01 Evaluating factor
exposures Portfolio analytics
01 Isolated risk and Evaluating factor exposures
contribution to risk We can decompose the risks within a selected portfolio into various underlying factors. Invesco
Vision includes multiple factor groups that allow investors to identify and evaluate the various risks
04 Historical &
with which they are faced. Figure 36 presents the economic factor groups considered and Figures
hypothetical scenario
37 and 38 present the Solvency II and NAIC factor groups. The following sections provide detail on
analysis
how to aggregate or decompose factor risks to various levels of granularity or relevant groupings.
Isolated risk and contribution to risk
Risk can be decomposed and viewed either in isolation or in terms of contribution to total risk.
In both cases, we need to first define the binary group inclusion matrix MG. It is a bit-mask-like
diagonal matrix whose entries are either a 1 (meaning inclusion in the group of interest) or 0
(meaning the factor is not included in the group of interest). As an example, consider the “Rates”
risk factor group matrix MG as detailed in the group factor map shown in Figure 35.
Figure 35: The “Rates” group matrix MG
Factor covariance matrix
Factor1 Factor2 … Shift Twist Butterfly … Factork-1 Factork
Factor1 0
Factor2 0
…
Shift 1
Twist 1
Butterfly 1
…
…
Factork-1 0
Factork 0
Source: Invesco, BarraOne.
With MG defined and Σf, the factor covariance matrix, we can compute the isolated risk for a
portfolio with B factor exposures and w weights as follows:
Similarly, we can calculate the contribution to risk as follows:
Here, the group contribution to risk is a percentage of the normalized total portfolio risk, so that all
of the group contributions to risk along with the specific risk add up to 100 percent.
Portfolio analytics 1
Figure 36: Economic factor risk groups
Defensive Growth Real Specific Currency
Total
Rates Inflation Credit Equity Commodity Real estate Private Hedge Fund Currency Specific
2
Shift Up Shift Up Swap spread Industry Energy Market value Large buyout Event driven Currency 1
Cons discr
Flatten Flatten Sovereign Base metals Income Small buyout Convert arb Currency 2
Cons staples
Mid Up Mid Up Agency Prec metals Late venture Fixed inc arb Currency 3
Energy
Eq Mkt
Implied Vol. Municipal Financials Agriculture Early venture Currency 4
Neutral
Corporate Health care Livestock Mezzanine Global macro Currency 5
Industrials
Emerging
Distressed Mgd futures Currency 6
mkts
Info Tech
Securitized Currency 7
Materials
Covered Telecom Currency 8
Utilities
Currency 9
Other
Currency 10
Style
Currency
Value Other
Carry
Small Size
Momentum
Low Vol.
Quality
Illiquidity
Other
Source: Invesco, BarraOne.
Figure 37: Solvency II factor group
Total
Rates Spread Equity Property Currency
Source: Invesco, Solvency II directive.
Figure 38: RBC factor groups
Total
Fixed Income Credit Non-Fixed Income
Source: Invesco, NAIC.
Example: Group factor analysis
We consider a simple example of our factor exposure analysis for a single fixed income instrument,
namely an investment grade bond issued by a telecommunications firm. In this case, there are only
eight factor exposures as shown below, and the specific risk of the bond is 7.97%.
Assuming that we wish to focus on the interest rate risks (indicated as GOV above), we first proceed
to create the bit mask as shown below. Factor entries corresponding to interest rate exposures are
indicated with a 1 while the corresponding group factor covariance matrix is shown in Figure 39.
Figure 39: Factor covariance matrix Σf (values X 10-6)
GOVSH GOVTW GOVBU SWPSH SWPTW SWPBU TELBBB CURUSD
GOVSH 50.63 6.82 -6.61 -2.00 1.45 0.06 -24.18 0.00
GOVTW 6.82 9.22 1.67 -1.16 -1.65 -0.75 1.27 0.00
GOVBU -6.61 1.67 3.35 -0.27 -0.56 -0.76 3.82 0.00
SWPSH -2.00 -1.16 -0.27 5.57 0.79 0.23 -2.55 0.00
SWPTW 1.45 -1.65 -0.56 0.79 2.91 0.42 -7.78 0.00
SWPBU 0.06 -0.75 -0.76 0.23 0.42 1.25 -1.66 0.00
TELBBB -24.18 1.27 3.82 -2.55 -7.78 -1.66 122.94 0.00
CURUSD 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Source: Invesco.
Portfolio analytics 3
Putting the basic ingredients together, we calculate both isolated interest rate risk and interest rate
contribution to risk as follows:
Historical and hypothetical scenario analysis
An understanding of how a portfolio might have performed historically during various geopolitical
and economic environments, as well as how it might perform in certain hypothetical scenarios that
could occur in the future, can be used to inform decisions regarding the navigation of potential
future market dynamics. Invesco Vision allows for these types of analyses providing detailed
decompositions that help to identify key drivers of risk within a portfolio.
• Historical scenario analysis
Modeling a portfolio’s performance during a historical period of interest is a straightforward
exercise. In general, if one considers a historical period of interest, during which the model risk
factors have returns of say rf1, rf2, through rfk, then using the known current exposures each asset
has to each of the factors βj,k, and the weights of each of the assets within the portfolio Wj, the
portfolio’s return for the period can be computed as follows:
It is important to note that this calculation relies on current factor exposures and that these values
could have been different during any actual historical period. Historical scenario analysis provides
valuable insight and yields the magnitude and direction of the portfolio’s return during periods of
interest. Invesco Vision covers various pre-determined historical scenarios of interest during which
a user can analyze their portfolio and observe its performance, including the 1970s oil crisis, the
1987 market crash, the Global Financial Crisis, and more recently Brexit.
Example: historical scenario
We consider a simple historical example for a portfolio holding a single fixed income instrument as
we did in the previous section. Here, we examine the 2010 European Bond Crisis spanning March
14 - May 26. During this time, we record the relevant factor returns as well as the corresponding
telecommunications firm bond factor exposures as shown in Figure 40.
Figure 40: The factor exposures, factor shocks and factor returns
2010 European Bond Crisis using telecommunications firm bond 4.682 ’46
Returns
Exposures Shocks
Factors (exposure x shock)
GOVSH 15.09 0.35% 5.27%
GOVTW 13.49 0.19% 2.62%
GOVBU 8.53 -0.02% -0.20%
SWPSH 15.09 0.03% 0.51%
SWPTW 11.87 0.00% 0.00%
SWPBU 8.80 0.00% 0.00%
TELBBB 15.10 -0.23% -3.42%
CURUSD 1.00 0.00% 0.00%
Total – – 4.79%
Source: Invesco, Barra One.
4
Based on the equations shown previously, the portfolio return can be computed as the sum-product
of the factor exposures and the factor shocks, which is 4.79%. It is impor tant to note that this
calculation is linear in nature and does not include any second order pricing effects such as
convexity. When we account for such effects, the return is estimated to be 4.48%.
• Hypothetical scenario analysis
Hypothetical portfolio analysis models various market movements that potentially could happen in
the future. There are two types of scenarios: uncorrelated, where changes are isolated to a some
specific factor, and correlated, where changes in factors are propagated across all other factors.
–– Uncorrelated scenario analysis
In an uncorrelated scenario analysis with β factor exposures, w weights, and where we assume
M factors are shocked by v, the portfolio return can be computed as follows:
–– Correlated scenario analysis
At the heart of correlated scenario analysis is the concept of conditional expectation. We assume
that our risk factor returns are distributed as a multivariate normal distribution and that the
factor returns are correlated as indicated by the factor covariance matrix. In this case the basic
recipe is to 1) prescribe a list of factor shocks, 2) propagate the factor shocks across the
remaining factors, and 3) compute the portfolio returns as the sum-product of the factor
exposures and the factor shocks.
To propagate the shock, we employ the following calculation:
where rfs are the originating factor shocks that are propagated to all factors rf. Σ2,1 is the covariance
between the shocked factors and all other factors and Σ1,1 is the covariance of the shocked factors
as shown in Figure 41.
Figure 41: The factor covariance matrix block form
Factor covariance matrix blocks
F1 F2 F3 F4 … FK-1 FK
F1
F2
Σ1,1 Σ1,2
F3
F4
Σ2,1 Σ2,2
…
FK-1
FK
Shocked factors All other factors
Source: Invesco, BarraOne.
With all factor shocks having been propagated we can proceed to compute the portfolio returns just
as we did in the historical case.
Invesco Vision allows users to consider a collection of hypothetical shocks. The shocks include
movements to global equities, US equities, and EAFE equities, US Treasuries, currency exchange
rates, oil, and gold. These can be viewed in correlated and uncorrelated terms.
Portfolio analytics 5
Example: Hypothetical scenario (uncorrelated and correlated)
Using the same single asset portfolio in the above examples, we consider the hypothetical scenario
in which we shock the US Treasury curve by a parallel 100-basis-point upward movement. As before,
there are eight non-zero factor exposures in this example.
In the uncorrelated case, the only factor exposure movements are the ones we have shocked,
GOVSH, GOVTW, and GOVBU, and all of the remaining factor shocks are identically zero. In the
correlated case, the three explicit interest rate shocks propagate and give rise to non-zero factor
shocks across the remaining five risk factors. All of these shocks, along with the factor exposures,
are summarized in Figure 42.
Figure 42: The factor exposures and the uncorrelated and correlated (propagated) factor shocks
Uncorrelated Correlated
Factors Exposures Shocks Returns Shocks Returns
GOVSH 15.09 -1.00% -15.09% -1.00% -15.09%
GOVTW 13.49 0.00% 0.00% 0.00% 0.00%
GOVBU 8.53 0.00% 0.00% 0.00% 0.00%
SWPSH 15.09 – – 0.05% 0.77%
SWPTW 11.87 – – -0.08% -0.93%
SWPBU 8.80 – – 0.04% 0.34%
TELBBB 15.1 – – 0.63% 9.44%
CURUSD 1.00 – – 0.00% 0.00%
Total – – -15.09% – -5.46%
Source: Invesco, BarraOne.
Similar to the previous result, the portfolio level return is just the sum-product of the factor
exposures and factor shocks. The portfolio return for the uncorrelated and correlated shocks
are -15.09% and -5.46%, r espectively. When we include full re-pricing that includes the impact
of convexity, the returns are -13.58% and -5.26%, r espectively.
6
Investment risks
The value of investments and any income will fluctuate (this may partly be the result of exchange rate fluctuations)
and investors may not get back the full amount invested.
Diversification and asset allocation do not guarantee a profit or eliminate the risk of loss.
Invesco Investment Solutions (IIS) develops Capital Market Assumptions (CMAs) that provide long-term estimates for the
behavior of major asset classes globally. The team is dedicated to designing outcome-oriented, multi-asset portfolios that
meet the specific goals of investors. The assumptions, which are based on 5- and 10-year investment time horizons, are
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comes to pass, actual returns could be significantly higher or lower than these estimates.
This information is not intended as a recommendation to invest in a specific asset class or strategy, or as a promise of
future performance. Refer to the IIS CMA methodology paper for more details.
Important information
All information is sourced from Invesco, unless otherwise stated. All data as of April 15, 2019 and is USD and hedged
unless otherwise stated.
This document has been prepared only for those persons to whom Invesco has provided it for informational purposes only.
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