PAK Study Manual Risk Neutral Dynamic Hedging

Transcription

PAK Study Manual
Foundations of CFE (CFE) Exam
Fall 2014 Edition
CTE
Hull-White
Ito’s Lemma
Agency Theory
Reinsurance
Risk Neutral
Corporate Finance
Stochastic Simulation
Efficient Market Hypothesis
Regime Switching Lognormal
Dynamic Hedging
Economic Capital
Behavioral Finance
Martingale
Value at Risk
Dividend
WACC
Delta
PAK Study Manual
for CFE Fall 2014
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PAK Study Manual for CFE Fall 2014
Frequent Answer Questions
Do You Need to Read the Source Readings?
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T=2.5 months
T=0
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Practice
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© 2014 PAK Study Manual
1
SOA CFE Exam (Study Manual)
F-107-13: A Market Cost of Capital Approach to Market Value Margins (by CFO Forum)
Key Points
SAMPLE
1.
Understand the economic balance sheet structure
2.
Understand how to use the market cost of capital approach to calculate the market value margin
1. Executive Summary
Two Common Approaches to Calculate the Market Value Margins (MVMs)
- Percentile approach
- Market cost of capital (MCoC) approach
Market Cost of Capital (MCoC) Appaoch
- The CRO Forum prefers the MCoC approach
- This approach works for both life and non-life business
Main Reasons for the CRO Forum’s Preference
1. Support appropriate risk management actions
2. Appropriate reflection of risk
3. Better response to a potential crisis
4. Easy to implement
5. Transparent, easily verifiable and understandable
6. Pass the “use test”
Figure 2: Economic Balance Sheet
Components of Market Consistent Value of Liabilities (MVL)
1. Expected PV of future LCFs
2. MVM for non-hedgeable risks (An explicit cost of risk for non-hedgeable risk)
MVL
-
It represents the market consistent value at which the liabilities could be transferred to a willing, rational,
diversified counterparty in an arms’ length transaction under normal business conditions
Note
“Arms’ Length Transaction” means the transaction made by two totally independent parties
2
MVM
-
It is the cost of risk (risk margin) in addition to the expected PV of future LCFs required to manage the
business on an ongoing basis
It is estimated by the PV of the cost of future capital requirements for non-hedgeable risks
Calculation of MVMs using a MCoC approach is straightforward given that the majority of the calculation
is prescribed under the standard SCR
2. Introduction
Purpose of This Paper
- The MCoC approach to MVMs is the CFO Forum’s preferred method
Calculate the Liability Value
- The assets and liabilities should be measured on a consistent basis for solvency purposes
o This basis should be market value
-
In order to determine the MVL, a MVM is added to the expected PV of future LCFs
o The CRO Forum does not agree with using the percentile method to calculate the MVM since it is
not consistent with the aim of introducing a risk based solvency assessment
o The MCoC approach to MVMs is the CFO Forum’s preferred method
3. Why Take a MCoC Approach to MVMs?
Two Common Approaches to Calculate the Market Value Margins (MVMs) for Non-Hedgeable Risks
1. Percentile approach
o Sufficient capital is needed to ensure that the liabilities can be met with a predefined confidence
level
2. Market cost of capital (MCoC) approach
o Sufficient capital is needed to be able to run-off the business
Main Reasons for the CRO Forum’s Preference
1. Support appropriate risk management actions
2. Appropriate reflection of risk
3. Better response to a potential crisis
4. Easy to implement
5. Transparent, easily verifiable and understandable
6. Pass the “use test”
Supports Appropriate Risk Management Actions
- MCoC approach
o It more appropriately differentiates between risks similar to the way in which capital markets
differentiate between risks (equity investment vs. equity option)
o
-
It ensures that the cost of risk is measured purely based on the economic cost of holding capital
to support non-hedgeable risks
 The cost of risk and any allowance for prudence are clearly separated
 The reserve reflects the best estimate of the cost of managing risk
 Margins for prudence should only be reflected in the capital held (SCR) and not in the
technical provision (MVL)
Percentile approach
o Prudence may be incorporated in both the reserves and capital which can lead to inefficient
management of risks and double counting of risks
3
Appropriate Reflection of Risk
- MCoC approach
o The MVMs will always reflect the risk inherent in the product
-
Percentile approach
o The MVMs will not always reflect the risk inherent in the product because there is no link
between the arbitrary percentile value chosen and the market price
o Also, the percentile approach does not refer to each risk type separately
Figure 1: Percentile vs. MCoC
Figure 1
- For the long tailed gamma distribution (“more-severity” risks: catastrophic risk, terrorist risk, etc)
o The percentile approach underestimates the price of non-hedgeable risk
-
For the normal distribution (regular insurance risks: mortality risk, etc)
o The percentile approach greatly overestimates the market price of risk
-
The MCoC approach ensures that insurers consider the tails of the distributions
o No consideration is given to the shape of the distribution using the percentile approach
Experiences of the Australian and the Swiss Regulators
- Australian regulators
o It is difficult to reasonably explain the spread of risk margins using the percentile approach
-
Swiss Regulators
o The MVMs under a MCoC approach appropriately reflect the risk inherent in the business
 The calculation is stable from period to period
 Life insurers writing mainly savings products have relatively small MVM since insurance
risk is small
 Life insurers writing risk products have relatively large MVM since they have a large
exposure to biometric risk and a long duration of the run-off portfolio
4
Response to a Potential Crisis
- MCoC approach
o It ensures that after a stressed event, the company will be able to appropriately remunerate either
a third party accepting the liability or new capital providers
o It achieves this through the SCR for non-hedgeable risks needed to support the liability in future
years and hence the MVM, which represents a provision for the cost of holding this capital
-
Percentile approach
o It implicitly forces the insurer to hold part of the capital needed to support the business in future
years in the form of a prudence margin
o This prudence margin will not suffice to run-off the liabilities with the level of confidence implied
by the SCR
o It does not ensure that there will be sufficient financial resources to cover future capital costs
needed to remunerate either a third party accepting this liability or new capital providers
Ease of Implementation
- Percentile approach
o It is complex (for small entities) to implement since it requires stochastic calculations over the
whole run-off period to determine the appropriate percentile on the distribution of liability values
-
MCoC approach
o It is easier to implement since there is only one unknown item (the SCR for non-hedgeable risks)
and this can be calculated with ease using the standard SCR
Transparent, Verifiable and Comprehensible
- MCoC approach
o Supervisors can easily replicate and verify the MVM calculation (using standard SCR and internal
model)
-
Percentile approach
o It is not so easily verified because it is not possible for the regulator to use the standard SCR to
benchmark the internal models used to determine the percentile value
o Australian regulators experienced a wide variation in the results from insurer to insurer , when
adopting a percentile approach
The “Use Test”
- The MCoC approach has been used for some 20 years
- It already passes the “use test” envisioned in the Solvency II framework
5
4. Theory Underpinning the MCoC Approach: The Economic (Solvency) Balance Sheet
Figure 2: Economic (Solvency) Balance Sheet
Note
In the general accounting framework, we know that assets = liabilities + equities.
Assume that assets = MVA and liabilities = MVL. Then equities = SCR + Excess capital
Economic (Solvency) Balance Sheet
- It distinguishes between the asset and liability values and the capital required for solvency purposes
- Capital requirements consider risks emanating from both sides of the balance sheet
- It does not explicitly identify the cost of capital
Market Value of Assets (MVA)
- The capital markets provide the MVA
Market Consistent Value of Liabilities (MVL)
- It is derived from the cost of managing the risks underlying the business on an ongoing basis
- It represents the market consistent value at which the liabilities could be transferred to a willing, rational,
diversified counterparty in an arms’ length transaction under normal business conditions
- Market values should be used where available to value the MVL
- Where market values are not available, the MVL must be explicitly calculated using market consistent
valuation techniques
6
Figure 3: Components Parts of the MVL Calculation
Expected Present Value of Future Liability Cash Flows
- It includes premiums, fees, policyholder claims, expenses and commissions
- The market consistent value of these future CFs may be determined as the cost of setting up a replicating
portfolio
Replicating (Hedge) Portfolio
- The portfolio of assets that most closely matches the corresponding liability CFs
- In the absence of arbitrage, and if the liability CFs could be matched exactly, the market consistent value
of the liabilities will exactly equal the market value of the replicating portfolio
Note
To the simplest form, the expected PV of future LCFs is the same as the reserve that we learnt in FAP. Remember the formula?
Reserve = PV(Expenses/Claims) – PV(Premiums)
Figure 4: Replicating Portfolio
7
MVM for Non-Hedgeable Risk
- The MVM is defined as the cost of risk (risk margin) in addition to the expected PV of future LCFs
required to manage the business on an ongoing basis
- It is only necessary to calculate an explicit MVM for non-hedgeable risks
Figure 5: Types of Risk Affecting the Liability Cash Flow
Hedgeable Risks
- A hedgeable risk is a risk which can be pooled or hedged using a replicating portfolio
o Hedging costs are implicit in the observed market price of those instruments
o Thus, it is not necessary to calculate an explicit MVM for hedgeable risks
Non-Hedgeable Risks
- Risks for which a deep and liquid market is not available are referred to as non-hedgeable
o They are risks for which a market price cannot be observed
-
To compensate an investor for the cost of taking non-hedgeable risks, an explicit MVM is demanded
o Under the MCoC approach, the MVM is the compensation required for the cost of holding capital
against non-hedgeable risks over the life of the policy
Figure 6: Flowchart for Determining the Appropriate Method for Calculating the MVL
8
Solvency Capital Requirement (SCR)
- It can be determined by the Standard Approach or the Internal Model Approach under Solvency II
- The projection of what the SCR is for non-hedgeable risk is needed to calculate the MVM
Cost of Capital (CoC)
- It refers to the capital charge on fully diversified capital held to cover non-hedgeable risks only
o This should not be seen as a total company CoC
Note
In the embedded value framework, the CoC means the cost of holding the capital needed to support the businesses of the whole company.
In the MVM calculation here, we just focus on the capital held to cover non-hedgeable risks only.
5. The MCoC Approach to MVMs
MCoC Approach to MVMs
- The MVM for non-hedgeable risk is calculated as the PV of the cost of future capital requirements for
non-hedgeable risks
Steps to Calculate the MVM for Non-Hedgeable Risks
1. Project the SCR for non-hedgeable risks
o Using the standard model or internal models, the projected SCR can be determined via an
underlying driver that is indicative of the risk level (e.g. PV of benefit) (See Appendix C)
o The required SCR at time 0 for non-hedgeable risk types would be set at 99.5% Value at Risk for
a holding period of one year and calculated net of full diversification effects (see Appendix B)
2. Calculate the capital charge
o The capital charge for non-hedgeable risks can be explicitly calculated by multiplying the future
SCR at each point in time by the CoC for non-hedgeable risk
o This cost of capital charge only applies to capital that is required for the non-hedgeable risk
3. Discount the projected capital charge
o The projected capital charge stream is then discounted at the risk free rate (swap rate) to get the
MVM
Figure 7: Calculating the MVM for Non-Hedgeable Risk
9
Illustrative Example
Data and Assumptions
- An insurer has sold 500 term policies and the premium on each policy is €50
- If the policyholder dies within a 5 year period, then a benefit of €1000 is paid
- The probability of dying within any one year is 1%
- The swap rate is 5% (yield curve is assumed to be flat)
- The CoC for non-hedgeable risks used to illustrate the approach in this example is 4%
Step 1: Project the SCR for Non-Hedgeable Risks
Time
No. of policies (start of year)
Premium (beginning of year)
Claims
PV Claims (Discounted at mid-year)
SCR
SCR as % of PV of benefits
0
500
25,000
5,000
21,764
2,176
1
495
2
490
3
485
4
480
4,950
17,729
1,773
4,901
13,543
1,354
4,851
9,199
920
4,803
4,687
469
5
0
= 10% x 21,764 = 10% x 17,729 = 10% x 13,543 = 10% x 9,199 = 10% x 4,687
10%
Step 2: Calculate the Capital Charge
Time
SCR
0
2,176
87
Capital Charge
Capital Charge / SCR
= 4% x 2,176
1
1,773
71
2
1,354
54
= 4% x 1,773 = 4% x 1,354
3
920
37
4
469
19
5
= 4% x 920
= 4% x 469
4%
Step 3: Discount the Capital Charge
Time
SCR
Capital Charge (mid-year)
0
2,176
87
85
Discounted Capital Charge
Total MVM
PV Liabilities
MVL
= 87/1.05^0.5
1
1,773
71
66
2
1,354
54
48
= 71/1.05^1.5 = 54/1.05^2.5
3
920
37
31
4
469
19
15
= 37/1.05^3.5
= 19/1.05^4.5
5
245
21,764
22,009
6. Frequently Asked Questions
Circularity
- The issue of circularity in the calculation arises because the MVM is calculated directly from the SCR
o But it also makes up part of the MVL and is assumed to be included in the SCR calculation
-
The relative size of the MVM in comparison to the total MVL is small so the impact of including the MVM
in the SCR calculation would be insignificant
o Thus, it is assumed that the MVM will have little or no effect on the SCR
Asset Liability Mismatching
- The acquiring insurer should not be compensated for any avoidable asset liability mismatch (ALM) taken
by the defaulting insurer
o ALM risk is a hedgeable risk so it is assumed that the acquiring insurer can swap back to the
replicating portfolio instantaneously and hedge any ALM taken by the defaulting insurer
10
Level of Risk Margin Relative to Overall Liability
- Some regulators and supervisors are concerned about the relatively low level of MVMs
- The explicit MVM only corresponds to the MVM charged for non-hedgeable risks
o Low margins are also observed using a percentile approach so this is not purely a MCoC issue
Tax
-
The liabilities and the MVM should be set on a pre-tax basis since this is analogous to how market values
are set for assets
Operational risk
- Since operational risk is a non-hedgeable risk, it should be included in the calculation of the MVM
Harmonization
- The MVM calculation is based on the projected capitals but various insurers have various internal models
o The CRO Forum has also formulated recommended approaches of benchmarking internal
models to address these concerns
o The proportion of the MVL that corresponds to the explicit MVM is relatively small so any
discrepancies will have a relatively small impact on the results
Total Portfolio vs. Different Business Line
- The MVM is calculated for each LoB for each risk type and then added to the best estimate liability value
o This will improve transparency and facilitate companies’ analysis of the risks they take
o The MVL is then aggregated to a company level
Non-Life Business
- The MCoC approach applies for non-life business
o For in-force business, a SCR for non-hedgeable risk will be set up for the current year to support
the insured risk
o In addition to this, SCR for the non-hedgeable reserve risks will be held after the expiration of the
contract
o This capital is held to cover the risk that reserves may not be sufficient to cover claims either
because ultimate claims or the pay out pattern had been incorrectly estimated
Small Entities
- The Swiss Solvency Field Test showed that this MCoC approach works well in smaller entities due to its
simplifying assumptions
- The standard SCR approach can be adopted to determine the SCR for non-hedgeable risks
Variation in CoC
- The CoC for non-hedgeable risks reflects the excess return over risk free rates that an acquiring company
would require to compensate them for the cost of holding capital to run-off the business
-
The CRO Forum suggests that the risk margin (MCoC) should not vary by risk type or business
o Since the SCR for non-hedgeable risk will differ between risk types, the MVM will automatically
differ under the MCoC approach thereby better reflecting risk
-
With regard to the reference market in which the products are sold, it is possible that the price of risk may
vary between countries due to uncertainty in the market
o Most of this uncertainty is due to the difficulties in hedging unwanted risk and will already be
factored into the SCR and MVM, as these risks are not hedgeable
11
Time Horizon
- One common misconception regarding the MCoC approach
o The MVM only considers one year worth of risk and everything after the first year is ignored
because the SCR is measured using a one-year shock approach
o This is incorrect
-
The SCR calculation does consider the expected LCFs over periods
o The SCR represents the change in liability value between expected future liability cash flows and
worst-case (99.5% percentile) cash flows
o These worst-case cash flows reflect not what we can observe in any one year but rather how far
off we can be in estimating our expected liability cash flows over their entire life
Appendix A. Expected Present Value of Future Cash Flows
Products without Optionality
- All cash flows that occur with certainty are totally risk free and can be replicated with risk-free assets of a
similar term
Diversifiable Risk
- The market consistent framework assumes that there is no reward for holding diversifiable risk
o This assumption implies that any cash flows that are not risk free but where risk is diversifiable
should be treated as risk free and hence discounted at the swap rate
o This assumption flows through to the treatment of the MVM for non-hedgeable risks
Non-Diversifiable Risk
- This assumption also implies that expected cash flows that are subject to non-diversifiable risk should be
discounted at a rate that reflects the risk inherent in the cash flow
o The risk-adjusted rate contains a risk premium that investors would demand for
o The greater the expected return driving such a risky cash flow, the greater the discount rate
applied to the cash flow
Certainty Equivalent Approach
- This is a form of risk-neutral valuation where the risky cash flows are adjusted and projected assuming
that all underlying assets earn the risk-free rate
Note
CFrisky ,t
1. Discount risky cash flows using risk-adjusted rate:
V0 = 
2. Discount risk-free cash flows using risk-free rate:
V0 = 
CFrisk free,t
3. Adjust risky cash flows and discount at risk-free rate:
V0 = 
CFrisky ,t − Z t
(1 + rf + rp )t
(1 + rf )t
(1 + rf )t
Products with Embedded Options or Guarantees
- For products with options/guarantees, using the certainty equivalent approach becomes a very complex
process due to the non-linear, asymmetric relationships with market returns that exist in these products
o It is necessary to determine the liability value using option pricing or Monte Carlo simulation
o In order to achieve market-consistency, both methods should be calibrated to reproduce the
prices of traded assets that best reflect the characteristics of the liabilities being valued
12
Appendix B. Diversification
Four Levels of Diversification
Diversification Levels
Level 1
(Within risk types)
Level 2
(Across risk types)
Level 3
(Across entities, within a given geography)
Example
Two or more insurance companies in the same geography within an
insurance Group
Level 4
(Across geographies or regulatory jurisdictions)
Two or more insurance companies in different geographies within an
insurance Group
Adding more unrelated risks to the portfolio
Combining two classes of insurance
Appendix C: Proxy Measures for Projecting Capital
Non-Hedgeable Risk Type
Mortality
Life
Morbidity
Insurance
Persistency
Products
Death Protection Survivorship Protection
Net amount at risk
N/A
Economic liability
PV of benefits
N/A
Economic liability
Savings
Accident & Health
Economic liability
N/A
Economic liability
Net amount at risk
Net amount at risk
Economic liability
Note
There is a long list in the reading. You should read it once. I personally think the life insurance line is more important so I put it here.
Practice Question
List the considerations of calculating the MVM
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Circularity
Asset Liability Mismatching
Level of Risk Margin Relative to Overall Liability
Tax
Operational risk
Harmonization
Total Portfolio vs. Different Business Line
Non-Life Business
Small Entities
Variation in CoC
Time Horizon
Past CFE/FETE SOA Questions Related To This Reading
SOA CFE Fall 2013 Q8 (Must Read)
13

PAK Study Manual Risk Neutral Dynamic Hedging

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