[PPT] - Enterprise Risk Management Through Strategic Allocation of Capital PowerPoint Presentation

SLIDE 1

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Enterprise Risk Management Through Strategic Allocation of Capital

Joint Work Jing Ai

University of Hawaii

Patrick Brockett Linda Golden

University of Texas at Austin

June 2-6, 2010

6th Samos Conference in Actuarial Science and Finance

1 / 33

SLIDE 2

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overview

Introduction to ERM and Literature Review Baseline One-Period Model The Choice of Risk Appetite A Numerical Illustration The Two-Period Model Discussions and Conclusion

2 / 33

SLIDE 3

Introduction Static Risk Appetite Numerical Dynamic Conclusion

What is ERM?

The concept of Enterprise Risk Management (ERM) Managing risks holistically rather than separately Unique features of ERM invovles

risk appetite inter-relations between risks risk prioritization alignment of strategic goals and risk considerations

The users of ERM: corporations, universities, and government

3 / 33

SLIDE 4

Introduction Static Risk Appetite Numerical Dynamic Conclusion

What is Driving ERM?

Compliance Sarbanes-Oxley Act (2002) COSO Framework (2004) Basel II Captial Accord (2006)

4 / 33

SLIDE 5

Introduction Static Risk Appetite Numerical Dynamic Conclusion

What is Driving ERM?

Compliance Sarbanes-Oxley Act (2002) COSO Framework (2004) Basel II Captial Accord (2006) Best Practice Standard Standard & Poor’s (2005, 2006)

A. M. Best (2007)

Standard & Poor’s (2007)

4 / 33

SLIDE 6

Introduction Static Risk Appetite Numerical Dynamic Conclusion

What is Driving ERM?

Compliance Sarbanes-Oxley Act (2002) COSO Framework (2004) Basel II Captial Accord (2006) Best Practice Standard Standard & Poor’s (2005, 2006)

A. M. Best (2007)

Standard & Poor’s (2007) Value Creation The ultimate goal of ERM is to create value for stakehoders

4 / 33

SLIDE 7

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Research Motivation

“Much ambiguity remains as to what ERM really is and how it should be implemented” (Towers and Perrin, 2006)

5 / 33

SLIDE 8

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Research Motivation

“Much ambiguity remains as to what ERM really is and how it should be implemented” (Towers and Perrin, 2006) The concept

5 / 33

SLIDE 9

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Research Motivation

“Much ambiguity remains as to what ERM really is and how it should be implemented” (Towers and Perrin, 2006) The concept Risk identification, risk assessment, and risk communication

5 / 33

SLIDE 10

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Research Motivation

“Much ambiguity remains as to what ERM really is and how it should be implemented” (Towers and Perrin, 2006) The concept Risk identification, risk assessment, and risk communication A quantified, implementable framework

5 / 33

SLIDE 11

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Research Motivation

“Much ambiguity remains as to what ERM really is and how it should be implemented” (Towers and Perrin, 2006) The concept Risk identification, risk assessment, and risk communication A quantified, implementable framework A desired approach A framework for operational decisions Capture important characteristics Flexible and adaptive

5 / 33

SLIDE 12

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Propose an ERM Framework

Research Question How to formulate a quantitative ERM framework which is amenable for implementation in a general firm?

6 / 33

SLIDE 13

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Propose an ERM Framework

Research Question How to formulate a quantitative ERM framework which is amenable for implementation in a general firm? Features of the proposed framework Facilitates the integration of and interactions between strategic goals, operational decisions, and risk considerations

6 / 33

SLIDE 14

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Propose an ERM Framework

Research Question How to formulate a quantitative ERM framework which is amenable for implementation in a general firm? Features of the proposed framework Facilitates the integration of and interactions between strategic goals, operational decisions, and risk considerations Explicitly captures “risk appetite”, “inter-relation”, and “risk prioritization” in the decision making process

6 / 33

SLIDE 15

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Propose an ERM Framework

Research Question How to formulate a quantitative ERM framework which is amenable for implementation in a general firm? Features of the proposed framework Facilitates the integration of and interactions between strategic goals, operational decisions, and risk considerations Explicitly captures “risk appetite”, “inter-relation”, and “risk prioritization” in the decision making process The dynamic framework allows the firm to account for the changing business environment

6 / 33

SLIDE 16

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Propose an ERM Framework

Research Question How to formulate a quantitative ERM framework which is amenable for implementation in a general firm? Features of the proposed framework Facilitates the integration of and interactions between strategic goals, operational decisions, and risk considerations Explicitly captures “risk appetite”, “inter-relation”, and “risk prioritization” in the decision making process The dynamic framework allows the firm to account for the changing business environment Provides a conceptual framework capable of facilitating more general ERM modeling

6 / 33

SLIDE 17

Introduction Static Risk Appetite Numerical Dynamic Conclusion

ERM Literature

Components of ERM Determination of ERM: Liebenberg and Hoyt (2003) (Quantile-based) risk measure: Dowd and Blake (2006) Risk appetite: Nocco and Stulz (2006) Practical publications: COSO (2004), IBM (2007)

7 / 33

SLIDE 18

Introduction Static Risk Appetite Numerical Dynamic Conclusion

ERM Literature

Components of ERM Determination of ERM: Liebenberg and Hoyt (2003) (Quantile-based) risk measure: Dowd and Blake (2006) Risk appetite: Nocco and Stulz (2006) Practical publications: COSO (2004), IBM (2007) Risk aggregation: Rosenberg and Schuermann (2006)

7 / 33

SLIDE 19

Introduction Static Risk Appetite Numerical Dynamic Conclusion

ERM Literature

Components of ERM Determination of ERM: Liebenberg and Hoyt (2003) (Quantile-based) risk measure: Dowd and Blake (2006) Risk appetite: Nocco and Stulz (2006) Practical publications: COSO (2004), IBM (2007) Risk aggregation: Rosenberg and Schuermann (2006) Chance-constrained programming approach (Charnes and Cooper, 1959, 1962, 1963)

7 / 33

SLIDE 20

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The Conceptual Framework

To achieve business target in light of a multitude of risk considerations (1) project risk ≤ project risk appetite (2) financial risk ≤ financial risk appetite (3) operational risk ≤ operational risk appetite (4) hazard risk ≤ hazard risk appetite (5) overall risk ≤ overall risk appetite (6) other considerations (e.g., budget constraint) by making appropriate operational decisions

8 / 33

SLIDE 21

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The General Idea

Decision makers set up and optimize the firm’s business target

9 / 33

SLIDE 22

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The General Idea

Decision makers set up and optimize the firm’s business target Risk considerations are traded off in stochastic constraints

Several major types of risks are considered Use Value-at-Risk (VaR) (quantile-based) risk measure

9 / 33

SLIDE 23

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The General Idea

Decision makers set up and optimize the firm’s business target Risk considerations are traded off in stochastic constraints

Several major types of risks are considered Use Value-at-Risk (VaR) (quantile-based) risk measure

Explicitly incorporate risk appetite, inter-relations between risks, and risk prioritization

9 / 33

SLIDE 24

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The Baseline Setting

Single period risk/return optimization for a general firm, with total capital C to allocate

10 / 33

SLIDE 25

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The Baseline Setting

Single period risk/return optimization for a general firm, with total capital C to allocate Two sets of investment opportunities

i = 1, ..., K real projects (P) (e.g., K manufacturing product lines) j = 1, ..., N financial assets (A)

10 / 33

SLIDE 26

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The Baseline Setting

Single period risk/return optimization for a general firm, with total capital C to allocate Two sets of investment opportunities

i = 1, ..., K real projects (P) (e.g., K manufacturing product lines) j = 1, ..., N financial assets (A)

Model assumptions

Random returns Direct loss from hazard risk is proportional to total capital Indirect loss from hazard risk is a percentage of direct loss Hazard risk is mitigated by insurance

10 / 33

SLIDE 27

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Objective Function

max [E(returns from projects)+E(returns from assets)-insurance cost]

11 / 33

SLIDE 28

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Objective Function

max [E(returns from projects)+E(returns from assets)-insurance cost] max

w(P)

i

,w(A)

j

,u K

i=1

w(P)

i

[1+E(r(P)

i

)]+

N

j=1

w(A)

j

[1+E(r(A)

j

)]−ud(1+θ)µ

11 / 33

SLIDE 29

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Objective Function

max [E(returns from projects)+E(returns from assets)-insurance cost] max

w(P)

i

,w(A)

j

,u K

i=1

w(P)

i

[1+E(r(P)

i

)]+

N

j=1

w(A)

j

[1+E(r(A)

j

)]−ud(1+θ)µ w(P)

i

: proportions of capital invested in projects r(P)

i

: random returns from projects

11 / 33

SLIDE 30

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Objective Function

max [E(returns from projects)+E(returns from assets)-insurance cost] max

w(P)

i

,w(A)

j

,u K

i=1

w(P)

i

[1+E(r(P)

i

)]+

N

j=1

w(A)

j

[1+E(r(A)

j

)]−ud(1+θ)µ w(P)

i

: proportions of capital invested in projects r(P)

i

: random returns from projects w(A)

j

: proportions of capital invested in assets r(A)

j

: random returns from financial assets

11 / 33

SLIDE 31

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Objective Function

max [E(returns from projects)+E(returns from assets)-insurance cost] max

w(P)

i

,w(A)

j

,u K

i=1

w(P)

i

[1+E(r(P)

i

)]+

N

j=1

w(A)

j

[1+E(r(A)

j

)]−ud(1+θ)µ w(P)

i

: proportions of capital invested in projects r(P)

i

: random returns from projects w(A)

j

: proportions of capital invested in assets r(A)

j

: random returns from financial assets µ: expected unit direct hazard loss; θ: indirect hazard loss d: insurance loading; u: proportion of total hazard risk insured

11 / 33

SLIDE 32

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Objective Function

max [E(returns from projects)+E(returns from assets)-insurance cost] max

w(P)

i

,w(A)

j

,u K

i=1

w(P)

i

[1+E(r(P)

i

)]+

N

j=1

w(A)

j

[1+E(r(A)

j

)]−ud(1+θ)µ w(P)

i

: proportions of capital invested in projects r(P)

i

: random returns from projects w(A)

j

: proportions of capital invested in assets r(A)

j

: random returns from financial assets µ: expected unit direct hazard loss; θ: indirect hazard loss d: insurance loading; u: proportion of total hazard risk insured

11 / 33

SLIDE 33

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Chance Constraints

Account for four major types of firm risks: project risk, financial risk, operational risk, hazard risk Serve as the tool to incorporate the firm’s risk appetite and risk prioritization decisions Overall liquidity/solvency likelihood constraint specifies the likelihood of being unable to meet obligations and intertwines all the decision variables together

12 / 33

SLIDE 34

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm

13 / 33

SLIDE 35

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm Project risk: P[projects returns ≤ threshold] ≤ (project risk appetite)

13 / 33

SLIDE 36

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm Project risk: P[projects returns ≤ threshold] ≤ (project risk appetite) P[project returns ≥ project investment×hurdle rate] ≥ (1-appetite) P[

K

i=1

w(P)

i

(1 + r(P)

i

) ≥ (

K

i=1

w(P)

i

)(1 + r(P) )] ≥ 1 − α1

13 / 33

SLIDE 37

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm Project risk: P[projects returns ≤ threshold] ≤ (project risk appetite) P[project returns ≥ project investment×hurdle rate] ≥ (1-appetite) P[

K

i=1

w(P)

i

(1 + r(P)

i

) ≥ (

K

i=1

w(P)

i

)(1 + r(P) )] ≥ 1 − α1

13 / 33

SLIDE 38

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm Project risk: P[projects returns ≤ threshold] ≤ (project risk appetite) P[project returns ≥ project investment×hurdle rate] ≥ (1-appetite) P[

K

i=1

w(P)

i

(1 + r(P)

i

) ≥ (

K

i=1

w(P)

i

)(1 + r(P) )] ≥ 1 − α1

13 / 33

SLIDE 39

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm Project risk: P[projects returns ≤ threshold] ≤ (project risk appetite) P[project returns ≥ project investment×hurdle rate] ≥ (1-appetite) P[

K

i=1

w(P)

i

(1 + r(P)

i

) ≥ (

K

i=1

w(P)

i

)(1 + r(P) )] ≥ 1 − α1

Select projects so that a pre-determined minimum project return can be secured with certain confidence

13 / 33

SLIDE 40

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Project Risk Constraint

Project risk concerns the real business part of the firm Project risk: P[projects returns ≤ threshold] ≤ (project risk appetite) P[project returns ≥ project investment×hurdle rate] ≥ (1-appetite) P[

K

i=1

w(P)

i

(1 + r(P)

i

) ≥ (

K

i=1

w(P)

i

)(1 + r(P) )] ≥ 1 − α1

Select projects so that a pre-determined minimum project return can be secured with certain confidence Selection is governed by project risk appetite α1

13 / 33

SLIDE 41

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Financial Risk Constraint

Different nature of project risk and financial risk

14 / 33

SLIDE 42

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Financial Risk Constraint

Different nature of project risk and financial risk Natural hedges

14 / 33

SLIDE 43

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Financial Risk Constraint

Different nature of project risk and financial risk Natural hedges Financial risk: P[asset returns ≥ financial investment×hurdle rate] ≥ (1-appetite) P[

N

j=1

w(A)

j

(1 + r(A)

j

) ≥ (

N

j=1

w(A)

j

)(1 + r(A) )] ≥ 1 − α2

14 / 33

SLIDE 44

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Financial Risk Constraint

Different nature of project risk and financial risk Natural hedges Financial risk: P[asset returns ≥ financial investment×hurdle rate] ≥ (1-appetite) P[

N

j=1

w(A)

j

(1 + r(A)

j

) ≥ (

N

j=1

w(A)

j

)(1 + r(A) )] ≥ 1 − α2

14 / 33

SLIDE 45

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Financial Risk Constraint

Different nature of project risk and financial risk Natural hedges Financial risk: P[asset returns ≥ financial investment×hurdle rate] ≥ (1-appetite) P[

N

j=1

w(A)

j

(1 + r(A)

j

) ≥ (

N

j=1

w(A)

j

)(1 + r(A) )] ≥ 1 − α2

14 / 33

SLIDE 46

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Financial Risk Constraint

Different nature of project risk and financial risk Natural hedges Financial risk: P[asset returns ≥ financial investment×hurdle rate] ≥ (1-appetite) P[

N

j=1

w(A)

j

(1 + r(A)

j

) ≥ (

N

j=1

w(A)

j

)(1 + r(A) )] ≥ 1 − α2 Could incorporate hedging policies in the decision framework (e.g., Caldentey and Haugh 2006)

14 / 33

SLIDE 47

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006)

15 / 33

SLIDE 48

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

15 / 33

SLIDE 49

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

P[op risk (projects)+op risk (financial) ≤ op risk limit] ≥ (1-appetite) P[γ1

K

i=1

w(P)

i

(1 + r(P)

i

) + γ2

N

j=1

w(A)

j

(1 + r(A)

j

) ≤ lop] ≥ 1 − α3

15 / 33

SLIDE 50

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

P[op risk (projects)+op risk (financial) ≤ op risk limit] ≥ (1-appetite) P[γ1

K

i=1

w(P)

i

(1 + r(P)

i

) + γ2

N

j=1

w(A)

j

(1 + r(A)

j

) ≤ lop] ≥ 1 − α3 General indicators for each business unit

15 / 33

SLIDE 51

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

P[op risk (projects)+op risk (financial) ≤ op risk limit] ≥ (1-appetite) P[γ1

K

i=1

w(P)

i

(1 + r(P)

i

) + γ2

N

j=1

w(A)

j

(1 + r(A)

j

) ≤ lop] ≥ 1 − α3 General indicators for each business unit Two different factors for each business unit

15 / 33

SLIDE 52

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

P[op risk (projects)+op risk (financial) ≤ op risk limit] ≥ (1-appetite) P[γ1

K

i=1

w(P)

i

(1 + r(P)

i

) + γ2

N

j=1

w(A)

j

(1 + r(A)

j

) ≤ lop] ≥ 1 − α3 General indicators for each business unit Two different factors for each business unit

15 / 33

SLIDE 53

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

P[op risk (projects)+op risk (financial) ≤ op risk limit] ≥ (1-appetite) P[γ1

K

i=1

w(P)

i

(1 + r(P)

i

) + γ2

N

j=1

w(A)

j

(1 + r(A)

j

) ≤ lop] ≥ 1 − α3 General indicators for each business unit Two different factors for each business unit

15 / 33

SLIDE 54

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Operational (OP) Risk Constraint

“The most important risk category” (Towers and Perrin 2006) Standardized approach in Basel Capital Accord II OP risk =

i(factori×general indicator in business unit i)

P[op risk (projects)+op risk (financial) ≤ op risk limit] ≥ (1-appetite) P[γ1

K

i=1

w(P)

i

(1 + r(P)

i

) + γ2

N

j=1

w(A)

j

(1 + r(A)

j

) ≤ lop] ≥ 1 − α3 General indicators for each business unit Two different factors for each business unit Interactions between project risk and financial risk

15 / 33

SLIDE 55

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Hazard Risk Constraint

Mitigate hazard risk by purchasing (costly) insurance One constraint to govern the financial consequences from uninsured proportion of the hazard risk

16 / 33

SLIDE 56

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Hazard Risk Constraint

Mitigate hazard risk by purchasing (costly) insurance One constraint to govern the financial consequences from uninsured proportion of the hazard risk Hazard risk: P[uninsured proportion×hazard risk≤hazard risk limit]≥(1-appetite) P[(1 − u)l′ ≤ m] ≥ 1 − α4

16 / 33

SLIDE 57

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Hazard Risk Constraint

Mitigate hazard risk by purchasing (costly) insurance One constraint to govern the financial consequences from uninsured proportion of the hazard risk Hazard risk: P[uninsured proportion×hazard risk≤hazard risk limit]≥(1-appetite) P[(1 − u)l′ ≤ m] ≥ 1 − α4 l′ = l × (1 + θ): the unit total hazard loss

16 / 33

SLIDE 58

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Hazard Risk Constraint

Mitigate hazard risk by purchasing (costly) insurance One constraint to govern the financial consequences from uninsured proportion of the hazard risk Hazard risk: P[uninsured proportion×hazard risk≤hazard risk limit]≥(1-appetite) P[(1 − u)l′ ≤ m] ≥ 1 − α4 l′ = l × (1 + θ): the unit total hazard loss u: proportion of total hazard risk insured

16 / 33

SLIDE 59

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Hazard Risk Constraint

Mitigate hazard risk by purchasing (costly) insurance One constraint to govern the financial consequences from uninsured proportion of the hazard risk Hazard risk: P[uninsured proportion×hazard risk≤hazard risk limit]≥(1-appetite) P[(1 − u)l′ ≤ m] ≥ 1 − α4 l′ = l × (1 + θ): the unit total hazard loss u: proportion of total hazard risk insured m: hazard risk limit

16 / 33

SLIDE 60

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Hazard Risk Constraint

Mitigate hazard risk by purchasing (costly) insurance One constraint to govern the financial consequences from uninsured proportion of the hazard risk Hazard risk: P[uninsured proportion×hazard risk≤hazard risk limit]≥(1-appetite) P[(1 − u)l′ ≤ m] ≥ 1 − α4 l′ = l × (1 + θ): the unit total hazard loss u: proportion of total hazard risk insured m: hazard risk limit

16 / 33

SLIDE 61

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default

17 / 33

SLIDE 62

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

17 / 33

SLIDE 63

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 64

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 65

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 66

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 67

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 68

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 69

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Overall Risk Constraint

An integration of all risk categories The firm needs to be able to repay the required obligations (when contingent borrowing is not possible) to avoid default P[(project: returns - op risk)+(financial: returns - op risk)

uninsured hazard risk - insurance ≤ obligation] ≤ (appetite)

P[(1 − γ1)

K

i=1

w(P)

i

(1 + r(P)

i

) + (1 − γ2)

N

j=1

w(A)

j

(1 + r(A)

j

) −(1 − u)l′ − (1 + d)u(1 + θ)µ ≤ c] ≤ ¯ α

17 / 33

SLIDE 70

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Deterministic Constraints

Budget constraint:

K

i=1

w(P)

i

+

N

j=1

w(A)

j

+ u(1 + d)(1 + θ)µ ≤ 1 Strategic constraint:

K

i=1

w(P)

i

≥ γ3 Range constraints: w(P)

i

≥ 0, w(A)

j

≥ 0, 0 ≤ ν ≤ 1

18 / 33

SLIDE 71

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The ERM Framework

max

w(P)

i

,w(A)

j

,u

E[project returns + financial returns - insurance] s.t. P[project returns≥hurdle rate]≥(1-α1) P[financial returns≥hurdle rate]≥(1-α2) P[op risk≤risk limit]≤ α3 P[uninsured hazard risk≤risk limit] ≤ α4 P[total returns - all risk - insurance≤obligation]≤ ¯ α budget constraint, strategic constraint, range constraints

19 / 33

SLIDE 72

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The ERM Framework

max

w(P)

i

,w(A)

j

,u

E[project returns + financial returns - insurance] s.t. P[project returns≥hurdle rate]≥(1-α1) P[financial returns≥hurdle rate]≥(1-α2) P[op risk≤risk limit]≤ α3 P[uninsured hazard risk≤risk limit] ≤ α4 P[total returns - all risk - insurance≤obligation]≤ ¯ α budget constraint, strategic constraint, range constraints

19 / 33

SLIDE 73

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The ERM Framework

max

w(P)

i

,w(A)

j

,u

E[project returns + financial returns - insurance] s.t. P[project returns≥hurdle rate]≥(1-α1) P[financial returns≥hurdle rate]≥(1-α2) P[op risk≤risk limit]≤ α3 P[uninsured hazard risk≤risk limit] ≤ α4 P[total returns - all risk - insurance≤obligation]≤ ¯ α budget constraint, strategic constraint, range constraints

19 / 33

SLIDE 74

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Computation of the Constraint Set

To convert from the VaR type risk constraint to a deterministic constraint for use in mathematical programming, we use the fact that if P[X ≥ t] ≥ 1 − α then t ≤ F −1(α)

20 / 33

SLIDE 75

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Computation of the Constraint Set

To convert from the VaR type risk constraint to a deterministic constraint for use in mathematical programming, we use the fact that if P[X ≥ t] ≥ 1 − α then t ≤ F −1(α) So for example the financial risk constraint

P[

N

j=1

w (A)

j

(1 + r (A)

j

) ≥ (

N

j=1

w (A)

j

)(1 + r (A) )] ≥ 1 − α2

becocmes

(1+r (A) )(W (A))Tι−(W (A))T(ι+E(r)(A)) ≤ Φ−1(α2)

(W (A))TΣ(A)W (A)

20 / 33

SLIDE 76

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Risk Appetite

The firm should articulate how risk appetite falls in line with strategic goals and risk culture (S&P 2005)

21 / 33

SLIDE 77

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Risk Appetite

The firm should articulate how risk appetite falls in line with strategic goals and risk culture (S&P 2005) Determine risk appetite in light of the credit rating target (Nocco and Stulz 2006)

21 / 33

SLIDE 78

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Risk Appetite

The firm should articulate how risk appetite falls in line with strategic goals and risk culture (S&P 2005) Determine risk appetite in light of the credit rating target (Nocco and Stulz 2006)

Strategic goals govern the risk appetite

21 / 33

SLIDE 79

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Risk Appetite

The firm should articulate how risk appetite falls in line with strategic goals and risk culture (S&P 2005) Determine risk appetite in light of the credit rating target (Nocco and Stulz 2006)

Strategic goals govern the risk appetite Credit rating target proxies for the firm’s strategic goals

21 / 33

SLIDE 80

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Risk Appetite

The firm should articulate how risk appetite falls in line with strategic goals and risk culture (S&P 2005) Determine risk appetite in light of the credit rating target (Nocco and Stulz 2006)

Strategic goals govern the risk appetite Credit rating target proxies for the firm’s strategic goals

determines the ability to raise capital and the cost of capital (West 1973) influences corporate policies and actual business strategies (Sufi 2007)

21 / 33

SLIDE 81

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Choice of Risk Appetite

Overall risk appetite parameter (¯ α)

target default probability implied by the credit rating target used in the overall risk constraint

f฀risk฀would฀be฀determined฀by฀comparing฀the฀costs฀associ-

ated฀with฀financial฀distress฀and฀the฀benefits฀of฀having฀a฀more฀ levered฀capital฀structure฀and฀taking฀on฀riskier฀projects.฀ To฀the฀extent฀that฀ratings฀are฀reliable฀proxies฀for฀financial฀health,฀companies฀can฀use฀a฀rating฀agency฀“transition฀ matrix”฀ to฀ estimate฀ the฀ amount฀ of฀ capital฀ necessary฀ to฀ support฀a฀given฀level฀of฀risk.฀The฀transition฀matrix฀shown฀ in฀Table฀1฀can฀be฀used฀to฀identify฀the฀frequency฀with฀which฀ companies฀moved฀from฀one฀rating฀to฀another฀over฀a฀certain฀ period฀(in฀this฀case,฀1920฀to฀2005).7฀For฀any฀rating฀at฀the฀ beginning฀ of฀ the฀ year฀ (listed฀ in฀ the฀ left-hand฀ column฀ of฀ the฀table),฀the฀column฀of฀numbers฀running฀down฀from฀the฀ heading฀“Baa”฀tells฀us฀the฀probability฀that฀a฀company฀will฀ end฀up฀with฀a฀Baa฀rating฀at฀the฀end฀of฀the฀year. Again,฀ let’s฀ assume฀ management฀ wants฀ the฀ probability฀of฀its฀rating฀falling฀to฀Baa฀or฀lower฀over฀the฀next฀year฀ to฀average฀around฀7%.฀To฀determine฀the฀probability฀of฀a฀ downgrade฀to฀or฀lower฀than฀Baa฀for฀a฀given฀initial฀rating,฀ we฀add฀up฀the฀probabilities฀of฀ending฀with฀a฀rating฀equal฀ to฀or฀lower฀than฀Baa฀along฀the฀row฀that฀corresponds฀to฀the฀ initial฀rating.฀The฀row฀where฀the฀probabilities฀of฀ending฀at฀ Baa฀or฀lower฀is฀closest฀to฀7%฀is฀the฀one฀corresponding฀to฀an฀ A฀rating.฀Consequently,฀by฀targeting฀an฀A฀rating,฀management฀would฀achieve฀the฀probability฀of฀financial฀distress฀that฀ is฀optimal฀for฀the฀firm.฀ In฀practice,฀however,฀the฀process฀of฀determining฀a฀target฀ rating฀ can฀ involve฀ more฀ considerations,฀ which฀ makes฀ it฀ more฀complicated.฀For฀example,฀Nationwide฀analyzes฀and฀ manages฀both฀its฀probability฀of฀default฀and฀its฀probability฀of฀ downgrade,฀and฀it฀does฀so฀in฀separate฀but฀related฀frameworks.฀ The฀company’s฀optimal฀probability฀of฀default฀is฀anchored฀to฀ its฀target฀Aa฀ratings฀and฀reflects฀the฀default฀history฀of฀Aa- rated฀bonds.฀By฀contrast,฀the฀probability฀of฀downgrade฀to฀ Baa฀or฀below฀is฀assumed฀to฀be฀affected฀by,฀and฀is฀accord- ingly฀managed฀by฀limiting,฀risk฀concentrations฀such฀as฀those฀ arising฀from฀natural฀catastrophes฀and฀equity฀markets. In฀ the฀ example฀ above,฀ the฀ company฀ is฀ assumed฀ to฀ maximize฀ value฀ by฀ targeting฀ a฀ rating฀ of฀ A.฀ As฀ we฀ noted฀ earlier,฀equity฀capital฀provides฀a฀buffer฀or฀shock฀absorber฀ that฀ helps฀ the฀ firm฀ to฀ avoid฀ default.฀ For฀ a฀ given฀ firm,฀ a฀ different฀probability฀of฀default฀corresponds฀to฀each฀level฀of฀ equity,฀so฀that฀by฀choosing฀a฀given฀level฀of฀equity,฀management฀is฀also฀effectively฀choosing฀a฀probability฀of฀default฀that฀ it฀believes฀to฀be฀optimal. As฀can฀be฀seen฀in฀Table฀1,฀an฀A฀rating฀is฀associated฀with฀ a฀probability฀of฀default฀of฀0.08%฀over฀a฀one-year฀period.฀ Thus,฀to฀achieve฀an฀A฀rating,฀the฀company฀in฀our฀example฀ must฀have฀the฀level฀of฀(equity)฀capital฀that฀makes฀its฀probability฀of฀default฀equal฀to฀0.08%.฀If฀we฀make฀the฀assumption฀ that฀ the฀ value฀ of฀ a฀ company’s฀ equity฀ falls฀ to฀ a฀ level฀ not฀ materially฀different฀from฀zero฀in฀the฀event฀of฀default,฀we฀ can฀use฀the฀probability฀of฀default฀to฀“back฀out”฀the฀amount฀

f฀equity฀the฀firm฀needs฀to฀support฀its฀current฀level฀of฀risk.

Although฀the฀probability฀of฀default฀is฀in฀fact฀a฀complicated฀ function฀ of฀ a฀ number฀ of฀ firm฀ characteristics,฀ not฀ just฀the฀amount฀of฀equity,฀the฀analytical฀process฀that฀leads฀ from฀the฀probability฀of฀default฀to฀the฀required฀amount฀of฀ capital฀ is฀ straightforward.฀ To฀ see฀ this,฀ suppose฀ that฀ the฀ company฀ becomes฀ bankrupt฀ if฀ firm฀ value฀ at฀ the฀ end฀ of฀ the฀fiscal฀year฀falls฀below฀a฀default฀threshold฀level,฀which฀ is฀a฀function฀of฀the฀composition฀and฀amount฀of฀the฀firm’s฀ debt.8฀Given฀this฀assumption,฀the฀firm฀needs฀the฀amount฀

f฀equity฀capital฀that฀will฀make฀the฀probability฀of฀its฀value฀

falling฀below฀the฀default฀threshold฀level฀equal฀to฀0.08%฀

Table 1 Transition Matrix from Moody’s Rating To: Rating From: Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.75% 7.26% 0.79% 0.17% 0.02% 0.00% 0.00% 0.00% Aa 1.32% 90.71% 6.92% 0.75% 0.19% 0.04% 0.01% 0.06% A 0.08% 3.02% 90.24% 5.67% 0.76% 0.12% 0.03% 0.08% Baa 0.05% 0.33% 5.05% 87.50% 5.72% 0.86% 0.18% 0.31% Ba 0.01% 0.09% 0.59% 6.70% 82.58% 7.83% 0.72% 1.48% B 0.00% 0.07% 0.20% 0.80% 7.29% 80.62% 6.23% 4.78% Caa-C 0.00% 0.03% 0.06% 0.23% 1.07% 7.69% 75.24% 15.69%

Average one-year rating transition matrix, 1920-2005, conditional upon no rating withdrawal. Source: Moody’s Default and Recovery Rates of Corporate Bond Issuers, 1920-2005, March 2006.

12 Journal of Applied Corporate Finance • Volume 18 Number 4 A Morgan Stanley Publication • Fall 2006

7. See footnote 2.
8. If all debt were due at the end of the year, the default threshold level would be the

principal amount of debt outstanding plus interest due. However, if debt matures later, firm value could fall below the principal amount of debt outstanding without triggering a

default. So, the default threshold level is lower than the principal amount of debt out-

standing when the firm has long-term debt.

22 / 33

SLIDE 82

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Choice of Risk Appetite

Overall risk appetite parameter (¯ α)

target default probability implied by the credit rating target used in the overall risk constraint

Individual risk appetite parameters (α′

is)

different appetites toward different risks reference point: financial distress probability

f฀risk฀would฀be฀determined฀by฀comparing฀the฀costs฀associ-

ated฀with฀financial฀distress฀and฀the฀benefits฀of฀having฀a฀more฀ levered฀capital฀structure฀and฀taking฀on฀riskier฀projects.฀ To฀the฀extent฀that฀ratings฀are฀reliable฀proxies฀for฀financial฀health,฀companies฀can฀use฀a฀rating฀agency฀“transition฀ matrix”฀ to฀ estimate฀ the฀ amount฀ of฀ capital฀ necessary฀ to฀ support฀a฀given฀level฀of฀risk.฀The฀transition฀matrix฀shown฀ in฀Table฀1฀can฀be฀used฀to฀identify฀the฀frequency฀with฀which฀ companies฀moved฀from฀one฀rating฀to฀another฀over฀a฀certain฀ period฀(in฀this฀case,฀1920฀to฀2005).7฀For฀any฀rating฀at฀the฀ beginning฀ of฀ the฀ year฀ (listed฀ in฀ the฀ left-hand฀ column฀ of฀ the฀table),฀the฀column฀of฀numbers฀running฀down฀from฀the฀ heading฀“Baa”฀tells฀us฀the฀probability฀that฀a฀company฀will฀ end฀up฀with฀a฀Baa฀rating฀at฀the฀end฀of฀the฀year. Again,฀ let’s฀ assume฀ management฀ wants฀ the฀ probability฀of฀its฀rating฀falling฀to฀Baa฀or฀lower฀over฀the฀next฀year฀ to฀average฀around฀7%.฀To฀determine฀the฀probability฀of฀a฀ downgrade฀to฀or฀lower฀than฀Baa฀for฀a฀given฀initial฀rating,฀ we฀add฀up฀the฀probabilities฀of฀ending฀with฀a฀rating฀equal฀ to฀or฀lower฀than฀Baa฀along฀the฀row฀that฀corresponds฀to฀the฀ initial฀rating.฀The฀row฀where฀the฀probabilities฀of฀ending฀at฀ Baa฀or฀lower฀is฀closest฀to฀7%฀is฀the฀one฀corresponding฀to฀an฀ A฀rating.฀Consequently,฀by฀targeting฀an฀A฀rating,฀management฀would฀achieve฀the฀probability฀of฀financial฀distress฀that฀ is฀optimal฀for฀the฀firm.฀ In฀practice,฀however,฀the฀process฀of฀determining฀a฀target฀ rating฀ can฀ involve฀ more฀ considerations,฀ which฀ makes฀ it฀ more฀complicated.฀For฀example,฀Nationwide฀analyzes฀and฀ manages฀both฀its฀probability฀of฀default฀and฀its฀probability฀of฀ downgrade,฀and฀it฀does฀so฀in฀separate฀but฀related฀frameworks.฀ The฀company’s฀optimal฀probability฀of฀default฀is฀anchored฀to฀ its฀target฀Aa฀ratings฀and฀reflects฀the฀default฀history฀of฀Aa- rated฀bonds.฀By฀contrast,฀the฀probability฀of฀downgrade฀to฀ Baa฀or฀below฀is฀assumed฀to฀be฀affected฀by,฀and฀is฀accord- ingly฀managed฀by฀limiting,฀risk฀concentrations฀such฀as฀those฀ arising฀from฀natural฀catastrophes฀and฀equity฀markets. In฀ the฀ example฀ above,฀ the฀ company฀ is฀ assumed฀ to฀ maximize฀ value฀ by฀ targeting฀ a฀ rating฀ of฀ A.฀ As฀ we฀ noted฀ earlier,฀equity฀capital฀provides฀a฀buffer฀or฀shock฀absorber฀ that฀ helps฀ the฀ firm฀ to฀ avoid฀ default.฀ For฀ a฀ given฀ firm,฀ a฀ different฀probability฀of฀default฀corresponds฀to฀each฀level฀of฀ equity,฀so฀that฀by฀choosing฀a฀given฀level฀of฀equity,฀management฀is฀also฀effectively฀choosing฀a฀probability฀of฀default฀that฀ it฀believes฀to฀be฀optimal. As฀can฀be฀seen฀in฀Table฀1,฀an฀A฀rating฀is฀associated฀with฀ a฀probability฀of฀default฀of฀0.08%฀over฀a฀one-year฀period.฀ Thus,฀to฀achieve฀an฀A฀rating,฀the฀company฀in฀our฀example฀ must฀have฀the฀level฀of฀(equity)฀capital฀that฀makes฀its฀probability฀of฀default฀equal฀to฀0.08%.฀If฀we฀make฀the฀assumption฀ that฀ the฀ value฀ of฀ a฀ company’s฀ equity฀ falls฀ to฀ a฀ level฀ not฀ materially฀different฀from฀zero฀in฀the฀event฀of฀default,฀we฀ can฀use฀the฀probability฀of฀default฀to฀“back฀out”฀the฀amount฀

f฀equity฀the฀firm฀needs฀to฀support฀its฀current฀level฀of฀risk.

Although฀the฀probability฀of฀default฀is฀in฀fact฀a฀complicated฀ function฀ of฀ a฀ number฀ of฀ firm฀ characteristics,฀ not฀ just฀the฀amount฀of฀equity,฀the฀analytical฀process฀that฀leads฀ from฀the฀probability฀of฀default฀to฀the฀required฀amount฀of฀ capital฀ is฀ straightforward.฀ To฀ see฀ this,฀ suppose฀ that฀ the฀ company฀ becomes฀ bankrupt฀ if฀ firm฀ value฀ at฀ the฀ end฀ of฀ the฀fiscal฀year฀falls฀below฀a฀default฀threshold฀level,฀which฀ is฀a฀function฀of฀the฀composition฀and฀amount฀of฀the฀firm’s฀ debt.8฀Given฀this฀assumption,฀the฀firm฀needs฀the฀amount฀

f฀equity฀capital฀that฀will฀make฀the฀probability฀of฀its฀value฀

falling฀below฀the฀default฀threshold฀level฀equal฀to฀0.08%฀

Table 1 Transition Matrix from Moody’s Rating To: Rating From: Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.75% 7.26% 0.79% 0.17% 0.02% 0.00% 0.00% 0.00% Aa 1.32% 90.71% 6.92% 0.75% 0.19% 0.04% 0.01% 0.06% A 0.08% 3.02% 90.24% 5.67% 0.76% 0.12% 0.03% 0.08% Baa 0.05% 0.33% 5.05% 87.50% 5.72% 0.86% 0.18% 0.31% Ba 0.01% 0.09% 0.59% 6.70% 82.58% 7.83% 0.72% 1.48% B 0.00% 0.07% 0.20% 0.80% 7.29% 80.62% 6.23% 4.78% Caa-C 0.00% 0.03% 0.06% 0.23% 1.07% 7.69% 75.24% 15.69%

Average one-year rating transition matrix, 1920-2005, conditional upon no rating withdrawal. Source: Moody’s Default and Recovery Rates of Corporate Bond Issuers, 1920-2005, March 2006.

12 Journal of Applied Corporate Finance • Volume 18 Number 4 A Morgan Stanley Publication • Fall 2006

7. See footnote 2.
8. If all debt were due at the end of the year, the default threshold level would be the

principal amount of debt outstanding plus interest due. However, if debt matures later, firm value could fall below the principal amount of debt outstanding without triggering a

default. So, the default threshold level is lower than the principal amount of debt out-

standing when the firm has long-term debt.

22 / 33

SLIDE 83

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Choice of Risk Appetite

Overall risk appetite parameter (¯ α)

target default probability implied by the credit rating target used in the overall risk constraint

Individual risk appetite parameters (α′

is)

different appetites toward different risks reference point: financial distress probability

Moody’s Rate Transition Matrix

f฀risk฀would฀be฀determined฀by฀comparing฀the฀costs฀associ-

ated฀with฀financial฀distress฀and฀the฀benefits฀of฀having฀a฀more฀ levered฀capital฀structure฀and฀taking฀on฀riskier฀projects.฀ To฀the฀extent฀that฀ratings฀are฀reliable฀proxies฀for฀financial฀health,฀companies฀can฀use฀a฀rating฀agency฀“transition฀ matrix”฀ to฀ estimate฀ the฀ amount฀ of฀ capital฀ necessary฀ to฀ support฀a฀given฀level฀of฀risk.฀The฀transition฀matrix฀shown฀ in฀Table฀1฀can฀be฀used฀to฀identify฀the฀frequency฀with฀which฀ companies฀moved฀from฀one฀rating฀to฀another฀over฀a฀certain฀ period฀(in฀this฀case,฀1920฀to฀2005).7฀For฀any฀rating฀at฀the฀ beginning฀ of฀ the฀ year฀ (listed฀ in฀ the฀ left-hand฀ column฀ of฀ the฀table),฀the฀column฀of฀numbers฀running฀down฀from฀the฀ heading฀“Baa”฀tells฀us฀the฀probability฀that฀a฀company฀will฀ end฀up฀with฀a฀Baa฀rating฀at฀the฀end฀of฀the฀year. Again,฀ let’s฀ assume฀ management฀ wants฀ the฀ probability฀of฀its฀rating฀falling฀to฀Baa฀or฀lower฀over฀the฀next฀year฀ to฀average฀around฀7%.฀To฀determine฀the฀probability฀of฀a฀ downgrade฀to฀or฀lower฀than฀Baa฀for฀a฀given฀initial฀rating,฀ we฀add฀up฀the฀probabilities฀of฀ending฀with฀a฀rating฀equal฀ to฀or฀lower฀than฀Baa฀along฀the฀row฀that฀corresponds฀to฀the฀ initial฀rating.฀The฀row฀where฀the฀probabilities฀of฀ending฀at฀ Baa฀or฀lower฀is฀closest฀to฀7%฀is฀the฀one฀corresponding฀to฀an฀ A฀rating.฀Consequently,฀by฀targeting฀an฀A฀rating,฀management฀would฀achieve฀the฀probability฀of฀financial฀distress฀that฀ is฀optimal฀for฀the฀firm.฀ In฀practice,฀however,฀the฀process฀of฀determining฀a฀target฀ rating฀ can฀ involve฀ more฀ considerations,฀ which฀ makes฀ it฀ more฀complicated.฀For฀example,฀Nationwide฀analyzes฀and฀ manages฀both฀its฀probability฀of฀default฀and฀its฀probability฀of฀ downgrade,฀and฀it฀does฀so฀in฀separate฀but฀related฀frameworks.฀ The฀company’s฀optimal฀probability฀of฀default฀is฀anchored฀to฀ its฀target฀Aa฀ratings฀and฀reflects฀the฀default฀history฀of฀Aa- rated฀bonds.฀By฀contrast,฀the฀probability฀of฀downgrade฀to฀ Baa฀or฀below฀is฀assumed฀to฀be฀affected฀by,฀and฀is฀accord- ingly฀managed฀by฀limiting,฀risk฀concentrations฀such฀as฀those฀ arising฀from฀natural฀catastrophes฀and฀equity฀markets. In฀ the฀ example฀ above,฀ the฀ company฀ is฀ assumed฀ to฀ maximize฀ value฀ by฀ targeting฀ a฀ rating฀ of฀ A.฀ As฀ we฀ noted฀ earlier,฀equity฀capital฀provides฀a฀buffer฀or฀shock฀absorber฀ that฀ helps฀ the฀ firm฀ to฀ avoid฀ default.฀ For฀ a฀ given฀ firm,฀ a฀ different฀probability฀of฀default฀corresponds฀to฀each฀level฀of฀ equity,฀so฀that฀by฀choosing฀a฀given฀level฀of฀equity,฀management฀is฀also฀effectively฀choosing฀a฀probability฀of฀default฀that฀ it฀believes฀to฀be฀optimal. As฀can฀be฀seen฀in฀Table฀1,฀an฀A฀rating฀is฀associated฀with฀ a฀probability฀of฀default฀of฀0.08%฀over฀a฀one-year฀period.฀ Thus,฀to฀achieve฀an฀A฀rating,฀the฀company฀in฀our฀example฀ must฀have฀the฀level฀of฀(equity)฀capital฀that฀makes฀its฀probability฀of฀default฀equal฀to฀0.08%.฀If฀we฀make฀the฀assumption฀ that฀ the฀ value฀ of฀ a฀ company’s฀ equity฀ falls฀ to฀ a฀ level฀ not฀ materially฀different฀from฀zero฀in฀the฀event฀of฀default,฀we฀ can฀use฀the฀probability฀of฀default฀to฀“back฀out”฀the฀amount฀

f฀equity฀the฀firm฀needs฀to฀support฀its฀current฀level฀of฀risk.

Although฀the฀probability฀of฀default฀is฀in฀fact฀a฀complicated฀ function฀ of฀ a฀ number฀ of฀ firm฀ characteristics,฀ not฀ just฀the฀amount฀of฀equity,฀the฀analytical฀process฀that฀leads฀ from฀the฀probability฀of฀default฀to฀the฀required฀amount฀of฀ capital฀ is฀ straightforward.฀ To฀ see฀ this,฀ suppose฀ that฀ the฀ company฀ becomes฀ bankrupt฀ if฀ firm฀ value฀ at฀ the฀ end฀ of฀ the฀fiscal฀year฀falls฀below฀a฀default฀threshold฀level,฀which฀ is฀a฀function฀of฀the฀composition฀and฀amount฀of฀the฀firm’s฀ debt.8฀Given฀this฀assumption,฀the฀firm฀needs฀the฀amount฀

f฀equity฀capital฀that฀will฀make฀the฀probability฀of฀its฀value฀

falling฀below฀the฀default฀threshold฀level฀equal฀to฀0.08%฀

Table 1 Transition Matrix from Moody’s Rating To: Rating From: Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.75% 7.26% 0.79% 0.17% 0.02% 0.00% 0.00% 0.00% Aa 1.32% 90.71% 6.92% 0.75% 0.19% 0.04% 0.01% 0.06% A 0.08% 3.02% 90.24% 5.67% 0.76% 0.12% 0.03% 0.08% Baa 0.05% 0.33% 5.05% 87.50% 5.72% 0.86% 0.18% 0.31% Ba 0.01% 0.09% 0.59% 6.70% 82.58% 7.83% 0.72% 1.48% B 0.00% 0.07% 0.20% 0.80% 7.29% 80.62% 6.23% 4.78% Caa-C 0.00% 0.03% 0.06% 0.23% 1.07% 7.69% 75.24% 15.69%

Average one-year rating transition matrix, 1920-2005, conditional upon no rating withdrawal. Source: Moody’s Default and Recovery Rates of Corporate Bond Issuers, 1920-2005, March 2006.

12 Journal of Applied Corporate Finance • Volume 18 Number 4 A Morgan Stanley Publication • Fall 2006

7. See footnote 2.
8. If all debt were due at the end of the year, the default threshold level would be the

principal amount of debt outstanding plus interest due. However, if debt matures later, firm value could fall below the principal amount of debt outstanding without triggering a

default. So, the default threshold level is lower than the principal amount of debt out-

standing when the firm has long-term debt.

Source: Nocco and Stulz. 2006. Enterprise Risk Management Theory and Practice. J. Corp. Fin.

22 / 33

SLIDE 84

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Choice of Risk Appetite

Overall risk appetite parameter (¯ α)

target default probability implied by the credit rating target used in the overall risk constraint

Individual risk appetite parameters (α′

is)

different appetites toward different risks reference point: financial distress probability

23 / 33

SLIDE 85

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Choice of Risk Appetite

Overall risk appetite parameter (¯ α)

target default probability implied by the credit rating target used in the overall risk constraint

Individual risk appetite parameters (α′

is)

different appetites toward different risks reference point: financial distress probability

Risk Prioritization Individual risk appetite parameters facilitate the quantification of risk prioritization decisions:

23 / 33

SLIDE 86

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Choice of Risk Appetite

Overall risk appetite parameter (¯ α)

target default probability implied by the credit rating target used in the overall risk constraint

Individual risk appetite parameters (α′

is)

different appetites toward different risks reference point: financial distress probability

Risk Prioritization Individual risk appetite parameters facilitate the quantification of risk prioritization decisions: smaller value indicates higher priority

f the risk

23 / 33

SLIDE 87

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Numerical Example: Baseline Setting I

Purposes Demonstrate how to implement the framework Demonstrate how to determine risk appetite parameters

24 / 33

SLIDE 88

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Numerical Example: Baseline Setting I

Purposes Demonstrate how to implement the framework Demonstrate how to determine risk appetite parameters Distributional assumptions Returns r(P)

i

and r(A)

j

: N

(E(r(P)) : E(r(A)))T, (PA)

24 / 33

SLIDE 89

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Numerical Example: Baseline Setting I

Purposes Demonstrate how to implement the framework Demonstrate how to determine risk appetite parameters Distributional assumptions Returns r(P)

i

and r(A)

j

: N

(E(r(P)) : E(r(A)))T, (PA)

Unit total losses from hazard risk l′ ∼ N((1 + θ)µ, (1 + θ)2σ2), 0 < µ < 1, σ > 0

24 / 33

SLIDE 90

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Numerical Example: Baseline Setting I

Purposes Demonstrate how to implement the framework Demonstrate how to determine risk appetite parameters Distributional assumptions Returns r(P)

i

and r(A)

j

: N

(E(r(P)) : E(r(A)))T, (PA)

Unit total losses from hazard risk l′ ∼ N((1 + θ)µ, (1 + θ)2σ2), 0 < µ < 1, σ > 0 Hazard risk is independent of the other risk categories

24 / 33

SLIDE 91

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Numerical Example: Baseline Setting II

Description of investment opportunities

Expected Return and Variance R&D Project Manufacturing Project Index Fund 3 Month T-Bill Expected return 0.3 0.1 0.12 0.038 Variance 0.3 0.003 0.03 0.00025 Correlation of the investments R&D Project Manufacturing Project Index Fund 3 Month T-Bill R&D Project 1 0.1 0.05

0.05

Manufacturing Project 0.1 1 0.05

0.05

Index Fund 1

0.1

3 Month T-Bill 1

25 / 33

SLIDE 92

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Parameter Values

Assume the firm has a target credit rating of A, which leads to 0.08% default probability and around 6% financial distress probability

f฀risk฀would฀be฀determined฀by฀comparing฀the฀costs฀associ-

ated฀with฀financial฀distress฀and฀the฀benefits฀of฀having฀a฀more฀ levered฀capital฀structure฀and฀taking฀on฀riskier฀projects.฀ To฀the฀extent฀that฀ratings฀are฀reliable฀proxies฀for฀financial฀health,฀companies฀can฀use฀a฀rating฀agency฀“transition฀ matrix”฀ to฀ estimate฀ the฀ amount฀ of฀ capital฀ necessary฀ to฀ support฀a฀given฀level฀of฀risk.฀The฀transition฀matrix฀shown฀ in฀Table฀1฀can฀be฀used฀to฀identify฀the฀frequency฀with฀which฀ companies฀moved฀from฀one฀rating฀to฀another฀over฀a฀certain฀ period฀(in฀this฀case,฀1920฀to฀2005).7฀For฀any฀rating฀at฀the฀ beginning฀ of฀ the฀ year฀ (listed฀ in฀ the฀ left-hand฀ column฀ of฀ the฀table),฀the฀column฀of฀numbers฀running฀down฀from฀the฀ heading฀“Baa”฀tells฀us฀the฀probability฀that฀a฀company฀will฀ end฀up฀with฀a฀Baa฀rating฀at฀the฀end฀of฀the฀year. Again,฀ let’s฀ assume฀ management฀ wants฀ the฀ probability฀of฀its฀rating฀falling฀to฀Baa฀or฀lower฀over฀the฀next฀year฀ to฀average฀around฀7%.฀To฀determine฀the฀probability฀of฀a฀ downgrade฀to฀or฀lower฀than฀Baa฀for฀a฀given฀initial฀rating,฀ we฀add฀up฀the฀probabilities฀of฀ending฀with฀a฀rating฀equal฀ to฀or฀lower฀than฀Baa฀along฀the฀row฀that฀corresponds฀to฀the฀ initial฀rating.฀The฀row฀where฀the฀probabilities฀of฀ending฀at฀ Baa฀or฀lower฀is฀closest฀to฀7%฀is฀the฀one฀corresponding฀to฀an฀ A฀rating.฀Consequently,฀by฀targeting฀an฀A฀rating,฀management฀would฀achieve฀the฀probability฀of฀financial฀distress฀that฀ is฀optimal฀for฀the฀firm.฀ In฀practice,฀however,฀the฀process฀of฀determining฀a฀target฀ rating฀ can฀ involve฀ more฀ considerations,฀ which฀ makes฀ it฀ more฀complicated.฀For฀example,฀Nationwide฀analyzes฀and฀ manages฀both฀its฀probability฀of฀default฀and฀its฀probability฀of฀ downgrade,฀and฀it฀does฀so฀in฀separate฀but฀related฀frameworks.฀ The฀company’s฀optimal฀probability฀of฀default฀is฀anchored฀to฀ its฀target฀Aa฀ratings฀and฀reflects฀the฀default฀history฀of฀Aa- rated฀bonds.฀By฀contrast,฀the฀probability฀of฀downgrade฀to฀ Baa฀or฀below฀is฀assumed฀to฀be฀affected฀by,฀and฀is฀accord- ingly฀managed฀by฀limiting,฀risk฀concentrations฀such฀as฀those฀ arising฀from฀natural฀catastrophes฀and฀equity฀markets. In฀ the฀ example฀ above,฀ the฀ company฀ is฀ assumed฀ to฀ maximize฀ value฀ by฀ targeting฀ a฀ rating฀ of฀ A.฀ As฀ we฀ noted฀ earlier,฀equity฀capital฀provides฀a฀buffer฀or฀shock฀absorber฀ that฀ helps฀ the฀ firm฀ to฀ avoid฀ default.฀ For฀ a฀ given฀ firm,฀ a฀ different฀probability฀of฀default฀corresponds฀to฀each฀level฀of฀ equity,฀so฀that฀by฀choosing฀a฀given฀level฀of฀equity,฀management฀is฀also฀effectively฀choosing฀a฀probability฀of฀default฀that฀ it฀believes฀to฀be฀optimal. As฀can฀be฀seen฀in฀Table฀1,฀an฀A฀rating฀is฀associated฀with฀ a฀probability฀of฀default฀of฀0.08%฀over฀a฀one-year฀period.฀ Thus,฀to฀achieve฀an฀A฀rating,฀the฀company฀in฀our฀example฀ must฀have฀the฀level฀of฀(equity)฀capital฀that฀makes฀its฀probability฀of฀default฀equal฀to฀0.08%.฀If฀we฀make฀the฀assumption฀ that฀ the฀ value฀ of฀ a฀ company’s฀ equity฀ falls฀ to฀ a฀ level฀ not฀ materially฀different฀from฀zero฀in฀the฀event฀of฀default,฀we฀ can฀use฀the฀probability฀of฀default฀to฀“back฀out”฀the฀amount฀

f฀equity฀the฀firm฀needs฀to฀support฀its฀current฀level฀of฀risk.

Although฀the฀probability฀of฀default฀is฀in฀fact฀a฀complicated฀ function฀ of฀ a฀ number฀ of฀ firm฀ characteristics,฀ not฀ just฀the฀amount฀of฀equity,฀the฀analytical฀process฀that฀leads฀ from฀the฀probability฀of฀default฀to฀the฀required฀amount฀of฀ capital฀ is฀ straightforward.฀ To฀ see฀ this,฀ suppose฀ that฀ the฀ company฀ becomes฀ bankrupt฀ if฀ firm฀ value฀ at฀ the฀ end฀ of฀ the฀fiscal฀year฀falls฀below฀a฀default฀threshold฀level,฀which฀ is฀a฀function฀of฀the฀composition฀and฀amount฀of฀the฀firm’s฀ debt.8฀Given฀this฀assumption,฀the฀firm฀needs฀the฀amount฀

f฀equity฀capital฀that฀will฀make฀the฀probability฀of฀its฀value฀

falling฀below฀the฀default฀threshold฀level฀equal฀to฀0.08%฀

Table 1 Transition Matrix from Moody’s Rating To: Rating From: Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.75% 7.26% 0.79% 0.17% 0.02% 0.00% 0.00% 0.00% Aa 1.32% 90.71% 6.92% 0.75% 0.19% 0.04% 0.01% 0.06% A 0.08% 3.02% 90.24% 5.67% 0.76% 0.12% 0.03% 0.08% Baa 0.05% 0.33% 5.05% 87.50% 5.72% 0.86% 0.18% 0.31% Ba 0.01% 0.09% 0.59% 6.70% 82.58% 7.83% 0.72% 1.48% B 0.00% 0.07% 0.20% 0.80% 7.29% 80.62% 6.23% 4.78% Caa-C 0.00% 0.03% 0.06% 0.23% 1.07% 7.69% 75.24% 15.69%

Average one-year rating transition matrix, 1920-2005, conditional upon no rating withdrawal. Source: Moody’s Default and Recovery Rates of Corporate Bond Issuers, 1920-2005, March 2006.

12 Journal of Applied Corporate Finance • Volume 18 Number 4 A Morgan Stanley Publication • Fall 2006

7. See footnote 2.
8. If all debt were due at the end of the year, the default threshold level would be the

principal amount of debt outstanding plus interest due. However, if debt matures later, firm value could fall below the principal amount of debt outstanding without triggering a

default. So, the default threshold level is lower than the principal amount of debt out-

standing when the firm has long-term debt.

α α α α α

( ) ( )

μ σ θ

γ γ γ

26 / 33

SLIDE 93

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Parameter Values

Assume the firm has a target credit rating of A, which leads to 0.08% default probability and around 6% financial distress probability Moody’s Transition Matrix

f฀risk฀would฀be฀determined฀by฀comparing฀the฀costs฀associ-

ated฀with฀financial฀distress฀and฀the฀benefits฀of฀having฀a฀more฀ levered฀capital฀structure฀and฀taking฀on฀riskier฀projects.฀ To฀the฀extent฀that฀ratings฀are฀reliable฀proxies฀for฀financial฀health,฀companies฀can฀use฀a฀rating฀agency฀“transition฀ matrix”฀ to฀ estimate฀ the฀ amount฀ of฀ capital฀ necessary฀ to฀ support฀a฀given฀level฀of฀risk.฀The฀transition฀matrix฀shown฀ in฀Table฀1฀can฀be฀used฀to฀identify฀the฀frequency฀with฀which฀ companies฀moved฀from฀one฀rating฀to฀another฀over฀a฀certain฀ period฀(in฀this฀case,฀1920฀to฀2005).7฀For฀any฀rating฀at฀the฀ beginning฀ of฀ the฀ year฀ (listed฀ in฀ the฀ left-hand฀ column฀ of฀ the฀table),฀the฀column฀of฀numbers฀running฀down฀from฀the฀ heading฀“Baa”฀tells฀us฀the฀probability฀that฀a฀company฀will฀ end฀up฀with฀a฀Baa฀rating฀at฀the฀end฀of฀the฀year. Again,฀ let’s฀ assume฀ management฀ wants฀ the฀ probability฀of฀its฀rating฀falling฀to฀Baa฀or฀lower฀over฀the฀next฀year฀ to฀average฀around฀7%.฀To฀determine฀the฀probability฀of฀a฀ downgrade฀to฀or฀lower฀than฀Baa฀for฀a฀given฀initial฀rating,฀ we฀add฀up฀the฀probabilities฀of฀ending฀with฀a฀rating฀equal฀ to฀or฀lower฀than฀Baa฀along฀the฀row฀that฀corresponds฀to฀the฀ initial฀rating.฀The฀row฀where฀the฀probabilities฀of฀ending฀at฀ Baa฀or฀lower฀is฀closest฀to฀7%฀is฀the฀one฀corresponding฀to฀an฀ A฀rating.฀Consequently,฀by฀targeting฀an฀A฀rating,฀management฀would฀achieve฀the฀probability฀of฀financial฀distress฀that฀ is฀optimal฀for฀the฀firm.฀ In฀practice,฀however,฀the฀process฀of฀determining฀a฀target฀ rating฀ can฀ involve฀ more฀ considerations,฀ which฀ makes฀ it฀ more฀complicated.฀For฀example,฀Nationwide฀analyzes฀and฀ manages฀both฀its฀probability฀of฀default฀and฀its฀probability฀of฀ downgrade,฀and฀it฀does฀so฀in฀separate฀but฀related฀frameworks.฀ The฀company’s฀optimal฀probability฀of฀default฀is฀anchored฀to฀ its฀target฀Aa฀ratings฀and฀reflects฀the฀default฀history฀of฀Aa- rated฀bonds.฀By฀contrast,฀the฀probability฀of฀downgrade฀to฀ Baa฀or฀below฀is฀assumed฀to฀be฀affected฀by,฀and฀is฀accord- ingly฀managed฀by฀limiting,฀risk฀concentrations฀such฀as฀those฀ arising฀from฀natural฀catastrophes฀and฀equity฀markets. In฀ the฀ example฀ above,฀ the฀ company฀ is฀ assumed฀ to฀ maximize฀ value฀ by฀ targeting฀ a฀ rating฀ of฀ A.฀ As฀ we฀ noted฀ earlier,฀equity฀capital฀provides฀a฀buffer฀or฀shock฀absorber฀ that฀ helps฀ the฀ firm฀ to฀ avoid฀ default.฀ For฀ a฀ given฀ firm,฀ a฀ different฀probability฀of฀default฀corresponds฀to฀each฀level฀of฀ equity,฀so฀that฀by฀choosing฀a฀given฀level฀of฀equity,฀management฀is฀also฀effectively฀choosing฀a฀probability฀of฀default฀that฀ it฀believes฀to฀be฀optimal. As฀can฀be฀seen฀in฀Table฀1,฀an฀A฀rating฀is฀associated฀with฀ a฀probability฀of฀default฀of฀0.08%฀over฀a฀one-year฀period.฀ Thus,฀to฀achieve฀an฀A฀rating,฀the฀company฀in฀our฀example฀ must฀have฀the฀level฀of฀(equity)฀capital฀that฀makes฀its฀probability฀of฀default฀equal฀to฀0.08%.฀If฀we฀make฀the฀assumption฀ that฀ the฀ value฀ of฀ a฀ company’s฀ equity฀ falls฀ to฀ a฀ level฀ not฀ materially฀different฀from฀zero฀in฀the฀event฀of฀default,฀we฀ can฀use฀the฀probability฀of฀default฀to฀“back฀out”฀the฀amount฀

f฀equity฀the฀firm฀needs฀to฀support฀its฀current฀level฀of฀risk.

Although฀the฀probability฀of฀default฀is฀in฀fact฀a฀complicated฀ function฀ of฀ a฀ number฀ of฀ firm฀ characteristics,฀ not฀ just฀the฀amount฀of฀equity,฀the฀analytical฀process฀that฀leads฀ from฀the฀probability฀of฀default฀to฀the฀required฀amount฀of฀ capital฀ is฀ straightforward.฀ To฀ see฀ this,฀ suppose฀ that฀ the฀ company฀ becomes฀ bankrupt฀ if฀ firm฀ value฀ at฀ the฀ end฀ of฀ the฀fiscal฀year฀falls฀below฀a฀default฀threshold฀level,฀which฀ is฀a฀function฀of฀the฀composition฀and฀amount฀of฀the฀firm’s฀ debt.8฀Given฀this฀assumption,฀the฀firm฀needs฀the฀amount฀

f฀equity฀capital฀that฀will฀make฀the฀probability฀of฀its฀value฀

falling฀below฀the฀default฀threshold฀level฀equal฀to฀0.08%฀

Table 1 Transition Matrix from Moody’s Rating To: Rating From: Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.75% 7.26% 0.79% 0.17% 0.02% 0.00% 0.00% 0.00% Aa 1.32% 90.71% 6.92% 0.75% 0.19% 0.04% 0.01% 0.06% A 0.08% 3.02% 90.24% 5.67% 0.76% 0.12% 0.03% 0.08% Baa 0.05% 0.33% 5.05% 87.50% 5.72% 0.86% 0.18% 0.31% Ba 0.01% 0.09% 0.59% 6.70% 82.58% 7.83% 0.72% 1.48% B 0.00% 0.07% 0.20% 0.80% 7.29% 80.62% 6.23% 4.78% Caa-C 0.00% 0.03% 0.06% 0.23% 1.07% 7.69% 75.24% 15.69%

Average one-year rating transition matrix, 1920-2005, conditional upon no rating withdrawal. Source: Moody’s Default and Recovery Rates of Corporate Bond Issuers, 1920-2005, March 2006.

12 Journal of Applied Corporate Finance • Volume 18 Number 4 A Morgan Stanley Publication • Fall 2006

7. See footnote 2.
8. If all debt were due at the end of the year, the default threshold level would be the

principal amount of debt outstanding plus interest due. However, if debt matures later, firm value could fall below the principal amount of debt outstanding without triggering a

default. So, the default threshold level is lower than the principal amount of debt out-

standing when the firm has long-term debt.

α α α α α

( ) ( )

μ σ θ

γ γ γ

26 / 33

SLIDE 94

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Parameter Values

Assume the firm has a target credit rating of A, which leads to 0.08% default probability and around 6% financial distress probability Moody’s Transition Matrix

f฀risk฀would฀be฀determined฀by฀comparing฀the฀costs฀associ-

ated฀with฀financial฀distress฀and฀the฀benefits฀of฀having฀a฀more฀ levered฀capital฀structure฀and฀taking฀on฀riskier฀projects.฀ To฀the฀extent฀that฀ratings฀are฀reliable฀proxies฀for฀financial฀health,฀companies฀can฀use฀a฀rating฀agency฀“transition฀ matrix”฀ to฀ estimate฀ the฀ amount฀ of฀ capital฀ necessary฀ to฀ support฀a฀given฀level฀of฀risk.฀The฀transition฀matrix฀shown฀ in฀Table฀1฀can฀be฀used฀to฀identify฀the฀frequency฀with฀which฀ companies฀moved฀from฀one฀rating฀to฀another฀over฀a฀certain฀ period฀(in฀this฀case,฀1920฀to฀2005).7฀For฀any฀rating฀at฀the฀ beginning฀ of฀ the฀ year฀ (listed฀ in฀ the฀ left-hand฀ column฀ of฀ the฀table),฀the฀column฀of฀numbers฀running฀down฀from฀the฀ heading฀“Baa”฀tells฀us฀the฀probability฀that฀a฀company฀will฀ end฀up฀with฀a฀Baa฀rating฀at฀the฀end฀of฀the฀year. Again,฀ let’s฀ assume฀ management฀ wants฀ the฀ probability฀of฀its฀rating฀falling฀to฀Baa฀or฀lower฀over฀the฀next฀year฀ to฀average฀around฀7%.฀To฀determine฀the฀probability฀of฀a฀ downgrade฀to฀or฀lower฀than฀Baa฀for฀a฀given฀initial฀rating,฀ we฀add฀up฀the฀probabilities฀of฀ending฀with฀a฀rating฀equal฀ to฀or฀lower฀than฀Baa฀along฀the฀row฀that฀corresponds฀to฀the฀ initial฀rating.฀The฀row฀where฀the฀probabilities฀of฀ending฀at฀ Baa฀or฀lower฀is฀closest฀to฀7%฀is฀the฀one฀corresponding฀to฀an฀ A฀rating.฀Consequently,฀by฀targeting฀an฀A฀rating,฀management฀would฀achieve฀the฀probability฀of฀financial฀distress฀that฀ is฀optimal฀for฀the฀firm.฀ In฀practice,฀however,฀the฀process฀of฀determining฀a฀target฀ rating฀ can฀ involve฀ more฀ considerations,฀ which฀ makes฀ it฀ more฀complicated.฀For฀example,฀Nationwide฀analyzes฀and฀ manages฀both฀its฀probability฀of฀default฀and฀its฀probability฀of฀ downgrade,฀and฀it฀does฀so฀in฀separate฀but฀related฀frameworks.฀ The฀company’s฀optimal฀probability฀of฀default฀is฀anchored฀to฀ its฀target฀Aa฀ratings฀and฀reflects฀the฀default฀history฀of฀Aa- rated฀bonds.฀By฀contrast,฀the฀probability฀of฀downgrade฀to฀ Baa฀or฀below฀is฀assumed฀to฀be฀affected฀by,฀and฀is฀accord- ingly฀managed฀by฀limiting,฀risk฀concentrations฀such฀as฀those฀ arising฀from฀natural฀catastrophes฀and฀equity฀markets. In฀ the฀ example฀ above,฀ the฀ company฀ is฀ assumed฀ to฀ maximize฀ value฀ by฀ targeting฀ a฀ rating฀ of฀ A.฀ As฀ we฀ noted฀ earlier,฀equity฀capital฀provides฀a฀buffer฀or฀shock฀absorber฀ that฀ helps฀ the฀ firm฀ to฀ avoid฀ default.฀ For฀ a฀ given฀ firm,฀ a฀ different฀probability฀of฀default฀corresponds฀to฀each฀level฀of฀ equity,฀so฀that฀by฀choosing฀a฀given฀level฀of฀equity,฀management฀is฀also฀effectively฀choosing฀a฀probability฀of฀default฀that฀ it฀believes฀to฀be฀optimal. As฀can฀be฀seen฀in฀Table฀1,฀an฀A฀rating฀is฀associated฀with฀ a฀probability฀of฀default฀of฀0.08%฀over฀a฀one-year฀period.฀ Thus,฀to฀achieve฀an฀A฀rating,฀the฀company฀in฀our฀example฀ must฀have฀the฀level฀of฀(equity)฀capital฀that฀makes฀its฀probability฀of฀default฀equal฀to฀0.08%.฀If฀we฀make฀the฀assumption฀ that฀ the฀ value฀ of฀ a฀ company’s฀ equity฀ falls฀ to฀ a฀ level฀ not฀ materially฀different฀from฀zero฀in฀the฀event฀of฀default,฀we฀ can฀use฀the฀probability฀of฀default฀to฀“back฀out”฀the฀amount฀

f฀equity฀the฀firm฀needs฀to฀support฀its฀current฀level฀of฀risk.

Although฀the฀probability฀of฀default฀is฀in฀fact฀a฀complicated฀ function฀ of฀ a฀ number฀ of฀ firm฀ characteristics,฀ not฀ just฀the฀amount฀of฀equity,฀the฀analytical฀process฀that฀leads฀ from฀the฀probability฀of฀default฀to฀the฀required฀amount฀of฀ capital฀ is฀ straightforward.฀ To฀ see฀ this,฀ suppose฀ that฀ the฀ company฀ becomes฀ bankrupt฀ if฀ firm฀ value฀ at฀ the฀ end฀ of฀ the฀fiscal฀year฀falls฀below฀a฀default฀threshold฀level,฀which฀ is฀a฀function฀of฀the฀composition฀and฀amount฀of฀the฀firm’s฀ debt.8฀Given฀this฀assumption,฀the฀firm฀needs฀the฀amount฀

f฀equity฀capital฀that฀will฀make฀the฀probability฀of฀its฀value฀

falling฀below฀the฀default฀threshold฀level฀equal฀to฀0.08%฀

Table 1 Transition Matrix from Moody’s Rating To: Rating From: Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.75% 7.26% 0.79% 0.17% 0.02% 0.00% 0.00% 0.00% Aa 1.32% 90.71% 6.92% 0.75% 0.19% 0.04% 0.01% 0.06% A 0.08% 3.02% 90.24% 5.67% 0.76% 0.12% 0.03% 0.08% Baa 0.05% 0.33% 5.05% 87.50% 5.72% 0.86% 0.18% 0.31% Ba 0.01% 0.09% 0.59% 6.70% 82.58% 7.83% 0.72% 1.48% B 0.00% 0.07% 0.20% 0.80% 7.29% 80.62% 6.23% 4.78% Caa-C 0.00% 0.03% 0.06% 0.23% 1.07% 7.69% 75.24% 15.69%

Average one-year rating transition matrix, 1920-2005, conditional upon no rating withdrawal. Source: Moody’s Default and Recovery Rates of Corporate Bond Issuers, 1920-2005, March 2006.

12 Journal of Applied Corporate Finance • Volume 18 Number 4 A Morgan Stanley Publication • Fall 2006

7. See footnote 2.
8. If all debt were due at the end of the year, the default threshold level would be the

principal amount of debt outstanding plus interest due. However, if debt matures later, firm value could fall below the principal amount of debt outstanding without triggering a

default. So, the default threshold level is lower than the principal amount of debt out-

standing when the firm has long-term debt.

Parameter Value

Risk Appetite Risk Limits

1

α

2

α

3

α

4

α α

( )

P

r

( )

A

r lop m 0.05 0.05 0.05 0.05 0.0008 0.2 0.01 Hazard Risk Op Risk Factors Strategic Factor Borrowed Capital μ σ θ d

1

γ

2

γ

3

γ

c 0.01 0.1 0.1 0.2 0.2 0.1 0.5 0.7

26 / 33

SLIDE 95

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Optimization Results

Optimization Results under the Baseline Setting

Decision Variable Optimal Value Investment in the R&D project 0.0558 Investment in the manufacturing project 0.6228 Investment in the index fund 0.0542 Investment in the Treasury bill 0.2546 The proportion of hazard risk insured 0.9479 Optimal Return 1.0806

27 / 33

SLIDE 96

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Risk Prioritization

No Prioritization vs. With Prioritization

Decision Variable No Prioritization All α’s = 0.05 With Prioritization α1 = 0.1, α4 = 0.01 Investment in the R&D project 0.0558 0.0991 Investment in the manufacturing project 0.6228 0.5197 Investment in the index fund 0.0542 0.0647 Investment in the Treasury bill 0.2546 0.3039 The proportion of hazard risk insured 0.9479 0.9625 Optimal Return 1.0806 1.0862

28 / 33

SLIDE 97

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Interactions Between Risks

High Op Risk Factors vs. Low Op Risk Factors

Decision Variable

High Op Factors (γ1=0.2, γ2=0.1) Low Op Factors (γ1=0.05, γ2=0.05)

R&D project 0.0558 0.0813 Manufacturing project 0.6228 0.9062 Index fund 0.0542 Treasury bill 0.2546 Hazard risk insured 0.9479 0.9479 Optimal Return 1.0806 1.1004

29 / 33

SLIDE 98

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Two-Period Setting

Risk/return optimization in a two-period planning horizon Three broad types of investment opportunities

Short-term (one-period) real projects: invest at the beginning of each period, return at end of each period Long-term (two-period) real projects: invest at the beginning of period 1, return at end of period 2 Financial assets

30 / 33

SLIDE 99

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Strategic Time Line

Time 0 Time 1 Time 2 C(0)

rj

(1) = E[Rj (1)| I0],

j = P1 or A rj

(2) = E[Rj (2)| I0],

j = P1 or A rP2 = E[R P2| I0]

E[C(1)]

31 / 33

SLIDE 100

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Strategic Time Line

Time 0 Time 1 Time 2 C(0)

rj

(1) = E[Rj (1)| I0],

j = P1 or A rj

(2) = E[Rj (2)| I0],

j = P1 or A rP2 = E[R P2| I0]

( )

1

1 1 j

P

w

,

l P

w

2 ,

( )

1

Ak

w , u(1)

( )

2

2 1 j

P

w

,

( )

2

Ak

w

, u(2)

E[C(1)]

31 / 33

SLIDE 101

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Strategic Time Line

Time 0 Time 1 Time 2

( )

1

1 1

ˆ j

P

r

,

( )

1

ˆ

Ak

r

sj

(2) = E[Rj (2)| I1],

j = P1 or A sP2 = E[R P2| I1]

C(1)

( ) *

1

1 1 j

P

w

,

( ) *

1

Ak

w

,

*

2l

P

w

,

* ) 1 (

u

31 / 33

SLIDE 102

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Strategic Time Line

Time 0 Time 1 Time 2

( )

1

1 1

ˆ j

P

r

,

( )

1

ˆ

Ak

r

sj

(2) = E[Rj (2)| I1],

j = P1 or A sP2 = E[R P2| I1]

C(1)

( ) *

1

1 1 j

P

w

,

( ) *

1

Ak

w

,

*

2l

P

w

,

* ) 1 (

u

( )

2

2 1 j

P

v

,

( )

2

Ak

v

, u(2)

31 / 33

SLIDE 103

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Strategic Time Line

Time 0 Time 1 Time 2

( )

2

2 1

ˆ

j P

s

,

( )

2

ˆAk s

,

l P

s

2

ˆ

( ) *

2

2 1 j

P

v

,

( ) *

2

Ak

v

,

* ) 2 (

u C(2)

( )

1

1 1

ˆ j

P

r

,

( )

1

ˆ

Ak

r

( ) *

1

1 1 j

P

w

,

( ) *

1

Ak

w

,

*

2l

P

w

,

* ) 1 (

u

31 / 33

SLIDE 104

Introduction Static Risk Appetite Numerical Dynamic Conclusion

The Dynamic Framework

Stage 1 (based on information set I0) max E[end-of-horizon total return] s.t. Each risk constraint for period 1 Each risk constraint for period 2 Budget constraint, Strategic constraint, Range constraint Stage 2 (based on information set I1) max E[end-of-horizon total return] s.t. Each risk constraint for period 2 Budget constraint, Strategic constraint, Range constraint

32 / 33

SLIDE 105

Introduction Static Risk Appetite Numerical Dynamic Conclusion

Discussions

Distributional assumptions

Elliptically symmetric distribution (convex programming) The copula method (search methods)

Risk management strategies Decision of risk appetite Other model extensions

Multi-stage model Other stochastic optimization models

33 / 33