Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Formula Sheet (2026)

Every Exam FAM formula you need on the test, grouped by topic, rendered with full math notation. 158 formulas across 10 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

158 Formulas
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2026 Syllabus
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All Exam FAM Formulas

Short-Term Insurance and Reinsurance Coverages 14 items
Limited expected value
E[X∧c]=∫0cS(x) dx=∫0c[1−F(x)] dxE[X \wedge c] = \int_0^{c} S(x)\,dx = \int_0^{c} [1 - F(x)]\,dx, c = censoring point, S(x) = survival function, F(x) = CDF of loss X
Layer expected payment with deductible and policy limit
E[min⁡((X−d)+, u)]=E[X∧(d+u)]−E[X∧d]E[\min((X - d)_+,\, u)] = E[X \wedge (d + u)] - E[X \wedge d], d = deductible, u = policy limit, X = loss
Expected insurer payment per loss with an ordinary deductible
E[(X−d)+]=E[X]−E[X∧d]E[(X - d)_+] = E[X] - E[X \wedge d], X = loss, d = deductible, E[X∧d]E[X \wedge d] = limited expected value at d
Expected payment per loss with deductible, coinsurance, and limit
E[YL]=α[E[X∧(d+u/α)]−E[X∧d]]E[Y^L] = \alpha\left[E[X \wedge (d + u/\alpha)] - E[X \wedge d]\right], d = deductible, α = coinsurance, u = limit, d+u/αd+u/\alpha = maximum covered loss
Per-payment mean from per-loss cost
E[YP]=E[X]−E[X∧d]S(d)E[Y^P] = \dfrac{E[X] - E[X \wedge d]}{S(d)}, d = deductible, S(d) = Pr(X>d), divide by survival not CDF
Uniform limited expected value
E[X∧c]=c−c22bE[X \wedge c] = c - \dfrac{c^2}{2b} for c≤bc \le b, c = limit, b = upper bound of Uniform(0,b)
Exponential limited expected value
E[X∧c]=θ(1−e−c/θ)E[X \wedge c] = \theta(1 - e^{-c/\theta}), c = limit, θ = exponential mean
Exponential expected cost per loss above a deductible
E[(X−d)+]=θe−d/θE[(X-d)_+] = \theta e^{-d/\theta}, d = deductible, θ = exponential mean; per-payment mean stays θ (memoryless)
Loss elimination ratio
LER(d)=E[X∧d]E[X] \text{LER}(d) = \dfrac{E[X \wedge d]}{E[X]} , X = loss, d = deductible, E[X∧d] = limited expected value, E[X] = expected loss
Loss elimination ratio under inflation with a fixed deductible
LER∗(d)=E[X∧d/(1+r)]E[X] \text{LER}^*(d) = \dfrac{E[X \wedge d/(1+r)]}{E[X]} , X = pre-inflation loss, d = fixed deductible, r = inflation rate
Surplus share cedant retention proportion
Cedant’s share=min⁡(1,RV)\text{Cedant's share} = \min\left(1, \frac{R}{V}\right), R = cedant's retention line, V = insured value; policies with V ≤ R are fully retained
Expected per-risk excess of loss reinsurer payment
E[Reins]=E[X∧(M+L)]−E[X∧M]E[\text{Reins}] = E[X \wedge (M+L)] - E[X \wedge M], X = loss, M = retention, L = layer limit, layer runs M to M+L
Per-occurrence catastrophe reinsurer payment
Rcat=min⁡((S−Mcat)+,Lcat)R_{cat} = \min((S - M_{cat})_+, L_{cat}), S = aggregate event loss, M_cat = cat retention, L_cat = cat layer limit, (·)_+ = positive part
Per-risk XOL reinsurer payment after quota share
R=min⁡(((1−c)X−M)+,L)R = \min(((1-c)X - M)_+, L), c = quota share cession, X = gross loss, M = XOL retention, L = layer limit, (·)_+ = positive part
Severity, Frequency, and Aggregate Models 34 items
Compound distribution — mean
E[S]=E[N]⋅E[X]E[S]=E[N]\cdot E[X]
S=X1+⋯+XNS=X_1+\cdots+X_N, NN=frequency, XiX_i=severity (iid, indep of NN)
Panjer recursion
fS(x)=11−a fX(0)∑y=1x(a+b yx)fX(y) fS(x−y)f_S(x)=\dfrac{1}{1-a\,f_X(0)}\sum_{y=1}^{x}\left(a+\dfrac{b\,y}{x}\right)f_X(y)\,f_S(x-y)
Applies to (a,b,0)(a,b,0) frequency class
Limited expected value — Pareto
E[X∧u]=θα−1[1−(θu+θ)α−1],α>1E[X\wedge u]=\dfrac{\theta}{\alpha-1}\left[1-\left(\dfrac{\theta}{u+\theta}\right)^{\alpha-1}\right],\quad\alpha>1
Mean excess loss — Pareto
e(d)=d+θα−1e(d)=\dfrac{d+\theta}{\alpha-1}
Pareto: excess loss variable is also Pareto
Lognormal k-th raw moment
E[Xk]=ekμ+k2σ2/2E[X^k] = e^{k\mu + k^2\sigma^2/2}, μ = log-scale mean parameter, σ = log-scale standard deviation, k = moment order; mean is eμ+σ2/2e^{\mu+\sigma^2/2}
Mean excess loss function
e(d)=E[X]−E[X∧d]S(d)e(d) = \frac{E[X] - E[X \wedge d]}{S(d)}, d = threshold, E[X∧d]E[X\wedge d] = limited expected value, S(d) = survival function at d
Variance of compound Poisson aggregate claims
Var(S)=λE[X2]\text{Var}(S) = \lambda E[X^2], λ = Poisson frequency mean, E[X2]E[X^2] = second raw moment of severity (not the variance)
Weibull hazard rate function
h(x)=τθ(xθ)τ−1 h(x) = \frac{\tau}{\theta}\left(\frac{x}{\theta}\right)^{\tau-1} , τ\tau = shape parameter, θ\theta = scale parameter, x = loss amount
Gamma coefficient of variation
CV=1α CV = \frac{1}{\sqrt{\alpha}} , α\alpha = gamma shape parameter; depends only on shape, unchanged by scaling or inflation
Finite mixture variance (within plus between)
Var(X)=∑wi Var(Xi)+∑wi(μi−μˉ)2\text{Var}(X) = \sum w_i\,\text{Var}(X_i) + \sum w_i(\mu_i - \bar{\mu})^2, w_i = component weight, Var(X_i) = component variance, μ_i = component mean, μ̄ = overall mixture mean
Power transformation CDF rule
FY(y)=FX(yτ)F_Y(y) = F_X(y^\tau) for Y=X1/τY = X^{1/\tau}, F_X = CDF of base variable X, F_Y = CDF of transformed variable Y, τ = power-transform parameter
Hazard rate function
h(x)=f(x)S(x) h(x) = \dfrac{f(x)}{S(x)} , f(x) = density, S(x) = survival function; decreasing = heavy tail, increasing = light tail, constant = exponential
Heavy-tailed distribution criterion
lim⁡x→∞etxS(x)=∞ for all t>0 \lim_{x \to \infty} e^{tx} S(x) = \infty \text{ for all } t > 0 , S(x) = survival function, t = positive constant; equivalent to MGF not existing for any t > 0
Zero-truncated probability function
pkT=pk1−p0 p_k^T = \frac{p_k}{1 - p_0} , p_k^T = zero-truncated probability, p_k = (a,b,0) probability of k, p_0 = probability of zero
Logarithmic distribution probability function
pk=−θkkln⁡(1−θ)p_k = \dfrac{-\theta^k}{k\ln(1-\theta)} for k=1,2,…k=1,2,\ldots, θ=a=β/(1+β)\theta = a = \beta/(1+\beta), no mass at zero
Negative binomial (a,b,0) recursion parameters
a=β1+β, b=(r−1)β1+βa = \dfrac{\beta}{1+\beta},\ b = \dfrac{(r-1)\beta}{1+\beta}, r = size parameter, β = scale parameter; geometric is r = 1 giving b = 0
Mean of an (a,b,0) class distribution
E[N]=a+b1−aE[N] = \frac{a+b}{1-a}, a and b = (a,b,0) recursion parameters (holds for Poisson and negative binomial)
Variance-to-mean ratio for the (a,b,0) class
Var(N)E[N]=11−a\frac{\text{Var}(N)}{E[N]} = \frac{1}{1-a}, a = (a,b,0) recursion parameter (ratio = 1 Poisson, < 1 binomial, > 1 NB)
Zero-inflated Poisson probability of zero
p0ZIP=π+(1−π)e−λp_0^{ZIP} = \pi + (1-\pi)e^{-\lambda}, π = inflation fraction with N=0 certainly, λ = Poisson mean
Collective risk model aggregate variance (process plus mixing)
Var(S)=E[N] Var(X)+Var(N) (E[X])2 \text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,(E[X])^2 , N = claim count, X = severity, S = aggregate loss
Individual risk model aggregate variance with fixed benefits
Var(S)=∑i=1nbi2 qi(1−qi) \text{Var}(S) = \sum_{i=1}^{n} b_i^2\, q_i(1-q_i) , b_i = fixed benefit, q_i = claim probability, n = number of policyholders
Lognormal moment-matching parameters for aggregate claims
σ2=ln⁡ ⁣(Var(S)(E[S])2+1), μ=ln⁡(E[S])−σ22\sigma^2 = \ln\!\left(\frac{\text{Var}(S)}{(E[S])^2} + 1\right),\ \mu = \ln(E[S]) - \frac{\sigma^2}{2}, μ, σ² = mean and variance of ln S, S = aggregate claims
Normal approximation for aggregate claim tail probability
Pr⁡(S>s)≈1−Φ ⁣(s−E[S]Var(S))\Pr(S > s) \approx 1 - \Phi\!\left(\frac{s - E[S]}{\sqrt{\text{Var}(S)}}\right), S = aggregate claims, s = threshold, Φ = standard normal CDF
Collective risk aggregate distribution by conditioning on N
Pr⁡(S=s)=∑n=0∞Pr⁡(N=n)⋅fX∗n(s) \Pr(S = s) = \sum_{n=0}^{\infty} \Pr(N = n) \cdot f_X^{*n}(s) , N = claim count, f_X^{*n} = n-fold convolution of severity pmf, f_X^{*0}(s)=1 if s=0 else 0
Discrete convolution of two independent variables
Pr⁡(S=s)=∑kPr⁡(X=k)⋅Pr⁡(Y=s−k) \Pr(S = s) = \sum_k \Pr(X = k) \cdot \Pr(Y = s - k) , S = X+Y, X and Y independent discrete, k ranges over values with both terms positive
Normal approximation stop-loss premium
π(d)=σS[ϕ(zd)−zd(1−Φ(zd))]\pi(d) = \sigma_S[\phi(z_d) - z_d(1 - \Phi(z_d))], zd=(d−μS)/σSz_d = (d - \mu_S)/\sigma_S, μS,σS\mu_S,\sigma_S = mean and SD of S, ϕ,Φ\phi,\Phi = standard normal pdf and cdf
Net stop-loss premium identity
π(d)=E[S]−E[S∧d]\pi(d) = E[S] - E[S \wedge d], π(d)\pi(d) = net stop-loss premium, S = aggregate claims, d = deductible, E[S∧d]E[S \wedge d] = limited expected value (retained amount)
Discrete recursive stop-loss premium
π(d)=π(d−1)−Pr⁡(S≥d)\pi(d) = \pi(d-1) - \Pr(S \geq d), π(d)\pi(d) = stop-loss premium at integer deductible d, S = integer-valued aggregate claims
Value at Risk
VaRp(X)=FX−1(p)=inf⁡{x:FX(x)≥p}\text{VaR}_p(X) = F_X^{-1}(p) = \inf\{x : F_X(x) \geq p\}, p = confidence level, F_X = CDF of loss X
Tail Value at Risk via stop-loss premium
TVaRp(X)=VaRp(X)+E[(X−VaRp(X))+]1−p\text{TVaR}_p(X) = \text{VaR}_p(X) + \dfrac{E[(X - \text{VaR}_p(X))_+]}{1 - p}, p = level, E[(X−d)+]E[(X-d)_+] = stop-loss premium at d = VaR
Exponential Value at Risk
VaRp(X)=−θln⁡(1−p)\text{VaR}_p(X) = -\theta \ln(1 - p), θ = mean of exponential, p = confidence level
Standard-deviation principle risk measure
ρ(X)=μ+kσ \rho(X) = \mu + k\sigma , μ = mean loss, σ = standard deviation of loss, k = loading factor
Subadditivity property of a risk measure
ρ(X+Y)≤ρ(X)+ρ(Y) \rho(X+Y) \leq \rho(X) + \rho(Y) , ρ = risk measure, X and Y = loss random variables; equality holds for comonotonic risks
TVaR integral representation in terms of VaR
TVaRp=11−p∫p1VaRu du \text{TVaR}_p = \frac{1}{1-p}\int_p^1 \text{VaR}_u\, du , p = confidence level, VaR_u = value at risk at level u
Parametric Estimation 5 items
MLE for exponential (\(\theta\) parameterization)
θ^=xˉ=1n∑i=1nxi\hat{\theta}=\bar{x}=\dfrac{1}{n}\sum_{i=1}^n x_i
(MLE=sample mean for exponential mean parameter)
Log-likelihood function for independent observations
ℓ(θ)=∑i=1nln⁡f(xi;θ)\ell(\boldsymbol{\theta}) = \sum_{i=1}^n \ln f(x_i; \boldsymbol{\theta}), f = density, x_i = observation i, θ = parameter vector, n = sample size
Exponential MLE with right-censored data
θ^=∑uncxi+(n−r)cr\hat{\theta} = \dfrac{\sum_{\text{unc}} x_i + (n-r)c}{r}, θ = mean, x_i = uncensored loss, c = censoring limit, n = total count, r = uncensored count
Observed information and approximate variance of the MLE
I(θ^)=−∂2ℓ∂θ2∣θ^,  Var(θ^)≈1I(θ^)I(\hat{\theta}) = -\frac{\partial^2 \ell}{\partial \theta^2}\big\vert_{\hat{\theta}}, \; \text{Var}(\hat{\theta}) \approx \frac{1}{I(\hat{\theta})}, ℓ = log-likelihood, θ̂ = MLE, I = observed information
Pareto MLE for the shape parameter with known scale
α^=n∑i=1nln⁡ ⁣(xi+θθ)\hat{\alpha} = \dfrac{n}{\sum_{i=1}^n \ln\!\left(\frac{x_i + \theta}{\theta}\right)}, α = shape, θ = known scale, x_i = loss i, n = number of losses
Introduction to Credibility 9 items
Buhlmann credibility estimate
μ^=ZXˉ+(1−Z)μ0\hat{\mu}=Z\bar{X}+(1-Z)\mu_0
Z=nn+k,k=vaZ=\dfrac{n}{n+k},\quad k=\dfrac{v}{a}
v=E[σ2(θ)],  a=Var(μ(θ))v=E[\sigma^2(\theta)],\;a=\text{Var}(\mu(\theta))
Buhlmann-Straub credibility estimate
Z=mm+k,k=vaZ=\dfrac{m}{m+k},\quad k=\dfrac{v}{a}
where m=∑imim=\sum_i m_i (total exposure)
Buhlmann credibility factor
Z=nn+k, k=E[process variance]Var(hypothetical means) Z = \frac{n}{n+k},\ k = \frac{E[\text{process variance}]}{\text{Var}(\text{hypothetical means})} , n = number of observations, k = credibility parameter; Z < 1 for finite n
Classical partial credibility factor
Z=min⁡ ⁣(n/n0, 1) Z = \min\!\left(\sqrt{n/n_0},\, 1\right) , n = observed exposures or claims, n₀ = full credibility standard, Z capped at 1
Credibility-weighted estimate
R^=Z⋅Xˉ+(1−Z)⋅M \hat{R} = Z \cdot \bar{X} + (1-Z) \cdot M , Z = credibility factor in [0,1], X̄ = individual experience, M = class estimate (complement)
Full credibility standard for Poisson claim counts
n0=(zpr)2n_0 = \left(\frac{z_p}{r}\right)^2, z_p = two-sided standard normal critical value for probability p, r = relative tolerance (e.g. 0.05)
Full credibility standard for average severity
n0=(zpr)2CV2n_0 = \left(\frac{z_p}{r}\right)^2 CV^2, z_p = critical value for probability p, r = relative tolerance, CV = coefficient of variation of severity (no +1)
Partial credibility factor (limited fluctuation)
Z=nn0Z = \sqrt{\frac{n}{n_0}}, capped at 1 when n≥n0n \geq n_0; n = observed claims, n_0 = full credibility standard; linear n/n0n/n_0 understates Z
Full credibility standard for aggregate losses
n0=(zpr)2(1+CV2)n_0 = \left(\frac{z_p}{r}\right)^2 (1 + CV^2), z_p = critical value, r = tolerance, CV = coefficient of variation of severity; "1" is Poisson frequency term
Pricing and Reserving for Short-Term Insurance Coverages 19 items
Bornhuetter-Ferguson ultimate loss
U=Actual Dev.+Expected UnreportedU = \text{Actual Dev.} + \text{Expected Unreported}
=C+(1−q) ELR⋅Premium= C + (1-q)\,ELR\cdot Premium
qq=reported fraction, ELRELR=expected loss ratio
Expected loss ratio ultimate losses
Ultimate=ELR×EP\text{Ultimate} = \text{ELR} \times \text{EP}, ELR = expected loss ratio, EP = earned premium for the accident year
Bornhuetter-Ferguson reserve credibility identity
RBF=1CDFRCL+(1−1CDF)RELRR_{\text{BF}} = \frac{1}{\text{CDF}}R_{\text{CL}} + \left(1-\frac{1}{\text{CDF}}\right)R_{\text{ELR}}, R = reserve by method, 1/CDF = reported (developed) fraction
Chain-ladder age-to-age development factor
fk→k+1=∑iCi,k+1∑iCi,kf_{k \to k+1} = \frac{\sum_i C_{i,k+1}}{\sum_i C_{i,k}}, C_{i,k} = cumulative loss for accident year i at age k, summed over years with data at both ages
Chain-ladder ultimate losses
Ultimatei=Ci,k×CDFk\text{Ultimate}_i = C_{i,k} \times \text{CDF}_k, C_{i,k} = cumulative loss at age k, CDF_k = product of remaining age-to-age factors times tail factor
Loss ratio
LR=LEP\text{LR} = \dfrac{L}{\text{EP}}, LR = loss ratio, L = incurred losses, EP = earned premium
Claim frequency and severity
Frequency=NE,Severity=LN\text{Frequency} = \dfrac{N}{E}, \quad \text{Severity} = \dfrac{L}{N}, N = number of claims, E = earned exposures, L = losses
Pure premium
PP=Frequency×Severity=LE\text{PP} = \text{Frequency} \times \text{Severity} = \dfrac{L}{E}, PP = pure premium, L = losses, E = earned exposures
Parallelogram method fraction at the new rate level
wnew=(1−f)22 w_{\text{new}} = \frac{(1-f)^2}{2} , f = fraction of the year elapsed before the rate change, annual policies with uniform writings
Combined pure premium trend
PP Trend=(1+ftrend)(1+strend)−1 \text{PP Trend} = (1 + f_{\text{trend}})(1 + s_{\text{trend}}) - 1 , f_trend = frequency trend rate, s_trend = severity trend rate
Trend factor for ratemaking
Trend Factor=(1+t)n \text{Trend Factor} = (1 + t)^n , t = annual trend rate, n = trend period in years (midpoint of experience year to midpoint of future policy period)
On-level premium factor
On-Level Factor=Current Rate LevelAverage Rate Level in Experience Year \text{On-Level Factor} = \frac{\text{Current Rate Level}}{\text{Average Rate Level in Experience Year}} , rate levels are cumulative and multiplicative across changes
Gross rate with fixed and variable expenses
Gross Rate=PP1−V−QT+F\text{Gross Rate} = \frac{\text{PP}}{1 - V - Q_T} + F, PP = L+LAE pure premium, V = variable expense ratio, Q_T = profit target, F = fixed expense per exposure
Permissible loss ratio
PLR=1−V−QT\text{PLR} = 1 - V - Q_T, V = variable expense ratio (% of premium), Q_T = underwriting profit target
Underwriting profit target with investment income offset
QT=Qoverall−QinvestQ_T = Q_{\text{overall}} - Q_{\text{invest}}, Q_T = underwriting profit target, Q_overall = overall required return, Q_invest = expected investment income
Combined loss cost and LCM rate change
Rate Change=(1+g) mnewmold−1 \text{Rate Change} = \frac{(1+g)\, m_{\text{new}}}{m_{\text{old}}} - 1 , g = loss cost change, m_new = new LCM, m_old = old LCM (changes multiply, not add)
Loss cost multiplier
LCM=11−V−QT \text{LCM} = \frac{1}{1 - V - Q_T} , V = variable expense ratio, Q_T = profit/contingency load; rate = advisory loss cost ×\times LCM
Indicated rate under the loss cost method
Rate=Indicated PP1−V−QT+F \text{Rate} = \frac{\text{Indicated PP}}{1 - V - Q_T} + F , PP = pure premium, V = variable expense ratio, Q_T = profit/contingency load, F = fixed expense per exposure
Rate change under the loss ratio method
Rate Change=Adjusted LRTarget LR−1 \text{Rate Change} = \frac{\text{Adjusted LR}}{\text{Target LR}} - 1 , Adjusted LR = developed trended L+LAE / on-level EP, Target LR = PLR = 1 - V - Q
Option Pricing Fundamentals 13 items
Black-Scholes call price
C=S0N(d1)−Ke−rTN(d2)C=S_0 N(d_1)-Ke^{-rT}N(d_2)
d1=ln⁡(S0/K)+(r+σ2/2)TσT,d2=d1−σTd_1=\dfrac{\ln(S_0/K)+(r+\sigma^2/2)T}{\sigma\sqrt{T}},\quad d_2=d_1-\sigma\sqrt{T}
Black-Scholes put price
P=Ke−rTN(−d2)−S0N(−d1)P=Ke^{-rT}N(-d_2)-S_0 N(-d_1)
Same d1,d2d_1,d_2 as call formula
Put-call parity
C−P=S0−Ke−rTC - P = S_0 - Ke^{-rT}
CC=call, PP=put, same KK and TT
Long put option payoff at expiration
max⁡(K−ST,0) \max(K - S_T, 0) , KK = strike price, STS_T = stock price at expiration; short put payoff is the negative
Long call option payoff at expiration
max⁡(ST−K,0) \max(S_T - K, 0) , STS_T = stock price at expiration, KK = strike price; short call payoff is the negative
Risk-neutral probability in the binomial model
p∗=erh−du−dp^* = \frac{e^{rh} - d}{u - d}, r = risk-free rate, h = period length, u = up factor, d = down factor
One-period binomial option price
C=e−rh[p∗Cu+(1−p∗)Cd]C = e^{-rh}[p^* C_u + (1-p^*) C_d], r = risk-free rate, h = period length, p* = risk-neutral probability, C_u/C_d = option payoff after up/down move
Black-Scholes d1 and d2 arguments
d1=ln⁡(S/K)+(r+σ2/2)TσT, d2=d1−σT d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}},\ d_2 = d_1 - \sigma\sqrt{T} , S = stock price, K = strike, r = risk-free rate, σ = volatility, T = time to expiry
Binomial replicating-portfolio call delta
Δ=Cu−CdSu−Sd \Delta = \frac{C_u - C_d}{S_u - S_d} , Cu, Cd = call payoffs in up/down states, Su, Sd = stock prices in up/down states
Black-Scholes call and put delta
Δcall=N(d1), Δput=N(d1)−1 \Delta_{call} = N(d_1),\ \Delta_{put} = N(d_1) - 1 , N = standard normal CDF, d1 = Black-Scholes first argument; call delta in [0,1], put delta in [-1,0]
Synthetic long stock from options and bond
S=C−P+Ke−rT S = C - P + Ke^{-rT} , S = stock price, C = call price, P = put price, K = strike, r = risk-free rate, T = time to expiry
American option put-call parity bounds
S−K≤CAm−PAm≤S−Ke−rT S - K \leq C_{Am} - P_{Am} \leq S - Ke^{-rT} , C = call price, P = put price, S = stock price, K = strike, r = risk-free rate, T = time to expiry
Put-call parity differenced across two strikes
(C50−P50)−(C55−P55)=(K2−K1)e−rT (C_{50} - P_{50}) - (C_{55} - P_{55}) = (K_2 - K_1)e^{-rT} , C = call, P = put at each strike, K_1<K_2 = strikes, r = rate, T = time to expiry
Long-Term Insurance Coverages and Retirement Financial Security Programs 5 items
Curtate future lifetime — mean
ex=∑k=1∞kpxe_x = \sum_{k=1}^{\infty} {}_k p_x
kpx=P(Tx>k){}_k p_x=P(T_x>k)
Complete future lifetime — mean
e˙x=∫0∞tpx dt\dot{e}_x = \int_0^{\infty} {}_t p_x\,dt
Defined benefit final average salary pension
B=a×Y×SˉB = a \times Y \times \bar{S}, B = annual pension, a = accrual rate per year, Y = years of credited service, Sˉ\bar{S} = final-period average salary
Final average salary for a DB pension
Sˉ=1k∑i=1kSi\bar{S} = \frac{1}{k}\sum_{i=1}^{k} S_i, Sˉ\bar{S} = final average salary, k = number of final years averaged, SiS_i = salary in final year i
Flat benefit pension formula
B=Y×FB = Y \times F, B = periodic pension, Y = years of service, F = flat dollar amount per year of service
Mortality Models 17 items
Survival function from force of mortality
tpx=exp⁡ ⁣(−∫0tμx+s ds){}_t p_x = \exp\!\left(-\int_0^t \mu_{x+s}\,ds\right)
Deferred mortality probability
t∣uqx=tpx−t+upx=tpx⋅uqx+t{}_{t|u}q_x = {}_{t}p_x - {}_{t+u}p_x = {}_{t}p_x \cdot {}_{u}q_{x+t}, ₜpₓ = survival t years, ᵤq = death prob over u years, x = age, t = defer, u = window
Gompertz law survival function
S0(x)=exp⁡(−B(cx−1)ln⁡c)S_0(x) = \exp\left(-\frac{B(c^x - 1)}{\ln c}\right), force μ_x = Bc^x, B > 0 = base level, c > 1 = age scaling, x = age
Conditional survival probability for a life aged x
tpx=S0(x+t)S0(x)=exp⁡(−∫0tμx+s ds){}_{t}p_x = \frac{S_0(x+t)}{S_0(x)} = \exp\left(-\int_0^t \mu_{x+s}\,ds\right), S₀ = survival from birth, μ = force of mortality, t = years, x = age
Life table survival probability for integer durations
tpx=lx+tlx{}_{t}p_x = \frac{l_{x+t}}{l_x}, l_x = number alive at age x, t = integer years, x = age
De Moivre complete expectation and variance
e˚x=ω−x2, Var(Tx)=(ω−x)212\mathring{e}_x = \frac{\omega-x}{2},\ \text{Var}(T_x) = \frac{(\omega-x)^2}{12}, ω = limiting age, x = current age, TxT_x uniform on (0,ω−x)(0,\omega-x)
Curtate expectation one-step recursion
ex=px(1+ex+1)e_x = p_x(1 + e_{x+1}), exe_x = curtate expectation of life at age x, pxp_x = probability of surviving one year from x, ex+1e_{x+1} = curtate expectation at age x+1
Constant force complete expectation and variance
e˚x=1μ, Var(Tx)=1μ2, ex=pxqx\mathring{e}_x = \frac{1}{\mu},\ \text{Var}(T_x) = \frac{1}{\mu^2},\ e_x = \frac{p_x}{q_x}, μ = constant force of mortality, pxp_x = 1-year survival, qx=1−pxq_x = 1-p_x
Second moment of complete future lifetime
E[Tx2]=∫0ω−x2t⋅tpx dtE[T_x^2] = \int_0^{\omega-x} 2t \cdot {}_{t}p_x\,dt, TxT_x = complete future lifetime, tpx{}_{t}p_x = t-year survival probability, ω = limiting age (note factor of 2)
n-year survival probability from select rates
np[x]=∏k=0n−1(1−q[x]+k) {}_{n}p_{[x]} = \prod_{k=0}^{n-1}(1 - q_{[x]+k}) , n = years survived, x = selection age, q = one-year death rate (select if k<d, ultimate if k≥d), d = select period
Constant force fractional-age survival probability
tpx=(px)t {}_{t}p_x = (p_x)^t , pxp_x = one-year survival probability at integer age x, t = fraction into the year, force =−ln⁡px=-\ln p_x
UDD fractional-age survival probability
tpx=1−t qx {}_{t}p_x = 1 - t\,q_x , 0≤t≤10\le t\le1, qxq_x = one-year death probability at integer age x, t = fraction into the year
UDD force of mortality at fractional age
μx+t=qx1−t qx \mu_{x+t} = \dfrac{q_x}{1-t\,q_x} , qxq_x = one-year death probability from start of integer-age interval, t = fraction into the year (force increasing)
Balducci (hyperbolic) fractional-age survival probability
tpx=px1−(1−t)qx {}_{t}p_x = \dfrac{p_x}{1-(1-t)q_x} , pxp_x = one-year survival, qxq_x = one-year death probability at integer age x, t = fraction into the year
UDD linear interpolation of select survivor counts within a year
l[x]+k+u=l[x]+k−u (l[x]+k−l[x]+k+1) l_{[x]+k+u} = l_{[x]+k} - u\,(l_{[x]+k} - l_{[x]+k+1}) , x = selection age, k = integer duration, u = fractional age 0≤u≤10\le u\le1, l = survivors under UDD
Select-ultimate boundary identity for survivor counts
l[x]+t=lx+t l_{[x]+t} = l_{x+t} for t≥d t \geq d , x = selection age, t = duration, d = select period length, l = number of survivors; brackets drop once duration reaches d
Select survival probability from life table counts
np[x]+s=l[x]+s+n/l[x]+s {}_{n}p_{[x]+s} = l_{[x]+s+n}/l_{[x]+s} , x = selection age, s = current duration, n = years survived, l = number of survivors at that selection age and duration
Present Value Random Variables for Long-Term Insurance Coverages 22 items
Whole life insurance APV (discrete)
Ax=∑k=0∞vk+1 kpx qx+kA_x = \sum_{k=0}^{\infty} v^{k+1}\,{}_k p_x\,q_{x+k}
Variance of whole life insurance
Var(Z)=2 ⁣Ax−(Ax)2\text{Var}(Z)={}^2\!A_x-(A_x)^2
2 ⁣Ax{}^2\!A_x evaluated at v∗=v2v^* = v^2 (i.e., rate i∗=(1+i)2−1i^*=(1+i)^2-1)
Term insurance APV (discrete, \(n\)-year)
Ax:n‾∣1=∑k=0n−1vk+1 kpx qx+kA^1_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^{k+1}\,{}_k p_x\,q_{x+k}
Annuity-due present value as a function of insurance present value
Y=(1−Z)/d Y = (1 - Z)/d , Y = annuity-due PV random variable, Z = insurance PV random variable, d = annual effective discount rate
Endowment insurance present value random variable
Z=vmin⁡(Tx,n) Z = v^{\min(T_x, n)} , v = discount factor, TxT_x = continuous future lifetime of (x), n = endowment term
Whole-life continuously increasing annuity under constant force
(Iˉaˉ)x=1/(μ+δ)2 (\bar{I}\bar{a})_x = 1/(\mu+\delta)^2 , μ = constant force of mortality, δ = force of interest
Pure endowment actuarial present value
nEx=vn⋅npx {}_{n}E_x = v^n \cdot {}_{n}p_x , v = discount factor, n = term, npx{}_{n}p_x = probability (x) survives n years
Woolhouse two-term m-thly annuity-due approximation
a¨x(m)≈a¨x−m−12m\ddot{a}_x^{(m)} \approx \ddot{a}_x - \dfrac{m-1}{2m}, a¨x\ddot{a}_x = annual whole life annuity-due, m = payments per year
UDD m-thly insurance APV from annual insurance
Ax(m)=ii(m) AxA_x^{(m)} = \dfrac{i}{i^{(m)}}\, A_x, i = effective annual rate, i(m)i^{(m)} = nominal rate compounded m-thly, AxA_x = annual insurance APV; ratio always > 1
Variance of a life annuity-due present value
Var(Y)=2Ax−(Ax)2d2\text{Var}(Y) = \dfrac{{}^{2}A_x - (A_x)^2}{d^2}, AxA_x = insurance APV, 2Ax{}^{2}A_x = insurance second moment at double force, d = effective discount rate (use δ2\delta^2 for continuous)
Covariance of insurance and annuity present values
Cov(Y,Z)=−Var(Z)d\text{Cov}(Y,Z) = -\dfrac{\text{Var}(Z)}{d}, Y = annuity-due PV, Z = insurance PV, d = effective discount rate; always negative (natural hedge)
Pure endowment factor
nEx=vn⋅npx {}_{n}E_x = v^n \cdot {}_{n}p_x , nEx{}_{n}E_x = pure endowment, vv = discount factor, nn = term, npx{}_{n}p_x = probability (x) survives n years
Term insurance shortcut from temporary annuity
Ax:n∣‾1=1−d⋅a¨x:n∣‾−nEx A^{1}_{x:\overline{n\vert}} = 1 - d \cdot \ddot{a}_{x:\overline{n\vert}} - {}_{n}E_x , Ax:n∣‾1A^{1}_{x:\overline{n\vert}} = term insurance, dd = discount rate, a¨x:n∣‾\ddot{a}_{x:\overline{n\vert}} = temporary annuity-due, nEx{}_{n}E_x = pure endowment
Endowment insurance decomposition
Ax:n∣‾=Ax:n∣‾1+nEx A_{x:\overline{n\vert}} = A^{1}_{x:\overline{n\vert}} + {}_{n}E_x , Ax:n∣‾A_{x:\overline{n\vert}} = endowment insurance, Ax:n∣‾1A^{1}_{x:\overline{n\vert}} = term insurance, nEx{}_{n}E_x = pure endowment
Whole life annuity splitting identity
a¨x=a¨x:n∣‾+nEx⋅a¨x+n \ddot{a}_x = \ddot{a}_{x:\overline{n\vert}} + {}_{n}E_x \cdot \ddot{a}_{x+n} , a¨x\ddot{a}_x = whole life annuity-due, a¨x:n∣‾\ddot{a}_{x:\overline{n\vert}} = temporary, nEx{}_{n}E_x = pure endowment, a¨x+n\ddot{a}_{x+n} = deferred annuity
Recursion for whole life insurance
Ax=vqx+vpxAx+1A_x = v q_x + v p_x A_{x+1}, v = annual discount factor, q_x = death probability at age x, p_x = 1−q_x, A_{x+1} = insurance value at age x+1
Single-age mortality change effect on insurance
ΔAx=v Δqx(1−Ax+1)\Delta A_x = v\,\Delta q_x (1 - A_{x+1}), v = discount factor, Δqx\Delta q_x = change in death probability at age x, A_{x+1} = insurance value at age x+1
Constant force whole life insurance
Aˉx=μμ+δ\bar{A}_x = \dfrac{\mu}{\mu + \delta}, μ = constant force of mortality, δ = force of interest
Recursion for whole life annuity-due
a¨x=1+vpxa¨x+1\ddot{a}_x = 1 + v p_x \ddot{a}_{x+1}, v = annual discount factor, p_x = survival probability at age x, a¨x+1\ddot{a}_{x+1} = annuity-due value at age x+1
Deferred whole life annuity-due
n∣a¨x=nEx⋅a¨x+n {}_{n\vert}\ddot{a}_x = {}_{n}E_x \cdot \ddot{a}_{x+n} , nEx {}_{n}E_x = n-year pure endowment, a¨x+n \ddot{a}_{x+n} = whole life annuity-due at deferred age x+n
Whole life insurance as term plus deferred insurance
Ax=Ax:n∣‾1+n∣Ax A_x = A^{1}_{x:\overline{n\vert}} + {}_{n\vert}A_x , Ax A_x = whole life APV, Ax:n∣‾1 A^{1}_{x:\overline{n\vert}} = n-year term APV, n∣Ax {}_{n\vert}A_x = n-year deferred whole life APV
Deferred whole life insurance via pure endowment
n∣Ax=nEx⋅Ax+n {}_{n\vert}A_x = {}_{n}E_x \cdot A_{x+n} , nEx {}_{n}E_x = n-year pure endowment, Ax+n A_{x+n} = whole life insurance APV at deferred age x+n
Premium and Policy Value Calculation for Long-Term Insurance Coverages 20 items
Insurance-annuity relationship
Ax=1−d a¨xA_x = 1 - d\,\ddot{a}_x
Aˉx=1−δ aˉx\bar{A}_x=1-\delta\,\bar{a}_x (continuous)
Net premium (benefit premium)
P=Axa¨xP=\dfrac{A_x}{\ddot{a}_x}
(equivalence principle: APV premiums = APV benefits)
Prospective policy value
tV=Ax+t−P a¨x+t{}_t V = A_{x+t} - P\,\ddot{a}_{x+t}
PV future benefits minus PV future premiums
Fully discrete whole life loss in linear form
L0=vKx+1(1+Pxd)−PxdL_0 = v^{K_x+1}\left(1 + \frac{P_x}{d}\right) - \frac{P_x}{d}, v = discount factor, K_x = curtate future lifetime, P_x = level premium, d = discount rate
Variance of loss under the equivalence principle
Var(L0)=2Ax−Ax2(1−Ax)2\text{Var}(L_0) = \frac{{}^{2}A_x - A_x^2}{(1-A_x)^2}, A_x = whole life APV, 2Ax{}^{2}A_x = second moment at rate (1+i)2−1(1+i)^2-1
General loss variance with loaded premium or expenses
Var(L0)=(1+Pd)2(2Ax−Ax2)\text{Var}(L_0) = (1+\tfrac{P}{d})^2({}^{2}A_x - A_x^2), P = annual premium, d = discount rate, A_x = whole life APV, 2Ax{}^{2}A_x = second moment
Future loss at issue
L0=PV(benefits)−PV(premiums)L_0 = \text{PV(benefits)} - \text{PV(premiums)}, L0L_0 = insurer's present-value loss at issue; positive means the insurer loses money
Equivalence principle net premium
Px=Axa¨x=1a¨x−dP_x = \frac{A_x}{\ddot{a}_x} = \frac{1}{\ddot{a}_x} - d, P = level net premium, A = insurance EPV, ä = annuity-due EPV, d = discount rate
Portfolio percentile principle premium
P=Ax+zα⋅SD(L0)/na¨xP = \frac{A_x + z_\alpha \cdot \text{SD}(L_0)/\sqrt{n}}{\ddot{a}_x}, z = percentile factor, SD(L₀) = std dev of unit loss, n = policies, A = insurance EPV, ä = annuity EPV
Increasing term insurance APV
(IA)x:n∣‾1=∑k=0n−1(k+1) vk+1 kpx qx+k(IA)^1_{x:\overline{n\vert}} = \sum_{k=0}^{n-1} (k+1)\, v^{k+1}\, {}_{k}p_x\, q_{x+k}, v = discount factor, ₖpₓ = survival to k, q = death prob, n = term
Gross premium with percentage-of-premium expenses
G=A+ea¨+E0(1−c)a¨G = \frac{A + e\ddot{a} + E_0}{(1-c)\ddot{a}}, G = gross premium, A = benefit EPV, e = renewal % expense, c = first-year %, E₀ = fixed expenses, ä = annuity EPV
Gross premium policy value
tVg=Ax+t:n−t∣‾+EPV(exp)−G a¨x+t:n−t∣‾ {}_{t}V^g = A_{x+t:\overline{n-t\vert}} + \text{EPV(exp)} - G\,\ddot{a}_{x+t:\overline{n-t\vert}} , A = endowment APV, EPV(exp) = future expenses, G = gross premium, a¨ \ddot{a} = annuity-due
Recursive net premium reserve relation
(tV+P)(1+i)=qx+t+px+t⋅t+1V ({}_{t}V + P)(1+i) = q_{x+t} + p_{x+t}\cdot {}_{t+1}V , P = net premium, i = interest rate, qx+t q_{x+t} = death prob, px+t p_{x+t} = survival prob, V = reserve
Full Preliminary Term reserve for whole life
tVFPT=t−1Vx+1 {}_{t}V^{\text{FPT}} = {}_{t-1}V_{x+1} for t≥1 t \geq 1 , tVFPT {}_{t}V^{\text{FPT}} = FPT reserve at duration t, t−1Vx+1 {}_{t-1}V_{x+1} = net reserve at duration t−1 for issue age x+1
Annuity-ratio net premium policy value
tVx=1−a¨x+ta¨x {}_{t}V_x = 1 - \frac{\ddot{a}_{x+t}}{\ddot{a}_x} , tVx {}_{t}V_x = reserve at time t, a¨x+t \ddot{a}_{x+t} = annuity-due at attained age, a¨x \ddot{a}_x = annuity-due at issue age
Re-valued net premium policy value
tVnew=Ax+tnew−Pxold a¨x+tnew {}_{t}V^{\text{new}} = A_{x+t}^{\text{new}} - P_x^{\text{old}} \, \ddot{a}_{x+t}^{\text{new}} , A = insurance EPV on new basis, P^old = premium locked at issue, ä = annuity-due EPV on new basis, t = duration
Substandard whole life insurance under constant addition to the force
Aˉx∗=Aˉx@δ+c+c aˉx@δ+c\bar{A}^*_x = \bar{A}_x^{@\delta+c} + c\,\bar{a}_x^{@\delta+c}, Aˉx∗\bar{A}^*_x = substandard insurance, terms evaluated at force of interest δ+c, c = extra force, δ = force of interest
Substandard whole life annuity under constant addition to the force
aˉx∗=aˉx@δ+c\bar{a}^*_x = \bar{a}_x^{@\delta+c}, aˉx∗\bar{a}^*_x = substandard annuity, aˉx@δ+c\bar{a}_x^{@\delta+c} = standard annuity at force of interest δ+c, δ = force of interest, c = extra force
Substandard survival probability under constant addition to the force
tpx∗=tpx⋅e−ct{}_{t}p^*_x = {}_{t}p_x \cdot e^{-ct}, tpx∗{}_{t}p^*_x = substandard t-year survival, tpx{}_{t}p_x = standard t-year survival, c = constant added to force, t = years
Substandard net premium shortcut under constant addition to the force
Pˉ∗=Pˉx@δ+c+c\bar{P}^* = \bar{P}_x^{@\delta+c} + c, Pˉ∗\bar{P}^* = substandard premium, Pˉx@δ+c\bar{P}_x^{@\delta+c} = standard premium at force of interest δ+c, c = extra force, δ = force of interest

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