.0

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Figures 2 and 3 present annual excess returns of the aggregate strategy from 1980-2005,

and the histogram of annual excess returns from 1950-2005. On an annual basis the

probability of loss over the entire period was 24%.

Figure 2

Annual Excess Returns - Aggregate Strategy 1980-2005

Average = 2.9%, Stdev = 3.8% , IR = 0.76

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Figure 3

Histogram of Annual Excess Returns 1950-2005

Probability of Loss = 24%

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-5 -4 -3 -2 - 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13

Returns %

Frequency%

Conclusion

We have outlined a dynamic VAR TAA strategy which is useful in controlling the risk

and expected losses of any balanced product. From our results it is evident that

controlling losses can improve returns and at the same time reducing risk. The attractivefeature of the strategy is that it is easy to implement and does not require assumptions

about the distribution of returns or estimating investors utility function.

In summary, the strategy provides pension fund managers with prescribed tactical tilts in

asset allocation which is consistent with their level of risk aversion. This approach can

be used as a stand alone strategy or can also be used in conjunction with the views of the

TAA manager.

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References

Arnott, R. and E. Sorenson, The risk premium and stock market performance, Journal

of Portfolio Management, Summer (1988),, p50-55.

Arnott, R., and R. Hendriksson, A disciplined approach to global asset allocation,

Financial Analyst Journal, (1989), p17-28

Alexander, G. and A. Baptista, 1999, Value at Risk and Mean Variance Analysis,

Working paper, University of Minnesota.

Basak, S. and A. Shapiro, 2001, Value at Risk based Risk Management: Optimal

Policies and Asset Prices, Review of Financial Studies, vol. 14, pp. 371-405.

Duffie, D. and J. Pan, 1997, An Overview of Value at Risk, Journal of Derivatives, , 4,

7-49.

Giavoronski, A. and G. Plug, Properties and computation of Value at Risk efficient

portfolios based on historical data, working paper, Norwegian University of Science and

Technology, (2001)

Huisman, R., Koedijk K and R Pownall, Asset Allocation in a value at Risk

Framework, Working paper, Erasmus University, (1999).

Jorion, P., 1997, Value at Risk: The New Benchmark for Controlling Risk, Irwin,

Chicago, Ill.

Nam, J. and B. Branch , Tactical Asset Allocation: Can it work?, Journal of Financial

Research, 1994, p.465-479.

Perold, A. and W. Sharpe, Dynamic strategies for asset allocation, Financial Analyst

Journal, (1988), p16-27

Watson Wyatt (2005), What if, memo on Tactical Asset Allocation.

Wang, c.,D. Shyu, C. Liao and M. Chen. 2004, A Model of Optimal Dynamic Asset

Allocation in a Value at Risk Framework, International Journal of Risk Assessment and

Management, vol. 4, p.

Weigel, E., (1991), The performance of tactical asset allocation , Financial Analyst

Journal, (1991), p63-70.

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