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: pinda@dec.euba.sk

Sylabus of the lectures

 

        Preloss financing: - Retention funding.

        Composite financing strategies:

- Full insurance and partial insurence.

- Insurance and risk reduction : - Risk reduction with coinsurance.

- Risk reduction with deductibles.

        Financial layering.

Identify Risk,Management Events:

Measure Capital Costs

Estimate Effects on CorporateEarnings and Cost of Capital

Immediate Investment Decision Contingent Investment Decision

LossReduction

No LossReduction

PostlossReinvestment

PostlossAbandonment

Financing Financing

PostlossFinancing

PrelossFinancingDept

Equity

Internal funds

Dept

Equity

Internal funds Insurance

Contingent Loans

Funding

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

For 25.0 from (2) and (7)

5.22.022.025.02.0125.0 RE and 5.175.220 EE .

Tab. 1

0 0.25 0.5 0.75 1

E(E) 17.4 17.5 17.6 17.7 17.8

For example 20,covLX

.64911.122012012100

,88486.112075.02075.02100

,18033.11205.0205.02100

,54751.102025.02025.02100

,102002002100

21

275.0

25.0

225.0

20

E

E

E

E

E

Tab. 2

LX,cov

E0 E25.0 E5.0 E75.0 E1

-20 10 10.54751 11.18033 11.88486 12.64911

-15 10 10.42832 10.95445 11.56503 12.24744

-10 10 10.30776 10.72381 11.2361 11.83216

-5 10 10.18577 10.48809 10.89725 11.40175

Tab. 3

LX ,cov E0 E25,0 E5,0 E75,0 E1

- 4 1 0 1 0 . 1 6 1 2 0 1 0 . 4 4 0 3 1 1 0 . 8 2 8 2 0 1 1 . 3 1 3 7 1

- 3 1 0 1 0 . 1 3 6 5 7 1 0 . 3 9 2 3 1 0 . 7 5 8 7 2 1 1 . 2 2 4 9 7

- 2 1 0 1 0 . 1 1 1 8 7 1 0 . 3 4 4 0 8 1 0 . 6 8 8 7 8 1 1 . 1 3 5 5 3

- 1 1 0 1 0 . 0 8 7 1 2 1 0 . 2 9 5 6 3 1 0 . 6 1 8 3 8 1 1 . 0 4 5 3 6

0 1 0 1 0 . 0 6 2 3 1 1 0 . 2 4 6 9 5 1 0 . 5 4 7 5 1 1 0 . 9 5 4 4 5

1 1 0 1 0 . 0 3 7 4 3 1 0 . 1 9 8 0 4 1 0 . 4 7 6 1 6 1 0 . 8 6 2 7 8

2 1 0 1 0 . 0 1 2 4 9 1 0 . 1 4 8 8 9 1 0 . 4 0 4 3 3 1 0 . 7 7 0 3 3

3 1 0 9 . 9 8 7 4 9 2 1 0 . 0 9 9 5 1 0 . 3 3 1 9 9 1 0 . 5 3 5 6 5

4 1 0 9 . 9 6 2 4 2 9 1 0 . 0 4 9 8 8 1 0 . 2 5 9 1 4 1 0 . 5 8 3 0 1

5 1 0 9 . 9 3 7 3 0 3 1 0 . 0 0 0 0 0 1 0 . 1 8 5 7 7 1 0 . 4 8 8 0 9

1 0 1 0 9 . 8 1 0 7 0 8 9 . 7 4 6 7 9 4 9 . 8 1 0 7 0 8 1 0 . 0 0 0 0 0

1 5 1 0 9 . 6 8 2 4 5 8 9 . 4 8 6 8 3 2 9 . 4 2 0 7 2 1 9 . 4 8 6 8 3

2 0 1 0 9 . 5 5 2 4 8 6 9 . 2 1 9 5 4 4 9 . 0 1 3 8 7 8 8 . 9 4 4 2 7

17,3

17,4

17,5

17,6

17,7

17,8

8,5 9 9,5 10 10,5 11 11,5 12 12,5 13

E(E)

)( E

=1

=0,75

=0,5

=0,25

=0

cov(X,L)=20 cov(X,L)=0 cov(X,L)=-20

Fig. 2

17,3

17,4

17,5

17,6

17,7

17,8

9,5 10 10,5 11 11,5 12 12,5 13 )(E

E(E)

=0,75

=0,5

=0,25

=0

=1 cov(X,L)=0 cov(X,L)=-20

Fig. 3

=0,75

=0,5

=0,25

=0

=1

)(E17,3

17,4

17,5

17,6

17,7

17,8

8,5 9 9,5 10 10,5 11

E(E)

cov(X,L)=20 cov(X,L)=0

Fig. 4

Deductible : - per loss deductible / is applied to each loss /,

- the cumulative deductibles / is applied to the annual total of losses /.

Settlement under policy with 20 000 deductibles

Per loss deductible Cumulative deductible

Loss Payment Cumulative loss Payment

16 000 0 173 000 173 000-20 000=153 000

33 000 33 000-20 000=13 000

124 000 124 000-20 000=104 000

Total settlement

117 000 153 000

Risk reduction with deductibles

Tab. 4

Fig. 5

Distribution of retained loss

Distribution of policy payment

x

xpxXE and x

xpXExX 22 ,

- the deductible,

D - the premium,

R - the actual risk management cost for the firm,

RE - the expected value of risk management costs,

2R - the variance of risk management cost,

LD

LDLR

if

if . ( 1 3 )

DlplplREll

Dlpllpllpllplllll

( 1 4 )

DlpllEDlpllpllll

,

l l

REDlpREDllpR 22 )( )( , ( 1 5 )

nlplmDl

1 , ( 1 6 )

m a n d n - t h e p a r a m e t e r s o f d e d u c t i b l e s .

Example 2 The loss distribution by Tab. 5

l 0 100 1000 10 000 20 000

p(l) 0.6 0.2 0.1 0.04 0.06

Tab. 5

T h e f i r m s e x p e c t e d 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 c o s t s a r e 00015XE ,

0002X , 0 and 5.0 nm . T h e d e d u c t i b l e a n b e s e t a t 0 , 1 0 0 , 1 0 0 0 , o r

00010 . T h e b u s i n e s s a n d r i s k m a n a g e m e n t c o m p o n e n t s o f e a r n i n g s a r e u n c o r r e l a t e d .

S o l u t i o n

REXEEE .

LXE 22 . ( 1 7 )

T h e d e d u c t i b l e 100 a n d 1720LE ,

lplDl 100

1005,1100

5202100000206.0100000104.010000011.05.1 ,

,560252021000002006.01000001004.010000011.0

72011001007201100

DlplRE

,400206.004.01.02.0560252021006.0560252020

100100100100100

22

222100

lpREDlpREDlRl

,4401210000015 REEE

59991.000200024002 222100100 XRE .

D RE R2 E(E) E

0 2 580 2 580 0 12 420 2 000

100 2 520 2 560 2 400 12 440 2 000.5999

1 000 2 250 2 470 153 600 12 530 2038.0383

5 000 2 100 2 420 4 090 640 12 580 2 844.405

10 000 900 2 020 8 847 600 12 980 3 584.3549

15 000 450 1 870 15 585 600 13 130 3 947.8602

20 000 0 1 720 25 143 600 13 280 5 398.4813

Tab. 6

12200

12400

12600

12800

13000

13200

13400

1000 2000 3000 4000 5000 6000

E(E)

Fig. 6

Financial Layering

Costs

Source

Actual cost

(AC)

Present expected value

(EC)

Insurance (1+ a) E(L) (1+ a) E(L)

New Issue b + (1+ c)L [PLAb + (1+ c)E(L)]( 1+ k1)-1

Internal Liquid Resources L E(L) (1+ k2)-1

Tab. 7

a, b, c, - the positive constants, (a c),

k1 , k2 - the risk - adjusted discount rates,

PL,A - the probability that a loss will arise that is above some threshold level A.

C

L0

C=L

L

C

0

C=L

C

11 k

C

Insurance New Issue

Fig. 7 Fig. 8

Internal Liquid Resources C

L0

C=L

11 k

C

Fig. 9

Two-layer financing methods

A

L

C

D G

0

Layer 1 Layer 2

Internal financing Insurance of

external financing

B

F

0 L

C

A

Layer 1 Layer 2

Internal financing New issues

Fig. 10

Three-layer financing method

Cover payments Loss

0 L0

L UL

U UL

Tab. 8

L 0

B

D

K

A

D

H

J

G

Layer 1 Layer 2 Layer 3

F

C

U

P

Fig. 11