3 capital forum by sutee

Aggregating Economic Capital Presentation to the Joint Regional Seminar 2011 Society of Actuaries Thailand Sutee Mokkhavesa Ph.D. Thursday 21 July 2011

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Economic capital and diversification

Given a random loss S :

: The total capital requirement, a probability of shortfall(p).

‘Economic default’ when S exceeds Joint Regional Seminar 2011

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Economic capital and diversification

E(S)

Figure 1: Expected loss, Unexpected loss, and Economic Capital.

Joint Regional Seminar 2011

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Economic capital and diversification

Calculating the Economic Capital(EC). (i) Determine EC per risk type. (ii) Aggregation.

Good aggregation techniques should allow for diversification effect to be incorporated.

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Economic capital and diversification Aggregating Economic Capital Given, two business units with risks(losses)

X1 and X2 Balance sheet capital is determined by the risk measure ρ (e.g. ρ[S] = ) An aggregation basis :

‘Merging decreases the shortfall risk’

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Economic capital and diversification

This also implies : ρ[X1 + X2] ≤ ρ[X1] + ρ[X2] Taking expectations is a linear operation : EC[X1 + X2] ≤ EC[X1] + EC[X2]

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Aggregation methods: Summing the individual capital Economic Capital of risk type i within business unit j :

The aggregate loss S:

A first possible way to aggregate EC :

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Aggregation methods: Summing the individual capital

All Xij are comonotonic :

Over estimate the required capital.

Safe strategy, but at cost of holding too much capital.

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Aggregation methods: Variance – Covariance matrix Assume Yk = Xij , The aggregate loss S:

Let ÎŁ be the variance-covariance matrix :

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Aggregation methods: Variance – Covariance matrix Correlation coefficient : Define correlation matrix Λ : EC of individual loss Yk : With: From normality assumption : (Quantile function) We get : Joint Regional Seminar 2011

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Aggregation methods: Variance – Covariance matrix • Assumes that the stand-alone losses Yk are multivariate normal distribution.

• The assumption of multivariate normality of random vector(Y1,Y2,…,Yn) express as follow : is normal distribute, for any set of real number α.

Mean : Standard deviation : Joint Regional Seminar 2011

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Aggregation methods: Variance – Covariance matrix The aggregate loss S Mean: Standard deviation: We find: From : We get: Joint Regional Seminar 2011

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Aggregation methods: Variance – Covariance matrix From :

Hence : Nb: 1. The random vector assumed to be multivariate normally distributed.

2. The individual economic capitals ECk = Joint Regional Seminar 2011

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Aggregating Economic Capital

Random Loss X1 X2 X3

-102 -1500

249 750

Random Loss Y1 Y2

-972

197

Y3

Mean

S.D.

• EC[X1+X2+X3]

?=

Mean

S.D.

-102 -1500

249 750

-972

197

EC[Y1+Y2+Y3]

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Aggregating Economic Capital • Normal distribution

• Non- Normal

Normal_X1

Normal_Y1

Normal_X2

Gamma_Y2

Normal_X3

Lognormal_Y3

Summation

Summation

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Aggregating Economic Capital Percentile 95.0% 99.0% 99.5%

Normal Non-Normal Difference -1358 MB -1729 MB 27.3% -1927 MB -2801 MB 45.3% -2156 MB -3235 MB 50.0%

Normal

Non - Normal

Joint Regional Seminar 2011

Mean S.D Skew Excess Kurtosis

: - 2.57 : 0.82 : - 0.03 : 0.04

Mean S.D Skew Excess Kurtosis

: - 2.57 : 0.93 : - 0.96 : 1.54

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Aggregating Economic Capital Risk Vs Uncertainty Normal distribution is good for modeling physical phenomena but can be inadequate when used to model phenomena generated by the human psychology such as movements of stock price and interest rates.

Normal distribution tend to under estimate extreme events.

It is imperative that we include diversification into the calculation.

Beware precision vs accuracy The results of the model is only as good as the quality of the underlying axioms and assumptions.

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References Dhaene, J., Goovaerts, M., Lundin, M. & Vanduffel, S. “Aggregating Economic Capital �, Belgian Actuarial Bulletin, Vol. 5, pp. 52-56, 2005 Taleb, Nassim Nicholas (2007), The Black Swan: The Impact of the Highly Improbable, Random House Pieter Klaassen and Idzard van Eeghen, (2009) Economic Capital: How It Works and What Every Manager Needs to Know, Elsevier Inc. Joint Regional Seminar 2011

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