Transcript Prezentace aplikace PowerPoint

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]
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 Corporate
Earnings and Cost of Capital
Immediate Investment Decision
Loss
Reduction
Contingent Investment Decision
No Loss
Reduction
Postloss
Reinvestment
Financing
Financing
Preloss
Financing
Postloss
Financing
Internal
funds
Dept
Postloss
Abandonment
Equity
Dept
Internal
funds Insurance
Equity
Tab. 1 Decision Framework for financial Risk Management
Funding
Contingent
Loans
Retention funds when alternative sources of finance have no transaction costs
V 1
- the present value of the firm at the beginnind the first period,
E E i  - the expected earnings for i ih year,
k
- the risk-adjusted discount rate,
rf

- the risk-free rate,
- the risk premium,


E ( Ei )
E ( Ei )
V (1)  

,

i
i
i 1 (1  k )
i 1 (1  r f   )
V (1) 
V ( 2) 
Using the CAPM
(1)
E V 2 
1  rf    ,
E V 3
1  rf    ,
(2)
V (3) 

E ( r )  r f   E ( rm )  r f
E V 4 
1  rf   

ect.
(3)
E rm  - the expected return of the market portfolio,
E r 
- the expected return of the business activity,

- the coefficient of the business activity,
   E ( rm )  rf  .
(4)
Substituing (4) in (2)
V (1) 
V (1) 

E V 2 
1  r f   E rm   r f


.

E V 2   E rm   r f cov V 2 , rm  /  m2
1  rf
(5)
.
(6)
cov V 2 , rm   cov V 11  r , rm   V 1 cov r , rm 
and
Substituing in (6)
V 1 


E V 2   E rm   r f V 1 
1  rf
.

cov r , rm 
 m2
.
Modify the express V 1 is valid (5)
V 1 
E V 2 
.
1  r f   E r m   r f


E V 2   E X 2  E L2 
F 1
F 1 1  E ra 
E(ra)
(7)
- the value of a retention fund at the beginning of the first period,
- the value of the fund immediately before the end of first period,
(8)
- the expected rate of return on fund assets.
EV 2 = E[X(2)] - F(1)(1+k) + F(1)[1+E(ra)] - E[L(2)] =
= E[X(2)] + F(1)[E(ra) - k] - E[L(2)].
(9)
Using (6) and (9) the value of the firm
V (1) 
E  X ( 2)  F (1)E ( ra )  k   E L ( 2)   cov  X ( 2)  F (1)( ra  k )  L ( 2), rm 
1  rf
where
λ = [E(rm)-rf]/σ2m .

,
(10)
Assume that F 1 is riskless
cov[X(2) + F(1)(ra - k) – L(2), rm] = cov [X(2), rm] +
+ F(1) cov(ra, rm) – F(1) cov(k, rm) – cov [L(2), rm],
(11)
 E  X (2)   cov X (2), rm 
V (1)  

1

r


f
 F (1) E (ra )  F (1) cov(ra , rm ) F (1)k  F (1) cov(k , rm ) 



1

r
1

r


f
f
 E L(2)   covL(2), rm 
 
.
1

r


f
(12)
Substitute in (10)
The value of the firm:
 The commercial non-risk management activities.
 The value of retention fund.
 The value of the firms loss exposure.
Retention funds when the alternative sources of finance have transaction costs
K
- the transaction costs from resorting to external financing,
Rn
- the probability of ruin of the fund,
E K  - the expected value of transaction costs,
Rn = Pr [L(2)>F(1)(1+ra)], E K   R n  K .
E(K) = Rn [F(2), L(2)] K.
(13)
E[V(2)] = E[X(2)] +F(1) [E(ra)-k] - E[L(2)] – RnK
V (1) 
E  X ( 2)  F (1)E ( ra )  k   E L ( 2)  Rn K

1  rf

 cov  X ( 2)  F (1)( ra  k )  L ( 2)  Rn K , rm 
1  rf
(14)
(15)
,
λ = [E(rm)-rf]/σ2m .
cov  X ( 2)  F 1 ra  k   L2   Rn K , rm  
 cov  X ( 2), rm   F 1 cov ra , rm   F 1 cov k , rm   cov L ( 2), rm   K cov Rn , rm 
.
(16)
Substituing in (15)

 E  X ( 2)   cov  X (2), rm 
  F (1) E (ra )  F 1 cov( ra , rm ) F (1)k  F 1 cov( k , rm ) 
V (1)  


 
1

r
1

r
1

r


f
f
f

 

 E L(2)   cov L(2), rm   Rn K  K cov( Rn , rm ) 
 
  
.
1  rf
1  rf

 

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.
(17)
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
EX R .
E
- the value of earnings,
X
- the value of earnings before deduction of risk management loss,
R
- the value of loss from risk management factors,
(1)
E  X  - the expected value of earnings before deduction of risk management loss,
E R  - the expected value of loss from risk management factors,
E E  - the expected earnings of a firm,
E E   E  X   E R  .
(2)
R  - the risk management cost,

- the uninsured proportion of loss,
1   
- the insured proportion of loss,
L
- the loss,
P 
- the insurance premium,
R    L  P  .
(3)
px 
- the probability of earnings,
pl 
- the probability of loss,
E  X    x px  ,
x
E L    l p l  ,
l
E R   
 R   pl     l  P   pl  
l

l
(4)
  l pl    P  pl    E L   P   .
l
l
 2  X    x  E  X 2 p  x  ,
x
 2 L    l  E L 2 p l 
l
,
 2 R      l  P     E L   P  2 p l  
l
  2  l  E L  p l    2 2 L 
2
l
.
(5)
The insurance premium will be calculated in relation to the expected value of claim payment
P   1    1  f
 E L   g ,
(6)
f, g - the positive constants to reflect the insures premium loadings.
Substituing in (3)
R    L  1   1  f  E L   g
E R    E L   1   1  f E L   g  1  f   f E L   g .
(7)
From (2) and (4) or (2) and (7)

2
E 
E E   E  X    E L   P   ,
(8)
E E   E  X   1  f   f  E L   g .
(9)
- the variance of the earning of the firm,
 E2  E  E  E  E

2 
(10)
Subtracting (1) and (2)
E  E E   X  E  X   R  E R    X  E  X   R  E R  ,


 E  X  E  X   2 E  X  E  X R  E  R   E R  E  R   
Substituing in (10)
 2  E   E  X  E  X 2  2 X  E  X R  E  R   R  E  R 2 
2
 D X   2cov X , R   DR  .
2
(11)
D(X), D(R), D(L) – the variance of X, R, L.
R  E R    L  P    E L   P    L  E L 




 2 E   E  X  E  X 2  2 E  X  E  X   L  E L   E  L  E L 2 




 E  X  E  X   2 E  X  E  X  L  E  L    2 E L  E L  
2
 D  X   2 cov  X , L    2 D  L 
2
.
  E    2  X   2 cov  X , L    2  2 L  ,
(12)
cov X , R  - the covariance between business earnings X and risk management cost
R 
cov X , L 
- the covariance between business earnings X and unisurenced losses L.
,
Example 1
Let E(X) = 20, E(L) = 2, 2  X   100 a  2 L   20 , 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   0.25 from (2) and (7)
ER0.25  1  0.2  0.25 0.2  2  0.2  2.5

E(E)
E E   20  2.5  17.5 .
and
0
0.25
0.5
0.75
1
17.4
17.5
17.6
17.7
17.8
Tab. 1
For example cov  X , L   20
 0 E   100 2  0   20  0 2  20  10,
 0.25 E   100 2  0.25  20  0.252  20  10.54751,
 0.5 E   100 2  0.5   20  0.52  20  11.18033,
 0.75 E   100 2  0.75  20  0.752  20  11.88486,
 1 E 
 100 2 1  20  12  20  12.64911.
 1 E 
cov  X , L 
 0 E 
 0.25  E 
 0 .5  E 
 0.75  E 
-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. 2
cov X , L 
 0 E 
 0 , 25 E 
 0 ,5 E 
 0 , 75 E 
-4
10
10.16120
10.44031
10.82820
11.31371
-3
10
10.13657
10.3923
10.75872
11.22497
-2
10
10.11187
10.34408
10.68878
11.13553
-1
10
10.08712
10.29563
10.61838
11.04536
0
10
10.06231
10.24695
10.54751
10.95445
1
10
10.03743
10.19804
10.47616
10.86278
2
10
10.01249
10.14889
10.40433
10.77033
3
10
9.987492
10.0995
10.33199
10.53565
4
10
9.962429
10.04988
10.25914
10.58301
5
10
9.937303
10.00000
10.18577
10.48809
10
10
9.810708
9.746794
9.810708
10.00000
15
10
9.682458
9.486832
9.420721
9.48683
20
10
9.552486
9.219544
9.013878
8.94427
Tab. 3
 1 E 
E(E)
cov(X,L)=-20
=1
cov(X,L)=0
cov(X,L)=20
17,8
17,7
=0,75
17,6
=0,5
17,5
=0,25
17,4
=0
  (E )
17,3
8,5
9
9,5
10
10,5
11
11,5
12
12,5
13
Fig. 2
E(E)
cov(X,L)=0
cov(X,L)=-20
17,8
=1
17,7
=0,75
17,6
=0,5
17,5
=0,25
17,4
=0
17,3
9,5
10
10,5
11
11,5
Fig. 3
12
12,5
  (E )
13
E(E)
cov(X,L)=20
cov(X,L)=0
=1
17,8
17,7
=0,75
17,6
=0,5
17,5
=0,25
17,4
=0
17,3
8,5
9
9,5
10
Fig. 4
10,5
11
  (E )
Risk reduction with deductibles
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
Total
settlement
124 000-20 000=104 000
117 000
153 000
Tab. 4
Fig. 5
Distribution of retained loss
Distribution of policy payment
E  X    x px 
x
and
 2  X    x  E  X 2 p  x  ,

- the deductible,
D 
- the premium,
R 
- the actual risk management cost for the firm,
ER  - the expected value of risk management costs,
 R2  
- the variance of risk management cost,
x
 L  D  
R    
  D  
E R   
if
L 
if
L 
.
(13)
 l pl     pl   D  
l
l
  l pl    l pl    l pl     l pl   D  
l 
l 
l 
(14)
l 
  l p l    l    p l   D    E l    l    p l   D  
l 
l
  R  
,
l 
 p(l ) l  D   E R    p(l )   D   E R 
2
l
2
,
(15)
l
D    1  m   l   p l   n
l 
m and n - the parameters of deductibles.
,
(16)
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
The firms expected earnings before deduction of risk management costs are E  X   15000 ,
  X   2 000 , m  0.5 and
n  0 . The deductible 
an be set at 0, 100, 1000, or
10 000 . The business and risk management components of earnings are uncorrelated.
Solution
E E   E  X   E R  .
  E    2  X    2 L  .
The deductible   100 and E L   1720 ,
(17)
D 100   1,5  l  100  p l  
l 100
 1.5 0.1 1 000  100  0.4 10 000  100  0.6 20 000  100  2 520 ,
E R 100   1 720   l  100  p l   D 100  1 720 
 0.1 1 000  100   0.04 10 000  100   0.06 20 000  100   2 520  2 560 ,
2
R    l  D 100   E R 100 2 p l    100  D 100   E R 100 2 p l  
 100
l 
 0  2 520  2 560  0.6  100  2 520  2 560  0.2  0.1  0.04  0.06   2 400 ,
2
2
E E   15 000 E R100  12 440,
2
R    2  X   2 400  2 000 2  2 000 .59991 .
 100 E    100
D 
E R 
 2 R 
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

  E 
Tab. 6
E(E)
13400
13200
13000
12800
12600
12400
12200
1000
2000
3000
4000
Fig. 6
5000
6000
Financial Layering
Costs
Actual cost
Present expected value
Source
(AC)
(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.
Insurance
New Issue
C
1  k1
C
C
C=L
C=L
C
L
0
L
0
Fig. 7
Fig. 8
Internal Liquid Resources
C
C=L
C
1  k1
L
0
Fig. 9
Two-layer financing methods
C
C
D
F
G
B
A
A
0

Layer 1
Internal financing
0
L
Layer 2

Layer 1
Insurance of
L
Layer 2
Internal financing New issues
external financing
Fig. 10
Three-layer financing method
Cover payments
Loss
0
0 L
L 
  L U
U 
L U
Tab. 8
C
F
K
J
P
D
G
D
H
B
A

0
Layer 1
U
Layer 3
Layer 2
Fig. 11
L