Some remarks on capital allocation by percentile layer

Abstract

Capital allocation by percentile layer is a relatively new method. There is a claim that this method will generate different capital allocations than three other popular methods: CoVaR, Alternative CoVaR and CoTVAR methods. It is also claimed that capital allocation by percentile allocates more capital to catastrophic perils that will cause more severe losses. We study these four methods formally. We show neither of the two claims holds in general. The results of this paper will provide actuaries and other financial risk analysts with valuable insights into capital allocation by percentile layer.

Appendix

In this Appendix, we provide more technical results of the model considered in this paper. We will derive two theorems which allow one to decide when the Percentile Layer method will allocate more capital to the first peril for two different target VaRs. However, our results do not indicate whether it is appropriate to do so. Indeed, we have not been able to find any general rule that can be used for such a decision. We believe that the decision should depend on the specific problem under consideration.

Theorem A.1

Let L ₁ and L ₂ be the loss amounts from two independent perils with 0 < L ₁ < L ₂. Suppose the target VaR is set at VaR(1 − p ₁ p ₂). Then the Percentile Layer method always allocates more capital to the first peril than CoTVaR method does if $0<p_2\leq \frac{1}{2},$ where p _i (i = 1, 2) is the probability that i th peril occurs.

Proof

From Table 2 and third line of Eq. (1) in Sect. 2, we see that the amounts of capital allocated to the first peril by CoTVaR method and the Percentile Layer are $\displaystyle \frac{p_1L_1L_2}{p_1L_1+L_2}$ and $\displaystyle \frac{p_1L_1[(p_1-p_1p_2+p_2)(L_2-L_1)+(L_1+L_2-p_2L_2)]}{(p_1-p_1p_2+p_2)(L_1+L_2)},$ respectively. Consider the ratio

$$\begin{aligned} & \frac{p_1L_1[(p_1-p_1p_2+p_2)(L_2-L_1)+(L_1+L_2-p_2L_2)]}{(p_1-p_1p_2+p_2)(L_1+L_2)}\bigg/\frac{p_1L_1L_2}{p_1L_1+L_2} \\ &= \frac{[(p_1-p_1p_2+p_2)(L_2-L_1)+(L_1+L_2-p_2L_2)](p_1L_1+L_2)}{(p_1-p_1p_2+p_2)L_2(L_1+L_2)}. & \end{aligned}$$

(2)

This ratio is greater than 1 if and only if

$$(p_1-p_1p_2+p_2)(p_1L_1+L_2)(L_2-L_1)+(L_1+L_2-p_2L_2)(p_1L_1+L_2)>\left(L_1L_2+L_2^2\right)(p_1-p_1p_2+p_2),$$

which is equivalent to

$$\begin{aligned} & (p_1-p_1p_2+p_2)(p_1L_1+L_2)(L_2-L_1)+(L_1+L_2-p_2L_2)(p_1L_1+L_2) \\ &\qquad -\left(L_1L_2+L_2^2\right)(p_1-p_1p_2+p_2) \\ &\quad= (p_1-p_1p_2+p_2)L_1(p_1L_2-p_1L_1-2L_2)+[L_1+L_2(1-p_2)](p_1L_1+L_2) \\ &\quad= \left(p_1^2L_1L_2-p_1^2L_1^2-2p_1L_1L_2\right)(1-p_2)+p_1p_2L_1L_2-p_1p_2L_1^2-2p_2L_1L_2 \\ &\qquad +p_1L_1^2+L_1L_2+p_1L_1L_2(1-p_2)+L_2^2(1-p_2) \\ &\quad= p_1^2L_1L_2-p_1^2L_1^2-2p_1L_1L_2-p_1^2p_2L_1L_2+p_1^2p_2L_1^2+2p_1p_2L_1L_2 \\&\qquad +p_1p_2L_1L_2-p_1p_2L_1^2-2p_2L_1L_2+p_1L_1^2+L_1L_2+p_1L_1L_2 \\ &\qquad -p_1p_2L_1L_2+L_2^2-p_2L_2^2 \\ &\quad =p_1^2L_1^2(p_2-1)+p_1L_1^2(1-p_2)+L_2^2(1-p_2) \\ &\qquad +L_1L_2\left(p_1^2-2p_1-p_1^2p_2+2p_1p_2-2p_2+1+p_1\right) \\ &\quad= p_1L_1^2(1-p_1)(1-p_2)+L_2^2(1-p_2)+L_1L_2\left[p_1^2(1-p_2)+(1-p_1)(1-2p_2)\right] \\ &\quad> 0. \\ \end{aligned}$$

Clearly, if $p_2\leq \frac{1}{2},$ then the last line of the equation is positive; hence the ratio in Eq. (2) is greater than 1. This establishes the theorem. □

Theorem A.2

Let L ₁ and L ₂ be the loss amounts from two independent perils with 0 < L ₁ < L ₂ and the target VaR be VaR(1 − p ₂). Then CoTVaR method allocates more capital to the second peril than the Percentile Layer method if p ₁ > p ₂, where p _i (i = 1, 2) is the probability that i th peril occurs.

Proof

We consider the ratio

$$\begin{aligned} &\frac{p_2L_1L_2}{p_1L_1+p_2L_2}\bigg / \frac{p_2L_1[L_2+L_1(1-p_1)]}{(p_1+p_1p_2+p_2)(L_1+L_2)} \\ =& \frac{(p_1-p_1p_2+p_2)L_2(L_1+L_2)}{(p_1L_1+p_2L_2)[(L_1+L_2)-p_1L_1]}. \\ \end{aligned}$$

This ratio is greater than 1 if and only if

$$\begin{aligned} (p_1-p_1p_2+p_2)L_2(L_1+L_2)-(p_1L_1+p_2L_2)[(L_1+L_2)-p_1L_1]>0. & \\ \end{aligned}$$

The left-hand side of this inequality equals

$$\begin{aligned} & (L_1+L_2)[L_2(p_1-p_1p_2+p_2)-p_1L_1-p_2L_2]+p_1L_1(p_1L_1+p_2L_2) \\ &\quad= (L_1+L_2)(p_1L_2-p_1L_1-p_1p_2L_2)+p_1L_1(p_1L_1+p_2L_2) \\ &\quad= p_1[(L_1+L_2)+p_1^2L_1^2+L_1(p_1L_1+p_2L_2)] \\ &\quad= p_1\left[(1-p_2)L_2^2-(1-p_1)L_1^2\right]. \\ \end{aligned}$$

Since L ₂ > L ₁, the quantity p ₁[(1 − p ₂)L ₂ ² − (1 − p ₁)L ₁ ²] is positive if p ₁ > p ₂. Thus, the CoTVaR method will allocate more capital to the second peril than the Percentile Layer method if p ₁ > p ₂. This completes the proof. □