2018-04-24 开胃学习数学系列 - Risk Capital Allocation

Bank's trading book

The trading book hierarchy is well defined and stable over time 结构定义良好而稳定
叶节点是最低级别的book，只包含单独的交易，但不包括book

Diversification benefits

w=∑iwi 是公司的全部投资组合

• wi is the notional vector of the i-th business unit，第i个业务部门的名义本金向量
• wi could represent a single trade in the ultimate granularity wi可以代表单个交易的最终间隔

c(⋅) is a cost function of a portfolio 投资组合的成本函数:

• c(w)=c(∑iwi) is the cost of the whole company 整个公司的成本
• c(wi) is the standalone cost of the i-th business unit of the company 公司第i个业务部门的独立成本
• risk capital is the most important cost functions, such as VaR, IRC

For most risk capital metrics: c(∑iwi)<∑ic(wi)

• because of the diversification/hedging benefits 多元化/对冲的好处
• total diversification benefits is therefore ∑ic(wi)−c(∑iwi) 总体分散后收益是Σic（wi）-c（Σiwi）

Allocation problem

How to divide the firm's total c(w) to individual business in an additive manner?
如何用累加方式将公司的总c（w）划分为单个业务？

• allocated cost ξi: c(∑iwi)=∑iξi by definition.
  分配成本ξi：c（Σiwi）=Σiξi
• The core of the allocation problem is the fair distribution of diversification benefits
  分配问题的核心是多样化收益的公平分配
• c(wi)−ξi is the diversification benefit allocated to the i-th business unit 分配给第i个业务单元的分散化收益
• In practice, allocation runs all the way down to individual trades
  在实践中，分配一直贯穿到单笔交易

Importance of allocation

• Business performance is measured by return on capital (ROC)
  业绩表现用资本回报率（ROC）衡量
• ROC is computed using allocated capital ξi, not the standalone capital c(wi)
  ROC使用分配资本ξi计算，而不是独立资本c（wi）
• certain business is only viable as part of a bank as ξi 某些项目只有在作为银行的一部分时，是可行的，因为ξi
• Business incentive is a critical consideration.
• allocated capital is actively managed by business heads分配资金由负责人主动管理
• allocation method directly affects trading desks' behaviour分配方法直接影响trading desk的行为
• allocation method should incentivize risk reduction 分配方法应该激励降低风险
• a business unit's action to reduce the firm's overall risk should also reduce its own allocation
  一个业务部门为降低公司整体风险的行动，也会减少自己的分配？？

Replacement cost

We use replacement cost as an example to illustrate allocation methods:

rc(w)=max(wTv,0)

• v is the PV vector of all tradable instruments (per unit notional)，所有可交易工具的PV矢量

• wTv is the portfolio's total PV，投资组合的总PV

• Replacement cost is the aggregated PV of the portfolio against a particular counterparty, floored at 0，重置成本是投资组合对特定交易对手的合计PV，0为底

• part of the Basel 3 leveraged balance sheet capital
  巴塞尔协议3杠杆资产负债表的一部分

• it is a measure of MtM loss if the counterparty defaults
  如果交易对手违约，这是衡量MtM损失的指标，这句话不理解
  重置成本是投资组合对特定交易对手的合计PV，位于0

A simple example

A bank has only three trade against a counterparty, their PVs are:
a=12,b=24,c=−24

The bank's total replacement cost is therefore:
rc(a+b+c)=max(12+24−24,0)=12

Diversification benefits:
rc(a)+rc(b)+rc(c)−rc(a+b+c)=12+24+0−12=24

Standalone allocation

The simplest allocation strategy, allocation proportional to the standalone cost:

• commonly used in practice
• not intrinsically additive, need the ad hoc scaling factor
• gives wrong business incentives (more on this later)

A model for fairness

In cooperative game theory, fairness is modeled by the following axioms:
在合作博弈论中，公平可以用以下公理axioms建模：

• efficiency: allocations sum up to total 分配加总 至 总和
• symmetry: if two units' contribution are identical to any arbitrary subsets of units, their allocations are the same
  对称性：如果两个单位的贡献与任意任意单位子集相同，他们的分配是相同的 (这句话不了解)
• linearity: allocation of sums equals to the sum of allocations
  线性：总和的分配等于分配总和(这句话不了解)
• null: if a unit has no contribution to any subsets, it has 0 allocation 如果一个单位对任何子集没有贡献，则它有0个分配

Shapley allocation is the only fair allocation method:

A unit's allocation equals to the average of its marginal contributions over all possible permutations 一个单位的分配等于所有可能排列的边际贡献的平均值
Developed in 1960s, Sir L. Shapley received the 2012 Nobel prize.

Shapley allocation example

• RC就是，相加后和0比较
• 第二列是累积RC，所以像第二行，算出来负数，第三个数要继续计算
• 留意一开始permutation，第三行把位置复原了
• 我不知道意义何在，但是就是这么计算了