Mildenhall Flashcards

Question

What's the worst pairing of X1 and X2 in terms of subadditivity?

Answer 1

not the comonotonic pairing given 10,000 pairs, we want the 100th largest to be maximized for 0.99 VaR crossed pairing is the solution: 1st with 100th, 2nd with 99th... etc works for any 2 nontrivial marginals values below p var DONT matter creates risk ONLY at p.

Answer 2

Suppose 2 iid exponentials with mean 1. X1 + X2 ~ Gamma(alpha = 2) Var xi = 1.204 varsum = 2.439 subadditive for p < 0.7.

Answer 3

mu + [sigma * phi(zp)] / (1 - p) phi = [e^-(z^2/2)] / sqrt(2pi)

Answer 4

[E[X] * dist_func(sigma - zp)] / (1-p)

Answer 5

g(x) doesn't involve powers of x alone or a = c(a) / c(a+1) * [1 - F(q(p): a+ 1)] / [1 - F(q(p): a] examples: gamma, generalized beta

Answer 6

Probability equivalent level of VaR and TVaR constant c such that TVaR(1 - c*epsilon) = VaR(1 - epsilon) for small epsilon c usually > 2.5

Answer 7

LOWER: 𝖢𝖳𝖤𝑝(𝑋) ∶= 𝖤[𝑋 ∣ 𝑋 ≥ 𝖵𝖺𝖱𝑝(𝑋)] UPPER: 𝖤[𝑋 ∣ 𝑋 ≥ 𝑞+(𝑝)]

Answer 8

Worst conditional expectation, aka highest E[X] over a subset of the sample space with at least probability 1- p 𝖶𝖢𝖤𝑝(𝑋) ∶= sup {𝖤[𝑋 ∣ 𝐴] ∣ 𝖯𝗋(𝐴) > 1−𝑝}

Answer 9

For discrete distributions when p coincides with a mass point

Answer 10

𝖵𝖺𝖱𝑝(𝑋)≤𝖢𝖳𝖤𝑝(𝑋)≤𝖶𝖢𝖤𝑝(𝑋)≤𝖳𝖵𝖺𝖱𝑝(𝑋)

Answer 11

𝖳𝖵𝖺𝖱𝑝(𝑋) = min𝑥 {𝑥 +(1−𝑝)−1𝖤[(𝑋 −𝑥)+]} 𝖵𝖺𝖱𝑝(𝑋) = argmin𝑥 {𝑥 +(1−𝑝)−1𝖤[(𝑋 −𝑥)+]}. TVaR balances the cost of providing capital against the cost of a shortfall

Answer 12

𝖤[(𝑋−𝑎)+]=(1−𝐹(𝑎))(𝖳𝖵𝖺𝖱𝐹(𝑎)(𝑋)−𝑎). 𝖳𝖵𝖺𝖱𝑝(𝑋) = 𝖵𝖺𝖱𝑝(𝑋)+ 𝖤[(𝑋 −𝖵𝖺𝖱𝑝(𝑋))+] / (1- p)

Answer 13

An insurance event that is potentially disastrous but plausible Events: E1.... EN

Answer 14

𝖰𝑘(𝐴) ∶= 𝖯(𝐴∩𝐵𝑘) / 𝖯(𝐵𝑘) 𝐵𝑘 ∶= 𝐵(𝐸𝑘) to be the set of all sample points where the insurance event 𝐸𝑘 occurs, i.e., those with the value 1 in the 𝑘th place.

Answer 15

𝜌𝑐(𝑋) ∶= max{𝐸𝖰𝟣[𝑋],…,𝐸𝖰𝗋[𝑋]}

Answer 16

1. Statistical uncertainty: 𝖯 is an estimate subject to the usual problems of estimation risk 2. Information uncertainty: 𝖯 is based on a limited and filtered subset of ambiguous information

Answer 17

Scenarios such as insureds being systematically misclassified, adverse selection, parameter error, take the expected losses of all and compute the maximum E[X]

Answer 18

● is intuitive and easy to communicate, ● can be used for capital and pricing, ● has properties as a risk measure that are aligned with rational risk preferences, and ● any measure with those properties is a 𝜌𝑐 for some set of probability scenarios.

Answer 19

p(X+c) = p(X) + c requires p to be in monetary units VaR, TVaR and a scenario loss at TI sd, variance, and higher order central moments aren't TI

Answer 20

p(0) = 0 risk of an outcome with no gain or loss is zero a risk is preferred to doing nothing if p(X) <= 0 (aka "acceptable")

Answer 21

If X<=Y for all outcomes then X >= Y (preferred over Y) all pc risk measures are monotone sd is not monotone.

Answer 22

if X <= c then p(X) <= c

Answer 23

p(X) >= E[X] not all pc have this property, generally larger Q means more likely it does

Answer 24

satisify MON and TI

Answer 25

p(lambda X) = lambda * p(X) scale invariance PH implies NORM since p(0) = p(0 x X) = 0 * p(X)

Answer 26

the difference in risk between random variables X and Y is at most the maximum of their outcome differences |p(x) - p(y)| <= MAX(|X - Y|)

Answer 27

𝜌(𝑋+𝑌) ≤ 𝜌(𝑋)+𝜌(𝑌) the risk of the pool is less than or equal to the sum of its parts mergers do not increase risk implies p(0) >= 0

Answer 28

PH and SA hold. sublinear pricing risk measures have a positive bid-ask spread ask price = p(X) bid price = -p(-X)

Answer 29

X and Y are COMON if X = g(Z) and Y = h(Z) for increasing functions g and h and a common variable Z different excess layers of a risk Z are comonotonic

Answer 30

p(X+Y) = p(X) + p(Y) if X and Y are comonotonic

Answer 31

p(X+Y) = p(X) + p(Y) if X and Y are independent Variance is an example

Answer 32

for some pricing applications, certain risks may have the same distribution but one is definitely more risky (florida hurricane risk vs auto liability) CAPM is not LI. same return distribution but correlation to the market is what matters.

Answer 33

monotone translation invariant positive homogeneous subadditive

Answer 34

coherent law invariant comonotonic additive

Answer 35

a combination of a pricing risk measure (p) and a capital risk measure (a) p(x) = p(X ^ a(x))

Answer 36

firms dont have diminishing marginal utility of wealth firm preferences aren't relative to a wealth level utility theory combines attitudes to wealth and risk not linear and thus expected utility isnt a monetary risk measure based on combination through mixing, not pooling, counter to insurance

Answer 37

a set of circumstances likely to result in insurance losses occurrence of a hurricane traffic jam

Answer 38

start with a seemingly reasonable risk measure and rationalize it by establishing it has properties desired (or argue against it) e.g. constant underwriting margin

Answer 39

use a rigorous economic theory to select a risk measure set up and solve an optimization problem (utility-based approaches)

Answer 40

start with a list of desirable properties and then determine which risk measures have those characteristics

Answer 41

Monotone and TI (intuitive) ability to be allocated diversification must be reflected Backtesting (consistent with observations) Explainable (sell to users) Elicitability (estimated via regression-like techniques) Robustness / continuity theoretic soundness and consistency MAD BEER (T)

Answer 42

1.The less that is known about the current estimate and its trend, the higher the risk margins should be. 2. Risks with low frequency and high severity have higher risk margins than risks with high frequency and low severity. 3. For similar risks, contracts that persist over a longer time-frame have higher risk margins than those of shorter duration. 4. Risks with a wide probability distribution have higher risk margins than risks with a narrower distribution. 5. To the extent that emerging experience reduces uncertainty, risk margins decrease, and vice versa.

Answer 43

(1) No mean. VaR is not subadditive. LLN doesn't apply. Impossible to insure. (2) Mean no variance: LLN applies by CLT doesnt (3) mean and var finitely many moments: LLN and CLT apply (4) all moments, sub exponential: decays slower than exponential e^kx * S(x) => inf for all k>0 (5) exponential tail (dividing line between thick and thin) (6) super exponential : thin-tailed e^kx*S(x) => 0 for all k > 0 (7) log concave density: proportional to -x^2 (sample averages tightly clustered around the sample mean) (8) bounded.

Answer 44

Intended purpose: The goal or question, whether generalized or specific, addressed by the model within the context of the assignment. Intended User: Any person whom the actuary identifies as able to rely on the model output.

Answer 45

Individual risk pricing: quoting and evaluation of market pricing classification rate making: setting profit margins and allocating cost of capital portfolio management: reinsurance purchase, ORSA, Capital: determine risk capital or evaluating held capital

Answer 46

Explainable: reasonable, transparent, and explainable basis Estimable: from market prices Computable Robust Allocation Optimal Diversification Law Invariant Theoretically sound: consistent with economic, financial theory, and behavioral considerations

Answer 47

Robustness to regulatory arbitrage: balance complexity against data available Simplicity and explainability (to regulators) Standardization and reliance on public data enables comparison Backtesting (was the portfolio managed to a risk measure tolerance?) Optimization against a regulatory standard

Answer 48

𝖤[𝑋]+𝑐𝖤[(𝑋−𝖤𝑋) ^ 21𝑋>𝖤[𝑋]]

Answer 49

𝖤𝑋+𝜆‖(𝑋−𝖤𝑋)+‖1

Answer 50

𝖤𝑋+𝑐‖(𝑋−𝖤𝑋)+‖ can also be from target tau (replace E[X] with tau)

Answer 51

𝖤[𝑋𝑒ℎ𝑋]∕𝖤[𝑒ℎ𝑋]

Answer 52

𝖤[(𝑋−𝖤𝑋)+]=‖(𝑋−𝖤𝑋)+‖1

Answer 53

Solve E[u(R - X)] = 0 for R

Answer 54

(1) Risk transfer: Pool their risk with other insureds and reap the benefits of diversification (2) Satisfying demand: satisfies certain statutory, regulatory, or contractual requirements in order for the insured to carry out an activity such as driving (3) Risk financing: seeks to finance uncertain future contingencies in an efficient manner (driven by accounting, regulation, and tax law) can make expenses tax deductible by making them an insurance deductible (e.g. SIRs, retro-pols, large ded)

Answer 55

sales, marketing, risk surveys, loss control, customer billing and support, underwriting and pricing, claims adjusting, risk bearing, and investment management services

Answer 56

(1) Controlling access to the pool through underwriting and pricing (2) Ensures the pool remains solvent by funding risk-bearing assets through the sale of liabilities

Answer 57

Customer functions can be handled by a direct writer, or independent agents and brokers MGUs can manage a risk pool themselves reinsurance and sidecar arrangements can provide capital services third party claims adjustment asset management services

Answer 58

underwriting is costly and requires an accumulation of private loss data insurer capital is costly and risk pools allow it to be used more efficiently long-lived risk pools signal competence, thus lowering the cost of capital

Answer 59

pay losses based on an explicit event outcome defined by an objective physical description easy to UW (doesn't depend on the insured characteristics) hard to ensure insured actually has a loss when claim is triggered basis risk: mismatch between subject loss and insurance recovery

Answer 60

objective event occurs AND it causes an economic loss to the insured indemnty payment is a function of the subject loss

Answer 61

(1) f(L) <= L (2) f(L) >= 0 (3) f(L) is monotonic (4) L - f(L) is monotonic (5) f is continuous

Answer 62

A franchise or diminishing deductible. Has jumps.

Answer 63

Deadline for filing claims against its assets

Answer 64

Class 1 = Admin expenses Class 2 = Guarantee association expenses Class 3 = insurance claims and UEP Class 4 = mortgate & financial guaranty claims and all other forms of insurance class 5 = federal government claims class 6 = debts due to employees class 7 = claims of other unsecured creditors including claims under reinsurance deals class 8 = claims of local governments class 11 = surplus notes, capital notes class 13 = shareholders

Answer 65

Capital is assets net of all policyholder liabilities (loss reserves and UEP reserves) Equity is the value of the owners residual interest in the firm assets net of all liabilities owed except the owners

Answer 66

Ratio of equitys market value to its accounting (book) value for publicly traded companies (stock insurers, not mutual insurers)

Answer 67

by law all stock companies must include a lowest priority ownership interest called common shareholders they bear the ultimate risk and receive the benefit of success governance rights and appoint the BOD and management can be paid dividends but not guaranteed franchise value is an important component of equity

Answer 68

(1) capital stock: par value of shares issued (2) additional paid-in capital: stock price excess over par (3) retained earnings: cumulative

Answer 69

owned by policyholders who have a noncertificated ownership interest collectively own the equity but dont have transfersble share certificates and can only own collectively, not individually (cant sell their interests)

Answer 70

blends characteristics of debt and equity. sits above common equity and below debt. pays a dividend but can be suspended without triggering default debt can trigger default.

Answer 71

(1) Principal - agent problem: investor and management interest misalignment (insurance is obaque and tough to monitor management) (2) no independent validation of insurance pricing (cat risk is an exception) (3) requires long-term commitment to a cyclical business (4) returns are left-skewed. (5) regulatory minimum capital standards can force an insurer into supervision before its technically insolvent which can restrict dividend payments (6) double taxation (corporate taxes and dividends are taxed)

Answer 72

(1) no market risk (2) diversifying, zero-beta asset class (not tied to financial market) (3) investors can validate loss costs and quantify risk potential independently (4) limited adverse selection (pricing based on detailed exposure data) (5) no principal-agent issues since cash flows are contractually defined (6) taxes: usually written in tax-free jurisdictions (7) no regulation: written in lightly regulated jurisdictions (8) limited tenor: 3-5 yr terms

Answer 73

all non-equity capital has an explicit cost combine all forms of capital by sumproduct the weights and costs cost of equity capital can come from an academic study or a peer study considering historical returns, volatility and P/B ratios reinsurance cost = ceded premium - expected recoveries and lost investment income debt and reinsurance expenses are tax deductible equity largely drives the WACC.

Answer 74

(1) Trade-off theory: argues debt and equity mix trades off the costs and benefits of each (notably expense of equity vs right of debt holders to force bankruptcy) (2) Pecking order theory: informational asymmetries b/w mgmt and owners makes equity more expensive, favoring retained earnings

Answer 75

(1) Statutory or regulatory standards (US NAIC, EU Solvency II) (2) Financial reporting (GAAP, IFRS) (3) Rating agencies (S&P, AM Best)

Answer 76

(1) Accounting conventions have real-world consequences (company default based on its accounts, regulators can take over an insurer as a conservator when accounts breach a certain threshold) (2) markets can remain irrational longer than you can remain solvent (models assume unlimited financing for insurers)

Answer 77

considers assets, liabilities, and own funds market valuation of assets and liabilities = best estimate + risk margin MCR = below which entity withdraws from the market SCR = estimated via standard formula, internal model, or both Free assets = amounts in excess of SCR one year stability criterion. 99.5% VaR

Answer 78

liquidation basis accounting regulators evaluate balance sheet to see if funds exist to pay current and future policyholder benefits non-admitted assets (non-liquid) are subtracted from assets liabilities are booked on a net basis

Answer 79

follows SAP for loss reserves, booked at mgmt best estimate on a nominal basis however, gross of reinsurance with reins recoverable as an asset allows acquisition expense to be deferred, unlike SAP

Answer 80

More market oriented than GAAP. loss reserves are valued on a discounted basis including a risk adjustment profits are earned over the payout period of reserves

Answer 81

An insurer charging the expected loss rate is guaranteed to fail in a finite length of time regardless of the starting surplus

Answer 82

P = log(E[e^(pi*X)]) / pi zero-utility for exponential function u(x) = (1 - e^(-pi * x)) / pi reminder: P solves E[u(P-X)] = 0

Answer 83

Income / Equity = U/P * P / Q + I / a * a / Q UW margin x UW leverage + Investment return * Investment leverage

Answer 84

a = Q + R (policyholder funds R) TR = I / a + R / Q * (I / a + U / R)

Answer 85

financial markets are complete, competitive, perfect, and arbitrage-free with pricing determined in general equilibrium competitive: many sellers and buyers and undifferentiated products perfect: no information or transaction costs, no bid-ask spread, ability to borrow or lend at the risk-free rate, no short-sale restrictions, no taxes complete: in a complete market there exist enough securities to replicate any set of future period cash flows by trading (replicating portfolio)

Answer 86

Pays 1 in a single future state w and 0 in all othe rstates any security can be built from a set of arrow-debreu securities

Answer 87

redundant: securities that can be determined from the value of other securities (options) non-redundant: fundamental (companys common stock)

Answer 88

opportunity for gain with no possible loss No arbitrage implies pricing operator p is linear p(aX + bY) = ap(X) + bp(Y) securites X,Y constants a,b implies no bid-ask spread

Answer 89

the market clears and supply equals demand no actor has incentive to trade to improve their position and all agree products are fairly priced

Answer 90

The following are equivalent: (1) Absence of arbitrage (2) existence of an optimal demand for some agent who prefers more to less (3) existence of positive linear pricing rule

Answer 91

(1) Classical: price fundamental securites like stocks or insurance policies; price based on supply and demand and rely on diversification and pooling (CAPM) (2) Derivative: price redundant securities like stock options, rely on replicating portfolios, dont rely on diversification or pooling (Black-Scholes)

Answer 92

A premium is fair if whenever a policy is issued the resulting equity value equals the equity invested in support of that policy if not satisified, not in equilibrium since investors have an incentive to write more or less insurance fair iff P = D (market value of liability, accounting liability + margin) company is indifferent as to whether a policy is written

Answer 93

(1) Value additivity: each policy evaluated stand-alone (prospective business separate from reserves) (2) Accounting is relevant only to the extent it impacts cash flows (3) Insurance is prone to double-counting and a good way to avoid it is to pick either the insured's or the insurers perspective

Answer 94

R = risk free rate + L / Q * (Rf - RL) / (1 + RL) * (1 - tax rate)

Answer 95

Rq = -k * rfr + beta_l * (market return - rfr) k is funds generating coefficient (unit of premium generates more than one unit of assets due to loss reserves) a = kP + Q beta_l: correlation of underwriting returns with market returns

Answer 96

P = L / (1 + rf) + phi*D * (tau * rf + (rs - rf)) / (1+rf)(1 - tau) phi = constant capital to reserves ratio D = market value of liabilities tau = tax rate rf = risk free rate rs = cost of capital

Answer 97

simplified DCF model with no taxes Premium = loss cost + cost of capital cost of capital = target ROE x capital constant coc is realistic if all capital is common equity usually use economic capital (aka risk capital) one year VaR P(a) = (E[X] + i*a) / (1 + i) official: P_a(X) = (E[X ^ a(X)] + i*a(X)) / (1+i) = v*E[X ^ a(X)] + delta * a(X) v = 1/(1+i) risk discount factor delta = i / (1+i) risk discount rate aka P(a) = v TVaR(0)(X^a) + delta * TVaR(1)(X^a) TVaR(0) = mean TVaR(1) = maximum risk neutral v percent of time, expect the worst delta percent

Answer 98

(1) Agency and informational costs: managers behave opportunistically, or insurers fail to control adverse selection (2) Double taxation of investment returns (3) regulation that allows insurer assets to be seized or temporarily controlled by a regulator if it fails minimum capital standards, and limit dividends frictional costs vary with geography and corporate legal form and level of capitalization

Answer 99

A bernoulli layer, where 1 is paid if X breaches the attachment point a.

Answer 100

How do we split the asset funding between policyholder premium and investor capital

Answer 101

f(a) = a - s(a) aka assets - losses (shared liability) insureds and investors have this shared liability to fund the assets in excess of the expected loss

Answer 102

i(a) = expected margin / investment M(A) / Q(A) [P(A) - S(A)] / [a - P(A)]

Answer 103

delta(a) = M(A) / [a - S(A)] v(a) = Q(A) / [a - S(A)]

Answer 104

g(s) = 1 - h(1 - s) g(s) + h(1-s) = 1 s = probability of insured loss 1 - s = probability of investor profit P(a) = g(S(a)) Q(a) = h(F(a))

Answer 105

function g: [0,1] => [0,1] (1) g(0) = 0, g(1) = 1 (2) g is increasing (3) g(s) >= s for all s

Answer 106

Concave functions are umbrella shaped if we draw a line between two points, the curve lies above g(ws1 + (1-w)s2) >= wg(s1) + (1-w)g(s2) If g is concave then -g is convex flip the inequality if we draw a line between two points, the curve lies below

Answer 107

(1) g is continuous everywhere except maybe at s = 0 (2) g is differentiable everywhere except at most countably infinite points (kinks) (3) g'(s) >= 0 where g' exists (4) left and right-hand derivatives of g exist everywhere on (0,1) (5) g is second-order differentiable almost everywhere (6) g is concave, thus g''(s) <= 0 where g''(s) exists (7) if g is differentiable then it is concave iff g' is decreasing

Answer 108

pg(X) = integral(0 to 1) q(p) * spectral weight function(p) dp spectral weight function = g'(1-p) >= 0 q(p) = VaRp(X)

Answer 109

iif g(1) - g(0) = 1 if g is not continuous then g(1) = 1

Answer 110

(1) P is coherent, comonotonic additive, and law invariant (2) P equals a spectral risk measure pg for a concave distortion function g (3) P(X) = g(0+) * sup(X) + integral(0 to 1) VaRp(X)phi(p) dp weighted average of VaRs (4) P(X) = g(0+) * sup(X) + integral(0 to 1) TVaRp(X)*(1-p)*phi'(dp) + sum(j) TVaRpj (X)(1-pj)Delta(j) (5) 𝜌𝑔(𝑋) = sup{𝖤[𝑋𝑍] ∣ 𝑍 ≥ 0,𝖤[𝑍] = 1,∫𝑞𝑍(1 −𝑡)𝑑𝑡 ≤ 𝑔(𝑠) for all 𝑠 ∈ (0,1)} (6) 𝜌𝑔(𝑋) = sup{𝖤𝖰[𝑋] ∣ 𝖰(𝐴) ≤ 𝑔(𝖯(𝐴)) for all events 𝐴}.

Answer 111

(1) Simulate uniform random variables ui (2) transform to gi = inverse(g) (ui) (3) apply inversion method to gi to simulate s inverse (gi) e.g. g(s) = s^ b | b < 1 gi = ui ^ (1/b) b = 0.5, u = 0.1 transformed to 0.1^2 = 0.01 90th percentile to 99th

Answer 112

(1) Pad the input by adding a zero outcome with probability zero if it doesnt exist already (2) sort the outcome ascending (3) group by xj and sum the corresponding pj (all xj need to be distinct) (4) decumulate probabilities to get S(X) (5) distort the survival function g(S(X)) (6) difference g(Sj) to compute risk adjusted probabilities (7) sumproduct to get pg(X) = sum Xj * delta g(Sj)

Answer 113

weighted average of 2 TVaRs 0 <= p0 <= p1 <= 1 and 0<=w<=1 biTVaR = (1-w)TVaR(p0) + wTVaR(p1) when a TVaR equals this, we call it the bi-TVaR associated with p* TVaR

Answer 114

g'(s) = (1-w) / (1 - p0) * 1[0, 1 - p0)(s) + w/(1-p1) * 1[0, 1 - p1)(s)

Answer 115

when p0 = 0 and p1 = 1

Answer 116

an empirical distortion defined as the concave envelope of a set of points {(𝑠𝑗,𝑔𝑗)} ∪ {(0,0),(1,1)}, where all 𝑠𝑗,𝑔𝑗 ∈ [0,1]. If the smallest 𝑠� is greater than zero the distortion function is continuous with finite slope at 𝑠 = 0. If there is a(𝑠𝑗 = 0,𝑔𝑗 > 0)itistreatedasalimitfromtheright.Ifthelargest𝑠𝑗 < 1withcorresponding 𝑔𝑗 < 1thedistortion has slope 0 < 𝑔′ ≤ 1 at 𝑠 = 1; otherwise it has slope 0.

Answer 117

g(s) = s^a for 0

Answer 118

g(s) = 1 - (1-s)^m m >= 1 larger m = greater risk aversion interpretation of applying powers to probabilities and not outcomes pg(X) = E[max(x1, ... xm)] aka MAXVAR

Answer 119

g(s) = norm.dist(norm.dist.inv(s) + lambda) lambda > 0 larger lambda = greater risk aversion operates on normal variables like an esscher transform alters the mean, doesnt touch the sd

Answer 120

at a minimum, must be defensible should be simple, transparent, evidently fair, objective, data-based and precedented four principal concerns of audience: 1. What is it; how does it work? 2. Why is this the best? 3. Who else uses it? 4. What can it do for me?

Answer 121

for any distortion function theres a point such that for all x lower than it g'(s(x)) <= 1 and all above g'(s(x)) > 1 meaning left-tail probs decrease and right-tail probs increase say an insurance policy pays out one dollar if below p and nothing if above p (clearly using the distortion the policy would be priced below E[X]) TVaR distortion is an extreme example of this because all probs below the threshold are 0.

Answer 122

highlights SRM don't define objective states and state prices (they are relative to risk being priced) basically just switch the order of events to make it make sense.

Answer 123

we consider fair price for V as part of portfolio X a fixed premium direct insurance policy is a variable premium ceded reinsurance policy by swapping premium and loss paying for reinsurance is the same as writing a direct policy at a loss (ceded prem > ceded loss) thus policy V makes a reasonable reinsurance contract (ceded recovery in all states but only requires ceded premium in good states)

Answer 124

Pi = li + delta(i) *(ai - li) unit premium = unit losses + unit discount rate * unit shared liability fixes the return, allocates assets => determines premium SRMs fix the return by layer and allocate premium to determine assets

Answer 125

risk-adjusted return on capital (RAROC): return varies with risk return on risk-adjusted capital (RORAC): return is constant when capital reflects risk

Answer 126

this is because all units have access to the entire insurers capital when needed

Answer 127

Given a functional p and a finite sum of RVs, an attribution of p is a functional taking the vector Xi to a vector of real numbers ai amount ai is the attribution of P(X) to unit i as part of X a(Xi: X) if sum of a = p(x) the attribution is additive and is called an allocation of p

Answer 128

(1) Should work at any granularity (2) should be decomposable: the allocation to a sum of random variables equals the sum of their allocations (3) it should be computed using a single, consistent formula

Answer 129

Endogenous: same risk measure is used to determine the total AND allocate it Exogenous: when the amount is stipulated by an external risk measure and allocated using an auxiliary measure (regulator or rating agency determines the capital)

Answer 130

ai = a(x) * E[Xi] / E[X] exogenous allocation using expected value as the auxiliary measure

Answer 131

ai = a(x) * a(xi) / sum a(xi) each unit evaluated on stand-alone basis and total is pro-rated to a(X) not influenced by dependence between Xi

Answer 132

exogenous version of the proportional allocation ai = a(x) * p(Xi) / sum p(Xi) not influenced by dependence between Xi

Answer 133

endogenous solves Sum a(Xi, p*) = a(X) for p* ai = a(Xi, p*) when p = VaR each unit has the same probability of default replacing actual dependence structure with comonotonic (worst-case scenario) dependence

Answer 134

allocates to each unit the capital that is reducing by removing that unit from the portfolio not additive (usually less than total)

Answer 135

based on marginal change in capital given marginal change in unit i written endogenous allocation equals directional derivative of a at X in direction of unit i only suitable for performance measurement as growing lines with higher return to capital ALWAYS improves average return doesnt account for possibility change in portfolio => change in coc

Answer 136

Atomic: each player is either 100% in or out Fractional or fuzzy otherwise Exact: a game with a continuum of players Core: the set of allocations that keeps everyone happy (satisfies the no-undercut property)

Answer 137

ci = sum(all sets dont include i) |S|! * (n - |S| - 1)! / n ! c(S U {i}) - c(S)

Answer 138

(1) its additive (in game theory its "efficient") (2) its symmetric, if two units i and j increase the cost of every S that contains neither i and j by the same amount they have the same cost (3) decomposable (linear) (4) homogeneous if c is (5) if c is subadditive no undercut property is met (6) allocates no capital to a constant risk (called null player)

Answer 139

to allocate to n unit you need to compute 2^n marginal impacts if a unit is subdivided, then the allocations assigned to OTHER units changes typically, it is not decomposable

Answer 140

generalized the shapley by allowing fractional participation and the euler allocation by dropping the positive homogeneous assumption a i = integral(0 to 1) partial derivative a with respect to xi (tx) dt computes average incremental capital as portfolio grows from zero, uniformly across units unique allocation satisfying linearity, decomposable, and either continuity or non-negativity when a is nondecreasing

Answer 141

p(X) = E[h(X) L(X)] h is an additive function co-measure E[h(Xi) L(X)] L is riskiness leverage function covariance = k 𝖤[(𝑋𝑖 − 𝖤𝑋𝑖)(𝑋 − 𝖤𝑋)] coTVaR average xi when X is greater than 1-p var coVaR E[xi | X = xp] Esscher: 𝖤[𝑋𝑖 𝑒𝑘𝑋]∕ 𝖤[𝑒𝑘𝑋].

Answer 142

solve VaRp(X) = E[X] + pi(x) SD(X) for constant pi(x) and use the scaled covariance allocation E[xi] + pi(x) cov(xi, x) /sd(x) exogenous method (when risk measure for total capital is hard to allocate)

Answer 143

an allocation should be fair (stakeholders consider the allocation reflective of reality and not unduly influenced by factors not germane to the problem intended to be solved) dont have too much focus on the tail vs the body when not rational can remedy this via a haircut allocation with a body-focused risk measure such as TVaR (0.75) be cognizant of professional actuarial standards recognize the scope of the larger task communication and persuasion sensitivity to stakeholder concerns simplicity, transparency, fairness, objectivity, and precedence

Answer 144

Xi | X <= a Xi * (a/X) | X > a

Answer 145

ex post: shares in default are determined after claims are known ex ante: shares are set at the beginning of the period based on expected loss (not used in practice since it can result in payment in default greater than actual loss)

Answer 146

Heavily biased towards tail risk SRMs can remedy this in reality, tail-risk is funded through cat bonds which are much cheaper

Answer 147

if multiple ties with k1, k2,.... equal rows there will be (k1!)*(k2!) .... alternatives

Answer 148

K(u) = (1 - u^2) +

Answer 149

M(i,j) = Beta(i,j)*g(Sj)) - a(i,j)*S(j) * [layer length] density * length [premium density - expected loss density] * layer length

Answer 150

M(i,j) / M(j) * Q(j) layer margin / overall cost of capital

Mildenhall Flashcards

(174 cards)