Abstract
In this chapter we introduce the valuation portfolio of Buchwalder–Bühlmann–Merz–Wüthrich. The valuation portfolio provides a systematic approach for replicating insurance liabilities by financial instruments, leaving only the non-hedgeable risks. The latter are replaced by so-called best-estimates. The valuation portfolio allows for the valuation of insurance cash flows, it describes the dynamical component called claims development result and it analyzes the inherit prediction uncertainties. This construction is supplemented by explicit examples in life and non-life insurance.
Keywords
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In the previous chapters we have discussed the valuation of cash flows for a given state price deflator and we have introduced the financial market of basis financial instruments \(\mathfrak{A}^{(i)}\), . Throughout we assume that this financial market is sufficiently rich containing at least all ZCBs \(\mathfrak{Z}^{(m)}\) with maturities , and hence also the bank account \(\mathfrak{B}\) (see Sect. 6.1). In the present chapter we address the full balance sheet of an insurance company. Therefore, we define the valuation portfolio (VaPo) which was introduced in Buchwalder et al. [28], Bühlmann–Merz [33] and Wüthrich et al. [168].
There is a general agreement that the balance sheet of an insurance company should be measured in a consistent way. Assets are usually measured with market values (where they exist) or in a marked-to-model approach where no deep and liquid market exists. The valuation of insurance liabilities is more tricky because there is no active market where insurance liabilities are traded. Therefore there are no market prices for insurance liabilities. For this reason we aim to calculate market-consistent values for insurance liabilities in a marked-to-model approach which we are going to develop below. The idea is to express insurance liabilities (liability side of the balance sheet) with the help of basis financial instruments \(\mathfrak{A}^{(i)}\), . That is, we map the insurance liabilities to a multidimensional VaPo in a vector space with basis financial instruments \(\mathfrak{A}^{(i)}\), , as basis. This VaPo is then compared to the real existing asset portfolio on the asset side of the balance sheet, see Fig. 7.1 and Table 1.1. Both the insurance liabilities expressed by the VaPo and the existing asset portfolio then live in the same vector space. Therefore, the two sides of the balance sheet become comparable and can be valued consistently. This allows to define solvency of an insurance company, i.e. it states under which crucial criteria an insurance company is able to successfully run its insurance business also under certain adverse stress scenarios.
Because we treat both sides of the balance sheet (assets and liabilities) simultaneously by the same method, this approach is called the full balance sheet approach. From a risk management point of view it is important that we show all exposures on the balance sheet, also those that appear on both sides (and disappear in a net view). It often happens that netting positions do not have exactly the same risk factors and one position may default where the other does not, and hence they are not exactly netting. Therefore, it is important to understand the entire exposure.
In this (economic) full balance sheet approach we use market values where available and market-consistent values where no market values are available (such that the entire valuation system is consistent). However, we would like to emphasize that market values do not have an absolute significance, and depending on the purpose other valuation methods may be preferred. This can easily be done by choosing an appropriate model for the state price deflator. However, it is important that the same method (and hence the same state price deflator) is applied to both sides of the balance sheet because otherwise there arise inconsistencies (which allow for accounting arbitrage over time). Summarizing, the valuation of all balance sheet positions should be consistent for the chosen state price deflator.
1 Construction of the Valuation Portfolio
Throughout this chapter we work under Assumption 6.3 (basic actuarial model) with probability distortion choice
for the VaPo construction. The valuation functional at time under this probability distortion choice is denoted by \(Q_{t}^{0}[\cdot]\), see (6.10). In Chap. 8 on the VaPo protected against insurance technical risk we will relax assumption (7.1) on the probability distortion φ T to obtain the general valuation functional Q t [⋅].
Remark
For cash flows X of product form (6.5) integrability of Λ (k) for all is sufficient for under Assumption 6.3 and (7.1), see also (7.9), below.
1.1 Financial Portfolios and Cash Flows
In this subsection, we briefly recall the essentials on financial portfolios introduced in Sect. 6.2. The financial market consists of basis financial instruments \(\mathfrak{A}^{(i)}\), , whose price processes are described by . These price processes are assumed to be \(\mathbb{A}\)-adapted, integrable and consistent w.r.t. φ as specified by Assumption 6.3 (basic actuarial model).
The VaPo construction assumes that an insurance liability cash flow can be expressed in terms of basis financial instruments \(\mathfrak{A}^{(i)}\), . Therefore, we consider financial portfolios \(\mathfrak{U}\) which are given by linear combinations of basis financial instruments \(\mathfrak{A}^{(i)}\), see (6.6). For the financial portfolio \(\mathfrak{U}\) is defined by
The price process of this financial portfolio \(\mathfrak{U}\) is (by linearity) given by and satisfies the following consistency property, this is an easy consequence of Corollary 6.5:
Corollary 7.1
Under Assumption 6.3 the price process of \(\mathfrak{U}\) satisfies for
Such financial portfolios \(\mathfrak{U}\) are used to represent insurance liability cash flows X. The choice of these financial portfolios needs to be done carefully. Many basis financial instruments \(\mathfrak{A}^{(i)}\) do not generate a natural cash flow. For example, if we buy a (non-dividend paying) stock \(\mathfrak{A}^{(i)}\) then the value of this stock develops as , but it does not generate a natural cash flow unless we sell that stock at a particular point in time. Henceforth, to describe the cash flow generated by a financial portfolio \(\mathfrak{U}\) we do not only need to determine the basis financial instruments \(\mathfrak{A}^{(i)}\) we would like to hold in our financial portfolio but we also need to specify when we are selling them at their actual prices \(A_{t}^{(i)}\). Therefore, we are going to denote the point in time at which we are selling the financial portfolio \(\mathfrak{U}\) with an upper index k, i.e. financial portfolio \(\mathfrak{U}^{(k)}\) is sold at time .
1.2 Construction of the VaPo
The VaPo is constructed in two steps and the monetary value for the VaPo is obtained in the third step. Throughout this chapter we assume φ T≡1, see (7.1).
Step 1 in the VaPo Construction
Choose an appropriate basis of financial portfolios \(\mathfrak{U}^{(k)}\), , and express the insurance liability cash flow in terms of this financial basis, i.e. express X by
with (i) \(\mathbb{T}\)-adapted insurance technical variables Λ=(Λ (0),…,Λ (n)) and (ii) the price processes of the financial portfolios \(\mathfrak{U}^{(k)}\), , are \(\mathbb{A}\)-adapted, integrable and consistent w.r.t. φ and the financial portfolio \(\mathfrak{U}^{(k)}\) is sold at time k, see Sect. 7.1.1. This gives the following mapping
The mapping (7.4) expresses the insurance liabilities in terms of financial portfolios \(\mathfrak{U}^{(k)}\), i.e. it maps cash flows X into the multidimensional vector space with financial basis \(\mathfrak{U}^{(k)}\), , see also (6.6) and (7.2).
Step 2 in the VaPo Construction
For a fixed point in time we replace the insurance technical liabilities Λ (k) by their best-estimates at time t. This gives the following VaPo mapping (under Assumption 6.3 and (7.1))
Remarks 7.2
(Asset-and-liability management (ALM))
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The VaPo mapping (7.5) maps the insurance liability cash flow X to a -measurable financial portfolio VaPo t (X). This financial portfolio VaPo t (X) replicates the expected insurance technical liabilities in terms of financial portfolios \(\mathfrak{U}^{(k)}\) and basis financial instruments \(\mathfrak{A}^{(i)}\), respectively (see (6.6) and (7.2)). As a consequence, it can be compared to any asset portfolio . The VaPo construction is achieved by setting the probability distortion φ T≡1 in the basic actuarial model Assumption 6.3, see (7.1).
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Integrability of Λ (k) implies that VaPo t (X) in (7.4) is well-defined.
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If the financial portfolio \(\mathfrak{U}^{(k)}\) is given by the following linear combination of basis financial instruments \(\mathfrak{A}^{(i)}\), , see (6.6),
with , then we obtain at time t the representations
(7.6)This VaPo can now directly be compared to the portfolio we hold on the asset side of the balance sheet. Assume that at time t it is given by
(7.7)Then we have an asset-and-liability mismatch if and VaPo t (X) differ, i.e. if for some
(7.8)Under inequality (7.8) the situation may occur that these expected liabilities are not covered by asset values at any point in time in the future, see also Fig. 7.1. When defining solvency we are going to be more specific about what we mean by “covered by asset values at any point in time in the future”. A first example was already presented in Sect. 5.2.2 and more analysis is done below.
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Formula (7.6) gives two different representations for the VaPo: the first line gives the cash flow representation, which highlights when the corresponding basis financial instruments are sold to generate a cash flow; the second line gives the instrument representation where we only care about the question how many of each basis financial instrument we need to buy to replicate the expected liabilities.
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So far we have not discussed monetary values, but we have only expressed expected liabilities in terms of financial portfolios. Monetary values are going to be calculated in the next step in the VaPo construction. We define the best-estimate value of the cash flow X to be equal to the value of the VaPo at time t.
Step 3: Monetary Value of the VaPo
In the last step we map the VaPo to monetary value at time t by
Remarks 7.3
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Pay attention to the valuation mapping \(Q^{0}_{t} [\cdot ]\) in (7.9), it assumes φ T≡1 under Assumption 6.3 (basic actuarial model).
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The construction is linear, i.e. it might be necessary to decouple cash flow X=X 1+X 2 in order to find appropriate financial portfolios \(\mathfrak{U}\), see Example 7.5.
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Mapping (7.9) attaches a monetary value to the expected insurance liabilities at time t. With (7.6) this monetary value can be rewritten as
which are the best-estimate discounted liabilities at time t. The first line is the cash flow representation, the second line the instrument representation, see also Remarks 7.2. We can now compare this value to the value of the asset portfolio at time t. It is given by
If we have
$$ {S}_t^{(t)} \ge Q^0_t [\mathbf{X} ], $$(7.10)then the expected liabilities are covered by asset values, this is the case in Fig. 7.1. Otherwise, if (7.10) is not fulfilled, the company has not got a sufficient amount of capital at time t for paying for all its expected liabilities. Requirement (7.10) is called the accounting condition, see also Sect. 9.2.1.
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If we have an ALM mismatch according to (7.8) but if the accounting condition (7.10) is fulfilled, then at time t we have the possibility to switch from the asset portfolio to the financial portfolio VaPo t (X) (and a non-negative excess capital) so that we have no ALM mismatch. A first example was already presented in Sect. 5.2.2. ALM mismatch is often wanted because such additional risks also give additional opportunities to earn extra money. Therefore, in practice, there is always an ALM mismatch. These additional risks will require additional protection so that solvency can be achieved. We will discuss this in Chap. 9, and we will also discuss the role of excess capital and best-estimates.
1.3 Best-Estimate Reserves
In the previous subsection we have defined the VaPo at time for a cash flow given by (7.3). If the insurance company is at time t and has already made all the payments X s for s≤t, then the outstanding liabilities are given by
Therefore, it will only build provisions for these outstanding liabilities X (t+1) at time t. The VaPo for these outstanding liabilities at time is given by
The best-estimate reserves for the outstanding liabilities X (t+1) at time t are defined by
It is important to realize that we choose conditional expectations for the definition of best-estimate reserves.
Define the VaPo for the cash flow X t =(0,…,0,X t ,0,…,0) at time t by
and its price at time t is given by .
Proposition 7.4
(Self-financing properties)
Under Assumption 6.3 with φ T≡1 we have for the outstanding liabilities of given by (7.3) (the first two statements need to be understood in vector notation)
Proof of Proposition 7.4
The linearity of the VaPo implies
which gives the first claim. The second claim follows from the tower property for conditional expectations (see Williams [159]) and the independence between \(\mathbb{A}\) and \(\mathbb{T}\). For the third claim we have
where we have used the choice φ T≡1 for the VaPo construction and that deflated price processes are \((\mathbb {P},\mathbb{F})\)-martingales. This completes the proof. □
Interpretation
Proposition 7.4 explains that the VaPo is in the average self-financing and gives an optimal financial portfolio to control ALM risk (in an L 2-sense), see Sect. 7.3.2. If we buy VaPo t (X (t+1)) at time at the price of , it generates the following value V t+1 at time t+1:
This value is compared to the best-estimate liabilities that we are facing at time t+1
The comparison defines the claims development result (CDR) at time t+1
see also Fig. 7.2. The CDR describes the difference between the available assets V t+1 and the required assets \(Q^{0}_{t+1}[\mathbf{X}_{(t+1)}]\) so that best-estimate liabilities at time t+1 are covered by asset values. The tower property for conditional expectations tells us that (see also (7.22) below)
which says that in the average we have the correct provisions. Moreover, any other (unbiased) asset portfolio provides a larger conditional variance (upon existence), which explains the terminology best-estimate reserves. We are going to further elaborate on this in Sect. 7.3, see Theorem 7.12 and Remarks 7.14.
2 Examples
We present three examples: (i) a life insurance endowment policy in Example 7.5, (ii) a life-time annuity in Example 7.6 and (iii) a non-life insurance run-off in Examples 7.7 and 7.8. These examples are going to be used throughout this book. Numerical illustrations are presented in Sect. 8.3.
2.1 Examples in Life Insurance
In this subsection we provide two explicit life insurance examples. The first example is similar to the endowment policy example provided in Wüthrich et al. [168]. The second example will consider a life-time annuity.
Example 7.5
(Endowment policy)
We start with a homogeneous portfolio of L x insured lives all aged x=50. These L x people sign at time 0 the following contract with a term of n=5 years:
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single initial premium installment Π at time 0;
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(yearly) death benefit is the index (financial portfolio) \(\mathfrak{U}\) with a minimal interest rate guarantee of r>0;
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survival benefit is the index (financial portfolio) \(\mathfrak{U}\).
We choose the final time horizon n=5, that is, we consider the basic discrete time model for points in time , and we make Assumption 6.3 (basic actuarial model) with fixed state price deflator φ. For the construction of the VaPo we assume, in addition, that φ T≡1, see (7.1).
The price process of financial portfolio \(\mathfrak{U}\) is given by (with initial value U 0=1). It is \(\mathbb{A}\)-adapted, integrable and φ-consistent. L x+t denotes the number of people alive at time t. The sequence is \(\mathbb{T}\)-adapted and non-increasing (and hence bounded and integrable). We define
These L x (initial) contracts generate the following cash flow X (for age x=50):
Time | Cash flow | Premium | Death benefit | Survival benefit |
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0 | X 0 | −L x Π | ||
1 | X 1 | D x+1 (U 1∨(1+r)1) | ||
2 | X 2 | D x+2 (U 2∨(1+r)2) | ||
3 | X 3 | D x+3 (U 3∨(1+r)3) | ||
4 | X 4 | D x+4 (U 4∨(1+r)4) | ||
5 | X 5 | D x+5 (U 5∨(1+r)5) | L x+5 U 5 |
Cash inflow (premium) is modeled with a negative sign, cash outflow (insurance benefits to the policyholders) is modeled with a positive sign and y∨z=max{y,z}.
Step 1 in the VaPo Construction
As described above, we need to choose the financial portfolios that replicate the insurance liabilities:
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The premium cash flow is simply cash value at time t=0. We denote the unit by \(\mathfrak{A}^{(0)}\) (this is a ZCB with maturity date m=0).
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The survival benefit cash flow is modeled by the financial portfolio \(\mathfrak{U}\).
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The death benefit cash flow for t=1,…,5 is the maximum of U t and (1+r)t (we have assumed initial value U 0=1). This maximum is modeled by
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the underlying financial portfolio \(\mathfrak{U}\) and
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a derivative on the underlying financial portfolio \(\mathfrak{U}\) modeled by a put option \(\mathfrak{P}^{(t)}\) which allows to sell the underlying financial portfolio \(\mathfrak{U}\) at time t at price (1+r)t whenever U t <(1+r)t, see also Sect. 5.3.
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This implies that we study a 7-dimensional vector space that is spanned by the financial portfolios
We assume that the financial market is sufficiently rich containing all these financial portfolios. In view of (7.4) we obtain the following mapping (cash flow representation and instrument representation)
The instrument representation immediately shows that there is no uncertainty in the number of underlying financial portfolios \(\mathfrak{U}\) the insurance company needs to buy, because every insured life receives exactly one of these financial portfolios no matter whether the person dies or survives the contract period. The only uncertainty lies in the number of put options \(\mathfrak{P}^{(1)}, \ldots, \mathfrak{P}^{(5)}\) the company needs to purchase.
Step 2 in the VaPo Construction
We need to determine how many of the financial instruments we need to hold at time . The VaPo at time t is given by, see (7.5),
and for the outstanding liabilities at time t=0,…,4 we obtain
The second equality follows from the tower property for conditional expectations and from and . Note that VaPo0(X (1)) are the total insurance liabilities of L x lives insured after they have paid the initial premium installment Π.
Step 3: Monetary Value of the VaPo
In the last step we then calculate the monetary value of the VaPo. In our example it is at time t=0 given by
where U t is the price of the financial portfolio \(\mathfrak{U}\) at time t and \(\mathrm{Put}_{t}(\mathfrak{U},(1+r)^{m},m)\) denotes the price of the put option \(\mathfrak{P}^{(m)}\) at time t with strike (1+r)m at time m. Since this price process of the put option also needs to be consistent w.r.t. φ it is for t≤m given by
From (7.12) we can also calculate the pure risk premium Π 0: choose Π in (7.12) such that the equilibrium principle \(Q^{0}_{0} [\mathbf{X}]=0\) is fulfilled. This provides
The pure risk premium Π 0 exactly corresponds to the expected value of the insurance benefits. The best-estimate reserves for the outstanding liabilities at time t=0,…,4 are given by
This describes the monetary run-off of the outstanding liabilities. In Sect. 8.3 we provide a numerical example.
Example 7.6
(Life-time annuity)
We assume that we have a homogeneous group of L x insured lives all having retirement age x=65. We would like to calculate a life-time annuity for these retirees. We assume that they receive an annual annuity payment a>0 and this annual annuity payment is adjusted to a well-defined inflation index. We again work under Assumption 6.3 (basic actuarial model).
We denote the survival probabilities by p x+t , t≥0, with p 121=0. That is, at age 121 every person insured has died with probability 1. The final time horizon is then given by n=120−65=55 which is the maximal remaining life-time after retirement at age 65. The people alive at time are denoted by L x+t (-measurable). The value of the inflation index at time t is denoted by I t with initial value I 0=1. The cash flow for this life-time annuity portfolio is then given by
Time | Cash flow | Annuity payment |
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0 | X 0 | |
1 | X 1 | L x+1 a I 1 |
2 | X 2 | L x+2 a I 2 |
⋮ | ⋮ | ⋮ |
55 | X 55 | L 120 a I 55 |
Step 1 in the VaPo Construction
We need to choose the appropriate financial portfolios. In this life-time annuity example we need to purchase a fund \(\mathfrak{I}\) whose value process is given by the inflation index . Another interpretation/choice is that we buy inflation-protected ZCBs for the different maturities. Since the latter is more involved we assume that the former financial portfolio \(\mathfrak{I}\) exists at the financial market (and henceforth is \(\mathbb{A}\)-adapted and fulfills Assumption 6.3 for the given state price deflator φ). This gives the mapping (instrument and cash flow representation)
Step 2 in the VaPo Construction
The VaPo for this inflation adjusted life-time annuity is then given by
The VaPo for the outstanding liabilities at time t=0,…,54 is given by
Step 3: Monetary Value of the VaPo
The best-estimate reserves for the outstanding liabilities at time t=0,…,54 are given by
This describes the monetary run-off of the outstanding liabilities for this life-time annuity portfolio. Below we see that this example involves longevity risk, i.e. if people live longer than expected we run out of the expected number of funds \(\mathfrak{I}\). A numerical example is provided in Sect. 8.3.
2.2 Example in Non-life Insurance
We consider a non-life insurance run-off. We remark that “non-life insurance” is the Continental European terminology for “general insurance” in the UK and “property & casualty insurance” in the US. We start with a brief discussion about the cash flows generated by non-life insurance contracts, for a detailed introduction we refer to Wüthrich–Merz [166].
To illustrate the problem we assume that the insurance company sells a non-life insurance contract which protects the policyholder against (random) claims within a fixed calender year, see Fig. 7.3. Assume that the company receives a premium Π at the beginning of this calender year. Hence the policyholder exchanges the premium Π against a contract which gives him a cover against well-specified random events (claims) occurring within that fixed time period (the specified calender year).
Assume that there is a claim within this fixed time period. In that case the insurance company will replace the financial damage caused by this claim (according to the insurance contract). Typically, the insurance company is not able to assess the claim immediately at the occurrence date due to:
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1.
Usually, there is a reporting delay (time gap between claim occurrence (accident) date and claim reporting to the insurance company). This time gap can be small (a few days), for example, in motor hull insurance, but it can also be quite large (months or years). In particular, in general liability insurance we can have large reporting delays: typical examples for large reporting delays are asbestos claims that were caused several years ago but are only noticed and reported today (because the resulting disease has only broken out today).
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2.
Usually it takes quite some time to settle a claim (time difference between reporting date and settlement date = claims closing). This is due to several different reasons, for example, for bodily injury claims we first have to observe the recovery process before finally deciding on the claim and on the compensation. Other claims can only be settled at court which usually takes some time until the final settlement takes place. In most cases a (more complex) claim is settled by several single claims payments: whenever a justified bill for that specific claim arrives it is paid by the insurance company, see Fig. 7.3.
Conclusion
Every claim generates claims cash flows. Our aim is to predict these claims cash flows and to put appropriate provisions aside so that we can fulfill all these claims payments (according to the insurance contract agreement).
In the life insurance examples, we have always assumed that we have homogeneous portfolios. In non-life insurance, homogeneous portfolios are constructed by, first, subdividing the whole non-life insurance portfolio into different lines of business (LoB), e.g. motor liability, motor hull, private property, commercial liability, etc. Second, these sub-portfolios are further subdivided by the accident date of the insurance claims.
We fix a LoB. X i,j denotes all payments of this LoB that correspond to accident year i∈{1,…,I} (occurrence date) and which are done in development year j∈{0,…,J} (settlement delay), see Table 7.1. We assume I=J+1 and that there are no further payments necessary after development year J, i.e. all claims are settled after development year J (after a maximal settlement delay of J years). The variables X i,0,X i,1,…,X i,J correspond to the payments done for all claims that have a fixed accident year i∈{1,…,I}. The nominal cumulative payments of accident year i after j development periods are given by
and C i,J is called the (nominal) ultimate claim of accident year i. All payments that are done in accounting (calendar) year k=1,…,I+J are given by the sum
these are the diagonals in Table 7.1. At time t we have payment observations in the upper loss development (or claims run-off) triangles and trapezoids, respectively,
and we need to predict and value the lower loss development triangles and trapezoids, respectively,
Our aim is to set up a stochastic claims reserving model that allows for predictions and valuations of .
Convention
Throughout the non-life insurance examples we use i for accident years, j for development years and k=i+j for accounting (calendar) years.
Example 7.7
(Non-life run-off)
The final time horizon is given by n=I+J=2I−1=2J+1 (for I=J+1). We assume that Assumption 6.3 (basic actuarial model) is fulfilled and as probability distortion we choose φ T≡1 for the VaPo construction, see (7.1). Then we study the cash flow X given by (7.15).
Step 1 in the VaPo Construction
We assume that the cash flow X is \(\mathbb{T}\)-adapted (and has integrable components). This implies that the ZCBs are the right financial instruments to replicate the cash flow X. We denote the ZCB with maturity k by \(\mathfrak{Z}^{(k)}\). Note that \(\mathfrak{Z}^{(k)}\) generates cash flow \(\mathbf{Z}^{(k)}=(0,\ldots, 0,1,0,\ldots, 0)\in\mathbb {R}^{n+1}\), where the payout 1 is at time k. In view of (7.4) we obtain the following mapping
The assumption that X is \(\mathbb{T}\)-adapted (and hence X is independent of \(\mathbb{A}\)) is often a restrictive strong one. One should be careful with this assumption especially in long-tailed LoBs (like third party liability insurance or workmen’s compensation insurance). But also for short-tailed LoBs that strongly depend on economic factors and unemployment rates (e.g. health insurance) the independence assumption may not be fulfilled. In all these cases we need to separate (independently) \(X_{k} = \varLambda_{k} \ U_{k}^{(k)}\), where \((U_{t}^{(k)})_{t\le k}\) is the price process of an inflation protected ZCB. One also needs to be careful about the correct inflation index. Claims inflation is often very different from the classical economic inflation defined via baskets of goods.
Remarks
(1) In Sect. 6.1 we have assumed the existence of ZCBs for all maturities . (2) Under Assumption 6.3 with φ T≡1 the assumption that every component of the \(\mathbb{T}\)-adapted random vector X is integrable is sufficient for .
Step 2 in the VaPo Construction
The non-life VaPo at time is obtained by
The VaPo of the outstanding liabilities at time t is given by
This exactly describes the best-estimate prediction of the inexperienced part of the loss development at time t, see Table 7.1 and .
Step 3: Monetary Value of the VaPo
The best-estimate reserves for the outstanding liabilities at time t are given by
where P(t,k) is the price of the ZCB \(\mathfrak{Z}^{(k)}\) with maturity k at time t≤k. This concludes Example 7.7.
Example 7.8
(Hertig’s [83] claims reserving model)
We give an explicit model for the calculation of the non-life run-off VaPo presented in Example 7.7. This example uses Hertig’s [83] claims reserving model. For an extended introduction to Hertig’s model we refer to Sect. 5.1 in Wüthrich–Merz [166] and Merz–Wüthrich [114] on the PIC claims reserving model.
Model Assumptions 7.9
We set Assumption 6.3, the existence of ZCBs \(\mathfrak{Z}^{(k)}\) for all maturities is assumed (see Sect. 6.1) and we assume
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(a)
X i,j are -measurable for i=1,…,I and j=0,…,J;
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(b)
X i,j are independent for different accident years i, X i,0>0 is integrable; and
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(c)
there exist parameters ϕ j and σ j >0 such that for cumulative payments
for j=0,…,J−1 and i=1,…,I.
Model Assumptions 7.9 imply that
Claims reserving models that have this property (7.17) are called chain-ladder models with chain-ladder factor f j (link ratio or age-to-age factor) defined by
Lemma 7.10
Choose t∈{1,…,n}. Under Model Assumptions 7.9 we have for i+j≥t (with 1≤i≤I∧t and 0≤j≤J−1)
where an empty product is set equal to 1.
Proof of Lemma 7.10
We iterate the tower property for conditional expectations which implies, see (7.17),
This proves the first claim. The second claim follows from the first claim with . This proves the lemma. □
Choose t≥I. The VaPo construction (7.16) then implies that we obtain the following VaPo for the outstanding liabilities at time t
The (discounted) best-estimate reserves at time t≥I are then given by
For the calculation of the discounted best-estimate reserves we use the market values P(t,i+j) at time t of the ZCBs \(\mathfrak{Z}^{(i+j)}\).
In classical statutory accounting one often needs to display nominal claims reserves for the outstanding liabilities in non-life insurance. This means that for statutory accounting we set P(t,k)≡1 which implies at time t≥I (we have set I=J+1)
The nominal reserves can be obtained by setting the financial deflator φ A≡1, which also provides a consistent valuation model. However, for this constant financial deflator model (statutory model) we obtain marked-to-model prices, which have not much to do with the prices we observe at the financial market.
We conclude with the following remark: the (discounted) best-estimate reserves were calculated under the assumption that the chain-ladder factors f j are known. In general, they are not known and need to be estimated. This adds an additional source of complexity and uncertainty to the problem. We treat this aspect in Sect. 10.2. A numerical example is provided in Sect. 8.3.
3 Claims Development Result and ALM
We work under Assumption 6.3 (basic actuarial model) with φ T≡1 and choose with product structure (7.3).
3.1 Claims Development Result
In Sect. 7.1.3 we have defined VaPo t (X (t+1)) for the outstanding liabilities X (t+1) at time t. The best-estimate reserves at time t for these outstanding liabilities are given by . Proposition 7.4 explains that if we buy this VaPo at time t then we have an asset strategy that is in the average self-financing. This indicates that the VaPo satisfies some optimality criterion concerning ALM risks. In this section we are going to analyze such ALM risks.
Assume we buy the financial portfolio VaPo t (X (t+1)) at time t at its price . This financial portfolio generates the following value at time t+1
This value V t+1 needs to cover the best-estimate liabilities that the company is facing at time t+1 which are given by
The claims development result (CDR) at time t+1 is defined by
Figure 7.2 illustrates the CDR at time t+1.
Remarks 7.11
-
The study of the CDR has a long tradition in insurance practice. Especially in non-life insurance, the profit and loss statement position “loss experience prior accident years” (which exactly corresponds to the CDR) can have a major influence on positive or negative earning statement results.
-
The time series of the CDRs, given by CDR t+1(X (t+1)), t≥0, allows for back testing of the question how a particular reserving method applies to the problem. For example, if the observed CDRs are always negative then the applied reserving method does not fit to the actual problem because it constantly underestimates the liabilities. That is, the time series should fluctuate around 0, see also (7.22).
-
The study of the CDR from a stochastic point of view is rather new. This study was mainly motivated by new solvency regulations and, independently at the same time, De Felice–Moriconi [51, 52], Böhm–Glaab [19] and Merz–Wüthrich [112, 113, 167] (and probably also others) have introduced this or similar notions. For further references we refer to AISAM–ACME [3], Ohlsson–Lauzeningks [122] and Bühlmann et al. [35].
-
The previously mentioned work has studied the CDR for nominal reserves in non-life insurance. Wüthrich–Bühlmann [165] study the CDR for discounted reserves , where the Vasicek model is used for the projection of ZCB prices. The resulting conclusions however should always be considered under the limitations mentioned in Sect. 9.4.6 below.
The tower property provides in the basic actuarial model
This says that in the average we have the correct provisions. Using the tower property once more we have (note that we have integrable financial price processes)
In the next section we are going to analyze the risk in the CDR as well as the risk resulting from an ALM mismatch.
3.2 Hedgeable Filtration and ALM
Assume we hold the following portfolio at time t on the asset side of the balance sheet
where we either express the asset portfolio in the financial portfolios \(\mathfrak{U}^{(k)}\) (cash flow representation, see Remarks 7.2) or, using (6.6), in the basis financial instruments \(\mathfrak{A}^{(i)}\) (instrument representation). We assume that this asset portfolio fulfills (with an equality sign) the accounting condition (7.10) at time t for the discounted best-estimate reserves, that is,
Thus, the expected outstanding liabilities are covered by asset values at time t. The asset portfolio generates at time t+1 value
This value needs to be compared to the best-estimate liabilities \(Q_{t+1}^{0}[\mathbf{X}_{(t+1)}]\) and to the value V t+1 that is generated by the VaPo at time t+1, see (7.20). In general, we have
i.e. the value generated by the asset portfolio does not match the value generated by VaPo t (X (t+1)) at time t+1. Using (7.22) we obtain
This explains that asset portfolios different from VaPo t (X (t+1)) do not necessarily match the expected value of the liabilities at time t+1. Of course, if we decide to hold an asset portfolio different from the VaPo, we would choose such that
which basically says that we aim to achieve an extra asset return over the VaPo by our asset portfolio . But as we will just see, this additionally expected asset return also generates additional (ALM) risks.
Usually, one uses different risk measures for analyzing these ALM risks. We further investigate risk measures in Chap. 9. For the present discussion we take the L 2-distance as a risk measure. The L 2-distance is probably the simplest choice that can be done for a first meaningful risk assessment. Thus, we consider the expected deviation
Formula (7.25) measures by how much the value of the asset portfolio may differ from the expected liabilities at time t+1 in the L 2-distance. Our aim is to minimize the ALM risk (7.25), of course, only in situations where it is well-defined. Therefore, we need to restrict Assumption 6.3.
Assumption 6.3 (B) In addition to Assumption 6.3 we assume that the price processes of all basis financial instruments \(\mathfrak{A}^{(i)}\), , are square integrable.
Theorem 7.12
Make Assumption 6.3 (B) and choose with decomposition (7.3). The asset portfolio VaPo t (X (t+1)) minimizes the ALM risk (7.25) among all -measurable portfolio choices and we obtain
The right-hand side is achieved by the portfolio choice .
The proof of Theorem 7.12 is given below and provides the following corollary.
Corollary 7.13
(ALM risk)
Under the assumptions of Theorem 7.12, the asset portfolio has the following ALM risk
with , see (7.20).
Remarks 7.14
(Best-estimate reserves)
-
Proposition 7.4 and Theorem 7.12 explain the terminology best-estimate reserves, i.e. the VaPo is in the average self-financing, see (7.23), and it minimizes the L 2-distance risk measure. The L 2-distance risk measure is always positive and we aim to minimize it. We remark that the L 2-distance is not a risk measure in the classical sense because it does not consider the trade-off between risk and reward, it only punishes risk taking. For other risk measures see Chap. 9 below.
-
In general, we do not assume square integrability, but it is a necessary condition in Theorem 7.12 for the optimality criterion interpretation.
-
Insurance practice states that the best-estimate should correspond to the probability weighted average of future cash flows taking account of the time value of money, see TP.2.1 in QIS5 [64]. This exactly corresponds to our definition.
-
Artzner–Eisele [4] choose a classical (conditional) risk measure ρ t as it will be introduced in Chap. 9. Then, they define the optimal replicating portfolio at time t relative to this conditional risk measure ρ t . Hence, optimality is always defined relative to a risk measure, see also Sect. 9.2, below.
In view of Theorem 7.12 we call the ALM optimal portfolio choice for the time period (t,t+1], here optimality is meant relative to the L 2-distance.
-
The question whether we switch from the ALM optimal portfolio choice given by VaPo t (X (t+1)) to an asset portfolio , which generates additionally expected asset return (7.24), will finally depend on the question, how much additionally expected return we want and how large the financial “punishment” is for not holding the ALM optimal portfolio. The L 2-distance risk measure as defined in (7.25) does not reward additionally expected asset return at all, therefore we will choose the ALM optimal portfolio as asset portfolio for this risk measure choice. In Chap. 9 we introduce other risk measures and we may deviate from the ALM optimal portfolio choice because they also reward extra expected return.
We define the filtration by and for
\(\mathbb{H}\) is called the hedgeable filtration, see also Malamud et al. [104]. This part of the risk can be hedged by an appropriate asset allocation which does the one-period roll-over from t to t+1. Note that , thus, the tower property for conditional expectations implies under the assumptions of Theorem 7.12
where the last equality follows from (7.22). We are left with the insurance technical risks from to which cannot be hedged by an appropriate asset allocation . That is, note that V t+1 is -measurable,
If we have no insurance technical risk at time t, i.e. all Λ (k) are -measurable, we obtain , see (7.27), which means that all (financial) risks are hedged by the asset portfolio . Therefore, the ALM optimal portfolio choice inherits no risks for deterministic insurance technical variables.
Proofs of Theorem 7.12 and Corollary 7.13
Let be the hedgeable filtration. Note that . With the tower property of conditional expectations we have
Next we note that the asset value \(S_{t+1}^{(t)}\) chosen at time t is -measurable. Moreover, the expected liability value generated by VaPo t (X (t+1)) is also -measurable. If we add and subtract this value, and if we use (7.22), we obtain
Using (7.23) we obtain
Note that the first term on the right-hand side of the above equality is positive and disappears for the portfolio choice , whereas the second term is equal to
and is minimized by the -measurable portfolio choice VaPo t (X (t+1)) which provides -measurable value V t+1. This completes the proof. □
3.3 Examples Revisited
In this subsection we revisit the examples from Sect. 7.2. As in Theorem 7.12 we make Assumption 6.3 (B), see Sect. 7.3.2.
Example 7.15
(Endowment policy, revisited)
We revisit Example 7.5 (endowment policy). In the endowment policy example the VaPo for the outstanding liabilities at time t=0,…,4 is given by
and the corresponding monetary value is given by
If we purchase this VaPo at time t it generates the following value at time t+1
which needs to be compared to the best-estimate liabilities at time t+1
This implies for the CDR at time t+1
Note that we rewrite q x+t+1 L x+t =(1−p x+t+1)L x+t . This provides that the CDR at time t+1 is given by
with
The variable β t+1 describes the change of best-estimate reserves if the observed number of people alive L x+t+1 differs from the expected number of people alive p x+t+1 L x+t . If more lives insured survive the time interval (t,t+1] we need to pay less put options \(\mathrm{Put}_{t+1}(\mathfrak{U},(1+r)^{t+1},t+1) =((1+r)^{t+1}-U_{t+1})_{+}\) on the one hand, and on the other hand we need to build additional reserves for future put options resulting in the first term of β t+1. Note that β t+1 belongs to the hedgeable σ-field . We immediately have, see also (7.23),
For the calculation of the underlying risk with the L 2-distance risk measure we have under Assumption 6.3 (B) (note that 0≤L x+t+1≤L x is bounded)
If the single lives are i.i.d. Bernoulli distributed with survival probability p x+t+1, this implies that the ALM optimal portfolio choice has non-hedgeable risk
Observe that this also involves a financial risk part due to the fact that we do not know at which price we need to purchase (or sell) additional put options at time t+1 for the insurance technical deviations. A numerical example is provided in Sect. 8.3. This finishes Example 7.15.
Example 7.16
(Life-time annuity, revisited)
In the life-time annuity Example 7.6 we have found the following VaPo for the outstanding liabilities at time t=0,…,54
The best-estimate reserves for the outstanding liabilities at time t are given by
If we purchase this portfolio at time t it provides at time t+1 the value
The CDR is then given by
with
The variable β t+1 describes the changes in the best-estimate reserves if the number of people alive L x+t+1 deviates from its expected value p x+t+1 L x+t . This implies similar to the previous example , and if we assume that the single lives are i.i.d. Bernoulli distributed with survival probability p x+t+1, this implies that the optimal ALM portfolio choice has non-hedgeable risk (see also (7.30) and Assumption 6.3 (B) in Sect. 7.3.2)
A numerical example is provided in Sect. 8.3. This finishes Example 7.16.
Example 7.17
(Hertig’s [83] claims reserving model, revisited)
We revisit Hertig’s [83] claims reserving model presented in Example 7.8. In Example 7.8 we have assumed that the non-life insurance payments X i,j satisfy Hertig’s [83] log-normal model given in Model Assumptions 7.9. The VaPo at time t≥I was then given by
with ZCBs \(\mathfrak{Z}^{(i+j)}\) as financial basis and with chain-ladder factors \(f_{l} = \exp\{\phi_{l} + \sigma_{l}^{2}/2\} +1\). The price for this VaPo at time t is
If we purchase this VaPo at time t it generates the following value at time t+1
where we have used that P(t+1,t+1)=1. Note that the middle line of the above formula are the expected payments in accounting year t+1 and the last line are the expected discounted payments in the accounting years after t+1. This value V t+1 should match the best-estimate liabilities at time t+1 given by
This implies that the CDR at time t+1 reads as
Note that we can rewrite the first term on the right-hand side of the CDR as follows
This describes how the observed cumulative payments C i,t−i+1 deviate from the expected cumulative payments C i,t−i f t−i , given the information . We define for i=t+1−J,…,I
note that β t+1−J,t+1=1. Then the CDR at time t+1 can be rewritten as
The terms in the brackets { ⋅ } are the risk bearing terms in accounting year t+1, i.e. C i,t−i+1 is predicted by C i,t−i f t−i . All the other terms belong to the hedgeable σ-field at time t≥I. In view of Lemma 7.10 (chain-ladder property) we know that we have unbiased claims predictions (in the average self-financing property) which implies (7.22) and (7.23) for the CDR. The ALM optimal portfolio VaPo t (X (t+1)) has for t≥I non-hedgeable risk given by
where we have used that C i,j in different accident years i are independent. Under the conditional log-normality assumption, see Model Assumptions 7.9, we obtain
where we have used Lemma 7.10 in the last equality. This implies that at time t≥I the non-hedgeable risk is given by
We see that this risk involves both insurance technical variables as well as financial variables because we do not know at which price we need to purchase (or sell) the additional financial instruments at time t+1 for insurance technical deviations.
Conclusions
The ALM optimal portfolio is given by the VaPo. This financial portfolio minimizes CDR risks (in an L 2-sense). It hedges all \(\mathbb{A}\)-adapted financial variables with the hedgeable filtration \(\mathbb{H}\) and it only leaves the non-hedgeable \(\mathbb{T}\)-adapted insurance technical risks. However, these insurance technical risks also involve financial risks because the insurance technical deviations need to be covered by financial instruments at actual market prices.
The VaPo is in the average self-financing, which means that in the mean we have correct provisions. However, since we consider stochastic processes there might still be shortfalls, e.g. we can observe adverse developments in the outstanding liability developments. That is, though the insurance company holds the ALM optimal portfolio it faces (insurance technical) risks that go beyond expected values. Therefore, a margin to protect against insurance technical risk is added to the best-estimate reserves. This margin stands for the risks beyond expected values and it makes best-estimate reserves into risk-adjusted reserves. These play the role of market prices for outstanding liabilities in a marked-to-model approach. This is exactly the subject of Chap. 8.
4 Approximate Valuation Portfolio
In the previous sections we have constructed the VaPo for simple examples. We have considered small homogeneous insurance portfolios and their liabilities were easily described by basis financial instruments and/or financial portfolios, respectively. In practice the situation is often more complicated. In particular, life insurance companies have complex high-dimensional insurance portfolios which usually involve embedded options and guarantees as well as management decisions. That is, the VaPo becomes highly path-dependent and the determination of the liability cash flows and the appropriate basis financial instruments is not straightforward. In such situations one often tries to approximate the VaPo by a financial portfolio. Here, we define an approximate VaPo (denoted by \(\text{VaPo}_{t}^{\mathrm{approx}}\)) which plays the role of a replicating portfolio. The use of an approximate VaPo induces ALM risks which need to be covered appropriately, see also Chap. 9.
Let us choose a filtered probability space and we assume that we have a given state price deflator . Our aim is to construct an approximate VaPo for an insurance liability cash flow .
In order to construct an approximate VaPo we choose a set of tradeable basis financial instruments \(\mathfrak{A}^{(0)}, \ldots, \mathfrak{A}^{(q)}\) from which we believe that they can replicate the insurance liabilities in an appropriate way and for which we can easily describe their price processes for i=0,…,q (of course, these price processes should be consistent w.r.t. φ). That is, we choose q+1 basis financial instruments for which we have a good understanding and which are traded in deep, liquid and transparent financial markets, see also remark in Sect. 6.1.
Our goal is to approximate the true (unattainable) VaPo of the outstanding liabilities X (t+1) at a fixed point in time denoted by
where we have used the linearity of the VaPo construction, see Proposition 7.4. For all single cash flows X k , k=t+1,…,n, we choose an asset allocation \(\mathbf{y}_{k}=(y_{0,k},\ldots, y_{q,k})' \in\mathbb{R}^{q+1}\) such that
approximates VaPo t (X k ). Note that the choice \(\mathbf{y}_{k}=\mathbf{y}_{k}^{(t)}\) also depends on t but for the sake of brevity we skip the superscript t. Definition (7.36) of the VaPo means that these financial instruments are all sold at time k in order to approximate the cash flow X k , that is, their sale generates a cash flow at time k which is given by
Hence, the VaPo for the outstanding liabilities X (t+1) is approximated by
The second last expression in (7.37) is again the cash flow representation and the last expression the instrument representation.
Our aim now is to choose \(\mathbf{y}\in\mathbb{R}^{(q+1)\times(n-t)}\) such that X (t+1) and Y (t+1) are “close together”. Of course, “close together” will depend on a distance function (or risk measure).
If there is no insurance technical risk involved and if \(\{\mathfrak{A}^{(0)},\ldots, \mathfrak{A}^{(q)}\}\) is a complete financial basis for the outstanding liabilities we can achieve
In general, we are not able to achieve (7.38) nor is it possible to evaluate the random vectors X (t+1) and Y (t+1) for all sample points ω∈Ω (often in life insurance it may take hours to evaluate X (t+1) for one single sample point ω because cash flows may be path-dependent including embedded options and guarantees as well as management decision). Therefore, one then chooses a finite set of so-called scenarios Ω K ={ω 1,…,ω K }⊂Ω and one evaluates the random vectors X (t+1) and Y (t+1) on these scenarios. This means we introduce a distance function on Ω K
and then the approximate VaPo at time t is determined by
For all k=t+1,…,n we define the approximate VaPo at time t by
This provides approximate VaPo for the outstanding liabilities at time t
Remark
It is important to realize that the \(\text{VaPo}^{\mathrm{approx}}_{t}(\mathbf{X}_{(t+1)})\) depends on
-
(a)
the choice of the financial instruments \(\mathfrak{A}^{(0)}, \ldots, \mathfrak{A}^{(q)}\),
-
(b)
the choice of the scenarios Ω K and
-
(c)
the choice of the distance function dist( ).
Based on the purpose of the approximate VaPo (e.g. profit testing, solvency, stress testing) these choices will vary and there is no obvious best choice.
Example 7.18
(Individual cash flow matching)
Assume that we want to match the entire outstanding cash flow X (t+1) as good as possible under an L 2-distance measure. We assume that there are positive -measurable weight functions \(\chi_{k} : \varOmega_{K} \to\mathbb{R}_{+}\) given for k=t+1,…,n and the distance function is defined by
For the weight functions χ k (⋅) we can make different choices. Often one wants to account for time values, therefore we choose the financial deflator φ A (under the basic actuarial model Assumption 6.3) and a normalized positive -measurable weight function \(p_{t} : \varOmega_{K} \to\mathbb{R}_{+}\) with \(\sum_{l=1}^{K} p_{t}(\omega_{l})=1\) and we define for k=t+1,…,n
The distance function is under these assumptions rewritten as
where denotes the expected value under the discrete probability measure which assigns probability weights p t (ω l ) to the scenarios ω l in Ω K .
The distance function tries to match the time values of each individual cash flow X k and Y k , k=t+1,…,n, as good as possible in the L 2-norm under the probability measure . Other approaches often work under other specific probability weights (such as risk neutral measures or forward measures) so that the discount factors become measurable at the beginning of the corresponding periods.
The optimal solution \(\mathbf{y}^{\ast}\in\mathbb{R}^{(q+1)\times(n-t)}\) is then found by minimizing the quadratic function
Note that in this individual cash flow matching example we can treat each accounting year k=t+1,…,n separately. We define the deflated random variables
The optimal solution \(\mathbf{y}^{\ast}\in\mathbb{R}^{(q+1)\times(n-t)}\) is then obtained by solving the following equations for each accounting year k=t+1,…,n
for all j=0,…,q. Because is a Hilbert space (\(\mathfrak{P}\) denotes the discrete σ-field), this last requirement means that
Note that we have chosen q+1 assets \(\mathfrak{A}^{(i)}\). The reason for this is that \(\mathfrak{A}^{(0)}\) should play a special role, namely let \(\mathfrak{A}^{(0)}\) be the cash value at time t, that is, we assume that \(A_{t}^{(0)} \equiv1\). Therefore we choose \(\mathfrak{A}^{(0)}\) such that it has price process
which explains that \(\mathfrak{A}^{(0)}\) is a ZCB \(\mathfrak{Z}^{(t)}\) with maturity t. If we use this selected asset for i=0, we obtain for each accounting year k=t+1,…,n the optimization requirements for \(\mathbf{y}_{k} \in\mathbb{R}^{q+1}\) given by
These are exactly the orthogonality conditions for credibility estimators, see Theorem 3.15 in Bühlmann–Gisler [32]. The first requirement guarantees that for all accounting years we do an unbiased portfolio choice \(\mathbf{y}_{k}\in\mathbb{R}^{q+1}\), relative to the probability measure , i.e. the optimal portfolio \(\mathbf{y}^{\ast}_{k}\in\mathbb{R}^{q+1}\) satisfies
The second requirement for the optimal portfolio choice \(\mathbf{y}^{\ast}_{k}\in\mathbb{R}^{q+1}\) becomes for j=1,…,q
Equations (7.43)–(7.44) are often called normal equations, see Corollary 3.17 in Bühlmann–Gisler [32]. The optimal portfolio \(\mathbf{y}^{\ast}_{k}\in\mathbb{R}^{q+1}\) is then obtained from these normal equations. With \(\mathbf{y}^{\ast}_{k}=(y^{\ast}_{0,k},\widetilde{\mathbf{y}}^{\ast}_{k})\) we have for all k=t+1,…,n
where \(\varSigma_{k} \in\mathbb{R}^{q\times q}\) is the covariance matrix (which should be non-singular) obtained from the vector \((\widetilde{A}_{k}^{(i)})_{i=1,\ldots, q}\) and \(\mathbf{c}_{k} \in\mathbb{R}^{q}\) is the covariance vector between \(\widetilde{X}_{k}\) and \((\widetilde{A}_{k}^{(i)})_{i=1,\ldots, q}\). Moreover, we obtain from (7.43)
where \(\mathbf{a}_{k}\in\mathbb{R}^{q}\) is the vector of expected values .
Remarks
-
We match each cash flow X k , k=t+1,…,n, individually by an optimal portfolio choice \(\mathbf{y}^{\ast}_{k}\in\mathbb{R}^{q+1}\).
-
The first normal equation (7.43) states the unbiasedness of the optimal portfolio choice \(\mathbf{y}^{\ast}_{k}\in\mathbb{R}^{q+1}\) w.r.t. the probability measure . One may additionally require for the probability measure
(7.45)(7.46)for all k=t+1,…,n and i=0,…,q. This then implies that the approximate VaPo given by \(\text{VaPo}_{t}^{\mathrm{approx}} (\mathbf{X}_{(t+1)} )\) satisfies the accounting condition for the outstanding liabilities X (t+1) at time t, i.e. we have, see (7.43),
-
The additional requirements (7.45)–(7.46) can be achieved by an appropriate choice of the probability weights p t (ω l ). Often this then provides good fits around best-estimate values but not necessarily in the tails of the distributions. The most common choice in practice is p t (ω l )=1/K, for all l=1,…,K, and ω l are chosen by a scenario generator under the real world probability measure \(\mathbb{P}\). If we are only interested in tail events, importance sampling may be more efficient for appropriate scenario generation.
However, we would like to emphasize that the scenarios ω l as such do not have a probability weight (as often mistaken in practice), only the attachment of the weights p t (ω l ) gives them a meaning in a probabilistic context.
-
Note that the probability measure (or the financial assets) should be chosen such that the covariance matrices Σ k are non-singular. Of course, this also requires some assumptions on the number K of scenarios ω l and the number q+1 of financial instruments (on finite sets Ω K ).
-
If we want to impose side constraints, for example no borrowing of cash value \(\mathfrak{A}^{(0)}\) for cash flow X k , i.e. y 0,k ≥0, then we consider the corresponding Lagrange or Kuhn–Tucker problem, see Ingersoll [87].
Example 7.19
(Time value matching)
This time we want to match the time value of X (t+1) as good as possible relative to an L 2-distance measure. For positive -measurable weight functions χ k (similar to (7.40)) we define the distance function
where denotes the expected value under the discrete probability measure , which assigns probability weights p t (ω l ) to the scenarios ω l in Ω K .
This distance function tries to match time value of entire cash flows X (t+1) and Y (t+1) as good as possible. Note that the difference is that we match the entire time value of X (t+1) in (7.47) whereas in (7.41) we match cash flows X k individually for k=t+1,…,n. This can also be seen when we formulate the corresponding quadratic minimization problems. The optimal solution \(\mathbf{y}^{\ast}\in\mathbb{R}^{(q+1)\times(n-t)}\) is now given by minimizing the quadratic function (recall definition (7.42))
This optimization problem is solved similarly to the one in Example 7.18.
As explained by Romanko et al. [135] one may introduce the notion of the sparse approximate VaPo: often the dimensionality and complexity of \(\text{VaPo}^{\mathrm{approx}}_{t}(\mathbf{X}_{(t+1)})\) is still too large in applications, especially if the total number q+1 of basis financial instruments \(\mathfrak{A}^{(i)}\) is large. Therefore, one chooses additional side constraints. For \(\mathbf{y} \in\mathbb{R}^{m}\) (with m=(q+1)(n−t)) we define the cardinality by
We can define the restricted portfolio set for some d∈{n−t,…,m} by
That is, we invest in at most d different basis financial instruments \(\mathfrak{A}^{(i)}\), and then the optimization problem reads as
Because the cardinality constraint often leads to involved optimization problems it is replaced by other conditions, for example an L 1-condition: for an appropriate η>0
and then one solves
Or one even solves the following optimization problem, which is similar to the previous one, choose ν>0 and
that is, we punish large values ∥y∥1.
We see that we have various different possibilities to choose an approximate VaPo and there is no comprehensive theory for good approximate VaPo choices. Absolutely crucial for good approximate VaPo choices are (a) the selection of the financial instruments and (b) the choice of the scenarios. For the selection of the financial instruments it may, for instance for European put options, even be important to not only choose the right maturities but also appropriate strike prices to get good fits. The choice of the scenarios should be such that they really reflect the part of the probability space that is of interest for the particular problem. Often a given sample is divided into two sub-samples. On the first sub-sample one solves the optimization problem and with the second sub-sample one performs an out-of-sample back test.
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Wüthrich, M.V., Merz, M. (2013). Valuation Portfolio. In: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31392-9_7
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DOI: https://doi.org/10.1007/978-3-642-31392-9_7
Publisher Name: Springer, Berlin, Heidelberg
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