Innovave financial instruments for mi&gang flood risks · Introduc&on The overall objec&ve is to...
Transcript of Innovave financial instruments for mi&gang flood risks · Introduc&on The overall objec&ve is to...
Innova&ve financial instruments
for mi&ga&ng flood risks
Fabio Castelli University of Florence
Marcello Galeo< Inter‐University Centre for Actuarial Sciences and Risk Management
Giovanni Rabi< University of Milano Bocconi
Introduc&on The overall objec&ve is to mi&gate the flood risk in a region by transferring
financial resources from damage covering to damage preven&ng, assuming that:
1. Some preven&ve measures exist whose cost is by far less than the cost of
covering the damages they prevent. 2. These preven&ve measures require
target investments (project costs), whose amount cannot be sustained by a
Central Public Administra&on (CPA) normal economic programming . 3. The
above CPA aversion to investment on preven&ve measures is even worsened (not
diminished!) by the frequent occurrence of catastrophes. In many cases, in fact, a
catastrophe occurrence triggers a nega&ve financial flow, in the sense that
resources previously allocated for preven&on projects are diverted toward
damage coverage. Points 1) and 2) are the main mo&va&on for issuing project
op&ons (which may be subscribed by local public administra&ons, LPA) to support
and finance risk preven&on measures. Points 2) and 3) are the main mo&va&ons
for accompanying the project op&ons with a second financial instrument, similar
to catastrophe bonds (which can be acquired by poten&al investors, PI), that: a)
reinforces the remunera&on of virtuous LPAs in terms of gains (not simply loss
reduc&ons) subsequent to risk reduc&on; b) assures the CPA for the damage
coverage in case of catastrophe, so that re‐alloca&on of resources is not required.
A case study: flood risk for Arno river
• Fabio Castelli, a professor of Civil
Engineering, has conducted a technical
study on the risk factors and costs
rela&ve to Arno river (recall the Florence
flood of 1966!)
• Parameters values of our model derive
from that study.
• Poten&al damages: urban micro‐scale
• Poten&al damages: urban micro‐scale
Danni non mone?zzabili (e.g. i beni culturali)
Cortesia di Bernardo Mazzan&
Autorità di Bacino del Fiume Arno
The basin scale
Poten&al damages and mi&ga&on costs
S?ma a scala comunale
(CIMA & AdB Arno, 2007)
Totale comuni bacino
dell’Arno:
Residenziale
0.26% 49÷103�€/����
Industriale‐commerciale
0.21% 12÷38
~0.4 ��↑2 ~4,000 residen&
Edifici
3.5 �€/anno
Cont. abitazioni
1.7 �€/anno
Commercio
0.5 �€/anno
Poten&al damages and mi&ga&on costs: from
urban to basin scale
Residenziale
0.26% 49�€/���� Tot
0.20% 13�€/����FI
Industriale‐commerciale
0.21% 12 Tot
0.04% 2FI
FI ~103 ��↑2 FI ~356,000 residen&
×1.8 S?ma rischio idraulico (beni
mone?zzabili) per il bacino
dell’Arno
~110 �€/anno
Poten&al damages and mi&ga&on costs:
from urban to basin scale
~110 �€/anno (+ non monet.+ indir.)
~200 �€
stralcio
~900 �€
Completamento
a Tr 200
Ammortamento su 50
anni con interesse al 4%
~41 �€/anno
Financial instruments
• There exists a vast literature on the topic “market
incen&ves for risk management” and many
instruments have been proposed (some&mes
adopted): catastrophe bonds, environmental op?ons,
project bonds etc..
• The pabern we suggest assumes a dynamic
interac&on among three types of agents: a central
public administra&on, CPA (state, region); some local
administra&ons, LPAs (municipali&es, districts) in
areas exposed to risk; a popula&on of possible
private investors, PIs.
• From the point of view of decision making, the
instruments we propose behave like risk measures
(or coverages), and the benefit of subscribing them,
for a given &me horizon, can be evaluated through a
comparison among suitable expected pay‐offs.
• The central public administra&on (CPA) fixes
as a target a maximal threshold of risk level
for the occurrence of a catastrophic event.
• The target must be reached through local
preven&ve works and the progress in its
achievement, within a given &me period, is
cer&fied by an independent Agency (or
Authority).
The model
Market incen+ves for environmental protec+on policies
Castelli and Galeo< (2013): first proposal of this combina&on of
financial instruments.
Resilience bonds (Swiss Re, 2015): CAT bond + project finance
Differences among the two proposals in mechanisms and finali&es:
1. the CPA is both sponsor and issuer of the nancial assets;
2. two dis&nct instruments, instead than one, addressed to
different popula&ons of agents (LPAs and PIs), whose
interac&on is substan&al to the effec&veness of the nancial
mechanism;
3. under suitable condi&ons, such interac&on leads to an op&mal
scenario (from the CPA's point of view), while in the resilience
bonds scheme such op&mal goal is never achieved.
On such a basis the CPA offers two one‐
period (e.g., yearly) contract proposals,
one addressed to local administra&ons
(LPA) and one to private investors (PI),
denoted, respec&vely, as project op)ons
and catastrophe bonds (see,e.g.,
Winkelvos et al. “Accuracy etc.” JRI, 2013)
Financial incen&ves
Project op&ons
• What we call project op&on is, in fact, an incen&ve‐deterrence pabern.
• The CPA proposes to LPAs a contract compelling the
underwriter to accomplish, in the given period, some determined preven&ve works of a cost CP.
• In case the fixed target of risk reduc&on in the region is
achieved, the subscribing (virtuous) LPA receives a full reimbursement plus a reward.
• If, instead, a catastrophe occurs, the non‐virtuous LPAs must
fully contribute to damage costs, while the virtuous ones will contribute so much the less as less numerous they are, in order not to discourage contract subscrip&on.
Project Op&ons: expected pay‐offs
[ ]PCxxHxxEP )(')1(
1µβ −−−=
Let us denote by EP1 and EP0 the expected pay‐offs (at the
end, say, of the year) of, respec&vely, virtuous and non‐
virtuous LPAs.
Let 1 be the normalized number of LPAs and x the number
(ra&o) of virtuous ones, 0≤x≤1.
Simplifying, the probability of a catastrophe, within the
period, will be given by H(x)= HT[1+c(1‐x)] and the
probability of reaching the pre‐fixed target will be set
equal to x.
In other words, by extreme simplifica&on, we could say
that the target is reached if x increases.
The following formulas hold:
PCxHEP )(''
0µ−=
1''',0 >>≈≈> µµµβ
Catastrophe Bonds (Cat‐Bonds)
The ahermath of Hurricane Andrew
The CPA offers to a popula&on of private investors
(PI), whose number is again normalized to 1, the
possibility of purchasing a type of one‐year, say,
zero‐coupon bond with an a_rac?ng return α in
case a catastrophe does not occur, while in the
opposite case no interest is paid and the capital
itself is reimbursed for a frac&on so much lower as
lower is the investors’ number.
Cat‐Bonds: expected pay‐offs
• Let us denote by y the number (ra&o) of possible investors
buying the cat‐bonds. In order to compare expected pay‐offs,
we pass, say, from a physical to a risk‐neutral probability
measure, so that we have to insert a risk‐premium.
• Moreover we pose equal to 1 the unitary price of cat‐bonds,
so that the works cost is also measured by cat‐bond uni&es.
We set the rela&ve cost of damage preven&on projects CP=ρ
0≤ε≤1, r>0 fixed return, π>0 risk premium
( ) ( )
rEC
yxHxHEC
=
−−−−=
0
1 1)()(1 πεα
• Let us set
• Parameter values (case of Arno river)
Mutually dependent incen&ves
( ) ( )ynxml
xd
yd
−+−+=
−=
+=
11
max
min
π
αα
ββ
α
β
( )
( ) ( )
( )xr
x
y
cHxH
HxH
T
T
−+=+
−=
+=
=
=
=
=+==
===
1015.003.0
06.008.0
06.004.1
85
2.0
33.0
50110
20011
π
α
β
µ
ε
ρ
Introducing the above financial instruments causes a
dynamic interac&on among CPA and the two popula&ons
of LPAs and PIs, which we can represent as an
evolu&onary game. For sake of simplicity we assume the
game to be con&nuous (the fixed period is instantaneous)
and the dynamics to be replica?ng (replicator dynamics),
meaning that in each popula&on those strategies spread,
at the expense of the others, whose pay‐off is higher than
the average one.
Dynamics
System equa&ons
By the above assump&ons the dynamics, taking place in the square [0,1]2, is given by the following equa&ons:
That system could in general exhibit up to a maximum of 12 equilibrium points and 4 abractors. However a reasonable choice of parameters, such as the one we adopted, notably reduces these possibili&es.
( )( )
( )( )01
.
01
.
1
1
ECECyyy
EPEPxxx
−−=
−−=
One instrument alone is not enough!
• We can assume the ul&mate aim of CPA to be
that all LPAs become virtuous, so that cat‐bonds
(contribu&ng to public debt) would be no more
necessary, i.e. that (1,0) be the only abractor of
the above system.
• However, as the following slide shows,
without issuing the cat‐bonds such a goal might
not be achievable, unless virtous LPAs were,
since the beginning, rather numerous.
Abractor
�=0
� =�(1−�)[��↓1 (�)−��↓0 (�)]
Abractor Repellor
x – LPAs underwri&ng Project Op&ons
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
� – Proj. Opts.
� – Cat‐Bonds
Abractor Saddle
� =�(1−�)[��↓1 (�,�)−��↓0 (�,�)]
� =�(1−�)[��↓1 (�,�)−��↓0 (�,�)]
P(1)=3; % c=P(1);
P(2)=0.33; % rho=P(2);
P(3)=0.2; % eps=P(3);
P(4)=0.005; % HT=P(4);
P(5)=0.08; % a1=P(5);
P(6)=0.06; % da=P(6);
P(7)=1.04; % b0=P(7);
P(8)=0.06; % db=P(8);
P(9)=0.02; % rl=P(9);
P(10)=0.015; % mm=P(10);
P(11)=0.0; % nn=P(11);
P(12)=82; % mu=P(12); % 82
P(13)=0; % k=P(13);
Cat‐bonds allow to bypass the block
Bifurca&ons
• With the chosen expressions the isoclines
are represented in the square (0,1)2 by the graphs , respec&vely, of y=f(x), having a maximum at x*, and
y=g(x) increasing.
• If the laber graph intersects the x‐axis to the leh of x* (but where f(x)>0), then, varying the parameter ε (i.e. the ra&o of capital reimbursed to
an investor in case of catastrophe), two bifurca&ons
may occur.
0 and 0 ==••
yx
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
• When ε increases the curve y=g(x) can intersect the
side x=1 of the square. When this happens, (1,1),
besides (1,0), is abrac&ng and there is a separatrix
between the two abrac&on basins.
Abractor
Abractor
� – Proj. Opts.
P(1)=3; % c=P(1);
P(2)=0.33; % rho=P(2);
P(3)=0.8; % eps=P(3);
P(4)=0.005; % HT=P(4);
P(5)=0.082; % a1=P(5);
P(6)=0.06; % da=P(6);
P(7)=1.04; % b0=P(7);
P(8)=0.06; % db=P(8);
P(9)=0.02; % rl=P(9);
P(10)=0.015; % mm=P(10);
P(11)=0.0; % nn=P(11);
P(12)=82; % mu=P(12); % 82
P(13)=0; % k=P(13);
Two abractors
Two abractors and one cycle
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cycle
Abractor
� – Proj. Opts.
P(1)=3; % c=P(1);
P(2)=0.2; % rho=P(2);
P(3)=0.34; % eps=P(3);
P(4)=0.005; % HT=P(4);
P(5)=0.08; % a1=P(5);
P(6)=0.06; % da=P(6);
P(7)=1.02; % b0=P(7);
P(8)=0.25; % db=P(8);
P(9)=0.025; % rl=P(9);
P(10)=0.025; % mm=P(10);
P(11)=0.0; % nn=P(11);
P(12)=68; % mu=P(12); % 82
P(13)=0; % k=P(13);
Vice‐versa if, star&ng from the ini&al value, ε decreases, an
interior abractor (besides (1,0)) eventually arises, whose basin is
bounded by a repelling cycle (subcri&cal Hopf bifurca&on).
Abractor
Comments and conclusions
The parameters ε, decided by the CPA and establishing the CAT‐bond value in case of catastrophe, and m, measuring the PIs risk aversion, are the cri&cal parameters.
In par&cular ε can't be too high (in order not to permanently increase CPA's debt) and can't be too low (if we want to encourage the PIs to buy CAT‐bonds and avoid
sub‐op&mal cases 3,4).
For a balanced ε op&mal scenarios can be directly produced.
Vice‐versa, if ε is too high, the CPA's goal can be achieved at two condi&ons:
• the number of virtuous LPAs at the beginning of the emission must be suffciently high;
• the quote of emibed CAT‐bonds must be carefully monitored by the CPA: it must be sufficiently high as to convince non‐virtuous LPAs to become virtuous, and at the same &me it must be sufficiently low in order to prevent the LPAs from
remaining virtuous only if an excessive bonus, financed by CAT‐bonds, is
awarded.