What if alpha is just polished beta?

On asset allocation and fund of funds.

15th September 2009

Abstract
What is the appropriate level of portfolio allocation towards fund of hedge

funds? The well-known core-satellite approach would give a number around 5%

or 10%, fund of hedge funds being the satellite allocation. The core allocation

should be given to often low-fee, passively managed, classical beta exposure

like equity and bonds. The core-satellite approach, however, is patently absurd

if the satellite mostly consist of beta exposure.

The aim of this article is two-folded. Firstly, we unveil the beta exposure of

fund of hedge funds with the application of a standard linear replication model.

Secondly, with the transparency of these replication portfolios we investigate

what the role fund of hedge funds should be in an investor's portfolio.

1
 What is the appropriate level of portfolio allocation towards hedge funds?

Many would give a number around 5% or 10%. The rational for this number

often steams from the core-satellite approach. This advocates that the core

allocation of the portfolio should be given to often low-fee, passively managed

equity and bond exposures. The satellite is the allocation given to actively

managed funds (e.g. hedge funds) which will ideally add uncorrelated and

absolute returns. To put it in a dierent asset management lingua, the core

is the beta allocation and the satellite is the alpha allocation. Fund of hedge

funds (hereafter fund of funds) with their claim of absolute returns naturally

belongs to the alpha allocation. This approach however is patently absurd if

the satellite mostly consist of beta exposure.

The objective of the paper is two-folded. With the application of a standard

linear replication model, we rst aim to unveil the beta exposure of fund

of funds. Secondly, with the transparency of these replication portfolios we

investigate what the role fund of funds should then be in an investor's portfolio.

We here choose to look at fund of funds since it is the primary investment

1
vehicle asset managers' use to access hedge funds . Few investors indeed dare

to pick their own hedge fund managers and quite often rely on the services of

fund of funds managers.

Several articles, e.g. Fung and Hsieh [2004] and Ennis and Sebastian [2003],

have provided evidence that fund of funds are exposed to a limited set of

market factors. Building on this other articles as Jaeger and Wagner [2005] and

Hasanhodzic and Lo [2007] use extracted hedge fund exposure to implement

portfolios which aim to replicate hedge fund returns. These latter portfolios

give a benchmark to fund of funds return and also give us a rough idea of

2
the beta exposure of fund of funds. This is where we begin to unveil the beta

exposure of hedge funds.

Methodology and data

Our approach to hedge fund replication is based on a parsimonious linear

regression model and follows to a large extent previous work in the eld. The

replication portfolio, or clone, is re-weighted at the end of each month (t)

over the sample period. The portfolio weights are estimated on a 24 month

rolling-window on in-sample data using a relaxed form of Sharpe's [1992] style

analysis (SSA) model. The portfolio is then held over the consecutive month

on out-of-sample data. The model to estimate portfolio weights is dened as

K
X
rt = wi fi,t + λt (1)
i=1
K
X
s.t. wi = 1 (2)
i=1

Where rt is the monthly returns of the fund of funds, K is the number of

factors, wi is the factor exposure towards the monthly return from factor fi,t ,

and λt is the part of the monthly returns, unexplained by the factors. The

constraint give a natural portfolio interpretation and unlike Sharpe's original

model this model allows for short-selling since it falls more in line with hedge

funds strategies. Hedge funds report their monthly returns to data vendors

with some weeks delay. To account for this we lag the in-sample data one

2
month .

As in Hasanhodzic and Lo (2007) we allow for leverage which adjust the

3
clone to yield an approximately equal volatility as its fund of funds. The

leverage factor, denoted γt , is simply the volatility of the fund of funds divided
3
by the volatility of the clone estimated on an ex ante basis .

Our approach to the selection of factors which are to explaining fund

of funds returns dier somewhat from the literature. Besides representing

plausible allocation of fund of funds we want the factors to be deeply liquid

and than almost to a fault a plain vanilla type of factor. The set of factors is

4
more precisely :

1. BXM, the CBOE S&P 500 BuyWrite total return index.

2. RSL2000, the Russell 2000 index return.
3. EAFE, the MSCI EAFE (Europe, Australasia, and Far East)
total return index.
4. BOND, the Barclays Capital U.S. Aggregate Corporate AA Bond
total return index.
5. CMDTY, the S&P Goldman Sachs Commodity total return index.

BXM and RSL2000 are supposed to capture US large and small-cap eq-

uity exposure, and EAFE captures equity exposure towards some parts of the

developed world from a US investor perspective. The BOND and CMDTY

both have natural interpretations as well as representing important exposure

of hedge funds. Table 1 present some summary statistics of the factors.

We have not included any credit, duration or forex factors since these dier

somewhat from the factors in the list above. These factors are constructed by a

long and short position in two dierent securities, often with the net investment

being zero. It is then not to be expected that these factors are leveraged in

the same fashion as an equity factor. A better specied replication model

4
 is therefore required to include these factors. Excluding these, however, from

the clone portfolio should rather bias the performance downward than upward.

Since we aim to have a simple replication solution, as to complement a standard

and traditionally balanced portfolio, we do not nd these long-short factors

appropriate.

Table 1  Summary performance statistics of the factors. (January 1990December 2008)

Performance statistics. (Annualized)

Mean S.D. Sharpe
CBOE S&P 500 BuyWrite 0.079 0.108 0.73
Russell 2000 0.057 0.204 0.28
MSCI EAFE 0.031 0.174 0.18
S&P GSCI CMDTY 0.039 0.219 0.18
Barclays U.S. agg. corp. AA bond 0.080 0.057 1.41

Correlation (%) matrix.

BXM RSL2000 EAFE CMDTY BOND
CBOE S&P 500 BuyWrite 100 76 60 18 15
Russell 2000 100 62 14 11
MSCI EAFE 100 17 17
S&P GSCI CMDTY 100 −2
Barclays U.S. agg. corp. AA bond 100

The hedge fund data sample we use is collected from the Hedge Fund

Research (HFR) database over the period January 1990 to December 2008.

We only collect fund of funds operating as of December 2008. Some fund of

funds are ltered out from the sample based on the criteria that they have to

be denominated in USD, report returns net of all fees, report AUM, and have

a track record which is longer than or equal to 37 month. For funds with the

same share class, the one with the longest history or domiciled in the US is

selected. These lters nally leave a sample of 885 fund of funds.

Including only operating fund of funds introduce survivorship and selection

bias. While acknowledging this, the main objective is to study the relative

5
performance of clones and fund of funds. As documented by among others

Fung and Hsieh [2000] and Liang [2000] the biases tend to introduce an upward

bias on average returns, which should imply that the linear replication model

is up for a more dicult task. Again, the objective of the article is to study

the relative performance between fund of funds and linear clones thus these

biases should aect them in similar ways.

HFR categorize each fund of funds in their database as either (sample

size in brackets) Conservative (224), Diversied (351), Market Defensive (29),

or Strategic (281). More specically, Conservative refers to fund of funds

which invest in low volatility strategies as equity market neutral, xed income

arbitrage, and convertible arbitrage. Diversied fund of funds invest in a broad

range of strategies. Market Defensive fund of funds invest in short-biased and

managed futures funds. Strategic is the most volatile category and is primarily

exposed towards emerging markets, sector specic, and equity hedge funds.

SSA vs. classic regression

The reason to favor SSA instead of classic linear regression is that it gives a

natural portfolio interpretation. However, as highlighted by ter Horst, Nijman,

and de Roon [2004], the constraint in (2) will give biased estimations of the

5
weights .

Assume the linear regression estimation as a reference point to the estima-

tion of fund of funds factor exposures. In order to get an appropriate portfolio

interpretation of this estimation the exposures have to be scaled linearly to

sum to 1. However, the benet of instead imposing a constraint will cause

the portfolio weights to only increase (decrease) the weight(s) to the factor(s)

6
 with the lowest (highest) variance (and covariance). Thus creating a portfolio

with lower risk than if the estimates from linear regression were to be used

(and scaled to sum to 1).

Table 2 give some empirical evidence of this. The table presents the es-

timated weights/exposures from linear regression and SAA. The dependent

variable is the HFRI fund of fund index which is explained by the previously

dened set of factors. For three of the factors the weights/exposures are similar

in magnitude. However for the two factors with the lowest volatility, BOND

and BMX, the weights and exposures from the two models varies signicantly.

Particularly for the BOND factor where the exposure is 0.26 but the weight is

0.62, more than twice as large.

Table 2 SSA vs. classic regression on the HFRI fund of funds index. (January 1990
December 2008)

ALPHA BXM RSL2000 EAFE CMDTY BOND

Factor exposure (Linear regression) 0.25 0.11 0.05 0.05 0.07 0.26
Portfolio weight (SSA) − 0.21 0.04 0.04 0.09 0.62

Clone performance
Given the simplicity of the replication model, the vanilla-style factors, and bi-

ased estimation of weights we conjecture that clones capture the trend rather

than individual performance. Hence it is more interesting to investigate prop-

erties of a basket of clones since large idiosyncratic deviations will cancel out.

Moreover, a basket of clones is close to common practice since investors tend

to invest into several funds of funds.

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 Table 3  Equal-weighted portfolios performance. (February 1992December 2008)

Performance summary
Annualized Monthly basis
Mean S.D. Sharpe VaRM5% ESM5% Min Max Median Kurt. Skew. ρ1 ρ2 TE
Funds 0.072 0.051 1.42 1.7 3.2 −6.0 5.1 0.7 7 −1 39.5∗ 23.5∗
All funds
Clones 0.062 0.073 0.84 3.1 5.0 −11.1 4.9 0.7 9 −1 16.2∗ −2.9 1.46
Funds 0.059 0.033 1.82 1.2 2.4 −5.2 2.5 0.7 16 −3 56.2∗ 37.4∗
Conservative
Clones 0.043 0.047 0.93 1.6 3.2 −8.1 4.7 0.5 17 −2 22.9∗ −4.5 1.12
Funds 0.072 0.052 1.40 2.0 3.4 −6.3 5.2 0.7 7 −1 37.7∗ 24.1∗
Diversied
Clones 0.065 0.074 0.88 2.9 5.0 −10.6 5.2 0.7 8 −1 14.3∗ −4.2 1.62
Funds 0.087 0.048 1.81 1.6 2.0 −2.6 4.4 0.8 3 0 1.6 −11.8
Market Defensive
Clones 0.068 0.081 0.84 2.9 5.6 −10.8 5.8 0.7 8 −1 18.3∗ 11.6 2.32
Funds 0.088 0.081 1.09 2.7 4.5 −8.4 9.1 0.8 6 0 36.8∗ 18.3∗
Strategic
Clones 0.084 0.118 0.71 5.4 7.7 −14.0 9.7 1.0 5 −1 11.4 −5.8 2.95

Correlation (%) towards some market indices. HFRI indices

SP500 BXM RSL2000 EAFE CMDTY BOND FX VIX HFRI FoF Cons. Div. M.D. Strat.
8

Funds 53 49 59 59 34 21 4 −40 89 97 90 95 61 93
All funds
Clones 69 67 66 65 39 57 −23 −58 69 63 60 59 28 63
Funds 49 50 49 54 40 19 −3 −37 75 84 90 80 47 77
Conservative
Clones 57 58 47 56 37 67 −28 −53 53 52 54 48 25 48
Funds 52 48 57 57 33 25 4 −39 88 97 90 94 62 92
Diversied
Clones 68 66 66 63 38 58 −22 −57 70 63 59 59 30 63
Funds 0 −4 9 16 21 30 −16 −2 31 45 38 43 85 34
Market Defensive
Clones 31 30 23 37 43 65 −21 −36 40 48 53 44 30 42
Funds 51 46 60 56 27 13 12 −38 88 94 82 92 55 94
Strategic
Clones 69 66 69 64 34 44 −22 −55 68 58 53 54 22 61
 In this section we have constructed equal-weighted, monthly re-balanced,

portfolios of funds of funds and of the replication solutions, based on either the

full sample or style classication. From an investor's perspective, the equal-

weighted basket of clones is easily implemented with portfolio allocations in

highly liquid assets.

Table 3 presents some summary performance statistics of these equal-

weighted portfolios. Considering the simplicity of the replication process the

distributional return properties of the linear clones are impressive. The return

dierential of portfolios of clones' and fund of funds is in the order of only

1% on an annualized basis. Yet with a higher volatility for the former, which

translates into a more signicant risk-adjusted performance dierential.

The level of tracking error (TE) between clone and fund of funds portfolios

indicate that any précis time-series properties are not captured by the linear

clones. However, we intend to investigate the properties of the fund of funds

vs. linear clones in a portfolio context. Thus the level of tracking error is not

our greatest concern but rather that the general return distributions can be

captured by a simple, linear replication model.

Figure 1 presents the cumulative performance of the equal-weighted port-

folios of clone, and fund of funds, and the S&P 500 index. The clone portfolio

seems to track the fund of funds portfolio fairly well and both has markedly

less volatility then the equity oriented S&P 500.

Turning to the lower panel in table 3, fund of funds and clone portfolios

have similar correlations towards equity and commodity. SSA's tendency to

bias the portfolio towards low-risk assets explains the large dierence in bond

index correlation between clones and fund of funds. As expected correlations

9
 Figure 1  Cumulative returns of clones and fund of funds.

towards the HFRI indices are lower for clone portfolios than fund of funds

portfolios.

The dismal performance of the Market Defensive clone should be seen in

light of the characteristic of the style. Market Defensive fund of funds are

focused on CTA and short biased hedge funds. The managed futures group of

hedge funds is not expected to be captured by our simple replication model

and plain vanilla set of factors.

The extent to what each factor contribute to the performance of clones

is presented in table 4. The rst panel presents the average allocation to

respective factors over the full sample period. There is little variation between

sub-categories. All portfolios constitute more than 50% on average towards

BOND. Any concerns that clone returns are due to leveraging are refuted by

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 Table 4  Portfolio allocation statistics. (%)
Turn
BXM RSL2000 EAFE CMDTY BOND Lvrg. over
All clones 25 7 10 4 54 100 9
Std. (21) (7) (12) (7) (18) (16) (8)

Min −14 −27 −14 −16 10 63 0

Max 78 20 35 28 87 141 44
Conservative 33 −0 5 5 57 68 6
(21) (6) (9) (5) (20) (18) (7)

Diversied 24 8 10 4 54 102 9
(20) (7) (12) (8) (19) (17) (8)

Market Defensive 17 −2 11 6 68 118 15

(30) (11) (12) (5) (25) (21) (18)

Strategic 20 14 17 2 47 136 20
(28) (12) (17) (16) (25) (42) (26)

the second, last column in table 4 which indicate low and often less than 100%

leverage levels. The last column presents the annual turnover rate which is

within a modest range. While this article do not account for trading cost, the

low turnover rate reassures that this should only have modest impact of the

clone performance.

A closer look at table 4 tells us that on average the equal-weighted portfolio

of clones of the Conservative fund of funds was only leveraged to around 70%.

Given the setting this implies that 30 % of the portfolio capital is held in cash

with no returns, which is not a plausible scenario. While the cost of leveraging

is neither accounted for, this is easily accessible at a low cost through margin

accounts.

In an undocumented empirical test we account for the issue of earnings

on cash or cost of leverage by making this accessible through the 3-month

6
Treasury bill rate . This increases the performance of the equal-weighted

clone portfolio of the Conservative style and puts it at par with its respective

11
fund of funds portfolio, only diering with a few basis points annually. On

the other hand it does modestly lower the returns of the other equal-weighted

clone portfolios, in particular the Market Defensive and the Strategic style

which have somewhat higher leverage.

The issue of including or excluding the Treasury bill rate as cost of leverage

brings us to another previously discussed topic of the set of factors which we use

to replicate the fund of funds. In our study we have chosen a broad, vanilla

style, and liquid set of factors. However this have come at the drawback

of some clones replicating fund of funds focusing on for example emerging

markets or xed income. As an example, we conducted the linear replication

using the same set of factors and including the 3-month Treasury bill rate.

As above, when it is included implicitly as earnings on cash, including it

explicitly in the set of factors improves the relative and absolute performance

7
of the Conservative clone equal-weighted signicantly . This highlights the

importance of the selection of factors in replicating hedge fund returns.

To conclude this section; keeping in mind the low turnover rate and the

vanilla type of factors, the model allows us to capture large parts of hedge

fund return distribution characteristics.

Portfolio allocation and linear clones

It should be obvious to the reader by now that fund of funds is signicantly

exposed to bond, equity and commodity risk. Even more, these extracted

exposures are able to fairly well replicate the average performance of fund of

funds. This poses some serious questions on which role fund of funds can play

12
in the investor's portfolio.

In this section we consider an investor which allocate between two portfo-

lios, a Traditional Portfolio and an Alternative Portfolio. The former portfolio

is a static, diversied, monthly re-weighted portfolio with the following allo-

cation:

Asset Allocation
S&P 500 40%
EAFE 20%
BOND 40%
Sharpe (Annual) 0.62

The Alternative Portfolio refers to one of the following three portfolios:

1. Equal-weighted portfolio of fund of funds

2. Equal-weighted portfolio of clones
3. Alpha portfolio: a portfolio long the fund of funds,
short the clone portfolio, and long the 3-month Treasury bill.

The essence of this last portfolio is to extract the return of fund of funds

that cannot be attributed to general market risk premiums, and often referred

to as alpha returns.

Allocation to the equal-weighted fund of funds portfolio is for a number of

reasons not feasible, to mention one: monthly re-weighting is impossible. The

equal-weighted clone portfolio is on the other hand easily implemented. As

mentioned before linear clones seem to be good at capturing the return trend of

fund of funds rather than a precise replication of each fund of funds. We would

therefore not expect any desirable or tractable eects on an individual fund

basis of exchanging fund of funds allocation to its clone. However, comparing

13
clone and fund of funds allocation as we describe it here should give good

indications of their exchangeability in a portfolio context.

We will test whether or not our three alternative portfolios should be in-

cluded to the strategic asset allocation decision process, based on the mean-

variance eciency criterion and using the Traditional Portfolio as a bench-

mark. Britton-Jones [1999] highlights the impact of sampling errors when es-

timating mean-variance ecient portfolio weights. Briey, Britton-Jones test

is based on the ordinary least square (OLS) regression of a constant vector 1

onto a set of asset returns R:

1 = Rw + e. (3)

The vector 1 shall be interpreted as the optimal return time-series with

zero variance and positive constant returns. The objective of the OLS min-

imization is consequently to nd the optimal factor loadings to be as close

tofrom a portfolio perspectiveoptimal positive return with zero variance.

Britton-Jones show in more detail that the OLS estimate of the factor loadings

ŵ = (R0 R)−1 R1 is indeed the unscaled portfolio weights of the mean-variance

tangent portfolio. The signicant benet of deriving the tangent portfolio

in this order is that the estimated weight ŵi has a t distribution under the

hypothesis that ŵi = 08 .

Table 5 presents the scaled (i.e. w̄i = ŵi /ŵ0 1) weights from the regression

in (3) and the associated standard errors (S.E.) and t-statistics. The rst and

second column presents the results with the fund of funds or clones portfolio

14
Table 5  Britton-Jones (1999) test of portfolio weights. (February 1992December 2008)

Fund of funds Clones Alpha Port.

All fund of funds

Weights (%) S.E. Weights (%) S.E. Weights (%) S.E.
Traditional Portfolio −9.86 12.5 −6.19 38.5 36.85∗∗ 7.2
(t-stat.) (−0.79) (−0.16) (5.14)

Alternative Portfolio 109.86∗∗ 21.3 106.19∗ 48.1 63.15∗∗ 11.6

(5.17) (2.21) (5.42)

Sharpe ratio 1 .47 0 .90 1 .53

Conservative
Weights (%) S.E. Weights (%) S.E. Weights (%) S.E.
Traditional Portfolio −8.22 6.0 −0.22 19.5 24.60∗∗ 4.8
(t-stat.) (−1.38) (−0.01) (5.17)

Alternative Portfolio 108.22∗∗ 15.3 100.22∗∗ 38.0 75.40∗∗ 10.6

(7.08) (2.64) (7.10)

Sharpe ratio 1 .90 0 .97 1 .91

Diversied
Weights (%) S.E. Weights (%) S.E. Weights (%) S.E.
Traditional Portfolio −9.13 12.9 −10.55 36.6 37.29∗∗ 7.5
(t-stat.) (−0.71) (−0.29) (4.97)

Alternative Portfolio 109.13∗∗ 21.6 110.55∗∗ 45.2 62.71∗∗ 12.2

(5.06) (2.44) (5.12)

Sharpe ratio 1 .45 0 .93 1 .46

Market Defensive
Weights (%) S.E. Weights (%) S.E. Weights (%) S.E.
Traditional Portfolio 10.68 7.5 22.93 25.6 44.99∗∗ 11.8
(t-stat.) (1.42) (0.90) (3.81)

Alternative Portfolio 89.32∗∗ 13.2 77.07∗∗ 29.9 55.01∗∗ 14.0

(6.75) (2.58) (3.92)

Sharpe ratio 1 .87 0 .93 1 .21

Strategic
Weights (%) S.E. Weights (%) S.E. Weights (%) S.E.
Traditional Portfolio 9.04 23.6 35.79 56.6 58.68∗∗ 14.8
(t-stat.) (0.38) (0.63) (3.96)

Alternative Portfolio 90.96∗∗ 25.8 64.21 44.2 41.32∗∗ 13.5

(3.52) (1.45) (3.06)

Sharpe ratio 1 .12 0 .80 1 .04

15
as the Alternative Portfolio. Weights are similar between these groups where

allocation is close to 100% for the Alternative Portfolio and consequently close

to zero allocation of the Traditional Portfolio. The dierences of weights are

most pronounced for the Strategic and Market Defensive, which is expected

from previous analysis. The hypothesis that the weight of the Traditional

Portfolio is equal to 0, and thus exclusion from a mean-variance ecient port-

folio, cannot be rejected at the 95% in any of the sample groups. The size

of weights as well as the t-statistics strongly suggest that it is sucient to

only allocate to the equal-weighted fund of funds or clone portfolio. Taken

together with previous results, the striking similarities of weights emphasize

the anity between linear clones and their respective fund of funds. The null

hypothesis of zero weights is rejected on the 99% level for fund of funds across

all styles. The signicance level to reject the null hypothesis is equally high

for most clone portfolio weights, except for the aggregate of all clones which

have a 95% signicance level and the Strategic which is not signicant.

Thus, from a mean-variance ecient perspective, rather than complement-

ing a portfolio of traditional assets, fund of funds and clones seems to replace

them. Beta exposure is suciently prevalent in fund of funds (and obviously

in the clones) that there is no need to keep the traditional long-only portfolio.

This conrms the result of the previous section. This also in contrast to the of-

ten used, and advocated approach, to include hedge fund investment through a

core-satellite approach where allocations to fund of funds often arrives around

5%.

The results in table 5 for the Alpha Portfolio shows that the t-statistics

in most cases reject any of the hypothesis that the weight of the Traditional

16
Portfolio or the Alpha Portfolio should be equal to 0. The sizes of the mean-

variance ecient weights are also dierent from before, where the size of the

two assets are in many cases close to equal.

The lower levels of Sharpe ratio for the clone portfolios relative to the fund

of funds portfolios goes in line with the ndings of the previous section. There

is a notable similarity of the level of Sharpe ratio between Alpha Portfolios and

fund of funds portfolios. Again, this stress that if the beta exposure of fund

of funds are not hedged, there is no need to give allocation to the long-only

Traditional Portfolio. At the same time, hedging the beta exposure seems to

imply the allocation conguration of the core-satellite approach.

Conclusion
The empirical results in this article suggest that linear fund of funds clones to

a large degree are interchangeable with its constituent fund of funds in an asset

allocation context. The pressing question is then why this is the case? The

answer is in the pudding, we believe. After fees, fund of funds in aggregate are

well diversied investment vehicles exhibiting modest levels of time-varying

risk exposure.

In aggregate linear clones captures the general performance of fund of funds

and they also give information to the allocation and market risks of fund

of funds. This transparency given by clones, and the results from the last

section, clearly reveals the shortcomings of using a core-satellite approach to

fund of funds investment. These are not sucient alpha generators to be

considered as a satellite asset. Given our results, it falls more natural to only

17
give allocation to the fund of funds; if not the market risk is hedged. Yet it has

to be emphasized that this analysis disregard signicant liquidity constraints,

opacity, and large fees associated with fund of funds investment. The two rst

proved very costly for investor during the credit crunch.

This suggests that funds of funds on aggregate do not contribute with new

sources of uncorrelated returns but rather classic, slightly time-varying, risk

exposure. This is by itself either a strong argument against the prevailing asset

allocation idea that fund of funds should only be given a marginal allocation or

a clear indication that fund of funds on average do not expose their investors

to enough pure alpha.

Endnotes
1 According to Hedge Fund Research Industry Report Q2-2008 the fund of funds
managed in aggregate 43% of all hedge fund assets.
2 I.e. at time t we estimate portfolio weights using data over [t − 25, t − 1]
√P
3 More precisely γ := σ = q , where σr is the volatility of the
25 2
rt k=2(r −r̄ ) /23
t−k t
t σ∗ 25 ∗ 2 t
rt
(r −r̄ ) /23
∗
P

fund and σr is the volatility of the clone.

k=2 t−k t
∗
t

4 All the data for these factors are downloaded from Datastream.

5 It is often incorrectly assumed that the exclusion of the intercept in SSA have a

big impact on the estimation process as compared to linear regression. Omitting the
intercept does in fact not change the solution at all, see Becker [2003]
6 These results are available on request.

7 These results are available on request.

8 See Britton-Jones [1999] Corollary 1, p 663

18
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