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Formulas Packet Page Directory Brosius ........................................................................................................................................................................................ 1 Clark ............................................................................................................................................................................................. 4 Feldblum .................................................................................................................................................................................... 8 Frachot ...................................................................................................................................................................................... 11 Goldfarb ................................................................................................................................................................................... 14 Hürlmann ................................................................................................................................................................................ 20 IAA ............................................................................................................................................................................. 24 Mack (1994) ........................................................................................................................................................................... 29 Mack (2000) ........................................................................................................................................................................... 33 Marshall ................................................................................................................................................................................... 34 McNeil ........................................................................................................................................................................................ 35 Patrik ......................................................................................................................................................................................... 37 Sahasrabuddhe .................................................................................................................................................................... 38 Shapland and Leong .......................................................................................................................................................... 42 Siewert ...................................................................................................................................................................................... 47 Teng and Perkins ................................................................................................................................................................ 50 Venter Copulas ..................................................................................................................................................................... 53 Venter Factors ...................................................................................................................................................................... 59 Venter Non­tail ..................................................................................................................................................................... 63 Venter Strategic Mgmt ..................................................................................................................................................... 67 Venter and Underwood ................................................................................................................................................... 70 Verrall ....................................................................................................................................................................................... 72 Exam 7 (2013) Brosius Formulas Link Ratio (LR) Method: ;
Budgeted Loss (BL) Method: ,
Least Squares Method: :
0;
0;
1 Simple Model: #
@
#
|
|
Recall (simple) Bayes’ Theorem: |
|
General Poisson‐Binomial Case: ~
,
@
1
1
Negative Binomial‐Binomial: :
1. ~
2.
,
@
Page 1 of 3 Packet Page: 1
Exam 7 (2013) 1
1
1
1
Brosius Formulas 1
Fixed Prior Case (BL method): Y = k for some value of k Fixed Reporting Case (LR method): Suppose a number d ≠ 0, such that the % of claims reported by year end is always d. Linear Approximation (development formula 1): Y = Ultimate losses; X = Reported Losses ,
;
Let: ,
;
;
Linear Approximation (development formula 2 – credibility weighted): |
|
. .
. .
Suppose a number d ≠ 0, such that E(X|Y=y) = dy 1
.
:
SPECIAL CASES: :
1
:
1
Page 2 of 3 Packet Page: 2
Exam 7 (2013) Brosius Formulas Linear Approximation (development formula 3 – caseload effect): :
1.
|
2.
0 1
Estimate EVPV and VHM: Page 3 of 3 Packet Page: 3
Exam 7 (2013) Clark Formulas Cumulative distribution function (CDF): %
1
:
@
= emergence pattern average
NOTE: see pg 456 & 457 for other variations of the formulas above Loglogistic: | ,
1
1
Weibull: | ,
1
Expected Loss Emergence: ExpLEAY;x,y = expected incremental loss dollars in accident year (AY) between ages x and y Cape Cod Method (generally preferred): ; ,
# of parameters = 3 (i.e. ELR, ω, θ) LDF Method: ; ,
# of parameters = n+2 (i.e. n AYs (1 ULT for each AY),ω, θ) Process Variance: Assume that the loss in any period has a constant ratio of Var/Mean. 1
,
Page 1 of 4 Packet Page: 4
Exam 7 (2013) Clark Formulas where n = # of data points p = # of parameters ActLE = actual incremental loss emergence Assume ActLE = c follows an ‘Over‐Dispersed Poisson’ (ODP) ODP Distribution: c = x*σ2 !
where λ = mean of standard Poisson Likelihood Function – Finding the “Best Parameters”: /
Pr
!
ln
ln
! Equivalent to maximizing: ln
Model #1: Cape Cod ln
,
where ActLE = actual loss in accident year i, development period t Prem = premium for AY i y = beginning age for development period t x = ending age for development period t 0
,
∑,
∑,
Page 2 of 4 Packet Page: 5
Exam 7 (2013) Clark Formulas Model #2: LDF ln
,
where ULT = ultimate for AY i ∑
0
∑
,
Parameter Variance: ,
,
,
,
,
,
1
,
Variance of Reserves: ∑
(IMO parameter variance of R is too complicated to be tested) Residuals (LDF Method): (continued onto page 4) Page 3 of 4 Packet Page: 6
Exam 7 (2013) Clark Formulas Adjustment for Difference Exposure Periods (2 Steps): 1. Calculate % of period exposed For AY: ,
12
1,
12
12
For PY: 1
1
2 12
1
2
2
,
12
12
,0
2
12
2. Calculate avg accident date of period that is earned For AY: ,
2
6,
12
12
For PY: 3
1
24
3
12
,
12
1
,
12
3. | ,
][
[
*
| ,
]
Variance in Discounted Reserves (Rd): .
where 1 Process variance of Rd: 1 Page 4 of 4 Packet Page: 7
Exam 7 (2013) Feldblum Formulas 97.5% VaR for ERM: 2
Φ
97.5%
,
Φ µ ,σ 1.96 97.5%
1.96
If 2 risks are correlated, 1.96
97.5%
2
Marginal vs Multivariate Distributions: Copulas join marginal distribution functions into multivariate distribution functions.
2
,
,
,
,
,
,
,
Person Product‐Moment Correlation: ∑
∑
∑
Kendall τ Statistic: Def: measures how many swaps (as a % of maximum) are needed to convert order of one series
into that of another
1
4
1
#
#
NOTE: ‐If 2 series are perfectly rank correlated (Q=0), τ=+1. ‐If inversely rank correlated (Q=N(N‐1)/2), τ=‐1. Page 1 of 3 Packet Page: 8
Exam 7 (2013) Feldblum Formulas Spearman’s Rank Correlation: 6∑
1
1
Χ‐Plots: 

Helps uncover patterns and select between dependencies and random fluctuations Constructed from the scatterplot of 2 variables. ,
,
1
. 1
, ,
. .

The χ‐coordinate is a measure of the distance between the empirical bivariate distribution
Hi and null hypothesis of independence FiGi. To calculate Hi,Fi and Gi, imagine a vertical line and horizontal line passing thru ith point: 1.
2.
3.
4.
1,2,3,4
,
3
2
3
3
4
4
,
,
0
4
,
,
0
0.5

0.5 The λ‐coordinate is a measure of the ‘signed’ distance of any point in the scatterplot to the
median coordinates of the bivariate distribution. Formal Definition of Copulas (2 Requirements): 0,1 ,
0,1 ,
,
,
1.
,
2.
,
,0
,1
,
,
0,
,
0,1
,
0 1,
. .
,
,
: 0 Page 2 of 3 Packet Page: 9
Exam 7 (2013) Feldblum Formulas Rectangle Probabilities: ,
1.
2.
3.
4.
,
,
,
,
,
,
,
,
,0
,0
,0
,0
0
0
0
0
,
Def:
,
,
,
,
,
,
,
Desired Probability = 1. – 2. – 3. + 4. = Full Rect – Left Rect – Bottom Rect + Corner Rect
Farlie‐Gumbel‐Morgenstern (FGM) Copula: ,
1
1
1
Page 3 of 3 Packet Page: 10
Exam 7 (2013) Frachot Formulas Severity Estimation: Severity Calibration: 
Maximize the log‐likelihood function (severity distribution = lognormal with parameters μ and σ max ℓ
,
,

# ; ℓ
, ,
Log‐likelihood of single dataset, correcting for reporting bias: ℓ

, ,
ℓ
,
, ,
ln
ln 1
; ,
The second term of the above formula corrects for reporting bias. Frequency Estimation: 
Assume Poisson distribution Reporting Bias Adjustment: 
unconditional frequency parameter: 1
; ,
Scaling function: 

Provides link between the # of events experienced by a bank and its business size. Possible scaling functions: o Square root (# of events of one bank is linked to sqrt(business size)) o Credibility (use credibility theory to adjust internal estimated frequencies) Capital Charge Computation: 
Define the aggregate loss as: N
Aggregate Loss
L
ζ # Page 1 of 3 Packet Page: 11
Exam 7 (2013) Three Definitions of Capital Charge: 1. (OpVAR): 0.1% 2. (OpVAR unexpected loss only): 3. (OpVAR above threshold H): ∑
0.1% 1
0.1% 1, 0, 1
Frachot Formulas Aggregation of Losses Across Risk Types: 


Consider 2 aggregate losses: .
,

Correlation upper bound between 2 aggregate losses: ,

Assuming severity distributions are lognormal: /
,
/
Confidence Interval: 1. Derive (or approx) the distribution of the underlying estimators (i.e. parameters) most important step 2. Simulate a large number of parameters from (1.) 3. For each (2.), compute capital charge. These capital charges form an empirical distribution. Frequency Parameter: 
If the # of events is assumed to be Poisson, then the maximum likelihood (ML) estimator of the annual # of events ( ) is the average # of events per year for the last T Years: ~

The accuracy of improves when the number of recorded years T grows, and with an order of magnitude of √ Page 2 of 3 Packet Page: 12
Exam 7 (2013) Frachot Formulas Severity Parameter: 

Use one of two methods to approx distribution: 1. Bootstrap methods 2. Gaussian (i.e. Normal) approximation Using method (2.): ̂
~
,Ω
Ω


The accuracy of severity estimators improves when the number of losses n grows, and with an order of magnitude of √ Fisher information matrix = take 2nd derivative of the log‐likelihood function (ℓ ) w/respect to . Capital Charge Accuracy: , ,
, ,
, ̂ ,
, ,
, ̂ ,
, ,

To give an idea of the accuracy of the capital charge estimate, we: has a much tighter distribution than , then 1. Graph all 3 ratios. If severity estimation is driving inaccuracy. 2. To access the accuracy of the capital charge estimate, estimate the value of c: 1
Data Sufficiency: We can back out # of losses needed to achieve an acceptable accuracy (since c depends on n). 1. Set c to an “acceptable” level that satisfies: 2. Solve for 1
3. Once we find , then we can calculate # of external losses needed= Page 3 of 3 Packet Page: 13
Exam 7 (2013) Goldfarb Formulas Dividend Discount Model (DDM): Two ways to apply method: 1. Dividends are forecasted for all future periods, then discounted using a risk‐adj.
rate.
2. Dividends are forecasted over a finite horizon, and a terminal value is used to reflect
the value of the remaining dividends beyond the horizon
Value of share of stock = disc. PV of expected future dividends:
1
1
1
.
When div. grow (in perpetuity) at constant rate g: When div. projected over finite horizon and then assumed to grow at g: 1
1
1
1
1
DDM: Dividend growth rates beyond forecast horizon: 1. Approach 1: (simple approach) use the growth rates during the forecast horizon to
extrapolate the future growth rates
2. Approach 2: base growth rates on:
a. Dividend payout ratio: proportion of earnings paid as dividends
b. Return on Equity (ROE): profit per dollar of reinvested earnings
1
Also, Dividend payout ratio = Div per Share / Earnings per Share
Page 1 of 6 Packet Page: 14
Exam 7 (2013) Goldfarb Formulas DDM: Determining the discount rate: Capital Asset Pricing model (CAPM):
DDM: Estimating risk‐free rate: 1. 90 Day T‐bills: free of both credit and reinvestment risk
2. Maturity Matched T‐Notes: term matches the average maturity of the CFs
3. T‐Bonds: best estimate for the long run avg short‐term yields (NOTE: subtract out liquidity prem)
DDM: Estimating equity market risk premium: 
Historically, averaged 6%‐8%
Discounted Cash Flow (DCF): Approach 1 ­ Free Cash Flow to the Firm (FCFF):
Approach 2 ­ Free Cash Flow to Equity (FCFE):
1.
2.
1.
. .
Page 2 of 6 Packet Page: 15
Exam 7 (2013) Goldfarb Formulas Applying FCFE Method: Growth Rates: 

To find FCFE reinvestment = use increase in required capital
Combine FCFE Reinv w/ROE to estimate g beyond horizon
‐ Free CF
t
Discount Rates: 
In practice, we assume same discount rate for DDM and FCFE methods.
Abnormal Earnings (AE): 

Separates book value (BV) of firm from value of future earnings.
Represents earnings excess of the investors’ required earnings.
1
Page 3 of 6 Packet Page: 16
Exam 7 (2013) Goldfarb Formulas Relative valuation using multiples: Pricing­Earnings (P­E) Ratio: 
aka Forward (or leading) P‐E ratio
.
Note: If we used EPS0  trailing P‐E Ratio.
Price­to­Book Value (P­BV) Ratio: 1
Assuming AE will eventually decline to 0 after n years:
1
1
1
1
Steps for valuing a firm with diverse operations: 1.
2.
3.
4.
5.
6.
Collect financial data by segment
Select peer companies
Choose multiples
Apply multiples for segment valuation
Calculate the total firm value = Sum 4.
Validate against other diversified insurers: using multiples for other diversified firms
(continued on page 5) Page 4 of 6 Packet Page: 17
Exam 7 (2013) Goldfarb Formulas Option Pricing Theory: Valuing equity as a call option: ,0


Equity can be viewed as a call option on the company assets, w/a strike price K=D=face
value of debt.
Equity holders have right to buy back assets of firm by paying D at time T. Otherwise,
they default (if VT<D).
Abandonment option: 
Project is terminated early and the investment sold for its liquidation value less
closing‐down costs o American put option on the value of the project, w/K=net liquidation
proceeds Expansion option: 
Scope of successful project is expanded to capture more profits o American call option on the gross value of the additional capacity, w/K=cost
of creating the capacity Contraction option: 
Opposite of expansion option o American put option on the gross value of the lost capacity, w/K=cost of
savings Option to defer: 
Measures value of holding off on a project until more info is known o American call option on the value of the project Option to extend: 
Extends the life of a project by paying a fixed amount o European call option on the asset’s future value Page 5 of 6 Packet Page: 18
Exam 7 (2013) Goldfarb Formulas Black­Scholes Formula: To determine real option value:
ln
2
√
√
Miscellaneous Items: 


Tangible BV  remove Goodwill
Cost of Capital = risk‐adjusted rate
If ROE=cost of capital NO abnormal earnings
Page 6 of 6 Packet Page: 19
Exam 7 (2013) Hürlmann Formulas Notation: ,
1
1
:
:
:
1;
1,
1
Formulas:
Total Ultimate Claims = ∑

Assuming that after n development periods all clms incurred in an origin period are
known and closed.
Cumulative Paid Claims = ∑
Page 1 of 4 Packet Page: 20
Exam 7 (2013) ∑
ith period clms reserve =
Hürlmann Formulas ;
2, … ,
∑
Total Claims Reserve = Expected loss ratio = representing the incremental amount of expected paid claims per
unit of premium in each development period
∑
∑
,
1, … ,
Expected value of burning cost (of total ult clms): ,
Loss ratio payout factor: ∑
∑
∑
Individual total ult claims: 

Obtained by grossing up latest cumulative pd clms for an origin period Considered “individual” since it depends on the individual latest clms experience of
origin period ,
Individual LR clms reserve: ,
,
Page 2 of 4 Packet Page: 21
Exam 7 (2013) Hürlmann Formulas Collective LR clms reserve: 
Considered “collective” since it depends on the portfolio claims experience of all
origin periods. ∑
∑
Collective TOTAL ult clms: ,
Credible Loss Ratio Claims Reserve: 1
Benktander Loss Ratio Claims Reserve: 1, … , Neuhaus Loss Ratio Claims Reserve: 1
Optimal Credibility: Assume:
1
2
,
,
. .
3
,
,
Page 3 of 4 Packet Page: 22
Exam 7 (2013) Hürlmann Formulas Optimal credibility weights:
,
,
,
Practical form of Zi*:
Use models for
,
,
and
,
MSE Formulas: 1
1
1
Iteration Rule: Arbitrary starting point
, then m=0,1,2,… ,
gives credibility measures… 1
1
Page 4 of 4 Packet Page: 23
Exam 7 (2013) IAA Formulas Reinsurance: Quota share (QS): 
Proportional; reinsurer pays a fixed percentage of each loss in exchange for that same percentage of premium Excess­of­loss (XOL): 
Non‐proportional; reinsurer pays the excess losses above a predefined threshold (known as the priority or attachment point) up to a certain limit 0,
,
Reinsurance credit risk: 
Net capital requirement for the cedant after reinsurance (assuming ρ is linear): 1
Appendix B: Case Study: Risk­based capital charge: %
Simulation Algorithm: 1.
2.
3.
4.
5.
6.
∑
,
, # 100
# 1 , 0,1 Page 1 of 5 Packet Page: 24
Exam 7 (2013) 7.
1 , ∑
IAA Formulas Basic Formulas: 


∑
∑ ∑
,

,
,

,
,
1
TVaR: 1. Calculate the parameters of lognormal that has same mean and variance as insurer’s aggregate loss distribution. 2. Calculate the VaR at α level 3. Calculate Λ Λ 1
Misc notes on b and c parameters: 

Adding more policies to the books should continue to reduce the variance of an insurer’s loss ratio o For single LOB, this doesn’t hold Variance is reduced when more policies are added, but there is a lower bound = 


b and c can be estimated using industry data b parameters affect correlation between LOB’s (i.e. inflation) c parameters affect correlation between individual policies w/in LOB Calc RBC with a factor based formula: 1. Supervisor must determine parameters for each LOB before applying reinsurance: a.
b.
2.
3.
4.
5.
6.
c.
after application of reinsurance Insurer calculates parameters: Insurer provided estimates of expected clm cts ( ) for each LOB Calc mean and var of aggregate loss distribution (with and without reinsurance) Calc RBC (Optional) Add a CAT PML component to RBC formula Page 2 of 5 Packet Page: 25
Exam 7 (2013) IAA Formulas Appendix D: Market Risk: Estimation of risk­free spot yields (Nelson­Siegel approach): /
1
Approx stdev for the value of bond: /
100
Stdev of value of foreign bond: 0.01 Stdev of exchange rate: 0.01
100
Stdev of foreign equity (if denominated in foreign currency): 100
0.01 Appendix E: Credit Risk: Default models: 
Current value of cash flow= 100 1
1
100
1
o
p is set using credit rating from various rating companies Page 3 of 5 Packet Page: 26
Exam 7 (2013) o
IAA Formulas R is set based on experience (typically around 40‐50%) Credit risk model to assess asset risk: 


Simple Credit Risk Model: 
Assuming one large principal payment at time T (no coupons): 
1
Variance of value change due to credit risk: 
If bonds are fully dependent (max risk): 
If bonds are fully independent: ,
Simple CR model, Generalized Formula: ,
2
,
,
1
,
2
, . .
0 Combined Market Risk and Credit Risk of a fixed‐income portfolio: ,
(continued on page 5) Page 4 of 5 Packet Page: 27
Exam 7 (2013) IAA Formulas Appendix H: Analytic Methods: Cumulant generating function: ln

As a series expansion: 2
3!
,
4!
,…
Base­line model: 
Normal distr cumulant generating function: 
If we apply this idea to aggregate risk, the multivariate normal distr serves as a base‐
line model. , , … , ~ 
,
Base­line capital requirement framework: 
If stdev is used as risk measure for each LOB I, capital required is multiple of stdev: 
Base‐line capital requirement for aggregate risk: ,

(above formula holds for TVaR) Page 5 of 5 Packet Page: 28
Exam 7 (2013) Mack (1994) Formulas Notation: 1
1
,
,
,
∑
,
2
∑
Chain‐Ladder (CL) Method: ,…,
,
,
,
1
,…,
,1
1 Average distance between forecast and actual: |
|
|
. .
. .
1
. .
1
∑
1
,
,
1
1
2 We need αI‐1 for the s.e. formula above: 1.
2.
3.
=0; if 1
1
,
,…,
,
,
Page 1 of 4 Packet Page: 29
Exam 7 (2013) Mack (1994) Formulas Confidence Intervals for CiI and Ri: Assuming a normal distribution, 95% C.I. = 2
. .
,
2
. .
Assuming a lognormal distribution = /
. .
Rearranging  ln
ln 1
2
. .
CI (lognormal) = /
,
Standard error (s.e.) of TOTAL reserve (R): :
. .
2
. .
. .
∑
Page 2 of 4 Packet Page: 30
Exam 7 (2013) Mack (1994) Formulas Minimizing fk parameter (b=fk): 1.
2.
∑
,
;
∑
∑
,
;
∑
,
,
∑
3.
1 ,
;
,
Check Assumption #1 (Development factors are uncorrelated): HOW? SPEARMAN’S RANK CORRELATION COEFFICIENT 1.
2.
,
.
1
.
,
,
3.
1
6
∑
: Note: we do not calculate Tk for k=1, I‐1, & I 4. Under null‐hypothesis that dev. factors are uncorrelated: 0 1
1
TEST Tk: A value Tk close to 0 indicates development factors are uncorrelated  DO NOT reject H0 5. To test, entire triangle, we need a global T to describe the overall correlation: ∑
∑
1
1
2
1
3 /2
(continued on page 4) Page 3 of 4 Packet Page: 31
Exam 7 (2013) Mack (1994) Formulas 6. Under null‐hypothesis that dev. factors are uncorrelated: 0 1
2
3 /2
TEST T: If T lies OUTSIDE of C.I., REJECT H0: 0.67
50% . .
2
3
2
0.67
,
2
3
2
Check Assumption #2 (Calendar Year Effects): .
1.
:" "
2.
3. Informal Test: ,
,
|1
" "
,
2
,
"*"
#
,
2
. #
,
. 4. Formal Test: .
50%
. .
: 1
2
1
2
1
1
4
2
2
1
1 TEST STATISTIC 
. .
2
,
2
,
. Page 4 of 4 Packet Page: 32
Exam 7 (2013) Mack (2000) Formulas BF Method: CL Method: Benktander Method (GB): 1
1
Esa Hovinen (EH): (application of credibility to reserves instead of ultimates) 1
Mean Square Error (MSE): 1
c = 0 (BF Method) c = pk (GB Method) c = c* (optimal cred that minimizes MSE) ,
2
Page 1 of 1 Packet Page: 33
Exam 7 (2013) Marshall Formulas Correlation bands (as defined in paper): 1.
2.
3.
4.
5.
Nil – 0% correlation Low – 25% correlation Med – 50% correlation High – 75% correlation Full – 100% correlation Formulas to remember: 
NOTE: OCL = outstanding claim liabilities : 2
: Page 1 of 1 Packet Page: 34
Exam 7 (2013) McNeil Formulas Elementary approaches to measuring Operational Risk: Basic­Indicator (BI) Approach: 

Banks must hold capital for opRisk equal to the average over three years of a fixed % (α) of positive annual gross income. o Exclude years w/negative or 0 AGI Risk capital in year t: 1
# ,0 
NOTE: Basel Committee suggests α = 15% Standardized (S) Approach: 1. Bank’s activities are divided into 8 business lines. a. Corporate finance b. Trading & sales c. Retail banking d. Commercial banking e. Payment & settlement f. Agency services g. Asset management h. Retail brokerage 2. Within each business line, gross income serves as a proxy for the scale of business operations (i.e. opRisk exposure) 3. Capital charge for each business line 4. Total Capital Charge= 1
3

,0 β ranges from 12‐18% Page 1 of 2 Packet Page: 35
Exam 7 (2013) McNeil Formulas Advanced Measurement (AM) Approaches: Def: risk capital is determined by a bank’s own internal risk‐measurement system; subject to approval and continual quality checking by a regulator; uses guidelines instead of “explicit” formulas. 
Total historical loss amount for business line b in year t‐i: , ,
, ,
,

Total loss amount for year t‐i: ,
AM Capital Charge: @ AM Capital Charge (Alternative): 
Because the joint distributional structure of losses for any given year is generally unknown, we will want to resort to simple aggregation of risk measures across loss categories: ,

This forms an upper bound for the total risk Page 2 of 2 Packet Page: 36
Exam 7 (2013) Patrik Formulas Cape Cod Method (Standard‐Bühlmann): ∑
∑
“used‐up” premium 1
ELRCC = SB/Cape‐Cod estimate of ELR IBNRCC(k) = IBNR @ year k RL(k) = reported reinsurance loss @ year k AdjPP(k) = Adjusted risk pure premium @ year k Rlag(k) = aggregate claim dollar report lag @ year k NOTE: see pg 456 & 457 for other variations of the formulas above Simple Credibility IBNR Estimate: 1
IBNRCL(k) = Chain‐ladder IBNR @ year k Z(k) = CF x Rlag(k) CF = Credibility Factor (0 < CF < 1) Page 1 of 1 Packet Page: 37
Exam 7 (2013) Sahasrabuddhe Formulas Notation: ,
,
,
,
,
@
,
0
∞ ,
Φ
Limited Adjustment Factors: Def: ratio of expected claims between layer La and Lb for exposure period I at the end of dev year j.
,
,
;Φ ,
;Φ ,
;Φ ,
;Φ ,
,
,
Gross‐up Factors: When there is no policy limit, pa=∞ and no deductible, da=0, S(a,b) simplifies to a factor to gross‐up
claims to a GUU basis.
Φ,
,
;Φ ,
;Φ ,
Claim Development Factors: 
F – represents the expected ratio of ult claims to claims at maturities prior to ult. (standard
age‐to‐ult link ratios from CL method).
,
,
,
Page 1 of 4 Packet Page: 38
Exam 7 (2013) Sahasrabuddhe Formulas Claim Size and Trend: 
Trend indices can be used to calculate the claim size model parameters for prior or future
exposure periods. Φ, ~ Φ
,
,
,
,
,
Development of Basic Limit Claims Development Pattern (exposure year n cost level): 

B = basic limit = threshold at which data is sufficiently credible for the purpose of
estimating claims development patterns. L = layer To adjust cumulative claims: i, |
;Φ
,
,
,
;Φ ,
,
Estimate development pattern:
,
,
,
Development Pattern for Any Layer and Cost Level: 
Cost Level = exposure period
To adjust claims for any layer X (given layer L):
,
|
,
;Φ ,
,
,
;Φ ,
,
,
(continued on page 3) Page 2 of 4 Packet Page: 39
Exam 7 (2013) Sahasrabuddhe Formulas Estimate development pattern:
;Φ ,
;Φ ,
,
,
;Φ ,
;Φ ,
,
;Φ ,
;Φ ,
,
;Φ ,
;Φ ,
,
,

,
,
,
Φ
Φ,
,
,
,
…
; T , , T , , Φn,
;Φ ,
,
;T, ,T , ,Φ
;Φ ,
,
Other Practical Uses: 1. Ignoring that dev patterns for limited claims vary w/cost level
2. Set cost = latest exposure period
3. Assume that claim size models are only available at the latest valuation date (rather than
any age)
,
,
;Φ ,
; Φi,
,
Rj(X,B): Def: ratio between limited expected values for layer X and B at the end of development interval j
NOTE: X = Layer 0 to X, i.e. L(0,X) and B = Layer 0 to B, i.e. L(0,B)
Assuming Rj(X,B)<1, R has the following properties:
1.
2.
,
a. At early maturities, there will be less dev in excess layer than at later maturities
,
lim
Page 3 of 4 Packet Page: 40
Exam 7 (2013) 3.

Sahasrabuddhe Formulas a. More dev associated w/the denominator of R (clms in layer B) than numerator of R
(clms in layer X).
b. U can be calculated as the product of R and the ratio of ult clm dev factors for layers
X and B.
,
These properties are violated when:
o If there is negative development (OR)
o If we assume that an excess layer may develop more quickly than working layer
Alternative calculation of R: ,
1
Calculation of Max Ratios of Basic Limit to Unlimited Claims: 
Max Ratio = represented by the case where all development in the unlimited layer occurs
above the basic limit
1. Divide claims:
Limited to B
Excess of B
Unlimited 2. Identities:
a.
. .
b.
c.
3. Maximum Ratio =
Prior to Ult Ba
Xa
Ca @ Ult Br
r
Cr ‘—–•‹†‡
/
/
Page 4 of 4 Packet Page: 41
Exam 7 (2013) Shapland and Leong Formulas Notation: ,1
,
,
,
,
,
1
,
,
,
,
1 ,
,
0
,
The Bootstrap Model: Development factors: assume that each AY has the same development factor (F(w,d)=F(d)):
∑
,
∑
,
1
,
. .
Ultimate claims: assume that each AY has a parameter representing its relative level:
̂
,
,
Level parameter (variation) for AYs: assume AYs are completely homogeneous, which means
c(1,d), c(2,d),…,c(n‐d+1,d) are generated by the same mechanism:
∑
,
1
Page 1 of 5 Packet Page: 42
Exam 7 (2013) Shapland and Leong Formulas Over‐Dispersed Poisson (ODP) model: ,
0
1,2, … ,
2, … , ,
,
,
,
,
,
,
, 1
Power z:

Used to specify the error distribution
o Normal: z = 0
o ODP: z = 1
o Gamma: z = 2
o Inverse Gaussian: z = 3
Estimating α and β:
Matrix Notation:
(see paper for Y, X, and A definitions)
Unscaled Pearson Residuals:
,
,
,
,
∑
,
#
#
2
#
1
Page 2 of 5 Packet Page: 43
Exam 7 (2013) Shapland and Leong Formulas Sample triangle of fitted incremental values: ,
,
,
Degrees of freedom (DF) adjustment factor:
Add process variƒ…‡ to the future incremental values: 1.
2.
,
,
Standardized Pearson Residuals:

DF adjustment does not create standardized residuals; hat matrix adj factor is needed
1
,
1-
,
,
,
,
,
,
∑
,
Calendar­year trends:
,
1,2, … ,
2, … ,
2, … ,
Page 3 of 5 Packet Page: 44
Exam 7 (2013) Shapland and Leong Formulas Negative Incremental Values: 
Modified log‐link function to solve GLM:
ln
0
ln
,
,
,
,
,
0
0
0
Additional modification if sum of incremental values in COLUMN < 0:
1. Find the largest negative in triangle (individual value or sum)
2. Set
1.
3. Adjust the log‐link function by making all incremental values positive:
,
ln
,
,
,
4. Using (3.) we can solve the GLM using one of the linear predictor forms:
o
,
o
,
o
,
5. Adjust the fitted incremental values:
,
,
Simplified GLM, using dev factors directly:
,
,
,
,
,
,
,
Negative values during simulation: Simulate negative values when using a gamma distribution:
OPTION 1:
,
OPTION 2:
,
,
,
,
,
,
,
2
,
Page 4 of 5 Packet Page: 45
Exam 7 (2013) Shapland and Leong Formulas Adjusting for Heteroscedasticity: Stratified Sampling:
1. Group development periods w/homogeneous variances
2. Sample w/replacement from the residuals in each group separately
Calculating Variance Parameters:
1. Group development periods w/homogeneous variances
2. Calc the stdev of residuals in each group in (1.)
3. Calc hetero‐adjustment factor for each group, i:
,
,
4. Multiply all residuals in group i by hi:
,
,
,
,
,
5. Now all groups have same stdev and we can sample w/replacement among all in (4.)
6. Original distribution of residuals has been altered, to correct:
,
,
,
Normality Test: P­value: If residuals are normally distributed, p‐value should be Žƒ”‰‡. (i.e. ε5%)
R2: If residuals are normally distributed, R2 should be close to 1.
AIC and BIC:
2
ln
ln
2
1
ln
If residuals are normally distributed, AIC and BIC should be small. Page 5 of 5 Packet Page: 46
Exam 7 (2013) Siewert Formulas Loss Ratio Approach: Per occurrence excess losses = Per aggregate excess losses = 1
Implied Development Approach: 1. Develop full coverage losses to ultimate 2. Develop deductible losses to ultimate by applying LDFs that reflect various inflation indexed limits 3. Ultimate Excess Losses = 1. – 2. Credibility Weighting/BF Development Approach: 1
: 1
: @ NOTE: when Z = 1/LDFt BF estimate Inverse Power Curve: 1


Select a time at which the projection should stop (ultimate age) Fit to age‐to‐age factors Page 1 of 3 Packet Page: 47
Exam 7 (2013) Siewert Formulas Relationship between Limited and Excess Development Factors: 1 2 1
1
Relationship between Limited and Excess Loss Dev to Unlimited Loss Dev: 1
3 Using (1) and (2) from section above, we deduce that: 4 Using (4) from above ∆
5 6 ∆ 1
Partitioned Expected Development: 1
1
1
Using formulas above, we deduce that: 1
1
1
1
Page 2 of 3 Packet Page: 48
Exam 7 (2013) Siewert Formulas Development Factors for Losses Excess of Aggregate Limits: Approach 1 ­ Collective Risk Model: 

Use Weibul to model severity Use Poisson to model frequency Approach ­ 2 NCCI Table M: Must adjust expected losses to reflect loss limits. 1
0.8
1
Service Revenue Steps to determine service revenue asset: 1.
2.
3.
4.
Determine ult. Deductible losses at the account level. Calculate: 1. – ult losses excess of aggregate limits Ult recoverables = [Apply loss multiplier to 2.] Total Service Revenue Asset = 3. + Aggregate Results – Any known recoveries Weibull: 1
Γ
1
α
Γ
1 , where Γ α
1
α
1
Γ
1
α
1;
x
β
Page 3 of 3 Packet Page: 49
Exam 7 (2013) Teng & Perkins Formulas Retro Reserve: Def: (Berry & Fitz) Ultimate premium deviation – premium deviation to date 2 methods to calculate retro reserve: 1. Projected ult. prem deviation – premium deviation to date 2. Estimate ult. prem using historical premium emergence pattern – current premium Notation: @ .
. . @ %
% .
. PDLD (Premium Development to Loss Development) Ratio: 1st PDLD Ratio: %
Page 1 of 3 Packet Page: 50
Exam 7 (2013) Teng & Perkins Formulas (BP/L1)xTM Term: %
%
%
1 2nd PDLD Ratio: Def: the incremental premiums developed between the 1st and 2nd retro adjustments divided by the incremental losses developed between these two adjustments. Empirical Approach: Two types of data needed: 1. Booked premium development 2. Reported loss development Empirical Retro Adjustments: 1. 1st retro prem computation = losses dev. thru 18 mths and prem. booked thru 27 mths 2. Each subsequent retro adj. will occur in annual intervals. PDLD Ratio: , 2 Page 2 of 3 Packet Page: 51
Exam 7 (2013) Teng & Perkins Formulas Cumulative PDLD (CPDLD) Ratios: Def: the average of the PDLD ratios in all subsequent retro periods (including the current adjustment period), weighted by the % of losses to emerge in each period. Interpretation: how much premium an insurer can expect to collect per dollar of loss to emerge Calculate Premium Asset: 1. CPDLD Ratios X Expected Future Loss Emergence = Expected Future Premium 2. 1. + Booked Prem from Prior Adj. = Estimate for Ultimate Premium 3. 2. – Current(or specified date) Booked Premium = Premium Asset Teng & Perkins Discussion – Feldblum Retro Premium: 1. Covers Incurred Losses, LAE, State taxes and other state assessments 2. Covers company expenses (u/w expenses & acquisition expenses), the insurance change, state taxes and other state assessments Basic premium
3. Insurance Charge = Difference between: a. Expected loss (to insurer) caused by max retro prem b. Expected gain (to insurer) caused by min retro prem 4. Retro Premium = .
Reserving Formula: To apply to entire book of business: 1 Graphical Representations: see pg. 283‐297 Page 3 of 3 Packet Page: 52
Exam 7 (2013) Venter Copulas Formulas Copulas We want: to express a join distribution F(x,y) as a function C of FX(x) and FY(y), where C is the copula Recall: ,
,
Def: ,
,
,
,
,
,
Basic Copulas: ,
1.
2.
3.
,
,
, M A X 0,
1 Note: W ≤ C ≤ M and W & M are called Fréchet lower and upper bounds Conditioning with Copulas: ,
,
,
|
Correlation: 4
1
Note: 1.
2.
3.
.
4
,
1 ,
,
,
,
,
;
0 1 1 Page 1 of 6 Packet Page: 53
Exam 7 (2013) Venter Copulas Formulas Frank’s Copula: 1 ln 1
,
0;
0, Simulating pairs u,v: 
Possible since C1 can be inverted. 1. Simulate u and p by random draws on [0,1], where p is considered a draw from conditional distribution of V|u | 2.
ln 1
1
1
3. X and Y can be simulated by inverting the marginal distributions: -
,
Gumbel Copula: 

More tail concentration than Frank’s Asymmetric more wt. in right tail (i.e. large losses) /
,
1
1
; 1 Simulating pairs u,v: 
Since C1 is NOT invertible, must use an alternative method. 1. Simulate 2 indep. uniform deviates (u & v) 2. Solve for s, using: ln
3. Then the pair (below) will have the Gumbel copula distribution: ,
(1-v)
, Page 2 of 6 Packet Page: 54
Exam 7 (2013) Venter Copulas Formulas Heavy Right Tail (HRT) Copula and Joint Burr 

HRT produces less correlation in left tail and more correlation in right tail. If X & Y are Burr distr., then a Joint Burr distr. is produced when “a” of both Burrs is the same as the HRT copula. ,
1
1
2
1
1
1
1
; 0 Burr Distribution: 1

1
NOTE: C1 IS invertible. Normal Copula: ;
, ;
,
.
.
; 0,1 ,
,
,
;
;
2 arcsin
,1
Simulating pairs u,v: Since C1 can be inverted, u & v can be simulated using conditional distribution: 1. Simulate p(u) from std normal 2. Simulate p(v) from C1 3. Apply std normal distribution to 1. and 2. to solve for u and v. Krep’s Partial Perfect Correlation Copula Generator: 1.
2.
3.
,
3 ,
,
,
; , ,
Page 3 of 6 Packet Page: 55
Exam 7 (2013) Venter Copulas Formulas h controls how often pairs are correlated and what parts of the distribution are correlated: 1.
2.
3.
4.
If h=0 in an interval, then independence occurs. If h=1 in an interval, then perfect correlation occurs. If h=p, then perfect correlation occurs in 100p% of the cases in that interval. If h=(uv)a, then more correlation occurs for larger values of u and v, as well as, smaller values of a. Simple Copula: h(u,v)=h(u)h(v) ,
1
,
PP Max Copula: 1.
1 0 0 2.
3.
: 1
1
4. Using 3. We can calculate C(u,v) 5.
1
,
,
PP Power Copula: 1
8
1
3
1
1
2
3
(continued on page 5) Page 4 of 6 Packet Page: 56
Exam 7 (2013) Venter Copulas Formulas Tail Concentration Functions: Define how much probability is in regions near (1,1) and (0,0) ,
|
,
, 0,1 1
|
1
,
, 2
1
,
0,1 Note: L is only useful below 0.50 and R is only useful above 0.50 Distinguishing Copulas in the Right Tail: Look to see whether R(1) = 0 or greater. R formulas for various copulas: Gumbel 2‐21/a HRT 2‐a PP Power 1/(1+a) PP Max 1‐a Cumulative tau: 1
4
,
,
,
4
,
1 Survival Function: 1
,
,
1
1
,
,
,
,
,
1
,
Page 5 of 6 Packet Page: 57
Exam 7 (2013) Venter Copulas Formulas Flipped Copula: ,
1
,1
,
,
1
1
,
,1
Clayton’s Copula: 

Heavy left tail concentration Correlates small losses /
,
1
2
1
/
1
Note: Flipped Clayton = HRT Flipped Gumbel: ,
/
1
1
1
Cumulative Conditional Mean: |
,
Note: Since E(V)=0.5, M(1)=0.5 Copula Distribution Function: ,
Joe’s BB1 Copula: 
Heavy in both tails ,
1
1
2
2
1
1
/
/
Page 6 of 6 Packet Page: 58
Exam 7 (2013) Venter Factors Formulas Notation: ,
,∞
,
1
,
,
1 ,
,∞ NOTE: AY @ 12 months as age 0  d=0, first AY in triangle is 0 w=0
Mack’s Assumptions 1.
2.
3.
,
1 |
;
,
,
1 |
,
,
, , ,
,
,
;
,
Implication 1: Significance of Factors |
|
2
0. Note: if f(d) predicts cumulative losses, test significance different than 1.
Implication 2: Superiority to Alternate Emergence Patterns To compare development methods use sum of squared errors (SSE) (adjusting for # of parameters).
Other test statistics:
Alternate Emergence Pattern 1­ Linear w/Constant:
, Ϊͳ
,
*If the constant is significant, this emergence pattern is more strongly supported than that
underlying the chain ladder.
Alternate Emergence Pattern 2 ­ Factor times Parameter: ,
Page 1 of 4 Packet Page: 59
Exam 7 (2013) ,2
Venter Factors Formulas 2
.
*To test if BF model against CL, we need to calculate SSE statistic for each and see which is
better.
Special case of BF = Cape Cod: 1. Set h(w)=h for all AY
2. Same parameters as CL (since nay change in h can be offset by
inverse changes in all f’s)
3. Needs stable loss exposure across all AYs
Alternate Emergence Pattern 3 ­ Diagonal Effects: ,
1
AY Effects
Trends:
1. CY effect, set
1
2. AY effect, set
1
3. For age parameters, set
,
1
1
1
1
,
1
:
1
Iterative BF Model: : 1.
0
1
.
.
2. Find h(w), by minimizing sum of squared residuals:
∑
,
∑
Page 2 of 4 Packet Page: 60
Exam 7 (2013) Venter Factors Formulas 3. Regression to find new f’s:
∑
,
∑
Iterative BF Model – Using Wtd Least Squares: Assuming variances of residuals are proportional to f(d)h(w):
,
∑
∑
,
∑
∑
Iterative CC Method: Assuming variances of residuals are constant over triangle:
∑
,
,
∑
,
Implication 3: Test of Linearity Use a scatter plot of incremental residuals (actual emerg. – expected emerg.) against previous
cumulative losses.
*If there are strings of positive and negative residuals in a row (non‐random pattern), then non‐
linear process may be indicated.
Implication 4: Tests of Stability (Residuals over Time) 1. Test 1: Plot incr. residuals against time (AY) a. If +/‐ strings in a row  non‐linear process 2. Test 2: Look at moving average of specific age‐to‐age factor a. If clear shift over time, then instability exists and should use a wtd avg. b. If moving avg shows large fluctuations around a fixed level, this does NOT mean we should focus on recent data broader range is better. 3. Test 3: The state‐space model compares the degree of instability of the observations around the current mean to the degree of instability in the mean itself over time (helps us determine whether to use all data OR wtd avg of more recent years) Page 3 of 4 Packet Page: 61
Exam 7 (2013) Venter Factors Formulas Implication 5: Correlation of Development Factors 1. Calculate sample correlation coefficient for all pairs of columns in the triangle, then count
how many are significant (i.e. at 10% level)
∑
2. Sample correlation (r)
∑
∑
3. Test statistic (T) =
Where T ~ t‐distributed w/n‐2 DF
*If T>t‐statistic for .90 @ n‐2 DF, then r is significant @ 10% level.
4. Test Improvement:
a. m=# of pairs of columns in triangle
i. Then, pairs w/significance at 10% follow binomial w/m @ 10% w/stdev =
0.3m0.5 (i.e. var=np(1‐p))
ii. Therefore, more than 0.1m+m0.5 significant correlations would strongly
suggest there is actual correlation within the triangle.
5. To correct for correlation: a. Replace the product of development factors: i.
, Implication 6: Significantly High or Low Diagonals 1. A diagonal = w+d
a. Multiplicative Diagonal Effect: i. Suppose w+d=7 has been estimated at 10% higher than normal. Then an
adjusted BF estimate of a cell:
1.10
,
7
ii.
,
,
b. Additive Diagonal Effect: i. Triangle with incremental losses organize data into matrix for multiple regression ii. 1st Col = string out triangle into column vector (except for age 0) iii. 2nd Col = cumulative losses @ age 0 corresponding to incremental loss entries @ age 1 (0‐to‐1 cumulative‐to‐incremental factor) iv. 3rd Col = cum. losses @ age 1 corresponding to incr. loss entries @ age 2 (1‐to‐2 cum. to incr factor) v. 4th Col = cum. losses @ age 2 corresponding to incr. losses entries @ age 3 (2‐to‐
3 cum. to incr factor) vi. Last 2 columns are dummy variables. 1. If incr. loss in col. 1 is on 2nd latest diagonal Dummy 1=1;Dummy 2=0 2. If incr. loss in col. 1 is on latest diag Dummy 1=0;Dummy 2=1 Page 4 of 4 Packet Page: 62
Exam 7 (2013) Venter NT Formulas Wang Transform: 

1
. .
Shifts the (1‐p) percentile of the std normal distr by λ, and then finds the probability for the
percentile using a t‐distribution. 1
Φ
1
Φ
Wang Transform for loss severity: ΦΦ
Esscher Transform: 1
.
/
Esscher Transform of density function f(y):
/
Transformed Mean:
/
Quadratic Transform: 1
1
Distortion Measure: Def: one that can be specified by a distribution function G(x) s.t. G(0)=0, G(1)=1 and G is non‐
decreasing.
NOTE: G[S(y)] itself is a survival function since it is non‐increasing and starts at 1 and goes
to 0.
Page 1 of 4 Packet Page: 63
Exam 7 (2013) 

Venter NT Formulas G transforms the probability of Y to another probability distribution.
A distortion measure is a transformed mean.
Density function of a transformed distribution:
Distortion Measure: Marginal Allocation: Def: allocating in proportion to the impact of the business unit on the risk measure.
Last­in marginal allocation:
Incremental marginal allocation: Marginal Decomposition: Def: an allocation method which results in incremental marginal impacts that add up to the whole
risk measure.
Proportional Allocation: Def: allocating a risk measure by calculating the risk measure on the company and each business
unit, and allocating by the ratio of the unit risk to the company risk.
∑
Suitable Allocation: Def: an allocation is considered suitable when proportionally increasing the size of a business unit
w/an above average return on capital results in an increase in the firm’s return on capital.

All marginal decompositions are suitable allocations
(continued on page 3) Page 2 of 4 Packet Page: 64
Exam 7 (2013) Venter NT Formulas Co‐measure: ∑
If risk measure =
|
∑
Then co‐measure =
|
TVaR Example:
Assume only one h and L, w/ F(Y) > α
1
|
:
:
|
RTVaR Example:
Use two sets of h’s and L’s, F(Y) > α
1
, |
|
|
:
|
,
:
|
Myers‐Read Allocation: Def: an additive marginal decomposition that requires that the value of the default put option as a
fraction of expected loss be the same for each business unit.
Page 3 of 4 Packet Page: 65
Exam 7 (2013) Venter NT Formulas Allocation by Layer: ,
1 ¢
¢ #
Allocation of Layer z to unit i: 1
,
Allocation of C to unit i: 1
,
WTVaR: Def: TVaR under transformed probabilities
|
1
Page 4 of 4 Packet Page: 66
Exam 7 (2013) Venter SM Formulas Economic Capital: Def: the amount needed to achieve a 1‐year probability of ruin below a target threshold (α). α is chosen such that 1/α is a large round number. ERM Risk Measures: 


VaR = at level α, VaR is 100(α) percentile of the probability distr TVaR = conditional mean of values greater than the VaR point Alternatives to VaR and TVaR: o WTVaR (Wtd TVaR) = adjusts prob distr to give more weight to larger losses o RTVaR (Risk­Adjusted TVaR) = TVaR plus some fraction of the conditional tail stdev Allocation of Risk: Standard: allocate economic capital to a unit based on risk, and then divide the unit profit by the allocated capital to determine the risk‐adjusted return. Proportional: VaR is computed for each unit and summed. Each unit’s share of the total is its share of economic capital. Marginal: charges each unit the additional capital it requires Incremental Marginal: each small bit of business of the unit would be charges w/the additional capital needed to keep the overall company risk measure the same with or without it. Incremental Marginal Decomposition: works for homogenous risk measures, ρ(Y) s.t. ρ(aY) = a ρ(Y) (incl. : Stdev, VaR, and TVaR) Def: For a risk measure, ρ(Y) (where Y is the sum of business units X1,…,Xn), the marginal decomposition r is defined for Xk by: lim
ρ Y
ρ Y
εX
Co­measure, r: | ρ Y
| : Page 1 of 3 Packet Page: 67
Exam 7 (2013) Venter SM Formulas Marginal co­measures: when a co‐measure is a marginal decomposition 1. MCM for VaR : 2. MCM for TVaR: | | ,
3. MCM for Stdev: Optimal Capital: Def: the capital strategy that maximizes the expected present value of CFs (i.e. dividends) to shareholders. Franchise value = market value – book value Tail‐based risk measures: 

Probability of default, VaR, TVaR, RTVaR, etc. Expected policy holder deficit (EPD) ,
1
Transforming probabilities of financial results: Spectral Measure: ,
|
Example: . 0, 1
, 1
Distortion Measure: 1. Starts w/a non‐decreaing function g with g(0)=1 and g(1)=1 , 2.
1
a. Denote
1
3. Intregral of survival function = mean. Therefore, ρ(y) is the expected value of Y using the distorted distribution F* (i.e. transformed mean) Compound Poission Process: 
Need to transform frequency and severity probabilities: 1
1
1
, 1 Page 2 of 3 Packet Page: 68
Exam 7 (2013) Venter SM Formulas Transforming probabilities of underlying events: 

For transformed process to work, it must be a martingale: expected value does not change over time. 2 choices for : : produces min martingale transform; hedging strategy 1.
designed to minimize the variance of the payoff risk. /
1: produces min. entropy martingale, minimizes the distance 2.
between two measures. Transformed Values: 1
*
1
/
] /
*
/
Note: transformed mean = λ* x E*[Y] = frequency x severity Return for risk taking: CAPM: is a transformed mean ,
,
:
Page 3 of 3 Packet Page: 69
Exam 7 (2013) Venter & Underwood Formulas Quantifying the value of risk transfer for insurers: Simple multiplier methods: 1. Quantify historical distress costs for distressed firms – as a percentage of pre‐distress capital (historically, 10‐23%) 2. (1.) X (Prob. of going into distress) = expected cost of distress a. NOTE: prob of distress adjusted to account for market risk reactions To risk­adjust distress probabilities: we can use firm’s bond rating (since market value of corporate bonds is based on default probabilities adjusted to reflect market risk) For insurers in distress: the market cap reaction = multiple of financial loss (could be estimated via internal models and then risk‐adjusted) Efficient frontier comparison: 1. Use simulation to compute probability distribution of financial results under each proposed reinsurance program. 2. Use (1.) to estimate probability of various levels of distress 3. Compute % of capital lost at various probability levels for each program = probability of ruin 4. To decide which option is most valuable, an efficient frontier is graphs (risk vs reward) a. Reward = net u/w profit (Retained Prem – Retained Loss – Retained Expense) b. Risk = probability of distress or VaR or TVaR c. A reinsurance program is inefficient if a less costly program, or linear combination of programs, gives a more favorable result at the selected threshold. i. Points on curve = efficient ii. Point below the curve = inefficient iii. Management discretion is always a factor in choosing the “best” program. Cost of allocated risk capital: Def: Quantifies the benefit of risk transfer in a single #. 1. Economic capital model is applied to the simulated results NET of reinsurance 2. Difference in cost of reinsurance programs (i.e. Net Cost of Reins = Reins Prem – Reins Recov – Re‹• Comm received) is compared to difference in cost of risk capital (capital measure at the selected threshold times cost of capital) 3. If savings in capital cost > net cost of reins, reinsurance adds value a. Savings in Capital Cost = (No Reins Capital Cost) – (Current Option Capital Cost) b. Example capital measures: VaR, TVaR, VaR/Capital, or TVaR/Capital Page 1 of 2 Packet Page: 70
Exam 7 (2013) Venter & Underwood Formulas Applying a simple model of firm value: Simplified Panning model: 
The risk adjustment is implemented by assuming that going into distress is fatal to the firm and eliminates all earnings from that point forward 1
1
1 1
1
To calculate E, we need: 1. Expected u/w profit 2. Investment income .
Page 2 of 2 Packet Page: 71
Exam 7 (2013) Verrall Formulas Notation: , . .: i 1 2 3 j 1 C1,1 C2,1 C3,1 2 C1,2 C2,2 3 C1,3 ∑
∑
,
1 ,
1
1 Mack’s Model: Non‐parametric approach and specifies only the first two moments for the cumulative losses ,
,
Over‐dispersed Poisson (ODP) – for incremental losses: :
ln
General ODP form: ȗ
Page 1 of 4 Packet Page: 72
Exam 7 (2013) Verrall Formulas CL Method ODP form: .
.
.
Over‐dispersed Negative Binomial – for incremental or cumulative losses: For Incremental: 1
,
1
,
1
,
For Cumulative: ,
Normal approx. to Negative Binomial Model: For Incremental: 1
,
,
For Cumulative: ,
,
Mean Square Error of Prediction (MSEP): √
. .
√
Page 2 of 4 Packet Page: 73
Exam 7 (2013) Verrall Formulas Over‐dispersed Negative Binomial (w/Diff LDFs in each row – incremental claims): 1
,
1
,
Case 1: LDF change in some rows: To illustrate, given a 10x10 triangle and the 2nd development factor for rows 8,9, and 10 = 1.50. 1,2, … ,
1;
2,4,5, … 1,2, … ,7 ,
,
,
Case 2: LDFs are based on 5 year weighted average: Divide data into 2 parts: 3,
2, … ,
1,2, … ,
1 4 Gamma Distribution (mean and var): Bayes for BF Method: Mean of incremental losses: 1
1
,
1
∑
∑
1
Gamma distribution as the improper distribution for column parameters (yi):
1
Page 3 of 4 Packet Page: 74
Exam 7 (2013) Verrall Formulas To find γi:
1 1
1
1
1
1
∑
,
ς
∑
1
∏
∑
for i = 3,…,n
Page 4 of 4 Packet Page: 75