Quantitative Research
October 2012
Analytics
Tactical Asset Allocation
With Macro Views:
From macro-forecasts to optimal portfolio construction
I. Introduction
Ravi Mattu
Managing Director
Vasant Naik
Executive Vice President
Mukundan Devarajan
Senior Vice President
Masoud Sharif
Vice President
PIMCO’s investment decisions are heavily influenced by our forecasts of the
likely path of the evolution of the global economy over the cyclical horizon and
the risks around our central scenarios. These evolving macroeconomic
forecasts (of growth rates of GDP, unemployment rates and other economic
indicators around the world) are anchored by discussions at our annual secular
forums, the quarterly cyclical forums and the Investment Committee (IC)
discussions following these forums. The gist of these deliberations can be
summarized in a set of probabilities for various economic outcomes defined
in terms of global growth and inflation over a cyclical horizon. For instance,
following the September 2012 cyclical forum, our outlook for real global GDP
growth over the next 12 months was in the range of 1.5% to 2% (below
the current rate of 2.2%).1 At the same time there was recognition of risks
around this base scenario. As our September 2012 Economic Outlook
mentions, “… while we do not expect recession across all developed countries
in 2013, we certainly would caution that the probability of a widespread
recession has increased, given the coordinated slowdown in global aggregate
demand we are witnessing across the world.”2 It was noted, however, that the
decision of the European Central Bank to commit its balance sheet to
unlimited but conditional purchases of short-term eurozone government
bonds “… substantially reduced the probability of depression like left-tail risk
over our cyclical horizon.” Such a layered and nuanced assessment of the
balance of probabilities of different macro-economic scenarios is a regular
and crucial part of the PIMCO investment process.
Another important input into our investment process is the forecast of returns
on the major investment choices we have under specific economic scenarios.
In our investment process, the sector-wise forecasts of asset returns conditional
on different macro scenarios emanate from teams that specialize in particular
market segments, while the IC defines the relevant macro-economic scenarios,
evaluates the balance of macro-economic risks and vets the scenario return
forecasts. In this way, the investment guidelines for our portfolio managers
Originally published by Ravi Mattu, Vasant Naik, Peter Matheos, Mukundan Devarajan and Masoud Sharif.
worldwide are the outcome of a process that takes its inputs
from both a top-down and a bottom-up view of the macroeconomic and market environment.
We can also utilize a framework for tactical asset allocation
that integrates the parameters codifying PIMCO’s cyclical
macro-economic outlook in a quantitative methodology for
optimal portfolio construction. We have applied elements
of this framework to advise our IC and individual portfolio
managers in their portfolio construction process. Our process
begins by specifying the probabilities of broad macro-economic
scenarios and then combines them with the estimates of
returns in different scenarios and the historical volatilities and
correlation among asset returns in an effort to create
optimal portfolios.
Our central premise is that the return forecasts for various
markets in different scenarios are best understood as
conditional averages of returns (i.e., averages conditional on
particular scenarios) rather than the only possible values of
returns in those scenarios. This is because a given scenario
represents a range of possible but related outcomes, and while
real growth and inflation shocks may be the most important
drivers of returns, other asset-specific factors ranging from
housing finance related policy variables in agency mortgages to
geopolitical supply shocks in oil markets help determine market
outcomes. Scenario probabilities together with the forecasts
of returns in various scenarios then allow us to compute the
estimated returns on various assets over the cyclical horizon.
To estimate risk we recognize that there are two sources of
uncertainty: the uncertainty about which scenario will be
realized and the uncertainty around conditional means of
returns within each scenario. In the context of the discussions
at the September 2012 Forum, there are two sources of risk
to the base case forecast of a deceleration in global economic
activity. The first and obvious risk is the possibility of markedly
less robust growth than expected (i.e., that a more pessimistic
scenario than the base case or that even a tail scenario occurs).
The second set of risks comes from the fact that the projection
of asset returns in a given scenario does not cover the full
range of probable outcomes even within that scenario. Our
risk estimates account for both of the above sources of risk.
2
OCTOBER 2012 | QUANTITATIVE RESEARCH
We estimate within-scenario risk from historical data while
the uncertainty caused by scenario shifts is derived from
variation in return forecasts across scenarios. In this way,
our methodology combines rich historical data on return
volatilities and correlations with the forward-looking views
emanating from our deliberations.
Finally, we use a mean-variance analysis to combine the
above-mentioned inputs and arrive at potentially optimal
overlays. In our optimization, we typically constrain the solution
to remain within reasonable bounds as well as to ensure that
the procedure does not attempt to leverage small differences
in returns by taking large long-short positions in highly
correlated assets. Liquidity-based constraints that require that
the solution move only gradually from existing positions that
are considered illiquid are also applied.
II. Return Estimation
How do we estimate returns on various assets that are the first
key input for the optimal overlay analysis? Let us look at a
hypothetical example. PIMCO has characterized the secular
outlook for the global economy as being in a “New Normal”
era of sub-trend growth with a non-negligible probability of
left-tail events. Consistent with this characterization and in
the context of our discussion at the September 2012 Cyclical
Forum, three scenarios can be considered probable over the
cyclical horizon, which we take to be one year:
1. Base: slow but continuing global growth and
moderate inflation
2. Pessimistic: outright recession
(without a financial, banking or sovereign crisis)
3. Tail: deep recession accompanied by a
financial/sovereign crisis
The first ingredient of the computation of estimated returns
is a set of scenario forecasts of returns on various asset
classes. See Figure 1a.
Scenario forecasts of returns on various assets should reflect
the macro behavior of these returns as well as valuation
considerations. Our example captures the property that
relatively riskless assets rally in the pessimistic and tail scenarios
while the risky assets outperform in the base case. Valuation
considerations also play an important role in these forecasts.
In the example below, it is assumed, for instance, that German
government bonds have a much more pronounced downside
in the base case than do U.S. Treasuries. This incorporates the
view that the ongoing sovereign crisis in the eurozone may
have led to an unusually large compression in German
government bond yields. This might normalize abruptly if there
were a definite move toward a meaningful resolution of the
European situation (an outcome more likely to be associated
with the base scenario than with the pessimistic or tail
scenarios). In addition, our example assumes that risky assets
(corporate bonds and equities in particular) are already pricing
in a substantial level of optimism, hence the upside in these
assets in the base case is limited while the loss potential,
especially in the tail scenario, is large.
The next ingredient that we need to compute estimated
returns is an assessment of the probabilities of the above
scenarios. We take these to be 0.55 (base scenario), 0.40
(pessimistic scenario) and 0.05 (tail scenario), in line with the
IC’s assessment of probabilities following the September 2012
Cyclical Forum discussions. These probabilities imply a
cautious assessment of the global economy over the cyclical
horizon. There is a substantial probability that either the
pessimistic or the tail scenario will occur. As we show later,
optimal portfolio overlays will reflect this view. If these
probability assessments change and the relatively benign
base case becomes more likely, the optimal overlays would
change too.
Given a set of scenario probabilities, it is straightforward to
compute return estimates for various asset classes. The
(unconditional) estimated returns are simply the probabilityweighted averages of scenario forecasts (Figure 1b).
III. Forecasting Volatilities
Our key observation is that estimates of asset return volatility
must account for the volatility of returns within each scenario
as well as the volatility across scenarios, which is due to the
uncertainty about which scenario will be realized. That is,
Var (asset returns) = E (Scenario variance) + Var (Scenario
forecast) where Var (-) denotes Variances and E (-) denotes
expected values.
We estimate the volatility of each asset under the base-case
(55% probability) and the pessimistic/tail scenarios (45%
probability) from historical data. We use the 10-year period
ending in June 2008 as a proxy for the base scenario and the
four-year period from July 2008 to September 2012 as a proxy
FIGURE 1A: CONDITIONAL ESTIMATED EXCESS RETURNS FOR KEY ASSETS (% P.A., AS OF 28 SEPTEMBER 2012)
(Estimated excess returns for government bonds, currencies, equities and oil are over short-term interest rates while those for credit and mortgage-backed
securities are over duration-matched Treasuries)
Base
Pessimistic
Tail
US Treasury 5-10y
-2.9
3.4
6.3
DE Government 5-10y
-10.2
4.0
8.8
US MBS
0.4
-0.1
-1.2
US Corp Financials
4.2
-7.4
-17.5
US Corp Non Financials
5.5
-0.8
-10.8
EUR (v. USD)
2.4
-12.5
-31.6
AUD (v. USD)
7.2
-7.9
-21.0
BRL (v. USD)
18.1
0.7
-11.7
US Equities
2.0
-19.9
-34.4
EM Equities
7.2
-13.6
-36.2
Oil
7.0
-23.0
-44.4
Source: PIMCO
Hypothetical example for illustrative purposes only.
QUANTITATIVE RESEARCH | OCTOBER 2012
3
for the pessimistic/tail scenarios. The estimation of volatilities
conditional on different economic regimes is justified by the
observation that both volatility and risky asset correlations are
elevated in stressed environments. Accordingly, our estimate of
the volatility of S&P 500 returns is 13.8% in the base scenario
versus 18.0% for the stressed scenarios. The impact of higher
equity volatility on diversified portfolios is magnified by an
increase in the correlation of these returns with the returns of
other risky assets in stressed periods. The correlation of S&P
500 returns with the excess returns (over duration-matched
Treasuries) of U.S. investment grade non-financials is estimated
at 0.53 for the base case and 0.65 for the stressed cases. The
increase in correlation during stress is even more pronounced
for oil, whose estimated correlation jumps from 0 to 0.7.
We add to this scenario-specific volatility the volatility
generated by the possibility of scenario switches (i.e., the
volatility caused by not knowing in advance whether the
base, pessimistic or tail scenario will be realized). Figure 2
illustrates this two-step estimation of volatility transparently.
The final volatility estimates are shown in column 5.
Comparing our estimate of the total volatility of each asset
class (column 5) with the estimates in column 4 (the volatility
implied from the return forecasts in the three possible
scenarios) shows the stark differences that appear in the
relatively more risky asset classes. For instance, the volatility of
scenario forecasts for oil is only 16.7% per year. This contrasts
with a historical volatility of over 30% in the previous 14 years
(columns 1 and 2). Our estimate for future volatility is 35.2%
per year. Similarly, the volatility of the EUR/USD currency pair
increases from 9.4% to 14.4%. By adjusting these volatilities
higher we lower our confidence in tilting the portfolio toward
being short oil or short the euro. Above Figure 2 are detailed
descriptions of our computations.
FIGURE 1B: ESTIMATED RETURNS FOR KEY ASSETS (% P.A., AS OF 28 SEPTEMBER 2012)
(Estimated excess returns for government bonds, currencies, equities and oil are over short-term interest rates while those for credit and mortgage-backed
securities are over duration-matched Treasuries)
Base
Pessimistic
Tail
55%
40%
5%
US Treasury 5-10y
-2.9
3.4
6.3
0.1
DE Government 5-10y
-10.2
4.0
8.8
-3.6
US MBS
0.4
-0.1
-1.2
0.1
US Corp Financials
4.2
-7.4
-17.5
-1.5
US Corp Non Financials
5.5
-0.8
-10.8
2.2
EUR (v. USD)
2.4
-12.5
-31.6
-5.3
AUD (v. USD)
7.2
-7.9
-21.0
-0.3
BRL (v. USD)
18.1
0.7
-11.7
9.7
US Equities
2.0
-19.9
-34.4
-8.6
EM Equities
7.2
-13.6
-36.2
-3.3
Oil
7.0
-23.0
-44.4
-7.6
Probability
Source: PIMCO
Hypothetical example for illustrative purposes only.
4
OCTOBER 2012 | QUANTITATIVE RESEARCH
Estimated excess returns
Column 1: We estimate volatility under the base scenario from
the historical volatility of excess returns over the 10-year period
ending in June 2008. We recognize that it is simplistic to
describe a decade that included periods of optimism about
productivity growth as well as the bursting of both the tech
and real estate bubbles as constituting a single regime. There
were periods in the decade where market sentiment turned
from exuberance to caution (reflecting the possibility of regime
switches). Therefore, to make the volatilities more consistent
with the base scenario, we remove the effect of extreme
observations by clipping the distribution of returns in the
bottom and top 5%. (That is, for each series individually the
returns in the bottom 5% are set to the 5th percentile return
while those in the top 5% of the distribution are capped at the
95th percentile.)
Column 2: The four-year period from July 2008 to September
2012 is used as a proxy for pessimistic/tail scenarios. This
period includes both the financial crisis of late 2008 and the
European sovereign debt crisis and is a reasonable proxy for
such scenarios. As before, we clip the top and bottom 5% of
the distribution. Given that the tail scenario is a low probability
extreme outcome, we do not attempt to estimate volatilities
separately in this scenario. Instead we use a composite
pessimistic/tail scenario.
Column 3: Average conditional volatility is computed from
the weighted average of scenario variances given in columns
1 and 2 where the weights are set equal to the probabilities
assigned to these scenarios.
Column 4: The additional volatility due to the possibility of
regime switches is estimated from the variance of scenariospecific return forecasts. When the forecasts differ a lot across
scenarios and it is highly uncertain which scenario will
materialize, this component will contribute more to the overall
volatility of returns. Box A presents an example that clarifies
the value of including forward-looking views in our estimates
of volatility.
FIGURE 2: VOLATILITY ESTIMATES FOR EXCESS RETURNS OF SELECT ASSET CLASSES (%, ANNUALIZED, AS OF 28 SEPTEMBER 2012)
1 Base Scenario
Volatility
2 Pessimistic/Tail
Scenario Volatility
3 Average Conditional
Volatility
4 Volatility of
Scenario Forecasts
(3) = 0.55*(1) + 0.45*(2)
2
2
5 Total volatility
(5)2 = (3)2 + (4)2
2
US Treasury 5-10y
1.6
3.3
2.6
3.3
4.2
DE Govt 5-10y
1.2
2.8
2.1
7.4
7.7
US MBS
0.6
1.0
0.8
0.4
0.9
US Corp Financials
2.9
4.5
3.7
6.7
7.6
US Corp Non Financials
2.8
4.5
3.7
4.2
5.6
EUR (v. USD)
8.4
13.4
11.0
9.4
14.4
AUD (v. USD)
9.9
17.1
13.6
8.7
16.1
BRL (v. USD)
12.9
15.4
14.1
9.7
17.1
US Equities
13.8
18.0
15.8
12.1
19.9
EM Equities
17.1
16.6
16.9
12.5
21.0
Oil
30.8
31.1
30.9
16.7
35.2
Source: PIMCO, Barclays, Bloomberg
Hypothetical example for illustrative purposes only.
See appendix for asset proxy information.
QUANTITATIVE RESEARCH | OCTOBER 2012
5
Column 5: Finally, we estimate total volatility from the
weighted scenario-specific volatility (“expectation of
conditional variance,” column 3) and the volatility across
scenarios (“variance of the conditional means,” column 4).
Empirical Regularities Reflected in Historical Estimates
It bears mentioning that the historical estimation that we
use for scenario-specific volatilities accounts for the empirical
regularities in the behavior of rates and spreads. In particular,
our estimates account for the observation that the volatilities of
changes in interest rates and credit spreads tend to depend on
their levels. Hence, our method models the dynamics of yields
and spread as a lognormal rather than a normal process, so
that changes in yields and spreads are proportional to the
current level of these variables. Thus the 1.6% return volatility
of U.S. five- to 10-year government bonds in the base
Box A: The Value of Combining Historical Experience With Forward-Looking Views: The Case of European
Sovereign Spreads
As the discussion in Section II makes clear, our estimates of risk parameters combine historical data with the volatility
generated in scenario forecasts. Since scenario probabilities and the forecasts themselves are forward-looking, our
framework incorporates a combination of historical experience and forward-looking assessments.
The value of such a combination can be easily seen in the case of European sovereign spreads (although peripheral
European government bonds are not included in the example presented in this article). Figure 3 below shows the time
series of the spread of generic 10-year Italian bonds over German bunds. The spread level and its volatility have increased
considerably in the last two years. However, even with this heightened spread uncertainty, the history does not include any
observation that would come close to what is being contemplated in the tail scenario in our example. This scenario includes
the possibility of a full-blown sovereign crisis in which case spreads and volatility could reach unprecedented levels. Our
estimates of total volatility take into account a forward-looking assessment of spread levels in this scenario and the volatility
introduced by this forecast. Thus, our framework uses a more realistic estimate of the risk of these assets than the case
where we rely on historical data alone.
FIGURE 3: SPREADS OF ITALY GOVERNMENT BONDS OVER GERMANY GOVERNMENT BONDS
(2006-2012, AS OF 28 SEPTEMBER 2012)
600
n IT - DE 10y spread
2011 average
500
2010 average
2012 average
BPS
400
300
200
100
0
Jan
‘06
Jul
‘06
Jan
‘07
Jul
‘07
Source: Bloomberg
6
OCTOBER 2012 | QUANTITATIVE RESEARCH
Jan
‘08
Jul
‘08
Jan
‘09
Jul
‘09
Jan
‘10
Jul
‘10
Jan
‘11
Jul
‘11
Jan
‘12
Jul
‘12
scenario would be approximately equivalent to a 3.2% return
volatility if initial yields were twice the current levels.
IV. Forecasting Correlations
Portfolio optimization also requires a view on return
correlations. If the scenario forecasts of returns were thought
of as the only possible outcomes of returns in the given
scenarios, the correlations that would result among various
assets would be extreme. When return forecasts are subjected
to the discipline of having to be created for economic scenarios
defined in terms of growth and inflation surprises, these
forecasts will appear to have been generated by a one-factor
model (“risk on/risk off”) even if different groups create them.
However, as a principal component analysis of returns on
macro-assets shows, the first principal component (which can
be identified as the risk on/off factor) explains less than 60%
of the return variation across broad asset classes, even in recent
episodes of financial stress. This bolsters the case for allowing
scenario-specific uncertainty in returns to be applied to risk
estimates as our method does. The methodology used in
estimating volatilities in Section III can be extended to
estimating correlations. As before, we measure correlations for
the base scenario using the clipped historical distribution of the
10-year period prior to June 2008 (Figure 4a) and those for
the composite pessimistic/tail scenario using the more recent
four-year period (Figure 4b). The fact that correlations tend
to increase in stressed environments comes through in these
estimates. It can be seen, for example, that over the 10-year
period ending in June 2008 U.S. equity (S&P 500) returns had a
zero correlation with oil. Over the subsequent four years this
return correlation was 0.7. Clearly, the decade prior to 2008
reflects a period where energy markets were driven by both
supply and demand fundamentals, while recent history has
been dominated by macro shocks that have affected global
demand. A similar story can be told about other asset pairs.
As before, we blend history with forward-looking correlations
embedded in the forecasts of returns in various scenarios.
We put a 55% weight on the normal-period covariance
FIGURE 4: CORRELATION MATRIX OF EXCESS RETURNS ON SELECTED ASSET CLASSES AS OF 28 SEPTEMBER 2012
a. Normal period: Jul 1998 – Jun 2008
US Treasury
5-10y
US Treasury 5-10y
1
US Corp Financials
US Corp
Financials
US Corp Non
Financials
EUR (v. USD)
US Equities
Oil
-0.35
-0.43
0.30
-0.39
0.05
1
0.85
-0.02
0.42
-0.01
1
-0.01
0.53
0.08
1
-0.07
0.09
1
-0.02
US Corp Non Financials
EUR (v. USD)
US Equities
Oil
1
b. Stressed period: Jul 2008 – Sep 2012
US Treasury
5-10y
US Treasury 5-10y
US Corp Financials
US Corp Non Financials
EUR (v. USD)
US Equities
Oil
1
US Corp
Financials
US Corp Non
Financials
EUR (v. USD)
US Equities
Oil
-0.45
-0.48
-0.02
-0.29
-0.30
1
0.91
0.49
0.68
0.46
1
0.46
0.65
0.49
1
0.68
0.52
1
0.70
1
Source PIMCO, Barclays, Bloomberg
QUANTITATIVE RESEARCH | OCTOBER 2012
7
matrix (that uses the values in Figure 4a) and a 45% weight
on the stressed-period covariance matrix (using the values
in Figure 4b) to obtain the covariance analog of column 3
of Figure 2. We then add the covariance matrix of scenario
forecasts to this weighted historical covariance matrix.
Applying a Shrinkage Procedure to Impart Robustness
in Correlation Estimates
The resulting variance-covariance matrix may still imply
spuriously high correlations between some assets, potentially
giving rise to instabilities in an optimization exercise. Hence,
as a final step we tilt the estimated correlation matrix toward
a matrix that assumes that all assets are uncorrelated. While
the assumption of uncorrelated assets is clearly counter-factual,
there is substantial evidence in financial and statistical studies
that a tilt of estimated correlations toward zero generates
valuable stability in portfolio analyses. Figures 5a and 5b
clearly show the moderating effect of applying this shrinkage
to our correlation matrix.
The correlation matrix that results from the shrinkage exercise
is the one we use for the optimization exercise. As can be seen
from the extract shown in Figure 5b, the correlation estimates
are intuitive but they should be treated as a starting point in
the optimization process. For instance, one may view the
return correlation of 0.74 between the excess returns of
financial and non-financial corporate bonds as too low over
the cyclical horizon. Such views could be incorporated into the
correlation matrix (with minimal adjustments to ensure internal
consistency or “positive-definiteness” of the resulting
covariance matrix) to better reflect these views.
V. Optimal Overlays
The prospective estimated returns on various assets and the
variance-covariance matrix of their returns represent the most
important inputs needed to analyze the optimal risk/reward
trade-off. To illustrate the interplay between risk and reward in
the determination of optimal overlays, we first take the case
FIGURE 5: COMBINED HISTORICAL AND FORWARD-LOOKING CORRELATION MATRIX AS OF 28 SEPTEMBER 2012
a. Before shrinkage
US Treasury
5-10y
US Treasury 5-10y
1
US Corp Financials
US Corp
Financials
US Corp Non
Financials
EUR (v. USD)
US Equities
Oil
-0.81
-0.75
-0.47
-0.63
-0.46
1
0.93
0.68
0.76
0.52
1
0.64
0.76
0.52
1
0.63
0.52
1
0.53
US Corp Non Financials
EUR (v. USD)
US Equities
Oil
1
b. After shrinkage
US Treasury
5-10y
US Treasury 5-10y
1
US Corp Financials
US Corp Non Financials
EUR (v. USD)
US Equities
Oil
Source: PIMCO, Barclays, Bloomberg
Hypothetical example for illustrative purposes only.
8
OCTOBER 2012 | QUANTITATIVE RESEARCH
US Corp
Financials
US Corp Non
Financials
EUR (v. USD)
US Equities
Oil
-0.65
-0.60
-0.38
-0.51
-0.37
1
0.74
0.55
0.60
0.42
1
0.51
0.61
0.42
1
0.50
0.42
1
0.43
1
when there are no constraints on the overlays to be
considered. Figure 6 presents this unconstrained optimal
overlay using the estimated returns of Figure 1b (column 4)
and the variance-covariance matrix estimated using the
method described in the previous sections. This portfolio
overlay maximizes the estimated mean alpha for a target
volatility of 200 bps per year.
The above results reveal a number of interesting insights.
1. The unconstrained optimum seeks to be overweight U.S.
duration where the estimated returns are positive. This
position is accentuated by the fact that there is an
offsetting underweight position in German duration. This
is driven by the negative estimated returns for German
government bonds as well as the high correlation between
U.S. and core-Euro rates.
2. There is a significant overweight in U.S. non-financial
credit (supplemented by an underweight in U.S. financial
credits). This combination is an attempt to take advantage
of the negative estimated returns of financials and the
significant correlation between financials and nonfinancial credits.
3. U.S. MBS are attractive. This results from the positive
estimated returns and also from the fact that the returns
of this asset class have only moderate correlations with
other assets.
4. An underweight position in equities is not surprising given
their negative estimated returns. Moreover, the model seeks
to overweight the Brazilian real (BRL) and Australian dollar
(AUD). Despite a higher estimated return, the underweight
in EM equities is larger than in U.S. equities. This is because
the BRL position is markedly positively correlated with EM
equity positions and a better risk-reward tradeoff is reached
by underweighting EM equities and being overweight
BRL (and AUD).
Imposing Constraints
While this simple exercise reveals interesting trade-offs
between competing trades in a convenient fashion, an
unconstrained optimization exercise is not a realistic one. In
practical situations, we must take into account the liquidity
of various markets, strategic positions in portfolios, as well as
mandate restrictions. The mean-variance optimization analysis
used above can be modified easily to take account of such
constraints. The overlay is constrained in terms of the
maximum and minimum exposures we can take in certain
asset classes. Equally importantly, we impose constraints that
limit the extent to which the optimizer can pick offsetting
overweight and underweight positions in highly correlated
assets. Again, the objective of the optimization exercise is
to select an overlay portfolio to maximize estimated returns
subject to a volatility of 200 bps per year. See Figure 7 for the
optimization constraints and the optimal overlay.
FIGURE 6: UNCONSTRAINED OPTIMAL OVERLAY AS OF 28 SEPTEMBER
2012: TARGET VOLATILITY 200BP
(See note for units)
Unconstrained optimal overlay
US Treasury 5-10y
1.2
DE Govt 5-10y
-1.7
Credit US Financials
-1.7
Credit US Non Financials
US MBS
2.6
0.5
EUR (v. USD)
-7.6%
AUD (v. USD)
2.6%
BRL (v. USD)
8.0%
US Equities
-2.1%
EM Equities
-6.1%
Oil
-1.0%
Estimated mean alpha (bp, p.a.)
361
Target volatility (bp, p.a.)
200
Note: Risk exposures to all assets other than Currencies, Equities and Oil are in years of
duration overweight (+) or underweight (-). Those of Currencies, Equities and Oil are in
percentage market value.
Source: PIMCO
Hypothetical example for illustrative purposes only.
QUANTITATIVE RESEARCH | OCTOBER 2012
9
With constraints, the framework suggests a smaller overweight
position in U.S. duration and a correspondingly smaller
underweight in German duration as there are restrictions
imposed on large opposite trades in correlated assets. The
overweight in MBS duration is maximal given the attractiveness
of its estimated returns and correlation properties. Moreover,
again because offsetting opposite positions are restricted,
both credit positions are overweights. However, there is a
bigger underweight in equities that acts as a partial hedge
to these credit positions. Also, the aggregate upper bound
in currencies is binding. That is, the model would want to
overweight BRL and AUD more (and underweight EUR more)
against USD if it could. A net short in these currencies is not
chosen to hedge the credit overweights.
VI. Conclusion
PIMCO’s framework synthesizes the forward-looking views
about asset returns in macro-scenarios considered likely at the
cyclical horizon into a set of optimal overlay positions, while
explicitly accounting for return volatilities and correlations
observed in historical data. Undoubtedly, such a framework is a
starting point to determining an optimal set of overlay trades.
It allows for views to be taken on its various inputs, which can
make its proposals richer and more representative of strategic
and macro considerations. In future extensions of our work,
we will include measures of risk other than variances (such as
conditional VaR) in the optimization procedure for cases where
they might be more appropriate.
FIGURE 7: OPTIMAL OVERLAY WITH CONSTRAINTS AS OF SEPTEMBER 2012: TARGET VOLATILITY 200BP
(See notes for units)
Constrained optimal overlay
US Treasury 5-10y
DE Govt 5-10y
0.3
-0.3
Credit US Financials
0.7
Credit US Non Financials
1.6
US MBS
1.0
EUR (v. USD)
-1.1%
AUD (v. USD)
1.2%
BRL (v. USD)
4.9%
US Equities
-2.9%
EM Equities
-7.1%
Oil
Constraints imposed
-1 – 1 year
-5% – 5%
-10% – 10%
-4.4%
Estimated mean alpha (bp, p.a.)
196
Target volatility (bp, p.a.)
200
Notes: 1. Risk exposures to all assets other than Currencies, Equities and Oil are in years of duration overweight (+) or underweight (-). Those of Currencies,
Equities and Oil are in percentage notional. 2. We impose constraints on matching over/underweight positions across buckets within Rates, Credit,
Currencies and Equities.
Source: PIMCO
Hypothetical example for illustrative purposes only.
10
OCTOBER 2012 | QUANTITATIVE RESEARCH
Appendix
Indexes used to compute historical volatilities and correlations (reported in Figures 2, 4
and 5)
n
n
n
n
n
n
n
n
n
n
n
1
2
U.S. Treasury 5-10y: Barclays US Treasury 5-10 year index monthly total returns, less
USD 1M funding (Source: Barclays Capital)
DE Government 5-10y: Bloomberg EFFA Germany 5-10 year index monthly total
returns, less EUR 1M funding (Source: Bloomberg)
US MBS: Barclays US Aggregate MBS Index monthly excess returns over durationmatched Treasuries (Source: Barclays Capital)
US Corp Financials: Barclays US Credit Financials Index monthly excess returns over
duration-matched Treasuries (Source: Barclays Capital)
US Corp Non Financials: Merrill Lynch US Corporate Non-Financial Index monthly
excess returns over duration-matched Treasuries (Source: Merrill Lynch Indices)
EUR: Monthly returns of a 1M forward contract on EUR/$, held to expiry (Source:
Bloomberg)
AUD: Monthly returns of a 1M forward contract on AUD/$, held to expiry (Source:
Bloomberg)
BRL: Monthly returns of a 1M forward contract on BRL/$, held to expiry (Source:
Bloomberg)
US Equities: MSCI US Equity Index monthly total returns, less USD 1M funding
(Source: Bloomberg)
EM Equities: MSCI EM Total Return Index monthly returns less monthly returns of
MSCI FX hedge index (Source: Bloomberg)
Oil: S&P GSCI Crude Index monthly returns (Source: Bloomberg)
Parikh, S., “PIMCO Cyclical Outlook: Building Rickety Bridges to Uncertain Outcomes,”
p. 4 (Table 1), PIMCO Economic Outlook, September 2012.
Ibid, p. 3
QUANTITATIVE RESEARCH | OCTOBER 2012
11
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