#  Risk contributions of lambda quantiles 111History: Earlier versions have
been presented at Birkbeck, University of London PhD Jamboree, the
Mathematical and Statistical Methods for Actuarial Sciences and Finance 2020
Conference (eMAF) (virtual), and the 10th General Advanced Mathematical
Methods for Finance (AMaMeF) 2021 Conference (virtual).

A. INCE∗,∤ ∤Corresponding author. Email<EMAIL_ADDRESS>I. PERI∗,†
and S. PESENTI‡
${\dagger}$Email<EMAIL_ADDRESS><EMAIL_ADDRESS>∗Department of Economics, Mathematics and
Statistics, Birkbeck, University of London, Malet Street, Bloomsbury, London
WC1E 7HX, UK
‡Department of Statistical Sciences, University of Toronto, 700 University
Avenue, Toronto, ON M5G 1Z5, Canada

(March 30, 2022)

###### Abstract

Risk contributions of portfolios form an indispensable part of risk adjusted
performance measurement. The risk contribution of a portfolio, e.g., in the
Euler or Aumann-Shapley framework, is given by the partial derivatives of a
risk measure applied to the portfolio profit and loss in direction of the
asset units. For risk measures that are not positively homogeneous of degree
1, however, known capital allocation principles do not apply. We study the
class of lambda quantile risk measures that includes the well-known Value-at-
Risk as a special case but for which no known allocation rule is applicable.
We prove differentiability and derive explicit formulae of the derivatives of
lambda quantiles with respect to their portfolio composition, that is their
risk contribution. For this purpose, we define lambda quantiles on the space
of portfolio compositions and consider generic (also non-linear) portfolio
operators.

We further derive the Euler decomposition of lambda quantiles for generic
portfolios and show that lambda quantiles are homogeneous in the space of
portfolio compositions, with a homogeneity degree that depends on the
portfolio composition and the lambda function. This result is in stark
contrast to the positive homogeneity properties of risk measures defined on
the space of random variables which admit a constant homogeneity degree. We
introduce a generalised version of Euler contributions and Euler allocation
rule, which are compatible with risk measures of any homogeneity degree and
non-linear but homogeneous portfolios. These concepts are illustrated by a
non-linear portfolio using financial market data.

###### keywords:

Lambda Quantiles; Capital Allocation; Risk Contribution; Lambda Value-at-Risk;
Euler Allocation

##  1 Introduction

Calculating firm-wide or portfolio-level risk is at the heart of modern
financial risk management. Financial institutions use risk measures to
determine economic capital, that is a capital buffer to absorb unexpected
losses during adverse market scenarios and to preserve solvency. However,
understanding how firm-wide or portfolio-level risk are formed and affected by
their respective constituents is of equal importance to risk management
processes. Determining contributions of assets or sub-portfolios to the
overall portfolio risk, or contributions of business lines to the firm-wide
risk enables market practitioners to make informed decisions on capital
allocations to protect each business line’s profitability and secure its
solvency.

In this paper we focus on lambda quantile risk measures, a class of law-
invariant risk measures that generalises the well-known risk measure Value-at-
Risk (VaR). Lambda quantiles were first proposed by Frittelli et al. (2014) to
overcome two of the most criticised aspects of VaR. First, VaR’s inability to
distinguish between different tail behaviours and second its failure to
capture extreme losses. Indeed, lambda quantiles have the ability to (a)
penalise heavy-tailed (portfolio) distributions and (b) identify extreme
losses dynamically, e.g., by recalibrating the lambda function of lambda
quantiles, see Hitaj et al. (2018). The key difference between VaR and a
lambda quantile is, that while $VaR_{\lambda}$ is the negative of a quantile
function at fixed level $\lambda$, a lambda quantile is the negative of a
generalised quantile at a level determined by a function – the so-called
lambda function.

Throughout, we consider generic, not necessarily linear, portfolio operators,
that are collections of linear and/or non-linear assets where both long and
short positions are permitted. Here, we consider portfolios consisting of a
random vector of asset profits and losses and a portfolio composition, a
vector containing the number of units of each asset. To calculate risk
contributions, we define lambda quantiles on the space of portfolio
compositions, a subset of $\mathbb{R}^{n}$, instead of the portfolio profit
and loss, the space of random variables. Using this novel domain for lambda
quantiles, we study how the portfolio’s risk – the lambda quantile of the
portfolio – is affected by changes in its composition. Specifically, we
address the question of what each asset’s contribution is to the overall
portfolio risk. These risk contributions quantify the extent of change in
portfolio risk due to changes in an asset’s exposure; an important metric in
portfolio rebalancing.

Lambda quantiles are the subject of extensive study in Burzoni et al. (2017),
Hitaj et al. (2018), and Corbetta and Peri (2018), where lambda quantiles are
referred to as Lambda Value-at-Risk. When defined on the set of probability
measures, lambda quantiles possess the properties of monotonicity and quasi-
convexity (Frittelli et al., 2014). Burzoni et al. (2017) study robustness,
elicitability, and consistency properties of lambda quantiles. A theoretical
framework for backtesting lambda quantiles is provided in Corbetta and Peri
(2018), who propose three backtesting methodologies. Moreover, Hitaj et al.
(2018) argue to estimate the lambda function of lambda quantiles using major
stock market indices, such as S&P500, FTSE100, and EURO STOXX 50, which
provides a dynamic macro approach to measuring market risk. The axiomatisation
and further properties of lambda quantiles are studied in Bellini and Peri
(2020). These previous studies on lambda quantiles have either defined lambda
quantiles on the space of probability measures or on the space of almost
surely finite random variables. For the purpose of risk contributions,
however, we define lambda quantiles on subsets of $\mathbb{R}^{n}$; the domain
of asset compositions of a portfolio. Defining lambda quantiles on the space
of portfolio compositions provides a natural way of comparing rates of change
in portfolio risk with respect to asset units. Understanding changes in
portfolio risk that may arise from portfolio rebalancing is highly important
from a performance measurement perspective and relevant for risk capital
allocation.

There exist a plethora of risk capital allocation methods that firms use for
risk management and performance measurement, see Balog et al. (2017) for a
review and comparison of risk capital allocation methods and their properties.
It should be noted that not all capital allocation methods are compatible with
a specific risk measure, and applicability is determined by the properties of
the risk measure in question. For example, the axiomatic approach taken in
Denault (2001) to define a coherent risk capital allocation principle derived
from the Aumann-Shapley value (Aumann and Shapley, 1974) only applies to
coherent risk measures (Artzner et al., 1999); a property lambda quantiles do
not possess. Furthermore, the Aumann-Shapley capital allocation rule
introduced in Tsanakas (2009), which was also inspired from the Aumann-Shapley
value, is defined for Gateaux differentiable risk measures on linear
portfolios. Explicit formulae of the Aumann-Shapley allocation rule is
provided, for the class of convex risk measures, in Tsanakas (2009). For
positive homogeneous (but not necessarily coherent) risk measures, the Euler
capital allocation (Patrik et al., 1999; Tasche, 1999; Denault, 2001; Tasche,
2007) can be used, which, on the space of coherent risk measures, coincides
with the Aumann-Shapley allocation. It is worth noting that both Denault
(2001) and Tasche (1999) arrive (for coherent risk measures) at the same
capital allocation rule using different theoretical approaches: the former
uses a game-theoretic approach whilst the latter the notion of risk-adjusted
performance measurement, a common practice for company internal economic
capital calculations. Kalkbrener (2005) provides an axiomatic approach for
sub-additive and homogeneous risk measures. The general class of non-Gateaux
differentiable but convex or quasi-convex risk measures is treated in Centrone
and Gianin (2018). These capital allocation rules are not applicable to this
study since they apply to linear portfolio operators, whereas we treat generic
(not necessarily linear) portfolio operators. While Pesenti et al. (2018)
define Euler allocation rules for non-linear portfolios, they only apply to
positive homogeneous risk measures with homogeneity degree equal to 1. The
homogeneity degree of lambda quantiles, as we show in this paper, is however a
function of the portfolio composition and the lambda function.

In this paper, we define risk contributions of lambda quantiles defined on the
space of portfolio compositions as the partial derivatives of lambda quantiles
with respect to asset units. We derive risk contributions of individual assets
to the overall portfolio risk, measured via the lambda quantile of the
portfolio composition. In doing so, we prove that lambda quantiles are
continuously partially differentiable in the space of portfolio compositions
using two independent methods which assume different properties. Furthermore,
we prove that lambda quantiles are continuously differentiable in smaller
subsets of $\mathbb{R}^{n}$, for a lambda function that may contain
discontinuities, as long as it is continuously differentiable within a
specific interval of $\mathbb{R}$.

Risk contributions calculated as directional derivatives of positive
homogeneous risk measures of degree 1 of portfolios with one unit per asset
are known as Euler contributions, where the assignment of capital using Euler
contributions is known as Euler allocation. We show in this paper that lambda
quantiles, scaled by a factor, can be written as a sum of their partial
derivatives scaled by number of assets. This property is then used to show
that lambda quantiles are homogeneous in the space of portfolio compositions,
with a homogeneity degree that depends on both the portfolio composition and
the lambda function. Only for the special case of a constant lambda function,
the lambda quantile reduces to the VaR and has a homogeneity degree of 1.
Therefore, the Euler allocation rule may not always be applicable to lambda
quantiles, since their homogeneity degree is not universally equal to 1. Due
to the variable nature of lambda quantiles’ homogeneity degrees, we introduce
a generalised Euler capital allocation rule, that is compatible with risk
measures of any homogeneity degree and non-linear but homogeneous portfolios.
We prove that the generalised Euler allocations of lambda quantiles have the
full allocation property. We further provide a financial application using
real market data that illustrates, for the case of non-linear portfolios, the
lambda quantile homogeneity degree as a function of both the lambda function
and the portfolio composition and the generalised Euler contributions of the
portfolio assets. Notice that this notion of variable homogeneity degree is in
favour of some criticisms that positive homogeneous risk measures of degree 1
defined on random variables have received. Föllmer and Schied (2002), for
example, argue that large position multiples may induce additional liquidity
risk, causing the portfolio risk to increase non-linearly compared to position
size.

This paper builds upon methods and results relating to risk contributions and
differentiability of VaR, whose literature is extensive and well established;
indicatively see Tasche (1999), Hallerbach (2003), Hong (2009), Tsanakas and
Millossovich (2016), Saporito and Targino (2020), and Pesenti et al. (2021).
Specifically, the papers of Tasche (1999), Hong (2009), and Tsanakas and
Millossovich (2016) provide a stepping stone for proving differentiability and
calculating risk contributions of lambda quantiles from a portfolio
performance measurement perspective, which are all novel pursuits in risk
measure theory. Indeed, for lambda quantiles to be partially differentiable,
we require additional smoothness assumptions. A first set of assumptions
relates to the invertibility property of the portfolio profit and loss and the
existence of an asset with a continuous density, similar to Tasche (1999). As
this may not always be satisfied for generic portfolio profit and loss, we
further prove our results using the condition that the portfolio profit and
loss possesses a (locally) Lipschitz continuous, akin to the assumptions in
Hong (2009).

The paper is organised as follows: Section 2 introduces the necessary notation
and definitions. In Section 3, we prove continuous partial differentiability
of lambda quantiles in subsets of $\mathbb{R}^{n}$ and derive explicit
formulae of risk contributions of lambda quantiles with respect to portfolio
compositions. In Section 4 we introduce the generalised Euler contributions
and generalised Euler allocation rule and prove that the generalised Euler
contributions of lambda quantiles fulfil the full allocation property. Section
5 is devoted to the study of the homogeneity properties of generic portfolio
operators. Section 6 illustrates the concept of homogeneity degree and risk
contributions of lambda quantiles on a non-linear portfolio using financial
market data.

##  2 Preliminaries

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. We denote by
$\mathcal{X}$ the set of random variables and by $\mathcal{X}^{n}$ for $n\geq
2$ the set of random vectors on that space, taking values in $\mathbb{R}$ and
$\mathbb{R}^{n}$ respectively. The joint probability distribution function of
$\bm{X}=(X_{1},\dots,X_{n})\in\mathcal{X}^{n}$ is represented by
$F_{\bm{X}}(\bm{x}):=\mathbb{P}(\bm{X}\leq\bm{x})$ for all
$\bm{x}\in\mathbb{R}^{n}$, where each $X_{1},\dots,X_{n}\in\mathcal{X}$. We
will use $\bm{X}_{-1}:=(X_{2},\dots,X_{n})\in\mathcal{X}^{n-1}$ and
$\bm{x}_{-1}:=(x_{2},\dots,x_{n})\in\mathbb{R}^{n-1}$ to indicate,
respectively, random and ordinary vectors with first components removed.
Define $\phi$ to be the density of the conditional probability distribution of
$X_{1}$ given $X_{2}=x_{2}\,\dots,X_{n}=x_{n}$. Also,
$U\subset\mathbb{R}\setminus\\{0\\}\times\mathbb{R}^{n-1}$ is a bounded set of
$n$-dimensional real vectors with at least one non-zero component, which we
set w.l.o.g. to the first component. Note that the choice of the first
component is arbitrary and $\phi$ could represent the density of the
conditional distribution of $X_{i}$ given
$X_{1}=x_{1},\dots,X_{i-1}=x_{i-1},X_{i+1}=x_{i+1},\dots,X_{n}=x_{n}$ for any
$i=1,\dots,n$, provided that the $i^{\text{th}}$ component of $U$ does not
contain zero.

In this paper, we treat a portfolio of $n$ assets. Random profits and losses
of assets are represented by $\bm{X}$ and the portfolio composition is given
by $\bm{u}\in U$.

######  Definition 1.

A mapping $g:U\times\mathcal{X}^{n}\rightarrow\mathcal{X}$ is called a
_portfolio operator_.

For fixed $\bm{X}\in\mathcal{X}^{n}$, we call the mapping
$g_{\bm{X}}:U\rightarrow\mathcal{X}$ such that
$g_{\bm{X}}(\bm{u})=g[\bm{u},\bm{X}]$ the _portfolio as a function of the
composition $\bm{u}$_ or _portfolio_ for short.

Finally, if the random vector $\bm{X}$ is realised, i.e.
$\bm{X}(\omega)=\bm{x}\in\mathbb{R}^{n}$ for some outcome $\omega\in\Omega$,
then we denote the portfolio using the mapping
$g_{\bm{x}}:U\rightarrow\mathbb{R}$ such that
$g_{\bm{x}}(\bm{u})=g[\bm{u},\bm{X}(\omega)]$, and call it the _realised
portfolio_.

A portfolio operator $g:U\times\mathcal{X}^{n}\rightarrow\mathcal{X}$ may
represent the mapping from a composition $\bm{u}$ and a profit and loss vector
$\bm{X}$ to the overall random portfolio’s profit and loss. For fixed
$\bm{X}\in\mathcal{X}^{n}$, $g_{\bm{X}}(\bm{u})$ can then be viewed as the
_portfolio profit and loss_. Note, that we do not require a portfolio to be
linear in $\bm{X}$, indeed, the main focus of this paper is on non-linear
portfolios. Finally, if the random vector $\bm{u}$ is realised, i.e.
$\bm{X}(\omega)=\bm{x}\in\mathbb{R}^{n}$ for some outcome $\omega\in\Omega$,
then $g_{\bm{x}}(\bm{u})=g[\bm{u},\bm{X}(\omega)]$, is the _realised portfolio
profit and loss_.

The portfolio operator $g$ is subject to stochastic variability because the
value taken by $\bm{X}$ at each outcome $\omega\in\Omega$ is random and we
assume that $g$ is independent of the probability distribution $F_{\bm{X}}$.
The portfolio operator is also subject to distributional variability because
we consider all random vectors in $\mathcal{X}^{n}$ – we are not restricted to
a class of random vectors of a specific distribution. If the random vector
$\bm{X}$, and hence its joint probability distribution $F_{\bm{X}}$, is fixed,
then the portfolio operator is only subject to stochastic variability and we
consider the portfolio $g_{\bm{X}}$. Note that $g_{\bm{X}}(\bm{u})$, for any
$\bm{u}\in U$, is a random variable, because $\bm{X}$ has not been realised.
Moreover, $g_{\bm{X}}(\bm{u})$ varies (deterministically) with the dynamics of
portfolio composition $\bm{u}$. As we must distinguish between the joint
probability distribution $F_{\bm{X}}$ and the probability distribution
function of the portfolio $Y:=g_{\bm{X}}(\bm{u})$, we denote the probability
distribution and density functions of the portfolio $Y$ by
$F_{Y}(y)=\mathbb{P}(g_{\bm{X}}(\bm{u})\leq y)$ and $f_{Y}(y)=dF_{Y}/dy$,
respectively, for all $y\in\mathbb{R}$.

The following example shows the conceptual difference between the portfolio
operator $g$ and the portfolio $g_{\bm{X}}$.

######  Example 1.

Let $\bm{X}=(X_{1},X_{2})$ and $\bm{u}=(u_{1},u_{2})$. Consider the portfolio
operator:

$g[\bm{u},\bm{X}]=u_{1}X_{1}+u_{2}X_{2}-\mathbb{E}[u_{1}X_{1}+u_{2}X_{2}].$

This operator represents the difference between actual and expected profits
and losses of a portfolio, or in other words, the unexpected profit and loss.
Even though the portfolio $g_{\bm{X}}$ has the same form as the operator $g$,
they are fundamentally different objects and we may choose to write
$g_{\bm{X}}$ as:

$g_{\bm{X}}(\bm{u})=u_{1}X_{1}+u_{2}X_{2}-\mu_{Y},$

where $\mu_{Y}$ is the mean of the random variable $Y:=u_{1}X_{1}+u_{2}X_{2}$
with fixed $\bm{X}$. This is because for a fixed composition
$\hat{\bm{u}}:=(\hat{u}_{1},\hat{u}_{2})$ and fixed $\bm{X}$, the distribution
of $\hat{Y}:=\hat{u}_{1}X_{1}+\hat{u}_{2}X_{2}$ is also fixed and therefore
the mean $\mu_{\hat{Y}}$ is a constant. On the other hand, if we do not fix
$\bm{X}$, then the expectation
$\mathbb{E}[\hat{u}_{1}X_{1}+\hat{u}_{2}X_{2}]$, that appears in the operator
$g[\hat{\bm{u}},\bm{X}]$, is a function of $\bm{X}$.

In practice, if the distribution of asset is known, practitioners are
interested in changing the portfolio composition $\bm{u}$ to achieve a higher
risk-adjusted profit and loss for their portfolio. The process of selecting
assets by comparing their expected profits and losses and contribution to
overall portfolio risk is known as risk-adjusted performance measurement. In
order to do this, one must know the per-unit contribution of each asset to the
overall portfolio risk, and, in particular, risk contributions that are
suitable for performance measurement.

In order to measure the risk, we use lambda quantiles that are traditionally
defined on distributions. However, the purpose of this paper is to calculate
the per-unit risk contribution of each asset to the overall portfolio risk.
Hence, as one of the novelties of this paper, we define lambda quantiles on
the set of the portfolio compositions $U$ and calculate partial derivatives of
lambda quantiles with respect to asset units. The partial derivative with
respect to asset units is the only definition of risk contribution that is
suitable for performance measurement (Tasche, 1999).

######  Definition 2.

The _lambda quantile_ $\rho_{\Lambda}:U\rightarrow\mathbb{R}$ with respect to
$g_{\bm{X}}(\bm{u})\in\mathcal{X}$ is defined as follows:

$\rho_{\Lambda}(\bm{u};g_{\bm{X}}):=-\inf\\{y\in\mathbb{R}\,|\,\mathbb{P}(g_{\bm{X}}(\bm{u})\leq
y)>\Lambda(y)\\}\,,$

where $\Lambda:\mathbb{R}\rightarrow[\lambda_{m},\lambda_{M}]$ is bounded such
that $0<\lambda_{m}\leq\lambda_{M}<1$ and referred to as the _lambda
function_.

The requirement that $\lambda_{m}$ and $\lambda_{M}$ are bounded away from 0
and 1, respectively, guarantees that the lambda quantile $\rho_{\Lambda}$
takes only finite values (Frittelli et al., 2014; Burzoni et al., 2017). In
Frittelli et al. (2014) the authors allow for $\lambda_{m}=0$ and
$\lambda_{M}=1$ in their definition of lambda quantiles on the set of
probability measures. However, the authors assume $\lambda_{M}<1$ within a
financial context and provide a discussion on the choices of $\lambda_{m}$
being strictly positive; we refer the reader to Chapter 4 and Remark 19 of
Frittelli et al. (2014) for a detailed discussion on choices of $\lambda_{m}$
and $\lambda_{M}$.

The lambda quantile at $\bm{u}$ is the negative of the smallest intersection
point of the distribution $F_{Y}$ and the lambda function $\Lambda$, provided
they are both continuous. Otherwise, it is the negative of the smallest point
from which the distribution $F_{Y}$ dominates the lambda function. As we work
with profits and losses, we use the right quantiles of a distribution function
and the negative of the right quantile. Left quantiles are typically used when
asset losses are considered, e.g., in an insurance context. The lambda
quantile $\rho_{\Lambda}(\bm{u};g_{\bm{X}})$ represents a positive amount to
be allocated to absorb losses. A positive amount is allocated only if the
profit and loss is negative (or loss is positive), i.e.
$-\rho_{\Lambda}(\bm{u};g_{\bm{X}})<0$, otherwise the risk measure suggests
there is a surplus of money which can be removed from the portfolio and still
ensures its solvency. For the time being, we assume that the lambda function
is bounded and we denote the derivative of the lambda function by
$\Lambda^{\prime}(x):=d\Lambda/dx$. We will, however, assume additional
properties in subsequent sections.

From Definition 2, we observe that the lambda quantile is a generalisation of
the quantile function, in that the lambda quantile is the negative of the
quantile function at a level determined by the lambda function $\Lambda$. If
the lambda function is a constant, i.e. $\Lambda(y)=\lambda\in(0,1)$ for all
$y\in\mathbb{R}$, then the lambda quantile simplifies to the well-known
_Value-at-Risk_ (VaR). In particular, the
$VaR_{\lambda}:U\rightarrow\mathbb{R}$ is given by:

$VaR_{\lambda}(\bm{u};g_{\bm{X}}):=-\inf\\{y\in\mathbb{R}|\mathbb{P}(g_{\bm{X}}(\bm{u})\leq
y)>\lambda\\}.$

Here, we view the VaR at (fixed) level $\lambda\in(0,1)$ as a function of
$\bm{u}$, which is in contrast to the typically definition of VaR as a
function of the random variable $g_{\bm{X}}(\bm{u})$. For simplicity, we write
$\rho_{\Lambda}(\bm{u})$ and $VaR_{\lambda}(\bm{u})$ when there is no
ambiguity on the portfolio $g_{\bm{X}}$.

We now provide examples of portfolio operators which we will use as running
examples. Operators (ii)-(iv) are studied in Major (2018), which we have
adapted to our notation.

######  Example 2.

Let $n=2$, $\bm{X}=(X_{1},X_{2})$, $\bm{u}=(u_{1},u_{2})$ and
$Y:=u_{1}X_{1}+u_{2}X_{2}$. The following are examples of portfolio operators
defined on the Cartesian product $U\times\mathcal{X}^{2}$:

  1. (i)

$g[\bm{u},\bm{X}]=u_{1}X_{1}+u_{2}X_{2}$

  2. (ii)

$g[\bm{u},\bm{X}]=u_{1}X_{1}+u_{2}X_{2}-\mathbb{E}[u_{1}X_{1}+u_{2}X_{2}]$

  3. (iii)

$g[\bm{u},\bm{X}]=\max\\{0,u_{1}X_{1}+u_{2}X_{2}-\mathbb{E}[u_{1}X_{1}+u_{2}X_{2}]\\}$

  4. (iv)

$g[\bm{u},\bm{X}]=u_{1}X_{1}+u_{2}X_{2}-VaR_{\lambda}(\bm{u};Y)$

  5. (v)

$g[\bm{u},\bm{X}]=u_{1}X_{1}+u_{2}X_{2}-\min\\{\max\\{u_{1}X_{1}+u_{2}X_{2}-D,0\\},L\\}$,
with $D,L>0$

  6. (vi)

$g[\bm{u},\bm{X}]=u_{1}^{\tau}X_{1}+u_{2}^{\tau}X_{2}-\rho_{\Lambda}(\bm{u};u_{1}^{\tau}X_{1}+u_{2}^{\tau}X_{2})$,
where ${\tau}\in\mathbb{R}$.

Portfolio operators (iii) and (v) are typical insurance portfolios. In
particular, Example 2 (v) is the loss for an insurance company after
reinsurance on $u_{1}X_{1}+u_{2}X_{2}$ with deductible $D$ and limit $L$.

Both the nature of assets in a portfolio and the context in which these assets
are used determine the operator $g$. For example, $g$ may be determined by the
pricing functions of assets. A simple linear portfolio of stock positions will
only involve profits and losses of these linear assets. However, a more
complex portfolio with both stock positions and stock options will include
option profit and loss that are calculated using an option pricing formula
(for example, the Black–Scholes formula), which is a non-linear function of
the underlying stock’s profit and loss. stock options or credit default swaps
are examples of non-linear assets.

A portfolio operator is referred to as a _linear operator_ if it is a linear
function of asset profits and losses, i.e. for all
$\bm{X},\hat{\bm{X}}\in\mathcal{X}^{n}$ and all $c_{1},c_{2}\in\mathbb{R}$ we
have
$g[\bm{u},c_{1}\bm{X}+c_{2}\hat{\bm{X}}]=c_{1}g[\bm{u},\bm{X}]+c_{2}g[\bm{u},\hat{\bm{X}}]$.
Clearly, only portfolio operators (i) and (ii) in Example 2 are linear. If
there is at least one position which is a non-linear function of an underlying
asset price, i.e. there is at least one non-linear asset, then the portfolio
is _non-linear_ when viewed as a function of the underlying asset price.
Therefore, whether or not the operators given in Example 2 represent linear
portfolios depend on what positions the $X_{i}$’s represent. For example, if
both $X_{1}$ and $X_{2}$ in Example 2 (i) represent profits and losses of
stocks, then this is an example of both a linear portfolio and linear
operator. If, on the other hand, $X_{1}$ is a stock’s profit and loss and
$X_{2}$ is the profit and loss of an option, then whilst the operator and the
portfolio viewed as a function of $\bm{X}$ are linear, the operator and the
portfolio are non-linear when viewed as a function of $X_{1}$ and the profit
and loss of the option’s underlying asset. If not otherwise stated, we assume
that a portfolio is linear if its corresponding portfolio operator is linear.

For the exposition, we require additional notation. Define
$(x_{1},\bm{x}_{-1}):=\bm{x}$, so that the realised portfolio, for all
$\omega\in\Omega$ with $\bm{X}(\omega)=\bm{x}$, becomes:

$g_{\bm{X}(\omega)}(\bm{u})=g_{\bm{x}}(\bm{u})=g_{(x_{1},\bm{x}_{-1})}(\bm{u})\,.$

Further, the partial derivatives of $g_{\bm{x}}$ with respect to $u_{i}$ and
$x_{1}$ are denoted by $\partial_{u_{i}}g_{\bm{x}}(\bm{u}):=\partial
g_{\bm{x}}/\partial u_{i}$ and $\partial_{x_{1}}g_{\bm{x}}(\bm{u}):=\partial
g_{\bm{x}}/\partial x_{1}$, respectively. Similarly, we denote the
$\mathbb{P}$-a.s. partial derivatives of $g_{\bm{X}}$ with respect to $u_{i}$
by $\partial_{u_{i}}g_{\bm{X}}(\bm{u}):=\partial g_{\bm{X}}/\partial u_{i}$.

We write $A_{1}\subset\mathbb{R}$ for the support of the random variable
$X_{1}$, i.e. $A_{1}\subset\mathbb{R}$ is the smallest closed set such that
$\mathbb{P}(X_{1}\in
A_{1})=\mathbb{P}(\\{\omega\in\Omega~{}|~{}X_{1}(\omega)\in A_{1}\\})=1$. We
say that $g_{\bm{X}}$ is _invertible with respect to $X_{1}=x_{1}$, for all
$X_{2}=x_{2},\dots,X_{n}=x_{n}$_, if for all $x_{1}\in A_{1}$, the function
$g_{(x_{1},\bm{x}_{-1})}$ is invertible with respect to $x_{1}$, for all
$x_{2},\dots,x_{n}$. We denote the inverse of $g_{(x_{1},\bm{x}_{-1})}$ by
$l_{(y,\bm{x}_{-1})}:U\rightarrow\mathbb{R}$ such that

$l_{(y,\bm{x}_{-1})}(\bm{u})=x_{1}\iff g_{(x_{1},\bm{x}_{-1})}(\bm{u})=y,$

for all $x_{1}\in A_{1}$, and its partial derivatives with respect to $u_{i}$
and $y$ are respectively
$\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u}):=\partial
l_{(y,\bm{x}_{-1})}/\partial u_{i}$ and
$\partial_{y}l_{(y,\bm{x}_{-1})}(\bm{u}):=\partial
l_{(y,\bm{x}_{-1})}/\partial y$. By Assumption 1 (i) below, the realised
portfolio $g_{\bm{x}}$ is strictly increasing in $x_{1}$ and thus in this case
the inverse $l$ is well-defined.

##  3 Differentiability and risk contributions of lambda quantiles

In this section, we study differentiability of lambda quantiles in the set
$U$. Derivatives of the lambda quantile risk measure have not been studied in
previous literature. Derivatives of the special case, namely the VaR measure,
however, have an extensive literature, see for example, Tasche (1999);
Gourieroux et al. (2000); Hallerbach (2003); Hong (2009); Tsanakas and
Millossovich (2016). Although these studies calculate derivatives of VaR, they
differ in both methods used and assumptions made in their respective settings.

Partial derivatives of risk measures with respect to asset units are crucial
in portfolio risk management as they represent the risk contribution of each
asset to the overall portfolio risk. This definition of risk contributions is
consistent with risk-adjusted performance measurement of portfolios (see
Definition 7 and Lemma 4 in the Appendix 8). This section extends the
literature to include partial derivatives of lambda quantiles and recover
previous results on VaR as special cases. In light of this motivation, this
section has two objectives. The first objective is to provide conditions under
which lambda quantiles are continuously partially differentiable in the set
$U$. The second objective is to calculate these partial derivatives
explicitly. There are several approaches one may follow to achieve the latter,
which ultimately depend on assumptions made regarding the portfolio
$g_{\bm{X}}$, the random vector $\bm{X}$, and the lambda function $\Lambda$.
Partial derivatives of lambda quantiles will be calculated using two different
approaches, each having its own set of assumptions.

In the first approach we generalise the treatment in Tasche (1999), who
calculated partial derivatives of $VaR_{\lambda}$ for linear portfolios, to
lambda quantiles for generic portfolios. We extend this method to take into
account the lambda function (instead of a fixed level $\lambda$) and to cover
non-linear portfolios (by defining lambda quantiles on generic portfolios).
These generalisations require additional assumptions for lambda quantiles to
be continuously differentiable in the set $U$.

In the second approach we utilise the closed-form representation of
probability sensitivities, proposed by Hong (2009). The probability
sensitivity corresponds, in our context, to the partial derivative of the
portfolio’s probability distribution function with respect to asset units.
Note that the two approaches mentioned above allow us to prove the same
property (continuously partially differentiable in $U$) of lambda quantiles.
Furthermore, partial derivatives of lambda quantiles are the same under both
approaches.

######  Assumption 1.

We say that Assumption 1 is satisfied if:

  1. (i)

$g_{(x_{1},\bm{x}_{-1})}(\bm{u})$ is strictly increasing in $x_{1}$, for all
$\bm{u}\in U$.

  2. (ii)

$g_{\bm{X}}(\bm{u})$ is $\mathbb{P}$-a.s. differentiable in $\bm{u}$, for all
$\bm{u}\in U$.

  3. (iii)

For fixed $\bm{x}_{-1}$, the density $y\mapsto\phi(y|\bm{x}_{-1})$ is
continuous in $y$.

  4. (iv)

The function $l_{(y,\bm{x}_{-1})}(\bm{u})$ is continuously differentiable with
respect to $y$ and $\bm{u}$.

  5. (v)

For fixed $\bm{u}$ and all $i=1,\dots,n$, the following maps are uniformly
bounded with respect to $y$:

$\displaystyle
y\mapsto\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})],$
$\displaystyle
y\mapsto\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})].$

  6. (vi)

For $g_{(x_{1},\bm{x}_{-1})}(\bm{u})$ strictly increasing in $x_{1}$, for each
$\bm{u}\in U$, assume:

$\mathbb{E}[\partial_{y}l_{(-\rho_{\Lambda}(\bm{u}),\bm{X}_{-1})}(\bm{u})\phi(l_{(-\rho_{\Lambda}(\bm{u}),\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]>\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u})).$

Note that Assumption 1 (i) implies that $g_{\bm{X}}(\bm{u})$ is
$\mathbb{P}$-a.s. invertible with respect to $X_{1}=x_{1}$, for all
$X_{2}=x_{2},\dots,X_{n}=x_{n}$. Although we assume that
$g_{(x_{1},\bm{x}_{-1})}(\bm{u})$ is strictly increasing in $x_{1}$ in this
paper, the results also hold for the strictly decreasing case, with some sign
changes. The proofs of the decreasing case are similar to those of the
increasing case and thus omitted. If Assumption 1 (i) to (vi) are fulfilled,
then by Lemma 1 below, Assumption 1 (vi) is equivalent to

$f_{Y}(-\rho_{\Lambda}(\bm{u}))>\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))\,.$
(1)

We require Equation (1) to ensure that the portfolio density adjustment (see
Definition 3) is well-defined and positive at the point
$y=-\rho_{\Lambda}(\bm{u})$. The portfolio density adjustment is fundamental
for the homogeneity degree and the risk contributions of lambda quantiles.

######  Example 3.

Here we discuss which of the portfolios in Example 2 fulfil Assumption 1. Note
that the corresponding portfolios of the portfolio operator (i), (ii), (iv),
and (vi) are of the form
$g_{\bm{X}}(\bm{u})=u_{1}^{\tau}X_{1}+u_{2}^{\tau}X_{2}+c$, for a constant
$c\in\mathbb R$ and $\tau\geq 0$. Thus, these portfolio operators satisfy,
under suitable condition on the joint distribution of $\bm{X}$, Assumptions 1.
The portfolio operators (iii) and (v) do not fulfil Assumption 1, however,
they satify Assumption 2 below.

######  Remark 1.

Assumption 1 (iii) and (iv) imply the following mappings are continuous for
fixed $\bm{x}_{-1}$ and $i=1,\dots,n$

$\displaystyle(y,\bm{u})\mapsto\partial_{y}l_{(y,\bm{x}_{-1})}(\bm{u})\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1}),$
$\displaystyle(y,\bm{u})\mapsto\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1}),$

and the following mappings are continuous, for all $i=1,\dots,n$,

$\displaystyle(y,\bm{u})\mapsto\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})],$
$\displaystyle(y,\bm{u})\mapsto\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})].$

The second set of assumptions relates to the probability sensitivity of Hong
(2009), which have been adapted to our setting. In contrast to the approach
taken in Hong (2009), we require Assumption 2 (vi) to account for the lambda
function in our treatment.

######  Assumption 2.

We say that Assumption 2 is satisfied if:

  1. (i)

$g_{\bm{X}}(\bm{u})$ is $\mathbb{P}$-a.s. differentiable in $\bm{u}$, for all
$\bm{u}\in U$.

  2. (ii)

There exists a random variable $m(\bm{X})$ with $\mathbb{E}[m(\bm{X})]<\infty$
such that for all $\bm{u},\bm{v}\in U$:

$|g_{\bm{X}}(\bm{u})-g_{\bm{X}}(\bm{v})|~{}\leq~{}m(\bm{X})\|\bm{u}-\bm{v}\|\quad\mathbb{P}\text{-a.s.}\,,$

where $\|\cdot\|$ denotes the Euclidean norm in $U$.

  3. (iii)

For all $\bm{u}\in U$, the random variable $g_{\bm{X}}(\bm{u})$ has a
continuous density denoted by $f_{Y}(y)$ in a neighbourhood of
$y=-\rho_{\Lambda}(\bm{u})$.

  4. (iv)

For the function $F:\mathbb{R}\times U\rightarrow[0,1]$ defined as
$F(y,\bm{u}):=\mathbb{P}(g_{\bm{X}}(\bm{u})\leq y)$, the partial derivatives
$\partial_{u_{i}}F(y,\bm{u})$ exist and are continuous in $\bm{u}$ and $y$ in
a neighbourhood of $y=-\rho_{\Lambda}(\bm{u})$, for all $i=1,\dots,n$.

  5. (v)

For all $\bm{u}\in U$ and for $i=1,\dots,n$, the following mappings are
continuous at $y=-\rho_{\Lambda}(\bm{u})$:

$y\mapsto\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}(\bm{u})=y].$

  6. (vi)

$f_{Y}(-\rho_{\Lambda}(\bm{u}))>\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))$ for
all $\bm{u}\in U$.

Assumption 2 (iii) implies that each distribution of the random field
$(g_{\bm{X}}(\bm{u}))_{\bm{u}\in U}$ is continuous in a neighbourhood of
$y=-\rho_{\Lambda}(\bm{u})$ for its respective $\bm{u}\in U$.

######  Example 4.

The portfolios corresponding to the portfolio operators in Example 2 (i),
(ii), (iii), and (v) clearly fulfil Assumptions 2 (i) and (ii). The remaining
portfolios in Example 2 do not in general fulfil Assumption 2 (ii), as lambda
quantiles and $VaR$ are not Lipschitz continuous.

Using Assumption 1, we demonstrate Lemmas 1 and 2, which we require to prove
Theorem 1 with condition 1 and Proposition 2. Lemmas 1 and 2 are
generalisations of Lemmas 3.2 and 2.2 in Tasche (2001), respectively, as they
apply to both linear and non-linear portfolios.

######  Lemma 1.

Suppose Assumption 1 (i)-(v) are satisfied. Then the function
$F:\mathbb{R}\times U\rightarrow[0,1]$ defined as
$F(y,\bm{u}):=\mathbb{P}(g_{\bm{X}}(\bm{u})\leq y)$ is partially
differentiable in $y$ and $u_{i}$, for $i=1,\dots,n$. The continuous
derivatives are given by:

$\displaystyle\frac{\partial F}{\partial y}(y,\bm{u})$
$\displaystyle=\mathbb{E}\left[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})\right],$
(2) $\displaystyle\frac{\partial F}{\partial u_{i}}(y,\bm{u})$
$\displaystyle=\mathbb{E}\left[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})\right].$
(3)

###### Proof.

We generalise the approach taken in the proof of Lemma 5.3 in Tasche (1999) to
prove that $F$ is continuously differentiable. Our method applies to a generic
random variable $g_{\bm{X}}(\bm{u})$ whilst the proof provided in Tasche
(1999) applies only to linear portfolios, that is to
$g_{\bm{X}}(\bm{u})=\sum_{i=1}^{n}u_{i}X_{i}$.

We first introduce the following integral using the density $\phi$ of the
conditional distribution of $X_{1}$ given $\bm{X}_{-1}=\bm{x}_{-1}$

$G(y,\bm{u},\bm{x}_{-1}):=\int_{-\infty}^{l_{(y,\bm{x}_{-1})}(\bm{u})}\phi(t|\bm{x}_{-1})dt.$
(4)

Note that $G$ can be written in the following form

$\displaystyle G(y,\bm{u},\bm{x}_{-1})$ $\displaystyle=\mathbb{P}(X_{1}\leq
l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{X}_{-1}=\bm{x}_{-1})$
$\displaystyle=\mathbb{P}(\\{\omega\in\Omega|X_{1}(\omega)\leq
l_{(y,\bm{x}_{-1})}(\bm{u})\\})$
$\displaystyle=\mathbb{P}(\\{\omega\in\Omega|g_{(X_{1}(\omega),\bm{x}_{-1})}(\bm{u})\leq
g_{(l_{(y,\bm{x}_{-1})},\bm{x}_{-1})}(\bm{u})\\})$
$\displaystyle=\mathbb{P}(\\{\omega\in\Omega|g_{(X_{1}(\omega),\bm{x}_{-1})}(\bm{u})\leq
y\\})$ $\displaystyle=\mathbb{P}(g_{\bm{X}}(\bm{u})\leq
y|\bm{X}_{-1}=\bm{x}_{-1}),$

and $F$ can be written in terms of $G$:

$F(y,\bm{u})=\mathbb{E}[\mathbb{P}(g_{\bm{X}}(\bm{u})\leq
y|\bm{X}_{-1})]=\mathbb{E}[G(y,\bm{u},\bm{X}_{-1})].$ (5)

We show that $F$ is continuously differentiable in $y$ and $u_{i}$, for
$i=1,\dots,n$, and that its derivatives can be computed by changing the order
of integration and differentiation on the right-hand side of (5). In order to
do this, we apply Lemma 3 (see Appendix 8) to the function $G:\mathbb{R}\times
U\times\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ to the components $y$ and
$u_{1},\ldots,u_{n}$. For this, we define $S_{y}:=U\times\mathbb{R}^{n-1}$,
$S_{u_{1}}:=U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}\times\mathbb{R}\times\mathbb{R}^{n-1}$,
and $S_{u_{j}}:=U\setminus\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{n-1}$
for $j=2,\dots,n$. Note that we distinguish $u_{1}$ from $u_{2},\ldots,u_{n}$
as $u_{1}$ cannot be zero.

For differentiability in the first component $y$, condition (i) of Lemma 3 is
satisfied as:

$\int_{S_{y}}|G(y,\bm{u},\bm{x}_{-1})|\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,d\bm{u}=\int_{U}\mathbb{E}[|G(y,\bm{u},\bm{X}_{-1})|]\,d\bm{u}=\int_{U}F(y,\bm{u})\,d\bm{u}<\infty.$

The finiteness follows from the observation that for any fixed $\bm{u}\in U$,
$F(y,\bm{u})=F_{Y}(y)$ is the distribution function of $Y=g_{\bm{X}}(\bm{u})$
and that $U$ is bounded. The same condition is satisfied for differentiability
of $u_{1}$ as:

$\displaystyle\int_{S_{u_{1}}}|G(y,\bm{u},\bm{x}_{-1})|\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-1}$
$\displaystyle=\int_{U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}}\mathbb{E}[G(Y,\bm{u},\bm{X}_{-1})]\,d\bm{u}_{-1}$
$\displaystyle=\int_{U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}]}\mathbb{E}[F(Y,\bm{u})]\,d\bm{u}_{-1}<\infty,$

where $\bm{u}_{-1}:=(u_{2},\dots,u_{n})\in
U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}$ and
$F_{\bm{X}_{-1},Y}:\mathbb{R}^{n-1}\times\mathbb{R}\rightarrow\mathbb{R}$ is
the distribution function of the joint probability distribution of
$\bm{X}_{-1}$ and $Y$. Similarly, for $u_{2},\dots,u_{n}$ we have:

$\displaystyle\int_{S_{u_{j}}}|G(y,\bm{u},\bm{x}_{-1})|\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-j}$
$\displaystyle=\int_{U\setminus\mathbb{R}}\mathbb{E}[G(Y,\bm{u},\bm{X}_{-1})]\,d\bm{u}_{-j}$
$\displaystyle=\int_{U\setminus\mathbb{R}}\mathbb{E}[F(Y,\bm{u})]\,d\bm{u}_{-j}<\infty,$

where for $j>1$, $\bm{u}_{-j}:=(u_{1},\dots,u_{j-1},u_{j+1},\dots,u_{n})\in
U\setminus\mathbb{R}$. For condition (ii) of Lemma 3, we differentiate $G$
partially with respect to $y$ and $u_{i}$, $i=1,\dots,n$ using (4):

$\displaystyle\frac{\partial G}{\partial y}(y,\bm{u},\bm{x}_{-1})$
$\displaystyle=\partial_{y}l_{(y,\bm{x}_{-1})}(\bm{u})\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1})\,,$
$\displaystyle\frac{\partial G}{\partial u_{i}}(y,\bm{u},\bm{x}_{-1})$
$\displaystyle=\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1}),$

which are all continuous for fixed $\bm{x}_{-1}$ by Remark 1. For condition
(iii) of Lemma 3, observe that the integral of $\partial G/\partial y$ in the
domain $S_{y}$ is continuous in $y$ by Remark 1:

$\displaystyle\int_{S_{y}}\frac{\partial G}{\partial
y}(y,\bm{u},\bm{x}_{-1})\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,d\bm{u}$
$\displaystyle=\int_{U}\mathbb{E}\biggl{[}\frac{\partial G}{\partial
y}(y,\bm{u},\bm{X}_{-1})\biggl{]}\,d\bm{u}$
$\displaystyle=\int_{U}\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]\,d\bm{u}.$

Similarly, by Remark 1 integral of $\partial G/\partial u_{1}$ in $S_{u_{1}}$
is continuous in $u_{1}$:

$\displaystyle\int_{S_{u_{1}}}\frac{\partial G}{\partial
u_{1}}(y,\bm{u},\bm{x}_{-1})\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-1}$
$\displaystyle=\int_{U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}}\mathbb{E}\biggl{[}\frac{\partial
G}{\partial u_{1}}(Y,\bm{u},\bm{X}_{-1})\biggl{]}\,d\bm{u}_{-1}$
$\displaystyle=\int_{U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}}\mathbb{E}[\partial_{u_{1}}l_{(Y,\bm{X}_{-1})}(\bm{u})\phi(l_{(Y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]\,d\bm{u}_{-1},$

and the integral of $\partial G/\partial u_{j}$ in $S_{u_{j}}$ is continuous
in $u_{j}$ for $j=2,\dots,n$:

$\displaystyle\int_{S_{u_{j}}}\frac{\partial G}{\partial
u_{j}}(y,\bm{u},\bm{x}_{-1})\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-j}$
$\displaystyle=\int_{U\setminus\mathbb{R}}\mathbb{E}\biggl{[}\frac{\partial
G}{\partial u_{j}}(Y,\bm{u},\bm{X}_{-1})\biggl{]}\,d\bm{u}_{-j}$
$\displaystyle=\int_{U\setminus\mathbb{R}}\mathbb{E}[\partial_{u_{j}}l_{(Y,\bm{X}_{-1})}(\bm{u})\phi(l_{(Y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]\,d\bm{u}_{-j}.$

For condition (iv) of Lemma 3, we show that for $\delta>0$, the following
integrals are finite:

$\displaystyle I_{y}:$
$\displaystyle=\int_{S_{y}}\int_{-\delta}^{\delta}\biggl{|}\frac{\partial
G}{\partial
y}(y+\theta,\bm{u},\bm{x}_{-1})\biggl{|}\,d\theta\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,d\bm{u},$
$\displaystyle I_{u_{1}}:$
$\displaystyle=\int_{S_{u_{1}}}\int_{-\delta}^{\delta}\biggl{|}\frac{\partial
G}{\partial
u_{1}}(y,\bm{u}+\theta\bm{e}_{1},\bm{x}_{-1})\biggl{|}\,d\theta\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-1},$
$\displaystyle I_{u_{j}}:$
$\displaystyle=\int_{S_{u_{j}}}\int_{-\delta}^{\delta}\biggl{|}\frac{\partial
G}{\partial
u_{j}}(y,\bm{u}+\theta\bm{e}_{j},\bm{x}_{-1})\biggl{|}\,d\theta\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-j},$

for $j=2,\dots,n$ where $(\bm{e}_{k})_{l}=1$ for $k=l$ and $0$ otherwise.
Recall that $\phi$ is a continuous density by Assumption 1 (iii) and
$\partial_{y}l_{(y,\bm{x}_{-1})}(\bm{u})\geq 0$ because $l_{(y,\bm{x}_{-1})}$
is strictly increasing in $y$. Therefore,

$\frac{\partial G}{\partial
y}(y+\theta,\bm{u},\bm{x}_{-1})=\partial_{y}l_{(y+\theta,\bm{x}_{-1})}(\bm{u})\phi(l_{(y+\theta,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1})\geq
0,$

for all $y\in\mathbb{R}$, $\theta\in(-\delta,\delta)$, $\bm{u}\in U$ and
$\bm{x}_{-1}\in\mathbb{R}^{n-1}$. We can also see the above inequality by
noting that $G$ is the conditional probability distribution of $Y$ given
$\bm{X}_{-1}$, and therefore its partial derivative w.r.t $y$ is a conditional
probability density. Now, observe that:

$\displaystyle I_{y}$
$\displaystyle=\int_{S_{y}}\int_{-\delta}^{\delta}\frac{\partial G}{\partial
y}(y+\theta,\bm{u},\bm{x}_{-1})\,d\theta\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,d\bm{u}$
$\displaystyle=\int_{S_{y}}(G(y+\delta,\bm{u},\bm{x}_{-1})-G(y-\delta,\bm{u},\bm{x}_{-1}))\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,d\bm{u}$
$\displaystyle=\int_{U}\mathbb{E}[G(y+\delta,\bm{u},\bm{X}_{-1})-G(y-\delta,\bm{u},\bm{X}_{-1})]\,d\bm{u}$
$\displaystyle=\int_{U}\biggl{(}F(y+\delta,\bm{u})-F(y-\delta,\bm{u})\biggl{)}\,d\bm{u}<\infty.$

To prove $I_{u_{1}}$ is finite, observe that since the integrand is positive,
we can change the order of integration by Tonelli’s theorem:

$\displaystyle I_{u_{1}}$
$\displaystyle=\int_{-\delta}^{\delta}\int_{S_{u_{1}}}\biggl{|}\frac{\partial
G}{\partial
u_{1}}(y,\bm{u}+\theta\bm{e}_{1},\bm{x}_{-1})\biggl{|}\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-1}\,d\theta$
$\displaystyle=\int_{-\delta}^{\delta}\int_{S_{u_{1}}}\max\biggl{\\{}\frac{\partial
G}{\partial u_{1}}(y,\bm{u}+\theta\bm{e}_{1},\bm{x}_{-1}),-\frac{\partial
G}{\partial
u_{1}}(y,\bm{u}+\theta\bm{e}_{1},\bm{x}_{-1})\biggl{\\}}\,dF_{\bm{X}_{-1},Y}(\bm{x}_{-1},y)\,d\bm{u}_{-1}\,d\theta$
$\displaystyle=\int_{-\delta}^{\delta}\int_{U\setminus\\{\mathbb{R}\setminus\\{0\\}\\}}\mathbb{E}\biggl{[}\mathbb{E}\biggl{[}\max\biggl{\\{}\frac{\partial
G}{\partial u_{1}}(y,\bm{u}+\theta\bm{e}_{1},\bm{X}_{-1}),-\frac{\partial
G}{\partial
u_{1}}(y,\bm{u}+\theta\bm{e}_{1},\bm{X}_{-1})\biggl{\\}}~{}\bigg{|}~{}Y=y\biggl{]}\biggl{]}\,d\bm{u}_{-1}\,d\theta<\infty,$

where we note that by Assumption 1 (v), the conditional expectation is
bounded. The proof of $I_{u_{j}}<\infty$ for $j=2,\dots,n$ follows the same
approach. Therefore, by Lemma 3 we conclude that $F$ is continuously partially
differentiable in $y$ and $u_{i}$ for $i=1,\dots,n$ with derivatives:

$\frac{\partial F}{\partial y}(y,\bm{u})=\mathbb{E}\biggl{[}\frac{\partial
G}{\partial y}(y,\bm{u},\bm{X}_{-1})\biggl{]}\quad\text{ and
}\quad\frac{\partial F}{\partial
u_{i}}(y,\bm{u})=\mathbb{E}\biggl{[}\frac{\partial G}{\partial
u_{i}}(y,\bm{u},\bm{X}_{-1})\biggl{]}.$

∎

######  Remark 2.

Lemma 1 also holds if $g_{(x_{1},\bm{x}_{-1})}$ is strictly decreasing in
$x_{1}$ with a change of sign of the partial derivatives in Equations (2) and
(3). The proof of the decreasing case follows using similar steps as the
increasing case, and noting that for the decreasing case:

$G(y,\bm{u},\bm{x}_{-1})=\mathbb{P}(X_{1}\leq
l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{X}_{-1}=\bm{x}_{-1})=1-\mathbb{P}(g_{\bm{X}}(\bm{u})\leq
y|\bm{X}_{-1}=\bm{x}_{-1}),$

and again $F$ can be written in terms of $G$:

$F(y,\bm{u})=1-\mathbb{E}[G(y,\bm{u},\bm{X}_{-1})].$

Finally, to prove that $I_{y}<\infty$, we note that:

$\biggl{|}\frac{\partial G}{\partial
y}(y+\theta,\bm{u},\bm{x}_{-1})\biggl{|}=-\frac{\partial G}{\partial
y}(y+\theta,\bm{u},\bm{x}_{-1}),$

since $\partial_{y}l_{(y+\theta,\bm{x}_{-1})}(\bm{u})\leq 0$ for all
$y\in\mathbb{R}$, $\theta\in(-\delta,\delta)$, $\bm{u}\in U$, and
$\bm{x}\in\mathbb{R}^{n-1}$.

######  Remark 3.

If Assumption 1 (i)-(v) are satisfied then, for any $\bm{u}\in U$, the random
variable $Y=g_{\bm{X}}(\bm{u})$ has a continuous probability density function
given by:

$f_{Y}(y)=\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})].$

To see this, note that for fixed $\bm{u}\in U$, $F$ and $F_{Y}$ are identical,
i.e. $F(y,\bm{u})=F_{Y}(y)$ for all $y\in\mathbb{R}$. The continuous partial
derivative of $F$ with respect to $y$ is then given in Lemma 1 as:

$\frac{\partial F}{\partial y}(y,\bm{u})=\frac{\partial F_{Y}}{\partial
y}(y)=f_{Y}(y)=\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})].$

Therefore, we see that Assumption 1 (vi) corresponds to the gradient of the
distribution function $F_{Y}$ being greater than that of the lambda function
at the point $y=-\rho_{\Lambda}(\bm{u})$.

The strictly decreasing case follows from the same argument with a change of
sign. Note that the density $f_{Y}(y)$ is indeed positive for the decreasing
case, because $l_{(y,\bm{x}_{-1})}$ is decreasing in $y$ and therefore
$\partial_{y}l_{(y,\bm{x}_{-1})}(\bm{u})\leq 0$ for all $y\in\mathbb{R}$,
$\bm{u}\in U$, and $\bm{x}_{-1}\in\mathbb{R}^{n-1}$.

######  Lemma 2.

If Assumption 1 (i)-(v) are satisfied then, for any $\bm{u}\in U$ and
$i=1,\dots,n$, we have:

$\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}(\bm{u})=y]=-\frac{\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]}{\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]}.$
(6)

###### Proof.

The proof method is inspired by the proof of Lemma 1 in Tsanakas and
Millossovich (2016). Our proof, however, considers a portfolio
$g_{\bm{X}}(\bm{u})$ on the set $U$, whereas Tsanakas and Millossovich (2016)
do not use asset units.

Consider the following expectation for an absolutely integrable function $k$,
i.e. $\int_{\mathbb{R}}|k(y)|\mathrm{d}y<\infty$, and fixed $\bm{u}$:

$\displaystyle\mathbb{E}[k(Y)\partial_{u_{i}}g_{\bm{X}}(\bm{u})]$
$\displaystyle=\mathbb{E}[\mathbb{E}[k(Y)\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}\bm{X}_{-1}]]$
$\displaystyle=\mathbb{E}\biggl{[}\int_{-\infty}^{+\infty}\partial_{u_{i}}g_{(x_{1},\bm{X}_{-1})}(\bm{u})k(g_{(x_{1},\bm{X}_{-1})}(\bm{u}))\phi(x_{1}|\bm{X}_{-1})dx_{1}\biggl{]}.$
(7)

We now apply a change of variable:

$x_{1}=l_{(y,\bm{x}_{-1})}(\bm{u})\iff y=g_{(x_{1},\bm{x}_{-1})}(\bm{u}).$ (8)

For any $\bm{u}\in U$ and $\bm{x}_{-1}\in\mathbb{R}^{n-1}$, we can write (8)
as:

$x_{1}=l_{(g_{(x_{1},\bm{x}_{-1})}(\bm{u}),\bm{x}_{-1})}(\bm{u})\iff
y=g_{(l_{(y,\bm{x}_{-1})}(\bm{u}),\bm{x}_{-1})}(\bm{u}).$ (9)

so that:

$\frac{\mathrm{d}x_{1}}{\mathrm{d}y}=\partial_{y}l_{(y,\bm{x}_{-1})}(\bm{u})|_{y=g_{(x_{1},\bm{x}_{-1})}(\bm{u})}=(\partial_{x_{1}}g_{(x_{1},\bm{x}_{-1})}(\bm{u})|_{x_{1}=l_{(y,\bm{x}_{-1})}(\bm{u})})^{-1}\,,$
(10)

where we used the representation of derivatives of inverse functions. Next, we
compute the partial derivative of the equation
$y=g_{(l_{(y,\bm{x}_{-1})}(\bm{u}),\bm{x}_{-1})}(\bm{u})$ in (9) with respect
to $u_{i}$, $i=1,\dots,n$, and note that the derivatives of the LHS are zero,
i.e. $\partial y/\partial u_{i}=0$ for $i=1,\dots,n$. For the RHS, we have:

$\partial_{u_{i}}g_{(l_{(y,\bm{x}_{-1})}(\bm{u}),\bm{x}_{-1})}(\bm{u})=\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})\partial_{x_{1}}g_{(x_{1},\bm{x}_{-1})}(\bm{u})|_{x_{1}=l_{(y,\bm{x}_{-1})}(\bm{u})}+\partial_{u_{i}}g_{(x_{1},\bm{x}_{-1})}(\bm{u})|_{x_{1}=l_{(y,\bm{x}_{-1})}(\bm{u})}.$

From this we deduce that:

$\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})\partial_{x_{1}}g_{(x_{1},\bm{x}_{-1})}(\bm{u})|_{x_{1}=l_{(y,\bm{x}_{-1})}(\bm{u})}=-\partial_{u_{i}}g_{(x_{1},\bm{x}_{-1})}(\bm{u})|_{x_{1}=l_{(y,\bm{x}_{-1})}(\bm{u})}\,.$
(11)

Using (10) and (11), our expectation in (3) now becomes:

$\displaystyle\mathbb{E}[k(Y)\partial_{u_{i}}g_{\bm{X}}(\bm{u})]$
$\displaystyle=\mathbb{E}\biggl{[}-\int_{-\infty}^{+\infty}k(y)\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})dy\biggl{]}$
$\displaystyle=\mathbb{E}\biggl{[}-\int_{-\infty}^{+\infty}k(y)\frac{\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})}{f_{Y}(y)}f_{Y}(y)dy\biggl{]}$
(12)
$\displaystyle=-\int_{-\infty}^{+\infty}k(y)\frac{\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]}{f_{Y}(y)}f_{Y}(y)dy$
(13) $\displaystyle=\mathbb{E}[k(Y)q(Y)],$

where:

$q(y)=-\frac{\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]}{f_{Y}(y)}.$

Notice that we have switched the order of integration and expectation to move
from (12) to (13). This can be justified by considering the following integral
on the product space $\mathbb{R}\times\mathbb{R}^{n-1}$:

$I_{k}:=\int_{\mathbb{R}\times\mathbb{R}^{n-1}}|k(y)\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1})\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})|\,dy\,dF_{\bm{X}_{-1}}(\bm{x}_{-1}).$
(14)

If the integral (14) is finite then changing the order of integrals in (13) is
justified by Fubini’s theorem. Observe that since the integrand is non-
negative, we can apply Tonelli’s theorem to (14):

$\displaystyle I_{k}$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{n-1}}|k(y)\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1})\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})|\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,dy$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{n-1}}|k(y)||\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1})\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})|\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,dy$
$\displaystyle=\int_{\mathbb{R}}|k(y)|\int_{\mathbb{R}^{n-1}}|\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1})\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})|\,dF_{\bm{X}_{-1}}(\bm{x}_{-1})\,dy$
$\displaystyle=\int_{\mathbb{R}}|k(y)|\biggl{(}\mathbb{E}\biggl{[}\max\biggl{(}\frac{\partial
G}{\partial u_{i}}(y,\bm{u},\bm{X}_{-1}),-\frac{\partial G}{\partial
u_{i}}(y,\bm{u},\bm{X}_{-1})\biggl{)}\biggl{]}\biggl{)}\,dy,$ (15)

where, as in the proof of Lemma 1, we used that:

$\frac{\partial G}{\partial
u_{i}}(y,\bm{u},\bm{x}_{-1})=\partial_{u_{i}}l_{(y,\bm{x}_{-1})}(\bm{u})\phi(l_{(y,\bm{x}_{-1})}(\bm{u})|\bm{x}_{-1}).$

By Assumption 1 (v), the expectation in the integrand of (15) is finite and
since $k$ is absolutely integrable, we conclude that $I_{k}<\infty$. Using the
explicit form of $f_{Y}$ from Remark 3, we conclude that:

$\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=y]=-\frac{\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]}{\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]}.$

∎

######  Remark 4.

Using Lemmas 1 and 2 and Remark 3, one can write the derivative of the
portfolio with respect to its composition as:

$\displaystyle\frac{\partial F}{\partial u_{i}}(y,\bm{u})$
$\displaystyle=\mathbb{E}[\partial_{u_{i}}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]$
$\displaystyle=-\mathbb{E}[\partial_{y}l_{(y,\bm{X}_{-1})}(\bm{u})\phi(l_{(y,\bm{X}_{-1})}(\bm{u})|\bm{X}_{-1})]\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=y]$
$\displaystyle=-f_{Y}(y)\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}(\bm{u})=y].$
(16)

Using Assumption 2, we demonstrate Proposition 1 which proves partial
differentiability of lambda quantiles without assuming $g_{\bm{X}}(\bm{u})$ is
$\mathbb{P}$-a.s. strictly increasing. Instead, we assume that the portfolio
has a continuous density in a neighbourhood of the lambda quantile. Assumption
2 and Proposition 1 are then used to prove Theorem 1 with condition 2.

######  Proposition 1.

Suppose Assumption 2 (iii), (iv) and (vi) are satisfied and $\Lambda$ is
continuously differentiable in a neighbourhood of $-\rho_{\Lambda}(\bm{u})$.
Then, $\rho_{\Lambda}$ is continuously partially differentiable in $U$ with
derivatives:

$\frac{\partial\rho_{\Lambda}}{\partial u_{i}}(\bm{u})=\biggl{(}\frac{\partial
H}{\partial y}(-\rho_{\Lambda}(\bm{u}),\bm{u})\biggl{)}^{-1}\frac{\partial
H}{\partial u_{i}}(-\rho_{\Lambda}(\bm{u}),\bm{u}),$

where $H(y,\bm{u}):=F(y,\bm{u})-\Lambda(y)$.

###### Proof.

Fix $\bm{u}\in U$. Then, $g_{\bm{X}}(\bm{u})$ is $\mathbb{P}$-a.s. a
continuous random variable in a neighbourhood of $-\rho_{\Lambda}(\bm{u})$.
Therefore, it holds that:

$F(-\rho_{\Lambda}(\bm{u}),\bm{u})=\Lambda(-\rho_{\Lambda}(\bm{u})).$

Then, $y=-\rho_{\Lambda}(\bm{u})$ is a solution of $H(y,\bm{u})=0$ for all
$\bm{u}\in U$, i.e. $H(-\rho_{\Lambda}(\bm{u}),\bm{u})=0$ for all $\bm{u}\in
U$. Note that $H$ is continuously partially differentiable in $y$ and $u_{i}$,
$i=1,\dots,n$, since by assumption, both $f_{Y}$ and $\Lambda^{\prime}$ are
continuous in the same neighbourhood of $-\rho_{\Lambda}(\bm{u})$. Also,
observe that:

$\frac{\partial H}{\partial
y}(y,\bm{u})\biggl{|}_{y=-\rho_{\Lambda}(\bm{u})}=f_{Y}(-\rho_{\Lambda}(\bm{u}))-\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))>0$

by Assumption 2 (vi). Applying the implicit function theorem to $H$ and using
Assumption 2 (iv), we conclude that $-\rho_{\Lambda}$ is continuously
partially differentiable in $U$ with derivatives:

$\frac{\partial(-\rho_{\Lambda})}{\partial
u_{i}}(\bm{u})=-\biggl{(}\frac{\partial H}{\partial
y}(-\rho_{\Lambda}(\bm{u}),\bm{u})\biggl{)}^{-1}\frac{\partial H}{\partial
u_{i}}(-\rho_{\Lambda}(\bm{u}),\bm{u}).$

∎

We now define the portfolio density adjustment which is important for both the
risk contributions and Euler decomposition of lambda quantiles. Also, we will
show that the portfolio density adjustment evaluated at the point
$y=-\rho_{\Lambda}(\bm{u})$ corresponds to the homogeneity degree of lambda
quantiles.

######  Definition 3.

For a continuous random variable $Y\in\mathcal{X}$ and continuously
differentiable lambda function, define the _portfolio density adjustment of
$Y$ with respect to $\Lambda$_ as the function
$\eta_{\Lambda,Y}:\mathbb{R}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ given by

$\eta_{\Lambda,Y}(y):=\frac{f_{Y}(y)}{f_{Y}(y)-\Lambda^{\prime}(y)},$ (17)

where we use the convention that $\frac{1}{0}=+\infty$.

######  Remark 5.

Observe that $\eta_{\Lambda,Y}(y)=1$ at a given $y$ if, and only if,
$\Lambda^{\prime}(y)=0$. Also, for a fixed lambda function, the portfolio
density adjustment $\eta_{\Lambda,Y}$ is law invariant with respect to the
random variable $Y$, that is, for random variables $Y_{1},Y_{2}\in\mathcal{X}$
that are equal in distribution, i.e. $Y_{1}\,{\buildrel d\over{=}}\,Y_{2}$, it
holds $\eta_{\Lambda,Y_{1}}(y)=\eta_{\Lambda,Y_{2}}(y)$ for all
$y\in\mathbb{R}$.

The following theorem states the conditions under which lambda quantiles are
continuously partially differentiable in the space of portfolio compositions
and provides closed form formulae of lambda quantile risk contributions. We
prove Theorem 1 using two different approaches, which correspond to the use of
Assumption 1 and Assumption 2. Also, note that the assumptions for the lambda
function are different for each approach.

######  Theorem 1.

Suppose either:

  1. 1.

$\Lambda$ is continuously differentiable on $\mathbb{R}$ and Assumption 1 is
satisfied,
or

  2. 2.

$\Lambda$ is continuously differentiable in a neighbourhood of
$y=-\rho_{\Lambda}(\bm{u})$ and Assumption 2 is satisfied.

Then, $\rho_{\Lambda}$ is continuously partially differentiable in $U$ with
partial derivatives:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=-\rho_{\Lambda}(\bm{u})],$
(18)

for $i=1,\dots,n$.

###### Proof of Theorem 1.

Proof using condition 1. By Lemma 1 and Remark 3, $Y=g_{\bm{X}}(\bm{u})$ is a
continuous random variable with a continuous probability density function
$f_{Y}$ and the partial derivatives $\partial_{u_{i}}F(y,\bm{u})$ are
continuous in $y$ and $\bm{u}$ for $i=1,\dots,n$. Furthermore, with Assumption
1 (vi), we invoke Proposition 1 to deduce that $\rho_{\Lambda}$ is
continuously partially differentiable in $U$ with derivatives:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})=\frac{\partial_{u_{i}}F(-\rho_{\Lambda}(\bm{u}),\bm{u})}{f_{Y}(-\rho_{\Lambda}(\bm{u}))-\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))},$

for $i=1,\dots,n$. By Remark 4, we note that:

$\partial_{u_{i}}F(-\rho_{\Lambda}(\bm{u}),\bm{u})=-f_{Y}(-\rho_{\Lambda}(\bm{u}))\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}(\bm{u})=-\rho_{\Lambda}(\bm{u})],$

which concludes the proof using condition 1, since:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})=-\frac{f_{Y}(-\rho_{\Lambda}(\bm{u}))}{f_{Y}(-\rho_{\Lambda}(\bm{u}))-\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}(\bm{u})=-\rho_{\Lambda}(\bm{u})].$

Proof using condition 2. By Proposition 1, $\rho_{\Lambda}$ is continuously
partially differentiable in $U$ with partial derivatives:

$\frac{\partial\rho_{\Lambda}}{\partial u_{i}}(\bm{u})=\biggl{(}\frac{\partial
H}{\partial y}(-\rho_{\Lambda}(\bm{u}),\bm{u})\biggl{)}^{-1}\frac{\partial
H}{\partial u_{i}}(-\rho_{\Lambda}(\bm{u}),\bm{u}),$

where $H(y,\bm{u}):=F(y,\bm{u})-\Lambda(y)$. By Theorem 1 of Hong (2009), we
have:

$\partial_{u_{i}}F(y,\bm{u})=-f_{Y}(y)\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}(\bm{u})=y],$

for $i=1,\dots,n$, which are continuous in a neighbourhood of
$y=-\rho_{\Lambda}(\bm{u})$ by Assumption 2 (iv). Furthermore, observe that:

$\frac{\partial H}{\partial y}(y,\bm{u})=f_{Y}(y)-\Lambda^{\prime}(y),$

which, again, is continuous in a neighbourhood of $y=-\rho_{\Lambda}(\bm{u})$
by Assumption 2 (iii) and by the assumption that $\Lambda$ is continuously
differentiable in a neighbourhood of $y=-\rho_{\Lambda}(\bm{u})$. We conclude
that the continuous partial derivatives of $\rho_{\Lambda}$ are given by:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})=-\frac{f_{Y}(-\rho_{\Lambda}(\bm{u}))}{f_{Y}(-\rho_{\Lambda}(\bm{u}))-\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}g_{\bm{X}}=-\rho_{\Lambda}(\bm{u})],$

for $i=1,\dots,n$. ∎

Theorem 1 with condition 2 is a generalisation of the quantile sensitivity of
VaR derived in Theorem 2 of Hong (2009) to the class of lambda quantiles.

For the special case of $VaR_{\lambda}$, we observe that the portfolio density
adjustment is equal to one, i.e. $\eta_{\lambda,Y}(y)=1$ for all
$y\in\mathbb{R}$, which leads to the following result.

######  Corollary 1.

Suppose $\Lambda(x)=\lambda\in(0,1)$ for all $x\in\mathbb{R}$. If Assumption 1
or Assumption 2 is satisfied, then $\rho_{\lambda}\equiv VaR_{\lambda}$ is
continuously partially differentiable in $U$ with partial derivatives:

$\frac{\partial VaR_{\lambda}}{\partial
u_{i}}(\bm{u})=-\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=-VaR_{\lambda}(\bm{u})],$
(19)

for $i=1,\dots,n$.

The Proof of Corollary 1 follows straightforwardly from Theorem 1, and is thus
omitted. Corollary 1 with Assumption 1 generalises Lemma 5.3 in Tasche (1999)
to generic portfolios $g_{\bm{X}}(\bm{u})$. Furthermore, even though (19) with
Assumption 1 is of the same form as the partial derivative given in Theorem 2
of Hong (2009) (except that here $\bm{u}$ is multivariate as opposed to one-
dimensional), the assumptions used to obtain these results differ. In Hong
(2009), the simulation output is assumed to be a continuous random variable.
Corollary 1 with Assumption 1, does not require this assumption, we do,
however, assume that at least one of the $X_{i}$ has a continuous density.

The following example shows that the risk contributions of $VaR_{\lambda}$ in
Tasche (1999), who considers linear portfolios, are a special case of those of
the lambda quantiles.

######  Example 5.

For the linear portfolio operator given in Example 2 (i), we fix the random
vector $\bm{X}$ to obtain the portfolio:

$g_{\bm{X}}(\bm{u}):=u_{1}X_{1}+u_{2}X_{2}.$

Then the lambda quantiles’ risk contribution of asset $i$ to the portfolio is
given by:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\mathbb{E}[X_{i}|u_{1}X_{1}+u_{2}X_{2}=-\rho_{\Lambda}(\bm{u})]\,,$

where $Y=u_{1}X_{1}+u_{2}X_{2}$ as given in Example 2 (i). If
$\Lambda(x)=\lambda\in(0,1)$ is a constant, then we retrieve partial
derivatives of $VaR_{\lambda}$ as obtained in Tasche (1999), Gourieroux et al.
(2000), and Hallerbach (2003):

$\frac{\partial VaR_{\lambda}}{\partial
u_{i}}(\bm{u})=-\mathbb{E}[X_{i}|u_{1}X_{1}+u_{2}X_{2}=-VaR_{\lambda}(\bm{u})].$

######  Example 6.

Consider the portfolio operator in Example 2 (vi) with $\tau=1$ such that
$g_{\bm{X}}$ is given by:

$g_{\bm{X}}(\bm{u})=u_{1}X_{1}+u_{2}X_{2}-\rho_{\Lambda}(\bm{u};Y),$

where $Y=u_{1}X_{1}+u_{2}X_{2}$. Then, the lambda quantile admits the
representation

$\rho_{\Lambda}(\bm{u};g_{\bm{X}})=\rho_{\Gamma}(\bm{u};Y)+\rho_{\Lambda}(\bm{u};Y),$

where $\Gamma(z):=\Lambda(z-\rho_{\Lambda}(\bm{u};Y))$ for all
$z\in\mathbb{R}$. Thus, the risk contributions of the lambda quantile for the
portfolio $g_{\bm{X}}(\bm{u})$, for $i=1,\ldots,n$, become

$\displaystyle\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u};g_{\bm{X}})$
$\displaystyle=\frac{\partial\rho_{\Gamma}}{\partial
u_{i}}(\bm{u};Y)+\frac{\partial\rho_{\Lambda}}{\partial u_{i}}(\bm{u};Y)$
$\displaystyle=-\eta_{\Gamma,Y}(-\rho_{\Gamma}(\bm{u};Y))\mathbb{E}[X_{i}~{}|~{}Y=-\rho_{\Gamma}(\bm{u};Y)]$
$\displaystyle\quad-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u};Y))\mathbb{E}[X_{i}~{}|~{}Y=-\rho_{\Lambda}(\bm{u};Y)].$

So far we proved in Theorem 1 that, under smoothness assumptions, lambda
quantiles are continuously partially differentiable in $U$. Next, we consider
differentiability in subsets of $U$. This is important for situations when
portfolio selection is restricted to specific classes of compositions, or in
other words, to subsets of $U$. In the following proposition, we use
Assumption 1 to prove that lambda quantiles are continuously partially
differentiable in subsets of $U$. This result allows for flexibility in the
choice of lambda function of lambda quantiles. Recall that in Theorem 1 with
condition 1, $\Lambda$ was assumed to be continuously differentiable in
$\mathbb{R}$. In Proposition 2, we only require continuous differentiability
of the lambda function within an interval, thus generalising to lambda
functions that may be discontinuous on $\mathbb{R}$.

Consider a subset $V\subset U$ such that for all $\bm{v}\in V$, the smallest
intersection point of $F$ and $\Lambda$ lies in the interval
$(\alpha,\beta)\subset\mathbb{R}$, i.e.
$-\rho_{\Lambda}(\bm{v})\in(\alpha,\beta)$ for all $\bm{v}\in V$. The
following result provides the necessary conditions to ensure lambda quantiles
are continuously partially differentiable in $V$, and hence allow us to
calculate risk contributions of lambda quantiles in $V$.

######  Proposition 2.

Assume that:

  1. (i)

$\Lambda$ is continuously differentiable in the interval
$(\alpha,\beta)\subset\mathbb{R}$;

  2. (ii)

Assumption 1 is satisfied;

  3. (iii)

$-\rho_{\Lambda}(\bm{v})\in(\alpha,\beta)$ for all $\bm{v}\in V\subset U$.

Then, $\rho_{\Lambda}$ is continuously partially differentiable in $V$, where
the partial derivatives are given by:

$\frac{\partial\rho_{\Lambda}}{\partial
v_{i}}(\bm{v})=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{v}))\mathbb{E}[\partial_{v_{i}}g_{\bm{X}}(\bm{v})~{}|~{}Y=-\rho_{\Lambda}(\bm{v})],$
(20)

for $\bm{v}\in V$ and $i=1,\dots,n$.

###### Proof.

The proof follows the same approach as that of Theorem 1 with condition 1. The
major difference is that we calculate partial derivatives with respect to $y$
in the interval $(\alpha,\beta)$ to ensure $\Lambda^{\prime}(y)$ exists and is
well defined. We further point out that for fixed $\bm{v}\in V$,
$-\rho_{\Lambda}(\bm{v})$ is the smallest intersection point of $F$ and
$\Lambda$ on $(\alpha,\beta)$, since they’re both continuous on this interval,
i.e. we have:

$F(-\rho_{\Lambda}(\bm{v}),\bm{v})=\Lambda(-\rho_{\Lambda}(\bm{v})),$

for all $\bm{v}\in V$. ∎

##  4 Euler decomposition and the generalised Euler allocation rule

In this section, we aggregate the risk contributions of lambda quantiles to
prove a relationship known as the _Euler decomposition_ for lambda quantiles.
The Euler decomposition is, for homogeneous risk measures in $U$ with
homogeneity degree 1, the property that the risk measure, scaled by its
homogeneity degree, can be written as a sum of its partial derivatives scaled
by the number of assets. We show that the homogeneity degree of lambda
quantiles is determined by the portfolio composition, the density function of
the portfolio, and the gradient of the lambda function, both evaluated at the
lambda quantile. This implies that lambda quantile homogeneity degree is not
constant over choices of portfolio compositions or lambda functions.
Furthermore, the homogeneity degree varies across different distributions of
the portfolio. This is in contrast to other risk measures, such as VaR, where
the homogeneity degree is constant.

In risk measure theory, the property of homogeneity is typically studied for
risk measures defined on the space of random variables. A risk measure defined
on random variables is positive homogeneous (of degree 1), if the risk of an
asset scales linearly, e.g. doubling the asset’s units doubles the position’s
risk. The positive homogeneity (of degree 1) property of risk measures forms
part of the definition of coherent risk measures, introduced in the seminal
paper by Artzner et al. (1999). However, the property of a risk measure having
a homogeneity degree of 1 has been questioned in Föllmer and Schied (2002).
They argue that large multiples of a position may introduce additional
liquidity risk and, therefore, the position’s risk and size may not increase
linearly.

In this paper, we study the homogeneity property of lambda quantiles on the
set $U$ and explore the relationship between asset units and portfolio risk
for lambda quantiles. Therefore, our treatment of the homogeneity property
should not be confused with homogeneity of risk measures defined on set of
random variables $\mathcal{X}$.

The Euler decomposition of a positive homogeneous (of degree 1) risk measure,
defined on the set of random variables, is known as the _Euler allocation
rule_ (Patrik et al., 1999; Denault, 2001; Tasche, 2007), which is one of the
most well established allocation methods in risk measure theory. This
allocation rule assigns economic capital to assets using directional
derivatives (in the direction of the asset) of positive homogeneous risk
measures. Furthermore, Euler allocation rule is used for portfolios with
linear risk aggregation or linear portfolio operators in Tasche (2007) and
Tsanakas (2009).

Our treatment considers a more general setup, where we consider a generic
portfolio operator and lambda quantiles, that are risk measures with non-
constant homogeneity degree.

######  Definition 4.

Let $\alpha\colon U\to\mathbb{R}$ be a function. We call a function
$r:U\rightarrow\mathbb{R}$ positively homogeneous of degree $\alpha(\bm{u})$,
if for all $\bm{u}\in U$ and $t>0$ such that $t\bm{u}\in U$, it holds that

$r(t\bm{u})=t^{\alpha(\bm{u})}r(\bm{u})\,.$ (21)

A function that is positively homogeneous of degree $\alpha(\bm{u})$ satisfies
an Euler-like theorem. Indeed, a differentiable function
$r:U\rightarrow\mathbb{R}$ is positively homogeneous of degree
$\alpha(\bm{u})$ if, and only if for all $\bm{u}\in U$ it holds that

$\sum_{i=1}^{n}u_{i}\,\frac{\partial r}{\partial
u_{i}}(\bm{u})=\alpha(\bm{u})r(\bm{u})\,.$ (22)

Note that if $\alpha(\bm{u})=\alpha\in\mathbb R$ is a constant, then we
recover the usual definition of positively homogeneous functions.

######  Definition 5.

Let $\tau\colon U\to\mathbb{R}$ and $E\in\mathcal{F}$ be an event. An operator
$g:U\times\mathcal{X}^{n}\rightarrow\mathcal{X}$ is said to be _$\mathbb{P}$
-almost surely positively homogeneous of degree $\tau(\bm{u})$_ in $U$ and in
the event $E$ if for all $\bm{X}\in\mathcal{X}^{n}$, and all $\bm{u}\in U$ and
$t>0$ with $t\bm{u}\in U$, we have:

$\mathbb{P}\left(\\{\omega\in
E:g[t\bm{u},\bm{X}](\omega)=t^{\tau(\bm{u})}g[\bm{u},\bm{X}](\omega)\\}\right)=1\,.$
(23)

If $E=\Omega$, we say $g$ is _$\mathbb{P}$ -a.s. positively $\tau$-homogeneous
for a function $\tau$ in $U$_.

Observe that if $g$ is $\mathbb{P}$-a.s. positively $\tau$-homogeneous for a
function $\tau$ in $U$ and in the event $E\in\mathcal{F}$, then $g_{\bm{X}}$
is also $\mathbb{P}$-a.s. positively $\tau$-homogeneous for a function $\tau$
in $U$ and in the event $E$ for all $\bm{X}\in\mathcal{X}^{n}$. Moreover, any
portfolio operator that is linear in $U$ is $\mathbb{P}$-a.s. 1-homogeneous in
$U$. For a non-linear operator $g$, however, the homogeneity property may not
hold for all $\omega\in\Omega$. In contrast, Definition 5 applies to functions
which map onto random variables, where there may exist outcomes $\omega$ for
which the operator $g$ is not homogeneous in $U$. Therefore, $\mathbb{P}$-a.s.
homogeneity is especially appealing to non-linear portfolio operators.

######  Theorem 2.

Suppose $\Lambda$ is continuously differentiable on $\mathbb{R}$, $g_{\bm{X}}$
is $\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$, for some function
$\tau\colon U\to\mathbb{R}$, and $g_{\bm{x}}(\bm{u})$ is differentiable in
$U$, for any fixed $\bm{x}\in\mathbb{R}^{n}$. If either Assumption 1 or
Assumption 2 is satisfied, then for all $\bm{u}\in U$, $\rho_{\Lambda}$
satisfies:

$\tau(\bm{u})\;\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\;\rho_{\Lambda}(\bm{u})=\sum_{i=1}^{n}u_{i}\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u}).$ (24)

###### Proof.

If $g_{\bm{X}}$ is $\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$ for
a function $\tau$ and $g_{\bm{x}}(\bm{u})$ is differentiable in $\bm{u}$, for
all $\bm{u}\in U$ and fixed $\bm{x}\in\mathbb{R}^{n}$, then by Equation (22)
it holds for almost all $\omega\in\Omega$ that

$\tau(\bm{u})\;g_{\bm{X}(\omega)}(\bm{u})=\sum_{i=1}^{n}u_{i}\partial_{u_{i}}g_{\bm{X}(\omega)}(\bm{u}),$
(25)

for all $\bm{u}\in U$. We note that (25) is equivalent to

$\tau(\bm{u})\;g_{\bm{X}}(\bm{u})=\sum_{i=1}^{n}u_{i}\partial_{u_{i}}g_{\bm{X}}(\bm{u})\quad\mathbb{P}\text{-a.s.},$
(26)

for all $\bm{u}\in U$. In (26), we have equivalence of two random variables in
a $\mathbb{P}$-a.s. sense, thus they have $\mathbb{P}$-a.s. equal conditional
expectations, that is

$\mathbb{E}[\tau(\bm{u})g_{\bm{X}}(\bm{u})~{}|~{}Y]=\mathbb{E}\biggl{[}\sum_{i=1}^{n}u_{i}\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y\biggl{]}\quad\mathbb{P}\text{-a.s.},$
(27)

where $Y:=g_{\bm{X}}(\bm{u})\in\mathcal{X}$. Note that under Assumption 1, we
apply Theorem 1 with condition 1, and under Assumption 2, we apply Theorem 1
with condition 2, to obtain partial derivatives of $\rho_{\Lambda}$. Recall
expression (18) from Theorem 1:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=-\rho_{\Lambda}(\bm{u})],$

for $i=1,\dots,n$. Note that the conditioning event
$Y=-\rho_{\Lambda}(\bm{u})$ in the expectation is the same for all $i$.
Therefore, summing the risk contributions scaled by the number of assets over
$i$ and using (27), we obtain

$\displaystyle\sum_{i=1}^{n}u_{i}\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})$
$\displaystyle=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\sum_{i=1}^{n}u_{i}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=-\rho_{\Lambda}(\bm{u})]$
$\displaystyle=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\mathbb{E}\biggl{[}\sum_{i=1}^{n}u_{i}\partial_{u_{i}}g_{\bm{X}}(\bm{u})~{}|~{}Y=-\rho_{\Lambda}(\bm{u})\biggl{]}$
$\displaystyle=-\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\mathbb{E}[\tau(\bm{u})\;g_{\bm{X}}(\bm{u})~{}|~{}Y=-\rho_{\Lambda}(\bm{u})]$
$\displaystyle=\tau(\bm{u})\;\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))\;\rho_{\Lambda}(\bm{u})\,,$

where the last equation holds by continuity of $F_{Y}$. ∎

From Theorem 2 we conclude that lambda quantiles are positively homogeneous of
degree given in the next proposition.

######  Proposition 3.

Let $g_{\bm{X}}$ be $\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$,
for a function $\tau\colon U\to\mathbb R$. Then, $\rho_{\Lambda}$ applied to
$g_{\bm{X}}$ is homogeneous in $U$ of degree
$\tau(\bm{u})\;\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))$. That is, the lambda
quantile $\rho_{\Lambda}$ is $\gamma$-homogeneous in $U$ for the function
$\gamma\colon U\to\mathbb{R}$, defined by
$\gamma(\bm{u}):=\tau(\bm{u})\;\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))$.

Theorem 2 and Proposition 3 has several interesting implications. To begin
with, the homogeneity degree of a lambda quantile is
$\tau(\bm{u})\,\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))$, a composition of
the homogeneity degree of the portfolio $\tau(\bm{u})$ and the portfolio
density adjustment $\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))$. Thus, the
homogeneity degree of $\rho_{\Lambda}$ of a linear portfolio operator (i.e.
$\tau(\bm{u})=1$) is $\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))$. It is
straightforward that the homogeneity degree of $\rho_{\Lambda}$ with a
constant lambda function, $\Lambda(x)=\lambda\in(0,1)$, and for a linear
portfolio operator is precisely the homogeneity degree of the $VaR_{\lambda}$
measure, that is 1. Indeed, for $\mathbb{P}$-a.s. 1-homogeneous portfolio
operators, $\rho_{\Lambda}$ is 1-homogeneous if, and only if,
$\Lambda^{\prime}(y)=0$ for all $y\in\mathbb{R}$. Note that for
$VaR_{\lambda}$ the homogeneity degree is independent of the portfolio
composition $\bm{u}$. This is in contrast to a non-constant $\Lambda$
function, in which case the lambda quantile homogeneity degree may differ for
each portfolio composition $\bm{u}\in U$. Moreover, for a fixed portfolio
composition $\bm{u}$, the homogeneity degree of lambda quantiles may change
for different choices of the $\Lambda$ function.

Next, we use Theorem 2 to define a new capital allocation rule, which
generalises the well-known Euler allocation. For a linear portfolio, risk
contributions calculated as directional derivatives of positive homogeneous
risk measures of degree 1 are known as Euler contributions (Tasche, 2007).
Furthermore, the assignment of capital using Euler contributions is known as
Euler allocation. Defining for Euler contributions is that they possess the
full allocation property, i.e. the sum of the Euler contributions over all
assets equals the risk measure itself. We propose a generalisation of Euler
contributions which satisfies the full allocation property and that is
compatible with $\gamma$-homogeneous risk measures, $\gamma\colon
U\to\mathbb{R}$, and generic portfolio operators, thus applicable to lambda
quantiles.

######  Definition 6.

Consider a portfolio $g_{\bm{X}}(\bm{u})$ and a risk measure
$\Phi:\mathcal{X}\rightarrow\mathbb{R}$ defined on the space of random
variables. Assume that the composition $\Phi\circ g_{\bm{X}}\colon
U\to\mathbb{R}$ is positively $\gamma$-homogeneous for a function
$\gamma\colon U\to\mathbb{R}$. Then, the functionals
$\psi_{i}^{\Phi}:\mathcal{X}\rightarrow\mathbb{R}$ defined by

$\psi_{i}^{\Phi}(g_{\bm{X}}):=\frac{1}{\gamma(\bm{1})}\;\frac{\partial\Phi}{\partial
u_{i}}\big{(}g_{\bm{X}}(\bm{u})\big{)}\biggl{|}_{\bm{u}=\bm{1}}\quad\text{for}\quad
i=1,\dots,n,$ (28)

are called _generalised Euler contributions_. Furthermore, we call the process
of allocating capital to sub-portfolios using generalised Euler contributions,
the _generalised Euler allocation rule_.

Euler contributions as defined by (Tasche, 2007) and the Euler allocation rule
(Patrik et al., 1999; Denault, 2001; Tasche, 2007) are special cases of
Definition 6 with $\eta=\tau=1$.

######  Proposition 4.

Suppose $\Lambda$ is continuously differentiable on $\mathbb{R}$, $g_{\bm{X}}$
is $\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$, for some
$\tau\colon U\to\mathbb R$, and $g_{\bm{x}}(\bm{u})$ is differentiable in
$\bm{u}$, for all $\bm{u}\in U$ and fixed $\bm{x}\in\mathbb{R}^{n}$. Also,
suppose that either Assumption 1 or Assumption 2 is satisfied. Then, the
generalised Euler contributions of the lambda quantile are given by:

$\psi_{i}^{\Lambda}(g_{\bm{X}})=-\frac{1}{\tau(\bm{1})}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{1})~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})]\quad\text{for}\quad
i=1,\dots,n.$ (29)

Furthermore, allocations $\psi_{i}^{\Lambda}(\cdot)$ define a generalised
Euler allocation rule for lambda quantiles with the full allocation property:

$\sum_{i=1}^{n}\psi_{i}^{\Lambda}(g_{\bm{X}})=\rho_{\Lambda}(\bm{1}).$ (30)

###### Proof.

The lambda quantile can be written as the composition
$\rho_{\Lambda}=\Phi_{\Lambda}\circ g_{\bm{X}}\colon
U\to\mathbb{R}\cup\\{+\infty\\}$, where we define $\Phi_{\Lambda}$ for fixed
$\bm{X}\in\mathcal{X}^{n}$ by
$\Phi_{\Lambda}(g_{\bm{X}}(\bm{u}))=\rho_{\Lambda}(\bm{u})$, for all
$\bm{u}\in U$. Moreover, the lambda quantile is positively homogeneous of
degree
$\tau(\bm{u})\;\eta_{\Lambda,g_{\bm{X}}(\bm{u})}(-\rho_{\Lambda}(\bm{u}))$ by
Proposition 3. We obtain, using Theorem 1 in the third equality, that

$\displaystyle\psi_{i}^{\Lambda}(g_{\bm{X}})$
$\displaystyle=\frac{1}{\tau(\bm{1})\,\eta_{\Lambda,g_{\bm{X}}(\bm{1})}(-\rho_{\Lambda}(\bm{1})))}\frac{\partial\Phi_{\Lambda}}{\partial
u_{i}}(g_{\bm{X}}(\bm{u}))\biggl{|}_{\bm{u}=\bm{1}}$
$\displaystyle=\frac{1}{\tau(\bm{1})\,\eta_{\Lambda,g_{\bm{X}}(\bm{1})}(-\Phi_{\Lambda}(g_{\bm{X}}(\bm{1})))}\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u})\biggl{|}_{\bm{u}=\bm{1}}$
$\displaystyle=-\frac{\eta_{\Lambda,g_{\bm{X}}(\bm{1})}(-\Phi_{\Lambda}(g_{\bm{X}}(\bm{1})))}{\tau(\bm{1})\,\eta_{\Lambda,g_{\bm{X}}(\bm{1})}(-\Phi_{\Lambda}(g_{\bm{X}}(\bm{1})))}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{1})~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})]$
$\displaystyle=-\frac{1}{\tau(\bm{1})}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{1})~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})].$

Observe that by the $\mathbb{P}$-a.s. $\tau$-homogeneity property of
$g_{\bm{X}}$, we can write

$\displaystyle\sum_{i=1}^{n}\psi_{i}^{\Lambda}(g_{\bm{X}})$
$\displaystyle=-\frac{1}{\tau(\bm{1})}\sum_{i=1}^{n}\mathbb{E}[\partial_{u_{i}}g_{\bm{X}}(\bm{1})~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})]$
$\displaystyle=-\frac{1}{\tau(\bm{1})}\mathbb{E}\biggl{[}\sum_{i=1}^{n}\partial_{u_{i}}g_{\bm{X}}(\bm{1})~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})\biggl{]}$
$\displaystyle=-\frac{1}{\tau(\bm{1})}\mathbb{E}[\tau(\bm{1})g_{\bm{X}}(\bm{1})~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})]$
$\displaystyle=\rho_{\Lambda}(\bm{1})=\Phi_{\Lambda}(g_{\bm{X}}(\bm{1})).$

∎

In applications, portfolio operators and their portfolios are typically
positively homogeneous of a constant degree, that is
$\tau(\bm{u})=\tau\in\mathbb{R}$. Thus, the multiplicative factor in the Euler
contributions $\frac{1}{\tau(\bm{1})}$ reduces to $\frac{1}{\tau}$.

######  Example 7.

For Example 2 (iii), the generalised Euler contributions of the lambda
quantile are given by

$\psi_{i}^{\Lambda}(g_{\bm{X}})=-\mathbb{E}\left[\max\left\\{0,X_{i}-\mathbb{E}[X_{i}]\right\\}~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1};g_{\bm{X}})\right]\,.$
(31)

For Example 2 (iv), the risk contributions of the lambda quantile are given by

$\displaystyle\psi_{i}^{\Lambda}(g_{\bm{X}})$
$\displaystyle=-\mathbb{E}[X_{i}~{}|~{}X_{1}+X_{2}=-\rho_{\Lambda}(\bm{1};g_{\bm{X}})+VaR_{\lambda}(\bm{1};X_{1}+X_{2})]$
(32)
$\displaystyle\quad-\mathbb{E}[X_{i}~{}|~{}X_{1}+X_{2}=-VaR_{\lambda}(\bm{1};X_{1}+X_{2})]\,.$
(33)

Euler allocations and their desirable properties are typically considered for
1-homogeneous risk measures defined on the space of random variables and
linear portfolio operators. An exception is Pesenti et al. (2021) who consider
non-linear but positively homogeneous portfolios and distortion risk measures
which are 1-homogeneous. If we consider the special case of a linear portfolio
operator, then the generalised Euler allocations of the lambda quantile fulfil
the properties of monotonicity and risklessness. Monotonicity is the property
that if $X_{j}\geq X_{i}$ $\mathbb{P}$-a.s., then the generalised Euler
contributions of $X_{j}$ is smaller than the contribution of $X_{i}$, i.e.
$\psi^{\Lambda}_{j}(g_{\bm{X}})\leq\psi^{\Lambda}_{i}(g_{\bm{X}})$. An Euler
allocation is called riskless, if $X_{j}$ is $\mathbb{P}$-a.s. constant,
$X_{j}=a$, $a\in\mathbb{R}$, then $\psi^{\Lambda}_{j}(g_{\bm{X}})=-a$. We
refer to Denault (2001) and Kalkbrener (2005) for a detailed discussion of
properties of Euler allocations for linear portfolios. For non-linear
portfolios, however, the monotonicity and the riskless property do not hold in
general nor are they desirable. Indeed consider the portfolio
$g_{\bm{X}}(\bm{u})=u_{1}X_{1}+u_{2}X_{1}X_{2}$, then the generalised Euler
contribution are

$\displaystyle\psi_{1}^{\Lambda}(g_{\bm{X}})$
$\displaystyle=-\mathbb{E}[X_{1}~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})\,$
(34) $\displaystyle\psi_{2}^{\Lambda}(g_{\bm{X}})$
$\displaystyle=-\mathbb{E}[X_{1}X_{2}~{}|~{}g_{\bm{X}}(\bm{1})=-\rho_{\Lambda}(\bm{1})]\,.$
(35)

Therefore, a stochastic ordering of $X_{1}$ and $X_{2}$ should not induce a
ordering of the risk contributions. Moreover, if $X_{2}=a$, for some
$a\in\mathbb{R}$, then the generalised Euler contributions to $X_{2}$ is given
by $\psi_{2}^{\Lambda}(g_{\bm{X}})=a\,\psi_{1}^{\Lambda}(g_{\bm{X}})\neq-a$.

##  5 Homogeneity of portfolio operators

In applications, portfolio operators and their portfolios are typically
positively homogeneous of a constant degree, that is $\tau(\bm{u})=\tau>0$.
Thus, for simplicity of exposition, we assume throughout this section that the
considered portfolios are positively $\tau$-homogeneous of constant degree
$\tau>0$. The central assumption for the Euler decomposition of lambda
quantiles is the $\mathbb{P}$-a.s. positively $\tau$-homogeneity of
$g_{\bm{X}}$. Thus, in this section, we study properties that ensure
$\mathbb{P}$-a.s. positively homogeneity in $U$ of generic portfolio
operators. For this, we first consider operators $g$ of the following additive
form to motivate some preliminary results:

$g[\bm{u},\bm{X}]=\mathfrak{a}[\bm{u},\bm{X}]+\mathfrak{b}(\bm{u},\bm{X}),$
(36)

where $\mathfrak{a}:U\times\mathcal{X}^{n}\rightarrow\mathcal{X}$ and
$\mathfrak{b}:U\times\mathcal{X}^{n}\rightarrow\mathbb{R}$. We refer to
$\mathfrak{a}$ as the _stochastic_ part of $g$ because it depends on a given
$\omega\in\Omega$ and $\mathfrak{b}$ as the _deterministic_ part of $g$
because it is a constant over all choices of $\omega\in\Omega$ (in Major
(2018), $\mathfrak{a}$ and $\mathfrak{b}$ are referred to as the pointwise and
constant functions respectively). In what follows and unless otherwise stated,
homogeneity of $g$ and $\mathfrak{a}$ is understood in the $\mathbb{P}$-a.s.
sense (see Definition 5), whereas homogeneity of $\mathfrak{b}$ and
$\rho_{\Lambda}$ is understood in the sense of Definition 4.

######  Proposition 5.

Suppose the portfolio operator $g$ can be written in the form (36). Then, $g$
is $\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$,
$\tau\in\mathbb{R}$, if $\mathfrak{a}$ is $\mathbb{P}$-a.s. positively
$\tau$-homogeneous in $U$ and $\mathfrak{b}$ is positively $\tau$-homogeneous
in $U$.

###### Proof.

If $\mathfrak{a}$ is $\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$,
then for almost all $\omega\in\Omega$ and for any $t>0$ and $\bm{u}\in U$ with
$t\bm{u}\in U$, we have:

$\displaystyle g[t\bm{u},\bm{X}](\omega)=$
$\displaystyle\mathfrak{a}[t\bm{u},\bm{X}](\omega)+\mathfrak{b}(t\bm{u},\bm{X})$
$\displaystyle=$ $\displaystyle
t^{\tau}\mathfrak{a}[\bm{u},\bm{X}](\omega)+t^{\tau}\mathfrak{b}(\bm{u,X})$
$\displaystyle=$ $\displaystyle t^{\tau}g[\bm{u},\bm{X}](\omega),$

and hence $g$ is $\mathbb{P}$-a.s. positively $\tau$-homogeneous. ∎

######  Proposition 6.

Suppose the portfolio operator $g$ can be written in the form (36) and
$\mathfrak{a}$ is given by:

$\mathfrak{a}[\bm{u},\bm{X}]=\sum_{i=1}^{n}u_{i}^{\tau}h_{i}(X_{i}),$

where $h_{i}:\mathcal{X}\rightarrow\mathcal{X}$ for $i=1,\dots,n$ and
$\tau\in\mathbb{R}$. Then, $g$ is $\mathbb{P}$-a.s. positively
$\tau$-homogeneous in $U$ if, and only if, $\mathfrak{b}$ is positively
$\tau$-homogeneous in $U$.

###### Proof.

We note that $\mathfrak{a}$ is $\mathbb{P}$-a.s. positively $\tau$-homogeneous
in $U$ because of the powers of the $u_{i}$’s. If $\mathfrak{b}$ is also
positively $\tau$-homogeneous, then $g$ is $\mathbb{P}$-a.s. positively
$\tau$-homogeneous by an argument similar to the proof of Proposition 5. For
the opposite case, assume $g$ is $\mathbb{P}$-a.s. positively
$\tau$-homogeneous. Then for any $t>0$ and $\bm{u}$ with $t\bm{u}\in U$ and
almost all $\omega\in\Omega$, we have:

$\displaystyle g[t\bm{u},\bm{X}](\omega)$
$\displaystyle=\sum_{i=1}^{n}(u_{i}t)^{\tau}h_{i}(X_{i}(\omega))+\mathfrak{b}(t\bm{u},\bm{X})$
$\displaystyle=t^{\tau}\mathfrak{a}[\bm{u},\bm{X}](\omega)+\mathfrak{b}(t\bm{u},\bm{X}).$

By our assumption, we can write:

$g[t\bm{u},\bm{X}](\omega)=t^{\tau}g[\bm{u},\bm{X}](\omega)=t^{\tau}(\mathfrak{a}[\bm{u},\bm{X}](\omega)+\mathfrak{b}(\bm{u,X})).$

Hence, $\mathfrak{b}(t\bm{u,X})=t^{\tau}\mathfrak{b}(\bm{u,X})$ and
$\mathfrak{b}$ is positively $\tau$-homogeneous in $U$. ∎

######  Corollary 2.

An operator of the form
$g=\mathfrak{a}[\bm{u},\bm{X}]+\rho_{\Lambda}(\bm{u})$, where $\mathfrak{a}$
is linear in $\bm{u}$ and $\Lambda$ is strictly increasing, is not homogeneous
in $U$.

The assumption that $\Lambda$ is strictly increasing in Corollary 2 implies
that the homogeneity degree of the lambda quantile is strictly greater than 1.
Therefore, by Proposition 5, operator $g$ is not positively homogeneous.

######  Example 8.

Consider the operators (i)-(v) from Example 2. These operators are
$\mathbb{P}$-a.s. 1-homogeneous in $U$ since the $\min$ and $\max$ functions
and the VaR are 1-homogeneous.
Operator (vi) from Example 2 is $\mathbb{P}$-a.s. positively homogeneous of
degree $\tau\in\mathbb{R}$ in $U$ if, and only if
$1=\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u}))$, which means that the
$\Lambda^{\prime}(-\rho_{\Lambda}(\bm{u}))=0$ is for all $\bm{u}\in U$.

The next result shows that if a deterministic variable
$\mathfrak{b}_{\bm{X}}(\bm{u})$ (of portfolio compositions), i.e. a positive
cash amount determined by asset units, is added to, or subtracted from, the
portfolio profit and loss, then the lambda quantile is reduced or increased,
respectively, by the same amount $\mathfrak{b}_{\bm{X}}(\bm{u})$. Indeed, the
lambda function is shifted by this deterministic variable. The following
result is related to the $\Lambda$-translation invariance property of lambda
quantiles in Frittelli et al. (2014), that is the cash additivity property of
lambda quantiles defined on the set of probability measures. However, it
should not be confused with the translation invariance property defined on
$U$, since we are not adding or subtracting from the portfolio composition
$\bm{u}$ but from $\mathfrak{a}_{\bm{X}}(\bm{u})$ instead.

######  Proposition 7.

For fixed $\bm{X}\in\mathcal{X}^{n}$, consider the portfolio:

$g_{\bm{X}}(\bm{u})=\mathfrak{a}_{\bm{X}}(\bm{u})+\mathfrak{b}_{\bm{X}}(\bm{u}),$
(37)

where $\mathfrak{a}_{\bm{X}}(u):=\mathfrak{a}[u,\bm{X}]$ and
$\mathfrak{b}_{\bm{X}}(u):=\mathfrak{b}(u,\bm{X})$. Also, define
$\Gamma(z):=\Lambda(z+\mathfrak{b}_{\bm{X}}(\bm{u}))$ for all
$z\in\mathbb{R}$. Then, for all $\bm{u}\in U$, it holds that:

$\rho_{\Lambda}(\bm{u};g_{\bm{X}})=\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})-\mathfrak{b}_{\bm{X}}(\bm{u}).$

###### Proof.

Observe that for all $\bm{u}\in U$, we can write:

$\displaystyle\rho_{\Lambda}(\bm{u};g_{\bm{X}})$
$\displaystyle:=-\inf\\{y~{}|~{}\mathbb{P}[\mathfrak{a}_{\bm{X}}(\bm{u})+\mathfrak{b}_{\bm{X}}(\bm{u})\leq
y]>\Lambda(y)\\}$
$\displaystyle=-\inf\\{y~{}|~{}\mathbb{P}[\mathfrak{a}_{\bm{X}}(\bm{u})\leq
y-\mathfrak{b}_{\bm{X}}(\bm{u})]>\Lambda(y)\\}$
$\displaystyle=-\inf\\{z+\mathfrak{b}_{\bm{X}}(\bm{u})~{}|~{}\mathbb{P}[\mathfrak{a}_{\bm{X}}(\bm{u})\leq
z]>\Lambda(z+\mathfrak{b}_{\bm{X}}(\bm{u}))\\}$
$\displaystyle=-\inf\\{z~{}|~{}\mathbb{P}[\mathfrak{a}_{\bm{X}}(\bm{u})\leq
z]>\Gamma(z)\\}-\mathfrak{b}_{\bm{X}}(\bm{u})$
$\displaystyle=\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})-\mathfrak{b}_{\bm{X}}(\bm{u})\,.$

∎

In Proposition 7, $\mathfrak{b}_{\bm{X}}(\bm{u})$ corresponds to the cash
amount determined by the asset units $\bm{u}$, which is added to the existing
portfolio $\mathfrak{a}_{\bm{X}}(\bm{u})$ to obtain the $g_{\bm{X}}(\bm{u})$
of the newly formed portfolio. As a result, the risk of the new portfolio,
i.e. $\rho_{\Lambda}(\bm{u};g_{\bm{X}})$, is obtained by subtracting
$\mathfrak{b}_{\bm{X}}(\bm{u})$ from the risk of the existing portfolio, i.e.
$\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})$. Note that the cash injection
causes the lambda quantile of existing and new portfolios to be calculated
using different, but related lambda functions, with the relationship given by
$\Gamma(z):=\Lambda(z+\mathfrak{b}_{\bm{X}}(\bm{u}))$.

######  Proposition 8.

Suppose $g_{\bm{X}}$ can be written in the form (37). If
$\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})$ and $\mathfrak{b}_{\bm{X}}$ are
positively homogeneous in $U$ with the same degree, then
$\rho_{\Lambda}(\bm{u};g_{\bm{X}})$ is positively homogeneous in $U$ with
degree:

$\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u};g_{\bm{X}}))=\eta_{\Gamma,Z}(-\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})),$
(38)

where $Y=g_{\bm{X}}(\bm{u})$ and $Z:=\mathfrak{a}_{\bm{X}}(\bm{u})$.

###### Proof.

Suppose $\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})$ and
$\mathfrak{b}_{\bm{X}}$ have homogeneity degree $\tau\in\mathbb{R}$. Then, by
Proposition 7 we can write the following for any $t>0$:

$\displaystyle\rho_{\Lambda}(t\bm{u};g_{\bm{X}})$
$\displaystyle=\rho_{\Gamma}(t\bm{u};\mathfrak{a}_{\bm{X}})-\mathfrak{b}_{\bm{X}}(t\bm{u})$
$\displaystyle=t^{\tau}\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})-t^{\tau}\mathfrak{b}_{\bm{X}}(\bm{u})$
$\displaystyle=t^{\tau}\rho_{\Lambda}(\bm{u};g_{\bm{X}}),$

which implies $\rho_{\Lambda}(\bm{u};g_{\bm{X}})$ is positively
$\tau$-homogeneous in $U$. Thus by Equation (22), both
$\rho_{\Lambda}(\bm{u};g_{\bm{X}})$ and
$\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})$ can be written in the form (24)
from Theorem 2. Therefore, the homogeneity degrees of
$\rho_{\Lambda}(\bm{u};g_{\bm{X}})$ and
$\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})$ are given by
$\eta_{\Lambda,Y}(-\rho_{\Lambda}(\bm{u};g_{\bm{X}}))$ and
$\eta_{\Gamma,Z}(-\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}}))$ respectively,
which proves our result. ∎

Proposition 8 in particular applies to Example 2 cases (ii), (iv), and (vi),
with $\mathfrak{a}_{\bm{X}}(\bm{u})=u_{1}X_{1}+u_{2}X_{2}$ for (ii) and (iv),
and with $\mathfrak{a}_{\bm{X}}(\bm{u})=u_{1}^{\tau}X_{1}+u_{2}^{\tau}X_{2}$
for example (vi). The components $\mathfrak{b}_{\bm{X}}$ are given by
$\mathfrak{b}_{\bm{X}}(\bm{u})=-\mathbb{E}[u_{1}X_{1}+u_{2}X_{2}]$,
$\mathfrak{b}_{\bm{X}}(\bm{u})=-VaR_{\lambda}(\bm{u};\mathfrak{a}_{\bm{X}})$,
and
$\mathfrak{b}_{\bm{X}}(\bm{u})=-\rho_{\Lambda}(\bm{u};\mathfrak{a}_{\bm{X}})$,
respectively.

######  Proposition 9.

Let $g_{\bm{X}}$ be of the form (37) and assume:

  1. (i)

Assumption 1 is satisfied for $\mathfrak{a}_{\bm{X}}$ and $\Gamma$;

  2. (ii)

$\Gamma$ is continuously differentiable on $\mathbb{R}$;

  3. (iii)

$\mathfrak{b}_{\bm{X}}$ is continuously differentiable by $u_{i}$ for
$i=1,\ldots,n$ for all $\bm{u}\in U$.

Then, $\rho_{\Lambda}(\cdot;g_{\bm{X}})$ is partially differentiable in $U$
with continuous derivatives given by:

$\frac{\partial\rho_{\Lambda}}{\partial
u_{i}}(\bm{u};g_{\bm{X}})=-\eta_{\Gamma,Z}(-\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}}))\mathbb{E}[\partial_{u_{i}}\mathfrak{a}_{\bm{X}}(\bm{u})|Z=-\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})]-\partial_{u_{i}}\mathfrak{b}(\bm{u}),$
(39)

for $i=1,\dots,n$.

###### Proof.

By Proposition 7, we can write

$\rho_{\Lambda}(\bm{u};g_{\bm{X}})=\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})-\mathfrak{b}_{\bm{X}}(\bm{u}).$

By Theorem 1 with condition 1, $\rho_{\Gamma}(\cdot;\mathfrak{a}_{\bm{X}})$ is
partially differentiable in $U$ with continuous derivatives given by:

$\frac{\partial\rho_{\Gamma}}{\partial
u_{i}}(\bm{u};\mathfrak{a}_{\bm{X}})=-\eta_{\Gamma,Z}(-\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}}))\mathbb{E}[\partial_{u_{i}}\mathfrak{a}_{\bm{X}}(\bm{u})|Z=-\rho_{\Gamma}(\bm{u};\mathfrak{a}_{\bm{X}})],$

for $i=1,\dots,n$. Since both $\rho_{\Gamma}(\cdot;\mathfrak{a}_{\bm{X}})$ and
$\mathfrak{b}_{\bm{X}}$ are continuously differentiable in $U$, we conclude
that $\rho_{\Lambda}(\cdot;g_{\bm{X}})$ is continuously partially
differentiable in $U$. ∎

As seen in Examples 2, the function $\mathfrak{a}$ is typically a polynomial
in $\bm{X}$ whilst $\mathfrak{b}$ an expectation or lambda quantile. A
portfolio operator may also be constructed via a function of these “building
blocks”. Thus, we consider $g$ to be a composition of a function
$f:\mathcal{X}\times\mathbb{R}\rightarrow\mathcal{X}$ with $\mathfrak{a}$ and
$\mathfrak{b}$, i.e. we consider operators of the form

$g[\bm{u},\bm{X}]=(f\circ(\mathfrak{a},\mathfrak{b}))(\bm{u,X})=f(\mathfrak{a}[\bm{u},\bm{X}],\mathfrak{b}(\bm{u,X})).$
(40)

Since $f$ acts on $\mathfrak{a}[\bm{u},\bm{X}]\in\mathcal{X}$ and
$\mathfrak{b}(\bm{u,X})\in\mathbb{R}$, positively homogeneity of the function
$f$ is discussed in $\mathcal{X}$ and $\mathbb{R}$, but not in $U$. The
function $f$ in this case may be implicitly positively homogeneous in $U$.

######  Proposition 10.

Suppose the function
$\mathfrak{a}:U\times\mathcal{X}^{n}\rightarrow\mathcal{X}$ is
$\mathbb{P}$-a.s. positively $\tau$-homogeneous in $U$ and
$\mathfrak{b}:U\times\mathcal{X}^{n}\rightarrow\mathbb{R}$ is positively
$\tau$-homogeneous in $U$. Also, suppose the function
$f:\mathcal{X}\times\mathbb{R}\rightarrow\mathcal{X}$ is $\mathbb{P}$-a.s.
positively $\nu$-homogeneous in both $\mathcal{X}$ and $\mathbb{R}$. Then, the
operator in Equation (40) is $\mathbb{P}$-a.s. positively homogeneous of
degree $\tau\nu$ in $U$.

###### Proof.

Noting that $\mathfrak{b}(\bm{u,X})$ is constant across outcomes
$\omega\in\Omega$, we can write the following for almost all $\omega\in\Omega$
and for any $t>0$ and $\bm{u}\in U$:

$\displaystyle g[t\bm{u,X}](\omega)$
$\displaystyle=(f\circ(\mathfrak{a},\mathfrak{b}))(t\bm{u,X})(\omega)$
$\displaystyle=f(\mathfrak{a}[t\bm{u,X}](\omega),\mathfrak{b}(t\bm{u,X}))$
$\displaystyle=f(t^{\tau}\mathfrak{a}[\bm{u},\bm{X}](\omega),t^{\tau}\mathfrak{b}(\bm{u,X}))$
$\displaystyle=(t^{\tau})^{\nu}f(\mathfrak{a}[\bm{u},\bm{X}](\omega),\mathfrak{b}(\bm{u,X}))$
$\displaystyle=t^{\tau\nu}f(\mathfrak{a}[\bm{u},\bm{X}](\omega),\mathfrak{b}(\bm{u,X}))$
$\displaystyle=t^{\tau\nu}(f\circ(\mathfrak{a},\mathfrak{b}))(\bm{u,X})(\omega)$
$\displaystyle=t^{\tau\nu}g[\bm{u},\bm{X}](\omega).$

Hence, $g$ is $\mathbb{P}$-a.s. positively homogeneous in $U$ of degree
$\tau\nu$. ∎

##  6 Applications to financial markets

In this section we present an application using real market data to illustrate
the lambda quantile’s homogeneity degree, its generalised Euler and risk
contributions on a non-linear but homogeneous portfolio.

###  6.1 Portfolio setup

We consider a non-linear portfolio where we measure the portfolio risk using
lambda quantiles; a study which has so far not been conducted in the
literature. For this purpose, we construct a portfolio consisting of an stock
and a European call option whose underlying asset is the same stock. Our
portfolio is non-linear since the profit and loss of an option is a non-linear
function of the underlying asset’s profit and loss.

We denote the prices of the stock and call option at the valuation date $t$ by
$S_{t}$ and $C_{t}$ respectively. Furthermore, we let
$\bm{u}_{P}=(u_{s},u_{c})$, where $u_{s}$ and $u_{c}$ represent the number of
units of stocks and call options in our portfolio respectively. The price of
the European call option at time $t\in[t_{0},T]$ is calculated using the
Black-Scholes formula, where $t_{0}$ is the option issue date and $T$ the
option maturity.

For this numerical application, we consider the stock of Exxon Mobil
Corporation (NYSE: XOM) and an option on the same stock with a maturity of 2
years ($T=2\times 250$ days), and strike at moneyness level $90\%$ of the
stock price. The daily stock market close prices for the period 1st January
2018 to 5th November 2021 have been sourced from Bloomberg. The option issue
date $t_{0}$ is 13th November 2018. In our implementation, we compute the
daily lambda quantiles of our portfolio for all days in the period
$[t_{0},T]$. At each valuation date $t\in[t_{0},T]$, the time-to-maturity of
the option is $T-t$ days. We assume that the number of asset units remains
constant over this period, unless otherwise stated. In the Black-Scholes
formula, we fix the annualised risk-free rate to be $2\%$ and use the
annualised standard deviation of historical Exxon Mobil Corporation stock log-
returns as the volatility. The value of our portfolio at time $t$,
$V_{t}(\bm{u}_{P}):U\rightarrow\mathcal{X}$, is thus given by:

$V_{t}(\bm{u}_{P})=u_{s}S_{t}+u_{c}C_{t},$

and the profit and loss of the portfolio,
$PL_{t}(\bm{u}_{P}):U\rightarrow\mathcal{X}$, from time $t-1$ to time $t$ is
defined by

$PL_{t}(\bm{u}_{P}):=V_{t}(\bm{u}_{P})-V_{t-1}(\bm{u}_{P})\,.$ (41)

Next, we rewrite the profit and loss in the same form as Example 2 (i). For
this, we define the profit and loss of the stock and call option at time $t$
by $X_{s,t}:=S_{t}-S_{t-1}$ and $X_{c,t}:=C_{t}-C_{t-1}$, respectively, such
that $\bm{X}_{t}:=(X_{s},X_{c})_{t}$, and

$PL_{t}(\bm{u}_{P})=g_{\bm{X}_{t}}(\bm{u}_{P})=u_{s}X_{s,t}+u_{c}X_{c,t}.$

Note that the $PL_{t}$ is a homogeneous function in $U$ of degree 1, thus the
lambda quantile applied to the portfolio is homogeneous with degree given in
Proposition 3. Partial derivatives of $PL_{t}$ with respect to asset units,
which appear in the partial derivatives of $\rho_{\Lambda}$ in (18), are given
by:

$\partial_{u_{s}}PL_{t}(\bm{u}_{P})=X_{s,t},\quad\partial_{u_{c}}PL_{t}(\bm{u}_{P})=X_{c,t}.$

To estimate the probability density and the distribution function (CDF) of the
portfolio profit and loss, we use the full valuation method based on
historical simulation of the underlying asset’s profits and losses combined
with kernel density estimation (KDE). We refer to Sironi and Resti (2007) for
details on how to implement the full valuation method. Specifically, we
consider 250 daily historically simulated profits and losses of the stock
corresponding to an in-sample period of 250 days. Then, we estimate the daily
portfolio densities and CDFs via KDE with a Gaussian kernel fitted to the 250
historically simulated portfolio profits and losses.

###  6.2 Estimation of the lambda function

100% stock portfolio

Equally weighted portfolio

Figure 1: Lambda function (blue) and portfolio CDF (red), where the $x$-axis
value of the intersection point is the negative of the lambda quantile, i.e.,
$-\rho_{\Lambda_{t}}(\bm{u}_{P})$. Both panels are at time 16th July 2020. The
left panel represents the portfolio with composition $u_{s}=100$ and $u_{c}=0$
and the right panel with composition $u_{s}=50$ and $u_{c}=50$.

A first approach in the literature to estimate the lambda function is given by
Hitaj et al. (2018) which has been generalised by Corbetta and Peri (2018). In
these works, the authors suggest to calibrate the lambda function via a data-
driven approach based on market benchmarks, such as stock market indexes in
the case of stock portfolios. The authors choose the maximum of the lambda
function equal to a maximum acceptable level $\lambda\in(0,1)$, so that in
this case the lambda quantile becomes comparable to $VaR_{\lambda}$. They
further set the minimum of the lambda function to be strictly larger than the
frequency of one observation, see Corbetta and Peri (2018) for a detailed
discussion on this requirement. The authors then obtain the lambda function as
a linear interpolation between points that are associated to left-tail order
statistics of the chosen market benchmarks. This approach thus treats the
lambda function as a proxy of the tail distribution of market benchmarks.
In this paper, we choose a parametric approach for the calibration of the
lambda function that maintains the original idea in the previous literature.
Specifically, we define the lambda function at time $t$, $\Lambda_{t}$, to
have an exponential growth in the interval $[x_{a,t},x_{b,t}]$ and to be
constant otherwise. The choice of the lambda function to be convex in
$[x_{a,t},x_{b,t}]$ is prudent, since compared to a concave interpolation, a
convex interpolation always results in a larger lambda quantile for any fixed
portfolio. Thus, the parametric form of the lambda function is given by:

$\Lambda_{t}(x)=1_{\\{x<x_{a,t}\\}}\lambda_{a}+1_{\\{x_{a,t}\leq x\leq
x_{b,t}\\}}\beta_{t}e^{\alpha_{t}x}+1_{\\{x>x_{b,t}\\}}\lambda_{b},\quad
x\in\mathbb{R},$ (42)

where $x_{a,t}<x_{b,t}$, $0<\lambda_{a}<\lambda_{b}<1$, and the coefficients
$\alpha_{t},\beta_{t}\in\mathbb{R}$ are chosen such that the lambda function
is continuous, i.e., they are given by:

$\displaystyle\alpha_{t}$
$\displaystyle=\frac{\log(\lambda_{a}/\lambda_{b})}{x_{a,t}-x_{b,t}},$
$\displaystyle\beta_{t}$
$\displaystyle=\lambda_{a}\exp{\biggl{(}-\frac{x_{a,t}}{x_{a,t}-x_{b,t}}\log{(\lambda_{a}/\lambda_{b})}\biggl{)}}.$

Note that we choose the minimum and maximum value of the lambda function,
i.e., $\lambda_{a}$ and $\lambda_{b}$, respectively, to be constant over time.
Point $x_{a,t}$ represents the threshold below which a risk manager assigns to
the portfolio the $VaR$ at the lowest confidence level
$\lambda_{a}:=\min_{x}\\{\Lambda_{t}(x)\\}=\Lambda_{t}(x_{a,t})$. Similar to
the previous literature, $x_{a,t}$ is chosen as the daily minimum over the
market index and the stock profits and losses for the given sample window.
This threshold is a technical requirement so that lambda quantiles are not
always lower than $x_{a,t}$, for all $t\in[t_{0},T]$. We fix
$\lambda_{a}=1.001\times\frac{1}{250}=0.4004\%$ as we require that
$\lambda_{a}$ is larger than the probability of one historically simulated
observation (in-sample period of 250 days). Point
$x_{b,t}:=\text{arg}\max\\{\Lambda_{t}(x)\\}$ corresponds to the threshold
above which a risk manager assigns to the portfolio the $VaR$ at the highest
acceptable confidence level
$\lambda_{b}:=\max_{x}\\{\Lambda_{t}(x)\\}=\Lambda_{t}(x_{b,t})$, here set to
$1\%$. Therefore $x_{b,t}$ is chosen, similar to the previous literature, as
an extreme left-tail statistic of a market index, here the $VaR_{1\%}$ of the
selected index. Note that both $\lambda_{a}$ and $\lambda_{b}$ are kept
constant for all time points $t\in[t_{0},T]$.

As our portfolio consists of stock positions, we chose the S&P500 index as the
market index, where daily market close prices have been sourced from Bloomberg
for the period 1st January 2018 to 5th November 2021. The lambda function is
calibrated on a daily basis, that is $x_{a,t}$ and $x_{b,t}$ are recalculated
daily using the previous 250 days. In Figure 1, we display estimated lambda
functions for two different dates. Finally, at any time $t$ we compute the
lambda quantile of our portfolio, here denoted
$\rho_{\Lambda_{t}}(\bm{u}_{P})$, as the negative of the smallest value on the
profit and loss axis corresponding to the intersection point between the
portfolio CDF and the lambda function using Matlab’s fzero solver.

In total, we have 499 daily estimates of lambda quantiles which correspond to
the 499 business days from option issuance date to $T-1=2\times 250-1$. Figure
2 shows the daily out-of-sample portfolio profits and losses and a comparison
between $-\rho_{\Lambda_{t}}(\bm{u}_{P})$ and $-VaR_{1\%,t}(\bm{u}_{P})$ of
the portfolio in consideration over time. As expected the lambda quantile is
more conservative than $VaR_{1\%}$.

Figure 2: Time series of portfolio profits and losses, negative of lambda
quantile and negative of $VaR_{1\%}$ of portfolio consisting of Exxon Mobil
Corporation stock and call option with $m=90\%$, $T=2$-years, $u_{s}=u_{c}=50$
and issue date of 13th November 2018.

###  6.3 Homogeneity degree

Following the calculation of the daily lambda quantiles of our portfolio, we
proceed by computing the daily homogeneity degree and show its variability
over both time and the choice of portfolio composition (note that once a
portfolio composition is chosen, we keep the composition fixed from inception
until option maturity). Even though we keep our portfolio composition
$\bm{u}_{P}$ fixed, we expect our portfolio’s homogeneity degree to show
variation over the life of the option contract since both the portfolio CDF
and lambda function are re-calibrated on a daily basis. Using Proposition 3,
recall that the homogeneity degree of the portfolio is equal to $\tau=1$, we
can write the lambda quantile homogeneity degree at $t$ as:

$\eta_{t}(\bm{u}_{P})=\frac{f_{t}(-\rho_{\Lambda_{t}}(\bm{u}_{P}))}{f_{t}(-\rho_{\Lambda_{t}}(\bm{u}_{P}))-\Lambda_{t}^{\prime}(-\rho_{\Lambda_{t}}(\bm{u}_{P}))},$

where $f_{t}$ is the probability density function of the portfolio profit and
loss $PL_{t}(\bm{u}_{P})$ estimated on a daily basis using KDE. From this
formula, it is evident that the homogeneity degree of lambda quantiles is
dictated by the portfolio density $f_{t}$ and the slope of the lambda function
$\Lambda$, both evaluated at the point $y=-\rho_{\Lambda_{t}}(\bm{u}_{P})$.
Since the lambda function is calibrated on a market index, the homogeneity
degree may change day-over-day. Furthermore, the homogeneity degree may also
change if a different portfolio composition is selected. Therefore, the
portfolio risk, measured using the lambda quantile, and portfolio composition
may not scale linearly under all market conditions, if at all.

In this numerical study, the homogeneity degree $\eta_{t}$ is well-defined
with $\eta_{t}\geq 1$ since by construction of the lambda function it holds
that
$f(-\rho_{\Lambda_{t}}(\bm{u}_{P}))>\Lambda^{\prime}(-\rho_{\Lambda_{t}}(\bm{u}_{P}))$.
This inequality is equivalent to Assumption 2 (vi) for our choice of portfolio
profit and loss $PL_{t}$. The derivative of the lambda function is easily
computed from (42) and given by:

$\Lambda_{t}^{\prime}(x)=1_{\\{x_{a,t}\leq x\leq
x_{b,t}\\}}\alpha_{t}\beta_{t}e^{\alpha_{t}x},\quad x\in\mathbb{R}.$

Figure 3: Time series of the homogeneity degree of the lambda quantile risk
measure applied on several portfolio compositions of Exxon Mobil Corporation
stock and call option with $m=90\%$, $T=2$-years, and issue date of 13th
November 2018.

Figure 3 displays the homogeneity degree of three different portfolio
compositions over time. From Figure 3, we see that, in the period between 13th
November 2018 and 6th November 2020, the homogeneity degree of all three
portfolio compositions is strictly larger than 1, which occurs when
$-\rho_{\Lambda_{t}}(\bm{u}_{P})\in[x_{a,t},x_{b,t}]$. In this case, the
lambda quantile of our portfolio is larger than the $VaR_{1\%}$ of the market
index. The variability of the homogeneity degree of each portfolio composition
is particularly visible during periods of high volatility of the Exxon Mobil
Corporation stock profits and losses, as at the end of 2018 and during the
initial stages of the Covid-19 pandemic. In contrast, the fluctuations of the
homogeneity degree are more moderate in 2019, a period of low volatility of
the Exxon Mobil Corporation stock profits and losses.

If $-\rho_{\Lambda_{t}}(\bm{u}_{P})\in[x_{a,t},x_{b,t}]$, then the portfolio
risk, measured using the lambda quantile, and portfolio composition scale with
a degree larger than one, i.e. it holds that

$\rho_{\Lambda_{t}}(k\bm{u}_{P})=k^{\eta_{t}}\rho_{\Lambda_{t}}(\bm{u}_{P})\quad\text{for
all }k>0.$ (43)

Therefore, for portfolio compositions for which the lambda quantile
$-\rho_{\Lambda_{t}}(\bm{u}_{P})\in[x_{a,t},x_{b,t}]$ is closer to $x_{b,t}$,
the higher is its homogeneity degree.

How the homogeneity degree is affected by different portfolio compositions is
displayed in Figure 3. The equally-weighted portfolio is displayed in blue.
The portfolio with 20 stocks and 80 options (displayed in yellow), has lower
homogeneity degrees for the 2-years under consideration. This is in contrast
to the portfolio with 80 stocks and 20 options (displayed in red) which has a
larger homogeneity degree.

The homogeneity degree of the lambda quantile is equal to 1, if the portfolio
CDF intersects the lambda function in its flat parts, i.e. when
$-\rho_{\Lambda_{t}}(\bm{u}_{n})\not\in[x_{a,t},x_{b,t}]$. In these cases, the
lambda quantile is equal to the $VaR$ at the lowest level $\lambda_{a}$, if
$-\rho_{\Lambda_{t}}(\bm{u}_{n})<x_{a}$, or at the highest level
$\lambda_{b}$, if $-\rho_{\Lambda_{t}}(\bm{u}_{n})>x_{b}$, and the portfolio
risk scales linearly with portfolio composition since:

$\rho_{\Lambda_{t}}(k\bm{u}_{P})=k\rho_{\Lambda_{t}}(\bm{u}_{P})\quad\text{for
all }k>0.$ (44)

We observe homogeneity degree 1 during the Covid-19 pandemic for the
composition of 100 stocks and 0 calls. The reader is referred to the left plot
of Figure 1 for an example. Here, the negative of the lambda quantile
$-\rho_{\Lambda_{t}}(\bm{u}_{n})$ is lower than $x_{a}$, the minimum over the
market index and the stock profits and losses of the previous 250 days. This
means that, the risk of the 100 stocks portfolio for this day is higher than
the worst case scenario over the previous 250 days.

###  6.4 Risk contributions and generalised Euler contributions

In this section, we calculate the daily generalised Euler contributions of
lambda quantile applied to the portfolio with 50 stocks and 50 options. The
generalised Euler contributions quantify how much each asset in the portfolio
contributes to the portfolio risk, measured by the lambda quantile risk
measure. Recall that for this portfolio we have a homogeneity degree of
$\tau=1$. Therefore, the generalised Euler contributions of the lambda
quantile at time $t$ are given, for $j\in\\{s,c\\}$, by:

$\displaystyle\psi_{j}^{\Lambda_{t}}(PL_{t}(\bm{1}))=$
$\displaystyle-\mathbb{E}[\partial_{u_{j}}PL_{t}(\bm{1})~{}|~{}PL_{t}(\bm{1})=-\rho_{\Lambda_{t}}(\bm{1})]=-\mathbb{E}[X_{j,t}~{}|~{}PL_{t}(\bm{1})=-\rho_{\Lambda_{t}}(\bm{1})]\,.$
(45)

We estimate the conditional expectations using the Nadaraya-Watson kernel
estimator, see e.g., Wand and Jones (1994), with a standard Gaussian kernel
smoothing function. Specifically, for each time $t$, we take a sample of the
historically simulated stock’s profits and losses using the past $N=250$
trading days. At time $t$, the samples, each of size $N$, consist of the
stock’s, option’s and portfolio’s profits and losses where the $i$-th sample
entry is from time $t-i$ and are given by:

$\displaystyle x_{s,t}^{i},\quad x_{c,t}^{i},\quad\text{and}\quad
y_{t}^{i}:=u_{s}x_{s,t}^{i}+u_{c}x_{c,t}^{i},$

where $i=1,\dots,N$. Therefore, we estimate the generalised Euler
contributions of the lambda quantile, for $j\in\\{s,c\\}$, by:

$\widehat{\psi}_{j}^{\Lambda_{t}}(PL_{t}(\bm{1}))=-\frac{\sum_{i=1}^{N}x_{j,t}^{i}\;\phi(\frac{-\rho_{\Lambda_{t}}(\bm{1})-y_{t}^{i}}{h_{t}})}{\sum_{i=1}^{N}\phi(\frac{-\rho_{\Lambda_{t}}(\bm{1})-y_{t}^{i}}{h_{t}})},$
(46)

where $\phi$ is the standard normal density and $h_{t}$ is the bandwidth given
by Silverman’s rule, see e.g., Pagan and Ullah (1999):

$h_{t}=1.06\times\sigma(y_{t}^{1},\dots,y_{t}^{N})\times N^{-1/5},$

where $\sigma(y_{t}^{1},\dots,y_{t}^{N})$ is the standard deviation of
portfolio profits and losses $y_{t}^{1},\dots,y_{t}^{N}$ in each sample.

Generalised Euler contributions

Mean simulated P&L

Figure 4: The left panel is a comparison between the lambda quantile
generalised Euler contributions of Exxon Mobil Corporation stock and call
option. The right panel displays a time series of mean of simulated profits
and losses of Exxon Mobil Corporation stock and call option. For both panels,
we have $m=90\%$, $T=2$-years, issue date of 13th November 2018 and
$u_{s}=u_{c}=1$.

The left panel of Figure 4 displays the time series of the generalised Euler
contributions for the stock and call option. At time $t$, the risk
contribution of the $j$-th asset represents how much the lambda quantile of
the portfolio increases for an infinitesimal increase in the number of units
of the $j$-th asset. We observe, that the generalised Euler contributions of
the stock ranges from 1.9064 to 4.2897, while that of the call option ranges
from 0 to 2.9161.

Note that the estimator of the generalised Euler contributions for the stock
and the call option only differ by the simulated profits and losses
$x_{j,t}^{i}$, $j\in\\{s,c\\}$ since the terms
$\phi(\frac{-\rho_{\Lambda_{t}}(\bm{1})-y_{t}^{i}}{h_{t}})$ are the same for
the stock and call option. Therefore, the generalised Euler contributions of
the stock are larger than the call option because the stock’s profit and loss
are more negative than that of the call option until the option price
approaches zero, see right panel of Figure 4 where we show the time series of
the average simulated profits and losses of the stock and the call option.
From a financial point of view, this can be interpreted as follows: if the
performance of asset 1 is poorer than that of asset 2, then increasing the
exposure to asset 1 increases the lambda quantile more compared to increasing
the exposure to asset 2.

Next, we consider the effect of varying the number of units of stocks and
options in the portfolio on the lambda quantile risk contributions. To do
this, we fix a valuation date and calculate risk contributions of the stock
and option by considering multiple portfolio compositions. Note that for each
risk contribution, the portfolio composition is kept constant from option
issuance until maturity. Figure 5 shows the lambda quantile risk contribution
as a function of the portfolio composition for three different dates.
Specifically, we plot:

$\frac{\partial\rho_{\Lambda_{t}}}{\partial
u_{j}}(\bm{u}_{P})=-\eta_{t}(\bm{u}_{P})\mathbb{E}[X_{j,t}~{}|~{}PL_{t}(\bm{u}_{P})=-\rho_{\Lambda_{t}}(\bm{u}_{P})]\,,$
(47)

for $\bm{u}_{P}=(u_{s},u_{c})=(10,90),(11,89),\dots,(90,10)$, and where the
conditional expectations is estimated using the Nadaraya-Watson kernel
estimator, and the homogeneity degree $\eta_{t}(\bm{u}_{P})$ is estimated as
described in Section 6.3.

We observe in Figure 5 that as the number of stocks in our portfolio
increases, the risk contribution of the stocks also increases. In contrast, as
the number of call options in our portfolio increases, the risk contribution
of the call options decreases. A different pattern is observed close to the
option’s maturity, right panel of Figure 5. In this case the risk
contributions of the call options become zero independently of the portfolio
composition. This is due to the fact that close to option maturity, the option
is out-of-the money and its price is zero. We further observe, from the left
to the right panel in Figure 5, that the values of the risk contribution of
both assets reduces as we approach option maturity.

11th November 2019

17th June 2020

5th November 2020

Figure 5: Lambda quantile risk contributions as of 11th November 2019 (left),
17th June 2020 (centre) and 5th November 2020 (right) of Exxon Mobil
Corporation stock and call option (right axis) as a function of portfolio
composition with $m=90\%$, $T=2$-years, and issue date of 13th November 2018.
On the $x$-axis, 10:90 represents the portfolio composition
$(u_{s},u_{c})=(10,90)$ and 90:10 represents the portfolio composition
$(u_{s},u_{c})=(90,10)$.

##  7 Conclusion

This paper presents a novel treatment of lambda quantile risk measures on
subsets of $\mathbb{R}^{n}$. We prove that lambda quantiles are differentiable
with respect to the portfolio composition, subject to smoothness conditions,
and derive explicit formulae for the derivatives. These partial derivatives
correspond to risk contributions of assets to the overall portfolio risk. We
further provide the Euler decomposition of lambda quantiles, i.e. the property
that lambda quantiles, scaled by a factor, can be written as sums of their
partial derivatives scaled by the number of assets. This decomposition
demonstrates that lambda quantiles are homogeneous risk measures in the space
of portfolio compositions. Our results further show that the homogeneity
degree of a lambda quantile is determined by the portfolio composition and the
lambda function. This allows us to treat homogeneity as a dynamic property
rather than constant and universal. Indeed, the lambda quantile homogeneity
degree varies across portfolio risk profiles, rather than remaining constant.
This contrasts the case of risk measures that have a constant homogeneity
degree, such as VaR. Due to the variable nature of lambda quantiles’
homogeneity degrees, we introduce a generalised Euler capital allocation rule,
that is compatible with risk measures of any homogeneity degree and non-linear
portfolios. We prove that the generalised Euler allocations of lambda
quantiles have the full allocation property. On a financial market portfolio,
we illustrate and provide interpretation of the homogeneity degree of a non-
linear portfolio and its generalised Euler allocation.

##  8 Auxiliary definitions and results

This appendix is a collection of results and definition relevant for the
exposition of the paper.

######  Lemma 3 (Theorem A.5.1 of Durrett (2019)).

Let $(S,\mathcal{S},\mu)$ be a measure space. Let $f$ be a complex valued
function defined on $\mathbb{R}\times S$. Let $\delta>0$, and suppose that for
$x\in(y-\delta,y+\delta)$ we have:

  1. (i)

$u(x)=\int_{S}f(x,s)\mu(ds)$ with $\int_{S}|f(x,s)|\mu(ds)<\infty$,

  2. (ii)

for fixed $s$, $\partial f/\partial x(x,s)$ exists and is a continuous
function of $x$,

  3. (iii)

$v(x)=\int_{S}\frac{\partial f}{\partial x}(x,s)\mu(ds)$ is continuous at
$x=y$,

  4. (iv)

$\int_{S}\int_{-\delta}^{\delta}|\frac{\partial f}{\partial
x}(y+\theta,s)|d\theta\mu(ds)<\infty$,

then $u^{\prime}(y)=v(y)$.

######  Definition 7 (Definition 4.2 of Tasche (1999)).

Let $U\neq\emptyset$ be a set in $\mathbb{R}^{n}$ and
$r:U\rightarrow\mathbb{R}$ be a function defined on $U$. A vector field
$\bm{a}:=(a_{1},\dots,a_{n}):U\rightarrow\mathbb{R}^{n}$ is called _suitable
for performance measurement_ with the function $r$ if the following conditions
are satisfied:

  1. (a)

for all $\bm{m}\in\mathbb{R}^{n}$ and $\bm{u}\in U$ with
$r(\bm{u})\neq\bm{m^{\prime}u}$ and $i\in\\{1,\dots,n\\}$ the inequality

$m_{i}r(\bm{u})>a_{i}(\bm{u})\bm{m^{\prime}u}$ (48)

implies that there exists an $\epsilon>0$ such that for all $t\in(0,\epsilon)$
we have

$g_{r,\bm{m}}(\bm{u}-t\bm{e}_{i})<g_{r,\bm{m}}(\bm{u})<g_{r,\bm{m}}(\bm{u}+t\bm{e}_{i}).$

  2. (b)

for all $\bm{m}\in\mathbb{R}^{n}$ and $\bm{u}\in U$ with
$r(\bm{u})\neq\bm{m^{\prime}u}$ and $i\in\\{1,\dots,n\\}$ the inequality

$m_{i}r(\bm{u})<a_{i}(\bm{u})\bm{m^{\prime}u}$ (49)

implies that there exists an $\epsilon>0$ such that for all $t\in(0,\epsilon)$
we have

$g_{r,\bm{m}}(\bm{u}-t\bm{e}_{i})>g_{r,\bm{m}}(\bm{u})>g_{r,\bm{m}}(\bm{u}+t\bm{e}_{i}),$

where $g=g_{r,\bm{m}}:\\{\bm{u}\in
U|r(\bm{u})\neq\bm{m^{\prime}u}\\}\rightarrow\mathbb{R}$ is the _profit and
loss function_ for $r$ for a fixed $\bm{m}\in\mathbb{R}^{n}$ defined by

$g_{r,\bm{m}}(\bm{u}):=\frac{\bm{m^{\prime}u}}{r(\bm{u})-\bm{m^{\prime}u}}\,.$

######  Lemma 4 (Theorem 4.4 of Tasche (1999)).

Let $\emptyset\neq U\subset\mathbb{R}^{n}$ be an open set and
$r:U\rightarrow\mathbb{R}$ a function partially differentiable in $U$ with
continuous derivatives. Also, let
$\bm{a}=(a_{i},\dots,a_{n}):U\rightarrow\mathbb{R}^{n}$ be a continuous vector
field. Then $\bm{a}$ is suitable for performance measurement with function $r$
if, and only if:

$a_{i}(\bm{u})=\frac{\partial r}{\partial u_{i}}(\bm{u})$

with $i=1,\dots,n$ and $\bm{u}\in U$.

## Funding

Silvana Pesenti would like to acknowledge the support of the Natural Sciences
and Engineering Research Council of Canada (NSERC) with funding reference
numbers DGECR-2020-00333 and RGPIN-2020-04289.

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