Aspects of natural hedging in the
Solvency II framework
Bertus Veurink
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: B. Veurink MSc
Student nr: 10681418
Email: abveurink@gmail.com
Date: January 13, 2015
Supervisor: Drs. R. Bruning AAG Second reader: Prof. dr. ir. M. H. Vellekoop
Natural hedging in a Solvency II Framework — Bertus Veurink iii
Abstract
A change in mortality rates has an opposite effect on the expected market value of annuities and term insurances. In calculations for the SCRlife two scenarios affect the
mortality rates, i.e. the calculation of longevity risk and mortality risk. This opposite mortality rate effect is only partly accounted for in the LTGA specifications.
That is why in this research the effect of combining these (and other) products for one insured into one portfolio for which the SCRlife, risk margin and SCR-ratio are
calculated. It is shown that all three quantities can benefit significantly from calculating them in this combined portfolio and that the Solvency II correlation of −0.25 between longevity risk and mortality risk is not adequate in the presented situation, because the anti-correlation needs to be stronger. We construct a base case portfolio composition for one insured, which is meant to reflect a realistic Dutch insurer’s situation, but then for only one insured. The results can be extended for more insureds with more than one life insurance product. The natural hedging effects of combining a multiple of insureds in one life portfolio is not considered, only the effect of more than one life product per insured. Assets are considered, which can also influence the result to some extent. Because we consider a full Solvency II balance sheet (based on LTGA-specifications), this yields a practical result, which can be translated to the situation of Dutch insurers. A projection is made in which the natural hedging potential in future projection years is made visible. To test the sensitivity of our result, the chosen input parameters are varied, to find out whether they have a large impact on the natural hedging potential. That is where we deviate from the base case.
It is found that these sensitivities mainly influence lapse risk and by consequence also the fraction of longevity risk and mortality risk in the SCRlife. This means that by
this indirect effect the natural hedging potential (measured in increase in SCR-ratio due to natural hedging) is sensitive to changes in the sum insured, yield and the mortality rates used in pricing.
Keywords Natural hedging, Longevity risk, Mortality risk, Solvency II, Life insurance, Risk Management, LTGA
Preface v
1 Outline 1
2 Introduction 3
2.1 Theoretical background . . . 3
2.1.1 Introduction . . . 3
2.1.2 Theoretical natural hedging . . . 4
2.1.3 Theoretical definition of research problem . . . 7
2.2 Relevant research . . . 7 3 Model description 11 3.1 Model . . . 11 3.1.1 Payment Cashflows . . . 11 3.1.2 Cost cashflow . . . 13 3.1.3 Premium cashflows . . . 14 3.2 Assumptions . . . 16 3.3 QIS-5/LTGA-model . . . 18 3.3.1 introduction . . . 18
3.3.2 Best Estimate valuation and Yield Curve . . . 18
3.3.3 SCR and Risk Margin . . . 19
3.3.4 Asset choices and valuation of BE at later projection years . . . 22
3.3.5 LTGA balance sheet . . . 24
4 Model Results 25 4.1 introduction . . . 25
4.2 Best Estimate model results . . . 25
4.3 Shock Scenarios and SCR . . . 27
4.4 Projection of SCR and Risk Margin . . . 29
4.5 Total market value balance sheet . . . 31
4.6 Implicit natural hedging in a combination of regular pension and partner pension and its impact . . . 35
4.7 Sensitivities in results . . . 36
4.7.1 Result sensitivity due to yield curve . . . 36
4.7.2 Result sensitivity due to different mortality rates in pricing than in valuation . . . 39
4.7.3 Result sensitivity due to a change in insured amounts . . . 41
5 Conclusion and Discussion 45 5.1 Conclusions . . . 45
5.2 Discussion . . . 47
Preface
After starting my actuarial career at ASR verzekeringen in the second half of 2011 along with that came the opportunity of studying for the job I am Performing. The roots of my education lie in Mathematics and Physics with a specialization in Physical Oceanogra-phy. The actuarial world is not clearly related, but of course the mathematical backbone consists of the same ideas, only different details arise in the practical implementation. In February 2012 this journey started with the premaster Actuarial Sciences at the University of Amsterdam. Now, almost three years later, an actuarial research is nearing its end, where learned actuarial insight and skills have been put to the test. This research started after two years of theoretical courses which were interesting from the practical job background I already had acquired. These were in itself doable next to the work I had to do right alongside of it for ASR. This was not for a small part caused by lavish study environment from the workspace. This enabled me to attend a lot of college hours. After this a thesis was also part of the curriculum. This was an entirely different expe-rience. Next to the regular work flow also effort needed to be put in to this extensive project, for which responsibility was entirely my own, with less clear weekly targets. This has been challenging. I felt discouraged by the need to prove my skills in an academic thesis once again. But, when I started and dived into the subject, it was interesting and pulled me in.
However, I again noticed that it is easy to go off track with the writing of your thesis. One really needs to focus on the answering of the research questions. For this, thanks goes to the thesis supervisor, Rob Bruning, who pointed out a multiple of times that a piece I wrote was interesting, but not concerning the answering of the research questions. He set me straight and advised me on how to go the right way in answering the research questions. This is no small task and I am grateful that he kept on giving me useful comments on the first manuscript versions.
Furthermore, thanks goes to my actuarial team manager, Willem Jan Bremmer and later Branko Steenvoorden, who allowed me to take on this challenge and provided me with time to do it. Useful peer review was provided by my colleague Gemma van de Sande, which I gladly acknowledge here. all these people provided me with the opportunity to make this thesis. This manuscript is what resulted from my strenuous effort in the created environment. I hope the reader will enjoy reading it.
Bertus Veurink
Utrecht, December 2014
Chapter 1
Outline
In this thesis we will investigate the natural hedging potential of combined pension products in a portfolio consisting of a funeral insurance product and a term insurance on the life of one insured. This recently became even more relevant, because of the recently published delegated acts (new solvency II guidelines). These state the following in article 137-138, which is about the calculation of the mortality risk and longevity risk in the life underwriting risk calculation, see [13, EIOPA]:
“The identification of insurance policies for which a(n) increase/decrease in mortality rates leads to an increase in technical provisions without the risk margin may be based on the following assumptions: multiple insurance policies in respect of the same insured person may be treated as if they were one insurance policy;”
So, where insurers do not already combine risk calculation for multiple policies of one insured into one calculation, our research may help to find the impact of what they are missing in solvability by not grouping their policies per insured. Note that in our further interpretation we take the LTGA (Long Term Guarantee Assessment) as the basis rule, see [11, EIOPA] and [12, EIOPA].
In the Dutch pension world a participant usually has a pension policy for himself and also carries an annuity for his/her partner in the case of his death. This combined product consists of a standard annuity, most regularly a pension product starting at the age of 65-67, and an annuity that starts directly at the death of the insured if the participant has a partner. The latter usually has a lower installment value. The widower annuity part (NP) of the combination product carries mortality risk, where the annuity part (OP) carries longevity risk. Furthermore a funeral insurance product and a term insurance carry mortality risk. We will need to specify all this, but intuitively it is clear that for lower ages of the insured the risk lies in the decease of the participant and at later ages it lies in living longer than expected. Quantifying this and looking at the age distribution will be the first focus of this research. Furthermore, we will look at the difference between gender and the choice of hedging longevity risk with other insurance products like term insurance products or funeral insurance products.
One person having more than one insurance policy at one insurer is a realistic scenario, for example because a lot of insurance companies insure the pension benefits for their employees themselves. Along with that, also package deals are offered, where insuring more policies with the insurer leads to a big discount in premium. Sometimes also employees mortgages can be obtained for a discounted price with the bank/insurer, usually combined with a term insurance, which leads to an extra insurance contract with the same insurer.
In the shock calculations for the SCR it seems logical to add the shocked cashflow (longevity shock for instance) of a young participant to that of an older one, where one of these participants might in fact have a negative shock value and the other a positive shock value for technical reasons. This can also be a form of natural hedging
inside homogeneous risk groups (which are only homogeneous to a certain extent) and this possibly does not fit the exact QIS 5 rules. This can be solved by only maintaining positive shock values in each scenario.
All of this will be investigated in a Solvency II framework, where the risk free rate, the SCR, the Risk Margin, the choice of assets (Own Funds) and eventually the SCR-ratio are of importance. Our work will be cut out for us, because all parameters may influence the outcome (mortality rates, yield curve, lapse rates, etcetera). That is why the set of assumptions and the research questions are so important. The central question which will be the topic of this research is the following.
If we construct a representative (pension) liability cashflow for one policyholder, with a smart mixture of longevity risk and mortality risk, by allowing natural hedging making the standard correlation of −0.25 implicitly more negative:
How much can we improve the SCR-ratio of a fictive insurer by constructing a model for the calculation of SCRlif e, for a portfolio of policies on the life
of one insured, where natural hedging of longevity risk and mortality risk is accounted for?
This leads to the following more specific sub questions:
• What is natural hedging and what is investigated in literature on the subject? • How is natural hedging currently accounted for in the Solvency II specifications? • How will the age of participants influence a combined product of an annuity and
a widow annuity? Does it make a difference if we consider paid-up premiums? • If we consider regular present insurance products in the Dutch market, what is
the realistic natural hedging potential for the policies for one insured?
• If we define a base case best estimate scenario, what kind of sensitivities are we concerned with and what is the impact of those on the base case?
The first subquestion concerns itself with the theoretical background of our investi-gation. we will look for what has already been published and will try to find some theoretical results that indicate natural hedging potential. In answering the second sub-question we will compare the result under standard Solvency II specifications to that which we obtain by allowing natural hedging. The last three subquestions lead to a more in-depth analysis of the problem and tests the validity of our base case result. We investigate one insured with more than one product to show results that will be appli-cable to a portfolio of more insureds with roughly the same characteristics by adding the SCR’s. This makes the research applicable for an insurer, for the insureds who carry more than one life coverage.
Chapter 2
Introduction
In this section we will introduce the subject of natural hedging longevity risk using products that carry mortality risk. We will determine the theoretical reasons (and def-inition) for natural hedging and where this research falls in the spectrum of already published articles.
The fact that in the Netherlands, as in the rest of the western world, the increasing life expectancy is increasing is a hot topic for debate. The topics, among others, are health care costs and pension funding. This is also true for life-insurers who sell these pension products. In short this risk of an increased life-expectancy implies a longer need to pay pension installments. This makes it a realistic risk. A natural way to hedge this risk is to sell products, for which the technical provision will drop when the mortality rates decrease. This has been known to be an effective measure, so let’s take a look at what results are found in literature and what we can derive theoretically.
We will divide this introduction in two subsections, firstly a theoretical background and secondly relevant research.
2.1
Theoretical background
2.1.1 Introduction
Where longevity risk in the Dutch insurance world is usually associated with annuity products, insurance products carrying mortality risk usually make one-time payments in the case of demise of the insured during the contract period. We know an actuarial relation between annuities and single payments, without taking into account mortality rates, which relates the two (easily proven by filling in definitions):
¯
An = 1 − δ · ¯an (2.1)
Here, ¯An is the current price for a single payment, payable in n years, ¯an is the current
price for an annuity with continuous payments starting now (adding up to one each period) and δ is the continuous force of interest. This means that for every annuity, payable for n years, we can find the matching single payment. In fact, we can estimate what the sum insured of a single payment insurance has to be to match an annuity. What this shows us is that the insured sum would need to be much higher for a single payment policy to match the best estimate reserve for an annuity. We can find a similar relation between annuity and life insurance if we include mortality rates. Note that in the results chapter, chapter 4, we will assume a non constant yield, which is given by a yield curve. Per maturity we will use a different interest rate.
2.1.2 Theoretical natural hedging
Define random variable T as the failure time (in our analogy the time of death) and define the survival function s(t) as follows:
s(t) = 1 − F (t) = P (T > t) (2.2)
So, s(t) gives us the probability that the failure time is larger than t (i.e. probability of survival up until t). On the other hand we can define the probability density function f (t) as a function of s(t), i.e. f (t) = −s0(t).
Now we can define random variables Y and Z, definitions are from [9, Wolthuis, ch.6-7]: Z = ( exp (−δ · T ), if T ∈ [0, n] exp (−δ · n), if T > n (2.3) Y = ( ¯ aT, if T ∈ [0, n] ¯ an, if T > n (2.4) We know that ¯An = exp (−δ · n). Now, because identity (2.1) holds for all n, hence also
for random variable T , we have that Z = 1 − δ · Y .Write expectations E[Z] = ¯Ax:n and
E[Y ] = ¯ax:n to find the relation between an annuity and a life product (works also for
limn→∞), which is also described in [9, Wolthuis, p.211,(29)]:
¯
Ax:n = 1 − δ · ¯ax:n (2.5)
We can derive another relation between a lifelong annuity and a lifelong life insur-ance. Write v(t) = exp (−δt) for some fixed interest δ and assume we have a constant insured amount of 1 for all times, which can be changed later on. Then we find:
¯ ax=
1 − ¯Ax
δ (2.6)
This means that if we create a combined portfolio of a life insurance and an annuity for the same insured we can theoretically nullify all life risk under our assumptions. Choose the insured amount for the life product to be 1/δ and find:
¯ ax+ ¯Ax· 1 δ = 1 − E[vT] δ + 1 δ · E[v T] = 1 δ (2.7)
This means that there is no stochastic component in the expected payout for this portfolio. To find prove of natural hedging we will investigate a combined portfolio of these two products will nullify the total variance. Define X = Y + Z ·1δ, with Y and Z as previously defined, then:
Var[X] = Var[Y + Z ·1 δ] = Var[Y + 1 δ + Y ] = Var[ 1 δ] = 0 (2.8)
We used that Z = 1 − δ · Y and that the variance of a constant is zero. This result means that for every choice of mortality rates the variance for this portfolio will be 0! If we think of the Solvency II framework, where we have to apply a mortality shock and longevity shock to calculate the SCRlife, we notice that if we shock both products
separately that the shocks are non-zero. If we would apply the shock to the combined portfolio, it is zero as previously shown. So what we learn is that even considering the −0.25 correlation between longevity risk and mortality risk, the Solvency II rules do not sufficiently allow calculation of the natural hedging potential for this portfolio. To describe this, a much stronger negative correlation is necessary.
Actually, we can also show the same result for the best estimate in a premium paying environment for stylized premium payments for the life contract, if we assume lifelong
Natural hedging in a Solvency II Framework — Bertus Veurink 5
premium payment. Let us start by assuming that the life insurance product is paid for by continuous constant premium payments of π per year. The equivalence principle imposes that the present value of premiums is equal to the present value of payments:
π · ¯ax= ¯Ax (2.9)
Using formula (2.6) we can write this relation to find the premium formula: π = 1
¯ ax
− δ (2.10)
Now we can fill in the premium formula and find for the same combination portfolio: ¯ ax+ ¯Ax· 1 δ = 1 +π δ ¯ ax = 1 + 1 ¯ ax· δ −δ δ · ¯ax= 1 δ (2.11)
However, we must note that in a premium paying situation variance also arises from the premium cashflows. Hence, the variance will not be 0 here as we did found in the paid-up scenario.
In reality we will have a combination of a term insurance and a deferred annuity (and probably more products). We can investigate whether there is proof of natural hedging in a combined portfolio consisting of a deferred annuity and a term insurance. We define the random variables Z0 and Y0, analogous to Z and Y , as follows:
Z0 = ( exp (−δ · T ), if T ∈ [0, n1) 0, if T ≥ n1 (2.12) Y0= ( 0, if T ∈ [0, n2) ¯ ax− ¯ax:n2, if T ≥ n2 (2.13) Note that n1 and n2 are allowed to differ. Now, we can define ¯A1x:n1 = E[Z
0] (1 indicates
that payment is made only in case of the demise of insured within n1years) andn2|¯ax=
E[Y0]. To find prove of natural hedging we are going to see if a combined portfolio of these two products has smaller variance than the contribution of the separate products. For this we will again use an analogy of [9, Wolthuis, p.161, Example 4]. Define X0 = Y0+ Z0·1δ to try the theoretical natural hedge we found before, then:
Var[X0] = Var[Y0] + Var[Z0] · 1 δ2 +
2
δ · Covar[Y
0, Z0] (2.14)
and:
Covar[Y0, Z0] = E[Y0Z0] − E[Z0] · E[Y0] = E[Y0Z0] − ¯A1x:n1 ·n2|¯ax (2.15)
The product (Y0· Z0) will be 0 from the definition of the random variables if n1≤ n2. In
this scenario the covariance term will be less than 0 and we indeed have that the variance of the combined product is smaller than the separate parts, which is an indication for natural hedging. If n1 ≥ n2 we find, by using [2, Promislow, p.196]:
E[Y0Z0] = Z n1
n2
exp(−δt) · exp (−n2t) − exp (−δt) δ
· f (t)dt (2.16) For n1 and n2 sufficiently close, we will still have a reducing variance effect by this
covariance term.
We can also look at the effect on the variance from the loss function if we consider premium payments instead of paid-up premiums for this combination portfolio. The loss function is defined as: (present value of future payments -/- present value of future premium income). We assume the premiums for our two policies to be constant and
payable for n1 and n2 years. For simplicity we also assume that n1 = n2. Now, we can
calculate the Loss functions (at the start of projection), from [9, Wolthuis, p. 223, p.294] with a small modification to make this applicable to a term insurance:
LA= ( exp (−δ · T ) − ¯P1· ¯aT , if T < n ¯ P1· ¯an, if T ≥ n (2.17) La= ( − ¯P · ¯aT, if T < n exp (−δ · n) · ¯aT −n − ¯P · ¯an, if T ≥ n (2.18) Here, the premiums are calculated from the equivalence principle, i.e. ¯P1= ¯A1
x:n/¯ax:n
and ¯P = vn·npx· ¯ax+n/¯ax:n. Finding the variance of a combination of these two policies
is more difficult, but we can try the previous combination: Var[1 δ · L A+ La] = Var[La] · 1 δ2 + Var[L a] + 2 δ · Covar[L A, La] (2.19)
And at the start of the insurance (at projection year t = 0) this reduces to:
Covar[LA, La] = E[LALa] − E[LA] · E[La] = E[LALa] − 0 (2.20) The last identity follows from the equivalence principle at the start of the insurance. We expect exactly enough premium to make insurance payments, so the expected loss is 0 for both insurances. Hence, we need to look at the product term. Let us try to find an exact result. We can work out the cross term for the period up until n using [2, Promislow, p.196] to find E[LALa] for T < n:
Z n 0 exp (−δt) · − ¯P ·1 − exp (−δt) δ + ¯P 1· ¯P · 1 − exp (−δt) δ 2! · f (t)dt (2.21) We can simplify this to:
Z n 0 ¯ P ¯P1 δ2 − ¯ P δ + 2 ¯P ¯P1 δ2 · exp (−δt) + ¯ P ¯P1 δ2 + ¯ P δ · exp (−2δt) · f (t)dt (2.22) For the period after n we can find the following expression which we need to add to the last one to find E[LALa]:
Z ∞ n ¯P1P¯ δ2 · 1 − exp (−δn) δ 2 − ¯P1·1 − exp (−δn) δ · exp (−δn) · 1 − exp (−δ(t − n)) δ ! ·f (t)dt (2.23)
This can be reduced to: 1
δ2
Z ∞
n
¯
P1P · (1 − 2 exp (−δn) + exp (−2δn)) − ¯¯ P1· (exp (−δn) − exp (−δt) + exp (−δ(t + n)))·f (t)dt (2.24)
From the total expression we can note that the combination of both first terms reduces to P ¯¯δP21, since
R∞
0 f (t)dt = 1. This means that the remaining integral terms need to
compensate this to yield a negative value. By using a first order Taylor-expansion of exp −δt, we can reduce the total expression (including the first term which we just pulled out) inside the integral sign to:
Z n 0 − ¯P · t · f (t)dt + Z ∞ n ¯ P1(2δn − 1) · f (t)dt (2.25)
Since values of f (t) are always positive the first integral is clearly negative and the second one is negative if δ < 2n1 , which might not be the case. This condition for δ will
Natural hedging in a Solvency II Framework — Bertus Veurink 7
not be satisfied for reasonable values of n. If we would include higher terms the even terms are positive and uneven terms are negative. We are not sure whether there is a clear answer here.
2.1.3 Theoretical definition of research problem
All of the previous research was into systematic risk, the risk that the realization of the probability function turns out to be worse than the expected value. We saw some proof of natural hedging. But, the shocks defined in the Solvency II specifications for the calculation of the capital SCRlifeare formulated in a non-systematic way. The longevity
shock is defined as a decrease of the mortality rates by 20% and the mortality rates is defined as an increase of these rates by 15% (more details follow in the model section). For an annuity product the first will require extra capital, but the second will not and for the term insurance this will have an opposite effect. Our research question focuses on how much the required capital of the sum of the two policies will drop if we combine them into one portfolio. Furthermore, how much more negative is the implied correlation than the prescribed correlation of −0.25 in the SCR-specifications. We can write our research question mathematically for some non-systematic risk scenario. We will assume that the mortality rates follow f (t) in best estimate and fm(t) in the mortality shock scenario and fl(t) in longevity risk scenario and we will mark other variables in shock variables using the same mark.
What our research comes down to is: Define capitals: Cnhm = Em[LA+ La] − E[LA+ La] and equivalent for longevity. Furthermore for the non-combined portfolios: Cm = Em[LA] − E[LA] and Cl = El[LA] − E[LA]. Note that under an increase in mortality rates the demise of the insured is expected to occur sooner, such that we have: El[L] > E[T ] > Em[T ]. For the mortality scenario this means that Em[La] drops (immediately clear from formula 2.18, where ¯ax drops) and this means that no capital is required for
this policy in the mortality scenario. The same holds for El[LA], which can be seen from formula 2.17. Because El[L] > E[T ] there is a larger probability that no payment needs to be made and the loss function decreases in the longevity scenario.
Now we can deduce, by applying the sum rule for expectations (or integrals), that Em[LA+La] < Em[LA] (since Em[La] < 0). This means that Cnhm < Cmand that natural hedging has a certain impact here. Analogously we can deduce that Cnhl < Cl. As stated, Solvency II rules prescribe a −0.25 correlation between these capital requirements. Now, we want to know the following:
enh = q
(Cl)2+ (Cm)2− 0.25 · 2 · Cl· Cm− (Cm
nh+ Cnhl ) (2.26)
This enh gives us the excess natural hedging potential, the decrease in capital for life-related risks due to non-systematic risks on top of the natural hedging that is accounted for in the standard formula (caused by the small negative correlation). The terms with subscript nh will not have interaction, since this has already been filtered out in the calculation of the capital. We already know that natural hedging will decrease both capital requirements (for longevity and mortality), but in our research we will find out if (and by how much) the natural hedging potential exceeds the correction term due to correlation between longevity and mortality risk proposed by Solvency II.
This is what we will investigate in the later sections, by using non-constant yield and by also including more risks and two more products, but the essence is the same as is given here.
2.2
Relevant research
Now let us look at some results from literature that are relevant to our subject. It is good to note that usually in the upcoming articles we will discuss, the best estimate
portfolio value is kept constant, but the fraction life insurance compared to annuities is changed in order to investigate the dynamics of the combination portfolio.
A fundamental research into natural hedging longevity risk using products that carry mortality risk is performed by Cox and Lin, see [1, Cox]. They hedge a pension product of a 65-year old policyholder by using a term-insurance product of a 35-year old policyholder. They find that the price of a combined portfolio performs best in both a situation of increase and decrease of mortality rates. They show that natural hedging is actually effective, because insurers who sell both annuity products and life products generally charge a lower premium. This can also have different reasons, for instance that large insurers, who will sell both products, have a scale economic advantage due to lower costs and this could also cause the lower premiums. Still, the first research result is a good motivation for us to continue our research. What we will establish is an even more certain hedge as we assume that one insured participates in both contracts. In this case we know for certain that a change in mortality rates affects both policies, because the same person is affected. What we will try to do is make this result more practical for insurers by investigating the effect of natural hedging, based on LTGA-rules, on the SCR-balance sheet.
In another article, [3, Wang], a profit function is developed and maximized and for this maximum profit an investment amount in term insurance and longevity bonds is returned. Also a Lee-carter model is used to predict the mortality rates. We also want to maximize profit using natural hedging, but will use the AG 2012-2062 generation table as input, so we do not develop our own Lee-Carter model and our measure of profit will be the SCR-ratio. Furthermore there is no liquid market for longevity bonds, certainly not for longevity bonds based on Dutch mortality rates. Therefore, we do not consider this a viable option and we will leave those out of our research. The same is true for all other longevity related assets that are sold in the market. They are usually illiquid or not applicable to the Dutch situation or both. Without further discussion we leave these assets out as option for hedging longevity risk, although we know that a lot more can be said about them.
If we return to article [3, Wang], we see that based on certain asset volatility as-sumptions the probability of default for an insurer is investigated and also calculated for different fractions of the total portfolio consisting of term insurances, next to the al-ready present pension (or annuity) liability. We think this is an interesting measure, but hard to objectively quantify, because an asset-portfolio always needs to be chosen. For our research, this is also a challenge. We need to make our results as asset-independent as possible to be valid in all cases or be clear about other investment strategies. That is what is done in [4, Gatzert], where an optimal hedge ratio is calculated for multiple investment strategies. That proves that assets do influence this optimal hedge ratio. in an article by [6, Wong], the same base result is found. Furthermore, a couple of life prod-ucts is tested for natural hedging effectiveness versus an annuity and also the optimal ratio annuity holders to life product owners. This is done by optimizing the 99% value at risk (VaR) for this ratio. The test is performed for three products, a single premium whole life insurance, a whole life insurance with a level premium and a level premium 30 year term insurance. Although the 99% VaR is highest for the term insurance, the combination portfolio in terms of lowest Value at Risk consists of approximately 80% annuities and 20% term insurance (composition measured in terms of premium values). The other products in combination with the annuity also have their optimum at around a fraction of 0.8, but the combination portfolio carries a higher VaR. Based on the simulations this would be the ideal portfolio composition.The article investigates more, though. How is the VaR affected if one or more of the portfolio products carries a profit loading? This might happen if the best estimate mortality rates differ from the mortal-ity rates used in pricing. It turns out that if both the term insurance and the annumortal-ity have the same profit loading nothing really happens, the 99% VaR decreases by the
Natural hedging in a Solvency II Framework — Bertus Veurink 9
same amount for all ratios. If we allow the profit loading for one of the two products we find that the optimal composition does not change for the term insurance, but does change for the whole life insurance. If profit loading is present on the annuity product it becomes beneficial to have more than a fraction of 0.8 of annuity in the portfolio and the other way around if profit loading is present on the whole life product.
Another approach is investigated by [7, Luciano]. In this article the researchers aim to immunize the portfolio of life products such that its value does not alter under changing mortality rates in the first and second order. This can be done by duration and convexity matching of the portfolio, a so-called Delta-Gamma hedge. Furthermore they investigate the hedge effectiveness per generation and gender, by allowing one generation group to hedge the other. The mortality risk which affects both predefined generations x and y can be isolated from the risk that only affects one generation. In this way a two generation problem can be defined and solved. An interesting property of this is that if the result from the Delta-Gamma hedge on generation x is unsatisfactory, if too many products of one kind need to be sold for the hedge or if these products are not available in practice, longevity risk can still be hedged for this generation by selling them according to the result from the cross-generational research. ”The market for generation x may be completed by generation y”, is how the authors characterize this.
In an extensive paper, [8, Stevens], we read about the effect of investment risk on the natural hedging potential. Separating investment risk and longevity risk can lead to an inadequate (optimal) choice of portfolio composition from a natural hedging perspective. The measure that is used to investigate this is the necessary capital to be solvable in 97, 5% of the scenarios. A lot of portfolio compositions are investigated, but the result that was most pronounced for our future research was found using a portfolio composition of a whole life insurance and and a single life annuity. If only liabilities are considered the optimal ratio of life insurance over annuity was around 28 (yielding the lowest necessary capital), but an investment strategy of investment in safe one year zero coupon bonds dropped this ratio to around 10! With riskier investment strategies this difference only increases. Furthermore the investment risk causes the required capital to increase as well. Other portfolio compositions, for instance by including a survivor annuity, also show differences in optimal ratio, but the effects are minor in comparison to the example that we discussed in detail. This really encourages us in wanting to consider assets and liabilities in our upcoming natural hedging model research. We need to realize that in our model research, to be discussed later, we choose an investment strategy and that another strategy will lead to a different outcome.
Most research into projection of mortality rates presumes to follow a simple Lee-Carter model. Our research will use the AG 2012-2062 generation table, which is based on the same kind of model. In [5, Zhu], authors warn that the choice for a parametric single factor model, like Lee-Carter, has a large impact on the results they find. Using higher order variations in mortality rates affects the performance of natural hedging. They find that not all longevity risk can be hedged, contrary to what others found using a Lee-Carter model for mortality rate projection. Using their non-parametrical model for mortality rates, they show that even if one would add term life insurance policies to their annuity portfolio the total economic capital does not decrease! The reason is that single factor models assume a dependency on age that might not be supported by the data and fitting nonparametric mortality forecasting model shows a variance in the estimators for the loss function of the annuity product, that cannot be hedged using a life product. If the economic capital would be calculated stochastically using this method, this implies less optimistic results for us. What we will use is the standard LTGA-model in which we determine our SCRlife deterministically, so for our scope (increase in SCR-ratio due
to natural hedging) there is no real problem here. But this is a warning that using the standard LTGA model to calculate natural hedging potential over a portfolio with more
than one insured might overestimate the real natural hedging potential!
There are numerous other papers on the subject, which all do not investigate the practical impact of natural hedging for an insurer on Solvency ratios and the projection of those. The reason might be that a lot of assumptions need to be made, something that we will have to be careful with later on. The core idea is always the same though: there are two kinds of longevity risk, systematic and non-systematic. The systematic longevity risk is caused by the fact that future mortality rates are uncertain and the systematic risk is caused by the fact that the mortality rate for one insured is a random variable. In a portfolio of annuities the latter can be diversified, but the first cannot. The solution to hedge this risk is by using a product with negative longevity risk.
In our model later on we will use the LTGA-shocks for longevity and mortality, but we have to realize that this only covers the non-systematic risk (risk that future mortality rates differ from what we predict). The systematic risk, risk of individual deviation from the predicted mortality rate (which is on average true), is ignored in our research as we will pretend that this is diversified to zero, once we add a lot of the policies we will investigate together.
Chapter 3
Model description
In the first section of this chapter we will describe the model along with its dynamics, which we will use to answer the research questions as described in the outline. The next section consists of the choice of parameters we made in the model to get realistic results. Later on, we will show how the choice of parameters might influence the outcomes. That is why we want to make them explicit here. Finally we will talk about the LTGA framework and its dynamics and then we can show what we precisely mean by natural hedging.
To decide whether a method based on natural hedging significantly changes the SCRlife as defined using Solvency II specifications, we need to calculate this first in a
LTGA prove setting and then in a setting where this framework entirely holds. The only deviation in the calculations is that the longevity shock and mortality shock are both calculated in such a way that the products which yield negative shocks compensate the ones yielding a positive shock.
The effect of this natural hedging becomes apparent in both the SCRlife and the
Risk Margin loading. This effect is the reason we want to look at the SCR-ratio instead of only the SCRlife. We need to include some representative assets to do this, which will
be specified further in the next section.
3.1
Model
The model that is used in answering the research questions is a cashflow projection model that can project premium, costs and installments (payouts). Four products are modeled and their effect on natural hedging potential will be investigated: pension, partner pension, term insurance and funeral expense insurance (one payout at death of insured). The first carries only longevity risk, whereas the second carries both mortality risk and longevity risk and the third and fourth carry only mortality risk. Assuming the insured is alive, we can view the situation from a risk perspective at later years. But first let us define the cashflow calculations. Our model is a discrete model.
3.1.1 Payment Cashflows
For the regular pension installments a payment is made on the pension date and yearly afterwards (in practice the pension date will be the 65th birthday). We modeled this as the probability of payment multiplied by the payable sum, which we can write as:
kpx· SIp· 1(x+k)≥xpension (3.1)
Here, we define the following variables:
• xpension is the pension age for the insured.
• For each projection year we calculate a payment cashflow value. k is the number of years that we are already have passed our projection. We start our projection at the reference date 31-12-2011, with k = 0. After one year of projection we reevaluate the payment cashflow at k = 1 and so on.
• x is the age of the insured at the start of the projection. For the sake of readabil-ity we assume that x is always an integer, although the model can also handle non-integer ages by interpolating the mortality rates and discounting accordingly (payments are always made on the birthday of the insured).
• kpx is the probability that a person of age x lives for the next k years.
• 1 is an indicator function which is 1 when the condition in the subscript is met and 0 otherwise
• The variable SIpis the insured sum that is paid at installment dates, the subscript
p denotes the pension product.
This gives a formula to evaluate at the end of each projection year. We want to build this in a cashflow model setting, but it is easy to see that by discounting the resulting cashflows using a constant interest rate we arrive at the more well known formula for a deferred annuity, for example given by [9, Wolthuis, p. 188], with the same definitions (assuming x and xpension to be integers, as we have done in formula (3.1)):
xpension−x|¨ax· SIp= SIp·
ω
X
s=xpension−x
vs·spx (3.2)
Here, we introduce v = 1+i1 and i is the interest rate.
Note: We assume a constant interest rate in this formula. We will do this for the rest of this section. In Chapter 4 we will produce results using a yield curve, though. This means a different interest rate for every maturity. We could quite easily incorporate this in all formulas in this chapter by replacing v at maturity s by vs= 1+i1s, with isthe
spot rate at maturity s. If we do this, the yield curve can be used instead of a constant interest rate. We choose not to write this subscript in all formulas for readability, but it should be present in market valuation of the cashflows.
The formula for the partner pension is a little bit more complicated. We define the age of the partner to be y and the owner of the partner pension has age x. We will make the assumption that the partner is of the opposite sex and that male partners are always 3 years older than their female partners. That is the same assumption as is commonly made in practice by insurers. On average this is close to reality and therefore we assume it here as well.
For each projection year, using the same definitions as previously, the value of the partner pension installment can be calculated as follows:
(1 −kpx) · (kpy) · SIp· ppf (3.3)
The definition of variables is as in (3.1), but we added a factor ppf for the amount to be received by the partner, which is usually lower than one, because partner pension installments are lower than regular pension installments. The values of these factors will be defined in section 3.2. Basically, this formula just states that an installment for partner pension is paid at the birthday of the owner of the partner pension, if the owner of the partner pension has died before that date and his/her partner is still alive.
This is equivalent to:
¨ ax|y· SIp· ppf = SIp· ppf · ω X s=0 vs·sqx·spy (3.4)
Natural hedging in a Solvency II Framework — Bertus Veurink 13
If we use a constant yield of 3% this will give us the paid-up value of this product, which we will need later on for the calculation of lapse risk.
The equation for funeral cost insurance is calculated as surviving for k years, but death has occurred at time k + 1. We can put this in formulas as follows:
kpx· qx+k· SIf · vk+1/2 (3.5)
The assumption is that on average if payment occurs this will be halfway the projection year, that is the reason we let the payment cashflow occur at that time. So to calculate the market value at the end of the year we apply the yield (for half a year, so v = (1+ik)),
with ikthe yield valid between time k and k +1. Note the different subscript in SIf,since
in principle we have a different insured sum for the funeral insurance than for the other insurances. In literature we find the method equivalent to, [9, Wolthuis, p. 147]:
¯ Ax· SIf = SIf · ω X s=0 vk+1/2·spx· qx+s (3.6)
The difference is that our product pays directly on the death of the insured. So each year the probability of payment is the same as in the cited reference, but the timing of payout should be halfway the projection year, under the assumption that on average a person dies halfway the year, which seems reasonable.
The liability cashflow for the term insurance is identical, the only difference is that the term insurance is temporary and that the insured sum is different and is defined as SIt.
There is no certain payout, such that after a certain age xterm has been reached, the
cashflow will return to zero to indicate that the insurance is expired. In formulas:
kpx· qx+k· SIt· vk+1/2· 1(x+k)<xterm (3.7)
and again from literature, [9, Wolthuis, p. 150-151], where nt= xterm− x:
SIt· ¯Axnt = SIt· nt
X
k=0
vk+1/2·kpx· qx+k (3.8)
To make all these cashflows comparable we need to value them all at the same time. We did this by inflating them to the end of the year (applying yield for the time left until the end of the year) from where we can easily discount the whole cashflow using the yield curve.
3.1.2 Cost cashflow
We modeled a cost cashflow per product in such a way that for the duration of the product a constant amount of costs is charged each year and this amount needs to be covered by the cost loading in the premiums. We constructed this such that about 8% cost loading per product is yielded. In the end this would mean that if we would have a constant 3% yield, no profit nor loss would be made on costs. For the funeral insurance and the pension product this is easy and the cost cashflow is given by the following formula, assuming that costs are timed halfway the projection year, which is on average true:
(k+0.5)px· costp/f· (1 + ci)k (3.9)
So as long as the insured is alive we have a constant amount of costs. For the term insurance we only have costs for the duration of the portfolio, this gives us the following formula for costs:
For the partner pension we need to take costs into account for as long as the insured and the partner are alive. This results in the following formula:
(k+0.5)pxy· costpp˙(1 + ci)k = ((k+0.5)px+(k+0.5)py−(k+0.5)pxy) · costpp· (1 + ci)k (3.11)
The definition of variables is as in (3.1), but we added the following components: • costpmeasures the amount of costs per year for the pension product, the subscript
f is for the costs for the funeral insurance, pp for the partner pension and t for the costs of the term insurance.
• ci is the cost inflation per year.
• kpxy is the probability that after k years at least one of the persons x and y is still
alive.
3.1.3 Premium cashflows
The premiums are constructed in such a way that the actual present value of payments is exactly covered for a fixed interest rate. This means that using this interest rate the present value of payments (costs included) is equal to the present value of premiums at the start of the insurance. There are two factors in the premium cashflow, the premium amount and the probability that the premium is paid.
For a given projection moment k of the insurance the payment will be paid at the next birthday, provided the insured has not reached pension age. We will assume for simplicity that the birthday of the insured exactly is on the projection day. Each year buildperc of the salary is built up as part of the pension. The price of this part of the pension increases as the insured becomes older, because of the fact that the pension date gets closer. This deferred annuity is bought each time a premium is paid. So this results in the premium amount, payable for as long as the insured is alive:
(
(xpension−x−k)|¨a(x+k)· Sal · buildperc · (1 + cload), if x + k < xpension
0, if x + k ≥ xpension
(3.12)
Here (xpension−x−k)|¨ax can be defined as follows, with ω the highest age in the mortality
table and v the one-year discount factor (in practice 0.97):
xpension−x−k|a¨x= ω X s=(xpension−x−k) vs·spx We define:
• Sal is the insured salary, the salary that is available for the buildup of pension. In the Dutch pension system this is usually not the whole salary, since part of the pension in practice is received from the government.
• buildperc is the percentage of the insured salary that is accrued every year to be paid out yearly after the pension age.
• cload is the percentage that needs to be added to the net premium to be able to cover the costs over the duration of the pension policy, for as long as the insured or the partner of the insured is alive.
The idea behind this payment schedule is that in buildperc1 years the pension scheme is fully paid, usually this takes 40 years, but we keep it variable. That would mean that the SIp, the pension payout is 80% of the insured salary if we assume buildperc = 2%.
Natural hedging in a Solvency II Framework — Bertus Veurink 15
The premium is paid for as long the insured is alive, so the eventual premium cashflow is:
(
(xpension−x−k)|¨a(x+k)· Sal · buildperc · (1 + cload) ·kpx, if x + k < xpension
0, if x + k ≥ xpension
(3.13)
For the situation that this insurance starts at the projection date and the SIp from
the previous paragraph equals the buildperc · (xpension− x) · Sal (in practice such that
insured sum SIp is 80% of the insured salary). We made a calculation, which shows that
the present value of the liability cashflow (using 3% constant yield as is common in the pricing of pension products) exactly equals that of the premium cashflow. This is of course what we want. Note however, that for a market valuation these cashflows will not be equal, because of the use of a market yield curve.
The cost loading cload is (based on 3% interest) calculated as follows, using NPV notation for the net present value of the cashflow:
N P V (cashflow pension premium) + N P V (cashflow costs) N P V (cashflow pension premium)
The cost loading for the partner pension is implicitly included in the premium cashflow for the regular pension. So, we only have to calculate a net premium for partner pen-sion. For this net premium, we made a distinction between risk premium and build-up premium. The risk premium consists of a premium that is paid for the situation that the policyholder dies before the age of xpension. The build-up premium for partner pension
works analogously to the standard pension premium and can be expressed as follows, using definitions as already used:
ω P s=0
(vs·spy·sqx) · Sal · buildperc · ppf, if x + k < xpension
0, if x + k ≥ xpension
(3.14)
Here,sqx = 1 −spx is the probability that in s years x has died. A payment is received
each year for a part of buildperc of the partner pension. To find the actual premium cashflow one must multiply bykpx, because we only calculated the premium amount up
to now. The new factor ppf is to correct for the fact that the partner pension usually carries a lower sum insured than the ordinary pension product. Note that implicitly we assume that our policy holder has a partner, but this is not necessarily the case. In the Dutch pension system usually a policyholder who built up partner pension can exchange this for a higher regular pension, see [10, Rijksoverheid]. This will have an effect on the distribution of mortality risk and longevity risk in our portfolio, so we assume a partner is present in our projections (or a choice to exchange partner pension has not yet been made).
The part of the present value of benefits for the partner pension that is not yet paid for needs to be paid using an additional premium (that becomes lower as more regular premiums are paid for partner pension). We can calculate this for each projection time k as follows:
(1 − px+k) · Sal · ppf · ¨ay+k·
payyrs − puyears
payyrs (3.15)
Here puyears ∈ [0, payears] is the amount of years that partner pension has already been built up and payyrs is the total of years premium needs to be paid for partner pension. Only for the part that has not been built up yet, we ask a risk premium. To find the actual premium cashflow one must multiply by kpx. By reasoning this way we
pension we did a calculation to check that the present value (using flat 3% yield) of the combined premium cashflows for partner pension equals that of the liability cashflow.
For the funeral cost insurance we use a risk premium and the assumption is that it is a new contract that needs premiums (at the same start date of the pension product). If the age of the insured is past the paid-up age we assume premiums are paid-up. The equivalence principle yields:
SIf · ¯Ax = N Pf · ¨axnf (3.16)
Here, N Pf is the constant premium for this insurance and nf is the years that premium
is going to be paid, see also [9, Wolthuis, p.220-221], which we define as nf = (xpuf− x),
where xpuf is the age at which the funeral insurance is paid up.
This yields for the premium cashflow at time k: N Pf·kpx =
SIf · ¯Ax·kpx
¨ axnf
(3.17) The premium cashflow for the term insurance is quite similar, because technically it is the same type of insurance as a funeral insurance, only temporary. So the equivalence principle is slightly different from (3.16) and is given by (note that premium is paid as long as the insurance is existent):
SIt· ¯Axnt = N Pt.¨axnt (3.18)
The premium cashflow at each time k is given by: N Pt·kpx=
SIt· ¯Axnt ·kpx
¨ axnt
(3.19) Here, we use N Ptas the constant net premium for the product, nt= xterm−x, with xterm
the age at which the term insurance stops, and the rest of the variables as previously defined.
3.2
Assumptions
• In principle, the reference date is variable, but always the end of a year. In our cal-culations the first reference date is 31 december 2011. This is the reference date for which the LTGA was performed. Since constructing the Countercyclical Premium (CCP) is a lot of work and require a lot of assumptions, we used this reference date. For this reference date the CCP is prescribed in the LTGA-specifications. • The mortality rates are taken from the Dutch sex-separated generation tables
(generatietafels) 2012-2062, so for each projection year a different column with mortality rates per age is used. We assume that our portfolio characteristics are consistent with the average mortality rates in the Netherlands. As we do not have any other information, this seems plausible.
• The age of the insured is flexible. Mortality rates are linearly interpolated if the age is not an integer. To obtain results we will consider a male insured of the age 25 at the start of his insurances. This is also the start of our model projection. • Payment of premium and installments is always on the birthday of the insurance
owner, except for the funeral cost insurance, which pays out directly when the insured dies. This might be unrealistic for the partner pension. That depends on the policy conditions, which we adapt to our convenience.
• We assume no indexation and increase in insured salary for the pension contract. This would be more realistic, but might be an expansion idea for the model.
Natural hedging in a Solvency II Framework — Bertus Veurink 17
• Cost loading is a loading on the regular pension premium, which reflects the costs for all four policies. This is constructed such that using a constant yield of 3% the present value of costs exactly corresponds to the present value of the cost loading in the premium. Using market valuation this means that a slight profit from costs is made, due to yield.
• The used yield curve is the DNB Swap Curve + 100% CCP + UFR (with last liquid point at maturity 20 years and convergence in 10 years to the UFR). The curve was prescribed by EIOPA, so we did not have to make any further assumptions there.
• Shock scenario calculations for longevity and mortality risk is done in the model by multiplying the mortality rates by the required factor. For yield-shocks we apply the shocked curves to find the shocks.
• Valuation of the portfolio on a later reference date (than 31 December 2011) is possible. We assume that the insured has not died in the time up until the new reference date. Using this assumption we can again calculate the SCR-ratio. • Model assumption for our base scenario is that pension premiums are paid for 40
years and pension age is 65, ppf is equal to 0.7. Furthermore the male insured is 3 years older than his partner and the other way around for a female. The sum insured is e10 000 for the pension product. This is on top of the Dutch AOW (pension provision for people older than 67) which is about e14 000 under current legislation at the moment of writing. The total pension income would be about average, maybe a little bit below that for a Dutch citizen, who has worked for 40 years. The funeral insurance has an insured sum of e10, 000, which is a decent amount for funeral costs for the foreseeable future. xpuf = 80, so premiums
are paid till the age of 80 years. For the term insurance the insured amount is e300, 000, corresponding with a mortgage on a house that is above average in price and the contract duration of the term insurance is until the age of xterm,
which we choose to be 70 years. This will be our base case situation, but we can certainly investigate sensitivities with respect to these assumptions for the insured sum.
• There is no solvability loading incorporated in the premium. Instead of this at the start of the projection an asset amount of e35 000 is assumed to be present to cover solvability problems that would arise from pricing on Best Estimate level, if we include all products. This is about the amount of assets we need at time k = 0 to be solvable for the four policies. Based on the same principle we made an asset allocation per policy and also per euro insured sum. This way we have our asset allocation ready, once we start changing the insured amounts. Furthermore, of course the premiums per policy are added to the asset allocation per policy, once they start coming in. In table 3.1 we see the details of this asset allocation. • Costs are equal in both stages of the pension contract (paying phase and receiving
phase) and are constructed such that we find about an 8% increase in premiums for cost loading, subject to a cost inflation of 2.5%. The values for yearly costs are found in table 3.2 based on the assumptions two bullets above. We made sure that if we change the insured sum that the costs change alongside this by scaling them. The premium is without commission included for the intermediary, who helps to close the contracts. These commissions in the premiums are less and less observed in the Dutch insurance market. Therefore we only consider administration costs for the insurer in our model. Note that we can only see the effect of costs clearly in the cost risk in the SCRlifemodule.
Table 3.1: Table of asset allocation under the assumptions of our base case
Asset allocation Type of insurance Asset allocation Insured sum per insured euro
Pension e27,000 e10,000 e2.700
Partner pension e3,800 e7,000 e0.543
Funeral e500 e10,000 e0.050
Term insurance e3,700 e300,000 e0.012
Table 3.2: Overview of yearly costs in combination with the effective cost loading in the premium, situation for our base case scenario values
Type of insurance Yearly costs Cost loading in premium
Pension e71 8.0%
Partner pension e14 8.2%
Funeral e2,5 7.5%
Term insurance e18 8.0%
Total e106 8.0%
3.3
QIS-5/LTGA-model
3.3.1 introduction
We have modelled our premium cashflows, our liability cashflows and cost cashflows and we will use these to calculate the Solvency Capital Ratio (SCR) in a Long Term Guarantee Assessment (LTGA) framework. This LTGA was a new step from EIOPA to enable market consistent insurance liability valuation and assessing the problems rising from long term guarantees in a current economic recession, namely overvaluing long term guarantees in a temporary bad economic climate. We will not discuss the validity of the elements of the calculation here, but we will use it, as it is the latest attempt in creating a uniform market valuation setting for all insurers. However, we will discuss how we modeled the framework’s SCR-components and the reasons for our choices. Furthermore, we will discuss where we make different choices than the standard formula and the reasons to make these differentiations.
We will also need to model some representative assets to show the behavior of our SCR-ratio in the standard LTGA-framework and in our improved framework, using natural hedging. A decrease in SCR and Risk Margin due to natural hedging has a double effect on the SCR-ratio in which they are both incorporated. That is the reason why we want to model assets in an otherwise liability based investigation.
3.3.2 Best Estimate valuation and Yield Curve
When we modeled our Best Estimate cashflows in sections 3.1.1 and 3.1.3 we already had in mind that this needs to be in accordance with [11, EIOPA, p.51]. Starting today we have to make a projection based on our knowledge of future probabilities, interest rates and so on. Because the extraction date for the LTGA was 31 December 2011, this will also be our start date for projection. The four insurance contracts that were described in 3.1.1 are all assumed to start on this day. We will model this such that we have a variable projection date and we can make the same valuation on a later time. It is interesting to see how our portfolio evolves (of course also each product can be looked at separately). The technical specifications state in [12, Eiopa, TP 2.1] that we have to take into account the time value of money. This discounting will be done in accordance with the base case LTGA scenario (scenario 1), which means adding 100% Countercyclical premium (CCP) to the curve,[12, EIOPA, p.6], and by using the UFR after 20 years of maturity. Because the basis curve, the convergence to UFR and the CCP
Natural hedging in a Solvency II Framework — Bertus Veurink 19
are all given, we will not worry about the interpretation too much. Also, no matching adjustment and transitional measures are used, because these do not serve the purpose of this investigation and needlessly complicate matters.
3.3.3 SCR and Risk Margin
Firstly we talk about the components of the SCRlife(concerning Life risks) and how these
are modeled in our cashflow model. Secondly, we talk about the operational risk and finally the market risks associated with the assets is discussed in the next subsection. The components of SCRlife are as follows, following the enumeration in [11, EIOPA,
p.179-190]:
Mortality risk
This is the increase of the Best Estimate under the assumption that the mortality rates are 15% higher than in the Best Estimate situation. Immediately an important LTGA notion is brought to our attention in [12, EIOPA, SCR 7.2], where we read:
“Mortality risk is the risk of loss, or of adverse change in the value of insurance liabilities, resulting from changes in the level, trend, or volatility of mortality rates,where an increase in the mortality rate leads to an increase in the value of insurance liabilities.” As stated in the outline, Chapter 1, the whole point of modeling products with opposite reactions to an in/decrease of mortality rates was, that if we know that one person carries risk on both such policies, the effect of the increase of the Best Estimate under this mortality shock for one product is compensated for by the decrease under the scenario of the other. This is because we know that once a change for this insured occurs on one policy, it can never be without effect on the other. We will do one calculation, where we follow this prescription exactly and one calculation where we allow negative values of mortality risk on one policy to compensate positive values on the other, for as long as the total mortality risk is non-negative. Later on, for the longevity risk we will do the same. All the other risks remain the same in both compared models. Once we get to the correlation matrices we discuss the model differences further. Apart from these two paths that we have created, our model does exactly as is prescribed by changing the mortality rates in this scenario according to the prescription, calculating a new Best Estimate and subtract the original Best Estimate to obtain the mortality risk, Lifemort.
Longevity Risk
This is the increase of the Best Estimate under the assumption that the mortality rates are 20% lower than in the Best Estimate situation. The same holds as for the mortality risk, in the sense that only policies with positive risks are to be included in the risk calculation. We will again calculate the paths described in the last subsection. The calculation of this risk is performed easily, by adapting the mortality rates for all policies and recalculating the Best Estimate and subtracting the original Best Estimate yields us the longevity risk, Lifelong.
Expense Risk
This is the increase of the Best Estimate under the assumption that the costs in the cost cashflow increase by 10% and the cost inflation increases by 1 percent point on top of the already present cost inflation. Clearly, this always leads to an increase of the Best Estimate and it is not hard to manipulate the Best Estimate cost cashflow. After calculating this, the Lifeexpense follows by subtracting the original Best Estimate.
Revision Risk
This is the increase of the Best Estimate under the assumption that due to legal cir-cumstances or the health of the insured the annuities have to pay more than accounted for in the Best Estimate. However, [12, EIOPA, SCR 7.66] states that such a risk only needs to be calculated for these annuities that can be altered by changes in the legal environment or health changes. This is not the case for our annuity, since it is a contract for a predetermined amount of money each year after pension date.
Morbidity/Disability Risk
This is the increase of the Best Estimate under the assumption that disability rates increase and recovery rates from disability decrease. In practice for a pension product this risk will have an effect on the Best Estimate, because usually in case of invalidity an insureds working ability is impaired and the premiums for his/her pension can no more be paid. In an insurance clause this may be taken care of by exempting an insured from paying pension premiums in the case of disability. We chose not to model this, as it is just a small special feature, which does not do too much from a Life risk perspective. Hence, we consider this risk to be negligible, this isl not entirely true.
Catastrophe Risk
This is the increase of the Best Estimate under the assumption that the first year mortality rate increases by 0.15 percentage points. For the rest of the projection years the Best Estimate mortality rates may be used. The Catastrophe Risk (CAT Risk) simulates some kind of catastrophe that could hit our insurance population and increases mortality rates for one year. For the calculation we use the simplification formula [12, EIOPA, SCR 7.80], which states that the CAT risk for life products is equal to:
Lif ecat= 0.0015 ·
X
i
CARi (3.20)
Here CARi is the capital at risk for the ith policy. Citing SCR 7.80:
“This capital at risk is CAR denotes the capital at risk of the policy i, meaning the higher of zero and the difference between the following amounts:
• The amount that the insurance or reinsurance undertaking would currently pay in the event of the death of the persons insured under the contract after deduction of the amounts recoverable from reinsurance contracts and special purpose vehicles; • The Best Estimate of the corresponding obligations after deduction of the amounts
recoverable form reinsurance contracts and special purpose vehicles;”
So, for the funeral insurance and the term insurance it is clear that CAR is just the difference between the sum insured and the Best Estimate. For the regular pension the capital at risk is zero, because in case of death of the insured no more payments have to be made. For the partner pension the capital at risk is the Best Estimate of the liabilities in case the insured dies. So we calculated the cashflow for this situation and calculated the Best Estimates for this case based on this cashflow. We subtract the Best Estimate and find the capital at risk for the partner pension.
Lapse Risk
This is the maximum decrease of the Own funds (definition in section 3.3.5) under certain assumptions, considering the behavior of insureds. Of these the mass lapse will always be the dominant in our model, so we will stick to that risk. Mass lapse concerns itself with the decrease in own funds in the situation that 70% of non-retail business is
Natural hedging in a Solvency II Framework — Bertus Veurink 21
lapsed and 40% of retail business. Pension products are considered non-retail, the rest of the products is retail. Most pension contracts are no individual contracts but are collective contracts and hence considered as non-retail.
Since the pricing of our product is done using a constant yield of 3% even at the start of our projection the lapse risk will not be equal to zero, as losing a client will mean losing a bit of profit. The yield curve + CCP will quickly rise above 3% and therefore we start with a Best Estimate that is lower than zero. This indicates a positive lapse risk, which we will discuss more elaborately in the result section. Since we reserve using a constant 3% yield, we will have to pay the insured the value of his contract calculated using a constant 3% yield. This means that the difference between this paid-up value and the market value is the exposure on which we have to calculate the lapse risk. Since the yield is larger than 3% for a lot of maturities, this will be a very big risk for us. Correlations and SCR
After the calculation of all these shock values, we need to know how these correlate and determine the amount of risk we carry in one number. This is defined as the Solvency Capital Requirement (SCR), the amount of capital needed to be “safe” from a risk management perspective. The before mentioned SCRlife can be calculated using the
correlations in figure 3.1. The correlation between longevity risk and mortality risk is the most interesting correlation for our research. Because we allow one to hedge the other we expect that in our natural hedging setting the correlation of −0.25 from the matrix will be without effect, since one of the risks will always be zero. Whether this really is the case will be discussed further in the results chapter, Chapter 4.
Figure 3.1: Correlation matrix of SCRlife
Market Risks
After obtaining the SCRlifethe basic SCR (or BSCR) follows after calculating the market
risks and the correlation with the SCRlifevalues. Correlation of market risk and life risk
is prescribed to be 0.25. These market risks are not the focus of our research, because our research interest lies mainly on the liability side. But, to give a semi-realistic view on the SCR-ratio effect of our natural hedging proposal we will incorporate some assets in our model. We will not give these too much attention though, but see them only as a way to calculate the SCR-ratio. In our model we will invest all our received premiums in 30-year zero coupon bonds and we will assume that these are liquid enough to sell at each time to make payments where necessary. In the pricing of these bonds we assume that market value is equal to a risk free valuation of its cashflow. For this to be true, they need to be at least AAA-rated bonds, so that risk free valuation is justified. Risk free valuation of bonds means that no CCP is added to the risk neutral curve, as in our view this CCP cannot be obtained in the asset market.
This leaves us with two market risks to be calculated: interest rate risk and CCP-risk. The last one is calculated as the present value of a complete loss of the CCP on the yield curve. Obviously, this only has an effect on the liabilities in our model. The other one, interest rate risk is calculated as the change of liabilities minus the change in assets if we apply a different yield curve for both. A Yield-up scenario and a Yield-down scenario is prescribed and the interest rate risk present is the maximum of these 2 scenarios
