11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance1 Profit...

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

1

Profit analysis via risk-adjusted performance

indicators in life insuranceRosa Cocozza, Università di Napoli Federico II

Emilia Di Lorenzo, Università di Napoli Federico IIAlbina Orlando, Consiglio Nazionale delle Ricerche

Marilena Sibillo, Università di Salerno


2

The research questions

The optimization of a structured insurance policy Which is the “optimal” price?

In a pure “state price” approach the “optimal” price is the arbitrage free since this is the equilibrium price

In a practical perspective the “optimal” price is one that can be appraised by policyholders that is “adequately profitable” to the insurer

Which factor influence the “optimal” price? The features of the policy The investment portfolio (replication alternatives) The return on capital at risk set by top management


3

The case analyzed: temporary variable annuity

The temporary variable annuity structure Pure Single Premium: U Office Premium: U(1+) Guaranteed annual rate: g (<risk-free rate)

Guaranteed minimal payment: R Additional Benefit indexed to the periodical performance of a

reference Stock Exchange Index Participation rate: (≤ 1) Duration: T years

Insurance portfolio annual OutFlow (portfolio installment):

1;01

0t

tpxtt ges

SMAXnpROF

m

t

txt

gep

UR

1


4

The investment coverage As shown by Brennan and Schwartz (1976;1979)

and Boyle and Schwartz (1977), the equity (index) linked policy with a minimum guaranteed rate of return is equivalent either to a plan providing a fixed benefit plus a call

option, or to a plan providing a benefit of the value of the

reference portfolio plus a put option

Asset Liability

Call Option Portfolio

Zero Coupon Bond Portfolio

Index Linked Provision (Portfolio)

Capital


5

The valuation approach

As shown by Cocozza (2005), the insurance business evolution can be described by means of the intermediation portfolio (economic value approach)

in order to gain an insight into the value of the business, since it accounts for the difference between asset and liability at any time of the portfolio cycle

the income flows (current earning approach) in order to gain an insight into the dynamics of the business, since it

accounts for the difference between the profit components periodically accrued

In order to compute risk-adjusted performance measures, we used both.


6

Specification of the valuation model The reference funds backing life insurance liabilities

contain both bond and call option In order to model properly the yearly returns, which

determine the readjustment of the contractual benefits, we have to model both interest rate risk and stock (index) market risk.

We adopt a two-factor diffusion model obtained by joining a one-factor CIR model for the interest rate risk and a Black-Scholes model for the stock index market risk


7

Interest Rate Uncertainty

The single source of uncertainty is the spot rate rt, which is a diffusion process described by the stochastic differential equation (Cox, Ingersoll and Ross, 1985)

where Zr(t) is a standard Brownian motion. As known, this specification assumes a mean-

reverting drift, with long term rate speed of adjustment , and a ”square root” diffusion, with volatility parameter . As is well-known, the CIR process implies non-central chi-square transition distribution for rt .

tdZrdtrdr rttt


8

The term structure of the interest rate Interest rate have been calibrated to Eonia (Euro OverNight

Index Average) and Euribor (Euro Interbank Offered Rate) rates on 11/04/2007

TERM STRUCTURE (11/04/2007)

3,60%3,80%4,00%4,20%4,40%4,60%

0 2 4 6 8 10

time (years)

Estimated Actual Mean

(rate) 4,513%

(force) 4,414%

0,10

5%

0,09


9

Stock Price Uncertainty

Also for the stock market we assume a single source of uncertainty, expressed by the stock index St; the diffusion process for the stock index is given by the stochastic differential equation

where Zs(t) is a standard Brownian motion with the property

Thus we have a geometric Brownian motion, with instantaneous expected return s and volatility s, which implies a lognormal transition density for St

dttdZtdZ Srt cov

tdZdtSdS Sstst


10

The valuation context

In the CIR model the preferences prevailing on the market (the market price of interest rate risk) are specified by the function:

under the CIR approach – which is a general equilibrium approach – it is shown that this form of the preference function avoids riskless arbitrage.

The market price of risk for the stock market has the classical form:

thus no additional parameter is needed in order to specify the preferences in this case.

tt

r rtrh ,

tt

S rtSh

,


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The strategic variables

Features of the policy Guaranteed rate of return: g with rf<g≤2% Participation rate:

The investment alternatives Perfect matching Imperfect matching (controlled risk assumption)

The return on capital at risk set by top management


12

The input data set The temporary index-linked annuity structure

Pure Single Premium: U = 2.325,00€ Loadings = 7% Office premium = 2.500,00€ Guaranteed annual rate: g = [0%; 0,5%; 1%; 1,5%; 2%] Guaranteed minimal payment: R calculated with actuarial methodology

on the basis of the different g-rates the mortality table IPS55 entry age of the insured = 40 years

Additional Benefit indexed to the performance of the SPMIB as quoted on the Italian Stock Exchange (Fixing Value Bloomberg)

Participation rate: = [70%; 80%; 90%; 100%] Duration: 10 years Number of policy initially sold = 1.000 homogeneous contracts Initial Insurer Capital = 100.000,00€ (=4%*U(1+)*1.000)


13

The cash flow mapping: Plain VanillaT Asset Flows Liabilities Flow Net

0 NF0

1 NF1

2 NF2

t NFt

m NFm

0

0111 ;0

11

s

essMAX

enpROF

g

gpxsp

0

202

222 ;01

1s

essMAX

enpROF

g

gpxsp

0

0;01

1s

ssMAX

enpROF

tt

tpxttsp

g

g

0

0;01

1s

essMAX

enpROF

mm

mpxmmsp

g

g

psp nUIF 1)0( UnPVPVOF ptctb ;0;00

g

gessMAX

esnpRIF pxsp

01

0

11 ;01

1

2

0220

22 ;01

1 g

gessMAX

esnpRIF pxsp

t

ttpxttspg

gessMAX

esnpRIF

0

0

;01

1

m

mmpxmmspg

gessMAX

esnpRIF

0

0

;01

1


14

The cash flow mapping: Path DependentT Asset Flows Liabilities Flow Net

0 NF0

1 NF1

2 NF2

t NFt

m NFm

0

0111 ;0

11

s

essMAX

enpROF

g

gpxsp

0

202

222 ;01

1s

essMAX

enpROF

g

gpxsp

0

0;01

1s

ssMAX

enpROF

tt

tpxttsp

g

g

0

0;01

1s

essMAX

enpROF

mm

mpxmmsp

g

g

psp nUIF 1)0( UnPVPVOF ptctb ;0ˆ;00

g

gessMAX

esnpRIF pxsp

01

0

11 ;01

1

g

gessMAX

esnpRIF pxsp

12

1

22 ˆ;0ˆ

11

g

gessMAX

esnpRIF tt

t

pxttsp

1

1

ˆ;0ˆ

11

g

gessMAX

esnpRIF mm

m

pxmmsp

1

1

ˆ;0ˆ

11


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The simulation of the final result Computation of the final result has been derived by numerical

methods, using Monte Carlo simulations for the bivariate process (path = 10000)

The risk sources are the interest rate (modeled on the basis of the CIR Process) and the price of the underlying index (modeled on the basis of a lognormal distribution) The intermediate net flows are reinvested at the corresponding

(expected) rates up to the expiration date Computation provide us with the Expected Profit E[P] at the end

of policy duration both for the plain vanilla replication (perfect matching) and the path dependent replication (imperfect matching)

The error term sets for the simulation procedure have been selected constant across different alternatives in order to maximize the comparison value


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The Expected ProfitPlain Vanilla Replication (perfect matching) 0,00% 0,50% 1,00% 1,50% 2,00%

70% 508.840,60 475.043,90 438.906,20 400.170,00 358.584,20

80% 428.302,10 401.041,90 371.273,50 338.701,50 303.037,40

90% 347.763,50 327.039,80 303.640,80 277.233,00 247.490,50

100% 267.224,90 253.037,70 236.008,20 215.764,50 191.943,70


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The Expected ProfitPath Dependent Replication (imperfect matching) 0,00% 0,50% 1,00% 1,50% 2,00%

70% 895.034,60 865.050,90 825.698,50 770.109,90 702.252,20

80% 869.666,60 846.764,20 813.321,90 761.490,00 695.799,20

90% 844.298,70 828.477,40 800.945,30 752.870,00 689.347,50

100% 818.930,60 810.190,60 788.568,70 744.250,10 688.584,50


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The CVaR 99%Plain Vanilla Replication (perfect matching) 0,00% 0,50% 1,00% 1,50% 2,00%

70% 577.337,42 539.738,48 499.535,06 456.440,89 410.176,60

80% 487.737,95 457.410,89 424.293,48 388.057,98 348.380,50

90% 398.138,49 375.083,35 349.051,89 319.673,10 286.584,40

100% 308.539,08 292.755,76 273.810,30 251.289,20 224.788,40


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The CVaR 99%Path Dependent Replication (imperfect matching)

0,00% 0,50% 1,00% 1,50% 2,00%

70% 1.033.148,60 994.601,98 944.859,45 879.547,20 801.954,85

80% 1.011.516,41 979.701,39 935.075,39 873.040,40 797.433,79

90% 990.333,63 965.062,03 925.733,02 866.861,07 793.082,85

100% 969.738,45 950.852,86 916.505,38 860.982,81 795.325,94


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The investment alternative

€ 0€ 100€ 200€ 300€ 400€ 500€ 600€ 700€ 800€ 900

€ 1.000

€ 0 € 200 € 400 € 600 € 800 € 1.000 € 1.200

CVaR (thousands)

Exp

ecte

d pr

ofit (

thou

sand

s)

Plain Vanilla Path Dependent


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Decision Making Criteria

The economic sustainability and/or profitability can be therefore appraised through the following:

Expcted Return on Capital = E[RoE] Risk-Adjusted Return on Risk-Adjusted Capital =

C.R.A.R.o.C. Economic Value Added = E.V.A.


22

The E[RoE]

The Expected Return on Capital can be evaluated by means of:

We concentrated on an average intertemporal measure, since we are concerned with the global profitability of the issue.

1

ˆ1][

1

m

CAPITALInitial

PROFITERoEE


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The E[RoE] Map:Plain Vanilla Replication

0,00% 0,50% 1,00% 1,50% 2,00%70%

80%

90%

100%

15,00%-20,00%

10,00%-15,00%

5,00%-10,00%

0,00%-5,00%


24

The E[RoE] Map:Path Dependent Replication

0,00% 0,50% 1,00% 1,50% 2,00%70%

80%

90%

100%

25,00%-26,00%

24,00%-25,00%

23,00%-24,00%

22,00%-23,00%

21,00%-22,00%


25

The C.R.A.R.o.C.

The Coherent Risk Adjusted Return on Risk Adjusted Capital can be evaluated by means of:

We concentrated on an average intertemporal measure, since we are concerned with the global profitability of the issue.

mCVaR

PROFITECRAROC

1ˆ


26

The C.R.A.R.o.C. Map:Plain Vanilla Replication

0,00% 0,50% 1,00% 1,50% 2,00%70%

80%

90%

100%

27,50%-28,00%

27,00%-27,50%

26,50%-27,00%


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The C.R.A.R.o.C. Map:Path Dependent Replication

0,00% 0,50% 1,00% 1,50% 2,00%70%

80%

90%

100%

27,50%-28,00%

27,00%-27,50%

26,50%-27,00%

26,00%-26,50%


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The profitability decision

In order to evaluate the profitability condition of the issue it is possible to compare the expected profit the return on the capital at risk () set by top management, that is the E.V.A.

Therefore any guaranteed rate g which is able to create a positive E.V.A. can be take into consideration and, among different alternatives, the one that offer the higher E.V.A. is preferable

Given the VA indicator, naturally those issues which are able to maximize the profit and reduce the CVaR are the best, but they have to respect the limit of the annual rate

We selected = 11% as a proxy of the threshold value. This is the average gross annual rate of return of the insurance companies listed on the Italian Stock Exchange over the last three years.

m

RaVC

m

PROFITEAddedValueEconomic

....ˆ


29

The E.V.A.Plain Vanilla Replication (perfect matching) 0,00% 0,50% 1,00% 1,50% 2,00%

70% 30.801,35 28.729,56 26.514,27 24.139,68 21.590,40

80% 25.864,22 24.193,13 22.368,28 20.371,53 18.185,31

90% 20.927,08 19.656,69 18.222,29 16.603,45 14.780,20

100% 15.989,94 15.120,25 14.076,31 12.835,34 11.375,09


30

The E.V.A.Path Dependent replication (imperfect matching) 0,00% 0,50% 1,00% 1,50% 2,00%

70% 53.565,33 51.907,81 49.702,86 46.415,89 42.329,18

80% 51.781,01 50.597,45 48.805,54 45.780,24 41.841,14

90% 49.981,06 49.278,00 47.892,86 45.133,19 41.347,32

100% 48.160,66 47.943,59 46.976,18 44.475,68 41.192,99


31

Conclusion and future research prospect The computation of Risk-

adjusted performance indicators can serve for

Pricing decision Portfolio decision “Production” decision

The computation through simulative models is prone to model risk

Other interest rate model, mainly multifactor models

Volatility surface to price the inner smile

GARCH model in option pricing

Stochastic mortality

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11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance1 Profit...

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