The paper is organized as follows: in Section 2 we introduce the mathemat- ical setup and recall the main results about representation of **set**-**valued** **risk** **measures** and their subdifferentials, in Section 3 we introduce the **Capital** **Allocation** problem in the **set**-**valued** context, define some **set**-**valued** **Capital** **Allocation** rules based on the directional derivative and on sub-differentials of a **set**-**valued** function, and study their properties. We also briefly investi- gate the case when we reduce to a scalar-**valued** **risk** measure. Moreover, by adopting Kalkbrener’s view ([24]), we also derive **set**-**valued** **risk** **measures** with suitable properties, by starting with a general **Capital** **Allocation** rule. Section 4 is devoted to some examples. Section 5 sums up and provides some conclusions.

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Studying and evaluating the **risk** faced by a financial or an insurance institu- tion has always represented challenge from both theoretical and practical points of view. Indeed, the good health of financial and insurance companies relies on hedging a given level of **risk** by identifying how much money should be kept aside in order to face worse case scenarios. This requires methods to readily quantify financial risks. In [ 6 ], the authors paved the mathematical ground to measure the **risk** of a company by using a solid mathematical construction. The authors defined the concept of a **risk** measure as a real-**valued** function on the space of random variables representing possible outcomes of a financial position. Such a function assigns to each financial model a numerical value quantifying the level of **risk** associated with it. They devised an axiomatic structure for such **measures** that is compatible with the **risk** management point of view in which risks are evaluated through weighted sums of a **set** of possible scenarios. A large amount of scientific work has appeared since their seminal work which not only gener- alizes the theory of **risk** **measures** but also brings the theory into the context of different applications.

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where ) D R ( L 1 , ..., L n can be considered to be a measure of diversification. Relation (10) is valid for all sub-additive **risk** **measures**, for instance. The standard deviation-based **risk** meas- ure (7), for instance, is globally sub-additive, the Value-at-**Risk** according to (8) is sub- additive as long ( L 1 , ..., L n ) follows a multivariate elliptical distribution (and α < 0 . 5 ) and the CVaR according to (9) is sub-additive for instance when ( L 1 , ..., L n ) possesses a (multi- variate) density function.

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view Euler method is the better one, as it can be calculated by the average return of the assets in the α% of the worst cases (regarding the whole portfolio). This can be done easily even if we have a large number of portfolio elements. However if we choose cost gap method the increments and correction factors can be easily calculated as long as we consider three or four assets but it becomes very complicated and time-consuming when we want to count it in case of more assets. Furthermore Euler method has other favor- able properties too when we use ES or other coherent **risk** measure. Buch and Dorfleitner [2008] showed that – using gradient method for **capital** **allocation** – there are links con- necting the axioms of coherent **measures** of **risk** and **capital** **allocation** rules:

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a concept, we have to search for another, more stable definition. For example, Begović (2007, p. 51) offers a very elegant solution. According to Begović, corruption can be defined as a behavior that spreads away from a certain norm; whereas the norm is defined as a **set** of legislative, public interest or public opinion criteria. This elegant solution has at least two major problems. First, there is an institutional problem: there are different judicial interpretations of corruption, which treats the notion of corruption the different way. A problem linked with this one is that the law system is a human – built, social sys- tem. This means that it is prone to promulgation of certain laws that are not favorable in diminishing corruption, but on the contrary, they aggravate it. Secondly, corruption can not be approached only from the judicial point of view. There has to be more sociological and economical explications.

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Summarizing we generate for both vectors of losses 10000 replications of size 1000 and 400 for Public data **risk** no. 1 and for Public data **risk** no. 2, respectively in order to obtain two vectors of length 10000 over which we can apply the **allocation** principle we are interested in. In the i− th iteration we sum up all the re-sampled points to get the i − th element of each vector and we repeat this procedure 10000 times as i moves from 1 to 10000. This way the non-identical length problem of the vectors is overcome. Now we have to allocate a total amount of exogenous **capital**

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When **risk** **capital** of a portfolio has been calculated based on a **risk** measure, an- other important task is to allocate it back to each **risk** component in the portfolio. The **allocation** of **risk** **capital** has its own challenges due to the nature of dependence structures of combined risks. There are many motives behind **risk** **capital** **allocation**. Firstly, by comparing different losses on **capital** for each component, it is often pos- sible to answer if a component is worth to keep or not. Secondly, as **risk** **capital** is defined as a **risk** measure of the whole company, one can assess the riskiness of each component’s position by splitting this **capital**, and compare one to another. In addi- tion, **risk** **capital** **allocation** provides a useful device for assessment of performance of managers, which can be linked to their compensations. Last but not least, insurers may want to use the **allocation** in pricing. A line with an excessive **capital** would have to produce a larger profit by increasing the product margin, see Valdez and Chernih (2003); Neil (2007). In the literature, many researchers have proposed a **set** of axioms that any desirable **allocation** method is expected to satisfy. For more details, see Denault (2001); Hesselager and Anderson (2002); Kalkbrener (2005). The following Axioms are adapted from Denault (2001).

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The **allocation** problem described above has received considerable attention in the lit- erature. Tasche (1999) considers **allocation** of **risk** **capital** to financial instruments in a portfolio, and argues that the only “appropriate” way to allocate **risk** **capital** for per- formance measurement purposes is to determine the marginal **risk** contribution of each investment. The marginal **risk** contribution is defined as the derivative of the aggre- gate **risk** **capital** with respect to the weight of the financial instrument in the portfolio. Denault (2001) instead proposes a game-theoretic approach to determine **risk** **capital** allocations for companies with multiple business divisions. He focuses on **risk** **capital** allocations that are ”fair” in the sense that no **set** of divisions is allocated more **risk** **capital** than the amount of **risk** **capital** that they would need to withhold if they were on their own. He shows that when business divisions are infinitely divisible, the only **allocation** that satisfies this fairness condition is the marginal **risk** contribution defined above. In game-theoretic terms, this **allocation** is referred to as the Aumann-Shapley value. For the special case where the **risk** measure is Expected Shortfall, the correspond- ing **allocation** rule is also referred to as Conditional Tail Expectation (CTE) rule (see, e.g., Overbeck 2000, Panjer 2002, and Dhaene et al. 2008, 2009).

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realized return 8 and not the last period’s TVPI (Berk and DeMarzo 2007). The geometric return is often called the time-weighted return because it values the timing of returns in the same way as the IRR. This measure considers only returns for one single unit invested, e.g., one stock or one dollar, and the gain or loss on that unit. This measuring method eliminates the effect of the size on cash flows from an individual investment, which also is **valued** by the IRR, but it serves as a better measure of return than the annualized TVPI for comparison with other asset types, such as public stocks (Bodie, Kane et al. 2009). The annual standard deviation for the sample is 51.2%. In the same period, the S&P 500 had an annual geometric average return of 15.9% and a SD of 12.3%. Chiampou and Kallett also find the standard deviation as an inappropriate measure for **risk**. One reason for this is due to the illiquidity of the market and the characteristics of a typical successful VC fund (Chiampou and Kallett 1989). For a typical successful VC fund, the internal rate of return can be illustrated by a J-curve (figure 5) because write-offs and management fees are acknowledged immediately in the portfolio, but the value of the firms in the portfolio (NAV’s) is **valued** at cost until a new round of financing (Berg-Utby 2010).

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Such a mapping is called a **capital** **allocation** principle. The relation (3) is sometimes called the full **allocation** property [27] since all of the overall **risk** cap- ital ρ(S ) (not more, not less) is allocated to the investment possibilities; [27] con- sider this property to be an integral part of the definition of an **allocation** principle. Given that a **capital** **allocation** can be carried out in a countless number of ways, additional criteria must be **set** up in order to determine the most suitable form of determining the mapping. A reasonable start is to require the allocated **capital** amounts K i to be “close” to their corresponding losses X i in some appro-

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To measure the signal of reward-to-**risk**, we develop the “Exit-to-Failure Ratio” (EXF), which is the ratio of the average dollars gained in the form of “net money-out” (gross realizations less investment) from exits, to those lost to failures or investments that do not achieve an exit, computed over rolling 12 month windows for each industry. EXF can intuitively be interpreted as the number of failures, on average, that are recuperated by an IPO or M&A exit. In comparison to alternative **measures** used in prior studies to assess the strength of the exit market (e.g., the volume of IPOs, or stock market performance), EXF captures both the upside potential and the downside **risk** of investments in addition to accounting for both forms of VC exit. Across the six major industries in our sample, nearly four failures on average are covered by an M&A or IPO exit, but this fluctuates from zero to over 12 in some periods. When the reward-to-**risk** signal (i.e., EXF) is perceived to be high (low), recent exits signal a more forgiving environment wherein VCs face a smaller (larger) penalty for mistakes, potentially leading to greater (lower) **risk** taking by VCs.

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It was shown by Mirman and Tauman (1982) that the Aumann-Shapley value yields equilibrium prices in a monopolistic production economy. Mo- tivated by this work, we generalise the example to a case where the different insurers choose the extent of coverage received from the pool, by expected utility maximization. This **set**-up is quite different from the equilibrium models usually found in the literature on **risk** sharing, for example Borch (1962), B¨ uhlmann (1980), Taylor (1995), Aase (2002). In these papers, mar- ket prices are obtained via a clearing condition, which is not applicable to the problem that we discuss. Finally, we provide a simple numerical example, where the pool offers stop-loss protection to the participating insurers.

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years and even allowed foreign-funded banks operating directly in China, there are limited data on Chinese banks' ratings. In fact, even for individual ratings--Moody's BFSRs, Moody only provides ratings services for 14 Chinese banks. Therefore, the sample of this paper focuses on the observations of Chinese commercial banks in the last decade. During this period, radical banking reform has been carried out in China, including establishing joint-venture units, initial public offerings of major state-owned banks, and full access of foreign financial institutions; therefore, the data **set** of this paper covers the period of the last decade from 2000 to 2011. In order to study the impact of state ownership on bank **risk**-taking behaviors, we use the dummy variable to measure whether government can control banks' operation or not. Therefore, the dummy variable is equal to 1 if the share of state and local government held is larger than 50%, and equal to zero otherwise. Similarly, we can also generate a dummy variable to denote foreign ownership. Different from government ownership, the dummy of foreign ownership is equal to one if only if foreign banks hold 100% of shares or a local subsidiary is owned by the foreign banks.

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This contribution relates to the use of **risk** **measures** for determin- ing (re)insurers’ economic **capital** requirements. Alternative sets of properties of **risk** **measures** are discussed. Furthermore, methods for constructing **risk** **measures** via indifference arguments, representation results and re-weighting of probability distributions are presented. It is shown how these different approaches relate to popular **risk** **measures**, such as VaR, Expected Shortfall, distortion **risk** **measures** and the ex- ponential premium principle. The problem of allocating aggregate eco- nomic **capital** to sub-portfolios (e.g. insurers’ lines of business) is then considered, with particular emphasis on marginal-cost-type methods. The relationship between insurance pricing and **capital** **allocation** is briefly discussed, based on concepts such as the opportunity and fric- tional costs of **capital** and the impact of the potential of default on insurance rates.

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ρ ( X i ) . It is the indicator to judge **allocation** principles used in Van Gerwald et al (2012). It shows that proportional rule may be the best, if we apply reduction indicator and RAROC indicator; but it doesn’t satisfy some properties required. As we know that, proportional rule (standard error) is not suitable for an **allocation** rule because of its flaws. As it is pointed, when the **risk** **capital** **allocation** is negative, the performance valuation can be meaningless. So we replace the value with a “*”, when there is a negative **risk** **capital**. The EBA based on two kinds of coherent **risk** **measures** may be the best. From the reduction indicator, EBA based on ES is a little higher than EBA based on IE. However, from the RAROC indicator, EBA based on IE is a little higher than EBA based on ES. With one word, combining the advantages of iso-entropic **risk** measure and EBA, the EBA based on IE is strongly recommended.

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A weaker requirement on **risk** **measures** than positive homogeneity / subadditivity is convexity, proposed by Deprez and Gerber (1985), who in- troduce convex **risk** **measures** and study them in the context of optimal **risk** exchanges. While convexity still acknowledges diversification, **risk** **capital** is no more scale-independent – in fact it is increasing per unit of exposure. Moreover, it is possible that for some portfolios pooling increases aggregate **risk**. Convex **risk** **measures** were introduced in the mathematical finance literature by F¨ ollmer and Schied (2002) and Frittelli and Rosazza Gianin (2002) and spurred a lively research area including dynamic generalisations (Detlefsen and Scandolo, 2005).

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In Chapter 2, Chapter 3 and Chapter 4, we will propose two **risk** **measures** and a new premium principle to price reinsurance contracts. One of the two **risk** mea- sures is a **risk** measure satisfying the property of tail-subadditivity, and the other is derived based on the weighted loss functions. In the literature, Young (2004) concluded three methods to attain premium principles, the “ad hoc method”, the “characterization method” and the “economic method”. If an actuary checks the desirable properties for a new premium principle, it is called the “ad hoc method”. Sometimes, they might firstly list the axioms for the potential premium princi- ple and then derive the principle following these axioms by the “characterization method”. Moreover, particular economic theory could be applied to determine premium principles and it is defined as the “economic method”. In fact, most of the premium principles are not derived by only one method, and we can combine these three methods together to find new premium principles.

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The problem of allocating **capital** to pooled portfolios of risky positions was studied for the case of **risk** **measures** based on distorted probabilities and previously obtained results were extended. The non-atomic core was defined as a generalisation of the fuzzy core in order to account for the potential formation of non-linear portfolios. Using the correlation order discussed in Dhaene and Goovaerts (1996) it was then shown that the **capital** **allocation** methodology derived by Tsanakas and Barnett (2002) is consistent with the non-atomic core property defined in this paper. Furthermore, correlation order gave us the means to formulate explicitly the effect of dependence on the **capital** allocated to a portfolio. Specifically, for two portfolios whose payoffs are equal in distribution, more **capital** is allocated to the one which is more correlated to the aggregate **risk** that its holder is exposed to.

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The **capital** **allocation** problem is that once the total **risk** **capital** for a multi-unit financial insti- tution (or a combination of different portfolios) is computed based on a specific **risk** measure, the **risk** **capital** must be allocated back to each business unit (or portfolio) in a consistent way which should recognize the benefit of diversification. Different **allocation** methods are avail- able; however, only some of them satisfy the nice properties which have economical meanings, while others do not. Those desirable properties were proposed as axioms of coherent alloca- tion (Denault, 2001) and fair **allocation** (Kim, 2007) (adapted from Valdez and Chernih (2003) which extended Wang’s 4 idea to the elliptical distribution class for **capital** **allocation** problem), respectively. They will be discussed in the next section.

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