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The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note, we elaborate on one specific formulation of a replicating portfolio problem. In contrast to the two most popular replication approaches, it does not yield an analytic solution (if, at all, a solution exists and is unique). Further, although convex, the objective function seems to be non-smooth, and hence a numerical solution might thus be much more demanding than for the two most popular formulations. Especially for the second reason, this formulation did not (yet) receive much attention in practical applications, in contrast to the other two formulations. In the following, we will demonstrate that the (potential) non-smoothness can be avoided due to an equivalent reformulation as a linear

Market-consistent valuation has gained increasing importance in the risk management of life insurance policies (e.g., Bauer et al. [

In hindsight, Pelsser [

As static replication is much more simplistic than dynamic replication, it has been an open question until quite recently, as to whether the approach of replicating portfolio can actually work. Although replication seems to resemble traditional immunization approaches, one major difference stands out: on the one hand, immunization works well in one period (as sensitivities to risk factors have been immunized), but leaves the portfolio un-immunized in the next period, which requires re-immunization (this is the main idea of dynamic delta hedging). On the other hand, replication matches cash flows, so there is hope that although one is not fully immunized in the first period, the immunization is still reasonable in the next period, as future cash flows are still replicated well. Very recent results have shown that replication indeed works under quite general setups, with efficiency comparable to the least squares Monte Carlo approach. The first theoretical foundations for replicating portfolios have been given in a series of papers by Beutner et al. [

The replicating portfolio approach represents a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of market risk capital. It is very specific to life insurance companies, and to our best knowledge so far, not used outside life insurance. We note that it could in theory also be used in banks for improving risk figure computation for hard-to-price portfolios

Of course, only considering financial instruments might fall short of matching actuarial risks like mortality or lapse risk. This is one of the well-known weaknesses of replicating portfolios, and usually leads to a remaining mismatch. However, as replicating portfolios are usually only applied within the market risk computation in Solvency II, this only represents a minor issue of replicating portfolios in practice.

As already briefly indicated above, there exist different choices for the specific matching problem. In Natolski and Werner [

_{SCF}

These two (and other) choices might differ in important properties:

Depending on the choice of the fitting criterion, similarly good fits of the fitting criterion lead to differing good or even only reasonably good approximations of the risk capital figure. In summary, terminal value matching provides better bounds than cash flow matching. Further, squared cash flow matching provides worse bounds than cash flow matching (to be defined below). Details on the exact relationships are provided in Natolski and Werner [

Depending on the choice of the fitting criterion, the solution of the corresponding optimization problems might pose difficulties. Although all formulations usually lead to convex problems, not all problems might be strictly convex, and thus may have non-unique optimal solutions. Further, non-smooth problems might be much harder to solve than smooth problems. Finally, some fitting criteria (for instance, those based on

It is expected by practitioners that the fair value of the replicating portfolio also matches the fair value of the liabilities. This has already been proven for the two most practically relevant criteria, but this question remains open for other criteria.

The analysis of the replication problems in Natolski and Werner [

For the above reasons, it might make sense to look for alternative criteria which have the potential to successfully address these issues. For this purpose, we consider a modified cash flow matching problem—called (_{CF}

Compared to the other two problems (_{SCF}_{CF}_{CF}

In the following, we will argue that on the contrary, there are good reasons to believe that (_{CF}_{SCF}_{CF}_{CF}_{SCF}

The rest of the paper is organized as follows. In

As mentioned in the introduction, this setup is taken from Natolski and Werner [

The final cash flow

The following three replication problems were considered in Natolski and Werner [_{CF}

The first formulation penalizes the deviation of cash flows by the

_{CF}) depends on the specific choice of the risk neutral measure. Thus, the optimal solution (i.e., the replicating portfolio) depends on this risk neutral measure. Similarly, the fair value of both the liability and the replicating portfolio also depend on the choice of the risk neutral measure. Only in the special case of perfect replication (i.e., optimal objective value of 0) is there no dependence on the choice of the measure. As all risk neutral measures for some numéraire are equivalent, the same (unique) replicating portfolio is optimal for all such measures.

For further analysis, let us introduce the symmetric matrices

These matrices represent the covariance matrices of the terminal values and the cash flows, respectively. They will play an important role in formulations of (weak) assumptions on the financial market. To obtain existence and uniqueness results (and more), the following few rather weak assumptions will have to be made. These are usually satisfied if financial assets for replication are chosen accordingly.

Assumption 3 can be easily

For problems (_{SCF}

The solutions to (_{SCF}

Under Assumption 3, the fair values of both optimal replicating portfolios equal the fair value of liability cash flows; that is:

For existence and uniqueness, the proofs exploit the convex quadratic structure of the two problems. Both replicating portfolios can be written as solutions to quadratic problems, and under the assumption of no redundant assets, the corresponding quadratic matrices are positive definite. Equality of fair values can then be deduced directly from the corresponding optimality conditions.

So far, problem (_{CF}

In what follows, we give a detailed proof of the existence and uniqueness of a solution based on the above assumptions. We also show that the optimal replicating portfolio indeed has the same fair value as the liability cash flows under surprisingly weak additional conditions. Further, we prove that for realistic problem instances (i.e., under certain additional assumptions), the numerical solution of (_{CF}_{CF}

Before we start to provide the results, we need a few technical preparations. Let us define:
_{CF}

It is easily seen that each

If

Obviously, if

As

We begin by showing that a solution to (_{CF}

_{CF}) possesses at least one optimal solution

As already observed,

By Minkowski’s inequality, it follows

Finally, it holds

Therefore, in total we have

As a continuous convex and coercive function,

Now, let us consider under which conditions we get uniqueness of the optimal replicating portfolio. Let us first consider the rather easy but practically most likely case that no liability cash flow at any time can be replicated:

Under this assumption,

Let

As no

By definition of

However, this means that in

Although for practical purposes the following result might not be as important as Theorem 2 (obtained under the stronger Assumption 4), there still remain some cases when Assumption 4 is violated. In particular, this includes all models where liability cash flows are predictable, for example, as in Grosen and Jørgensen [

Let us assume that

As

In the general case without Assumption 4, no first- or even second-order method can be used to directly solve the cash flow replication problem (_{CF}

Let us start with the unconstrained formulation of the cash flow replication problem

Now, we have

Finally, let us consider the equality of fair values. The standard Fermat–Torricelli problem as in Nam [

However, although the structure of (_{CF}_{CF}_{CF}_{CF}

_{CF}) satisfies

Condition (3) implies that for any

By examining the directional derivatives of the objective function _{CF}

As

As

_{CF}) and (RP_{SCF}) are equivalent. In particular, fair values of the replicating portfolios are the same and equal to the fair value of liability cash flows. Going one step further, Theorem 4 now illustrates that for matching fair values it is already sufficient if cash payments equal to the numéraire can be generated at each time separately. This can be considered a rather weak condition, which is usually fulfilled in practical settings if the replication instruments are chosen accordingly.

So far, problem (_{CF}

Problems (_{SCF}

Problems (_{SCF}

Problem (_{CF}

In the above analysis, we have shown that all other reasons besides the first one go up in smoke after a detailed investigation of (_{CF}_{SCF}

However, due to the long maturity of life insurance contracts (usual models range from 40 to 60 years final maturity), numerical issues cannot be avoided. Usually, the matching error of certain maturities significantly dominates all other maturities. In this case, only the dominating maturities are properly matched. From a statistical point of view (robust statistics, robust least squares), it is well known that this can be partially mitigated if one moves from the square of the _{CF}

We picked up on the replication problems presented in Natolski and Werner [_{SCF}_{CF}_{CF}_{CF}_{CF}_{SCF}

The authors, Jan Natolski and RalfWerner, contributed equally to this research paper.

The authors declare no conflict of interest.

In Natolski and Werner [

A recent paper by Broadie et al. [

All results of this exposition remain true with one obvious exception: if instead of the risk neutral measure the real world measure is chosen. Naturally, the result on the fair value in

We use tilded variables to express the fact that the variable is discounted.

Let us note that in Theorem 4 we will need a somewhat stronger condition; however, this condition can be satisfied exactly along the same lines as Assumption 3, and can thus also be characterized as rather weak condition.

A function

Replicating portfolio approach: Each node at