Mildenhall Ch 8: Classical Portfolio Pricing Theory Flashcards
1
Q
Briefly explain the risk transfer driver for insurance demand
A
Insured purchases insurance to shift their risk to insurer.
They do this by pooling their risk averse indivuals
1
Q
Identify 3 drivers of insurance demand
A
- Risk transfer
- Satisfying demand
- Risk financing
2
Q
Name a line of business for which risk transfer driver is important
A
Short Tail P&C lines
Inter-insured, intratemporal
3
Q
Briefly explain the Satisfying demand driver for insurance demand
A
Insured purchases insurance to satisfy statutory, regulatory or contractual requirements.
4
Q
Name a line of business for which satisfying demand is an important driver.
A
In order to drive an auto in USA, drivers must have 3rd-party liability insurance.
Accounts for at least 60% of global P&C premium.
5
Q
Briefly explain the risk financing driver au insurance demand
A
Insured purchases insurance to finance uncertain future contingencies in an efficient manner.
Requirements are driven by accounting, regulation and tax law.
6
Q
Name a line of business for which risk financing is an important driver.
A
Large employers have LDD workers compensation plans where employer pays for all claims under deductible & insurer covers claims above deductible.
7
Q
List 4 forms of supply of insurance services
A
- Sales
- Marketing
- Risk surveys
- Loss control
- Risk bearing
- Pricing
- Underwriting
- Customer billing & support
- Investment management services
These services are bundled in different ways
8
Q
Describe 2 critical functions when managing risk pool
A
- Controlling access to the pool through UW and pricing (Pool Co.)
- Ensuring pool is solvent by funding risk-bearing assets through sale of liabilities (Capital Co.)
9
Q
Identify 5 ways insurance services/functions can be handled
A
- Some insurance companies offer all of these services in-house
- Customer-facing functions are included with direct writer, but farmed out to independent agents or broker for other types of writers
- Managing general agents (MGAs) might only provide UW & pricing services
- Some insurers might use third-party claims adjusters
- Some insurers manage assets in-house, while others use third-party firms to manage assets
10
Q
Define insurance contract
A
Insurance contracts must be written such that claims are clear and objective, easy to adjust and discourage fraud.
The insured should NEVER profit from insurance.
It pays no more than subject loss.
11
Q
Describe 2 ways insurance contracts can trigger
A
- Parametric insurance contracts
Policies pay based on an objective event outcome, such as hurricane intensity and landfall or earthquake magnitude and epicenter. - Dual trigger contracts
Policies pay based on an objective event outcome AND the event that caused an economic loss to insured.
Indemnity payment is a function of the underlying subject loss suffered by insured.
12
Q
Name and advantage and disadvantage of Parametric insurance contracts
A
Advantage:
These policies are easy to underwrite since they do not depend on characteristics of insured.
Disadvantage:
These policies are difficult to design in such a way that they eliminate potential for an insured to profit from a claim.
13
Q
Define basis risk
A
Mismatch between insured’s subject loss and insurance recovery.
Risk that loss payout will be greater than sponsoring insurer’s actual losses (negative net losses = profit)
14
Q
Define indemnity function f()
A
Combined limits, deductibles, etc.
Must satisfy the following conditions:
1. 0 =< f( ) =< L
2. F(L) i monotonic function of L
3. L-f(L) is monotonic function of L
4. f is continuous
15
Q
Briefly explain why franchise deductible fails to meet indemnity function requirements underlying dual trigger contracts
A
Under a franchise deductible d, if L smaller than d, f(L) = 0.
If L > d, f(L) = L instead of L-d
Thus, f is not continuous (jumps at d) and L-f(L) is not monotone (increases until d and then decreases to 0)
16
Q
Briefly explain why excess of loss covers meet the indemnity function requirements underlying dual-trigger contracts
A
XOL indemnity function is f(L) = (L-a)+
Clearly, 0 =< f(L) =< L, f is continuous and monotonic since it is zero before a then increases after a.
L-f(L) is monotone since it increases to a before a and stays there after.
17
Q
Contrast assets & liabilities
A
Assets are things you own or are owed.
Assets are created when a tangible or intangible product or promise is received in exchange for cash or other consideration paid.
Ex: a ceded reinsurance contract creates an assets
Liabilities are things you owe.
Liabilities are created when a promise is made in exchange for cash or other consideration received.
Amounts of cash received for liability equals its market value.
Ex: an insurance policy is a promise to pay losses in exchange for premium.
18
Q
True or False?
All assets are reported in financial statements.
A
False, not all assets are reported.
Ex: brand value or ownership of renewals are not included
19
Q
True or False?
Financial reporting takes a more comprehensive approach to booking liabilities than assets.
A
True
20
Q
Explain the payment order upon liquidation
A
When an insurer is put into liquidation, all policies are cancelled and insurer assets are used to pay all of its stakeholders.
This process is managed by a receiver.
Claims against the assets are paid out in priority order:
1. Highest priority items to be paid (senior priority) are receiver expenses
2. First material items paid are insurance claims from policyholders
3. Then, debt holders are paid
4. Finally, shareholders receive what is left
All claimholders are paid on pro-rata basis.
21
Q
When can we say an insurer is bankrupt?
A
When equity is wiped out and there is zero residual value for equity holders.
22
Q
When can we say an insurer is default?
A
When debt holders are wiped out as well (Assets < policy holder liabilities)
23
Q
Differentiate between capital, surplus and equity
A
Capital (aka policyholder surplus in US) equals assets net of liabilities owed to policyholders.
Equity equals assets net of liabilities owed to all parties EXCEPT owners.
Equity is also defined as owners residual value.
Both equity and capital depend on the specific accounting standards used (statutory, GAAP, IFRS, etc)
24
Describe 4 types of insurance capital
1. Common equity
Common shareholders bear the ultimate risk and receive benefit of success.
They control the insurer & profit the most if successful.
2. Reinsurance capital
If premium collected by reinsurer (assets) is greater than ceded losses and expenses (liabilities), then there is a positive residual value transferred to reinsurer.
3. Debt capital
When subordinated to policyholder claims, it creates capital.
In the US, this is known as surplus note.
Since they cannot issue shares, mutual companies use surplus notes to raise capital.
4. Preferred Equity
Blends characteristics of debt and equity.
25
Identify 3 categories of common equity
1. Capital stock
Par value of shares issued (applies to stock companies)
2. Additional paid-in capital
XS amount paid-in over par value.
Occurs when shares are sold at price higher than par (applies to stock companies)
3. Retained earnings
Cumulative undistributed earnings (applies to all insurers: stock companies, mutual, reciprocals, etc)
26
True or False?
Suspension of dividend can trigger default.
False,
Dividends can be suspended without triggering default.
27
Place the 3 types of insurance capital in terms of priority
1. Debt
2. Preferred
3. Common
28
Which types of insurance capital are tax deductible
1. Reinsurance capital
2. Debt capital
29
Which type of insurance capital has repayment obligations
Debt capital
30
Which type of insurance capital can trigger default
Debt capital
31
Identify 4 reasons why equity capital is expensive
1. Principal-agent problem
Investors (principal) and management (agent) do not have perfectly aligned incentives.
Insurance is opaque and difficult for investors to monitor management.
2. No independent validation of insurance pricing
Insurance is not expertizable, CAT risk is an exception.
3. Equity requires a long-term commitment to a cyclical business.
Public securities markets provide an exit but valuations may follow UW cycle.
4. Returns are left-skewed.
Investors prefer right-skewed returns with big upside such as tech stocks.
5. Regulatory minimum capital standards can force an insurer into supervision before it is technically insolvent.
In this case, regulator can restrict dividend payments and other capital withdrawals.
6. Double taxation
Insurers corporate profits and dividend distributions are taxed. This essentially taxes investors twice who hold income producing assets in an insurer.
32
Identify a popular type of of reinsurance capital
CAT bond
CAT bonds offer a lower a lower cost of insurance capital since it addresses reasons for expensiveness of equity capital.
33
Identify 4 ways in which CAT bonds address high cost of equity capital
1. CAT bonds are not equity and do not have market risk
2. CAT bonds are diversifying, zero-beta (independent of financial market) asset class
3. Catastrophe pricing can be validated
4. There is limited adverse selection
5. There are no principal-agent problems since CAT bonds CFs are contractually defined.
6. CAT bonds are typically written in tax-free jurisdictions
7. CAT bonds are lightly regulated
8. CAT bonds have defined, 3-5y terms (ie no long-term commitment)
34
Provide a major disadvantage of CAT bonds
CAT bonds are illiquid and trade in a thin market.
35
Briefly explain how cost of debt, reinsurance and equity capital can be estimated
For debt capital, explicit cost is debt coupon rate.
For reinsurance capital, estimable cost is ceded premiums less expected recoveries plus lost investment income.
Cost of equity can be estimated from historical studies of P&C insurer returns, CAPM or Fama-French models, or peer valuation studies to determine return necessary to support target price-to-book ratio.
36
Describe a method that combines the cost of all 3 forms of capital into now single figure.
Weighted Average Cost of Capital (WACC)
It is the weighted average of the various forms of capital.
Target ROE drives WACC since it is usually the most significant proportion of capital.
37
Describe 2 theories of optimal capital structure
1. Trade-off Theory
Debt & equity mix trades off costs & benefits of each capital type.
It compares the expense of equity and right of debt holders to force bankruptcy.
2. Pecking-order Theory
Informational asymmetries between management and owners makes equity more expensive, in which case retained earnings are the favourable form of financing.
38
Identify the 3 most important accounting standards
1. Statutory Standards
2. Financial Standards
3. Rating Agency Standards
39
Describe 2 types of Statutory Standards
1. US NAIC Statutory Accounting Principles (SAP)
Regulators evaluate BS to determine if enough funds exist to pay current and future PH benefits.
SAP is a liquidation basis accounting framework and focuses on BS.
Assets that cannot be readily converted to cash to pay PH claims are illiquid and considered non-admitted assets in BS.
Liabilities are booked on a net basis
Policy acquisition expenses are earned when written (no deferral)
2. EU Solvency II
Assets are based on market valuations and liabilities consist of best estimate plus risk margin.
Assets minus liabilities produces own funds, which consists of 3 parts:
a. Minimum Capital Requirement (MCR) is the lowest acceptable capital level before firm must exist market.
b. Solvency Capital Requirement (SCR) is a soft floor and reflects required capital.
If a firm drops below SCR, regulators can step in to address the issue.
SCR corresponds to 1y VaR at 99.5% of the basic own funds.
SCR is estimated via standard formula promulgated by regulators and internal model formulated by insurer, or both.
c. Free assets which are own funds in XS of SCR.
40
Describe 2 types of Financial Standards
1. Generally Accepted Accounting Principles (GAAP)
Used in multiple jurisdictions
US GAAP follows SAP treatment of loss reserves (book management's best estimate with no discounting or risk adjustment).
On BS, US GAAP presented loss reserves gross of reinsurance (unlike SAP) and books reinsurance recoverable as separate assets.
This differs from SAP where loss reserves are printed on BS net of reinsurance.
Unlike SAP, which immediately recognizes acquisition expenses, US GAAP allows deferral.
2. International Financial Reporting Standards (IFRS)
IFRS focuses on market values.
Loss reserves are valued on a discounted basis and include a risk adjustment.
Profits on insurance policy are earned over period of reserves rather than fully earned at end of policy period.
41
Describe Rating Agency Standards and provide examples.
Ex: Standard & Poor's, AM Best
Typically use adjusted statutory or GAAP financials in their internal capital models.
Adjustments include removal of goodwill and addition of discount in loss reserves when reserves are booked on an undiscounted basis.
Explicit quantification of CAT risk using model evaluation.
Net, after-tax losses at a 100-to-250 years return period are used.
42
Which standards produce the most conservative company valuation.
Statutory Standards since they exclude some assets and disallow discounting (US SAP) or add risk loads on liability side (Solvency II)
43
Identify 2 reasons why adjustments for accounting cannot be ignored
1. Accounting conventions have real world consequences
A company defaults based on its accounts, not a hypothetical risk BS.
Min capital standards are based on statutory accounting standards, not a hypothetical risk BS,
This means regulators can intervene when a company's statutory capital drops below minimum level, even if the company is technically still solvent.
Statutory accounting standards can directly impact CF and hence economics.
2. Models assume unlimited financing.
In reality, insurer might not be able to raise capital upon default.
44
Define Classical PCP
A classical PCP is a function that maps a loss distribution to a premium.
Since premium is based on underlying loss distribution, classical PCP are law invariant.
The simplest classical PCP is the expected value since it is fair and consistent.
However, expected value is not universally adopted because of the Fundamental Theorem of Risk Theory.
45
Explain the Fundamental Theorem of Risk Theory.
Says that an insurer charging the expected loss rate will fail in a finite length of time regardless of the starting surplus.
For this reason, positive risk loads are included.
46
How do you calculate the following PCP:
1. Net Premium
2. Constant loading premium
3. Maximum loss premium
4. VaR premium
5. TVaR premium
6. Variance premium
7. Standard deviation premium
8. Esscher
9. Exponential
10. Dutch
11. Fischer
1. P = E(X)
2. P = E(X) + pi*E(X)
3. P =. E(X) + pi*max(X)
4. VaR_pi(X)
Fails positive loading = major drawback
5. P = TVaR_pi(X)
6. P = E(X) + pi*V(X)
7. P = E(X) + pi*SD(X)
8. P = E(X*exp(pi*X))/E(exp(pi*X))
9. P = ln(E(exp(pi*X)) / pi
10. P = E(X) + pi*E((X-E(X)+)
11. P = E(X) + pi*E(((X-E(X))+)^p)^1/p
47
Define Classical Risk Theory
Classical risk theory determines combinations of starting assets and risk loads consistent with stability criterion.
Stability criterion establishes a constraint.
48
Explain top-down pricing
Begins by pricing an insurer's total portfolio s.t. agg premium satisfies stability criterion.
Ex: price s.t. prob(ruin) = 5%
A bottom-down approach would focus on pricing individual risks rather than portfolio in total.
This approach neglects to check whether tot premium obtained by aggregate policy premiums satisfies reasonable stability criterion.
49
Defne Arrow-Debreu Security
Pays 1 in single future state w and 0 otherwise.
50
Define incomplete market
Means there are future CFs that cannot be replicated by trading securities.
51
Define fair system of regulation and compensation
A fair system of regulation is one in which stockholders get what they pay for, no more or no less.
Fair compensation means company is indifferent to writing policy or not.
52
List the 5 assumptions underlying CAPM and Black-Scholes Models
1. Competitive
There are many small sellers and buyers and undifferentiated products.
2. Perfect
There are no information or transaction costs and no bid-ask spread.
There are no restrictions on short sales and taxes.
3. Complete
There are enough securities to replicate any set of future period CFs by securities trading.
4. Arbitrage-Free
There is no potential for a risk-free gain on an initial investment of 0.
Means pricing operator is linear.
5. General Equilibrium
Supply equals demand, no trader has an incentive to trade to improve their position and everyone agrees all products are fairly priced.
There is no arbitrage when prices are in general equlibrium
53
Define arbitrage
Money making machine.
It is an opportunity for a gain in some possible state of the world with no change of loss, for zero initial investment.
54
Explain the Fundamental Theorem of Asset Pricing
Says that the following are equivalent:
1. Absence of arbitrage
2. Existence of an optimal demand for one individual who prefers more to less
3. Existence of a positive linear pricing rule.
55
Describe the 2 classes of general equilibrium models for securities pricing
1. Classical General Equilibrium
Prices fundamental securities like stocks or insurance contracts.
Classical models price securities based on supply and demand and rely on diversification and pooling to manage risk.
CAPM is the primary example of classical model.
2. Derivative General Equilibrium
Prices redundant securities like stock options.
Derivative models price securities based on the cost of replicating portfolio and do not rely on diversification or pooling to mange risk.
Derivative models have non need for insurers or capital since replicating portfolios provide theoretically unlimited protection. To provide untilimited protection through pooling, insurer would need unlimited capital.
Black-Scholes is the primary example of derivative model
56
Define fundamental securities
Fundamental securities are securities that can add to set of CFs an investor can generate.
Fundamental implies not redundant.
57
Define redundant securities
Redundant securities are securities that do not add to set of CFs an investor can generate.
Its securities whose value can be determined from value of other securities.
58
Identify 3 market imperfections that motivate existence of insurers
1. Transaction costs
2. Frictional costs of capital
3. Market frictions
59
Describe the transactions costs market imperfection motivating existence of insurers
In a perfect market, insureds would purchase insurance directly from investors.
However, this does not happen due to existence of UW expenses.
In the event that an insured tried to purchase insurance directly from an investor, it's unlikely that investor would have UW expertise to properly evaluate insured.
As a result, insurance intermediaries and/or insurers are used to perform this task.
60
Describe the frictional costs of capital market imperfection motivating existence of insurers
3 primary kinds:
1. Agency and informational costs
Agency refer to principal-agent problem in which manager incentives do not align with owners.
Informational costs refer to an insurer's failure to control adverse selection and moral hazards of insureds due to information asymmetry.
2. Double-taxation
Insurer's corporate profits and dividends are taxed.
3. Regulation
Can allow insurer assts to be seized or temporarily controlled by regulator if insurer fails min capital standards.
Can also restrict investment opportunities and payments to investors.
61
Explain how frictional costs of capital can be incorporated in Portfolio CCoC Pricing
P^fric_a(X) = E(X lim to a(X)) + 2*a(X) / (1 + i*)
(1+i*) = (1 + i)*(1 + tau)
tau is constant frictional cost per period.
62
Describe the market frictions imperfection motivating existence of insurers
Bid-ask spreads are an example.
There are driven by information symmetry.
For ex: UWriters may price in positive loadings even when there is no risk due to uncertainty around adverse selection.
63
Discuss 2 fundamental flaws in assumptions underlying CAPM and Black-Scholes.
1. To assume complete and arbitrage free markets is not only unrealistic, but it assumed a solution to the hard part of the problem because we are given prices.
2. It is equivalent to supposing existence of a magic linear pricing functional that prices any securities. Since prices are linear, there is no diversification benefit and insureds can transfer risk directly into securities market, which means no need for insurers at all.
Modern portfolio pricing theory gives a pricing solution based on more realistic assumptions.
64
True or False?
Investment Income has always been considered in ratemaking.
Historically in the US, investment income was not considered in ratemaking.
In the 1970s, things started to change and investment income is now a critical piece in US ratemaking.
65
Describe Ferari's model of total return
Ferari decomposed total return on equity (TR) into UW and Inv Inc returns as follows:
TR = Income/Equity
TR = (U+I)/Q
TR = (U/P)*(P/Q) + (I/a)*(a/Q)
TR = (I/a) + (R/Q)*(I/a + U/R)
U = UW Income
P = Premium
I = Investment Income
Q = Equity
a = assets
R = loss reserves and portion of UPR
66
Calculate UW Margin
U/P
67
Calculate UW Leverage
P/Q
68
Calculate Investment Margin
I/a
69
Calculate Investment Leverage
a/Q
70
How can you calculate a if not given
a = Q + R
71
Complete the sentence:
The optimal capital structure seeks to maximize....
TR, which considers both UW and Investment variability
72
What's the conclusion from Ferari's model
Policyholder-fund (R/Q) magnifies net cost of policyholder funds (I/a + U/R)
73
Comment on timing risk and amount risk of insurance loss payments
Insurance loss payments have both timing risk and amount risk, which may be dependent risks since larger claims take longer to settle.
Timing risk tends to be fairly small since payment pattern often follow predictable claim settlement process.
74
Differentiate between timing risk and average timing.
A payment's average timing is related to time value of money and is always present.
Timing risk refers to uncertainty in payment of a claim.
75
Explain Myers-Cohn Fair Premium condition
A premium is fair whenever a policy is issued, the resulting equity value equals equity invested in support of that policy.
If premium increases equity, it is unfair to PH because it transfers wealth to investors.
If premium decreases equity, it is unfair to investors because it transfers wealth to PH.
Fair rates mean that prices are in equilibrium.
Otherwise, investors would have incentive to write more or less insurance.
76
Describe the Discounted Cash Flow (DCF) model
DCF model discounts each CF in a project at an appropriate risk-adjusted interest rate.
Some CFs are positive (ex. recurring revenue) and others are negative (ex. initial investment)
If total DCF is positive, project should be undertaken.
77
True or False?
When applying DCF model to insurance policy, we consider equity, reinsurance and debt capital
False,
We consider equity capital and ignore reinsurance and debt capital
78
Calculate premium for Ins. Co. using DCF formula
P = L/(1+R_L) + tau*R_f*(P+Q)/(1+R_f) + tau*P/(1+R_f) - tau*L/(1+R_L)
L = expected losses
P = premium
Q = capital invested to support policy
R_L = risk-adjusted discount rate for losses
R_f = risk-free rate
tau = tax rate
In words, DCF equals PV of loss plus tax on investment income and tax on UW income.
79
Calculate YE value of Ins. Co.
V1 = P*(1+R_f) + Q*(1+R_f) - L - tau*(P-L) - tau*R_f*(P+Q)
80
Calculate the return on equity R for Ins. Co.
R = (V1-Q)/Q = R_f + (L/Q)*(R_f-R_L)*(1-tau)/(1+R_L)
It is the sum of risk-free rate on equity and UW return as a leveraged after-tax spread.
81
True or False?
R_f - R_L is typically positive
True since risk adjustment on losses is normally negative (i.e. R_L smaller than R_f)
82
Discuss an argument for using risk-free rate for investment income in DCF model.
Proponents of risk-free rate argue that premium profit margins should not depend on actual investment portfolios, but only on risk-free rate and systematic UW risk.
Since PH do not share investment risks of insurer, they should pay the same premium regardless of firm's investment strategy.
83
Discuss an argument for using anticipated market return for investment income in DCF model.
Proponents of anticipated market return argue that adding investment risk increases probability of default for PH and changes their expected loss recoveries.
Hence, PH premiums should reflect anticipated market returns.
84
Define the Internal Rate of Return (IRR) model
IRR model is from investors viewpoint.
It estimates all CFs to or from investor and then compute discount rate (IRR) required to produce a PV = 0.
If IRR higher than investor hurdle rate, project should be undertaken.
85
State an advantage and a disadvantage of IRR model
Advantage:
It does not require to determine discount rates for each CF.
Disadvantage:
It requires a hurdle rate to make final decision
86
Discuss the IRR model CFs for Ins. Co.
At time t=0:
Investors supply required surplus phi*D
phi is a constant capital-to-reserves ratio
D is market value of liabilities
Equity flow = - phi*D
At time t=1
1. Investors receive after-tax UW profit of (1-tau)*(P-L) and after-investment income of (1-tau)*(P+phi*D)*R_f
Investment income assumes risk-free was earned on collected premium and required surplus at time 0
2. Since policy is expired, investors receive back initial surplus of phi*D
Equity flow = (1-tau)*(P-L) + (1-tau)*(P+phi*D)*R_f + phi*D
87
Calculate IRR rate
IRR = (1-tau)*(P-L+R_f*(P+phi*D)) / phi*D
88
Calculate the fair premium for Ins. Co. based on IRR model
P = L/(1+R_f) + phi*D*(tau*R_f+(R_s-R_f))/(1+R_f)*(1-tau)
89
Explain the 5 Cummins conclusions on DCF and IRR models
1. Ratemaking is prospective, thus should stand alone from legacy business and reserves.
2. Investment returns should estimate yields expected during policy term. We should use anticipated return for the firm rather than risk-free rate.
3. Regulation and accounting are relevant only if they affect CFs.
4. It is difficult to determine appropriate CFs associated with writing a new policy.
5. Both models depend on liability discount rate, or equivalently, a cost of capital.
90
Define the portfolio constant cost of capital (Portfolio CCoC) pricing
Portfolio CCoC is based on DCF model with no taxes.
Premium = Loss Cost + Cost of Capital
CoC = Target Return on Capital + Amount of Capital
Amount of capital is typically based on economic capital (aka risk capital) which the capital that guarantees payment of liabilities with high decree of confidence.
Pbar_a(X) = (E(X lim to a(X)) + i*a(X))/(1+i)
i is constant CoC
v = 1/(1+i)
d = i/(1+i)
Pbar_a(X) = v*E(X lim to a(X)) + d*a(X)
Pbar_a(X) = v*TVaR0(X lim to a) + d*TVaR1(X lim to a)
TVaR0 = mean
TVaR1 = max
91
If we ignore default adjustment, calculate premium under Portfolio CCoC
Pbar_a(X) = (E(X) + i*a(X))/(1+i)
92
Briefly explain why default adjustment (limit to a(X)) is often ignored in practice.
Since PH deficits are usually very small.
93
Is Portfolio CCoC classical or modern?
Portfolio CCoC is neither classical nor modern.
It is not classical because it considers capital, whereas classical PCPs ignore capital.
It is not modern since it fails sub-additivity, thus not a modern PCP.
