ROE and
Fama-French
7/26/2016

Setting the Stage

The Analysis

The Results

Conclusions
2
Setting the Stage
•
Insurance companies require capital in order to function.
•
Raising and holding capital costs money.
•
The cost of capital is a necessary part of the insurance rate. Per
ASOP 30:
“Property/casualty insurance rates should provide for all expected costs,
including an appropriate cost of capital associated with the specific risk
transfer.”
3
Setting the Stage
What is an appropriate cost of capital? Per the Supreme
Court:
“[T]he return to the equity owner should be commensurate with returns on
investments in other enterprises having corresponding risks. That return,
moreover, should be sufficient to assure confidence in the financial integrity
of the enterprise, so as to maintain its credit and to attract capital.”*
*Hope Natural Gas, 320 U.S. at 603 (citations omitted)
4
Setting the Stage
In Actuarial Pricing, we are used to having to project
future, unknown quantities, but the cost of raising and
maintaining capital is more difficult to quantify than most
other expenses.
Q: How do you know if you are accurately estimating the
company’s cost of capital?
5
Setting the Stage
Cost of capital discussions often suffer from lack of
context.
• Why should we target the return we are targeting?
• What capital should that return be based on? And why?
6
Setting the Stage
Allstate Example:
• Allstate is a publicly owned company that raises capital by
issuing shares of stock (and bonds).
• Investors expect a return on their investment.
• The return is expected on their investment.
• The return should be commensurate with the risk.
The goal: determine the return to shareholders (i.e. on the market value of the company) that
is commensurate with the risk of investing in Allstate.
7
Setting the Stage
Income to Shareholders is provided through Underwriting
Income and Investment Income.
Underwriting
Income
Income to
Shareholders
Income to
Company
Investment
Income
8
Setting the Stage
Pricing calculations work in reverse, by determining the
appropriate income to Shareholders, then determining the
needed Underwriting Income.
Underwriting
Income
Income to
Shareholders
Income to
Company
Investment
Income
9
Setting the Stage
We need to determine what return on the market value (MV) of Allstate is commensurate
with the risk.
• There are many methodologies that have been developed to measure returns on
MV.
• Markowitz portfolio theory from the 1950s developed into the Capital Asset
Pricing Model (CAPM) in the 1960s:
𝑟𝑖 = (𝑟𝑚 −𝑟𝑓 ) ∗ 𝛽𝑖 + 𝑟𝑓
• In the 1990s, CAPM expanded into various multi-factor models, including the
Fama-French Three-factor Model (FF3F):
𝑟𝑖 = (𝑟𝑚 −𝑟𝑓 ) ∗ 𝛽𝑖 + 𝛽𝑠 ∗ 𝜋𝑠 + 𝛽ℎ ∗ 𝜋ℎ + 𝑟𝑓
10
Setting the Stage
A common approach* for calculating the betas is a two-step process:
• Step one – using Simple Linear Regression (SLR) calculate for each company a beta
relative to the market based on 60 months of company-specific data:
Company A: 𝑟𝐴 = (𝑟𝑚 −𝑟𝑓 ) ∗ 𝛽𝐴 + 𝛽𝑠−𝐴 ∗ 𝜋𝑠 + 𝛽ℎ−𝐴 ∗ 𝜋ℎ + 𝑟𝑓
Company B: 𝑟𝐵 = (𝑟𝑚 −𝑟𝑓 ) ∗ 𝛽𝐵 + 𝛽𝑠−𝐵 ∗ 𝜋𝑠 + 𝛽ℎ−𝐵 ∗ 𝜋ℎ + 𝑟𝑓
Company C: 𝑟𝐶 = (𝑟𝑚 −𝑟𝑓 ) ∗ 𝛽𝐶 + 𝛽𝑠−𝐶 ∗ 𝜋𝑠 + 𝛽ℎ−𝐶 ∗ 𝜋ℎ + 𝑟𝑓
etc.
Note: at this point, all of the betas are company-specific.
*This is the approach outlined in “ESTIMATING THE COST OF EQUITY CAPITAL FOR PROPERTY-LIABILITY INSURERS” by J.
David Cummins and Richard D. Phillips, The Journal of Risk and Insurance, 2005, Vol. 72, No. 3, 441-478
11
Setting the Stage
• Step Two – Use the betas calculated in Step One as the dependent variable in a
system of linear equations. Using the market betas as an example:
𝛽𝐴 = 𝜔𝐴.𝑋 𝛽𝑋 + 𝜔𝐴.𝑌 𝛽𝑌 + 𝜔𝐴.𝑍 𝛽𝑍
𝛽𝐵 = 𝜔𝐵.𝑋 𝛽𝑋 + 𝜔𝐵.𝑌 𝛽𝑌 + 𝜔𝐵.𝑍 𝛽𝑍
𝛽𝐶 = 𝜔𝐶.𝑋 𝛽𝑋 + 𝜔𝐶.𝑌 𝛽𝑌 + 𝜔𝐶.𝑍 𝛽𝑍
In this example, X, Y, and Z represent industries, and ω represents the percentage of
the firm’s sales in a given industry.
Note: after Step Two, the calculated betas are now industry betas. This is called the
“Full-information Beta” (FIB) approach.
12
Setting the Stage
Q: How do we know if a methodology is any good?
• There is significant debate in academic literature regarding whether the CAPM
framework holds.
• Articles defending specific methodologies generally use regression output, such as pvalues, to show that certain parameters are statistically significant, and therefore the
methodology holds.
• As a test, I checked all of the p-values from the first-run regression (SLR on a
company-specific basis) to see how many of them were below 5% or 10%:
Market
Size
Value
Betas
311,564
311,564
311,564
P < .05
29.4%
13.8%
12.5%
P < .10
36.6%
21.0%
19.3%
13
Setting the Stage
Q: How do we know if one methodology is better than another?
• The approach used to defend individual methodologies in literature (using
regression output) is akin to fitting a model on a Training data set.
• Modern-day model building typically does not stop there – after building a
model on a Training data set, the model is then run on a Test data set to see
how it performs.
• Using this approach, we can see how well different methodologies perform
relative to each other.
14
The Analysis
Testing
• Available data included monthly stock returns from 1989 through 2013.
• The Training/Testing of the methodologies was designed to mirror our actual
Pricing process – calculate the parameters based on N consecutive years of data;
use the calculated parameters to predict a company’s risk premium in the
following chronological year.
• Example:
Calculation data
Test data
2000
2001
2002
2003
2004
2005
• Note: this is not a test of predicting the stock market, but rather a test of
predicting a company’s return given the market/size/value return in that future
period.
15
The Analysis
•
•
Cummins/Phillips Methodology (Base Scenario)
• First regression run – Simple Linear Regression (SLR) on 60 months of returns for each individual
company, using the Sum Beta approach
• Second regression run – Full-Information Beta calculation solving a system of Linear Equations, weighted
by market value, for industry betas using Seemingly Unrelated Regression (SUR)
Additional Options
• Regression Techniques
• Industry factors
• Hierarchical model
• Bayesian model
• Regression + Shrinkage
• Quantile regression
• Detail Tweaks
• Weighting
• Months of data
• Capping
• Sum Beta
• Seemingly Unrelated Regression
16
The Analysis
• Pooling
• When data can be categorized at multiple levels – in this case, by company and by
industry – decisions need to be made regarding what level of pooling will be done
in the regression analysis.
• No pooling – data is not pooled by industry at all, and regression is run only at the
individual-company level.
• Complete pooling – data is not kept at the individual-company level at all, and
regression is run entirely at the industry level.
No Pooling
Partial Pooling
Complete Pooling
SLR
Hierarchical
Models
Bayesian
Models
FIB (sort of)
SLR - Industry Factors
• Note that CAPM testing has always been done using portfolios (not necessarily by
industry) in order to minimize estimation error.
17
The Analysis
• Regression Detail Tweaks
•
•
•
Weighting – for regressions that involve some pooling of the data, weighting the data by
some measure (such as market value) might provide greater accuracy.
Months of data
• From the 2013 Ibbotson SBBI Valuation Yearbook: “The amount of history included in
beta calculations done by commercial beta services is fairly consistent at five years.
Using five years of data is a rather arbitrary decision that attempts to use as much data
as possible without including irrelevant historical data. Using five years of data would
ideally cover a number of different economic scenarios such as expansion and
contraction in the economy.”
• In my experience, most people use five years, and some use fewer than five. I’ve
never seen anyone use more than five.
Capping - Betas could be capped at some quantile to help eliminate the crazies.
18
The Analysis
• Measurement Criteria
• Accuracy
•
•
After calculating the betas on the Training data set, we can calculate predictions on the
Test data set and compare to the actual value.
Accuracy can be measured based on Mean Squared Error or Mean Absolute Error on
the Test predictions.
• Stability
•
•
•
In ratemaking, we not only care about accuracy, but also the stability of our
projections.
• Stability can be measured based on the average absolute change in estimated betas
and/or returns from one year to the next.
Unfortunately, stability and accuracy tend to be opposed to each other.
The Actuary needs to decide if the desire is to be most accurate, most stable, or have some
blend of the two.
19
The Results
Given all of the options mentioned above, there are too many scenarios to look
at in one shot. However, we can make decisions on some of the smaller
adjustments first:
•
Sum Beta – the Sum Beta approach is commonly used today, but in a comparison
across methodologies, the use of this approach does not appear to impact accuracy
much, but it results is much less beta stability.
Sum Beta Methodology Impact
Change in
Change in
Avg ABS Error Beta Stability
FIB
0.002
0.073
SLR (CAPM)
0.001
0.060
SLR (FF3F)
0.004
0.127
Hierarchical Model
0.000
0.034
Better
Same
Worse
20
The Results
•
Seemingly Unrelated Regression (SUR) – The SUR approach is only needed when
solving systems of linear equations. This is only relevant in the Full Information Beta
(FIB) approach. It appears to have minimal impact.
Seemingly Unrelated Regression
Change in
Change in
Avg ABS Error Beta Stability
FIB
(0.000)
(0.001)
•
Weighted Regression – weighting is more meaningful in methodologies that involve
some level of pooling, though in most cases it appears to have minimal, and mixed,
impact:
Weighted Regression
Change in
Change in
Avg ABS Error Beta Stability
FIB
0.001
0.025
SLR - Industry Parameters
0.000
0.022
Hierarchical Model
0.000
(0.030)
Better
Same
Worse
21
The Results
•
Years of Data – Despite common practice, using more than 5 years of data was
uniformly better, with 10 years slightly outperforming 8 years.
Years of Data
5-yr vs. 8-yr
5-yr vs. 10-yr
Change in
Change in
Change in
Change in
Avg ABS Error Beta Stability Avg ABS Error Beta Stability
FIB
0.000
(0.032)
0.000
(0.038)
SLR (FF3F)
(0.001)
(0.116)
(0.001)
(0.146)
SLR - Industry Parameters
(0.000)
(0.034)
(0.000)
(0.042)
Hierarchical Model
0.000
(0.042)
0.000
(0.058)
Quantile
(0.001)
(0.116)
(0.001)
(0.146)
SLR + Shrinkage
(0.000)
(0.079)
(0.000)
(0.100)
Better
Same
Worse
22
The Results
Beta Capping – Testing showed that capping betas at some quantile (to rein in the tail
values) improved the results in all cases. Of the quantile values that I tried, 90%/10%
appeared to perform best.
Beta Estimates - SLR 10-yr
25000
Beta Capping
FIB 10-yr
SLR 10-yr
Quantile 10-yr
SLR IND 10-yr
SLR + Shrinkage 10-yr
Hierarchical 10-yr
Change in
Change in
Avg ABS Error Beta Stability
(0.000)
(0.016)
(0.001)
(0.040)
(0.001)
(0.058)
(0.000)
(0.015)
(0.001)
(0.030)
(0.000)
(0.028)
20000
15000
Frequency
•
Frequency
10000
5000
0
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 More
Market Beta
Better
Same
Worse
23
The Results
Stability vs. Accuracy
0.500
0.450
FF3F SLR (5-yr) - Step 1 of Base
Scenario
0.400
Avg Beta Change
It can be helpful to compare
methodologies by charting the
measures of accuracy and
stability on two axes of a graph.
In this graph, lower on the yaxis and left on the x-axis are
better.
FF3F SLR
0.350
FIB
Quantile
0.300
SLR Industry
0.250
Ridge
Bayesian
0.200
FIB - Base Scenario
0.150
Quantile - 10-yr Capped
0.100
Lasso - 10-yr Capped
Hierarchical
CAPM
MKT
CAPM - 10-yr Capped
Ridge - 10yr Capped
Lasso
Vasicek
0.050
0.0900
0.0920
0.0940
0.0960
Market
0.0980
0.1000
0.1020
0.1040
0.1060
0.1080
0.1100
Avg ABS Diff
24
The Results
Things of note on the previous chart:
•
•
•
•
•
On average, there is nothing less accurate than the Base Scenario.
With the benefit of additional years of data and capping, methodologies that use companyspecific data can be both more accurate and more stable than the current approach.
Both CAPM and simply using the Market Betas perform surprisingly well.
The methodologies that incorporate industry information perform surprisingly poorly.
• This could be related to how we currently identify industry information in the data.
The previous two points, plus the fact that the best-performing methodologies are those
that incorporate parameter shrinkage, long data horizons, and extreme-event capping,
suggest that over time company returns regress towards the market means, and many
methodologies tend to over-fit based on short-term data.
However, the results on the whole portfolio are not identical to the results on a specific company. As
a larger, more-stable company, Allstate is probably more likely to be fit and predicted accurately by
models (see next slide).
25
The Results
Stability vs. Accuracy - Allstate
0.2000
FIB - Base Scenario
0.1800
Avg Beta Change
0.1600
FF3F SLR
0.1400
FIB
Quantile
0.1200
SLR Industry
0.1000
0.0800
Ridge
Quantile 10-yr
Bayesian
Hierarchical
FF3F SLR - 10-yr Capped
CAPM
0.0600
Hierarchical - 10yr Capped
MKT
0.0400
Lasso
Ridge - 10-yr Capped
Vasicek
0.0200
0.0450
0.0460
0.0470
0.0480
0.0490
0.0500
0.0510
0.0520
0.0530
Market
0.0540
0.0550
Avg ABS Diff
26
The Results
Things of note on the previous chart:
• Market Betas or CAPM perform the worst, by far.
• The current method also leaves opportunity for improvement.
• For Allstate-specific data, the gains in accuracy are minimal, but the
stability can be improved greatly.
• Allstate does appear to track better with industry information than the
average company does.
• The best-performing methodologies are still those that incorporate
shrinkage.
27
The Results
The results also appear to be very consistent across years in terms of relative performance:
Average Absolute Error
0.1800
0.1600
0.1400
0.1200
Ridge
0.1000
Quantile
Bayesian
0.0800
Hierarchical
FIB (Base Scenario)
0.0600
0.0400
0.0200
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
28
The Results
Still, the methodologies can provide a range of answers for Allstate:
Calculated Allstate Cost of Equity
20%
18%
16%
Calculated Cost of Equity
14%
SLR (5-yr)
Bayes (10-yr) Capped
12%
Ridge (10-yr) Capped
FIB - Base Scenario
10%
Hierarchical (10-yr) Capped
8%
SLR (10-yr) Capped
Quantile (10-yr) Capped
6%
CAPM (10-yr) Capped
4%
2%
0%
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
29
Conclusions
Final Thoughts:
• We can easily improve on the base scenario.
• The best methods appear to use company-specific information, along
with a 10-year time horizon and Beta capping.
• Using the average of multiple methodologies can help safeguard against
the particularities of any one approach being overly impacted by some
data quirk.
• If averaging methodologies, it makes sense to select ones that are
conceptually different from each other.
• This analysis is just a starting point. There are many other modeling
techniques that could be tested.
30