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\fancyhead[CO]{Designing Sound Deposit Insurances}
\fancyhead[CE]{Hirbod Assa, Ramin Okhrati}
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\centerline{\textit{The $4^{rd}$ FINACT-IRAN Conference, IPM, Tehran, July 29-31, 2017.}}
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{\Large \bf Designing Sound Deposit Insurances}
\vspace*{0.5cm}
{\small Ramin Okhrati}\index{Okhrati, Ramin}\footnote{Speaker}\\
{\small Assistant professor, University of Southampton, Southampton, UK}\\[2mm]
{\small Hirbod Assa}\index{Assa, Hirbod}\\
{\small Assistant professor, University of Liverpool, Liverpool, UK}\\[2mm]
%{\small First-name Last-name}\index{Last-name, First name}\\
%{\small MSc Student, ..... University, City, Country}
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%%%%%%%%%%%%% abstract %%%%%%%%%%%%%
\begin{abstract}
Deposit insurances were blamed for encouraging the excessive
risk-taking behavior during the 2008 financial crisis. The main reason
for this destructive behavior was ``moral hazard risk'', usually caused
by inappropriate insurance policies. While this concept is known and
well-studied for ordinary insurance contracts, yet needs to be further investigated for insurances on financial positions. In this paper, we set
up a simple theoretical framework for a bank that buys an insurance
policy to protect its position against market losses. The main objective
is to find the optimal insurance contract that does not produce the
risk of moral hazard, while keeping the bank's position solvent. In
a general setup we observe that an optimal policy is a multi-layer
policy. In particular, we obtain a closed form solution for the optimal
insurance contracts when a bank measures its risk by either Value
at Risk or Conditional Value at Risk. We show the optimal solutions
for these two cases are two-layer policies.
\end{abstract}
\keywords{Deposit insurance, solvency, risk measure and premium, Black-Scholes model, moral hazard, VaR, CVaR, stop-loss
}
\subject{G11, G13, G22}
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\section{Introduction}
An important lesson from the 2008 financial crisis is that an underestimated
moral hazard risk can be destructive. Whilst this fact was widely
known in insurance, it is rather new in the banking
industry. Moral hazard is a phenomenon in which some agents take excessive
risk because they know that the potential
costs of taking further risk will be covered by other agents and/or
the government. %Kenneth Arrow (e.g.,\cite{Arrow:1971}) was among the first who discussed the risk of moral hazard as an inevitable risk caused by altering a policyholder’s incentives.
An extensive use of deposit insurances in the financial sector
caused excessive risk-taking behavior. In the years prior to the 2008
financial crisis, financial institutions, like banks and in particular
hedge funds, bought deposit insurances in order to protect their investments.
As a result, financial institutions could venture riskier investments,
by transferring the big losses to the insurance companies; see \cite{Collins:1988}, \cite{Dowd:1996} and \cite{Freixas/Rochet:2008}.
This was a reason for huge risk-taking behaviors which caused big
losses in 2008.
In a banking system, moral hazard is a result of the absence of enough
prudential policies. While, the minimum capital requirement is aimed
to partly prevent the excessive risk-taking behavior by putting banks’
equity at risk, it can also encourage further risk-taking behavior. After 2008, it is proven
that neither these measures nor any other prudential regulatory law
can prevent another crisis unless the excessive risk-taking behavior
is controlled, see \cite{Dowd:2009}.
The key to prevent the risk of moral hazard is that the financial system must
not be used for reckless gambling inspired by
excessive risk-taking behavior. In general, there are two methods to
reduce the excessive risk-taking behavior. First, introducing ex-ante
policies which enforce banks to bear part of any loss they impose
to the system, and second, introducing ex-post policies which penalizes
the excessive risky behaviors.
In this paper, we have chosen to set an ex-ante policy. In this approach,
the risk of moral hazard is reduced by setting contracts that both the insurer and the insuree are being made partially responsible for the losses. We consider
a bank that seeks an optimal insurance contract that does not produce
any risk of moral hazard, while also keeping the bank's position solvent.
By adopting a complete market model as in \cite{Merton:1977},
where the author treated a deposit contract as an option, we will
characterize the optimal contracts. Ultimately, we use $\alpha$-percent
Value at Risk and Conditional Value at Risk for the minimum capital
requirement\footnote{As recommended in the Basel II accord and Solvency II},
and we observe that the optimal contracts are two-layer polices whose upper
and lower retention levels are completely determined.
Our work is important from two perspectives: first, we introduce a
mathematical framework to design deposit insurances that cannot impose
the risk of moral hazard to the financial system. Second, we use actuarial mathematics methods that are rather new in related
finance and banking problems. A similar approach is applied in \cite{Assa/Banking/2015},
where a risk management under prudential policies is discussed. The problem of insurance
and re-insurance design with no risk of moral hazard is very well
studied in the literature of actuarial science, for instance in \cite{Cheung/Sung/Yam/Yung:2014} and more recently in \cite{Assa:2015}.
The rest of the paper is organized as follows: Section 2 introduces
some mathematical notion and a model set-up for a bank balance
sheet. In Section 3, general optimal solutions are discussed.
\section{Model Set-up}
Let $(\Omega,\mathbb{P},\mathcal{F})$ be a complete probability
space, where $\Omega$ is the set of all scenarios, $\mathbb{P}$
is the physical probability measure, and $\mathcal{F}$ is a $\sigma$-
field of measurable subsets of $\Omega$. We denote the set of all
random variables by $L^0=L^{0}\left(\Omega,\mathcal{F}\right)$.
Furthermore, $\mathbb{E}$ represents the mathematical expectation with
respect to $\mathbb{P}$.
In this paper, we assume that contracts (policies) are issued at $t=0$, let say
the beginning of a policy year, and liabilities are settled at $t=T$, the maturity time or the
end of the year. A random variable represents losses of a policy and for any $X\in L^{0}$, the cumulative distribution
function associated with $X$ is denoted by $F_{X}$. The constant risk-free interest rate is denoted by $r\ge0$.
Let us consider a bank with an initial capital\footnote{For technical reasons, we assume the value of $b$ at time $T$ and
discount it to make it comparable to today's value.} $e^{-rT}b$, and a non-negative loss random variable $\mathcal{L}\ge0$
at time $T$. By buying an optimal insurance contract, the bank wants to hedge its global position by transferring part of its losses to
an insurance company. If we denote the insurance policy by a non-negative
random variable $I$ at time $T$, it has to satisfy $0\le I\le\mathcal{L}$.
The value of the insurance policy is given by a premium function $\pi:\mathcal{D}\to\mathbb{R}$
at time $0$, where $\mathcal{D}\subseteq L^{0}$ is the domain of
$\pi$. Therefore, the bank's global loss position is composed of
four parts: the initial capital at time $0$ i.e., $e^{-rT}b$, the
global loss i.e., $\mathcal{L}$, the insurance policy i.e., $-I$,
and the premium payed for the insurance policies at time 0, i.e.,
$\pi\left(I\right)$ which accumulates to $e^{rT}\pi\left(I\right)$ at time $T$.
Therefore, a simplified balance sheet of the bank's position at time
$T$ is given as follows
\begin{table}[!h]
\begin{centering}
\begin{tabular}{|c|c|c|c|}
\hline
Equity & Liability & Total Balance & Total Loss\tabularnewline
\hline
\hline
$b+I-e^{rT}\pi\left(I\right)$ & $-\mathcal{L}$ & $b+I-e^{rT}\pi\left(I\right)-\mathcal{L}$ & $e^{rT}\pi\left(I\right)+\mathcal{L}-b-I$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{The bank's balance sheet at time $T$.}
\end{table}
The bank is solvent if its global position is solvent. To measure
the solvency we use a risk measure; for instance, Value at Risk (VaR)
or Conditional Value at Risk (CVaR). We recall that VaR is recommended in the Basel II accord for the banking system and also
in the Solvency II for the insurance industry. In this paper, $\varrho$
denotes the risk measure recommended by regulator. The bank is solvent
if its capital $b$ is adequate for the solvency i.e., $\varrho\left(e^{rT}\pi\left(I\right)+\mathcal{L}-b-I\right)\le0$.
In other words, the position $e^{rT}\pi\left(I\right)+\mathcal{L}-b-I$
does not produces any risk. Therefore, an optimal decision for the
bank is to buy the cheapest insurance contract i.e.,
\begin{equation}
\begin{cases}
\min\pi(I),\\
\varrho(e^{rT}\pi\left(I\right)+\mathcal{L}-b-I)\le0,\\
0\le I\le\mathcal{L}.
\end{cases}\label{eq:main-first}
\end{equation}
\\
Next, we use a more
specific model for the bank's asset. Our paper follows an approach
similar to \cite{Merton:1977}, assuming that the market is composed of a risk-free asset
and the risky asset that is modeled by a geometric Brownian motion. If the asset price at time $t\in\left[0,T\right]$ is denoted by $S_{t}$, then we assume
that it follows:
\[
\begin{cases}
dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t},\\
S_{0}>0,
\end{cases}
\]
where $W_{t}$ is a standard Wiener
process, and the constants $\mu$ and $\sigma$ are respectively the drift and volatility.
%It is also known that the solution to this SDE is the geometric Brownian
%motion given by
%\[
%S_{t}=S_{0}\exp\left(\left(\mu-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}\right).
%\]
We suppose that the bank's loss is a non-negative and non-increasing
function of its assets value. In mathematical terms, $\mathcal{L}=L\left(S_{T}\right)$,
where $L:\mathbb{R}\to\mathbb{R}_{+}\cup\left\{ 0\right\} $ is a
non-increasing function. A natural example is losses due to negative
returns
\begin{equation}
L_{n}(x)=\begin{cases}
e^{rT}S_{0}-x, & \text{ if }x\le e^{rT}S_{0},\\
0, & \text{ if }x>e^{rT}S_{0}.
\end{cases}\label{eq:Loss}
\end{equation}
\textbf{\textit{Assumption 1}}. We assume that there is no risk of moral
hazard. If we suppose that both the bank and insurance loss random variables
are non-decreasing functions of the global loss variable, then the risk of moral hazard is ruled out, as both sides are exposed to any
increase in the global loss, see for example \cite{Bernard/Tian:2009}. Therefore we assume that $I=f(\mathcal{L})$
where both $f$ and $\text{id}-f$ are non-decreasing (here $\mathrm{id}$
denotes the identity function).
\begin{definition}
A distortion risk measure $\varrho_{\Pi}$ (or simply $\varrho$)
is a mapping from $\mathcal{D}_{\Pi}$ to $\mathbb{R}$ defined as
\begin{equation}
\varrho_{\Pi}(X)=\int_{0}^{1}\mathrm{\mathrm{VaR}}_{t}(X)d\Pi_{\varrho}(t).\label{eq:Choquet_form}
\end{equation}
where $\Pi_\varrho:[0,1]\to[0,1]$ is a non-decreasing and c\'adl\'ag function
such that $\Pi(0)=1-\Pi(1)=0$.
\end{definition}
\textbf{\textit{Assumption }}2. We suppose that $\varrho$ is a distortion risk measure and it satisfies the following regularity condition
\begin{equation}
\lim_{n\to\infty}\varrho(X\wedge n)=\varrho(X)\text{ for all random variables }X\text{ in } L^0.\label{eq:continuity}
\end{equation}
The main results are presented in the next section, but first, we need to introduce some notation. Let $B=b-\varrho(\mathcal{L})$,
$\Phi^{\varrho}\left(t\right):=1-\Pi_{\varrho}\left(t\right),$ $\Phi_{\mathcal L}^{\varrho}\left(t\right):=\Phi^{\varrho}\left(F_{\mathcal L}\left(t\right)\right), \Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right) =N\left(\frac{(\mu-r)\sqrt{T}}{\sigma}+\frac{\log\left(\frac{L^{-1}\left(t\right)}{S_{0}}\right)-\left(\mu-\frac{\sigma^{2}}{2}\right)T}{\sigma\sqrt{T}}\right)
$ ($N$ is standard normal distribution), and
\[
\theta^{*}:=\underset{\mathrm{\theta\ge0}}{\mathrm{argmin}}\left(\int_{0}^{\infty}\left(\Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right)+\theta\left(\Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right)-\Phi_{\mathcal{L}}^{\varrho}\left(t\right)\right)\right)_{+}dt+B\theta\right),
\]
\\
note that here $\left(x\right)_{+}=\max\left\{ x,0\right\} $.
\section{Main results}
\begin{theorem}
\label{thm:main}If Assumptions 1 and 2 hold, and if $m=\mu-r\ge0$, the
optimal solution to \eqref{eq:main-first} is given by $I=f^{*}(\mathcal{L})$,
where $f^{*}(x)=\int_{0}^{x}h^{*}(t)dt,$ and
\begin{enumerate}
\item If $\theta^{*}>0$
\begin{align*}
h^{*}\left(t\right) & =\begin{cases}
1, & \text{ if }\Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right)<\frac{\theta^{*}}{1+\theta^{*}}\Phi_{\mathcal{L}}^{\varrho}\left(t\right),\\
0, & \text{ if }\Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right)>\frac{\theta^{*}}{1+\theta^{*}}\Phi_{\mathcal{L}}^{\varrho}\left(t\right),
\end{cases}\\
\text{ and } & \int_{0}^{\infty}\left(\Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right)-\Phi_{\mathcal{L}}^{\varrho}\left(t\right)\right)h^{*}(t)dt=B,
\end{align*}
\item If $\theta^{*}=0$
\[
h^{*}(t)=0.
\]
\end{enumerate}
\end{theorem}
\begin{corollary}
\label{cor:2}If in addition to Assumptions 1, 2 and $m=\mu-r\ge0$, the
following condition holds
\[
\forall\theta\ge0,\left\{ 0\le t<\mathrm{esssup}\left(\mathcal{L}\right)\vert\Phi_{\mathcal{L}}^{\bar{\pi}}\left(t\right)=\frac{\theta}{1+\theta}\Phi_{\mathcal{L}}^{\varrho}\left(t\right)\right\} \text{ \text{is of Lebesgue measure zero},}
\]
then the optimal solution to problem \eqref{eq:main-first} is
given by
\[
h^{*}=1_{\left\{ \Phi_{\mathcal{L}}^{\bar{\pi}}<\frac{\theta^{*}}{1+\theta^{*}}\Phi_{\mathcal{L}}^{\varrho}\right\} }.
\]
\end{corollary}
The next theorem shows that for the particular risk measure $\mathrm{VaR}_{\alpha}$, the optimal deposit insurances are indeed
stop-loss policies. A similar result can be obtained for $\mathrm{CVaR}_{\alpha}$.
\begin{theorem}\label{thm:21}
If $\varrho=\mathrm{VaR}_{\alpha}$, $m=\mu-r\ge0$, and the assumptions
of Corollary \ref{cor:2} hold, then the optimal insurance contract
$I$ is a two-layer policy with upper retention level $u=F_{\mathcal{L}}^{-1}\left(\alpha\right)=\mathrm{VaR}{}_{\alpha}\left(\mathcal{L}\right)$
and a lower retention level $l$ given as a solution to
\begin{equation}
\bar{\pi}\left(\min\left\{ \mathcal{L}-l,0\right\} \right)+b=\bar{\pi}\left(\min\left\{ \mathcal{L},\mathrm{VaR}_{\alpha}(\mathcal{L})\right\} \right).\label{eq:policy-VaR}
\end{equation}
\end{theorem}
\begin{remark}
In contrast with \cite{Merton:1977}, where it is assumed that an
insurance contract is a put option, our assumptions lead to the contracts
that are bounded from above.
\end{remark}
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\end{thebibliography}
\end{document}