University of Economics, Faculty of Informatics Dolnozemská cesta 1, 852 35 Bratislava Slovak Republic Financial Mathematics in Derivative Securities and Risk Reduction Insurance and Risk Reduction, Financial Layering Ass. Prof. Ľudovít Pinda, CSc. Department of Mathematics, Tel.:++421 2 67295 813, ++421 2 67295 711 Fax:++421 2 62412195 e-mail: [email protected]
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University of Economics, Faculty of Informatics Dolnozemská cesta 1, 852 35 Bratislava Slovak Republic Financial Mathematics in Derivative Securities and.
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University of Economics, Faculty of Informatics
Dolnozemská cesta 1, 852 35 Bratislava
Slovak Republic
Financial Mathematics in Derivative Securities and Risk Reduction
Tab. 1 Decision Framework for financial Risk Management
Retention funds when alternative sources of finance have no transaction costs
1V - the present value of the firm at the beginnind the first period,
iEE - the expected earnings for i ih year,
k - the risk-adjusted discount rate,
rf - the risk-free rate,
- the risk premium,
11 )1(
)(
)1(
)()1(
ii
f
i
ii
i
r
EE
k
EEV
, ( 1 )
fr
VEV
1
21 )( , ( 2 )
fr
VEV
1
32 )( ,
fr
VEV
1
43 )( e c t .
U s i n g t h e C A P M
fmf rrErrE )()( ( 3 )
mrE - the expected return of the market portfolio,
rE - the expected return of the business activity, - the coefficient of the business activity,
fm rrE )( . ( 4 )
S u b s t i t u i n g ( 4 ) i n ( 2 )
fmf rrEr
VEV
1
2)1( . ( 5 )
f
mmfm
r
rVrrEVEV
1
/,2cov2)1(
2. ( 6 )
mmm rrVrrVrV , cov 1,1 1cov,2cov a n d
2
, cov
m
mrr
.
S u b s t i t u i n g i n ( 6 )
f
fm
r
VrrEVEV
1
121
.
M o d i f y t h e e x p r e s s 1V i s v a l i d ( 5 )
fmf rrEr
VEV
1
21 .
222 LEXEVE ( 7 )
1F - t h e v a l u e o f a r e t e n t i o n f u n d a t t h e b e g i n n i n g o f t h e f i r s t p e r i o d ,
arEF 11 - t h e v a l u e o f t h e f u n d i m m e d i a t e l y b e f o r e t h e e n d o f f i r s t p e r i o d , ( 8 )
E ( r a ) - t h e e x p e c t e d r a t e o f r e t u r n o n f u n d a s s e t s .
2VE = E [ X ( 2 ) ] - F ( 1 ) ( 1 + k ) + F ( 1 ) [ 1 + E ( r a ) ] - E [ L ( 2 ) ] =
= E [ X ( 2 ) ] + F ( 1 ) [ E ( r a ) - k ] - E [ L ( 2 ) ] . ( 9 )
U s i n g ( 6 ) a n d ( 9 ) t h e v a l u e o f t h e f i r m
f
maa
r
rLkrFXLEkrEFXEV
1
),2())(1()2(co v)2()()1()2()1(
, ( 1 0 )
w h e r e
λ = [ E ( r m ) - r f ] / σ 2m .
A s s u m e t h a t 1F i s r i s k l e s s
c o v [ X ( 2 ) + F ( 1 ) ( r a - k ) – L ( 2 ) , r m ] = c o v [ X ( 2 ) , r m ] +
+ F ( 1 ) c o v ( r a , r m ) – F ( 1 ) c o v ( k , r m ) – c o v [ L ( 2 ) , r m ] , ( 1 1 )
S u b s t i t u t e i n ( 1 0 )
.
1
),2(cov)2(
1
),cov()1()1(
1
),cov()1()()1(
1
),2(cov)2()1(
f
m
f
m
f
maa
f
m
r
rLLE
r
rkFkF
r
rrFrEF
r
rXXEV
( 1 2 )
The value of the firm:
The commercial non-risk management activities.
The value of retention fund.
The value of the firms loss exposure.
Retention funds when the alternative sources of finance have transaction costs
K - the transaction costs from resorting to external financing,
nR - the probability of ruin of the fund,
KE - the expected value of transaction costs,
R n = P r [ L ( 2 ) > F ( 1 ) ( 1 + r a ) ] , KRKE n .
E ( K ) = R n [ F ( 2 ) , L ( 2 ) ] K . ( 1 3 )
E [ V ( 2 ) ] = E [ X ( 2 ) ] + F ( 1 ) [ E ( r a ) - k ] - E [ L ( 2 ) ] – R n K ( 1 4 )
f
na
r
KRLEkrEFXEV
1
)2()()1()2()1( ( 1 5 )
f
mna
r
rKRLkrFX
1
,)2())(1()2(cov, λ = [ E ( r m ) - r f ] / σ 2
m .
mnmmmam
mna
rRKrLrkFrrFrX
rKRLkrFX
,cov),2(cov,cov1,cov1),2(cov
,21)2(cov
. ( 1 6 )
S u b s t i t u i n g i n ( 1 5 )
(17) .
1
),cov(
1
),2(cov)2(
1
),cov(1)1(
1
),cov(1)()1(
1
),2(cov)2()1(
f
mnn
f
m
f
m
f
maa
f
m
r
rRKKR
r
rLLE
r
rkFkF
r
rrFrEF
r
rXXEV
The value of the firm:
• The value from commercial non-risk management activities.
• The value or cost from establishing a fund and investing its assets minus the cost of raising capital to
finance the fund.
• The ( negative ) value contributed by the loss exposure.
• The ( negative ) contribution to value arising from the prospect of incurring transaction costs for
unfunded loses.
Composite financing strategies
• Full insurance and partial insurance
ACV – the actual cash value is the measure of the direct ownership claim of an individual
property. These claims represent rights the income generated from corporate
investments and rights to share in the residual value of the firm.
Fig. 1 Loss distribution and Proportionale Coinsurance
Risk Reduction with Coinsurance
RXE . ( 1 )
E - t h e v a l u e o f e a r n i n g s ,
X - t h e v a l u e o f e a r n i n g s b e f o r e d e d u c t i o n o f r i s k m a n a g e m e n t l o s s ,
R - t h e v a l u e o f l o s s f r o m r i s k m a n a g e m e n t f a c t o r s ,
XE - t h e e x p e c t e d v a l u e o f e a r n i n g s b e f o r e d e d u c t i o n o f r i s k m a n a g e m e n t l o s s ,
RE - t h e e x p e c t e d v a l u e o f l o s s f r o m r i s k m a n a g e m e n t f a c t o r s ,
EE - t h e e x p e c t e d e a r n i n g s o f a f i r m ,
REXEEE . ( 2 )
R - t h e r i s k m a n a g e m e n t c o s t ,
- t h e u n i n s u r e d p r o p o r t i o n o f l o s s ,
1 - t h e i n s u r e d p r o p o r t i o n o f l o s s ,
L - t h e l o s s ,
P - t h e i n s u r a n c e p r e m i u m ,
PLR . ( 3 )
xp - t h e p r o b a b i l i t y o f e a r n i n g s ,
lp - t h e p r o b a b i l i t y o f l o s s ,
x
xpxXE ,
l
lplLE ,
ll
lpPllpRRE ( 4 )
PLElpPlplll
.
x
xpXExX 22 ,
l
lpLElL 22 ,
l
lpPLEPlR 22
LlpLEll
2222 . ( 5 )
T h e i n s u r a n c e p r e m i u m w i l l b e c a l c u l a t e d i n r e l a t i o n t o t h e e x p e c t e d v a l u e o f c l a i m p a y m e n t
gLEfP 11 , ( 6 )
f , g - t h e p o s i t i v e c o n s t a n t s t o r e f l e c t t h e i n s u r e s p r e m i u m l o a d i n g s .
S u b s t i t u i n g i n ( 3 )
gLEfLR 11
gLEffgLEfLERE 111 . ( 7 )
F r o m ( 2 ) a n d ( 4 ) o r ( 2 ) a n d ( 7 )
PLEXEEE , ( 8 )
gLEffXEEE 1 . ( 9 )
E2 - t h e v a r i a n c e o f t h e e a r n i n g o f t h e f i r m ,
22 EEEEE ( 1 0 )
S u b t r a c t i n g ( 1 ) a n d ( 2 )
RERXEXRERXEXEEE ,
S u b s t i t u i n g i n ( 1 0 )
222 2 RERRERXEXXEXEE
22 2 RERERERXEXEXEXE
RDRXXD ,cov2 . ( 1 1 )
D ( X ) , D ( R ) , D ( L ) – t h e v a r i a n c e o f X , R , L .
LELPLEPLRER
222 2 LELELELXEXEXEXEE
222 2 LELELELXEXEXEXE
LDLXXD 2,cov2 .
LLXXE 222 ,cov2 , ( 1 2 )
RX ,cov - t h e c o v a r i a n c e b e t w e e n b u s i n e s s e a r n i n g s X a n d r i s k m a n a g e m e n t c o s t R ,
LX ,cov - t h e c o v a r i a n c e b e t w e e n b u s i n e s s e a r n i n g s X a n d u n i s u r e n c e d l o s s e s L .
Example 1
Let E(X) = 20, E(L) = 2, 1002 X a 202 L , the insurance premium is (6) for
f = 0.2 a g = 0.2. Analyse the expected levels of earnings with respect to the standard
deviations, at different levels of insurance. Solution