Illiquidity premium of alternative investments

Illiquidity premium of alternative
investments
FACULTY OF ECONOMICS AND BUSINESS
Mathias Wambeke
r0632420
Master of Financial and Actuarial Engineering
Promoter: Prof. Wim Schoutens
Academic year 2017-2018

Illiquidity premium of alternative
investments
abstract
In this master paper, the illiquidity premium of equity, property and mortgage loans is examined. None
of these illiquidity premia are currently considered in the Solvency II long-term guarantee measures.
We analyze equity expected returns and determine the part of expected returns remunerating for
illiquidity. We furthermore simulate the balance sheet of an insurance undertaking and verify whether
discounting with an equity illiquidity premium preserves a 1/200 probability of ruin. We also examine
the relationship between illiquidity premia and expected and realized returns of property investments.
Finally, we suggest an approach for taking into account the illiquidity premium of mortgage loans
under Solvency II.
FACULTY OF ECONOMICS AND BUSINESS
Mathias Wambeke
r0632420
Promoter: Prof. Wim Schoutens
Master of Financial and Actuarial Engineering
Academic year 2017-2018

3
Table of Contents
1 Introduction.....................................................................................................................................5
2 Illiquidity premium: literature review and impact on Solvency regimes..........................................7
2.1 Equity illiquidity premium .....................................................................................................7
2.2 Property illiquidity premium................................................................................................10
2.3 Mortgage loan illiquidity premium ......................................................................................10
2.4 Illiquidity premium: impact on Solvency II..........................................................................11
2.5 Illiquidity premium: impact on the Insurance Capital Standard (ICS) ................................13
2.5.1 Three bucket approach............................................................................................13
2.5.2 Revised Own Assets with Guardrails (OAG 2.0) approach.....................................14
3 Expected equity returns and illiquidity..........................................................................................15
3.1 Expected equity returns: dividend discount model ............................................................15
3.1.1 Introduction ..............................................................................................................15
3.1.2 Data and descriptive statistics .................................................................................16
3.1.3 Method and results ..................................................................................................18
3.1.4 Discussion and preliminary conclusion....................................................................20
3.2 Expected equity returns: Merton (1974) ............................................................................22
3.2.1 Introduction ..............................................................................................................22
3.2.2 Analysis....................................................................................................................23
3.2.3 Discussion and preliminary conclusion....................................................................25
4 Realized equity returns and illiquidity...........................................................................................26
4.1 Data and descriptive statistics ...........................................................................................26
4.2 Dependence structure equity returns vs. spreads .............................................................29
4.3 Methodology and results....................................................................................................34
4.3.1 Simulation approach 1: assets solely composed of shares.....................................34
4.3.2 Simulation approach 2: assets composed of shares and fixed income ..................36
4.4 Discussion and preliminary conclusion..............................................................................40
5 Property illiquidity premium ..........................................................................................................42
5.1 Data and descriptive statistics ...........................................................................................42
5.2 Expected returns and illiquidity ..........................................................................................44
5.3 Realized returns and illiquidity ...........................................................................................45
6 Mortgage loan illiquidity premium.................................................................................................48
6.1 Mortgage loan valuation and pricing..................................................................................48
6.2 Mortgage loan illiquidity premium under Solvency II .........................................................50

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7 General conclusion.......................................................................................................................54
Appendix 1: data series..........................................................................................................................56
Appendix 2: regression diagnostic tests.................................................................................................57
Appendix 3: alternative regression specifications ..................................................................................60
Appendix 4: replication of the Solvency II Volatility Adjustment.............................................................61
Basic risk-free interest rate curve ......................................................................................61
Currency volatility adjustment............................................................................................62
Appendix 5: copulas equity return vs. corporate bond spread ...............................................................67
Appendix 6: empirical copula US data ...................................................................................................76
List of figures ..........................................................................................................................................77
List of tables ...........................................................................................................................................78
References .............................................................................................................................................79

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1 Introduction
The illiquidity premium is a topic best known in the sphere of corporate and sovereign bonds. A broad
consensus exists in academic literature that bonds remunerate for illiquidity and illiquidity risks, see
e.g. Dick-Nielsen et al. (2012), Kim (2017) or van Loon et al. (2015). This illiquidity premium becomes
most apparent during times of crisis.
Long-term investors, such as insurers, are able to capture this illiquidity premium. When insurers hold
bonds against predictable liabilities, they are more certain to keep their investments until maturity.
Insurers are then not exposed to changes in the illiquidity premium of their assets and can earn this
illiquidity premium by simply holding their bonds until maturity. This illiquidity premium has an impact
on the valuation of predictable insurance liabilities. Indeed, because insurers can capture this illiquidity
premium, it seems sensible to incorporate this illiquidity premium into the discount rate to valuate
predictable insurance liabilities.
These findings are also recognised in the Solvency II valuation rules. Under the Solvency II long-term
guarantee measures, insurers are allowed to discount insurance liabilities with the volatility adjustment
or matching adjustment. Both measures take into account the illiquidity premium of fixed income
investments. However, the illiquidity premium of alternative investments, such as equity, property, or
mortgage loans, is set to zero in the Solvency II regime.
In this master paper, the assumption of zero illiquidity premia for alternative investments will be
challenged. Intuitively, it is not clear why the remuneration of alternative investments would not include
an illiquidity premium. It is difficult to imagine that illiquidity shocks only affect bond markets, whereas
equity and property markets would be completely isolated and unaffected from changes in illiquidity.
Many investors such as insurers or pension funds, who are active in bond markets, also invest in
equity. Most companies that are publicly traded on equity markets, also seek funding from publicly
traded bonds. If equity and bond markets mainly consist of the same market participants, it is likely
that they must share similar liquidity dynamics. Property investments cannot be bought or sold
immediately and trades involve significant search and transaction costs. Hence, it seems unlikely that
property investments would not remunerate for illiquidity, whereas bond yields do include an illiquidity
premium.
There is a general lack of academic research in the field of illiquidity premia for alternative
investments. Research on the illiquidity premium for property or mortgage loans is close to non-
existing, whereas the prevailing academic research on equity illiquidity premia is not compatible with
the methods applied under Solvency II.
It is of utmost practical importance to have an adequate understanding of the illiquidity premium of
alternative investments. Under the current Solvency II regime, own funds are heavily impacted when
illiquidity premia increase. The asset side of insurers then declines due to illiquidity shocks, while this
is not corrected on the liability side. Not recognizing a potential illiquidity premium of alternative
investments may thus introduce a great volatility of Solvency II own funds. Having a better
understanding of the illiquidity premium of alternative investments is important in order to prepare the
long-term guarantee review described in article 77f of the Solvency II Directive. This is also relevant
when considering future alignment with the measures of the IAIS Insurance Capital Standard. Allowing
for a potential illiquidity premium of alternative investments could alleviate the volatile asset valuations
and could thus incentivize insurers to invest in socially useful equity and property projects.
This master paper is structured as follows: chapter 2 discusses the current literature on illiquidity
premia and its impact on valuation under the Solvency II regime and the Insurance Capital Standard.
Chapter 3 analyzes expected equity returns and determines the part of expected returns remunerating
for illiquidity. In chapter 4 we simulate the balance sheet of an insurance undertaking and verify
whether discounting with an equity illiquidity premium preserves a 1/200 probability of ruin. Chapter 5
provides similar analyses for property investments. In chapter 6, the valuation of mortgage loans is

6
discussed and a method for taking into account the mortgage loan illiquidity premium under Solvency
II is presented. Since data for property investments and mortgage loans are only scarcely available,
chapters 5 and 6 are aimed at providing a general view on the potential illiquidity premia, rather than a
detailed quantitative analyses. Chapter 7 concludes.

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2 Illiquidity premium: literature review and impact on Solvency
regimes
Illiquidity can have an significant impact on the pricing of securities. The illiquidity premium is an
excess yield that is required for an illiquid asset compared to a hypothetical, liquid asset with otherwise
equal features. Over the last years, an important amount of literature has been published concerning
the estimation of illiquidity premia, in particular for bonds. In this chapter, an overview will be provided
of recent literature on the illiquidity premium for equity, property and fixed income. The most important
methods to determine illiquidity premia will be summarized. We will also present how illiquidity premia
are considered in the valuation of technical provisions under the Solvency II regime and the Insurance
Capital Standard.
2.1 Equity illiquidity premium
The illiquidity premium of equity investments has been an important topic in academic literature over
the past years. Literature often refers to the liquidity-adjusted capital asset pricing model (LCAPM),
which can be summarised as follows:
𝐸(𝑅𝑖,𝑡 − 𝑅𝑓,𝑡) = 𝐸(𝐶𝑖,𝑡) + 𝜆𝛽𝑖
1
+ 𝜆𝛽𝑖
2
− 𝜆𝛽𝑖
3
− 𝜆𝛽𝑖
4
𝜆 = 𝐸(𝑅 𝑀,𝑡 − 𝐶 𝑀,𝑡 − 𝑅𝑓,𝑡)
𝛽𝑖
1
=
𝐶𝑜𝑣(𝑅𝑖,𝑡, 𝑅 𝑀,𝑡)
𝑉𝑎𝑟(𝑅 𝑀,𝑡 − 𝐶 𝑀,𝑡)
𝛽𝑖
2
=
𝐶𝑜𝑣(𝐶𝑖,𝑡, 𝐶 𝑀,𝑡)
𝛽𝑖
3
=
𝐶𝑜𝑣(𝑅𝑖,𝑡, 𝐶 𝑀,𝑡)
𝛽𝑖
4
=
𝐶𝑜𝑣(𝐶𝑖,𝑡, 𝑅 𝑀,𝑡)
Where 𝑅𝑖 is the return of share 𝑖, 𝑅 𝑀 is the market return, 𝑅𝑓 the risk-free return and 𝐶𝑖 is the relative
trading cost (amount of trading costs divided by the share price). The beta’s can be interpreted in the
following manner:
 𝛽𝑖
1
denotes the traditional market risk as included in the ordinary CAPM, adjusted for trading
costs in the denominator.
 𝛽𝑖
2
is liquidity risk caused by the comovement of share illiquidity with the general market illiquidity.
𝛽𝑖
2
is positively related to expected returns in the LCAPM. It reflects the compensation for holding
a share whose illiquidity rises when the general market is highly illiquid.
 𝛽𝑖
3
captures the comovement of share returns and general market illiquidity. 𝛽𝑖
3
is negatively
related to expected returns in the LCAPM. It reflects the lower expected return for a share whose
return tends to be higher when the general market is illiquid.
 𝛽𝑖
4
reflects the comovement of share illiquidity and general market returns. 𝛽𝑖
4
is negatively
related to expected returns in the LCAPM. It reflects the lower expected returns for shares that
tend to be more liquid in a down market.
The estimation method for the equity illiquidity premium can be broadly summarized in the following
manner. First, an illiquidity measure is calculated for each share in the stock market. Different choices
are possible as illiquidity measure: e.g. the Amihud (2002) illiquidity measure, Holden’s (2009)
effective tick proxy, the zero return proportion by Lesmond et al. (1999) etc. The shares are then
sorted into different portfolios based on this illiquidity measure. Next, the betas of the LCAPM are
estimated for each portfolio. The illiquidity premium is then estimated as:

8
𝑇𝑜𝑡𝑎𝑙 𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑙𝑒𝑣𝑒𝑙 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 + 𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑙𝑒𝑣𝑒𝑙 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 =
𝐸(𝐶 𝑃25,𝑡) − 𝐸(𝐶 𝑃1,𝑡)
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑
𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝜆(𝛽 𝑃25
2
− 𝛽 𝑃1
2
) − 𝜆(𝛽 𝑃25
3
− 𝛽 𝑃1
3
) − 𝜆(𝛽 𝑃25
4
− 𝛽 𝑃1
4 )
Where it is assumed that shares are sorted into 25 portfolios, with P25 indicating the most illiquid
portfolio and P1 indicating the most liquid portfolio.
Alternatively, the equity illiquidity premium can also be estimated by sorting shares into different
portfolios according to their illiquidity measure and calculating the difference in return between the
most liquid and the most illiquid portfolio. This illiquidity premium, often denoted by IML, illiquid-minus-
liquid, can then be further analysed as the dependent variable of a regression model with Fama and
French (1993) risk factors:
𝐼𝑀𝐿 𝑡 = 𝛼𝐼𝑀𝐿 + 𝛽1 𝑅𝑀𝑡 + 𝛽2 𝑆𝑀𝐵𝑡 + 𝛽3 𝐻𝑀𝐿 𝑡 + 𝜀𝑡
Where 𝛼𝐼𝑀𝐿 is denoted as the risk-adjusted illiquidity premium, 𝑅𝑀𝑡 is the excess market return,
𝑆𝑀𝐵𝑡 is the size factor and 𝐻𝑀𝐿 𝑡 is the book-to-market factor. The values obtained for 𝛼𝐼𝑀𝐿 are most
often very similar to the values of 𝐼𝑀𝐿 𝑡. The table below summarizes the methods and results of
different publications regarding the equity illiquidity premium:
Table 1: Literature overview – equity illiquidity premium
Author (year) Illiquidity premium Market Method
Amihud et al.
(2015)
9.61%–9.80% Global Return on illiquid-minus-liquid shares or
the risk-adjusted illiquidity premium from
a six-factor model. Shares are sorted
according to the Amihud (2002) illiquidity
measure.
13.26%–13.93% Emerging
6.79%–6.95% Developed
Hagströmer
et al. (2013)
1.74%–2.08% US Sum of the illiquidity level premium and
the illiquidity risk premium. Shares are
sorted according to Holden’s (2009)
effective tick proxy.
Kim et al.
(2014)
4.20%–4.44% US Difference in alpha return of a Fama-
French factor model between the most
illiquid and most liquid portfolio. Shares
sorted by their pre-raking 𝛽𝑖
2
or 𝛽𝑖
4
of the
LCAPM.
Lee (2011) 2.82%–4.62% Global Difference in alpha return of a Fama-
French factor model between the most
illiquid and most liquid portfolio. Shares
sorted by their pre-raking 𝛽𝑖
2
or 𝛽𝑖
4
of the
LCAPM.
6.65%–6.98% Emerging
In sum, an extensive body of academic research exists which finds large illiquidity premia for equity
investments. However, the methods considered in these papers are not very relevant for the illiquidity
premia that insurers use in the valuation of their liabilities. The illiquidity premia obtained in the current
academic research cannot be used for discounting insurance liabilities for the following reasons:
 Most importantly, current academic research defines equity illiquidity premia as the returns of
highly illiquid portfolios relative to highly liquid portfolios. Such a definition may lead to very high
illiquidity premia, e.g. 9.61%–9.80% estimated by Amihud et al. (2015) for the global market.
However, the return of highly illiquid relative to highly liquid portfolio is of little interest for insurers’

9
valuation purposes. The illiquidity premia currently defined in insurance solvency regimes are
based on general market portfolios, not on highly illiquid sub-portfolios.
 The highly illiquid portfolios considered in current research have a very low market capitalisation.
Furthermore, per definition, these highly illiquid portfolios can only be bought after paying
considerable transaction costs. This makes it unfeasible to generate returns equal to the illiquidity
premia obtained in academic research. As an illustration, Acharya and Pedersen (2005) provide
the following costs and market capitalisations for 25 illiquidity-sorted portfolios:
Table 2: Portfolio characteristics in Acharya and Pedersen (2005)
Portfolio 𝐸(𝐶𝑖,𝑡) (%) Market cap ($bn)
1 0.25 12.5
3 0.26 2.26
5 0.27 1.20
7 0.29 0.74
9 0.32 0.48
11 0.36 0.33
13 0.43 0.24
15 0.53 0.17
17 0.71 0.13
19 1.01 0.09
21 1.61 0.06
23 3.02 0.04
25 8.83 0.02
 In current academic literature, equity illiquidity premia are estimated by referring to proxies for
trading costs, e.g. the Amihud (2002) illiquidity measure, proxies for bid-ask spreads etc. In
Solvency II however, illiquidity premia for bonds are not estimated through trading costs. Under
Solvency II, illiquidity premia are roughly estimated as the excess return that long-term investors
could earn by holding their assets until maturity.
 The majority of academic literature on equity illiquidity premia focuses on the cross section of
equities. For insurers’ valuation purposes, however, it would be more interesting to focus on the
time series of illiquidity premia for a general market portfolio.
 When the equity illiquidity premia in academic research are transformed to equity illiquidity premia
for an average portfolio, the illiquidity premia obtained are often unrealistically low. As an
example, Hagströmer (2013) provides estimations of illiquidity level premium and illiquidity risk
premium over time. By interpolating these results, using the portfolio characteristics reported in
Acharya and Pederson (2005), the illiquidity premium of an average portfolio is obtained. This is
drafted in the figure below. These average equity illiquidity premia are however substantially
below AA bond illiquidity premia as estimated under the methods of Solvency II. This average
equity illiquidity premium thus does not appear to be economically meaningful.
 The methods applied in academic literature are often too complex to be applied in practice.
Ideally, simple proxies (e.g. of part of expected returns or a reference to bond illiquidity premia)
should be put forward such that the equity illiquidity premia can be easily integrated in the
quarterly reporting by insurance undertakings.

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Figure 1: Equity illiquidity premium compiled from Hagströmer (2013), Acharya et al. (2005)
The equity illiquidity premium was calculated by adding the realized illiquidity cost and illiquidity risk premium in Hagströmer
(2013), multiplied by a factor based on Acharya et al. (2005) to take into account average portfolio characteristics, rather than
the most illiquid – most liquid portfolio. Option adjusted spreads are obtained from the BofAML US Corporates BBB and AA
rated indices.
2.2 Property illiquidity premium
Illiquidity premium estimations for real estate only received very limited attention in academic research
up to now. An interesting quantitative estimation of the property illiquidity premium has been made by
Schweizer et al. (2013), who focus on Open-ended Property Funds (OPF). These OPF shares are
redeemable at their net asset value, and thus offer perfect daily liquidity, as long as the fund holds at
least 5% in cash. Redemptions are suspended when this 5% threshold is surpassed. When
redemptions are suspended, these vehicles offer an average discount to net asset value of 6%, which
is partly explained by the authors as an illiquidity premium.
Property investments are an important asset class for insurance undertakings. According to the most
recent Solvency II statics published by EIOPA (2018), direct property investments represent €8,6bn for
Belgian solo insurance undertakings, or €237,8bn for insurance groups of the European Economic
Area. Given the importance of property investments and the lack of research concerning the property
illiquidity premium, this topic will be further examined in this master paper.
2.3 Mortgage loan illiquidity premium
To the best of our knowledge, mortgage loan illiquidity premia have not yet been discussed in
academic literature. Literature on the illiquidity premium for bonds, on the hand, is very well
developed. Some of the estimation methods and results for bond illiquidity premia, summarised below,
may also be relevant for the illiquidity premium of mortgage loans:
Reduced-form models
Reduced-form models are most often regressions where bond spreads are regressed on illiquidity
proxies. Possible illiquidity proxies are the bid-ask spread, the Amihud (2002) illiquidity measure, the
number of zero trading days etc. As an example, van Loon et al. (2015) construct an illiquidity
measure based on bid-ask spreads and use this as an explanatory variable in a regression model for
bond spreads. The illiquidity premium obtained for A rated bonds is 0 to 50 bps in 2004-2008, 50-240
bps in 2008-2010 and 25-140 bps in 2010-2015.
0%
1%
2%
3%
4%
5%
6%
7%
8%
1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
Equity illiq premium AA spread BBB spread

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CDS basis
A credit default swap (CDS) is a contract whereby a protection buyer pays a fee and in return receives
a compensation when a certain entity defaults. Duffie (1999) shows that the CDS premium is equal to
the spread of a corporate floating-rate note (FRN) over a default-free FRN. Empirical research
however often finds a negative difference between the CDS premium and the corporate bonds spread
(the CDS basis). This negative CDS basis is attributed to illiquidity effects. As an example, Kim (2017)
estimates the illiquidity premium as the difference between CDS premia and corporate bond yield
premia from May 2009 to June 2011. The illiquidity premium estimates are 8 bps for AA rated bonds,
15 bps for A ratings, 57 bps for BBB ratings and 134 bps for BB rated bonds.
Structural models
Structural models of default are related to the work of Merton (1974), in which the probability of default
is described based on debt structure. Structural models of default often lead to an underestimation of
credit spreads, which can be attributed to the existence of an illiquidity premium. As an example,
Webber et al. (2007) estimates bond illiquidity premia based on the Merton model and finds illiquidity
premia of 20-130bps for investment grade USD bonds over the period 1997-2007.
Despite the very extensive literature on illiquidity premia for fixed income, there does not appear to be
any research on the illiquidity premium specifically for mortgage loans. Mortgage loans can
nevertheless be an important investment for insurers, e.g. the Solvency II statistics published by
EIOPA (2018) show that Belgian insurers have invested €18,9bn in loans and mortgages to
individuals. The mortgage loan illiquidity premium thus appears to be an important gap in literature;
this topic will be further discussed in this master paper.
2.4 Illiquidity premium: impact on Solvency II
Extensive literature is available on the existence of illiquidity premia for fixed income investments. It
can then be asked whether insurance liabilities may be discounted using similar illiquidity premia. The
viewpoint of the insurance industry, also presented in CEIOPS (2010a) and EIOPA (2013) can be
summarised in the following manner:
 A large share of insurance products (group life and pension, fiscal retail life, disability claims
reserves, annuities...) offer predictable liabilities. These contracts may not be surrendered before
their maturity or at least not without severe surrender penalties.
 These predictable, illiquid insurance liabilities are generally covered by bonds with corresponding
maturities.
 Ample evidence exists that bond spreads compensate, at least partly, for illiquidity. Such an
illiquidity premium becomes most apparent during times of crisis.
 Insurance undertakings holding bonds against predictable liabilities can be more certain that they
will be able to hold their bonds to maturity; i.e. insurance undertakings are not exposed to forced
sales in such a situation. Under these circumstances, insurers are able to capture the illiquidity
premium of fixed income investments over their entire maturity. It is therefore sensible to discount
predictable insurance liabilities, which are matched by assets compensating for illiquidity, with an
illiquidity premium.
 For predictable insurance contracts that are, discounting with an illiquidity premium is fully market
consistent. Indeed, when illiquidity premia are likely to be observed in financial markets and
insurance liabilities are transferred from one insurance undertaking to another, the transfer value
will depend on the predictability of the insurance contracts. Predictable insurance liabilities will
allow the insurance undertaking to capture the illiquidity premium on its assets, which will affect
the transfer value of liabilities.

12
Under Solvency II, it is allowed to take into account an illiquidity premium when valuating technical
provisions, by using either the matching adjustment or the volatility adjustment.
The matching adjustment is described in articles 77b and 77c of the Solvency II Directive. The
matching adjustment is calculated as the spread of a portfolio of fixed income assets of the insurance
undertaking over the risk-free rate, less the fundamental spread. This fundamental spread is
calculated as the sum of the probability of default and the cost of downgrading. The matching
adjustment is however not widely used in continental Europe, because of its restrictive conditions:
 The matching adjustment portfolio can only constitute of fixed income assets. This portfolio
should be managed under a strict asset-liability matching regime.
 The matching adjustment portfolio must be ring fenced. This implies, amongst others, that it is
prohibited to take into account any diversification benefits between the matching adjustment
portfolio and any other portfolio of the insurer.
 The insurance obligations must be highly predictable and must comply with a very severe set of
requirements. In practice, the matching adjustment essentially only applies to annuities. Some of
the requirements for the insurance obligations are: no future premiums, limited mortality risk, no
disability-morbidity risk, nor any lapse risk.
The volatility adjustment, defined in article 77d of the Solvency II Directive, is calculated as:
𝑉𝐴 𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 = 0.65 ∗ [𝑤𝑔𝑜𝑣 ∗ (𝑚𝑎𝑥(𝑆𝑔𝑜𝑣, 0) − 𝑅𝐶𝑔𝑜𝑣) + 𝑤𝑐𝑜𝑟𝑝 ∗ (𝑚𝑎𝑥(𝑆𝑐𝑜𝑟𝑝, 0) − 𝑅𝐶𝑐𝑜𝑟𝑝)]
Where the following notation is used:
 𝑤𝑔𝑜𝑣 is the ratio of the value of government bonds to the total value of the euro reference
portfolio
 𝑆𝑔𝑜𝑣 (before risk correction) is the average spread on government bonds included in the euro
reference portfolio
 𝑅𝐶𝑔𝑜𝑣 is the risk correction corresponding to the portion of the spread 𝑆𝑔𝑜𝑣 that is attributable
to a realistic assessment of the expected losses, unexpected credit risk or any other risk
 𝑤𝑐𝑜𝑟𝑝 is the ratio of the value of bonds other than government bonds, loans and securitisations
to the total value of the euro reference portfolio
 𝑆𝑐𝑜𝑟𝑝 (before risk correction) is the average spread on bonds other than government bonds,
loans and securitisations included in the euro reference portfolio
 𝑅𝐶𝑐𝑜𝑟𝑝 is the risk correction corresponding to the portion of the spread 𝑆𝑐𝑜𝑟𝑝 that is attributable
to a realistic assessment of the expected losses, unexpected credit risk or any other risk.
The volatility adjustment does not take into account the own assets of the insurance undertaking. The
volatility adjustment is based on a currency reference portfolio, reflecting the average asset allocation
of insurers to cover their obligations denominated in that currency. It is only in exceptional
circumstances that the currency volatility adjustment, for products sold in a particular country, will be
adjusted for the higher spreads observed in that country. Contrary to the matching adjustment, the
volatility adjustment does not take into account the full spread over the risk-free rate, but only 65% of
the spread over the risk-free rate, less the fundamental spread. The volatility adjustment is however
applicable without any severe conditions and is thus very widely used by European insurers, contrary
to the matching adjustment.
It is important to note that neither the matching adjustment, nor the volatility adjustment, allow for an
illiquidity premium of alternative investments. The matching adjustment portfolio can only consist of
fixed income assets of which the cash flows cannot be changed by the issuers of the assets or any
third parties. This means that equity, property or mortgage loans (unless they offer a make-whole
clause) are excluded from the matching adjustment portfolio.

13
The volatility adjustment is based on a Euro Area reference portfolio which includes bonds, equity,
property, and mortgage loans. However, a positive illiquidity premium is only assigned to the sub-
portfolios of government bonds and corporate bonds. Alternative investments can make up an
important part of the Euro area reference portfolio, but their illiquidity premium is set to zero.
2.5 Illiquidity premium: impact on the Insurance Capital Standard (ICS)
The Insurance Capital Standard (ICS), developed by the International Association of Insurance
Supervisors (IAIS), is a global capital standard with the final aim of achieving comparable outcomes
across jurisdictions. The ICS is applicable to Internationally Active Insurance Groups (IAIGs),
insurance groups which are active in multiple jurisdictions, with total assets of more than USD 50bn, or
gross written premiums of more than USD 10bn. The IAIS has issued the ICS 2.0 technical
specifications in May 2018, which are currently used by IAIGs for confidential reporting.
The ICS 2.0 technical specifications allow to discount insurance liabilities with an illiquidity premium. In
the sections below, we focus on two particular methods: the “three bucket approach” and the “revised
Own Assets with Guardrails” (OAG 2.0) approach.
2.5.1 Three bucket approach
The standard method of discounting with an illiquidity premium is the “three bucket approach”,
summarized in table 3.
Table 3: ICS three bucket approach
Top bucket Middle bucket General bucket
Liability criteria
Only annuities
No future premiums
No surrender options
Some conditions of the
top bucket are relaxed:
contractually defined
future premiums allowed;
immaterial surrender
options allowed
No conditions
Ring-fencing criteria
1
Ring-fenced required Ring-fenced required
No ring-fencing
requirement
Asset-liability
management (ALM)
criteria
No material asset-liability
mismatch
Assets and liabilities
matched according to
duration bands
2
No ALM criterion
Asset portfolio
composition used in the
calculation of the illiquidity
premium
Own assets
Own assets, where
spreads are based on
bond indices
Representative portfolio
per currency
Application ratio
3
100% 90% 80%
Comparable Solvency II
discount method
Matching Adjustment
No comparable Solvency
II discount method
Volatility Adjustment
The eligible assets for each of the three buckets are solely composed of fixed income securities.
Mortgage loans with a call option, equity, or property investments are all excluded from the calculation
of the illiquidity premium.
1
Ring-fencing indicates that the assets and liabilities of the particular bucket should be identified and managed
separately, without being used to cover losses arising from other business
2
Duration bands are periods of 3 years in which assets and liabilities should be matched.
3
The application ratio is the part of the adjusted spread (i.e. illiquidity premium) that is used to discount insurance
liabilities. discount rate = risk free rate + application ratio * adjusted spread

14
2.5.2 Revised Own Assets with Guardrails (OAG 2.0) approach
Besides the three bucket approach, the ICS 2.0 technical specifications also allow to apply the
“revised Own Assets with Guardrails” (OAG 2.0) illiquidity premium. The revised Own Assets with
Guardrails method is a proposal from the insurance industry, whereby the illiquidity premium is based
on the own assets of the insurance undertaking, without major restrictions on asset eligibility, and
using a 100% application ratio. Compared to the three bucket approach, the revised Own Assets with
Guardrails approach allows to include an illiquidity premium for a broader range of fixed income
assets, including (mortgage) loans with call options and convertible notes. The OAG 2.0 also includes
an illiquidity premium for a list of equity and alternative long-term duration assets, including:
 Equities
 Hedge Funds
 Private equity
 Real estate (for investment purposes)
 Infrastructure (equity like)
 Other alternative long duration (equity like) assets
The illiquidity premium for these equity and alternative long duration assets is equal to the illiquidity
premium of a BBB bond, subject to a 200bps long-term average equity spread cap. This BBB illiquidity
premium is based on bond indices, corrected for default risk. An additional quantitative guardrail
regarding the use of equity and alternative long duration assets is to multiply the BBB bond illiquidity
premium with:
𝑚𝑖𝑛 (1,
𝑙𝑜𝑛𝑔 𝑡𝑒𝑟𝑚 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑟𝑎𝑡𝑖𝑜
𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝑎𝑠𝑠𝑒𝑡𝑠 𝑟𝑎𝑡𝑖𝑜
)
where:
𝑙𝑜𝑛𝑔 𝑡𝑒𝑟𝑚 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 =
𝑠𝑢𝑚 𝑜𝑓 𝑢𝑛𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤𝑠 > 12 𝑦𝑒𝑎𝑟𝑠
𝑡𝑜𝑡𝑎𝑙 𝑢𝑛𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑 𝑙𝑖𝑎𝑏𝑖𝑙𝑡𝑖𝑦 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤𝑠
𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝑎𝑠𝑠𝑒𝑡𝑠 𝑟𝑎𝑡𝑖𝑜 =
𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦 𝑎𝑛𝑑 𝑜𝑡ℎ𝑒𝑟 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝑙𝑜𝑛𝑔 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑠𝑠𝑒𝑡𝑠
𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑙𝑙 𝑒𝑙𝑖𝑔𝑖𝑏𝑙𝑒 𝑎𝑠𝑠𝑒𝑡𝑠
Hence, this quantitative guardrail restricts the use of the BBB illiquidity premium for short-term
liabilities or for asset portfolios with an excessive amount of equities or alternative investments.
The publications and technical specifications concerning the (revised) Own Assets with Guardrails
approach do not provide any justification for assuming a BBB bond illiquidity premium for equity and
alternative investments. Hence, in this master paper, we will further explore the link between bond
illiquidity premia and illiquidity premia for alternative investments. We will research the relationship
between fixed income and alternative investment returns, and verify whether it is prudent to discount
liabilities, backed by alternative assets, with a bond illiquidity premium.

15
3 Expected equity returns and illiquidity
In this chapter, we will analyse expected equity returns and determine whether these expected returns
can be explained through illiquidity measures.
3.1 Expected equity returns: dividend discount model
3.1.1 Introduction
The expected, or required stock return is the value of 𝑟𝑒 that satisfies:
𝑃𝑡 = ∑
𝔼[𝐷𝑡+𝑇]
(1 + 𝑟𝑒) 𝑇
∞
𝑇=1
where 𝑃𝑡 is the current stock price and 𝐷𝑡+𝑇 is the dividend at time 𝑡 + 𝑇. Assuming dividends grow at
a constant rate 𝑔, the formula above simplifies to:
𝐷𝑡+1
𝑃𝑡
= 𝑟𝑒 − 𝑔
Following Koutmos (2015), we write 𝐷1 = 𝐸𝑡+1 ∗ (1 − 𝑅𝑅) where 𝐸𝑡+1 are the earnings at time 𝑡 + 1
and 𝑅𝑅 is the earnings retention rate. We also impose that 𝑔 = 𝑅𝑂𝐸 ∗ 𝑅𝑅 where 𝑅𝑂𝐸 is the return on
equity. The formula above then becomes:
𝐸𝑡+1 ∗ (1 − 𝑅𝑅)
𝑃𝑡
= 𝑟𝑒 − 𝑅𝑂𝐸 ∗ 𝑅𝑅
We also know that, in a steady state economy, 𝑟𝑒 = 𝑅𝑂𝐸. The formula above thus simplifies to:
𝐸𝑡+1
𝑃𝑡
= 𝑟𝑒
Hence, we find that, under general conditions, the expected stock equals the earnings yield.
Many publications have considered similar measures of expected stock returns and have researched
their relationship between expected returns, realized returns and measures of risk. E.g. Koutmos
(2015) finds a cointegration relationship between expected returns and various measures of volatility.
Li et al. (2013) find that expected returns strongly predict realized returns, especially at longer horizons
of 2 to 4 years. Pastor et al. (2008) construct expected stock returns at country level and find a
positive risk-return relation. Lee et al. (2009) find that expected returns are positively related to world
market beta, idiosyncratic volatility, financial leverage, and book-to-market ratios, and are negatively
related to firm size.
Many publications on the equity illiquidity premium such as Amihud et al. (2015), Hagströmer et al.
(2013) or Kim et al. (2014) are based on measures of realized stock returns. In this chapter, however,
the illiquidity premium will be based on measures of expected stock returns, also denoted as “required
return” or “implied cost of capital”. We use expected stock returns, instead of realized stock returns for
the following reasons:
 Expected returns display information about the economic outlook, risk preferences, liquidity
conditions, and prospects for other investments. Ex post realized returns, on the other hand, are
per definition heavily influenced by economic conditions, news, risks, and liquidity conditions that
were unknown ex ante. Realized returns hide expectations about future returns and thus hide the
factors that investors use to price stocks ex ante.
 Illiquidity premia are best known in the sphere of fixed income markets. Insurers and insurance
regulators have, up to now, only used fixed income illiquidity premia for the valuation of insurance
liabilities. Fixed income or bond illiquidity premia calculations are based on bond yields (expected

16
returns), and are never based on the first differences of their prices (realized returns). Bond yields
reflect the discount factor for which the bond price is equal to the current value of the future
coupon and principal payments. This bond yield also reveals investor’s future expectations and
perceived risks. Expected stock returns can be defined in an analogous manner, namely the
discount factor for which the equity price is equal to the current value of the future dividend
payments. If we want to make meaningful comparisons between bond illiquidity premia and equity
illiquidity premia, we have to make sure equity illiquidity premia are based on a measure of
expected stock returns, and not on realized stock returns.
3.1.2 Data and descriptive statistics
As a measure of equity expected returns, we choose the earnings yield of the Euro Stoxx 50 index.
More precisely, we use the reciprocal of the variable “Adjusted Positive Price/Earnings” (INDX_ADJ_
POSITIVE_PE) from Bloomberg. The adjusted positive price/earnings are calculated as the last price
divided by the positive earnings per share. The index positive earnings per share (EPS) are the index
EPS calculated excluding negative trailing 12 month equity earnings. EPS are the trailing 12 month
EPS before extraordinary Items.
As an alternative measure of equity expected returns, we also use the dividend yield of the Euro Stoxx
50 index. More precisely, we extract the variable “Dividend 12 Month Yld – Gross” (EQY_DVD_YLD_
12M) from Bloomberg. The dividend 12 month yield – gross is calculated as the gross trailing 12
month dividends per share divided by the last price. All cash dividend types are included in this yield
calculation.
As a measure for illiquidity, we calculate the illiquidity premium of BBB rated 10 year corporate bonds:
𝐵𝐵𝐵 10 𝑦𝑒𝑎𝑟 𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝐵𝐵𝐵 10 𝑦𝑒𝑎𝑟 𝑦𝑖𝑒𝑙𝑑 − 10 𝑦𝑒𝑎𝑟 𝑠𝑤𝑎𝑝 − 𝑟𝑖𝑠𝑘 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
This definition of “illiquidity premium” corresponds to the calculations implicit in the Solvency II volatility
adjustment and matching adjustment. Indeed, the Solvency II volatility adjustment and matching
adjustment are essentially calculated as a bond spread minus a risk correction. The formula above
specifically applies this definition, implicit in Solvency II, to BBB 10 year corporate bonds. We choose
BBB corporate bonds as these are proposed as an illiquidity premium for equities under the “own
assets with guardrails” approach of the Insurance Capital Standard. We choose the 10 year maturities,
the longest maturity available, as equities are generally considered to be long-term investments,
supposed to back long-term insurance liabilities. We obtain the annual yield of the iBoxx € Corporates
BBB 10+ index from the Markit website and obtain the 10 year EUR swap rate from Bloomberg. We
calculate the risk correction as the weighted average of the 10 year BBB financial and non-financial
fundamental spreads published by EIOPA at year-end 2017. The weights we choose are the
weightings of financials (13%) and non-financials (87%) in the iBoxx € Corporates BBB 10+ index at
year-end 2017. We thus obtain a risk correction of 0.77%.
We also use the negative CDS basis as an alternative illiquidity measure:
−𝐶𝐷𝑆 𝑏𝑎𝑠𝑖𝑠 = 𝑐𝑜𝑟𝑝 𝑏𝑜𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 − 𝑠𝑤𝑎𝑝 − 𝐶𝐷𝑆 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
The negative CDS basis has been used in previous literature as an illiquidity proxy, see e.g. Kim
(2017) or Longstaff et al. (2005). The basic idea is that, by covering corporate bonds with their
corresponding CDS contracts, a portfolio is obtained where default risk is eliminated and only illiquidity
risk remains. Advantages of this approach include that it is “model free” and relatively easy to
calculate. An important caveat is however that CDS contracts may also bear counterparty risk and
thus do not completely eliminate default risk. Such counterparty risk may have been concern during
e.g. the global financial crisis in the years 2008-2009, but with the EMIR reforms and central clearing
obligations, this counterparty risk may now be of lesser importance. We measure the CDS premium
through the Markit iTraxx EUR Generic index, downloaded from Bloomberg. This CDS index
comprises 125 equally weighted credit default swaps on investment grade European corporate

17
entities. We also use the annual yield of the iBoxx € Corporates index from the Markit website and the
5 year swap rate from Bloomberg in the calculation of the negative CDS basis.
Table 4 presents the different variables used in the regression analysis.
Table 4: Descriptive statistics
Descriptive statistics of the Euro Stoxx 50 earnings yield (Earn), Euro Stoxx 50 dividend yield (Div), iBoxx € Corporates BBB
10+ annual yield (BBB10), BBB 10 year illiquidity premium (BBBilliq), Markit iTraxx EUR Generic index (ITRX), negative CDS
basis (CDSbasis) and the 1, 5, and 10 year swap rates.
Earn Div BBB10 BBBilliq ITRX CDSbasis Swap1 Swap5 Swap10
Obs 245 245 189 174 160 160 234 234 234
Minimum 4.310 1.400 1.790 0.125 0.204 0.087 -0.266 -0.155 0.265
Maximum 13.030 7.963 8.500 4.127 2.020 2.957 5.382 5.709 5.950
1
st
Quartile 6.190 2.654 3.310 0.663 0.525 0.292 0.395 0.955 1.750
3
rd
Quartile 8.440 3.966 5.980 1.562 1.060 0.649 3.529 4.057 4.473
Mean 7.433 3.346 4.819 1.279 0.828 0.583 2.076 2.688 3.250
Median 7.320 3.347 5.260 1.074 0.741 0.418 2.095 2.969 3.548
Variance 2.700 1.324 2.827 0.740 0.171 0.270 2.962 2.992 2.703
Stdev 1.643 1.151 1.681 0.860 0.414 0.519 1.721 1.730 1.644
Skewness 0.554 0.606 -0.198 1.390 0.648 2.731 0.218 -0.148 -0.265
Kurtosis -0.068 1.344 -0.754 1.658 -0.158 7.661 -1.216 -1.297 -1.195
Figure 2 displays the times series of the two illiquidity proxies: the BBB 10 year illiquidity premium
(=BBB 10 year corporate bond spread minus risk correction) and the negative CDS basis (=corporate
bond spread minus CDS premium). Both series display clear peaks around the financial crisis
(beginning of 2009) and the European sovereign debt crisis (year-end 2011).
Figure 2: Illiquidity proxies (%)
The BBB 10 year illiquidity premium (BBBilliq) is the BBB 10 year corporate bond spread minus the risk correction. The negative
CDS basis (CDSbasis) is the corporate bond spread minus the CDS premium.
Table 5 presents the correlation matrix of the variables considered in the cointegration analysis.
0
1
2
3
4
5
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018
BBBilliq CDSbasis

18
Table 5: Correlation matrix
Correlation matrix of the Euro Stoxx 50 earnings yield (Earn), Euro Stoxx 50 dividend yield (Div), BBB 10 year illiquidity premium
(BBBilliq), Markit iTraxx EUR Generic index (ITRX), negative CDS basis (CDSbasis) and the 1, 5, and 10 year swap rates.
Earn Div BBBilliq ITRX CDSbasis Swap1 Swap5 Swap10
Earn 1.000 0.328 0.550 0.525 0.515 0.204 0.164 0.163
Div 0.328 1.000 0.866 0.895 0.825 -0.445 -0.468 -0.463
BBBilliq 0.550 0.866 1.000 0.891 0.843 -0.114 -0.061 -0.002
ITRX 0.525 0.895 0.891 1.000 0.640 -0.175 -0.120 -0.050
CDSbasis 0.515 0.825 0.843 0.640 1.000 0.095 0.178 0.232
Swap1 0.204 -0.445 -0.114 -0.175 0.095 1.000 0.966 0.932
Swap5 0.164 -0.468 -0.061 -0.120 0.178 0.966 1.000 0.992
Swap10 0.163 -0.463 -0.002 -0.050 0.232 0.932 0.992 1.000
3.1.3 Method and results
Table 6 presents the unit root tests for the variables that will be subsequently used in the regression
analysis. The first columns of table 6 display the Augmented Dickey-Fuller (ADF) tests, the last
columns of table 6 display the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. The null hypothesis of
the ADF test is that the variable has a unit root. The null hypothesis of the KPSS test is that the
variable is level or trend stationary. As shown in table 6, all variables are non-stationary. This means
that a regression in the level of these variables is only allowed when the variables are cointegrated.
Table 6: Unit root tests
ADF test
p-value
ADF test
KPSS test
p-value
KPSS test
Earnings yield -3.2139 0.08773 1.7234 <0.01
Dividend yield -2.7974 0.2438 0.62706 0.02018
Swap 5 years -3.165 0.09618 4.838 <0.01
Swap 10 years -2.3545 0.4282 3.9994 <0.01
BBB 10 year illiquidity premium -2.9572 0.1765 0.49976 0.04172
-CDS basis -3.4897 0.04563 0.57275 0.02528
In order to verify whether the illiquidity proxies have a significant impact on expected stock returns, we
estimate the following regressions:
𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑦𝑖𝑒𝑙𝑑 = 𝛼 + 𝛽1 ∗ 𝑆𝑤𝑎𝑝10 + 𝛽2 ∗ 𝐵𝐵𝐵 10 𝑦𝑒𝑎𝑟 𝑖𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 (1)
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 = 𝛼 + 𝛽1 ∗ 𝑆𝑤𝑎𝑝5 + 𝛽2 ∗ (−𝐶𝐷𝑆 𝑏𝑎𝑠𝑖𝑠) (2)
The coefficients are estimated with least squares, and we subsequently check for heteroscedasticity,
autocorrelation and normality of the residuals. We also perform functional form or model
misspecification tests. The results of these diagnostic tests are presented in detail in annex 2. Newey-
West errors are used as we find heteroscedasticity and autocorrelation in the residuals. The results of
these regressions are presented in table 7. The Phillips-Ouliaris cointegration test shows that, for the
two regression models, we can reject the null hypothesis that the variables are not cointegrated.
The regression results demonstrate that the illiquidity proxies have a highly significant influence on
equity expected returns. The 𝛽2 coefficients are significant at the 1% level and are thus significantly
different from 0. The 95% confidence interval for 𝛽2 in regression model (1) is [0.546, 1.285] and is

19
thus not significantly different from 1. This means that the BBB 10 year illiquidity premium has a similar
impact on equity expected returns compared to the impact that the 10 year illiquidity premium has on
BBB 10 year corporate yields. The 95% confidence interval for 𝛽2 in regression model (2) is [1.299,
1.689] and is thus significantly higher than 1.This means that equity expected returns (proxied by the
dividend yield) are heavily impacted by illiquidity (proxied by the negative CDS basis).
The adjusted R² for both regression models is higher than 60%. Hence, even though our models are
fairly simple and only apply two explanatory variables, they are able to explain more than 60% of the
variation in expected equity returns.
Table 7: Regression results
This table presents the results of the results of (1) regressing earnings yield against the BBB 10 year illiquidity premium and (2)
regressing dividend yield against the negative CDS basis. The last column displays the Phillips-Ouliaris cointegration test.
Values in parentheses are the t values. (*) , (**) and (***) denote significance at the 10%, 5% and 1% level at least, respectively.
Regression
model
𝛼 𝛽1 𝛽2 Adj. R² PO test
(1)
4.69349 0.63266 0.91577 0.6290 -27.536**
(13.3759)*** (6.1263)*** (4.8892)***
(2)
3.221601 -0.089604 1.494034 0.6996 -31.687**
(21.6763)*** (-1.8971)* (15.1356)***
The coefficient for the 10 year swap rate in model (1) is 0.63 and significant at the 1% level. The
coefficient for the 5 year swap rate in model (2) is negative and is not significant at the 5% level. It is
not clear why these coefficients for swap rates (as a proxy for risk-free rates) are much lower than 1,
or even negative. Appendix 3 presents regression model (1) with alternative specifications for the
swap rate. It appears that, with a swap rate of a shorter maturity, the adjusted R² increases (up to
70%) and the coefficient for the BBB 10 year illiquidity premium increases (up to 1.05), but the
coefficients for the swap rates do not meaningfully change. It is left for future research to examine why
expected stock returns are not highly aligned with risk-free rates.
We also present a model where earning yields are regressed on the negative CDS basis in appendix
3. An important caveat is however that the null hypothesis of no cointegration in the Phillips-Ouliaris
test could only be rejected at the 10% level and not at the 5% level. Hence, it is probably less
appropriate to draw conclusions from this model. In any way, the results of this regression are overall
very similar to models (1) and (2): the coefficient of the illiquidity proxy is 1.25 and significant at the 1%
level, the adjusted R² is 0.57.
The figures below present the part of equity expected returns that can be explained by the illiquidity
proxies. Figure 3 plots the earnings yield together with 𝛽2 ∗ BBB 10 year illiquidity premium. Figure 4
plots the dividend yield together with 𝛽2 ∗ (-CDS basis). As shown in the figures below, the illiquidity
proxies explain a large part of expected stock returns, especially during the financial crisis in the years
2008-2009 and during the European sovereign debt crisis in the years 2011-2012. Figures 3 and 4
display that the expected stock returns and illiquidity proxies are tightly connected, as also suggested
by the cointegration of between these variables.

20
Figure 3: Earnings yield and illiquidity (%)
Euro Stoxx 50 earnings yield together with 𝛽2 ∗ BBB 10 year illiquidity premium (BBBilliq) i.e. the part of earnings yield that can
be explained by illiquidity
’
Figure 4: Dividend yield and illiquidity (%)
Euro Stoxx 50 dividend yield together with 𝛽2 ∗ (-CDS basis) i.e. the part of dividend yield that can be explained by illiquidity
3.1.4 Discussion and preliminary conclusion
Previous literature on equity illiquidity premia has focussed on realized stock returns. We argue that
this is inappropriate for our purposes since realized stock returns hide investor’s expectations which
are used ex ante in stock price valuation. The use of realized stock returns also contrasts with the
widespread use of bond yields (i.e. expected returns) in determining the fixed income illiquidity premia.
We expand the existing literature on equity illiquidity premia through a cointegration analysis of
expected stock returns on proxies for illiquidity. We find that the BBB 10 year illiquidity premium has a
significant influence on expected stock returns (proxied by the earnings yield). We also find that the
impact of this illiquidity premium on expected stock returns is of comparable magnitude to the impact
of the BBB 10 year illiquidity premium on expected bond returns. We obtain similar results when
regressing the dividend yield (as a proxy for expected stock returns) on the negative CDS basis (as a
proxy for illiquidity).
A possible criticism on the analysis presented in this chapter could be that our measures of expected
stock returns are not sufficiently forward looking. Indeed, our measures of earnings yield and dividend
yield are essentially the amounts of past dividends or earnings divided by the current stock price.
Ideally, we should have used estimates of future dividends or earnings. However, we have chosen not
0,00
2,00
4,00
6,00
8,00
10,00
12,00
14,00
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018
Earnings yield BBBilliq
0
1
2
3
4
5
6
7
8
9
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016
Dividend yield CDSbasis

21
to use such future estimates due to the lack of publicly available dividend or earnings growth forecasts
with a sufficiently long horizon.
Furthermore, even though our measures of expected stock returns could be criticized for not being
sufficiently forward looking, it should be noted that the same criticism could hold for the illiquidity
premia implicit in the Solvency II volatility adjustment and matching adjustment. Indeed, the illiquidity
premia currently applied in Solvency II are based on a stable risk correction, which does not change
based on forward looking estimates of default or states of the economy.
Finally, we argue that dividend or earnings yields based on the past 12 months data do not differ
heavily from forward looking estimates of dividend or earnings yields. As an illustration, figure 5
compares the dividend yield estimates of the Euro Stoxx 50 based on dividends of the past 12 months
(actual), dividend estimates of the current year (Y Est) and dividend estimates two years in the future
(Y+2 Est). Data are obtained from Bloomberg. It appears that the three series are tightly connected
and display peaks and troughs at the same time periods.
Figure 5: Dividend yield estimates (%)
Dividend yield estimates of the Euro Stoxx 50 based on dividends of the past 12 months (actual), dividend estimates of the
current year (Y Est) and dividend estimates two years in the future (Y+2 Est). Source: Bloomberg.
The regression models presented in this chapter are parsimonious and easy to understand. However,
one could also argue to include more variables that proxy for illiquidity or other risks. Other useful
proxies for illiquidity include:
 Intermediary equity, cf. Muir (2013)
 Stock market turnover, cf. Dick-Nielsen et al. (2012)
 Broker-dealer leverage, cf. Adrian et al. (2014)
 Primary dealer capital ratio, cf. He et al. (2017)
 Spread between on the run vs. off the run treasury bonds, cf. Goyenko et al. (2014)
Proxies for fundamental risks could include e.g. stock market volatility, implied volatility or VIX, the
variance risk premium (i.e. difference between implied and realized variance, cf. Bollerslev, et al.
2009), macro-economic variables such as GDP growth, inflation, term spread or the uncertainty in the
estimation of macro-economic forecasts.
However, the inclusion of some of these variables into our regressions would probably go beyond the
problem statement at hand. The main goal of this chapter was to verify whether a part of equity returns
are linked to proxies for illiquidity that are frequently used in fixed income literature and that would fit in
the Solvency II framework. The purpose of this chapter was not necessarily to disentangle expected
equity returns into different risks or disentangle illiquidity (risk) vs. other risks. After all, the illiquidity
premium implicit in the Solvency II volatility adjustment or matching adjustment is not calculated by
separating illiquidity vs. fundamental risks. Under Solvency II, the fundamental risks are already taken
into account into the Solvency Capital Requirement (SCR): e.g. listed equities generally have a 39%
capital requirement, BBB 5 year corporate bonds generally have a 12,5% capital requirement. Since
the fundamental risks are already taken into account in the SCR, the illiquidity premia implicit in the
0
2
4
6
8
10
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018
SX5E Index Actual SX5E Index Y Est SX5E Index Y+2 Est

22
Solvency II calculations do not separate illiquidity vs. other risks. The illiquidity premia implicit in the
Solvency II volatility adjustment and matching adjustment only account for fundamental (default) risks
by subtracting a fixed risk correction from the bond spread. The same reasoning could hold for the
equity illiquidity premium: as fundamental risks are already taken into account in the SCR, the main
question is whether part of equity returns may be attributed to illiquidity, as discussed in this chapter.
3.2 Expected equity returns: Merton (1974)
3.2.1 Introduction
In this section, we present an alternative way to determine equity expected returns. Following the
framework of Merton (1974), we assume that firms are leveraged with a fixed amount of debt 𝐵𝑡. Firms
default when their asset value 𝐴 𝑡 becomes lower than their debt value 𝐵𝑡. Hence, stock 𝑆𝑡 and debt 𝐵𝑡
values are both contingent claims on the same assets 𝐴 𝑡 and must therefore depend on comparable
risk characteristics.
Following Campello et al. (2008) we can write the stock return at time 𝑡, 𝑅 𝑆,𝑡 as:
𝑅 𝑆,𝑡 =
𝑑𝑆𝑡
𝑆𝑡
= (
𝜕𝑆𝑡
𝜕𝐴 𝑡
) (
𝐴 𝑡
𝑆𝑡
) (
𝑑𝐴 𝑡
𝐴 𝑡
)
where 𝑆𝑡 is the stock value at time 𝑡 and 𝐴 𝑡 is the asset value at time 𝑡. We also know that the
expected stock premium satisfies
𝐸𝑡[𝑅 𝑆,𝑡] − 𝑟𝑡 = −𝑐𝑜𝑣(𝑚 𝑡, 𝑅 𝑆,𝑡)
where 𝑚 𝑡 is the stochastic discount factor. Hence, we obtain:
𝐸𝑡[𝑅 𝑆,𝑡] − 𝑟𝑡 = −𝑐𝑜𝑣 (𝑚 𝑡,
𝜕𝑆𝑡
𝜕𝐴 𝑡
𝐴 𝑡
𝑆𝑡
𝑑𝐴 𝑡
𝐴 𝑡
) (1)
We can write a similar formula for the bond return at time 𝑡, 𝑅 𝐵,𝑡:
𝑅 𝐵,𝑡 =
𝑑𝐵𝑡
𝐵𝑡
= (
𝜕𝐵𝑡
𝜕𝐴 𝑡
) (
𝐴 𝑡
𝐵𝑡
) (
𝑑𝐴 𝑡
𝐴 𝑡
)
where 𝐵𝑡 is the bond value at time 𝑡 and 𝐴 𝑡 is the asset value at time 𝑡. We also know that the
expected bond premium satisfies
𝐸𝑡[𝑅 𝐵,𝑡] − 𝑟𝑡 = −𝑐𝑜𝑣(𝑚 𝑡, 𝑅 𝐵,𝑡)
which can be rewritten as
𝐸𝑡[𝑅 𝐵,𝑡] − 𝑟𝑡 = −𝑐𝑜𝑣 (𝑚 𝑡,
𝜕𝐵𝑡
𝜕𝐴 𝑡
𝐴 𝑡
𝐵𝑡
𝑑𝐴 𝑡
𝐴 𝑡
) (2)
By dividing (1) by (2), we obtain
𝐸𝑡[𝑅 𝑆,𝑡] − 𝑟𝑡
𝐸𝑡[𝑅 𝐵,𝑡] − 𝑟𝑡
=
𝜕𝑆𝑡
𝜕𝐵𝑡
𝐵𝑡
𝑆𝑡
And finally, we obtain a conditionally linear relationship between expected equity premium and
expected bond premium:
𝐸𝑡[𝑅 𝑆,𝑡] − 𝑟𝑡 =
𝜕𝑆𝑡
𝜕𝐵𝑡
𝐵𝑡
𝑆𝑡
(𝐸𝑡[𝑅 𝐵,𝑡] − 𝑟𝑡)
Hence, the expected equity premium equals the expected bond premium multiplied by the elasticity of
the equity market value vs. bond market value. This simple formula again demonstrates the relation

23
between expected equity returns and bond spreads, as also demonstrated in the section 3.1.3, or as
suggested in the ICS 2.0 own assets with guardrails approach, cf. section 2.5.2.
3.2.2 Analysis
In this sub-section, we aim to calculate the expected bond premium, as well as the elasticity between
stock and bond values, in order to obtain a series of expected equity returns.
Our measure of bond yields is the annual yield of the iBoxx € Corporates index, downloaded from the
Markit website. We obtain data from January 1999 up to June 2018. We obtain the corporate spread
by subtracting the 5 year swap rate, obtained through Bloomberg. We need to correct the corporate
spreads for expected defaults. We first examine the relationship between our series of corporate
spreads and 5 year cumulative default rates, as presented in table 8. No clear relationship between
the two series appears, and the correlation between our measures of corporate spreads and default
rates is even negative at -0.485. Therefore, we choose not to model the relationship between
corporate spreads and defaults and rather assume a fixed correction for expected defaults. We use
the S&P (2018) average 5 year cumulative default rate for investment grade European corporates
(0.36%), which equals an annualized rate of 0.0721%.
Table 8: Corporate spread and default rates
Corporate spreads are calculated as the iBoxx € Corporates annual yield minus the 5 year swap rate. Default rates are the 5
year cumulative global corporate default rates among all investment grade ratings obtained from S&P (2018).
Year 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
Default 2.33 2.12 2 1.09 0.33 0.64 1.2 1.01 1.2 1.17 0.63 0.09 0.09 0.22 0.15
Spread 0.834 0.695 0.87 0.9485 1.228 0.748 0.682 0.545 0.465 1.125 4.11 1.6 1.759 3.016 1.544
In sum, our measure of expected bond premia is:
𝐸𝑡[𝑅 𝐵,𝑡] − 𝑟𝑡 = 𝑐𝑜𝑟𝑝𝑜𝑟𝑎𝑡𝑒 𝑦𝑖𝑒𝑙𝑑 − 5 𝑦𝑒𝑎𝑟 𝑠𝑤𝑎𝑝 − 0.0721%
Next, we need to measure the elasticity of stock market value with respect to bond market value.
Campello et al. (2008) calculate monthly elasticities at firm level by dividing the stock return by the
bond return and regressing this ratio on firm leverage, stock volatility, and risk-free rates. These
regressions however result in low R² and thus leave a large part of the variation in elasticity
unexplained. The regression results may also be biased due to extremely high or low elasticity
observations (when the bond return is close to zero) or negative elasticity observations. We therefore
choose to measure an average elasticity directly from observations of stock and bond returns.
We measure equity returns as the 1 year percentage return of the Euro Stoxx 50 gross return index,
obtained from Bloomberg. The Euro Stoxx 50 gross return index is only available since January 2001.
We thus replicate the value of the gross return index during the years 1999-2000 by multiplying the
returns of the Euro Stoxx 50 net return index with a fixed percentage. We measure bond returns as the
1 year percentage return of the iBoxx € Corporates Total Return Index Level. We choose 1 year
returns, rather than monthly returns, in order to obtain a more stable elasticity series. As bond returns
are often close to 0%, the elasticity series inevitably shows large peaks and troughs, as displayed in
figure 6. We therefore calculate a trimmed mean of the elasticity series, excluding the top 5% and
bottom 5% observations, yielding an average value of 2.07. This trimmed mean (2.07) is relatively
close to the median value of the elasticity series (1.71).

24
Figure 6: Elasticity stock market vs. bond market values
Elasticity is calculated as the 1 year percentage return of the Euro Stoxx 50 gross return index divided by 1 year percentage
return of the iBoxx € Corporates Total Return Index Level.
Figure 7 displays the main results of this section. The equity risk premium is calculated as the
elasticity of the equity market value with respect to the market value, multiplied by the debt risk
premium. The equity expected return is calculated as the equity risk premium, added to the 5 year
swap. The 5 year swap was chosen as a risk-free rate measure for reasons of consistency with our
bond risk premium, where the 5 year swap was also used as a risk-free rate.
Figure 7: Equity expected return based on Merton (1974)
Debt risk premium = iBoxx € Corporates index annual yield – 5 year swap – 0.0721% (expected default). The equity risk
premium is calculated as the elasticity of the equity market value with respect to the market value, multiplied by the debt risk
premium. The equity expected return is calculated as the equity risk premium, added to the 5 year swap.
The expected stock returns, displayed in figure 7, show quite some resemblance with the Euro Stoxx
50 earnings yields and dividend yields displayed in figures 3 to 5. Just like the earnings yields and
dividend yields, the expected stock returns in figure 7 first display a fairly stable path up to the year
2007, followed by two clear peaks at the financial crisis of 2008-2009 and the sovereign debt crisis of
year-end 2011. In general, the Euro Stoxx 50 dividend yields and earnings yields are the respective
lower and upper bounds of the expected stock returns in figure 7. In more recent years, the expected
stock returns, displayed in figure 7, decreased significantly, even below the observed dividend yield.
This may indicate that the elasticity to be applied for more recent years should be higher than the
overall trimmed mean of 2.07 used in this section. However, the high volatility in observed elasticity
makes it difficult to prove any change in the elasticity regime.
-100
-50
0
50
100
150
200
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018
0
2
4
6
8
10
12
14
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
Debt risk premium Equity risk premium Equity expected return

25
3.2.3 Discussion and preliminary conclusion
In this section, we provided a straightforward formula for the equity risk premium, based on the
framework of Merton (1974) and the research of Campello et al. (2008). It was demonstrated that the
equity premium equals the debt premium, multiplied by the elasticity of the equity market value with
respect to the debt market value. Hence, we provided analytical evidence for the intuition that spreads
and equity risk premia should be linked. This model is also a rationalization of the ICS 2.0 own assets
with guardrails approach (cf. section 2.5.2) where a BBB spread is used to discount liabilities backed
by equity investments. This model also provides an explanation for our cointegration analyses (cf.
section 3.1), where we found a significant relationship between expected stock returns and BBB
corporate bond illiquidity premia. A major advantage of the model presented in this section is that it is
easily understandable and only relies on a limited number of data series.
The equity risk premium model presented in this section relies on the elasticity of equity market values
with respect to the bond market values. This elasticity series however often presents negative values
and is be highly volatile. Due to this high volatility, it becomes difficult to distinguish different elasticity
regimes over time. We were therefore unable to provide a conclusive explanation for the low expected
stock returns observed in recent years. This is an important disadvantage of the model presented in
this section.
In this section, we used a trimmed mean to calculate the elasticity of the equity market values with
respect to the bond market values. Rather than using an average elasticity level, it could be argued to
work with a (regression) model to explain the variations in the elasticity series over time. However, due
to negative elasticity observations, or extremely high or low elasticity values, such models can be
highly biased and thus are unlikely to provide added value. A trimmed mean is a simple measure and
has the advantage of leading to a stable elasticity value.

26
4 Realized equity returns and illiquidity
In the previous chapter, we analyzed the impact of illiquidity proxies on expected stock returns. We
demonstrated that equity expected returns are highly related to measures of illiquidity. In general, we
found that the equity illiquidity premium is of comparable magnitude to the BBB corporate bond
illiquidity premium. In this chapter, we examine how prudent it would be for insurers to take into
account this illiquidity premium when valuing technical provisions. Under Solvency II, insurers are
required to have sufficient own funds to cover a 1 year 99.5% Value-at-Risk (VaR). Therefore, in this
chapter, we examine the probability of ruin for an insurer that applies the equity illiquidity premium to
discount future obligations. We simulate values of the equity illiquidity premium together with equity
returns and apply these to a Solvency II balance sheet in run-off. We then examine whether the
Solvency II capital requirements provide a sufficient buffer in order to allow the inclusion of an equity
illiquidity premium.
4.1 Data and descriptive statistics
Our series of equity returns are based on the MSCI world gross return index. We obtain monthly index
values in local currency as of year-end 1969 from the MSCI website. CEIOPS (2010b) has previously
used the MSCI world index in its advice on the Solvency II equity capital requirements. Hence, we
have chosen the same index in order to make sure our simulated equity returns are aligned with the
Solvency II calibrations.
The Solvency II capital requirement for global equities is 39%, cf. article 169§1(b) of the Solvency II
Delegated regulation. A symmetric adjustment (SA) is added to the 39% capital charge with the aim of
alleviating capital requirements in downturns and increasing the capital requirements in times of bullish
equity markets. The symmetric adjustment is described in article 172 of the Solvency II Delegated
Regulation and is calculated as:
𝑆𝐴 = 𝑚𝑎𝑥 [−10% ; 𝑚𝑖𝑛 [+10% ;
1
2
∗ (
𝐶𝐼 − 𝐴𝐼
𝐴𝐼
− 8%)]]
where SA is the level of the symmetric adjustment, CI is the current level of the equity index and AI
denotes the average of the equity index over the last 36 months. We choose the MSCI world price
index in order to calculate the symmetric adjustment.
Figure 8 displays the level of the MSCI world gross return index (left scale) together with the levels of
the symmetric adjustment (right scale).
Figure 8: MSCI world index and symmetric adjustment (SA)
We obtain the annual yield of the iBoxx € Corporates index and the iBoxx € Corporates BBB index
from the Markit website. We use a series of monthly values starting at the beginning of 1999 up to
-15%
-10%
-5%
0%
5%
10%
15%
0
1.000
2.000
3.000
4.000
5.000
6.000
7.000
1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
MSCI World SA

27
June 2018. In order to calculate the corporate bond spreads, we subtract the 5 year EUR swap rate
downloaded from Bloomberg.
For robustness checks, we also use equity or spread data from the United States. US data may be
interesting because it is often available with a significantly longer horizon compared to EU data. We
obtained monthly values of Moody's Seasoned Aaa and Baa Corporate Bond Yields from FRED.
These series start in January 1919 and we obtained data up to year-end 2017. We calculate the
corporate bond spread by subtracting Aaa from Baa corporate bond yields. We choose the S&P500
total return index to measure US equity returns. As the S&P500 total return index is only available
since January 1988, we replicate the total return index based on the values of the S&P500 price index
and S&P500 dividend yield as follows:
𝑆𝑃𝑋𝑇𝑡 = 𝑆𝑃𝑋𝑇𝑡−1 ∗
𝑆𝑃𝑋𝑡
𝑆𝑃𝑋𝑡−1
∗ (1 + 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 𝑡)
1
12
Where 𝑆𝑃𝑋𝑇𝑡 is the replicated value of the S&P500 total return index at month t, 𝑆𝑃𝑋𝑡 is the S&P500
price index at month t and 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 𝑡 is calculated as the amount of dividends distributed over
the past 12 months divided by the S&P500 price index.
In our simulations of fixed income assets, we also need a series of default probabilities. We use the
historical default rates of investment-grade and speculative-grade global issuers from table 1 in S&P
(2018). Yearly default rates are calculated as the weighted average of investment grade defaults
(weight of 95%) and speculative grade defaults (weight of 5%). The 5% weight was chosen such that
the average historical default rate (0.29%) is close to the risk correction of the Volatility Adjustment
fixed income portfolio (0.30%). We previously discussed that long-term default rates appear
disconnected from spread observations (cf. table 8). It is therefore assumed in our simulations that
defaults are independent from spreads or other variables. The default rates applied in the simulations
of the total fixed income portfolio are presented in table 9.
Table 9: Historical default rates (%)
Yearly default rates calculated as the weighted average of investment grade defaults (weight of 95%) and speculative grade
defaults (weight of 5%). Source: S&P (2018) table 1.
Year
Default
rate
Year
Default
rate
Year
Default
rate
Year
Default
rate
1981 0.03% 1991 0.69% 2001 0.71% 2011 0.12%
1982 0.39% 1992 0.31% 2002 0.87% 2012 0.13%
1983 0.23% 1993 0.13% 2003 0.35% 2013 0.12%
1984 0.33% 1994 0.15% 2004 0.13% 2014 0.07%
1985 0.22% 1995 0.22% 2005 0.10% 2015 0.14%
1986 0.43% 1996 0.09% 2006 0.06% 2016 0.24%
1987 0.14% 1997 0.18% 2007 0.05% 2017 0.12%
1988 0.19% 1998 0.32% 2008 0.58%
1989 0.44% 1999 0.44% 2009 0.81%
1990 0.54% 2000 0.54% 2010 0.15%
The assumption of 5% speculative grade issuers and the resulting default rates can be considered
prudent since:
 The historical default rates are based on global issuers rather than European issuers. European
default rates have been significantly lower compared to global issuers over the past decades.
 We assume no recovery rate, i.e. loss given default (LGD) = 100%
 In our simulations, we also apply the corporate default rates also to the sovereign debt portfolio
 The corporate reference portfolio of the Volatility Adjustment only has 3% speculative grade
issuers.

28
In this chapter, we will also use a replication of the Solvency II Volatility Adjustment (VA). Our
replication of the Solvency II VA is largely based on the analysis of Wambeke (2017), which is
presented in appendix 4 for information purposes. Wambeke (2017) exactly follows the Solvency II
methodology for the replication of the VA. In this paper, however, we will calculate two modified
versions of the Solvency II VA, as explained in the paragraphs below.
In the calculations of the Solvency II VA, the risk-corrected currency spread is multiplied by 65%. This
factor of 65% is usually denoted as the “application ratio”. In this paper, we will also use a modified
version of the VA with an application ratio of 100%, which we denote as “VA100%”. This series is
equal to the risk-corrected currency spread calculated by Wambeke (2017).
The Solvency II volatility adjustment is based on a currency reference portfolio, in which particular
weights are given to “government” and “other” bonds. Assets which are not a part of the government
or other portfolio are assumed to have a zero illiquidity premium. The portfolio of “other” bonds is
further decomposed into rating and sector (financial or non-financial). Table 10 presents the reference
portfolio characteristics of the Solvency II VA at year-end 2017.
Table 10: Volatility Adjustment reference portfolio weights
Central Govts Other assets
27.4% 43.8%
Financial Non-financial
Rating AAA AA A BBB BB AAA AA A BBB BB
Allocation 23% 13% 20% 8% 2% 3% 7% 11% 12% 1%
Duration 7.7 7.0 5.6 4.8 4.8 7.0 8.5 6.1 5.6 4.8
The reference portfolio presented in table 10 was also used by Wambeke (2017) in his replication of
the VA. In this paper, we calculate another modified version of the VA assuming that all assets which
were not a part of the “government” or “other” bonds are equities. We assume a BBB illiquidity
premium for these equities, in line with the analysis of the previous chapter and in line with the “Own
Assets with Guardrails” approach of the ICS. We assume that the BBB illiquidity premium for equities
can be split into a 17% BBB financial and 83% BBB non-financial illiquidity premium
4
. Hence, we
recalculate the VA, increasing the proportion of “other” bonds to 72.6% and increasing the proportion
of BBB financial and BBB non-financial bonds in the “other” bond category. This portfolio composition
is summarized in table 11. We assume a 100% application ratio and denote the resulting VA series as
“VABBB100%”.
Table 11: VABBB100% reference portfolio weights
Central Govts Other assets
27.4% 72.6%
Financial Non-financial
Rating AAA AA A BBB BB AAA AA A BBB BB
Allocation 14% 8% 12% 12% 1% 2% 4% 7% 40% 1%
Duration 7.7 7.0 5.6 4.8 4.8 7.0 8.5 6.1 5.6 4.8
4
The weight of 17% is based on the average composition of the iBoxx € Corporates index during the years 2008-
2011. We have chosen the years 2008-2011 as this period displays particularly high spreads, and thus are most
important for determining portfolio weights.

29
Figure 9 displays the time series of the VA (as replicated by Wambeke, 2017), the VA100% (VA series
with a 100% application ratio), and VABBB100% (VA series with a 100% application ratio and a BBB
illiquidity premium for alternative investments).
Figure 9: Volatility Adjustment replications
Table 12 presents the descriptive statistics of the main variables used in this chapter.
Table 12: Descriptive statistics
Descriptive statistics of the MSCI world gross 1 year returns (MSCI), € BBB corporate bond spread (BBB spread, in
percentages), € corporate bond spread (Corp spread, in percentages), VA series with a 100% application ratio (VA100%), VA
series with a 100% application ratio and a BBB illiquidity premium for alternative investments (VABBB100%) and the overall
default rate (Default)
MSCI BBB spread Corp spread VA100% VABBB100% Default
Nobs 571 231 234 149 149 37
Minimum -0.420 0.540 0.380 -0.001 -0.002 0.000
Maximum 0.512 5.822 4.662 0.030 0.053 0.009
1. Quartile 0.011 1.059 0.781 0.001 0.002 0.001
3. Quartile 0.212 2.181 1.478 0.008 0.014 0.004
Mean 0.106 1.774 1.244 0.006 0.010 0.003
Median 0.131 1.445 1.031 0.004 0.007 0.002
Variance 0.026 1.057 0.565 0.000 0.000 0.000
Stdev 0.163 1.028 0.751 0.007 0.011 0.002
Skewness -0.584 1.877 2.194 1.352 1.661 1.014
Kurtosis 0.412 3.880 5.746 1.739 2.944 -0.019
4.2 Dependence structure equity returns vs. spreads
In this section, we discuss the dependence structure of stock returns (MSCI world gross returns at
different horizons) versus spread or illiquidity premia measures. Discovering this dependence is an
essential preparation for our Solvency II balance sheet simulations. In the previous chapter, we found
evidence that measures for illiquidity (illiquidity premia derived from spreads) are closely linked to
expected stock returns. In this section, we want to analyze whether illiquidity premia are linked to
realized stock returns. Discovering a particular form of dependence, or a lack of dependence, will
determine the outcome of our simulations. If we do not find a positive dependence at the upper tail of
the illiquidity measures (i.e. if highly positive spreads or illiquidity premia are not linked to higher equity
-1%
0%
1%
2%
3%
4%
5%
6%
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017
VA VA100% VABBB100%

30
returns), our simulations of the Solvency II balance sheet will show that it is not appropriate to take into
account an illiquidity premium for equity investments.
We proceed as follows. We first determine the empirical cumulative distribution function of the equity
return and spread series. We then fit this data to different copulas, by using the function fitCopula()
from the R package copula. We consider the Gumbel, Frank, Clayton, Gaussian (normal) and
Student copulas. We then calculate the Akaike Information Criterion (AIC) for the different fitted
copulas and visually inspect the plots of random simulations of the fitted copulas. We select the copula
that minimizes the AIC for the different horizons of equity returns.
We note that that analysing copulas is identical for bond spreads or bond illiquidity premia (calculated
as the bond spread minus the risk correction). We thus use these terms interchangeably in this
section.
Figures 10, 11 and 12 display the fitted copulas for 6 year MSCI world returns
5
versus BBB spreads,
Corporate bond spreads and the Volatility Adjustment, respectively. All figures display a scatterplot of
the raw data (titled “data”), a plot of the pseudo-observations (uniform-transformed variables, titled
“emp copula”), and also display plots of 800 random simulations of the fitted copulas (titled “Gumbel”,
“Frank”, “Clayton”, “Norm” and “t”).
The plots of the empirical data and pseudo-observations display a clear tail dependence: high spreads
are linked to high stock returns. This finding is most apparent for the 6 year stock returns but can also
be found for the other return horizons. There does not appear to be a significant lower tail
dependence: low spread regimes can be linked to high or low future stock returns. This indicates that
the Gumbel copula, which implies a high upper tail dependence and no lower tail dependence, is likely
to be a good fit for our data. This is confirmed by the AIC values presented in tables 13, 14 and 15.
The Gumbel copula minimizes the AIC and thus presents the best fit, whereas the Clayton copula
(which presents a positive lower tail dependence and no upper tail dependence) presents the highest
AIC and thus the worst fit.
The results for the BBB spread, Corporate spread or Volatility Adjustment reported in tables 13-15 are
quite similar. In all cases, the Gumbel copula is favoured compared to the other copula models. The
Gumbel copula appears to display a higher dependence for the corporate spread data (parameter
estimate of 3.33) compared to the BBB spread series (parameter estimate of 3.16). The Gumbel
copula displays a very high dependence for the Volatility Adjustment data (parameter estimate of
5.18). However, it should be noted that the volatility adjustment is only available for a short horizon
(since 2004). The copulas fits are thus based on only 94 overlapping monthly observations, which
makes the parameter estimates less trustworthy.
5
Figures 10-12 display the 6 year MSCI world net return +39%. Hence, this is actually a series of own funds for
insurer, solely investing in shares, after a 6 year horizon.

31
Figure 10: Copula BBB spread vs. 6 year stock returns
BBB rated corporate bond spreads are displayed on the horizontal axis. 6 year MSCI world net returns (+39%) are displayed on
the vertical axis. This figure displays the raw data (titled “data”), pseudo-observations (uniform-transformed variables, titled “emp
copula”), and plots of 800 random simulations of the fitted copulas (titled “Gumbel”, “Frank”, “Clayton”, “Norm” and “t”).
Table 13: Copula AIC and parameters – BBB spread vs. 6 year returns
Top row displays the Akaike Information Criterion. The bottom row presents the estimated parameters, for the t copula the
parameter 𝜌 is displayed
Gumbel Frank Clayton Normal t
AIC -234.353360 -199.2399 -83.150645 -184.467138 -187.993757
parameter 3.163322 10.1468 1.378842 0.839019 0.8508184

32
Figure 11: Copula Corporate spread vs. 6 year stock returns
Corporate bond spreads are displayed on the horizontal axis. 6 year MSCI world net returns (+39%) are displayed on the
vertical axis. This figure displays the raw data (titled “data”), pseudo-observations (uniform-transformed variables, titled “emp
copula”), and plots of 800 random simulations of the fitted copulas (titled “Gumbel”, “Frank”, “Clayton”, “Norm” and “t”).
Table 14: Copula AIC and parameters – Corp spread vs. 6 year returns
AIC -252.453006 -218.97292 -94.087803 -194.386795 -212.224536
parameter 3.328614 10.91878 1.558436 0.8457864 0.8714885

33
Figure 12: Copula Volatility Adjustment vs. 6 year stock returns
The Volatility Adjustment is displayed on the horizontal axis. 6 year MSCI world net returns (+39%) are displayed on the vertical
axis. This figure displays the raw data (titled “data”), pseudo-observations (uniform-transformed variables, titled “emp copula”),
and plots of 800 random simulations of the fitted copulas (titled “Gumbel”, “Frank”, “Clayton”, “Norm” and “t”).
Table 15: Copula AIC and parameters – Volatility Adjustment vs. 6 year returns
AIC -217.906192 -217.10399 -93.403958 -175.6770 -190.779657
parameter 5.175792 20.85993 2.640326 0.9272733 0.9470668
Appendix 4 presents the plots and AIC estimates for the copulas of corporate bond spreads versus
stock returns at different horizons. For the majority of data series, the Gumbel copula presents the
best fit. We therefore choose the Gumbel copula in our simulations for all stock return horizons. Table
16 summarizes the Gumbel copula parameter estimates for the different spread series and stock
return horizons. The parameters estimates for the VA series are often very high, but these are based

34
on a very limited number of observations. Therefore, they will not be considered further in our
simulations.
Table 16: Gumbel copula parameters
Fitted parameters for the Gumbel copula, at different MSCI world stock return horizons.
1 year 2 year 3 year 4 year 5 year 6 year 7 year 8 year 9 year
BBB spread 1.2050 1.5071 1.4911 1.6867 2.3976 3.1633 2.0255 1.4967 1.3019
Corp spread 1.2279 1.6032 1.6231 1.7201 2.3812 3.3286 2.2272 1.6487 1.4625
VA 1.1673 1.6708 2.0320 2.3970 3.3225 5.1758 2.6521 2.3672 3.2188
Table 17 displays the values of Kendall's tau and Spearman's rho for the fitted Gumbel copulas.
Values are presented for BBB spread and corporate spread series, and at different stock return
horizons. Overall, the dependence appears highest for the corporate spread series, and at 6 and 7
year stock returns.
Table 17: Tau and rho of Gumbel copulas
Values of Kendall's tau and Spearman's rho for the fitted Gumbel copulas, at different MSCI world stock return horizons.
1 year 2 years 3 years 4 years 5 years 6 years 7 years 8 years 9 years
𝜏
BBB 0.1701 0.3365 0.3294 0.4071 0.5829 0.6839 0.5063 0.3319 0.2319
Corp 0.1856 0.3762 0.3839 0.4186 0.5800 0.6996 0.5510 0.3934 0.3162
𝜌
BBB 0.2524 0.4790 0.4697 0.5707 0.7713 0.8635 0.6899 0.4730 0.3405
Corp 0.2749 0.5313 0.5412 0.5849 0.7684 0.8757 0.7378 0.5535 0.4525
Next to simulations based on the fitted Gumbel copulas, we will also consider simulations based on
empirical copulas, meaning that we will draw from the historical pseudo-observations. Simulating from
the empirical copulas may be useful, because some stock return horizons (e.g. 1year or 9 years), do
not appear to match very well with any of the theoretical copulas considered. This may indicate that
some series have a special dependence structure that cannot be readily mapped with the
conventional copula models. The empirical copula incorporates the actual dependence form
automatically and may thus be a better choice compared to the parametric models that could be prone
to misspecification.
4.3 Methodology and results
4.3.1 Simulation approach 1: assets solely composed of shares
In this subsection, we will simulate a Solvency II balance sheet where all assets are assumed to be
shares, modelled by the historical returns of the MSCI world gross return index. We set up a Solvency
II balance sheet where the initial SCR ratio (i.e. the ratio of own funds to the Solvency Capital
Requirement) is 100% i.e. the insurance undertaking respects the Solvency II capital requirement, but
does not hold any excess own funds.
The probability of ruin is compared under two discounting approaches. Either insurance liabilities are
solely discounted with the swap rates observed at year-end 2017, or liabilities are discounted with the
swap rates of year-end 2017, to which the BBB illiquidity premium is added. This BBB illiquidity
premium is simulated based on the historical observations of BBB corporate spreads minus a risk
correction of 0.75%.
4.3.1.1 Empirical copula
In this first simulation, we model the dependence structure though the empirical copula of stock
returns versus EUR BBB corporate spreads. Stock returns are simulated from the historically observed

35
returns from the MSCI world gross return index, where we take into account the effect of the Solvency
II symmetrical adjustment. That is, we construct a historical series of Solvency II own funds:
𝑆𝐼𝐼 𝑂𝐹𝑡+𝑇 = (
𝑀𝑆𝐶𝐼𝑡+𝑇
𝑀𝑆𝐶𝐼𝑡
− 1) + 39% + 𝑆𝐴
where 𝑆𝐼𝐼 𝑂𝐹𝑡+𝑇 denotes the Solvency II own funds at the end of the simulation horizon 𝑇,
𝑀𝑆𝐶𝐼 𝑡+𝑇
𝑀𝑆𝐶𝐼 𝑡
denotes the gross return of the MSCI index, 39% is the standard Solvency II capital requirement for
global equities cf. article 169§1(b) of the Solvency II Delegated Regulation and 𝑆𝐴 is the symmetrical
adjustment cf. article 172 of the Solvency II Delegated Regulation. Table 18 displays the simulation
results.
Table 18: Probability of ruin – empirical copula
The table below presents the probability of ruin for an insurer discounting liabilities with a swap + BBB illiquidity premium (top
row) or discounting liabilities solely with the swap rate (bottom row). Probabilities of ruin are given for horizons of 1 year up to 9
years.
BBB illiq 0.78% 0.41% 0.00% 0.00% 0.00% 0.00% 0.00% 0.61% 0.00%
swap 0.78% 0.41% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
The table below presents the probability of ruin for an insurer discounting liabilities with a swap + BBB illiquidity premium, given
that the simulation starts in a high spread regime.
Spread
regime
>2.5% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
>3.0% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
>3.5% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
The results displayed in table 18 indicate that it is quite prudent to apply a BBB illiquidity premium for
equity investments. The probabilities of ruin hardly change when discounting with swap + illiquidity
premium rather than solely discounting with the swap rates.
The simulation results at the 8 year horizon do indicate a probability of ruin >0.5%. These cases of
negative Solvency II own funds appear due to a particular historical observation where the MSCI world
has a return of -27.7%. This negative return is simulated together with an illiquidity premium of 0.92%.
This negative stock return together with a modest illiquidity premium <1% should actually not result in
negative own funds at the end of the 8 years simulating horizon. However, the return of -27.7% was
historically observed together with a symmetric adjustment of -5.3%. As we assume that the insurance
undertaking starts with a 100% SCR ratio and does not hold excess own funds, the simulation starts
with a capital charge of 39%-5.3%. The symmetric adjustment dos not appear to work well in this
case, as a negative symmetric adjustment is observed together with highly negative stock returns. If
the symmetric adjustment had been 0% at this point in time, the simulation would have led to positive
own funds at the end of the 8 year simulation horizon.
The bottom panel of table 18 displays the probabilities of ruin under the condition that the simulations
start in a high spread regime. These statistics are presented to verify whether any of the positive
probabilities of ruin presented in the upper panel are due to the application of a relatively high illiquidity
premium. As presented in the bottom panel of table 18, none of the negative own funds are caused by
applying a high illiquidity premium.
The copula used in this section is based on relatively limited time series spanning 20 years. Therefore,
as a sensitivity analysis, we repeat the same simulation in appendix 6 with a copula based on US
data, spanning a history of 99 years. The results remain essentially unchanged.

36
4.3.1.2 Gumbel copula
In this subsection, we apply a random simulation based on the fitted Gumbel copula for BBB spreads.
In section 4.3.1.1, we pointed out that the symmetrical adjustment may have an inappropriate impact
on capital requirements and the resulting own fund simulations. Therefore, in this subsection, we
assume a zero symmetrical adjustment in our own fund simulations. Results are presented in table 19.
Table 19: Probability of ruin – Gumbel copula
The table below presents the probability of ruin for an insurer discounting liabilities with a swap + BBB illiquidity premium (top
row) or discounting liabilities solely with the swap rate (bottom row). Probabilities of ruin are given for horizons of 1 year up to 9
years.
BBB illiq 0.36% 0.86% 0.82% 0.00% 0.00% 0.00% 0.00% 0.01% 0.41%
swap 0.23% 0.61% 0.76% 0.00% 0.00% 0.00% 0.00% 0.00% 0.24%
The table below presents the probability of ruin for an insurer discounting liabilities with a swap + BBB illiquidity premium, given
that the simulation starts in a high spread regime.
Spread
regime
>2.5% 0.40% 0.21% 0.07% 0.00% 0.00% 0.00% 0.00% 0.07% 0.56%
>3.0% 0.59% 0.11% 0.00% 0.00% 0.00% 0.00% 0.00% 0.10% 0.89%
>3.5% 0.53% 0.15% 0.00% 0.00% 0.00% 0.00% 0.00% 0.14% 0.75%
The probabilities of ruin displayed in table 19 remain relatively low and are comparable to the results
displayed in table 18, where we made use of the EU empirical copula. We observe some probabilities
of ruin >0.5% in the upper panel of table 18, but these probabilities of ruin of >0.5% also appear when
we solely discount with the swap rate.
The bottom panel of table 19 displays a relatively high probability of ruin for high spread regimes at a
horizons of 1 and 9 years. These default probabilities did not appear when using the empirical copula,
as presented in table 18. This may indicate that the Gumbel copula does not capture the dependence
structure very well for the high spread regimes at these horizons.
4.3.2 Simulation approach 2: assets composed of shares and fixed income
In this section, we assume that the assets of the Solvency II balance sheet are composed of equity
and fixed income investments. In line with the VA reference portfolios of year-end 2016 (cf. table 10),
we assume that 71.2% of assets are fixed income, the remainder of the portfolio is assumed to consist
of equity investments. As in the previous section, equity returns are simulated from the historically
observed returns of the MSCI world gross return index.
The Solvency Capital Requirement (SCR) is calculated according to the standard formula, for which
calibration are given in the Solvency II Delegated Regulation. Modified durations of the corporate bond
portfolio are calculated by taking into account the durations of the VA reference portfolio at year-end
2016, divided by the corporate bond yields of the Markit iBoxx financial and non-financial indices at
year-end 2016. The SCR spread is calculated by multiplying the modified durations by the standard
spread shocks of article 176§3 of the Solvency II Delegated Regulation. The SCR equity is calculated
by multiplying the equity market value (i.e. 28.8% of the balance sheet total) by the standard 39%
equity shock of article 169§1(b) of the Solvency II Delegated Regulation. We pointed out in section
4.3.1.1 that applying the symmetrical adjustment (SA) may have inappropriate consequences on
capital requirements and the resulting own funds simulations. Therefore, in this section, we will not
adjust the 39% capital charge with the SA. No other SCR (sub-) modules are taken into account. The
SCR spread and SCR equity are added by taking into account the standard correlation of 75%, cf.

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