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SIGHT
The Equity Imperative
IMPROVING ACTIVE
RISK BUDGETING
Investors recently have begun to rethink how they take and manage risk in their portfolios,
and many have reduced their allocations to managers that use high levels of active risk in
favor of index strategies. Those investors taking this approach clearly do not believe they are being
appropriately compensated for the risks their active managers are taking, leading them to
limit or eliminate active risk from their portfolios. This approach is reflective of the misconception
that overall portfolio risk can be managed best by allocating a significant portion of the portfolio
to index strategies, while incorporating managers with a high level of active risk is the primary
way to generate outperformance. In reality, taking a different approach will often better position
investors to generate outperformance and target objectives that are not performance related.
How? Using strategies that provide strong risk-adjusted returns by focusing on compensated risk
factors, such as value, low volatility and momentum. In fact, strategies that take on high levels
of active risk tend to be relatively inefficient and often result in low risk-adjusted performance.
The following research reveals a better way to maximize risk-adjusted returns.
May 2014
We hope you enjoy the latest presentation from Northern Trust’s Line of Sight. By providing research, findings,
analysis and insight on the effects and implications of our changing financial landscape, Line of Sight offers the
clarity you need to make better informed decisions. The Equity Imperative research series provides new insights
around the evolving equity landscape to help you navigate a better route to achieving your investment objectives.
northerntrust.com | Improving Active Risk Budgeting | 1 of 12
The works of the greatest minds in finance reveal a common prescription: to achieve the
best results, investors should maximize risk-adjusted returns1. Harry Markowitz2 led the way
when he showed that a portfolio is only efficient if it has the highest possible return for a
given level of risk. William Sharpe3 formalized the relationship between equilibrium risk
and return in the celebrated capital asset pricing model (CAPM), while Eugene Fama
and Kenneth French4 demonstrated ways to improve returns per unit of risk with style
factors. The academics’ mantra: optimize risk-adjusted returns.
Do investors actually take this advice? When asked, most investors we work with say
they do. However, a quick review of their asset and risk allocations often suggests otherwise.
The reality is that while investors may want an efficient portfolio and can obtain an
efficient portfolio in principle, many obstacles stand in the way of achieving this goal.
THE BIAS TOWARD ISOLATING RISK
Research suggests investors may have mental biases that prohibit them from behaving
perfectly rationally when allocating risk across their portfolios. Of primary importance
is a natural tendency to isolate risk in small pockets of the portfolio, while keeping the
remainder for lower risk strategies. Behavioral economist Richard Thaler5 suggests this
cognitive bias is essentially a mental containment strategy that leads to the implicit but
ultimately false notion that total risk is somehow reduced if it is concentrated.
The problem is perhaps most acute with active risk, which often is characterized in
portfolios by significant concentration. For example, most active investors maintain at least
some passive exposure in their portfolios, suggesting a segregation of assets into high and
low active risk. Those that don’t, typically employ varied levels of active exposure across
allocations – also indicating heterogeneity and, thus, relative concentration in the total
active risk budget. Indeed, risk isolation and concentration is so pervasive in the active
space that it goes by several formal names, such as core-satellite, core-completion or barbell.
No matter what it is called, following this type of strategy immediately causes something
of a paradox. In a mental effort to control risk through concentration, investors allocate
small portions of their portfolios to risky strategies. However, to meaningfully outperform
a benchmark, the active risk taken by this fraction of total assets must be quite high. And
managers with such high levels of active risk tend to have lower risk-adjusted returns.
While active risk concentration may be intuitively appealing, the approach ultimately is
inefficient and fails to achieve desired investment objectives. Most importantly, it fails
to maximize risk-adjusted returns (i.e., information ratios) and, thus, is not consistent with
established financial theory.
Instead, an optimal active risk budget should:
1. Target managers with the highest risk-adjusted returns, and
2. Distribute risk relatively evenly throughout the portfolio without
distinct concentration.
The result is an investment allocation that meaningfully outperforms a stated benchmark
while significantly reducing the probability of underperforming.
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The reality is that while investors
may want an efficient portfolio
and can obtain an efficient
portfolio in principle, many
obstacles stand in the way of
achieving this goal.
DIMINISHING RISK-ADJUSTED RETURNS
A phenomenon firmly established in the financial literature is that the information
ratios of long-only managers tend to decrease as their active risk increases. This is the
direct result of long-only managers’ limited ability to express negative views on stocks
while their expression of positive views is generally unrestricted. This asymmetry becomes
more pronounced as active risk increases and causes the managers’ excess return per unit
of risk (tracking error) to decline.
For example, active equity managers overweight and underweight stocks based on each
stock’s expected excess returns. Managers that have a long-only constraint and, hence, cannot
short stocks, are limited in their ability to underweight a specific stock by the stock’s
allocation in the benchmark portfolio. At low tracking-error levels, this constraint may not be
overly restrictive, and will still allow managers to express a full range of positive and negative
views. However, as tracking error increases, the limitation on underweighting securities
has a bigger impact and the managers’ active weights will begin to deviate from their
forecast of expected excess returns. In other words, long-only managers can overweight
their favorite names as much as they like, but can only fully underweight their least
favorite names in a few cases where the benchmark weight is large enough to accommodate
the desired underweight. Since the overweights and underweights must balance each other
out, the no-short constraint effectively hinders a manager’s ability to express a strong bullish or
strong bearish view.
This result was demonstrated empirically by Alford, Jones and Winkelmann6. Using
data on 1,052 large-cap U.S. equity managers over 13 years, they broke managers into two
groups based on tracking error. The first group, which they called Structured Managers,
contained managers with an annual tracking error between 100 basis points (bps) and
300 bps. The second group, known as Traditional Managers, was composed of managers
with an annual tracking error between 500 bps and 1500 bps. The authors ignored managers
whose tracking error fell between 300 bps and 500 bps because they felt these managers
were too difficult to classify. (According to the authors, the results were not sensitive to
the exclusion of these managers.) Their results are shown in Exhibit 1 (on page 4).
DIMINISHING RISK-ADJUSTED RETURNS: THE FUNDAMENTAL LAW OF ACTIVE MANAGEMENT
The concept of diminishing risk-adjusted returns was
formally proven in the landmark paper by Clarke, de Silva and
Thorley7. Their expression for the information ratio is now
widely known as the Fundamental Law of Active Management:
IR = IC * TC * √N
(IR is the information ratio, IC is the information coefficient, TC
is the transfer coefficient and N is the number of active bets taken.)
For an active equity manager the information coefficient
(IC) reflects the correlation between forecasted returns and
realized returns. This is a measure of the investment manager’s
skill in predicting stock returns. More importantly, the transfer
coefficient (TC) is the correlation between the portfolio’s active
weights and the forecasted returns. The information ratio is
maximized when the transfer coefficient (which could be as
high as 1 or as low as -1) is as large as possible.
By limiting the manager’s ability to express his or her views,
the no-short constraint effectively limits the transfer coefficient.
As tracking error increases, the correlation between active
weights and expected returns is reduced. Thus, the transfer
coefficient is reduced causing total information ratio to decline,
all things being equal. (While it is beyond the scope of this
paper, hedge funds do not have this “no short” constraint, so
they are potentially in a better position to express their views
within their long/short portfolios.)
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EXHIBIT 1: TRACKING ERROR AND PERFORMANCE — EMPIRICAL RESULTS
Managers with higher tracking error tended to have lower information ratios.
INFORMATION RATIO
0.5
0.4
0.44
0.3
0.2
0.28
0.26
0.1
0.0
0.05
0.27
0.07
–0.02
–0.1
–0.2
Average
Median
Top Quartile
Structured Managers (Low Tracking Error)
STRUCTURED MANAGERS
(LOW TRACKING ERROR)
–0.16
Bottom Quartile
Traditional Managers (High Tracking Error)
AVERAGE
MEDIAN
TOP
QUARTILE
BOTTOM
QUARTILE
ACTIVE RETURN (bps)
43
52
92
– 4
TRACKING ERROR (bps)
209
221
266
147
0.26
0.28
0.44
– 0.02
INFORMATION RATIO
TRADITIONAL MANAGERS (HIGH TRACKING ERROR)
ACTIVE RETURN (bps)
53
53
201
– 120
TRACKING ERROR (bps)
821
767
971
619
INFORMATION RATIO
0.05
0.07
0.27
– 0.16
Source: Northern Trust Quantitative Research using data from Alford, Jones and Winkelmann.
Data is gross of fees.
While high tracking-error managers have higher active returns, their risk-adjusted
performance clearly lagged that of low tracking-error managers, just as predicted by the
fundamental law of active management (see Diminishing Risk-Adjusted Returns: The
Fundamental Law of Active Management, on page 3). The average information ratio of
the low tracking-error group is 0.26 while that of the high tracking-error group is just
0.05. The respective medians are 0.28 and 0.07. Our analysis of data over the last 10 years
reaffirms these findings, showing that lower tracking-error managers tend to have higher
information ratios than higher tracking-error managers.
Perhaps even more interesting is the dispersion in performance. The top-quartile
structured managers had an active return of 92 bps while the bottom quartile had an
active return of negative 4 bps. In contrast, the top quartile Traditional Manager outperformed
by 201 bps but the bottom quartile underperformed by 120 bps. While these results are
consistent with the tracking error classification, it does highlight that manager selection is
much more important for high tracking-error Traditional Managers because the range of
potential outcomes is much wider. Further, the severity of potential underperformance is
clearly reduced with low tracking-error Structured Managers.
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It appears the natural tendency to concentrate risk may have led investors down the
wrong path. Instead of enhancing the active risk budget, concentration has caused them
to pursue investments that do not maximize risk-adjusted returns.
IMPROVING RETURNS AND REDUCING RISK
If risk concentration is synonymous with high tracking error and, therefore, lowinformation-ratio managers, can it possibly be part of an optimal risk budgeting structure?
Borrowing from Markowitz’s idea of the efficient frontier, we can look at an optimal
risk budget using an efficient frontier of active risk. Each point on this frontier would
represent the maximum excess return per unit of tracking error. This would show us the
highest information ratio attainable for a given level of risk8.
Much like the Markowitz efficient frontier, the active efficient frontier (the grey line
in Exhibit 2) is concave due to the Fundamental Law of Active Management and the
diminishing information ratio phenomenon discussed previously. As you move up and
to the right along the frontier, both expected excess returns and tracking error increase,
but the information ratio declines because tracking error increases at a faster rate. For
example, in Exhibit 2 the portfolio labeled “A” on the efficient frontier has a lower expected
excess return and tracking error than the portfolio labeled “B,” but portfolio A (or any point
to the left of portfolio B on the active efficient frontier) has a higher information ratio.
PORTFOLIO EXCESS RETURN
EXHIBIT 2: EFFICIENT FRONTIER OF ACTIVE MANAGEMENT
C
A
Levered Portfolio A
Levered Portfolio B
B
PORTFOLIO TRACKING ERROR
Source: Northern Trust Quantitative Research
One of Markowitz’s great insights was that managers can use leverage to achieve higher
portfolio returns without necessarily increasing risk or, conversely, a portfolio’s risk may
be reduced without affecting return. This concept applies equally to the active portfolio,
as Exhibit 2 demonstrates. Applying leverage to any portfolio would generate a new linear
frontier that connects the origin at the bottom left corner with the portfolio’s point on the
original frontier. For example, by applying leverage to portfolio A we obtain a new efficient
frontier, as shown in portfolio C, that extends above the original frontier (the green
line) as you move to the right of portfolio A. Similarly, portfolio B can be levered so that
its new efficient frontier (red line) extends above the original frontier to the right of B.
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Note that the new linear efficient frontier formed by leveraging portfolio A, with
its high information ratio, always achieves a higher return per unit of risk than the linear
frontier formed from the low-information-ratio portfolio B. For example, portfolio A
could be levered to achieve portfolio C, which has the same tracking error as portfolio
B, but with much higher expected excess returns. Markowitz concluded that if leverage
is available, low information ratio portfolios should be expressly avoided because they are
inefficient from a risk/return perspective.
Of course, if the modern portfolio theory results are to be anything more than just
academic, we must reconcile the use of leverage in its framework with an active risk
budgeting approach. If investors cannot use or choose not to use leverage, can they still
benefit from this concept?
IMPLICIT LEVERAGE
To illustrate the use of implicit leverage in a risk budgeting framework, consider a simple
scenario in which we are deciding between two active managers. One has a high
information ratio of 0.7, but a lower expected excess return of 2.0% (Manager A).
The other has a higher expected excess return of 4.0% but a lower information ratio
at 0.4 (Manager B). Which of these managers should we select?
Exhibit 3 (on page 7) details the excess return and tracking error metrics for the two
options. Tracking error is the expected tracking error of the total equity allocation relative
to the benchmark index. A traditional risk concentration strategy might place 20% of
the total equity allocation with the active manager and the remaining 80% in something
with less active risk, such as indexing. Manager B’s expected excess return is 0.80%, with
an expected total equity allocation tracking error of 2.00%. These results are highlighted
under the Low IR section.
However, Manager A can achieve the same expected portfolio excess return with significantly
less tracking error by simply increasing the active allocation from 20% to 40% and decreasing
the index allocation from 80% to 60%. At this allocation Manager A also provides an
expected total equity allocation excess return of 0.80%, but with an expected total equity
allocation tracking error of just 1.14% – which is 43% less risk than Manager B’s tracking
error of 2.00%. These results are highlighted in the High IR section, in Exhibit 3.
Alternately, suppose we wanted to target the same 2.00% total equity allocation
tracking error as Manager B in the first example. What level of excess return could we expect
to achieve by using Manager A? As can be seen under the High IR section, a 2.00% total
equity allocation tracking error would yield an expected excess return of 1.40%. To achieve
this we simply must raise the active allocation from 20% to 70% as shown in the yellow
highlight. This increase in excess return from 0.80% (Manager B) to 1.40% (Manager A)
represents an increase of 75%.
In short, our allocation decision acts as implicit leverage within our active risk budget. By
increasing the active allocation we can effectively “lever up” managers with higher information
ratios and lower expected excess returns to obtain higher total equity allocation excess returns
and/or a lower tracking error. In this sense we can move up or down along the capital
allocation line, achieving better risk adjusted results than with our lower information ratio/
higher alpha manager (Manager B).
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If the modern portfolio theory
results are to be anything more
than just academic, we must
reconcile the use of leverage in
its framework with an active risk
budgeting approach.
EXHIBIT 3: INCREASING THE USE OF MANAGERS WITH HIGH INFORMATION RATIOS
TO IMPROVE EFFICIENCY
Increasing the portion of the portfolio allocated to the high information ratio strategy and reducing the
portion allocated to index strategies has the same effect as levering the high information ratio strategy
portion of the portfolio without actually using leverage.
HIGH IR
(MANAGER A)
ASSET ALLOCATION
LOW IR
(MANAGER B)
INDEX
ACTIVE
PORTFOLIO
EXCESS
RETURN
0%
100%
2.00%
2.86%
4.00%
10.00%
10%
90%
1.80%
2.57%
3.60%
9.00%
20%
80%
1.60%
2.29%
3.20%
8.00%
30%
70%
1.40%
2.00%
2.80%
7.00%
40%
60%
1.20%
1.71%
2.40%
6.00%
50%
50%
1.00%
1.43%
2.00%
5.00%
60%
40%
0.80%
1.14%
1.60%
4.00%
70%
30%
0.60%
0.86%
1.20%
3.00%
80%
20%
0.40%
0.57%
0.80%
2.00%
90%
10%
0.20%
0.29%
0.40%
1.00%
100%
0%
0.00%
0.00%
0.00%
0.00%
PORTFOLIO
TRACKING
ERROR
PORTFOLIO
EXCESS
RETURN
PORTFOLIO
TRACKING
ERROR
Source: Northern Trust Quantitative Research
Note these results hold generally. If we assume our two active managers realize an
information ratio of 0.4 and 0.7 for this example, then the higher information ratio manager
will always be able to generate more efficient results using leverage, whether explicit or
implicit. Exhibit 4 (on page 8) details the same total equity allocation spectrum except
that we used the data from the Alford, Jones and Winkelmann9 research. Keeping our
20% allocation to active, to Manager B would yield an expected total equity allocation
excess return of 0.11% with 1.64% of tracking error. With leverage we could either hold
expected total equity allocation excess return constant and reduce tracking error to
0.63% or hold the tracking error constant and increase the expected excess return to
0.34%. This represents a 62% reduction in risk or a 300% increase in expected excess
return. Of course, adjusting the allocation between active and passive makes possible
other combinations of risk reduction or increased excess return as well.
northerntrust.com | Improving Active Risk Budgeting | 7 of 12
EXHIBIT 4: EMPIRICAL RESULTS: INCREASED ALLOCATION TO HIGH INFORMATION
RATIO MANAGERS
Using data from research by Alford, Jones and Winkelmann in 2003, we see that an increased allocation
to the high information ratio manager can improve efficiency without sacrificing excess returns.
HIGH IR
(MANAGER A)
ASSET ALLOCATION
LOW IR
(MANAGER B)
INDEX
ACTIVE
PORTFOLIO
EXCESS
RETURN
0%
100%
0.43%
2.09%
0.53%
8.21%
10%
90%
0.39%
1.88%
0.48%
7.39%
20%
80%
0.34%
1.67%
0.42%
6.57%
30%
70%
0.30%
1.46%
0.37%
5.75%
40%
60%
0.26%
1.25%
0.32%
4.93%
50%
50%
0.22%
1.05%
0.27%
4.11%
60%
40%
0.17%
0.84%
0.21%
3.28%
70%
30%
0.13%
0.63%
0.16%
2.46%
80%
20%
0.09%
0.42%
0.11%
1.64%
90%
10%
0.04%
0.21%
0.05%
0.82%
100%
0%
0.00%
0.00%
0.00%
0.00%
PORTFOLIO
TRACKING
ERROR
PORTFOLIO
EXCESS
RETURN
PORTFOLIO
TRACKING
ERROR
Source: Northern Trust Quantitative Research, using data from Alford, Jones and Winkelmann
BENEFITS OF HIGH IR/LOW TRACKING ERROR MANAGERS
Perhaps more importantly, moving toward higher IR managers increases the likelihood of
beating the benchmark, which is, after all, most active investors’ stated objective. Using
the data from original example of Managers A and B, Exhibit 5 details the probability of
underperforming the benchmark across various periods.
EXHIBIT 5: PROBABILITY OF UNDERPERFORMING A BENCHMARK
PROBABILITY OF
UNDERPERFORMANCE
Using a more efficient portfolio with a high information ratio manager reduces the probability of
underperforming over various time periods.
40%
35%
30%
25%
20%
15%
34.5%
24.2%
18.6%
10%
5%
0%
24.4%
11.3%
10.3%
5.9%
1 YEAR
3 YEARS
Manager A
(High IR)
5 YEARS
1.3%
10 YEARS
Manager B
(Low IR)
Source: Northern Trust Quantitative Research
While this analysis does not include the impact of fees, a review of fees likely would only magnify these findings;
typically low IR/high alpha managers have materially higher fees than high IR/lower alpha managers.
8 of 12 | northerntrust.com | Improving Active Risk Budgeting
Note that differences in portfolio tracking error lead to material differences in
underperformance probabilities. The higher IR manager has a significantly lower
probability of underperforming across all time horizons. For example, at the three-year
horizon – the period at which most managers’ performance is evaluated – the lower IR
manager has nearly a 25% chance of underperforming the benchmark while the higher
IR manager has just a 10% chance. It should also be noted that the analysis above does
not include the impact of fees. A review of fees would likely only magnify these findings.
To better understand the dynamics of this underperformance, we prepared a Monte
Carlo simulation study – widely used in modeling phenomena with input uncertainty – of
both high IR and low IR managers. Using the tracking error, IR and alpha from Managers A
(high IR) and B(low IR) in our scenario, we simulated 100,000 return paths spanning
10 years10. A sample of these paths is shown in Exhibits 6 and 7. As you can see, a $1
investment at the begining of the period with Manager A (shown in Exhibit 6) yields a
much narrower distribution of outcomes over any period relative to Manager B (shown in
Exhibit 7). Most importantly, note that the frequency with which each manager falls below
$1 is much lower with Manager A, the higher IR manager, than with Manager B.
EXHIBIT 6: HIGHER INFORMATION RATIO MANAGERS (MANAGER A) HAVE A
NARROWER RANGE OF PROBABLE RESULTS
VALUE OF $1 INVESTED
$3.00
$2.50
$2.00
$1.50
$1.00
$0.50
$0.00
0
1
2
3
4
5
6
7
8
9
10
YEAR
Source: Northern Trust Quantitative Research
VALUE OF $1 INVESTED
EXHIBIT 7: LOWER INFORMATION RATIO MANAGERS (MANAGER B) HAVE A
WIDE DISPERSION OF PROBABLE RESULTS
$3.00
$2.50
$2.00
$1.50
$1.00
$0.50
$0.00
0
1
2
3
4
5
6
7
8
9
10
YEAR
Source: Northern Trust Quantitative Research
northerntrust.com | Improving Active Risk Budgeting | 9 of 12
Again, these results hold generally. Using the data from Alford, Jones and Winkelmann’s
research, we computed underperformance probabilities in Exhibit 8. Note that the
differences between lower IR managers (Traditional Managers) and higher IR managers
(Structured Managers) are even more extreme. At a three-year horizon, the lower IR
managers still have more than a 45% probability of underperforming a benchmark
while the higher IR managers have slightly more than an 11% likelihood. Note also
that while underperformance probabilities decline steadily over time for higher IR
managers, the reduction is significantly less extreme for lower IR managers. At the 10-year
horizon, lower IR managers still have a 42% chance of underperforming; that probability is
reduced to just above 1% for higher IR managers.
EXHIBIT 8: EMPIRICAL RESULTS: PROBABILITY OF UNDERPERFORMING A BENCHMARK
PROBABILITY OF
UNDERPERFORMING
A BENCHMARK
50%
40%
47.4%
45.5%
44.3%
41.9%
30%
20%
24.2%
10%
0%
11.3%
1.3%
5.9%
1 YEAR
3 YEARS
Low IR/High Alpha — Traditional
5 YEARS
10 YEARS
High IR/Low Alpha — Structured
Source: Northern Trust Quantitative Research using data from Alford, Jones and Winkelmann.
So while many investors believe that risk concentration gives them the best of both worlds – risk
control and the opportunity for outperformance – the reality is that it actually produces a
portfolio with much higher risk and a significantly lower probability of outperformance
than one focusing on higher IR/lower alpha managers. In this sense risk concentration is
clearly inefficient and ultimately ineffective at delivering desired investment outcomes.
PUTTING AN EFFICIENT RISK BUDGET INTO PRACTICE
So ultimately, the evidence shows that the long-held belief that risk concentration is a
viable component of an efficient active risk budget is actually counterproductive. A more
effective strategy is for investors to maximize their risk-adjusted returns by creating an
efficient active risk budget. By targeting high IR managers and using the natural leverage
implicit in the active risk budgeting process, investors can still “dial up” or “dial down”
alpha targets to meet their needs.
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If the key to achieving an efficient active risk budget is to target high IR managers,
how can investors identify these managers? Northern Trust’s research, detailed in the paper
“Understanding Factor Tilts” shows that portfolios with tilts toward risk factors such as size,
value and momentum, have outperformed their benchmarks in domestic, international
and emerging markets, and do so with lower risk. This verifies the results of Carhart’s11
research, which showed that managers employing portfolio tilts toward compensated
risk factors tend to have higher information ratios than other active managers. In particular,
Carhart reveals that managers focused on fundamental “stock picking” (i.e., high active share/
high conviction strategies that are typical active candidates) commonly have negative alpha
and, hence, negative information ratios, while compensated risk factor tilt strategies have
consistently positive alpha and, thus, higher information ratios.
THINK DIFFERENTLY, MAXIMIZE RISK-ADJUSTED RETURN
While investors may have a natural bias toward strategies that isolate risk, this often
leads to portfolios with higher levels of overall risk than many investors realize. At best, it
is an inefficient strategy; at worst it can result in portfolios failing to meet their performance
goals. Fortunately, research shows that there is a better approach: using an efficient active
risk budget.
By stepping outside the comfort zone of a traditional risk-concentration strategy and
seeking an efficient active risk budget instead, investors will have a more even distribution of
risk across the portfolio. This approach can increase returns, decrease risk and raise the
likelihood of outperforming a benchmark – the ultimate goals of any active risk strategy.
LEARN MORE
If you would like to learn more about how an efficient active risk budget might benefit
your equity portfolio, please contact your local Northern Trust relationship manager or
visit northerntrust.com.
northerntrust.com | Improving Active Risk Budgeting | 11 of 12
Endnotes:
1.Specifically, returns per unit of risk (return/risk).
2.Harry Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7,
No. 1, 77-91, March, 1952.
3.William F. Sharpe, “Capital Asset Prices – A Theory of
Market Equilibrium Under Conditions of Risk,” The Journal of
Finance, Vol. XIX, No. 3, 425-442, September 1964.
4.Eugene F. Fama and Kenneth B. French, “The Cross-Section of Expected Stock Returns,” The Journal of Finance, Vol. 47,
No. 2, 428-465. June 1992.
5.Richard Thaler, “Mental Accounting Matters,” Journal of
Behavioural Decision Making, 183-206, 1999.
6.Andrew Alford, Robert C. Jones, and Kurt Winkelmann,
“A Spectrum Approach to Active Risk Budgeting,” Journal of
Portfolio Management, Vol. 30, No. 1, 2003, 49-60.
7.Clarke, de Silva and Thorley, “Portfolio Constraints and the Fundamental Law,” Financial Analysts Journal, Vol. 61, No. 5, 70-83, September 2005.
8.Note that for the total equity allocation to have zero tracking
error there would be no active risk taken and, hence, the expected excess return relative to the benchmark would also
be zero.
9.Alford, Jones and Winkelmann, 49-60.
10.Annual returns are assumed to be identically and independently normally distributed.
11.Mark M. Carhart. “On Persistence in Mutual Fund Performance,” Journal of Finance, Vol. 52: 57–82, 1997.
IRS CIRCULAR 230 NOTICE: To the extent that this material concerns tax matters, it is not intended to be used and
cannot be used by a taxpayer for the purpose of avoiding penalties that may be imposed by law. For more information
about this notice, see northerntrust.com/circular230.
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