[PPT] - On the Determination of Capital Charges in a Discounted Cash Flow PowerPoint Presentation

SLIDE 1

On the Determination of Capital Charges in a Discounted Cash Flow Model

Eric R. Ulm Georgia State University

SLIDE 2

Motivation

Solvency II

– Required Assets determined on a consolidated basis – Assets allocated to the lines of business on a marginal basis – Division into “Reserves” and “Capital” is line by line – Do Capital Charges on capital and change in reserves cancel for performance analysis of line managers?

SLIDE 3

Motivation

Multiple Candidates for Reserves:

– U.S. Statutory Reserves; – U.S. GAAP Reserves; – U.S. Tax Reserves; – Fair Value of Liabilities; – Assets at a somewhat conservative solvency standard (Solvency II uses 75%); – Expected Loss under the realistic measure discounted at the risk-free rate.

SLIDE 4

Performance Evaluation

(50,000) 50,000 100,000 150,000 200,000 20 40 60 80 Year Income less Capital Charges

Tax Reserves Evaluation Reserves

SLIDE 5

Discounted Cash Flow Model

Myers and Cohn (1987)
Cummins (1990)
Taylor (1994)

– Assumes reserves are “technical reserves”, i.e. discounted value of expected losses – Free parameter is “capital”, i.e. assets = capital + technical reserves.

Overview in Cummins and Phillips (2000)

SLIDE 6

Examples Single Premium / Single Loss

1

CF

i

CF

1 T

CF

T

CF

T

A P CF

, ) ( , 1 , , 1 , 1

) 1 (

T f T T T

V r A P A A CF

1 i T-1 T

) ( , 1 ) ( , , 1 , , 1

) 1 (

T i T i f T i T i T i i

V V r A A A CF

) ( , 1 , 1 , 1

) 1 ( ) 1 (

T T f T T T T T T

V r A L A CF

SLIDE 7

Examples Single Premium / Single Loss

1 1 1 ) ( , , 1

) 1 )( 1 ( ) 1 )( 1 ( ) ( ) 1 )( 1 (

T i T i i T i i f T i T T

x y y V x y r y A x y L E P

1 x y

1 1 1 ) ( , ,

) 1 )( 1 ( ) 1 )( 1 ( ) ( ) 1 (

T i T i i T i i f T i T f T

x y y V x y r y A r L E P

– Equivalently

T i i T f T T i

A r L E A

, ,

) 1 ( ] [

) ( , ) ( ,

) 1 ( ] [

T i i T f T T i

V r L E V

SLIDE 8

Examples Single Premium / Single Loss

Evaluation Reserves and capital
solves for (by induction)

) ( , e T i

V

) ( , , e T i T i

V A

) ( ) 1 ( ) 1 ]( [

) ( , 1 , 1 ) ( , 1 ) ( , 1 , 1

x V A V V r A L E

e t t t t t t e t t f t t T

x V L E x r x A x L E V

T T T f T T T e T T

1 ) ] [ ( 1 )] 1 ( [ ) 1 ( ] [

) ( , 1 , 1 ) ( , 1

) ( ) 1 (

) ( , , ) ( , 1 ) ( , ) ( , 1 ) ( , ,

x V A V V V V r A

e T i T i T i T i e T i e T i f T i

1 1 ) ( , ) ( , 1 1 1 , ) ( ,

) 1 ( ) ( ) 1 ( )] 1 ( [ ) 1 ( ] [

T i j i j T j T j T i j i j f T j i T T e T i

x V V x r x A x L E V

P V e

T ) ( ,

SLIDE 9

Examples Single Premium / Single Loss

More intuitively ...

1 1 ) ( , ) ( , 1 1 1 , ) ( ,

) 1 ( ) ( ) 1 ( )] 1 ( [ ) 1 ( ] [

T i j i j T j T j T i j i j f T j i T f T e T i

x V V x r x A r L E V

SLIDE 10

Examples Multiple Premium / Multiple Loss

Nonstochastic P and A (i.e. losses are

uncorrelated and premiums paid with certainty ...

Otherwise replace with in the

premium equations, and with in the reserve equation. Make similar substitutions for

1 1 1 ) ( 1 1 1

) 1 )( 1 ( ) 1 )( 1 ( ) ( ) 1 )( 1 ( ) 1 (

T i T i i i i f i T i i i T i i i

x y y V x y r y A x y L E x P

i

A

] [

i

A E

j

A

] | [

i j

A E

i

P

SLIDE 11

Examples Multiple Premium / Multiple Loss

More intuitively ...
Defining
Practically, the and often depend on

the premiums.

i

A

1 1 1 ) ( 1 1

) 1 )( 1 ( ) 1 )( 1 ( ) ( ) 1 ( ) 1 (

T i T i i i i f i T i i f i T i i f i

x y y V x y r y A r L E r P

i T i j i j f j T i j i j f j i

A r P r L E A

1 1 1

) 1 ( ) 1 ( ] [

) ( 1 1 ) (

) 1 ( ) 1 ( ] [

i T i j i j f j T i j i j f j i

V r P r L E V

) ( i

V

SLIDE 12

Examples Multiple Premium / Multiple Loss

) 1 ( ) 1 ]( [ ) 1 (

1 f i i i

r A L E P

) (

) ( ) ( 1 ) ( ) ( 1 ) (

x P V A V V V V

i e i i i i e i e i

1 1 ) (

) 1 ( ) 1 ( ] [

T i j i j j T i j i j j e i

x P x L E V

1 1 ) ( 1 ) ( 1 1 1

) 1 ( ] [ ]) [ [( ) 1 ( )] 1 ( [

T i j i j j j j j T i j i j f j

x P V L E V x r x A

1 1 ) (

) 1 ( ) 1 ( ] [

T i j i j f j T i j i j f j e i

r P r L E V

1 1 ) ( ) ( 1 1 1

) 1 ( ) ( ) 1 ( )] 1 ( [

T i j i j j j T i j i j f j

x V V x r x A

SLIDE 13

Solvency II Context One Period

In one year assets are

and liabilities are 0. Solve

to find
Premium is with

P L r A

f

) 1 ( )] 1 ( 1 [

1 1 ,

995 . ) 1 ( )] 1 ( 1 [ Pr

1 1 ,

P L r A

f

)] 1 ( 1 [ ) 1 (

995 . 1 ,

1

f L

r P A

) 1 ( ]) [ ( ] [

1 995 . 1

1

f L

r R L E L E P

x r x R

f

1 ) 1 (

SLIDE 14

Solvency II Context Multiple Period

Last period is similar:
Other periods require the determination of
Key insight: , the premium which would be

charged at time i to cover the loss at time t, must be and this premium can be found from the previous analysis.

Find the market values recursively.

)] 1 ( 1 [ ) 1 (

) ( , 1 995 . , 1 f T T L T T

r V A

T

] [ ] [

] [ T i i T

L MV L MV

T i

P, ~

] [

T i L

MV

SLIDE 15

Solvency II Context Multiple Period

) 1 ( ] [ ) 1 ( ] [ ] [

) ( , 1 1 1 f t j T j f T j t j

r V L MV r r L MV L MV 1 1 1 ) 1 )( 1 ( x y x x r

1 1 1 ) ( , 1 995 .

) 1 ( ) 1 ( 1 ) 1 ( ]) [ ( ] [ ] [

T i T i f T i T T f T L T T

r r V r r r R L E L E L MV P

t

2 1 995 . ) ( , ) ( , 1 ,

1 ) 1 ( ]) [ ( ] [ ) ( )] 1 ( 1 [ Pr

i T i T f T L T T i T i f T i

r r R L E L E V V r A

t

995 . ) 1 ( ) 1 (

1 2 2 1 ) ( , T i j i j i j f t j

r r V r

Assets from

SLIDE 16

Solvency II Context Multiple Period

)] 1 ( 1 [ ) ( 1 )] 1 ( 1 [ ) 1 ( ]) [ ( ] [

) ( , ) ( , 1 2 1 995 . , f T i T i i T f i T f T L T t i

r V V r r r R L E L E A

t

)] 1 ( 1 [ ) 1 ( ) 1 (

1 2 2 1 ) ( , f T i j i j i j f T j

r r r V r

Assets

SLIDE 17

Solvency II Context Multiple Period

ann

r i T f f f i T T e T i

s R r R r x r r r x L E V

| 2 2 ) ( ,

) 1 ( 1 1 ) 1 )]( 1 ( 1 [ ) 1 ( ] [  

ann t

r i T f f i T T L

s R x r r x R L E

| 2 2 995 .

1 1 ) 1 )]( 1 ( 1 [ ) 1 ( ]) [ (  

1 2 1 | 1 ) ( , 1 ) ( , ) ( , 1

) 1 )]( 1 ( 1 [ ) 1 ( ) 1 )]( 1 ( 1 [ ) (

T i j f i j r i j T j T i j i j f T i T i

r r x s R V r x r V V

ann

 

Evaluation Reserves

1 1 1 1 ) 1 ( ) 1 )( 1 (

f f ann

r y r x r r

SLIDE 18

Tax Reserves are Eq. Principle reserves

at 7%.

Guess
Assets are

Examples Two Period Loss

400 ] [

1

L E 500

995 .

1

L

500 ] [

2

L E 700

995 .

2

L

% 6

f

r

% 10 x

1

P P

) 1 ( 1 ) 1 (

1 ) ( 1 ) ( 995 .

1

f i i i i L i

r MVL P V V A

i

SLIDE 19

Examples Two Period Loss

Market Value of Liabilities are
sets

1 1 1

) 1 )( 1 ( ) ( ) 1 )( 1 (

T i j i j f j T i j i j j i

x y r y A x y L E MVL

1 1 1 ) (

) 1 ( ) 1 )( 1 (

T i j i j j T i j i j j

x P x y y V

5 430.910689

MVL

SLIDE 20

Examples Two Period Loss

Balance Sheet Items for Two Premium Two Loss Example

Time

Capital 0.00 0.00 491.69 75.85 0.00 60.78 0.00 1 48.31 7.52 601.13 129.43 50.22 120.00 51.07

i

) ( i

V

) ( i

V

i

A

i

A

) (e i

V

i

MVL

i

CF

) (e i

V

Cash Flow Cash Income Change in Capital Charges (60.78) 1 (53.15) 56.30 (50.22) (6.08) 2 132.00 (38.22) 50.22 (12.00)

Income Statement Items for Two Premium Two Loss Example

SLIDE 21

Examples Two Year Term Life

$100,000 face, 1000 identical individuals
is binomial with probability 0.025
is binomial with probability 0.02

020 .

x

q

025 .

1 x

q

1000 P

980 ] [ 1 P E

)] 1 ( 1 [ ) 1 ( ) (

1 995 . | 1 1

1 2

f N L

r N N A

1 2 | N

L

1

N

5078876 . 320 05 . 312 , 412 , 2 ] [

1

A E

SLIDE 22

Examples Two Year Term Life

) 1 )( 1000 ( * 000 , 100 1000 )] 1 ( 1 [

1 1

N r A MVA

f

) 1 ( ) ]( [ ) 1 ( ] | [

1 1 1 1 2 1

y r y N A N y N L E MVL

f

99.5% solvency at N1 = 968
Determine

87683598 . 579 48 . 501 , 237 , 4 A

20 . 185 , 2

82 . 338 , 161 ] [

) ( 1 e

V E

SLIDE 23

Examples Two Year Term Life

i

] [

) ( i

V E

] [

) ( i

V E

] [

i

A E

] [

i

A E

] [

) (e i

V E

] [

i

MVL E

Expected Capital 0.00 0.00 2,970,357.36 923,348.50 0.00 785,162.09 0.00 1 0.00 (169,829.39) 3,112,684.37 801,363.61 161,338.82 809,854.19 233,516.71

Balance Sheet Items for Term Life Example Income Statement Items for Term Life Example

i

] [

i

CF E

] [

) (e i

V E

Cash Flow Expected Cash Income Change in Expected Capital Charges (785,162.09) 1 53,824.11 239,855.03 (161,338.82) (78,516.21) 2 890,839.61 (80,353.40) 161,338.82 (80,985.42)

SLIDE 24

Examples Whole Life

We need ,
Assume
Solve for

995 . |

i i N

D

] | Pr[

i j

N N

1 1 1 1 1 1

] [

i i i i i i

N d N c N MVL

) 1 ( 1 ~ ~ ) 1 ( 000 , 100 ] [

995 . | 1 ) ( ) ( 1 995 . | 995 . | f N D i i i i i N D i N D i i

r N c V N V N N A

i i i i i i

] [ ] [ ) 1 ( 1

995 . | 1 i i i i f N D i i i

N b N a r N d N

i i

k ki k ki i i N k i k i k

N f N e N A N N N A E

i

1000

] | Pr[

SLIDE 25

Examples Whole Life

Finally (whew ...)
Then
gives

1 1 1 1 1

) 1 )( 1 ( ) ]( [ ) 1 )( 1 ( 000 , 100 ] [

t i j i j f i ij t i j i j j x i x i j i i i

x y r y N e x y q p N N MVL

1 1 1 ) (

) 1 ( ) 1 )( 1 ( ) ]( [ ) 1 )( 1 ( ~

t i j i j i x i j i t i j i j f i ij t i j i j j i x i j i

x p N x y r y N f x y V p yN

] [ ] [

i i i i

N d N c

] 1000 [ MVL

] 1000 [ ] 1000 [ d c

SLIDE 26

Examples Whole Life

Assumptions Premium Equivalence Principle $1203.30 Sol 99.5%, Tax EP 6% $1234.95 Sol 99.5%, Tax EP 6.5% $1272.80 Sol 99.5%, Tax CRVM 6.5% $1301.37 Sol 99%, Tax EP 6% $1233.50 Sol 95%, Tax EP 6% $1229.28

1980 CSO on 40 year old.

SLIDE 27

Examples Whole Life

i

] [

i

CF E

] [

) (e i

V E

Cash Flow Expected Cash Income Change in Expected Capital Charges (883,845) 1 48,144 1,030,638 (942,253) (88,385) 2 110,037 1,059,478 (967,070) (92,409) 3 57,235 1,087,511 (996,865) (90,646)

i

] [

) ( i

V E

Expected Cash Income Change in Adjusted Expected Capital Charges Adjusted Income 1 1,030,638 (973,497) (88,385) (31,244) 2 1,059,478 (1,002,048) (89,284) (31,854) 3 1,087,511 (1,032,237) (84,024) (28,749)

SLIDE 28

Examples Whole Life

Expected Balance Sheet Items 5,000,000 10,000,000 15,000,000 20,000,000 25,000,000 30,000,000 35,000,000 10 20 30 40 50 60 Tax Rsvs Evaluation Rsvs Market Values Required Assets

SLIDE 29

Examples Whole Life

Performance Evaluation

(50,000) 50,000 100,000 150,000 200,000 20 40 60 80 Year Income less Capital Charges

Tax Reserves Evaluation Reserves

SLIDE 30

Examples GMDB

) ( ) 1 ( ) ( 1

1 2 1 1

d N S d N e q q S MVL

t i t r t t i

f

) 1 ( ) 1 (

t i t t

S q

t t r S d

t i

2 ] ) 1 ( ln[

2 1

t t r S d

t i

2 ] ) 1 ( ln[

2 2

SLIDE 31

Examples GMDB

) 1 (

) 995 . ( 995 .

1 1

N i s

e S

i

)] 1 ( 1 [ ) 1 )( , 1 ( ) ( ) 1 ( ) (

995 . 995 .

1 1

f s s i

r qMax MVL q S A

i i

Solvency criterion gives

)] 1 ( 1 [ )] ( ~ ) 1 ( ) ( ~ [

995 . ) ( ) (

1

f i s i

r S V q S V

i

SLIDE 32

Examples GMDB

Assume
from option pricing theory.
“x” is the free parameter, and equations

give:

% 6

f

r

% 34

% 2 q

% 10

0009422 .

% 07 . 12 x

SLIDE 33

Examples GMDB

0.04
0.02

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.0 0.5 1.0 1.5 2.0 2.5 Stock Value Assets Eval Rsv Capital MVL