Exam FAM - SOA

Put call parity equation

c(t) - p(t) = St - Ke^-rt

convert discrete insurance to continuous insurance using UDD

contAx = i/delta Ax for all but endowment where the pure endowment isn't converted.

How to convert a formula for A to a formula for the second momment

delta becomes 2delta v becomes v^2 i becomes 2i+i^2 and d is treated like i (it becomes 2d + d^2

What does it mean when a life is age rated 5 years

they're rated at 5 years older than they are to reflect their poorer risk status i.e if they're 42 they're rated as if they're 47

tqx in terms of fx Fx Sx

tqx = Fx(t)

u|t q x

u years deferred probability of dying within t years

Variable & Fixed Expense Ratio

variable expenses/Premium and Fixed expenses/Premium

MLE Notes when there's a deductible

when you're setting up L (x) make sure to use f(x + d) / (S(d)) this is bc P(A|B) = P(A & B) / P(B)

the better term insurance and annuity formula. REMEMBER ME

Ax + nEx * Ax+n

Just a friendly reminder that the exhibit has notes for zero modified formulas

Have a gangsta day, handsome!

annuity certain

(1 - v^n )/ d

Log Transformed confidence interval for H(t)

(H(t)/Us , H(t)*Us) Where Us = e^(z of 1+p/2 *CV of H(t))

Log Transformed confidence interval for S(t)

(S(t)^1/Us , S(t)^Us) Where Us = e^(z of 1+p/2 * sqrt(var[S(t)]) /(S(t) * ln(S(t)) )

Interim Reserve Exact method for time t+s where s is the additional period <=1

(Vt + P)* (1 + i ) ^s = sqx+t * v^(1-s) + spx+t * t+sV where the goal is generally t+sV This does work if s=1

Variance of the loss random variable

(b + P/delta)^2 * (2Abar x - Abarx^2) REMEMBER THE FIRST BIT IS SQUARED

Risk neutral probability of a stock price going down

(u - e^delta* t)/(u-d)

SULT Tricks for the second moment of tEx

(v^t)^2 * tpx and (v^t)^2 * lx+t / lx and v^t * tEx

Parts of automobile coverage

- Third party bodily injury - medical payments (tort jurisdictions) or personal injury projection (no-fault jurisdictions) - Unidentified, Uninsured, underinsured motorists protection - Collision and other than collision

Delta of a put

-Phi(-d1)

Permissible Loss Ratio

1 - V - Qt where V is variable expenses and Qt is target profit and contingencies ratio

4 Rules of coherence

1 Translation Invariance rho(c+x) = rho(x) + c 2 Postivie Homogeneity rho(cx) = c*rho(x) 3 subadditivity rho(x + y) <= rho(x) + rho(y) 4 monotonicity rho(x) <= rho(y) if P(x<y) = 1 (if x is strictly less than y the function returns a value for x lower than it returns the value for y) Note VaR fails subadditivity but TVar is fine

Parallelogram method

1 determine timing and amt of rate changes 2 calculate cumulative rate level index for each rate group (the product of all the relevant 1*rate change) 3 calculate portion of premium in each square that corresponds to each rate level 4 calculated the weighted average factor for each year 5 calculate the on level factor (the final factor / that year's weighted factor) then use this on level factor to reprice that year's premium

relationship between annuities and insurance

Ax = 1-d(annuity immediate on x) second moment Ax = 1-(2d-d^2) * (second moment annuity) if it's continuous Ax = 1-delta * (annuity immediate on x) second moment Ax = 1-(2delta) * (second moment annuity)

What makes risk insurable? (how many)

6 Economically Feasible Economic Value is calcuable loss is definite loss is random exposures in a rate class are homogenous exposure units are spatially and temporally independent

f(x) in terms of other things

= tpx * mu x+t

On Level premium when you have rating factors in addition to trend like age and risk class

Base level premium * trend ^ number of years * current rating factors

UDD Converting annuities to m-ly annuities

Below the SULT tables is the common values when i = 0.05 and formulas for alpha and beta if i isn't 0.05 (or the frequency is odd AF) m-ly whole life annuity = alpha(m)*annual version - beta(m) m-ly n year term annuity = alpha(m) * annual version - beta(m)*( 1 - nEx) m-ly n year deferred whole = alpha (m) * annual version - beta(m) * nEx

Bernoilli is a special case of ________ where ______ equals ________

Bernoulli is a special case of Binomial where n equals 1

Nelson Aalen Estimator Variance of S(t)

H(t) = sum( d sub j/ r sub j) S(t) = exp( - H(t) ) Var(S(t)) = S(t)^2 * Var(H(t))

Replicating portfolio value

Delta S + B where Delta = Vu - Vd / S(u-d) B = e-rh * ( uVd - dVu) / (u-d) Note for a put it's B - Delta S but just always check the value makes sense

expense policy value

EPV(expenses) - EPV(expense loading) (where expense loading is G-P or just Pe if you need to find that for the solution)

Loss Elimination Ratio (LER)

E[X /\ d]/E[X]

E[Y super L] with a franchise deductible

E[X] - E[X /\ D] + d*S(d)

E[Y] per loss super formula if d = deductible u = policy Limit alpha = coinsurance r = inflation rate m = maximum covered loss (and the formula for m)

E[Y] per loss = alpha * (1+r) * (E[x/\ m/(1+r)] - E[X/\d/(1+r]) where m = (u / alpha) + d

What type of insurance is popular as microinsurance in developing nations

Endowment insurance

Term life for n years using SULT

Endowment insurance on age x:n - pure endowment x:n whole life at age x - pure endownment of n years * whole life at age x+n

Pure premium method of ratemaking

Expected Losses + expected fixed expenses / Permissible Loss ratio where PLR is 1 - V - Qt where V is variable expenses and Qt is target profit and contingencies ratio

Exponential is special case of _______ where _______ equals __________

Exponential is special case of Gamma where alpha equals 1

f of x given j<x<k

Fx(x) / pr(j<X<k)

Geometric is a special case of ________ where ______ equals ________

Geometric is a special case of Negative Binomial where alpha (the first parameter) equals 1 Therefore sum of n geometric is a geometric with alpha = n

what to do with censored data in KM or NA estimators of survival

Include them in the risk pool r and then take them out at the value specified (so if it's at least 50 include it in r but when you get a d>50 it's removed from r and isn't included in the ds)

Varying Insurance & annuities (Discrete)

Increasing Whole Life = SUM from k=0 to infinity of (K+1)*v^(k+1)*k|qx if it's term then you're summing from k=0 to k=n-1 (remember it's kurtate where 0 is the first year and n-1 is the final) if it's decreasing it's the same formula but with (n-k) rather than (k+1) for annuities it's tpx not k|qx (probability of surviving not probability of surviving then dying) Recursive formula for increasing whole life vqx + vpx(Ax+1 +(IA)x+1)

Ratemaking: Loss Ratio Method

Indicated Average Rate Change = (Losses per exposure + Fixed Expenses) / (1 - Variable Expenses - Qt aka desired profit) the denominator is also known as the permissible loss ratio Losses per exposure is also called Pure Premium

Rate Making Pure Premium Method

Indicated Average Rate Change = (average loss per exposure + average fixed expenses) / (1 - variable expenses - profit)

Continuous Ax integral value

Integral from 0 to t of v^t npx force of mortality generally gonna be integral of force of mortality * e^(delta + force of mortality)*t

Black Scholes note: What is the volatility of the log-return on the stock?

It's the *standard deviation* used in the formulas. It's not the variance or a manipulated version of that.

loss ratio method of ratemaking

Losses + Fixed expenses / Permissible Loss ratio where PLR is 1 - V - Qt where V is variable expenses and Qt is target profit and contingencies ratio

Incurred losses for Accident year or plan year (Pricing and reserving)

Losses Paid + Reserve

Incurred losses for calendar year (pricing and reserving)

Losses paid + change in reserve

Gompertz law relationship to Makeham's law

No A terms in either mu or tPx remember! Both are on the sheet below the SULT

P(A|B)

P(A and B)/P(B)

Special distribution shortcuts if there's a deductible

Pareto( alpha, theta + deductible) exponential is unchanged exponential(theta) bc of Memoryless property Uniform(0, b-d) assuming d>a

Credibility Premium

Pc = M + Z(xbar - M) Where M is the manual premium and Z is sqrt( n/needed n for full credibility) if it's exposures use actual exposures if it's claims use actual claims)

zero modified Pn and E[N]

Pn Modified = ((1-Pm0)/(1-p0))*Pn E[N] Modified = ((1-Pm0)/(1-p0))*E[N]

zero truncated Pn and E[N]

Pn truncated = (1/(1-p0))*Pn E[N] truncated = (1/(1-p0))*E[N]

Unearned premium for CY

Premium Written - Premium Earned + Premium Unearned the previous year's end

Kaplan Meier estimator

Product( 1 - d sub j / r sub j) for all t <=T

Bornhuetter-Ferguson Method of reserving estimates

R = Premium earned * Loss ratio ultimate losses × (1- 1/Chain-ladder age to ultimate factor)

Reserves are added to what when determining ultimate losses and why

Reserves are added to paid losses (not incurred) since we are trying to estimate the ultimate paid amounts. IBNP amounts are included in the reserve so we would be double counting them.

Black-Scholes Model of a call option

S(t) * phi(d1(t)) - Ke-rt*phi(d2(t)) d1 = (ln(St/K) + (r + sigma^2/2)*(T-t)) / (sigma root(T-t)) d2 = d1 - sigma*root(T-t)

S(x) in terms of H(x)

S(x) = e^-H(x) (the cumulative hazard function)

How many ADLs must a policyholder be unable to perform in order to receive LTC benefits? and what are they?

at least 2 ADLs the ADLs are bathing, transferring (to a chair), eating, toileting, Dressing, continence

Tips for finding value of an insurance/annuity in terms of other annuities

Stay organized, keep writing down relationships until something clicks. adding a zero or multiplying by one might help!

What is covered by marine insurance

The following risks are eligible for coverage under inland marine insurance: Domestic shipments made by railroad, motor vehicle, or ship on inland waterways. Provision is made for insuring goods transported by air, mail, parcel post, express, armored car, or messenger. Instrumentalities of transportation and communication, such as bridges, tunnels, piers, wharves, docks, communication equipment, and movable property. Personal property floater risks used for coverage of construction equipment, personal jewelry and furs, agricultural equipment, and animals.

Woolhouse approximation notes (also if force isn't available)

The formulas are on the exhibits below the SULT if it's two term approximation then don't use the final bit with (delta + force of mort) if it's a n year term then you need to use this relationship ax:n = ax - nEx * ax+n * if force isn't available it's ~~ -(ln(px-1) + ln(px)/2 JUST NEED ONE YEAR EVEN IF IT"S 10 years *

For surplus share treaty reinsurance that covers Y% in excess of $X who pays what?

The primary insurer pays $X + (1-Y%)*(Loss -X) the Y% is the amount COVERED by the reinsurance contract

When a question tells you that VaR level p% (x) = some value what does that mean?

The value is the pth percentile of the random variable x. aka Fx(the value) = p%

Structured Settlement top down vs bottom up

Two approaches can be used to determine the annuity payments in a structured settlement: Top-down approach: Determine an appropriate lump sum amount, then convert that amount to an annuity Bottom-up approach: Determine an appropriate income stream, then calculate the EPV of the payments The bottom-up approach is more suitable for structured settlements as their objective is to match the pre- injury income of the IP.

Uniform is a special case of ________ where ______ equals ________

Uniform is special case of Beta where the first two parameters equal 1

Variance of uniform distribution

Var(x) = (b - a)^2 / 12

Variance of an aggregate model

Var[S] = E[N]*Var[X] + Var[N]*E[X]^2

Replicating portfolio (Delta)

Vu - Vd / S(u-d)

Are waiting periods common for LTC contracts? how many days if so?

Yes, and 90 days or so is common

mean excess loss function

e(d)= E[X-d | X>d] = (E[X] -E[X^d])/(1-F(d))

Replicating portfolio (B)

e-rh * ( uVd - dVu) / (u-d)

recursive formula for the expectation of life

ex = p(1+ex+1) ex = 1px + 2px(1 + ex+2) ex = 1px + 2px + 3px(1+ex+3)

e∘x when you have S(x)

e∘20=∫0 to ∞ tp20 dt = ∫0 to ∞ S0(20+t)S0(20) dt which should simplify nicely

h(x) in terms of other functions of x

f(x)/S(x)

fx(t) in terms of force of mortality

fx(t) = Sx(t)* force = tPx * force

coinsurance on homeowners insurance (Formula)

if coinsurance requirement isn't met (amount that needs to be covered ) then the benefit is = L*(amount covered/(coinsurance * value at time of loss)) |f the factor applied to loss is greater than 1 then it's just L

MLE notes on incomplete data

if it's left truncted f(x)/S(d) if it's right censored at u use S(u) if it's grouped use P(a<x<=b) generally with S(a) - S(b) or F(b) - F(a)

What does a standard fire policy cover

if the proximate cause was a fire or lightning strike it covers at least one of 1 personal p

If you have a short "death benefit is policy value" whats a trick you can use

if you have premium by equivalence you should know the policy value at beginning and end so you can use the recursive formula to get two formulas for P

varying insurance (relationship between increasing and decreasing)

increasing term + decreasing term = term with benefit N+1 increasing continuous term + decreasing continuous term = continuous term with benefit amount = n

Varying Insurance & annuities (continuous)

increasing whole life = integral from 0 to infinity of tv^t*tpx*force of mortality of x+t if it's term then it's the same but integrating to n if it's decreasing then it's (n-t) rather than t for annuities it's the same but it's not times the force of mortality

PV of whole life with constant interest and force of mortality

integral from 0 to infinity of mu*e^-(mu+delta)

continuous ax integral value

integral from 0 to of v^t npx generally gonna be integral of e^(force + delta)*t

what does this mean: For a fully discrete 15-pay whole life insurance of 200,000 on (50), you are given

it means that it's a 200,000 life insurance policy on someone aged 50 and that person pays 15 premiums for only the first 15 years (rather than paying for their whole life)

c*lognormal(mu, sigma^2)

lognormal(mu+ln(c), sigma^2)

force of mortality in terms of other functions of x and t

mu x+t = f(x)/S(x) = derivative with respect to time of -ln(S(x)) and -derivative of tpx / tpx you usually either have a formula of tpx or can get it by using lx's so tpx = t*lx+1 + (1-t) lx / lx (if t>0.5 otherwise flip it). simplify this so tpx = 1 + t(lx+1 - lx)/lx = tpx = 1 + tqx therefore derivative of tpx with respect to time is qx

Variance of n independent whole life policies with benefit = b

n*b^2*(2Ax - Ax^2) bc this is the var(1000A1 + 1000A2.... 1000An) => 1000^2var(A1) + 1000^2var(A2) +..... cov(A1,A2)..... the covariance terms drop out and you just get the sum from j=1 to n of 1000^2var(Aj)

Full credibility for claims

nc = ((z of 1+p/2) / k)^2 *(Variance of N / expected N + (CV of X)^2 )

relationship between full credibility amount by exposures and full credibility amount by claims

nc = ne * E[N]

Full credibility formula for EXPOSURES

ne = ((z of 1+p/2) / k)^2 * (CV of S)^2 where CV of S is the CV of the aggregate model

c*Normal(mu, sigma^2)

normal(c*mu, (c*sigma)^2 )

c*force of mortality + k

npx^c * e^-kn

convert second moment of continuous insurance to second moment of discrete

second moment of continuous whole life = (i^2 + 2i)/(2*delta) all times the second moment of a discrete whole life insurance

SULT tricks for tEx

tEx = v^t * tpx = v^t * lx+t / lx t+hEx = tEx * hEx+t (aka 30E(15) = 20E15 * 10E35

What is a semicontinuous version of insurance (as opposed to continuous or discrete)

the benefits are paid continuously but the premiums are discrete

u|t q x means what

the u year deferred probability a life aged x dies in time (u,u+t)