Part 2
Topic articles

6 Tracking error

6.1 Introduction

Expected tracking error is used to define the magnitude of the deviations from the benchmark index of the Government Pension Fund Global (GPFG) and the Government Pension Fund Norway (GPFN) permitted to Norges Bank and Folketrygdfondet. The Ministry has stipulated a one percentage point limit on expected tracking error in the mandate for the management of the GPFG. The stipulated limit for the GPFN is three percentage points. This topic article analyses factors that influence tracking error, including the impact of the number of securities in the benchmark index.

6.2 Factors that influence tracking error

6.2.1 Model

A simple simulation model is used to shed light on how characteristics of indices, portfolios and the market may influence tracking error. The purpose is to illustrate how certain factors influence tracking error, rather than to provide an exact estimate of tracking error.

The model simulates developments in an equity portfolio and an equity index over ten periods. All equities included in the index carry the same weight in the first period. It is assumed that the equity portfolio comprises only 90 percent of the equities included in the index. This implies that each equity carries somewhat higher weight in the equity portfolio than in the index.

The return on all equities is drawn from a statistical probability distribution with a standard deviation of 35 percent. The model is structured such as to make the average correlation between equities about 0.2. This is more or less in line with the average standard deviation and correlation over the last three years for the equities included in the Benchmark Index of the Oslo Stock Exchange. The technical calculation assumptions represent a simplification when compared to how equity prices actually develop. The return, the standard deviation and the correlation of the equities in a global portfolio will all be influenced by, inter alia, country risk, sector risk, foreign exchange risk and systematic risk factors.

Developments in the portfolio and the benchmark index will vary for each simulation, which implies that tracking error also varies. The model has been reiterated one hundred times, and the average tracking error for all simulations has been calculated.

6.2.2 The number of securities in the benchmark index

The number of securities in the benchmark index may have a major impact on the calculation of tracking error. When the number of companies increases, the deviations from the index are spread across more equities, and hence there is less of an impact from company-specific fluctuations. The benchmark index for the Norwegian equity investments of the GPFN is the Benchmark Index of the Oslo Stock Exchange, which comprises just over 50 equities. In comparison, the benchmark index for the GPFG comprises about 7,500 equities.

Figure 6.1 shows how the number of companies in an equity index can influence the calculation of tracking error, under the above assumptions. A portfolio with a benchmark index comprising 50 equities will under these assumptions have a tracking error of 1.8 percentage points. Correspondingly, a portfolio with a benchmark index of 7,500 equities will have a tracking error of 0.1 percentage points.

Source Ministry of Finance.

6.2.3 Overlap between portfolio and benchmark index

The simulation model is structured such as to make the overlap, or the identical portion of the portfolio and the index, 90 percent in the first period. In comparison, the overlap between the Norwegian equity investments of the GPFN and the benchmark index was just in excess of 90 percent at yearend 2014, whilst the overlap between the equity investments of the GPFG and the benchmark index was in excess of 80 percent at the same point in time. When overlap increases, there are either fewer deviations from the benchmark index, or individual deviations are smaller when taken in isolation. Figure 6.2 shows that tracking error declines when overlap increases.

Source Ministry of Finance.

6.2.4 Market volatility

The average volatility of the equities included in the Benchmark Index of the Oslo Stock Exchange has been about 35 percent over the last three years. Average volatility was more than 50 percent during the financial crisis. High market volatility also results in large tracking error. Figure 6.3 shows the changes in calculated tracking error when the standard deviation of the distribution from which the equity returns are drawn is increased.

Source Ministry of Finance.

6.2.5 Correlation

High correlation between equities will normally reduce tracking error. When correlation is high, equity prices will largely move in the same direction. This implies that the consequences of selecting one stock over another are small, since there is a high probability that their development will be approximately the same. Figure 6.4 shows the changes in tracking error when average correlation between individual equities increases.

Equity correlation normally increases during periods of high market volatility. Average correlation for the equities included in the Benchmark Index of the Oslo Stock Exchange was about 0.2 over the period from December 2006 to December 2009. This is slightly higher than the correlation over the period from January 2012 to December 2014. Between different industrial sectors, on the other hand, correlation differences were larger in the two periods, with higher correlation during the financial crisis than during the three-year period until the end of 2014.

Source Ministry of Finance.

6.2.6 Index changes

The composition of indices is changed on a regular basis. The composition of the Benchmark Index of the Oslo Stock Exchange is, for example, altered every six months. Such index changes may give rise to tracking error between a portfolio and its benchmark index.

In order to prevent any impact on tracking error, an asset manager may trade in the equities entering or exiting the index on the same day as the index is changed. If large sums are channelled into trading equities within a short space of time, equity prices may be affected. The equity prices at which one trades will in such case be less favourable than if the trades were spread over a longer period of time. Other market participants may also exploit known information on how an asset manager with a large portfolio changes the composition of its investments upon index changes, which may give rise to large indirect transaction costs. Consequently, the preference for low tracking error needs to be weighed against the level of transaction costs.

The GPFN is a relatively large investor in the Norwegian stock market, and the value of the Norwegian equity portfolio corresponds to about 10 percent of all equities included in the Benchmark Index of the Oslo Stock Exchange. Figure 6.5 shows the number of days it will on average take to purchase or sell index weights of the equities included in the Benchmark Index. The figure is based on the GPFN daily trading in an amount corresponding to 15 percent of the trading volume for each equity. In the most liquid company it will take more than 30 days to trade the equities, whilst it will take more than 1,400 trading days in the least liquid company. The figure shows that it is challenging for an investor of the GPFN’s size to adjust its portfolio to index changes within a short space of time.

Source Oslo Stock Exchange, Folketrygdfondet and the Ministry of Finance.

6.2.7 Changes to index deviations and market volatility

The limits on expected tracking error for the GPFN and the GPFG apply to a portfolio comprising both equities and bonds. Figure 6.6 shows calculated tracking error for the GPFN as at yearend 2014. A simple model is used to illustrate effects based on the equity and fixed-income portfolio of the GPFN. The model is structured such as to enable adjustment of over- and underweights in Norwegian equities, whilst the fixed-income portfolio and the Nordic equity investments are based on actual developments in GPFN investments.

Source Folketrygdfondet and the Ministry of Finance.

Based on market developments over the years 2012–2014, expected tracking error for the GPFN is estimated at about 0.5 percentage points as at yearend 2014.

Applying equity and bond price developments for the years 2007 to 2009, result in an increase in estimated expected tracking error to 1.0 percentage point.

At yearend 2014, the overlap between the Norwegian equity investments of the GPFN and the benchmark index was about 91 percent, up from 83 percent at yearend 2008. If one doubles, as a technical calculation exercise, the deviations in the Norwegian equity portfolio at yearend 2014, the overlap between the portfolio and the benchmark index is at more or less the same level as at the end of the financial crisis. Market developments over the years 2012–2014 mean that tracking error for such a portfolio would have been about 1.1 percentage points for the Fund as a whole. Given market developments during the financial crisis, estimated tracking error would have been close to 2.1 percentage points.

7 Statistical analyses of performance in the Government Pension Fund Norway

7.1 Introduction

This topic article provides a more detailed analysis of Folketrygdfondet’s performance in managing the GPFN, cf. the discussion in section 3.2. The emphasis is on shedding light on the following issues:

• Has Folketrygdfondet achieved excess return over time?

• Which factors have contributed to the excess return?

• What have been the implications of asset management for the ratio between risk and return in the GPFN?

The analyses have a special focus on the Norwegian equity and fixed-income portfolios of the GPFN, since these constitute 85 percent of the benchmark index defined by the Ministry of Finance.

The GPFN has a long time horizon for its investments. Performance is therefore evaluated over long periods. The analysis emphasises the period since 2007, when a new framework for the management of the GPFN was established. Since a review of Folketrygdfondet’s asset management was conducted in 2010, analyses of performance over the last four years are also presented.

Folketrygdfondet’s management of the GPFN has delivered excess return over the period 1998–2014, as well as in the sub-periods 2007–2014 and 2011–2014, cf. table 7.1. Both the equity and fixed-income management have contributed to the excess return.

Statistical tests can be used to analyse whether the excess return was coincidental or can be attributed to Folketrygdfondet’s management. Table 7.1 shows that there is a high probability that the excess return can be attributed to Folketrygdfondet’s management, and that the performance is not a coincidental occurrence.1

7.2 Contributions to excess return

Financial research has shown that tilting investments towards assets with certain characteristics such as low market value, relatively low pricing and low liquidity, has historically delivered a higher return than a market-weighted portfolio. Such characteristics are often labelled factors, and systematic tilting of investments towards such assets is called factor strategies. It is more uncertain whether factor strategies have generated excess return in the Norwegian market.2

Return fluctuations as a result of factor strategies may deliver negative excess return relative to the benchmark index for extended periods of time, and such deviations may be amplified during periods of market slumps. Consequently, significance of factors in generating excess returns may say something about the risk profile of the GPFN. The risk associated with security selection is, on the other hand, company-specific and more readily reduced by spreading the deviations from the index over a number of securities.

By calculating the extent to which excess return has fluctuated in correlation with factors, one can seek to gain insight into the significance of such factors in determining achieved risk and return. Correlation may be the result of specific factor strategies, but may also reflect the sum total of a number of company-specific security selections.

The analyses below make use of four factors that were also applied in reviewing Folketrygdfondet’s asset management in 2010, cf. Report No. 15 (2010–2011) to the Storting – The Management of the Government Pension Fund in 2010.3 Two of the factors used in the analysis, size and value, relate to the stock market.4 The size factor measures the difference in returns between companies with low and high market value. The value factor measures the difference in returns between companies with high book value relative to market value and companies with low book value relative to market value. The factors credit and term are used for the bond market.5Credit is a measure of the difference in returns between more risky corporate bonds and government bonds. Term measures the difference in returns between government bonds with a long time to maturity and short-term treasury bills.

Table 7.2 shows that the excess return in the Norwegian equity portfolio has been negatively correlated with the size factor over the period from 1998. Consequently, it would appear that Folketrygdfondet has invested more in equities with high market value and less in equities with low market value compared to the benchmark index. The same tendency can be attributed to asset management since 2011. However, correlation with factors has declined over the period since 2007 as a whole. Varying degrees of correlation over time may indicate that Folketrygdfondet has exploited time variation in stock market factor premiums.

Folketrygdfondet notes, in a letter of 16 December 2014, that it has historically been emphasised that the risk taking in the equity portfolio shall be dominated by company-specific characteristics. Equity management aims to generate excess return by, inter alia, investing in so-called quality companies and avoiding high-risk companies. Such a strategy may result in a larger portion of companies with high market value. The correlation with the size factor in table 7.2 is in conformity with the declared strategy of Folketrygdfondet. The strategy may, at the same time, entail a larger portion of companies with relatively low equity prices, thus resulting in a positive correlation with the value factor. However, the excess return has registered a correlation that is negative or close to zero with the value factor over the analysed periods, and significantly negative over the period since 2011.

Folketrygdfondet notes that its equity management may also result in a larger portion of equities with low return volatility and lower liquidity than the benchmark index. No quantitative analysis of these factors has been carried out due to a lack of data in the Norwegian market.

The findings in table 7.2 indicate that the Norwegian fixed-income portfolio has historically involved periods of interest rate and credit risk that differ significantly from the benchmark index. The findings also show that the importance of these factors has varied over time, which would be in conformity with a strategy of exploiting time variations in bond market factors. The deviations from the benchmark index are consistent with a larger portion of corporate bonds in the GPFN since 2007.

The overall significance of factors in explaining excess return may say something about the risk profile of the GPFN. The factors size, value, credit and term can explain about 20 percent of the fluctuations in the excess return on the GPFN since 2007. The limited significance of factors suggests, in line with Folketrygdfondet’s assessments in its letter of 16 December 2014, that company-specific security selection is an important part of the asset management strategy.

7.3 The relationship between risk and return

The risk in the GPFN is predominantly determined by the benchmark index defined by the Ministry. Volatility in the benchmark index can explain more than 99 percent of the volatility in the GPFN over the period since 2007, cf. table 7.3.

Consequently, volatility resulting from deviations from the benchmark index represents a small fraction of overall volatility in the GPFN. Different deviations from the benchmark index may, at the same time, have different risk profile implications. No single measure captures all aspects of risk. The absence of one single measure makes it appropriate to use several methods and approaches. Investors may also differ in their risk preferences and in the weight they attach to different risk measures. Table 7.4 presents several different risk measures and the ratio between risk and return in the GPFN.

Standard deviation is a commonly used risk measure, and says something about the distribution of returns around the mean. The same weight is attached to returns that are both lower and higher than the mean. Calculating standard deviation for those periods when returns have been negative only, provides an indicator of so-called downside risk. Such a risk measure assumes that investors only attach weight to the probability of loss. Table 7.4 shows that both the standard deviation in the GPFN and the downside risk have been lower than for the benchmark index over the period since 2007.

The Sharpe ratio is a measure of the relationship between return in excess of the risk-free rate, on the one hand, and risk as measured by standard deviation, on the other hand. Consequently, the Sharpe ratio is a measure of the compensation for carrying risk, and shows the return achieved for each percentage point of risk undertaken.6 If the Sharpe ratio is higher for the GPFN than for the benchmark index it may suggest that asset management has served to improve the ratio between risk and return. The findings in table 7.4 show that the Sharpe ratio is higher for the GPFN than for the benchmark index in all periods.

Active management strategies may, in general, involve a risk of periods of large losses. It is commonly assumed that investors wish to limit such risk and that they prefer more smoothly distributed returns. Figure 7.1 shows the distribution of returns on the GPFN since 2007, as compared to a normal distribution with the same average and standard deviation. The figure shows that losses are incurred more often and are larger than would be suggested by a normal distribution. A so-called adjusted Sharpe ratio seeks to take account of such return asymmetries, to obtain an estimate of the relationship between risk and return that attaches more weight to such losses.7 Table 7.4 shows that asset management has improved the ratio between risk and return, also when taking such risk into consideration.

¹ The figure shows monthly return frequency since 2007 in percent (blue), and a normal distribution with the same average and standard deviation (grey).

Source Folketrygdfondet and the Ministry of Finance.

7.4 References

Johnsen, T. (2011). Evaluation of active management for the Government Pension Fund Norway (In Norwegian only. Norwegian title: Evaluering av aktiv forvaltning for Statens pensjonsfond Norge), www.regjeringen.no/spf

Nagy, Z., Sørensen, L. Q. (2010). Report on active management of the Norwegian Government Pension Fund – Norway, www.regjeringen.no/spf

Næs, R., Skjæltorp, J. A., Ødegård. B. A. (2008). Which factors drive price developments on the Oslo Stock Exchange (In Norwegian only. Norwegian title: Hvilke faktorer driver kursutviklingen på Oslo Børs), Norsk Økonomisk Tidsskrift, 123, 36-81.

Pezier, J., White, A. (2006). The Relative Merits of Investable Hedge Fund Indices and of Funds of Hedge Funds in Optimal Passive Portfolios, ICMA Centre Discussion Papers in Finance, Henley Business School, Reading University.