Revisiting Optimal Investment Strategies of Value-Maximizing Insurance Firms

We study capital management and investment decisions of a value-maximizing<br>insurance firm with a broad ownership base in a discrete-time setting. We<br>highlight that the valuation measure used to determine the value of the cash flows to<br>shareholders should reflect two economically sound requirements: market-consistency<br>and indifference to idiosyncratic risk. We provide a rigorous construction of this economic<br>valuation measure and use it to derive the optimal capital-management and<br>investment strategies that realize the economic value of the firm. Our objective is to<br>shed light on the controversial question of whether insurers should invest in liquidly-traded<br>risky assets. Decomposing firm value into net tangible value, default option<br>value, and franchise value, we find that whether or not taking investment risk is<br>optimal essentially depends on how the tradeoff between the impact of investment<br>risk on the owner’s option to default and on the firm’s franchise value resolves. A<br>variety of numerical examples illustrates how changes in the regulatory and financial<br>environment can result in materially different optimal investment strategies.


Introduction
This paper revisits questions related to the capital and investment strategies of publicly-traded, limited-liability insurance firms with a diffuse shareholder base. By capital strategy we mean how much dividends the firm should pay and how much capital it should raise and by investment strategy how much investment risk it should take. 1 Of these, the most controversial, both amongst academics and practitioners, is the question of whether insurance firms should take investment risk at all. In reality, most insurance firms do engage, to varying degrees, in risk taking on the asset side. Indeed, some practitioners view insurance firms as little more than investment funds that are leveraged through the sale of insurance strategies. In his letters to shareholders in the annual reports of Berkshire Hathaway (see, e.g. Berkshire Hathaway Inc. (2016)), Warren Buffet explains the attractiveness of the insurance business as follows: "... insurers receive premiums upfront and pay claims later. In extreme cases, such as claims arising from exposure to asbestos, payments can stretch over many decades. This collect-now, pay-later model leaves P/C companies holding large sums -money we call "float" -that will eventually go to others. Meanwhile, insurers get to invest this float for their own benefit".
Regardless of what insurers actually do, academic research has tried to provide a normative answer to what insurers should do. In this respect, Froot (2007) states that "financial intermediaries should shed all liquid risks in which they have no ability to outperform and devote their entire risk budgets toward an optimally diversified portfolio in exposures where they have an edge." On the other extreme, some authors, e.g. Højgaard and Taksar (2004), conclude that full investment in the risky asset is optimal. In this paper, we present a more nuanced picture, so it is interesting to understand what is driving these absolute and antipodal answers in Froot (2007) and Højgaard and Taksar (2004). Both of these papers seek, as we do, to identify the optimal investment strategy of a value-maximizing insurance firm, where firm value corresponds to the present value of cash flows to shareholders. Thus, the specification of cash flows to shareholders and of the present value rule will ultimately determine which investment strategies are optimal. As we will argue below, the model in Froot (2007) suffers from a misspecification of cash flows to shareholders because it ignores the option to default, 2 and the model in Højgaard and Taksar (2004) from a misspecification of the present value rule because it values cash flows using expectations with respect to the "physical" probability measure.

The valuation measure
The present-value rule advocated in this paper is in line with a general shift in perspective witnessed over the past decades which favours economic over traditional accounting approaches. 3 The rationale for our advocacy relies on the following argument. In the absence of financial frictions, a diffuse shareholder base is typically associated with shareholders being indifferent to idiosyncratic risk because it can be diversified; see e.g. Mayers and Smith (1982). For a company that is not exposed to financial-market risk, this results in a present-value rule in which the physical probability is the valuation measure. However, for financial firms, or more generally, for firms whose cash flows to shareholders depend on financial-market prices, care needs to be taken when selecting the appropriate valuation measure. Indeed, the valuation measure must be simultaneously consistent with both financial-market prices and the firm's indifference towards idiosyncratic risk: it needs to reproduce market prices when applied to the payoffs of traded assets and coincide with expected cash flows discounted at the risk-free rate when applied to cash flows that contain only diversifiable risk. Since our framework is intrinsically one of incomplete markets, there are infinitely many valuation measures that are consistent with market prices. One of our contributions is to show that only one of them is consistent with indifference to idiosyncratic risk. We call it the economic valuation measure. 4 The valuation of cash flows to shareholders then entails taking expected values with respect to this special measure and discounting them at the risk-free rate. Indeed, the economic valuation measure is the only valuation measure that should be used when determining the present value of cash flows to shareholders of any publicly-traded firm with a diffuse shareholder base and, as a special case, of cash flows of an investment fund investing in liquidly-traded assets that has a diffuse ownership base. 5 6 The insistence on market consistency is not without justification. The choice of the wrong valuation measure does not only produce wrong firm values but results in wrong decision making. For instance, using the physical probability as the valuation measure makes the market seem to undervalue the risky asset due to its higher expected return. This creates a clear bias towards a risky investment strategy which is easily seen to be problematic. This is most effectively illustrated taking the example of an investment fund investing in liquidly-traded assets. Clearly, the value of such a fund should always be exactly equal to the market price of the assets it holds. However, using a present value rule based on the physical probability, this would be true only when the fund's assets are risk free. Were the fund to be invested in the risky asset, the use of such a present value rule would yield a value that is higher than the market value of the fund's assets.

Capital and investment strategies
We use the economic valuation framework described above to provide insight into the capital and investment management decisions of value-maximizing insurance firms in a discrete-time, dynamic framework. The controls at the manager's disposal are the decision to either liquidate the firm or continue operations, the amount of dividends to pay, the amount of capital to raise, and the amount of investment risk to take. In particular, premia are not a control variable. In fact, in order to focus on the capital and investment decisions of the firm, we assume that, in each period, it sells the same portfolio of insurance strategies. Our model incorporates several well-documented frictions that are typical for the environment in which insurance companies operate: i) carry costs of capital, i.e. deadweight costs associated with holding capital within the firm, which may include double-taxation, as well as agency and financial-distress costs; 7 ii) cost of raising new capital, which we assume to be fixed, and iii) minimum regulatory capital requirements that the firm must satisfy in order to operate. It is well known 8 that these financial frictions generate costs at the corporate level which ultimately affect the cost of taking risk and thus the choice of optimal strategies. 9 Each choice of capital and investment strategies generates a stream of cash flows to shareholders, obtained by netting dividend payments and capital injections at each date. The manager's task is to select an optimal strategy, i.e. a strategy that maximizes added value. 10 We establish the existence of deterministic, stationary optimal capital and investment strategies by applying the dynamic-programming principle. As a result, firm value is described by a value function, depending exclusively on the current level of equity capital. The focus on added value implies that raising capital is meaningful only if doing so adds value, i.e. only if the value of the firm increases by more than the amount raised, and that paying out dividends makes sense only if keeping these funds within the firm does not add value. These simple observations allow us to provide a comprehensive description of the capital strategies of the firm. First, since holding capital is costly, the firm never recapitalizes unless capital falls below the regulatory minimum. 5 We stress here that the above argument does not apply to, say, a sole ownership. Indeed, in that case, the preferences of the sole owner will drive decision making. 6 A recurrent assumption in the finance literature is the risk neutrality of all agents (e.g. Décamps et al. (2011), Gryglewicz (2011) and Hugonnier and Morellec (2017)). Under this assumption, the use of the physical probability measure to value financial firms is justified. However, in an economy in which agents are risk neutrals the risk premium for risky assets should logically be equal to zero for consistency reasons. 7 See e.g. Froot and Stein Froot and Stein (1998). 8 See Mayers and Smith (1982), Froot andStein (1998) or Froot (2007). 9 For the effects of financial frictions on pricing see Koijen and Yogo (2015) and Koijen and Yogo (2018). 10 Added value is the value of the firm over and above the value of invested capital.
Second, there exists a liquidation barrier, such that the firm is liquidated whenever capital falls below it and recapitalized whenever capital lies between it and the regulatory minimum. Third, there exists an upper dividend barrier such that any capital in excess of it is paid out as dividends. The upper dividend barrier is also the capital level at which added value for the firm is maximal. As a consequence, whenever the firm is recapitalized, it is recapitalized to this level. Fourth, if raising capital is costless, the upper dividend barrier coincides with the regulatory minimum. Finally, at capital levels between the regulatory minimum and the upper dividend barrier we may encounter either intermediate dividend payments, so-called "dividend bands", or no dividend payments at all. We provide examples exhibiting each of these behaviors in Section 5.
What about the investment strategy? If the value function were concave, the firm would never take investment risk and if it were convex, the firm would seek maximum exposure to investment risk. 11 However, as we show in Section 4, the value function is typically neither concave nor convex. As a consequence, we cannot hope to have a universal answer as to the optimal amount of investment risk to take (see Section 5). To illustrate the critical mechanism driving the optimal investment strategy, we identify three different components of firm value: net tangible capital, 12 the value of the default option, and franchise value. 13 We show that risky investments have no impact on net tangible value, and affect default option value and franchise value in opposite directions. Indeed, by increasing the likelihood and size of a potential default, risky investments increase the default option value. On the other hand, risky investments typically increase the likelihood of liquidation, which leads to a loss of franchise value, or the likelihood of recapitalization, which is costly. Both of these effects have a negative impact on firm value. Interestingly, it is also possible that, at low capital levels, risky investments increase the likelihood of the firm reaching capital levels at which franchise value is significantly higher. In this case, given that no recapitalization costs are incurred to reach more value-adding capital levels, risky investments may be seen as a substitute for costly capital raising. Our analysis shows that the optimal investment strategy will depend on how all these tradeoffs resolve.
Our work shows that if we account for the limited liability when specifying cash flows to shareholders and use the right valuation measure, value-maximizing insurance firms will exhibit widely different optimal behaviours depending on the environment in which they operate. The range of optimal behaviours is illustrated by the examples in Section 5, which show how investment risk impacts the two key value components: default option value and franchise value.

Embedding in the literature
To date, most of the literature on optimal investment strategies for insurers does not consider a value-maximizing insurer but mainly focuses on an insurance firm that either maximizes the utility of surplus, 14 minimizes the probability of ruin, 15 or takes decisions within a mean-variance framework. 16 By contrast, our approach is based on an insurer that maximizes value, defined as the market-consistent value of the stream of dividends net of capital injections. It was de Finetti (1957) who first suggested that an insurer should aim to maximize the present value of dividend streams rather than minimize the probability of ruin, which had thus far been the approach in the actuarial literature. De Finetti's work spawned a vast body of literature on dividend-distribution problems. 17 Since the bulk of this literature did not consider investments in a risky asset, the market consistency of the valuation rule was never an issue. More problematic is that the only two papers dealing with investments in a risky asset we know of, Højgaard and Taksar (2004) and Azcue and Muler (2010), also failed to adopt a market-consistent approach to valuation. Both these papers work in a continuous-time setting, allow investments in financial markets of Black-Scholes type, and rely on the dynamic-programming principle. Under the realistic assumption that the risk-free rate of return is strictly smaller than the expected rate of return on the risky asset under the physical probability measure, the results in Højgaard and Taksar (2004) and Azcue and Muler (2010) imply that it is always optimal to invest part of the available funds in the risky asset. In particular, in Højgaard and Taksar (2004) full investment in the risky asset is optimal even though the value function is concave. 18 A crucial factor contributing to these results is the fact that their cash-flow valuation is not market consistent. Indeed, their valuation rule consists in taking the expected cash flows under the physical probability and discounting them at a constant dividend discount rate that exceeds the risk-free rate. Since for this valuation procedure to be market-consistent, the discount rate would need to depend on the particular investment strategy, this creates an evident bias towards investing in the risky asset.
The series of papers Froot et al. (1993), Froot and Stein (1998), and Froot (2007 provides another relevant point of comparison in the existing literature. In particular, Froot (2007) derives the optimal financial strategies of an insurer under a market-consistent valuation. While their model accounts for deadweight cost of capital, it does not contemplate the possibility of firm default as we do. In this setting, Froot (2007) concludes that insurers should not take liquid investment risk. The gist of the argument is that taking liquid investment risk does not create value per se and that, by taking investment risk, insurers may end up having less capital with which to exploit value-creating opportunities in the future. The result seems plausible as long as the limited-liability option of the firm has no value. In the more realistic case where there is a possibility of default, this logic is less compelling because risky investments can increase the value of the default option. 19

The Model
We work in a discrete-time, infinite-horizon setting with dates indexed by n ∈ N. 20 For every n ∈ N \ {0}, the period starting at date n − 1 and ending at date n is denoted by [n − 1, n]. We consider a limited-liability insurance firm with a diffuse shareholder base contributing the firm's equity capital. To be allowed to operate, the regulator requires the firm requires to have a minimum amount of equity capital. At the beginning of each period, the firm decides whether to stay in business or to liquidate. If it decides to liquidate, any remaining equity capital is returned to shareholders as a dividend and the firm ceases to exist. If it decides to continue operating, the firm may pay dividends but needs to make sure that minimum capital requirements are met. This may possibly require recapitalization. The firm then sells a standardized insurance portfolio of one-period strategies in exchange for a premium and decides how to invest collected premiums and capital in assets that are traded in a frictionless and arbitrage-free market. At the end of the period, investment returns, insurance losses are realized and insurance claims are settled. Thus, to steer the firm, the manager has four controls at his disposal: i) the decision 17 See e.g. Schmidli (2008), Albrecher andThonhauser (2009), andAvanzi (2009). 18 See Footnote 11. 19 Note that Froot et al. (1993), Froot and Stein (1998), and Froot (2007 explicitly mention that they consider a default-free firm. However, in their analytical model the possibility of default cannot be ruled out because it assumes normal returns. 20 We use the standard notation N = {0, 1, 2, . . . }. to continue operations or to liquidate the firm, possibly defaulting if assets do not suffice to cover claims; ii) the amount of dividends to pay (if any) to shareholders; iii) the amount of new capital (if any) to raise from existing owners; 21 and iv) the amount of investment risk (if any) to take. As a result, the insurer's balance sheet consists of insurance obligations and equity on the liability side (in particular, there is no debt) and investments in liquidly traded securities on the asset side. 22 Finally, we allow for carry costs for capital held within the firm and for costs when raising new capital. These two financial frictions are described further below in this section.

The financial market, insurance losses and the valuation measure
We proceed to describe the underlying probabilistic model in broad terms. A more detailed and technically rigorous description can be found in the Online Appendix. The probability space Ω, F, P described below represents uncertainty in the economy with P being the 'physical' probability measure. If G is a sub-σ-algebra of F and X : Ω → R is a random variable, E P [X] denotes the expectation of X with respect to P and E P [X| G] the conditional expectation of X with respect to P and G. Finally, for any p ≥ 0, L p (Ω, F, P) denotes the space of p-integrable random variables and · p,P the corresponding norm. Stochastic processes are denoted by X = X t , t ≥ 0 . Inequalities and equalities between random variables are always in the "Palmost sure" sense.

The financial market
Even though the insurer takes decisions at discrete dates, we assume that financial markets trade continuously. From a modeling perspective, this is consistent with the fact that insurance firms take decisions, such as rebalancing their investment portfolio, at a significantly lower frequency than trading takes place in financial markets. We assume a Black-Scholes market in which a risk-free money market account and a risky security, e.g. an index, are traded. The moneymarket account pays a deterministic, instantaneous interest rater > 0, i.e. one unit of currency follows the process B = B t , t ≥ 0 , where B t = e tr , t ≥ 0.
The risky security has initial price s 0 > 0 and its price process S = S t , t ≥ 0 follows a geometric Brownian motion with drift µ >r and volatility σ > 0, i.e.
where W = W t , t ≥ 0 is a standard P-Brownian motion. The flow of information in the financial market is described by the market filtration F W = F W t , t ≥ 0 . 23 We denote by F W the smallest σ-algebra containing all the σ-algebras in F W , i.e. F W = σ( t≥0 F W t ). For every finite maturity T > 0, the Black-Scholes model is arbitrage-free and complete. 24 As a result, every cash flow X ∈ L 2 (Ω, F W T , P) maturing at date T has a unique, well-defined 21 To avoid having to deal with dilution effects, we only allow existing shareholders to inject capital. This is akin to a rights issue where incumbent shareholders purchase all newly-issued equity pro rata to their existing share holdings. 22 We assume that both assets and liabilities are denominated in a single, fixed currency. All payoffs, prices and values are silently understood to be in this common currency. 23 The market filtration is slightly larger than the raw filtration generated by W . The raw filtration is the smallest filtration F W such that W is F W -adapted. See the Online Appendix for further details. 24 Completeness implies that every cash flow X ∈ L 2 (Ω, F W T , P) maturing at date T > 0 is F W -replicable, i.e. there exists a dynamic, (F W t , 0 ≤ t ≤ T )-predictable, self-financing trading strategy, whose value at time T coincides with X. The market price of X at date 0 ≤ t ≤ T corresponds to the value at date t of any replicating strategy. The fact that no arbitrage opportunities exist implies that, for a nonzero, nonnegative X, this price will be strictly positive. market price at every date t ≤ T . In the Online Appendix we show that there exists a unique probability measure P * defined on F W , 25 the pricing measure, such that the market price at date t ≤ T of any cash flow X ∈ L 2 (Ω, F W T , P) maturing at an arbitrary date T is given by For every T > 0, the probability measure P * is equivalent to P when restricted to F W T but not on F W . 26

Insurance losses and the insurer's filtration
The standardized portfolio of one-period strategies sold by the insurer is characterized by a fixed aggregate premium p and an i.i.d. sequence L = (L n , n ∈ N \ {0}) of nonzero, nonegative random variables, where L n ∈ L 2 (Ω, F, P) represents aggregate losses over the period [n − 1, n]. We assume that, for every n ∈ N \ {0}, L n is independent of the σ-algebra F W , i.e. independent of F W n . The random variables represent the insurance risk. R = (R n , n ∈ N \ {0}) is an i.i.d sequence of P-square-integrable, centered random variables that are independent of F W . The insurer has at his disposal the information contained in the market filtration but, as time elapses, he also learns about realized insurance losses. This means that he has access to the information contained in a larger filtration F = F t , t ≥ 0 , that we call the insurer's filtration. We assume that F is the smallest filtration such that F t contains F W t and information of the insurance losses σ(R n ; n ≤ t). We set F := σ( t≥0 F t ).

The extended financial market
The insurance firm has an exposure both to financial market risk and to insurance risk. Thus, cash flows to shareholders are, in general, adapted to F rather than F W . As a result, we need to extend the Black-Scholes economy described above keeping the same traded assets but replacing the flow of information F W by the insurer's filtration F. After passing from F W to F, the market remains arbitrage free for every finite maturity but is no longer complete. As a result, a replicable cash flow X ∈ L 2 (Ω, F T , P) maturing at date T > 0 has a well-defined market price π t,T (X) ∈ L 2 (Ω, F t , P) at every date t prior to maturity. 27 Incompleteness with respect to the enlarged filtration F leads to an infinite number of market-consistent valuation measures, i.e. of probability measures Q defined on F such that π t,T (X) = E Q e −(T −t)r X|F t for every replicable cash flow X maturing at date T > 0. To choose one of these measures in order to value cash flows to shareholders, i.e. the cash flow stream of dividends net of the cost of capital injections, we need to require more than just market consistency. 25 The standard theorems in mathematical finance only imply the existence of probability measures P * T defined on FT for each fixed maturity T > 0. Thus, we need to establish the existence of a single probability measure that works for all maturities. A rigorous construction of P * is provided in the Online Appendix. 26 The lack of equivalence of the market valuation measure and the physical probability on (Ω, F W ) results from the infinite time horizon, but is of no consequence in our setting because we only consider cash flows that have fixed, albeit arbitrarily long, maturities. Hence, we only need equivalence on (Ω, F W T ) for every T > 0. 27 Note that, in the new market, replicating strategies are allowed to be (Ft, t ∈ [0, T ])-predictable. For instance, if at date n the market goes up, one may choose a different strategy depending on whether the insurance has to settle a small or a large loss. This genuinely enlarges the set of self-financing trading strategies and results in a larger set of replicable cash flows. In particular, because F is an enlargement of F W , all cash flows in L 2 (Ω, F W T , P) remain replicable in the new economy and have the same price as in the original Black-Scholes economy. For this reason we continue to use the same symbol for the market price of a cash flow.

The economic valuation measure
With a diffuse shareholder base and in the absence of financial market risk, a firm manager is typically assumed to behave in a risk-neutral manner because shareholders are able to diversify all risk. 28 In this case, the present value of cash flows to shareholders should be determined by taking their expectation with respect to P and discounting these at the risk-free rate. This approach leads, however, to serious inconsistencies when cash flows to shareholders depend on financial market risk, as in our setting. The reason is that P is not a market-consistent valuation measure: under P the expected return of the risky asset is higher than the risk-free rate. Consequently, computing risk-free discounted expectations using P would make the risky asset appear underpriced. This is equivalent to the manager "second guessing" the market, which, given the manager's value mindset, would result in an exclusive demand for the risky asset. As mentioned in the introduction, the same issue also arises in frameworks where dividends are discounted at a "dividend discount rate" that is independent of the investment strategy of the insurer, as in Højgaard and Taksar (2004) or Azcue and Muler (2010). From the preceding discussion, it follows that the insurer's valuation measure must meet two key requirements. First, it must be market consistent, i.e. when applied to cash flows generated by market instruments it must reproduce their prices, and , second, when applied to cash flows that are independent of financial markets it must coincide with expected cash flows with respect to the physical probability discounted at the risk free rate. In the following theorem we establish the existence and uniqueness of such a valuation measure. For the proof see the Online Appendix. 29 Theorem 1.1. There exists a unique probability measure Q * defined on (Ω, F), the economic valuation measure, such that (i) Q * is equivalent to P on F T for every T > 0, but not on F.
(iii) Under Q * , the sequence L is i.i.d. with the same distribution as under P and is independent of F W .
(iv) For every 0 ≤ t < T and every replicable cash flow X ∈ L 2 (Ω, F T , P) maturing at date T , market consistency holds, i.e.
If X ∈ L 2 (Ω, F T , P) is a cash flow maturing at date T > 0 and 0 ≤ t < T , we define its economic value at date t as i.e. it extends the functional in (3) beyond just replicable cash flows. 30 The projection formula (2) implies that the economic value of a general cash flow X ∈ L 2 (Ω, F T , P) with maturity T can be computed as follows: 28 Indifference to diversifiable risk, does not imply indifference to possible side effects of taking diversifiable risk. Indeed, taking risk may generate costs due to financial frictions or benefits if part of the risk is externalized. Thus, even a risk neutral firm may be led to behave in a risk averse or a risk seeking way depending on whether costs outweigh benefits or the other way around. 29 To our knowledge, the explicit construction of the economic valuation measure is new. It can be shown that, in the finite-time horizon case, the probability measure Q * T coincides with the so-called variance optimal measure. See, e.g. Delbaen andSchachermayer (1996), Schweizer (1996), and the references therein. 30 We emphasize that the economic justification for using the economic valuation measure critically depends on the fact that the insurance firm has a diffuse shareholder base.
its replicable, or hedgeable, component and E X = X − E P [X|F W T ] is an error term that is uncorrelated with the financial market and, hence, has zero economic value; ii) compute the unique market price π 0,T (H X ) of the replicable component of X, which yields the economic value of X at date 0.
Economic value of discrete-time cash flow streams Cash flows to shareholders are F-adapted sequences X = X n , n ∈ N , where X n denotes the cash flow due at date n ∈ N. If X is bounded in L 1 (Ω, F, Q * ), 31 then its economic value at date 0 can be defined in a natural way as the sum of the economic values of the individual components where r := er − 1 denotes the one-period risk-free rate of return.

Admissible strategies and capital dynamics
The manager's objective of maximizing the economic value of the firm can be analyzed using dynamic-programming techniques. At any date n ≥ 0 the firm may be either in a state m ∈ R representing the firm's capital if the firm is still active or in the state * if the firm has been liquidated. Hence, the state space is X := R ∪ { * }. The "cemetery state" * is assumed to be an absorbing state, i.e. once the firm reaches that state it remains there forever. A (financial) strategy 32 is an F-adapted process S = z, κ, λ, δ , where, for every n ∈ N: i) δ n ∈ {0, 1} with 0 indicating liquidation (which sends the firm into state * ) and 1 continuation; ii) z n ≥ 0 is the amount of dividends to be paid; iii) κ n ≥ 0 is the amount of capital to be raised, and iv) λ n ∈ [0, 1] is the proportion of capital to be invested in the risky asset. The random vector δ n , z n , κ n corresponds to financing operations at date n and the random variable λ n to the investment strategy at date n.
The case in which the initial state of the firm is * is devoid of interest since the firm then remains inactive for ever. In this case, by convention, the only possible strategy S is defined by δ n = 0, z n = 0, κ n = 0 and λ n = 0 for every n ≥ 0. Assume now that the initial state of the firm is m ∈ R and that the manager has chosen strategy S. Set M m,S 0 = m and, for n ≥ 0, denote by M m,S n the state of the firm at the end of the period [n − 1, n] under S. The liquidation time is the stopping time τ m,S := inf n ≥ 0 | δ n = 0 representing the date at which the company is liquidated according to strategy S. If the firm is not liquidated, to be able to operate, the firm must satisfy minimum regulatory capital requirements given by M reg ≥ 0. 33 Hence, for every n ≥ 0, any strategy must satisfy 31 This means that the components of X belong to L 1 (Ω, F, Q * ) and there exists a constant B > 0 such that Xn 1,Q * ≤ B for every n ≥ 0. Note that by Theorem 1.1, for every n ∈ N, we have that Xn ∈ L 1 (Ω, F, Q * ) whenever Xn ∈ L 2 (Ω, Fn, P). 32 Our main reference for dynamic programming is Hernández- Lerma and Lasserre (1999) where what we call a strategy is called a deterministic policy. 33 We allow for the possibility that Mreg = 0 in which case the insurer can operate with zero capital, i.e. without putting own capital at risk. Moreover, we note that assuming that Mreg is fixed, is a justifiable approximation because the insurer can only underwrite a fixed standardized book of business.
When the firm is liquidated, the manager is only allowed to return any remaining capital to shareholders, i.e. we have δ n = 0, z n = max{M m,S n , 0}, κ n = 0 and λ n = 0, on {τ m,S = n} , where we have set λ n = 0 by convention. Moreover, since the cemetery state * is absorbing we must have δ n = 0, z n = 0, κ n = 0, and λ n = 0 on {τ m,S < n} , where we again have set λ n = 0 by convention.
For the remainder of the paper, when the initial state of the firm is m ∈ R, any strategy is always assumed to satisfy Conditions (5), (6), and (7).

End-of-period capital dynamics
Let m ∈ R be the initial state of the firm and assume that M m,S n = * . If δ n = 1, i.e. if the manager decides to continue operations during the period [n, n + 1], the amount z n ≥ 0 of dividends to pay and the amount κ n ≥ 0 of capital to raise must be chosen. 34 When the firm writes new business, it collects the premium p which we write as is the (actuarially) fair premium and q ≥ 0 is the margin (over the fair value). After selling insurance, the capital in the firm increases to M m,S n − z n + κ n + p. However, holding capital is costly due to the presence of carry costs, 35 which are accounted for by scaling down the amount of capital by a fixed factor γ ∈ (0, 1). Thus, capital available for investments amounts to γ (M n − z n + κ n + p). We refer to 1 − γ as the carry cost of capital. We assume that the manager always matches liabilities when investing the fair premium, i.e. l is invested in the money-market account, but has the choice of investing γ (M n − z n + κ n + q) in a mix of the risk-free and the risky assets, with λ n ∈ [0, 1] being the portion invested in the risky asset. 36 Noting that, since l(1 + r) = E Q * [L n+1 ], the total return resulting from the investment of l in the money-market account after netting claims L n+1 is precisely R n+1 , the firms capital at the end of the period [n, n + 1] is given by is the one-period excess return on the risky asset. Note that E Q * [ρ n+1 |F n ] = 0 for n ≥ 0. In summary we have that end-of-period capital at date n + 1 is given by  (2007), Goolsbee (1998) and Hann et al. (2013). Note that, in the literature, carry costs are also referred to as deadweight or frictional costs. 36 Since λn ∈ [0, 1] the insurer can only take long positions in the money-market account and the risky asset.
This amounts to disallowing leverage.

Cash flows to shareholders and admissible strategies
Raising capital is assumed to incur a fixed cost C ≥ 0, e.g. legal fees, that is independent of the amount κ > 0 being raised. Hence, for the firm to receive the amount κ ≥ 0 of capital, shareholders have to inject κ + C(κ) where C(κ) := C · 1 {κ>0} . Under strategy S, the cash flow to shareholders at date n ≥ 0 is The cash flow stream CF S := (CF S 0 , CF S 1 , . . . ) is referred to as cash flows to shareholders under the strategy S.
We now argue that the firm's manager can focus on strategies satisfying Properties (S1)-(S3) specified below. In the Online Appendix, we provide a formal proof that there is no loss in generality in doing so. 37 Here we are content to explain why restricting the attention to these strategies makes economic sense.
Assuming the firm's initial state is m ∈ R, the first defining property of an admissible strategy is that the firm should never simultaneously raise capital and pay dividends, i.e. we require that P(z n · κ n = 0) = 1 for every n ≥ 0. (S1) This property makes economic sense because, when raising capital is costless, there is no discernible benefit to raising capital and using it to pay dividends or paying dividends only to have raise capital to be able to operate. In contrast, when raising capital is costly, this procedure is clearly disadvantageous.
The second defining property is that the firm should not raise capital unless regulatory requirements are not met: The rationale for this property is that, unless it is absolutely necessary to be able to meet regulatory requirements, raising any amount K of capital can always be delayed to the next period and, by doing so, the firm can avoid incurring the carry costs (1 − γ)K and save the interest, rC on the cost of raising it.
The final defining property is that, at any date, the firm should avoid holding to much capital above the regulatory minimum. More precisely, This property is reasonable because dividends should be paid at the latest when the cost of carrying the excess above the regulatory minimum, (1 − γ)(M n − z n + κ n − M reg ) (incurred at the start of the period) exceeds the discounted cost of raising capital at the end of the period C/(1 + r).
Strategies satisfying Properties (S1)-(S3) are said to be admissible for the initial state m ∈ R. The set of admissible strategies for m is denoted by S(m). Since S = (0, 0, 0, 0) is the only possible strategy when m = * , it makes sense to call S = (0, 0, 0, 0) admissible for * and to set S( * ) = {(0, 0, 0, 0)}. From now on we restrict the manager's choices to admissible strategies. For convenience, we will write statements such as "for S ∈ S(m)" instead of the more cumbersome "for m ∈ R and S ∈ S(m)".
The following proposition establishes that admissible strategies are bounded in L 1 (Ω, F, Q * ) and, thus, admit an economic value. The proof can be found in the Online Appendix.
Proposition 2.1. There exists a constant B > 0 such that, for every S ∈ S(m) and n ∈ N \ {0} In particular, for every strategy S ∈ S(m), the cash flow stream CF S received by shareholders is bounded in L 1 (Ω, F, Q * ). 38

Decision functions and stationary strategies
Given the stationary nature of our model, i.e. that the parameters of our model are time independent, it is particularly interesting to search for strategies that result from applying at each date a decision rule prescribing a "one-step strategy" that depends only on the state of the firm at that date. For any m ∈ X , the set S 0 (m) of all admissible, one-step strategies at date 0 consists of all tuples s = (z, κ, λ, δ) such that (z 0 , κ 0 , λ 0 , δ 0 ) = s for some strategy S ∈ S(m). Given any one-step strategy s ∈ S 0 (m) we set Thus, the random variable M m,s 1 represents the capital of the firm at the end of period [0, 1] provided the manager implements s at date 0.
A decision function is a measurable mapping 39 such that D(m) ∈ S 0 (m) for every m ∈ X . Given a decision function D we can define the process (M m,D n ; n ≥ 0) by setting M m,D 0 = m and for n ≥ 1. The associated strategy S D is defined by setting for every n ≥ 0 It is clear that S D is admissible, i.e. S ∈ S(m). Strategies that can be generated in this way are said to be stationary.

Firm value, added value, and optimal strategies
Whenever the firm's manager adopts a specific strategy S ∈ S(m), the corresponding cash flows to shareholders admit an economic value by Proposition 2.1. The manager's task is to select a strategy with the highest possible economic value. This section is devoted to providing precise definitions of firm value and firm added value and establishing the existence of "optimal" strategies that are stationary. The proofs of the results in this section are found in Appendix A.
38 Since the constant B does not depend on m ∈ R, Inequality (10) cannot be true for n = 0. Indeed, if m > 0, we may choose a strategy satisfying z0 = m making the left-hand side arbitrarily large as m tends to infinity. 39 In Appendix A we show that X can be naturally viewed as a metric space. Measurability should be understood as being with respect to the corresponding Borel σ-algebra.

Firm value and added value
The value of a strategy S ∈ S(m) is denoted by V S (m) and corresponds to the economic value of cash flows to shareholders CF S , i.e.
To introduce the notion of "added value" it is useful to separate the impact of financing operations and the "productive" use of capital. To this effect, consider a strategy S ∈ S(m) for which δ 0 = 1. Note that the capital that is being put to productive use is not m but m − z 0 + κ 0 . Denote by S + the admissible strategy in S(m − z 0 + κ 0 ) defined by We clearly have value due to capital put to productive use within the firm . ( Note that z 0 − κ 0 − C(κ 0 ) is the cash flow to shareholders resulting from financing operations at date 0. It is natural to define the added value of S ∈ S(m) with δ 0 = 1 as To obtain the total cost of reaching the capital level m − z 0 + κ 0 (m), we need to take the cost of raising capital, C κ 0 , and add max{−m, 0}, which represents the amount of outstanding liabilities the firm needs to settle before being allowed to continue operations. Added value of S ∈ S(m) with δ 0 = 0 is set at 0. It is immediate to see that for every S ∈ S(m) we have Finally we note that the value of S depends only on added value after financing operations and on whether capital had to be raised, i.e.
Firm value and firm added value are defined assuming the firm makes optimal use of its resources and will be seen to correspond to the value, respectively added value, of an "optimal" admissible strategy.
Firm value given initial capital m ∈ X is defined as The map V : X → [0, ∞) is called the value function. Firm added value given initial capital m ∈ X is defined as The map AV : X → [0, ∞) is called the added-value function. Note that we always have The following result collects some important monotonicity properties of the value and addedvalue functions.
Proposition 3.1. The value function V : X → R satisfies the following properties: (iv) The added-value function AV : X → R takes only nonnegative values and is increasing, but not strictly increasing, on R.
Remark 3.2. At any capital level m, both the value function V (m) and the added-value function AV (m) are decreasing in the recapitalization cost C, decreasing in the carry cost of capital 1 − γ, and increasing in the margin q.

Existence of stationary optimal strategies
A key question is whether, for every m ∈ X , there exists an optimal strategy, i.e. an admissible strategy A stationary optimal strategy is an optimal strategy S * ∈ S(m) such that for some decision function D * , in which case we say that the decision function D * is optimal.
Given the stationary nature of our model, it is reasonable to expect that stationary optimal strategies exist: a decision that is optimal at a particular date should also be optimal at any other date as long as capital before financing operations is the same. Our next result establishes the existence of an optimal stationary strategy.

Capital management
We can now provide a comprehensive picture of the capital management decisions of the firm, i.e. of its optimal liquidation, dividend and capital-raising strategies.

The Dividend Principle
The following Dividend Principle for optimal strategies is intuitive: as long as a unit of capital adds extra value when left in the firm, 40 the manager should not pay out dividends. On the other hand, if an additional unit of capital does not add value, then it should be returned to shareholders. To formalize this, it is useful to introduce notation for the intervals on which AV is constant. For every m ∈ R, we denote by J(m) the largest interval containing m such that AV (m) = AV (m) for everym ∈ J(m). Note that J(m) may be degenerate and have m as its only element and that there is at most a countable number of J(m)'s that are not degenerate. Because the function AV is right continuous, if J(m) is non degenerate, it must be closed to the left and open to the right unless AV is continuous at its right end.
Our next result shows that we can always choose the optimal decision function in Theorem 3.3 to be aligned with the Dividend Principle as formalized in Property (23).
Proposition 3.4 (Dividend Principle). The decision function D * (·) in Theorem 3.3 can always be chosen in such a way that, for every m ∈ R, where d(m) := inf J(m).
The nondegenerate intervals J(m) are called dividend bands. Whenever capital belongs to a band, the firm pays dividends down to the band's left end. Proposition 3.4 says that we can always choose an optimal decision function whose dividend strategies are of band type. If there is only one band we say that the dividend strategy is of barrier type. For the remainder of this paper we assume that D * (·) is an optimal decision function satisfying the above Dividend Principle.
Note that Property (23) implies that the firm is liquidated if and only if AV (m) = 0, in which case V (m) = max{m, 0}. To make the problem studied in this paper worthwhile, we make the following Nontriviality Assumption:

Liquidation and upper-dividend barriers
There are two critical thresholds associated with the added-value function. The first one is the upper-dividend barrier By the Dividend Principle (23), when capital levels are above M , the firm pays dividends down to M because any capital in excess of M does not add value. Moreover, the monotonicity of the added-value function immediately implies that added value is maximal at M , i.e. AV (M ) ≥ AV (m) for every m ∈ R.
An implication of the above result is that costly capital raising should be the only reason for an insurance firm to choose to operate above the regulatory minimum. As illustrated by the examples in Section 5, no general statement about the investment strategy can be made when raising capital is costly. These examples also show that, when M > M reg , the region between M reg and M may contain multiple dividend bands.
The second critical threshold is the liquidation barrier The next proposition shows that the firm is liquidated whenever capital is below M . It also shows that, when the firm recapitalizes, it always does so up to M , i.e. up to the level at which added value is greatest. Indeed, recapitalizing to any other level would result in less added value at the same cost. Hence, for the recapitalization option to have any value, i.e. for M < M reg to hold, the cost of raising capital must be lower than added value at the upper-dividend barrier.
Proposition 3.6. The liquidation barrier satisfies: Note that the firm is liquidated when capital is at M if M < M reg but not if M = M reg . This is because, recapitalizing adds zero value in the first case and continuing operations always adds value in the second case by the Nontriviality Assumption (24). Observe also that the recapitalization option is always valuable if raising capital is costless. The next result establishes that, ceteris paribus, there exists a maximal cost of raising capital above which the recapitalization option is worthless.
Proposition 3.7. Consider firms that are identical with the exception of their recapitalization costs. Denote by AV C the added-value function of the firm with recapitalization cost C ≥ 0 and by M C and M C the corresponding upper-dividend and liquidation barriers, respectively. There exists C > 0 such that M C < M reg if and only if C < C. Moreover, C ≤ AV 0 (M 0 ).

Decomposition of value and added value
To identify the drivers of firm value, we decompose it into components attributable to financing operations, business of the current period, the limited liability option of shareholders, and the ability to write profitable new business in the future. Assume m is a capital level at which the firm continues to operate, i.e. δ * (m) = 1, and set By Equation (22), where the impact of financing operations FO(m) is defined as and reflects how dividend payments and recapitalization affect firm value.
value added by future operations Net tangible capital corresponds to the value of internal funds available at the end of the period assuming no default is possible. It includes the profit margin from business from the current period and is given by where we have used Expression (26) and the fact that The firm's default option value reflects the limited liability of shareholders. Its value is The firm's franchise value reflects its ability to collect economic rents, i.e. to add value, from future operations and is given by With this terminology we immediately obtain the following decomposition of firm value.
Proposition 3.8. Assume that starting capital m is such that δ * (m) = 1. Then, To obtain the added-value counterpart of the above decomposition, we first define economic profit at the start of the period to be Economic profit at the start of the period corresponds to the margin q adjusted for costs incurred, which include: the carry cost of capital (1−γ)(m−z * (m)+κ * (m)+q); the cost of raising capital C(κ * (m)); and the cost of not defaulting on claims max{−m, 0}. The following decomposition of added value is just a restatement of Proposition 3.8 in terms of added value.
We can further split franchise value to highlight the value of the recapitalization option. Noting that the set {M < M * 1 < M reg } corresponds to those states in which the firm can continue to operate but only after recapitalizing, it makes sense to define the firm's franchise value due to recapitalization as where the equality follows from Proposition 3.6. The second term after the equality sign reflects the cost of settling outstanding liabilities when recapitalizing from negative capital levels. On the other hand, {M * 1 ≥ M reg } corresponds to the states in which the firm can continue writing business without having to recapitalize so that franchise value from internal resources is Note that FV (m) = RFV (m) + IFV (m).

Capital buffers and policyholder protection
Assume initial capital m is such that δ * (m) = 1. Net tangible capital NTC(m), defined by Expression (28), corresponds to the amount of "physical" capital available at date 0 to absorb unexpected losses from the insurance portfolio. However, policyholders may effectively enjoy more protection than that provided by net tangible capital. Indeed, whenever M < 0 and M < m < 0, instead of defaulting, the firm chooses to recapitalize. This means that policyholders can view max{−M , 0} as an additional layer of protection over and above net tangible capital. We highlight that whether or not the effective capital buffer is higher than what regulatory minimum requires is driven purely by the manager's desire to maximize value and not by any concern for policyholder safety. Examples in Section 5 illustrate that in the relation M ≤ M reg ≤ M the inequalities can be strict or not, in all possible combinations depending on the environment in which the insurer operates.

Convexity of the value function and investment strategies
In this section we focus on optimal investment strategies. For this reason, we restrict our attention to situations where the firm is not liquidated. Moreover, we need only consider capital levels at or above M reg , since the amount of capital available for investments is always determined after financial operations. For all m ≥ M reg and λ ∈ [0, 1] we set M m,λ 1 := γ(m − z * (m) + q)(1 + r + λρ 1 ) + R 1 , which corresponds to end-of-period capital if the manager adopts the investment strategy λ. Denoting by ess inf(X) the essential infimum of the random variable X, we have that P(M m,λ 1 ≥ ess inf(R 1 )) = 1 holds for every m ≥ M reg and λ ∈ [0, 1]. Hence, for all practical purposes, properties of the value function V matter only on the interval (ess inf(R 1 ), ∞). If the value function is affine on (ess inf(R 1 ), ∞), then the firm is indifferent between taking investment risk or not. By contrast, if the value function is convex but not affine on this interval, then the firm seeks maximal investment risk exposure. The situations in which the value function exhibits these two behaviours are characterized in Proposition 4.2, if R 1 ≡ 0, and in Proposition 4.4, if R 1 ≡ 0.

No insurance losses
In the limiting case where there are no insurance losses, i.e. R 1 ≡ 0, the firm receives the margin q without assuming any obligation. This, however, requires holding at least the minimum regulatory capital and incurring the corresponding carry cost. Therefore, operating the firm is only worthwhile if the effective, per-period margin γq exceeds the corresponding carry cost of capital (1 − γ)M reg or, equivalently, if the per-period upfront profit γq − (1 − γ)M reg is strictly positive.
Moreover, the recapitalization option is valuable, i.e. M < M reg , if and only if Finally, (i) If C = 0 or M reg = 0, then the value function is affine on [0, ∞). In this case, the set of optimal investment strategies is the entire interval [0, 1].
(ii) If C > 0 and M reg > 0, then the value function is neither convex nor concave on [0, ∞).
In this case, the set of optimal investment strategies is the closed interval [0, λ] with Proposition 4.2 shows that, if the firm pursues a risk-free investment strategy, end-of-period capital never falls below M reg and the firm receives the amount γq − (1 − γ)M reg perpetually without ever incurring recapitalization costs. Indeed, Formula (34) is just the present value of a perpetuity entitling shareholders to receive γq − (1 − γ)M reg at the beginning of every period. This applies to any investment strategy under which end-of-period capital cannot fall under M reg . If M reg = 0, this is true for all investment strategies. If M reg > 0, this is true only if the firm takes a sufficiently small amount of investment risk. This is precisely what the condition λ < λ ensures.
Proposition 4.2 can be used to derive an upper bound for the value of the firm in the general case when insurances losses are not zero. The value of an insurance firm exposed to insurance losses is clearly bounded from above by the value of a firm that obtains the amount l + q in every period and carries no insurance risk. 41 Hence, when R 1 ≡ 0, we have for every m ≥ 0: This proves the following necessary condition for the Nontriviality Assumption (24) to be satisfied when insurance losses are nonzero.
It follows that, for a fixed 0 < γ < 1, the regulatory capital requirement cannot be arbitrarily large.

Nontrivial insurance losses
We now turn to the general case where insurance losses are not identically zero, i.e. where R 1 ≡ 0. We start with the case of costless recapitalization. (ii) If M > ess inf(R 1 ), then the value function is convex, but not affine, on (ess inf(R 1 ), ∞). Whenever the firm is not liquidated, the only optimal investment strategy is λ * ≡ 1.
In case of costly recapitalization, the value function is neither convex nor concave.
The two preceding propositions show that the value function is convex if and only if recapitalization is costless, i.e. C = 0. In this case, the firm takes full investment risk when M > ess inf(R 1 ). When M ≤ ess inf(R 1 ) the firm is indifferent towards investment risk.
The case of costless recapitalization can be of interest even in an environment where raising capital is costly. Indeed, one can think of an insurer that operates within a holding that keeps capital at the holding level. Once the holding has raised capital and incurred the corresponding costs, no additional costs are incurred when injecting it into the subsidiaries. In addition, the case of costless recapitalization provides a benchmark for the case where recapitalization is costly. Denote by M C the upper-dividend barrier, by V C the value, and by AV C the added value of the firm when C ≥ 0. Then, for m ∈ R, V C (m) ≤ V 0 (m) and AV C (m) ≤ AV 0 (m), which proves the next result. This result shows that, regardless of the recapitalization costs, AV 0 (M reg ) is the maximal added value to which shareholders can aspire given the regulatory threshold M reg .
A remark on concavity and optimal investment strategies Propositions 4.2, 4.4, and 4.5 show that V is never concave on (− ess inf(R 1 ), ∞), unless it is affine, in which case all investment strategies are optimal. Even though in our concrete setting we never have concavity, in any market-consistent setting in which the value function happens to concave but not affine, it is easy to prove that it is optimal to avoid all investment risk. 42 As already discussed in the introduction, the fact that Højgaard and Taksar (2004) exhibit instances where the value function is strictly concave and it is optimal to invest in the risky asset is due to the lack of market consistency of their valuation rule, which creates a bias towards risky investments.

Examples
Optimal strategies depend on the financial and the regulatory conditions in which the insurer operates. When there is no insurance risk (R 1 ≡ 0) or recapitalization is costless (C = 0), these strategies can be fully characterized (see Proposition 4.2 and Proposition 4.4). When recapitalization is costly and insurance risk is nontrivial, a whole range of financial and investment strategies can occur. In particular, whether or not investing in the risky asset is optimal cannot be established without a narrower model specification, as the following examples illustrate. 43 That being said, our numerical examples suggest that the firm will tend to seek investment risks in regions where the value function is locally convex and to avoid it in regions in which it is locally concave.

How to read the figures
It is helpful to bear two things in mind when reading the figures. First, as per our convention, whenever capital before financing operations m is smaller or equal to M , we have set λ * (m) = 0, although, strictly speaking, the firm has no capital to invest because it is liquidated. Second, whenever capital before financing operations m is such that the firm continues to operate, we have λ * (m) = λ * (m − z * (m) + κ * (m)) because the amount relevant for investments is capital after financing operations. In particular, whenever M < M reg and m ∈ (M , M reg ), then λ * (m) = λ * (M ) and whenever m ≥ M , λ * (m) = λ * (M ). Similarly, if M = M reg , then the firm always operates at the regulatory minimum and the investment strategy for all capital levels m at which the firm continuous to operate is λ * (M reg ).

Financial and investment strategies in different financial environments
We start introducing a base case that we use as a benchmark. In the base case, it is optimal to hold capital above the regulatory minimum (M > M reg ) and the recapitalization option is 42 The proof of this fact relies on the conditional version of Jensen's inequality and on the market consistency of the valuation rule. 43 The numerical examples are computed using the "value iteration method", which relies on the approach used to prove Theorem 3.3, i.e. on applying the contraction principle to find a fixed point of the operator T , defined by in an appropriate function space.
valuable (M < M reg ). Moreover, depending on the level of capital, various levels of exposure to investment risk are optimal. We illustrate how these optimal strategies change with the cost of recapitalization C, the insurance margin level q, the carry cost of capital 1−γ, and the minimum regulatory capital requirement M reg .
Throughout this section we consider a risk-free rate r = 5%, a volatility of the risky asset σ = 10%, and a loss distribution constructed as follows: the densities of the N (0, 0.01) and N (3, 0.5) distributions 44 are truncated outside the interval [0, 6], added up, and normalized to obtain a new density with support [0, 6]. The resulting loss distribution is a smooth approximation of a Bernoulli random variable taking the values 0 and 3 with equal probabilities. 45

Base case
For the base case we specify the following parameters: C = 0.1, q = 10%l = 0.142, 1−γ = 0.005 and M reg = 0.25. Figure 1 superposes the graphs of the added-value function AV , the dividend strategy (left vertical axis), and the investment strategy (right vertical axis); all regarded as functions of initial capital m. For our choice of C, M = −2.4 < M reg , which implies that the recapitalization option is valuable. The optimal divided strategy is of barrier type with M ≈ 3.6. The optimal investment strategy is of bang-bang type, i.e. the firm chooses either full exposure to investment risk (λ * = 1) or no investment risk at all (λ * = 0). Interestingly, risky investment is optimal precisely at capital levels where the added value function is strictly convex. The optimal dividend strategy is of barrier type: The firm pays any capital in excess of 3.6 as dividend and pays no dividends when m ∈ (M reg , 3.6). The optimal investment strategy prescribes maximal investment risk when AV is convex and no investment risk otherwise.

Changing the cost of recapitalization
We now increase the recapitalization cost from C = 0.1 in the base case to C = 2.5. Figure 2 shows that M = M reg so that recapitalization is no longer valuable. The prohibitive recapitalization cost makes it optimal to increase the capital buffer above M reg . Indeed, the upper-dividend barrier is now M = 7.4, which is significantly higher than the 3.6 of the base case. The dividenddistribution strategy is richer than in the base case. The firm dividends out any capital in excess of M , does not pay dividends in the interval (1.3, M ), and pays dividends down to M reg when capital belongs to the dividend-distribution band (M reg , 1.3]. This pattern is a consequence of the shape of the loss distribution and the fact that the firm never recapitalizes: at levels of capital at which falling below M reg and having to liquidate is likely, there would be no benefit to compensate the carry cost of holding capital above the regulatory minimum. By contrast, at higher levels of capital, it does make sense to keep the excess within the firm because it helps avoid liquidation. The investment strategy is no longer of bang-bang type. Indeed, it is optimal to take full investment risk when capital lies in the interval (1.47, 1.62), to only take partial investment risk when capital lies in the interval (2.75, 3.1), and not to take any investment risk otherwise. As in the base case, investment risk is taken precisely when capital belongs to intervals in which the added value function is strictly convex. . The optimal investment strategy prescribes taking maximal investment risk when m ∈ (1.47, 1.62), partial investment risk when m ∈ (2.7, 3.1) and no investment risk otherwise.

Changing the margin
We now decrease the margin from q = 10%l = 0.142 in the base case to q = 5%l = 0.071. Figure 3 shows that after decreasing q it is no longer attractive to hold capital above the regulatory minimum, i.e. M = M reg , resulting in a dividend strategy of barrier type. This is because, with a lower margin, there is less added value to protect. The reduction in added value also makes recapitalization less attractive, as can be seen from the increase of the liquidation barrier from M = −2.4 in the base case to M = −1.4. As long as the firm operates, capital after financial operations remains always at the same level, with the amount γ(M reg + q) available for investment. As a result, the optimal investment strategy does not depend on m and, in this case, requires taking full investment risk, i.e. λ * ≡ 1. The reason is that, by taking investment risk, the firm stands to gain more from the increase in value of the default option than what it can potentially lose in terms of added value.
Remark 5.1. If, together with decreasing q to 0.071, we increase the cost of raising capital to C = 2.5, recapitalization becomes too costly, that is M = M reg . In such a setting the firm operates with capital at the regulatory minimum M reg and is liquidated as soon as capital falls below M reg . The optimal dividend strategy is of barrier type and no capital in excess of M reg is kept within the firm. The optimal investment strategy prescribes maximal investment risk. .

Changing the carry cost of capital
Next we consider the effect of increasing the carry cost of capital from 1 − γ = 0.005 in the base case to 1 − γ = 0.03. Figure 4 shows that, with this particular increase in 1 − γ, holding capital becomes so costly that it is no longer optimal to keep any capital above the regulatory minimum, i.e. M = M reg , implying a dividend strategy of barrier type. As in the preceding example, at capital levels where the firm is not liquidated, capital after financial operations remains at the same level and the optimal investment strategy does not depend on m. Here, it is also optimal to take full investment risk, i.e. λ * ≡ 1. Clearly, the higher carry cost of capital, reduces the amount of value the firm can add, leading to an increase of the liquidation barrier M from −2.4 in the base case to −1.83. The optimal dividend strategy is of barrier type and no capital in excess of M reg is kept within the firm. The optimal investment strategy prescribes maximal investment risk.

Changing the minimal regulatory capital requirement
Finally, we increase the minimal regulatory capital requirement from M reg = 0.25 in the base case to M reg = 3. Having to incur carry costs on a higher level of capital reduces the firm's potential to add value and results in an increase of the liquidation barrier M from −2.4 in the base case to −2.2 as shown in Figure 5. Note that capital above M reg is held to avoid having to incur recapitalization costs. Since, at any capital level, an increase in M reg makes the likelihood of having to recapitalize higher, the upper-dividend barrier M also increases from 3.6 in the base case to 6. However, due to the presence of carry costs, M does not increase by the same amount as M reg . It is optimal to take full investment risk at capital levels between M reg and 4. As capital increases from 4 to 4.2, the proportion of investments in the risky asset gradually decreases from 1 down to 0.

Risky investments can also support franchise value
The previous section shows that increasing the value of the default option can justify pursuing risky investment strategies; see, for instance, Cases 5.1.3 and 5.1.4. In particular, Proposition 4.4 establishes that, when recapitalization is costless and there are states in which it is worthwhile defaulting, taking maximal investment risk to increase the value of the default option is the only optimal choice. However, increasing the value of the default option is not the only possible reason for investing in the risky asset. To illustrate this, we consider a firm for which, despite defaulting never being optimal, it is nevertheless optimal to take investment risk. We obtain a firm with these features by increasing the margin from q = 10%l = 0.142, in the base case, to q = 20%l = 0.284. 46 In this case, M = 3.29 and the optimal dividend strategy is of barrier type. The upper-dividend barrier is M = 6. The optimal dividend strategy is of barrier type: The firm pays any capital in excess of 6 as dividends and pays no dividends for m ∈ (M reg , 6). The optimal investment strategy prescribes maximal investment risk when m ∈ (M reg , 4), partial investment risk when m ∈ (4, 4.2) (gradually dropping from maximal to no investment risk), and no investment risk when m ≥ 4.2.
In particular, no dividends are paid between M reg and M . We now apply the decomposition of firm value in Proposition 3.
value of outstanding liabilities when capital is negative Observe that any increase in the value of the default option is offset by the value of outstanding liabilities in those instances where the firm could have defaulted but chooses to recapitalize. This is particularly interesting because it shows that any gain from investing in the risky asset cannot be attributed to an increase in the default option value but rather to a boost in franchise value. Figure 6 shows 47 this is what occurs at levels of capital between M reg = 0.25 and 1.4 where taking maximal investment risk is optimal.

Conclusions
We have studied a dynamic model for a value-maximizing insurance firm that takes decisions on its capital and investment strategies, including the possibility of liquidation. In a first step we developed a rigorous economic valuation framework for insurers with a broad ownership base. Because in an insurance firm cash flows to shareholders generally depend on financial-market returns, it is necessary to use a market-consistent valuation measure to assess their value. Using discounted expected cash flows with respect to the "objective" probability measure, as often done in the literature, would mean second guessing market prices and creating a bias towards risky investments due to their higher expected returns and result. This would result in an incorrect firm value and suboptimal investment strategies. On the other hand, due to the firm's broad ownership base, the manager should be indifferent to idiosyncratic risk and act as a risk-neutral investor whenever valuing cash flows that do not depend on financial-market returns. We proved the existence of a unique economic valuation measure that is both market consistent and captures indifference towards idiosyncratic risk. Using this measure, the economic value of the firm is the NPV of cash flows to shareholders under an optimal strategy. We used dynamic-programming techniques to prove the existence of stationary optimal strategies.
To analyze investment strategies we derived a decomposition of firm value and added value that isolates the four sources of value: the impact of financing operations, net tangible capital, default option value, and franchise value. The investment strategy only impacts the default option value and the franchise value and, typically, in opposite directions. Hence, whether or not to take investment risk is the result of how this trade-off resolves. We provide a full description of optimal investment strategies when either recapitalization is costless or there is no insurance risk. When recapitalization is costly and insurance risk nontrivial, we find that optimal investment strategies can cover the full range from risk-free to maximally risky investments depending on the particular constellation of the financial and regulatory environment within which the firm operates. Interestingly, investment in risky assets may not only be driven by the desire to increase firm value by boosting the option to default. Indeed, they may also serve as a surrogate for recapitalization by helping the firm reach capital levels at which franchise value is greater.

Appendix
This Appendix provides proofs for the results in the paper with the exception of the existence and uniqueness of the economic valuation measure, which are relegated to the Online Appendix. In the Online Appendix we also substantiate the claim that the firm manager can focus exclusively on admissible strategies.
iii) This follows directly from i) and ii). iv) This follows from i) -iii) since, for every m ∈< R, we have AV (m) = V (m) − max{m, 0} by the definition of AV through Expression (18).
Proof of Theorem 3.3. Note that X = R ∪ { * }, equipped with the disjoint union topology, 48 is a separable, completely metrizable space. Define, for every a > 0, the weight function We search for a solution to the dynamic-programming equation (20) within the space B a (X ) of Borel-measurable functions f : X → R satisfying f a < ∞, where f a := sup x∈X |f (x)/w a (x)|. For brevity we introduce the impact of financing operations m ∈ X , s ∈ S 0 (m) .
We start with the three parts of Assumption 8.5.1 in Hernández-Lerma and Lasserre (1999) i) We first establish that the map m → S 0 (m) is compact valued and upper semicontinuous. By definition, S 0 (m) is compact for all m ∈ X . Theorem 17.20 in Aliprantis and Border (2006) guarantees that upper semicontinuity of m → S 0 (m) is equivalent to sequential closedness of its graph. To see that this is the case, consider a convergent sequence ((m n , s n ) , n ≥ 0) with elements in the graph of S 0 and denote its limit by (m, s). There are two possibilities: (a) If m = * (equivalently δ = 0), for n large enough, we must have m n = * and s n = (0, 0, 0, 0).
ii) Next we must show that, for every m ∈ X , the map s → c(m, P ) is upper semicontinuous. This follows directly from the lower semicontinuity of the map κ → C(κ) = C · 1 (0,∞) (κ).
iii) Finally, we prove that, for all m ∈ X , the map s → E Q * [w a (M m,s 1 )] is continuous on S 0 (m).
In the case m = * , we have s → E Q * [w a (M * ,s 1 )] ≡ 1, which is clearly continuous. Otherwise, consider s ∈ S 0 (m) and note that Given that the right-hand side of the above expression is integrable, the dominated convergence theorem implies the continuity of the map s → E Q * [w a (M m,s 1 )]. To establish Assumption (8.5.2) in Hernández-Lerma and Lasserre (1999) we show that there exist a > 0, c ≥ 1, and β ∈ [1, 1 + r) such that, for all m ∈ X , Since the function f (m) := (|m| + C)/w a (m) is bounded, there exists c ≥ 1 such that f (m) ≤ c for all m. This implies that |c(m, s)| ≤ cw a (m) holds for all s ∈ S 0 (m) and, therefore, that Condition (A3) is satisfied. If s ∈ S 0 (m) is such that δ = 0 then M m,s 1 = * . It follows that w a (M m,s 1 ) = 1 ≤ w a (m) ≤ βw a (m) for any choice of β ∈ [1, 1 + r) and, hence, that Condition (A4) is satisfied. On the other hand, if δ = 1, then The right-hand side is Q * -integrable, hence there exist a ≥ 1 such that E Q * [max{Y − a, 0}] ≤ r 2 . As a result, Together with the fact that w a ≥ 1, this implies that E Q * w a M m,s 1 ≤ βw a (m) for all s ∈ S 0 (m), which yields Condition (A4).
Finally, we verify Assumption 8.5.3 in Hernández- Lerma and Lasserre (1999) by proving that the map s → E Q * [u(M m,s 1 )] is continuous for every m ∈ X and every bounded, continuous function u : X → R. If m = * , then the only admissible strategy s is (0, 0, 0, 0) and continuity of s → E Q * [u(M * ,P 1 )] is trivial. For m ∈ R and s ∈ S 0 (m), the map ω → u(M m,s 1 (ω)) is measurable. Moreover, for every ω ∈ Ω, the map s → u(M m,s 1 (ω)) is continuous. Consider a sequence (s n , n ∈ N) ⊂ S 0 (m) converging to some s, which must belong to S 0 (m) by compactness. By the assumptions on u, the sequence u(M m,sn 1 ), n ∈ N is uniformly bounded. The result now follows by direct application of the dominated convergence theorem.
Step 2. Given that our problem satisfies Assumptions 8.5.1-8.5.3 in Hernández-Lerma and Lasserre (1999), we immediately have the following: • The value function V is the unique solution in B a (X ) to the fixed point problem • From Expression (8.5.3) in Hernández-Lerma and Lasserre (1999), we have that V (thus AV ) is upper semicontinuous and, as V and AV are nondecreasing, this is equivalent to continuity from the right.
It follows that D * is an optimal decision function.
Proof of Proposition 3.4. We verify that we can take D * to satisfy the Dividend Principle (23).
If m = * , then the only possible strategy is (0, 0, 0, 0). If m ∈ R, then we may replace D * (·) by Note that, by definition of d, we have This implies the optimality of the decision function D * .
Proof of Proposition 3.5. We know from the defining Property (S3) of admissible strategies that, if capital is above The proof of Proposition 3.7 requires the following lemma establishing that the value function depends continuously on the recapitalization costs.
Lemma A.1. Denote by V C the value function of the insurance firm with recapitalization cost Proof. Fix m ∈ R and denote by S C = (z C , κ C , λ C , δ C ) an optimal strategy when refinancing costs are C ≥ 0. Assume first that 0 ≤ C 1 ≤ C 2 .
where we have used that, when C 1 ≤ C 2 , the strategy S C 1 is also admissible for the firm with recapitalization costs C 2 . The above estimate is equivalent to |V C 1 (m) − V C 2 (m)| ≤ 1 r |C 1 − C 2 |. If C 2 ≤ C 1 then we exchange the roles of C 1 and C 2 to obtain the same estimate.
Proof of Proposition 3.7. We use the notation of Lemma A.1 and denote by M C the liquidation barrier for the firm with recapitalization cost C ≥ 0. By Proposition 3.6, M C < M reg if and only if AV C (M ) > C. By Remark 3.2 we have AV C (M C ) ≤ AV 0 (M 0 ). Hence, if C ≥ AV 0 (M 0 ) we automatically have C ≥ AV C (M C ) and, hence, M = M reg . That the range of recapitalization costs at which the firm recapitalizes is an interval is clear: if a firm has an incentive to recapitalize, then any firm that is an identical but has lower recapitalization costs will have an even greater incentive to do so. By Proposition 3.6, M < 0 if C = 0. Hence, this interval is not empty. We show that it is right open. Indeed, Lemma A.1 implies that if AVĈ(M ) >Ĉ holds for someĈ, we will have AV C (M ) > C for any C ≥ 0 that is sufficiently near toĈ.

Appendix B Proofs of results in Section 4
Proof of Lemma 4.1. We start with a preliminary remark. As there are no insurance losses, the value of the default option is zero and the decomposition of added value of Corollary 3.9 yields Using that the added-value function attains its maximum at M and that M reg ≤ M , Expression (B1) implies that AV (M ) ≤ γq − (1 − γ)M reg + 1 1+r AV (M ). It follows that If the Nontriviality Assumption (24) holds, i.e. AV (M reg ) > 0, then Inequality (B2) immediately implies that γq − (1 − γ) M reg > 0 must hold.
Because we need it in the proof of Proposition 4.2 below, we point out that we have actually proved that M = M reg and that the strategy S is optimal. Indeed, Inequality (B2) and Expression (B3) yield Proof of Proposition 4.2 In the preceding proof we showed that M = M reg and that paying dividends down to M reg and pursuing a risk-free investment strategy is always optimal. We also provided the explicit Expression (34) for AV (M reg ), from which Condition (35) immediately follows. Below we will use that, for every m ≥ M reg , we always have If M reg > 0, then, for every m ∈ [0, M reg ), we have that (i) Note that if C = 0, then M < M reg . From identities (B4) and (B5) we immediately obtain that, whenever M reg = 0 or C = 0, V is affine on [0, ∞).
Note that whenever the firm is not liquidated it operates with capital equal to M reg . As a result optimal strategies are fully determined by the investment strategy. Set M λ 1 = γ(M reg + q)(1 + r + λρ 1 ) + R 1 for every λ ∈ [0, 1]. Then, λ * ∈ [0, 1] is optimal if and only if 1]. Note that, as λ → V (M λ 1 (ω)) is convex for every fixed ω, the function f is also convex.
(i) If M ≤ ess inf(R 1 ), then we already know that V is affine on [ess inf(R 1 ), ∞) and the firm is never liquidated. It follows that and, hence, f is constant on [0, 1] and every λ ∈ [0, 1] is optimal.
(ii) If M > ess inf(R 1 ), then we already know that V is convex but not affine. Moreover, we know that ess inf(R 1 ) < M implies that 0 < P(M 1 1 < M ) < 1. In particular, the probability of recapitalization and the probability of default are both strictly positive whenever taking maximal investment risk. Applying the conditional version of Jensen's inequality, we obtain Here, we have used that ρ 1 and R 1 are Q * -independent. The above inequality cannot be an almost sure equality. To see this, set E = {M 0 1 ≤ M } and note that E ∈ σ(R 1 ). Assume first that P(E) > 0. Because ρ is unbounded above, we have that {M 1 1 > M } ∩ E has strictly positive probability. This means that V (M 1 1 )1 E is positive and non zero and, as a result, is also positive and nonzero. Since V (M 0 1 )1 E ≡ 0 on E, Inequality (B8) is not an almost sure equality. Assume now that P(E) = 0. Then, recalling that P(M 1 1 < M ) > 0, we have This concludes the proof that Inequality (B8) is not an almost sure equality. It follows that Since f , as a convex function, cannot first rise and then fall, this implies that λ * = 1 is a global maximum for f .  This online appendix gathers some of the more technical proofs of the paper. The first section is devoted to proving the existence and uniqueness of the economic valuation measure and the second to substantiating the claim in the paper that the firm manager may discard from consideration all strategies that are not admissible.

The economic valuation measure
We provide a rigorous construction of the underlying filtered probability space (Ω, F, F, P), the insurance loss process L = (L n , n ∈ N \ {0}), the price process S = (S t , t ≥ 0), and the valuation measure Q * used to compute the economic value of the insurance firm.

Construction of the probability space
Set Ω 1 := C(R + , R) and Ω 2 := R N 0 , where R N 0 is the set of sequences x = (x j , j ∈ N) such that x 0 = 0. The coordinate processes W = ( W t , t ≥ 0) on Ω 1 and L = ( L n , n ∈ N) on Ω 2 are defined by W t (f ) := f (t) and L n (x) := x n for every f ∈ Ω 1 , t ∈ R + , x ∈ Ω 2 , and n ∈ N.
We denote the σ-algebras generated by the coordinate processes W and L by F W = σ( W ) and F L = σ( L), respectively.
We equip (Ω 1 , F W ) with the Wiener measure P 1 so that the process W becomes a Brownian motion on (Ω 1 , F W , P 1 ). Furthermore, we let P 2 be the unique probability measure on (Ω 1 , F L ) making L an i.i.d. sequence of random variables with a prescribed common distribution corresponding to the distribution of insurance losses. With this notation, we define the probability space (Ω, F, P) by setting Ω := Ω 1 × Ω 2 , F := F W ⊗ F L , and P := P 1 ⊗ P 2 .
On this space we consider the processes W = (W t , t ≥ 0) and L = (L n , n ∈ N) defined by setting W t (ω) := W t (f ) = f (t) and L n (ω) := L n (x) = x n for ω = (f, x) ∈ Ω, t ≥ 0 and n ∈ N. Note that F is the smallest σ-algebra containing the σ-algebras F W := σ(W ) and F L := σ(L). By definition, F W and F L are P-independent σ-algebras or, equivalently, W and L are P-independent families of random variables.
The sequence L of i.i.d. random variables represents the insurance losses to which the insurance company is exposed. The evolution of the money-market account is given by the deterministic process B = (B t , t ≥ 0) defined by B t = e tr , t ≥ 0, for a fixedr ≥ 0. The risky security has initial price s 0 > 0 and its price process S = (S t , t ≥ 0) follows a geometric Brownian motion with drift µ >r and volatility σ > 0, i.e.
For t ≥ 0, we set The "raw" filtrations F W = ( F W t , t ≥ 0) and F L = ( F L t , t ≥ 0) are the smallest filtrations summarizing market information and insurance loss information, respectively. The filtration F = ( F t , t ≥ 0) summarizes the total information that is available to the insurer. However, to be able to exploit the rich theory of stochastic processes, this filtration needs to be appropriately augmented.
Before looking at the particular augmentation we record the following lemma on the raw filtered measurable spaces (Ω, F W , F W ), (Ω, F L , F L ). This technical result will be important for the construction of the probability measures P * and Q * . (1) For every t ≥ 0, F t is countably generated.
(3) Let (ω n , n ∈ N) ⊂ Ω be a sequence in Ω and denote by A n (ω n ) = A ∈ F n , ω n ∈ A the smallest set in F n containing ω n . If N n=0 A n (ω n ) = ∅, then n∈N A n (ω n ) = ∅.
Proof. The proof mimics the proof of Proposition 4.4 in Najnudel and Nikeghbali (2011) and is therefore omitted.

The natural augmentation
Denote by (Ω, F u , F u , P) the usual augmentation of F. 1 For each T > 0, the filtered probability space (Ω, F u T , (F u t ; 0 ≤ t ≤ T ), P) satisfies the usual conditions. As a result, the standard results for arbitrage free markets imply the existence of a pricing measure Q u, * T that is defined on F u T ; see e.g. Musiela and Rutkowski (2005). The question we address here is: can we find a probability measure Q u, * defined on F u that coincides with Q u, * T on F u T for every T > 0? If we use the usual augmentation of F, the answer is negative; see Najnudel and Nikeghbali (2011) to see what can go wrong. On the other hand, the answer is positive if we replace F u by the so-called natural augmentation F = (F t ; t ≥ 0) defined further below and first introduced by Bichteler (2002) and, independently, by Najnudel and Nikeghbali (2011). The natural augmentation will be large enough so that (Ω, F T , (F t ; 0 ≤ t ≤ T ), P) satisfies the usual conditions, implying the existence of pricing measures Q * T defined on F T for each maturity T > 0, and small enough so that there exists a probability measure Q * defined on F which coincides with Q * T on F T .
of events, such that E n ∈ F n and P(E n ) = 0 for all n ∈ N.
The filtered probability space (Ω, F, F, P) is said to satisfy the natural conditions if F 0 contains all its σ-negligible sets and F is right continuous. (2011), a filtered probability space (Ω, F, F, P)

By Proposition 2.4 of Najnudel and Nikeghbali
always admits a smallest filtration G ⊃ F such that (Ω, F, G, P) satisfies the natural conditions.
G is called the natural augmentation of F. 2 We are now in a position to specify the information structures of the financial market and the insurer used in Section 1 of the paper: the natural augmentation of the filtration F W , F L , and F, respectively, is the financial market filtration F W , the insurance losses filtration F L , and the insurer's filtration F.
The following result is a straightforward consequence of the construction of F as a product and its augmentation F. Lemma 1.3. The filtrations F W and F L are independent under P. Moreover, for every t ≥ 0 we have where S t is the set consisting of all intersections A ∩ B with A ∈ F W t and B ∈ F L t .

The two market models
We consider two market models. In both models, the same assets are traded, i.e. the risky asset with price process S, and the money-market account evolving according to the process B. The two models differ only in the class of trading strategies that are allowed. In the first model, which we will simply refer to as the "Black-Scholes market model", trading strategies take into account only the information available in the market, i.e. trading strategies are assumed to be F W -predictable. The underlying probability space is (Ω, F W , F W , P). In the second model, referred to as the "extended market model", when devising their trading strategies, traders are allowed to additionally incorporate the information on insurance losses, i.e. trading strategies are F-predictable. As a result, the underlying probability space is now (Ω, F, F, P). While, for finite maturities, both models turn out to be arbitrage free, the first model is complete and the second is not. We now proceed to construct equivalent market measures in both of them. For every T > 0, we set Construction of the pricing measure P * for the Black-Scholes model We now establish that in the Black-Scholes market with infinite time horizon, there exists a single pricing measure that can be used to price cash flows at all maturities.
Proposition 1.4. There exists a unique probability measure P * defined on (Ω, F W ), the pricing measure, such that (i) P * is equivalent to P on F W T for every T > 0 but not on F W . (ii) For every T > 0, L 2 (Ω, F W T , P) is a subset of L 1 (Ω, F W T , P * ). (iii) For every 0 ≤ t < T and every cash flow X ∈ L 2 (Ω, F W T , P) maturing at date T , market consistency holds, i.e. (3) Proof. We first show the existence of the finite-horizon pricing measures P * T for every T > 0. We fix T > 0 and note that S is a geometric Brownian motion defined on (Ω, F W T , P) that is F W T -adapted. By Proposition 2.5 of Najnudel and Nikeghbali (2011), the filtered probability space (Ω, F W T , F W t , P) satisfies the usual conditions. As a consequence, we can apply the classical results on the Black-Scholes model. Consequently, by Lemma 3.1.3 in Musiela and Rutkowski (2005), there exists a unique equivalent martingale measure, i.e. a probability measure P * T , defined on (Ω, F W T ) that is equivalent to P on F W T and such that (e −tr S t , 0 ≤ t ≤ T ) is a P * T -martingale. Its density is given by Moreover, by Corollary 3.1.3 in Musiela and Rutkowski (2005), every cash flow X ∈ L 2 (Ω, F T , P) with maturity date T is replicable and has, at each date 0 ≤ t ≤ T , the unique replication price We now proceed to prove the individual statements of the proposition.
(i) We start by noting that (P * T , T > 0) is a coherent family of probability measures, i.e. P * T | Fs = P * s whenever 0 ≤ s < T . Indeed, for A ∈ F W s , we have π s,T (er T 1 A ) = er s 1 A and, hence, P * s (A) = π 0,s (e sr 1 A ) = π 0,s (π s,T (e Tr 1 A )) = π 0,T (e Tr 1 A ) = P * T (A) , Lemma 1.1 and Corollary 4.10 of Najnudel and Nikeghbali (2011) now yield the existence of a where L 2 (Ω, F W T , P) ⊥ is the orthogonal complement of L 2 (Ω, F W T , P) in L 2 (Ω, F T , P). In fact, Q * T is the only probability measure on (Ω, F T ) with these two properties. Indeed, we can use (11) and (12), as well as the tower property for conditional expectation to easily conclude that, for (ii) Hölder's inequality and the fact that Z T ∈ L 2 (Ω, F W T , P * ) imply that L 2 (Ω, F W T , P) ⊂ L 1 (Ω, F W T , Q * ) for every T > 0. Since Identity (13) continues to hold if we replace Q * T by Q * and P * T by P * , we obtain the desired projection formula. (iii) For any interval I ∈ R, set A = {L n ∈ I}. Then, as 1 A and Z T are independent under P, we have Q * (A) = E P [Z T X] = P(A). This implies that, under Q * , the sequence L is i.i.d.
with the same distribution as under P. A similar argument shows that L remains independent of F W under Q * .
(iv) This immediately follows from Expression (14) and the fact that Q * coincides with Q * T on F T .

Admissible strategies
Recall that the state space is X := R ∪ { * }. The "cemetery state" * is an absorbing state, i.e. once the firm reaches that state it remains there forever, and represents a liquidated firm.
A state m ∈ R represents the firms capital. A (financial) strategy is an F-adapted process S = δ, z, κ, λ , where, for every n ∈ N: (i) δ n ∈ {0, 1} with 0 indicating liquidation and 1 continuation; (ii) z n ≥ 0 is the amount of dividends to be paid; (iii) κ n ≥ 0 is the amount of capital to be raised; and (iv) λ n ∈ [0, 1] is the proportion of capital to be invested in risky assets.
The random vector δ n , z n , κ n corresponds to financing operations at date n and the random variable λ n to the investment strategy at date n.
We assume that the initial state of the firm is m ∈ R and that a strategy S has been chosen. representing the date at which the company is liquidated according to strategy S. If the firm is not liquidated, to be able to operate, the firm must satisfy minimum regulatory capital requirements given by M reg ≥ 0. Hence, for every n ≥ 0, any strategy must satisfy When the firm is liquidated, the manager is only allowed to return remaining capital, if any, to shareholders, i.e. we have δ n = 0, z n = max{M m,S n , 0}, κ n = 0 and λ n = 0, on {τ m,S = n} , where we have set λ n = 0 by convention. Moreover, since the cemetery state * is absorbing we must have δ n = 0, z n = 0, κ n = 0, and λ n = 0 on {τ m,S < n} , where we again have set λ n = 0 by convention.
For every n ≥ 0 we have where is the one-period excess return on the risky asset. Note that E Q * [ρ n+1 |F n ] = 0 for n ≥ 0.
Clearly, M m,S = M m,S n , n ∈ N is an F-adapted process.
Under strategy S, the cash flow to shareholders at date n ≥ 0 is The sequence CF S := (CF S 0 , CF S 1 , . . . ) is referred to as cash flows to shareholders under the strategy S.

Economic value of cash flow streams to shareholders
Assume the cash flow stream to shareholders is described by a sequence X = X n , n ∈ N such that X n ∈ L 2 (Ω, F n , P) for every n ≥ 0. Recalling that L 2 (Ω, F n , P) ⊂ L 1 (Ω, F, Q * ) by Theorem 1.5, we also assume that X is bounded in L 1 (Ω, F, Q * ). 3 The economic value Π * 0 (X) of X at date 0 can be defined in a natural way by where, r := er − 1 denotes the one-period risk-free rate of return. Similarly, the economic value of X at any date N > 0 is defined as We show further down that the firm's manager can focus on "admissible strategies", defined below, and that these strategies satisfy the above boundedness condition. However, to prove that the manager can discard strategies that are not admissible we need to compare them with strategies that are not necessarily bounded. To this effect, we need to relax the notion of value at a date N > 0 in line with the following two observations. The first observation is that value is forward looking, i.e. the value of a cash flow stream at date N depends only on the cash flows maturing at later dates. The second observation is that, at date N , we only need to determine economic value "locally", i.e. for the state that has occurred at that date.
Taking these observations into account we arrive at the following definition. We say that the cash flow stream X admits an economic value at date N > 0 whenever there exists an F N -localizing sequence at date N D k , k ∈ N , 4 such that X D k is a bounded sequence in L 1 (Ω, F, Q * ) for every fixed k ∈ N. Here, X D k denotes the localization of X to D k which is given by In particular, the economic value of X D k at date N is well defined by Expression (21) with For such a cash flow stream we define its economic value at date N as the Note that the random variable Π * N (X) is integrable if X is bounded in L 1 (Ω, F, Q * ) but may otherwise fail to be integrable. 5

Admissible strategies
Any strategy S must satisfy (15), (16), and (17). An admissible strategy for the initial state m ∈ R is a strategy S satisfying, for every n ∈ N, the following additional conditions: (S1) z n · κ n = 0. The set of admissible strategies for the initial state m ∈ X is denoted by S(m). The economic interpretation of the above conditions is discussed in Section 2 of the paper. The following proposition shows that the cash flow streams generated by admissible strategies admit an economic value at any date N ≥ 0.

The discrimination principle
Before showing that the firm's manager can focus on admissible strategies, we need to establish a basic principle that allows to say when a cash flow stream to shareholders associated with a given strategy is "better" than that associated with another strategy.
Definition 2.2. A cash flow stream X = X n , n ∈ N is said to be value enhancing whenever there exists N ≥ 0 such that (i) X n = 0 for n = 0, . . . , N − 1 and (ii) X admits a positive economic value Π N (X) at date N (A positive number if N = 0 and a positive random variable if N > 0).
Note that, in particular, the zero cash flow stream is value enhancing. Clearly, obtaining a value-enhancing cash flow for free is attractive for any value-oriented manager. Given two cash flow streams X and Y , we write X Y whenever X n ≥ Y n holds for every n ∈ N and Xñ = Yñ for at least oneñ ∈ N. This partial order, together with the notion of a value-enhancing cash flow stream, allows us to establish a preference relation among strategies: Definition 2.3. Let S and S be two strategies. We say S is preferable to S (we write S S) if there exists a value-enhancing cash flow stream X such that CF ( S) − CF (S) ≥ X. If CF ( S) − CF (S) X, then S is said to be strictly preferable than S (we write S S).
All the results in this section rely on the following guiding principle: Discrimination Principle: If for a strategy S there exists a strategy S such that S S, then the manager may discard S without sacrificing value.

Why admissible strategies suffice
In Theorem 2.6 we will establish that, without loss of generality, when looking for optimal strategies, the firm's manager can focus on admissible strategies. For convenience, if m ∈ R and S ∈ S(m), we write (M n ; n ≥ 0) instead of (M m,S n ; n ≥ 0). We start by showing that we can always assume (S1).
Lemma 2.4. Let m be the starting capital. For every strategy S there exists a strategy S S satisfying (S1).
Proof. Define S by setting z n = max{z n − κ n , 0}, κ n = max{κ n − z n , 0}, λ n = λ n , and δ n = δ n for every n ≥ 0. Given that z n − κ n = z n − κ n , for every n ≥ 0, these two strategies have the same capital process. Moreover, by definition, z n · κ n = 0 for every n ≥ 0. Denote by τ the liquidation time, which is clearly the same for both strategies and note that CF S − CF S = (C( κ 0 ) − C(κ 0 ), C( κ 1 ) − C(κ 1 ), . . . ) where D n = {z n ≥ κ n > 0} corresponds to the set on which the insurer simultaneously raises capital and pays dividends under strategy S but only pays dividends under strategy S. Because 0 is value-enhancing, we conclude that S S.
The next result shows that one can always assume (S1) and (S2).
Lemma 2.5. Let m be the starting capital. For every strategy S there exists a strategy S S satisfying (S1) and (S2).
Proof. By Lemma 2.4, we may assume without loss of generality that z n · κ n = 0 for every n ≥ 0.
Set E = {M N ≥ M reg , κ N > 0} ∈ F N and define a new strategy S which delays capital raising by one period at date N and is otherwise identical to S. On E we set z N = 0 and z N +1 = z N +1 − κ N +1 − γκ N (1 + r + λ N ρ N +1 ) + κ N = 0 and κ N +1 = z N +1 − κ N +1 − γκ N (1 + r + λ N ρ N +1 ) − Using the notation ( M n ) = (M m, S n ) we have on E: Moreover, while end-of-period capital at date N + 1 differs for S and S on E, starting capital for the next period coincides, i.e.
Hence, we may define S to be identical to S except on E where we have changed the dividend and capital processes at dates N and N + 1. Note that S still satisfies (S1).
Note that{κ N +1 > 0} is contained in { κ N +1 > 0} and set F = { κ N +1 > 0} \ {κ N +1 > 0}. For the difference between the cash flow stream to shareholders corresponding to these two strategies To see that S S, we need to show that is a value-enhancing cash flow stream. To this effect, set D k := {κ N ≤ k} ∈ F N for every k ≥ 0 so that (D k ) is a localizing sequence. Note that, for each fixed k ≥ 0, the sequence (1 D k X n , n ≥ N ) is bounded in L 1 (Ω, F, Q * ). Therefore, X admits an economic value at time N and for each k ≥ 0 we have It follows that X is a value-enhancing cash flow stream, so that S S.
We have now constructed a strategy S with the required properties up to date N . We may proceed inductively and construct a strategy that is strictly preferable to S and has the required properties at all dates.
We are now in a position to prove the main and final result.
Theorem 2.6. Let m be the starting capital. For every strategy S there exists a strategy S S satisfying (S1), (S2) and (S3).
Proof. By Lemma 2.5 we may assume without loss of generality that S satisfies (S1) and (S2).
Assume P M n − z n + κ n > C (1 − γ)(1 + r) + M reg > 0 for some n ≥ 0 and let N be the smallest N ≥ 0 for which (1−γ)(1+r) } and define a new strategy S which, on E, pays dividends down to M reg at date N and raises capital at date N + 1 to compensate. Otherwise, the strategy is identical to S. Hence, we first define the dividend and capital process at dates N and N + 1 on E as follows: As in the previous proof, end-of-period capital at date N + 1 differs for S and S on E, but starting capital for the next period is identical, i.e.
Hence, we may define S to be identical to S except on E where we have changed the dividend and capital processes at dates N and N + 1. It is easy to see that S still satisfies (S1) and (S2).
Moreover, it is easy to see that { κ N > 0} = {κ N > 0}. For the difference between the cash flows to shareholders corresponding to these two strategies we then have is a value-enhancing cash flow stream. To see this set D k := {κ N ≤ k} ∈ F N for every k ≥ 0 so that (D k ) is a localizing sequence. Note that, for each fixed k ≥ 0, the sequence (1 D k X n ) is bounded in L 1 (Ω, F, Q * ). Therefore, X admits an economic value at time N and for each k ≥ 0 we have 1+r on E and that E Q [ρ N +1 |F N ] = 0. It follows that X is a value enhancing cash flow stream, so that S S.