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Free SOA Exam SRM (Statistics for Risk Modeling) Formula Sheet (2026)

Every Exam SRM formula you need on the test, grouped by topic, rendered with full math notation. 94 formulas across 5 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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2026 Syllabus

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All Exam SRM Formulas

Basics of Statistical Learning 10 items

Bias-variance tradeoff

E [(y - f^(x))^{2}] = Bias^{2} (f^) + Var (f^) + σ_{ε}^{2}

Irreducible error

σ_{ε}^{2}

cannot be reduced

k-fold cross-validation error

C V_{(k)} = \frac{1}{k} \sum_{j = 1}^{k} MSE_{j}

Each fold serves as validation once

Expected test MSE decomposition

E [(y_{0} - f^(x_{0}))^{2}] = Var (ε) + [Bias (f^(x_{0}))]^{2} + Var (f^(x_{0}))

— Var(ε) = irreducible noise, Bias² = squared model bias, Var(f̂) = model variance

Mallows Cp statistic

C_{p} = \frac{1}{n} (RSS + 2 d σ^^{2})

— n = sample size, RSS = residual sum of squares, d = number of predictors,

σ^^{2}

= residual variance estimate

Mean squared error (regression)

MSE = \frac{1}{n} \sum_{i = 1}^{n} (y_{i} - f^(x_{i}))^{2}

— n = sample size, y_i = actual,

f^(x_{i})

= predicted response

Misclassification rate

Err = \frac{1}{n} \sum_{i = 1}^{n} I (y_{i} \neq = y^_{i})

— n = sample size, I = indicator function, y_i = actual class,

y^_{i}

= predicted class

LOOCV closed-form estimate for least-squares linear regression

CV_{(n)} = \frac{1}{n} \sum_{i = 1}^{n} (\frac{y _{i} - y ^ _{i}}{1 - h _{i}})^{2}

— n = sample size, y_i = observed, ŷ_i = fitted, h_i = leverage of obs i

Logistic regression logit model

lo g [p / (1 - p)] = X β

— p = P(Y=1|X), X = predictor matrix, β = coefficient vector, log-odds linear in X

Linear regression model

Y = X β + ε

— Y = numeric response, X = predictor matrix, β = coefficient vector, ε = error term with mean 0

Expected prediction error (squared loss)

E [(Y - f^(X))^{2}]

— Y = true response,

f^(X)

= predicted response, E = expectation over (X, Y)

Linear Models 37 items

R-squared

R^{2} = 1 - \frac{S S _{res}}{S S _{tot}} = 1 - \frac{\sum ( y _{i} - y ^ _{i} ) ^{2}}{\sum ( y _{i} - y ˉ ) ^{2}}

Adjusted R-squared

R ˉ^{2} = 1 - \frac{( 1 - R ^{2} ) ( n - 1 )}{n - p - 1}

p

=number of predictors (excludes intercept)

F-statistic (regression)

F = \frac{( S S _{tot} - S S _{res} ) / p}{S S _{res} / ( n - p - 1 )}

Tests

H_{0} :

all slope coefficients are zero

Variance inflation factor (VIF)

VIFj​=1−Rj2​1​
Rj2​=R2 from regressing Xj​ on all other predictors
VIF>5–10 indicates multicollinearity

AIC (ordinary linear model)

AIC=n1​(RSS+2dσ^2)
d=model size; σ^2 from the full model, held fixed; lower is better
SRM scope: ordinary linear models only. The form (-2ln L + 2d) is out of scope.

BIC (ordinary linear model)

BIC=n1​(RSS+ln(n)dσ^2)
Same d and σ^2 as AIC. Penalizes complexity harder once n>7
SRM scope: ordinary linear models only. The form (-2ln L + dln n) is out of scope.

Ridge regression penalty

Minimize:

\sum (y_{i} - y^_{i})^{2} + λ \sum_{j = 1}^{p} β_{j}^{2}

L_{2}

penalty; shrinks but does not zero out coefficients

LASSO penalty

Minimize:

\sum (y_{i} - y^_{i})^{2} + λ \sum_{j = 1}^{p} ∣ β_{j} ∣

L_{1}

penalty; produces sparse solutions (exact zeros)

Elastic net penalty

Minimize:

\sum (y_{i} - y^_{i})^{2} + λ_{1} \sum ∣ β_{j} ∣ + λ_{2} \sum β_{j}^{2}

Combines LASSO (

L_{1}

) and Ridge (

L_{2}

)

Studentized residual

r_{i} = \frac{e _{i}}{σ ^ 1 - h _{ii}}

— e = raw residual, σ̂ = residual SE, h = leverage; |r| > 2 flags outlier

Durbin-Watson statistic

D W = \frac{\sum _{i = 2}^{n} ( e _{i} - e _{i - 1} ) ^{2}}{\sum _{i = 1}^{n} e _{i}^{2}} \approx 2 (1 - ρ^)

— e = residual, n = sample size, ρ̂ = lag-1 residual autocorrelation

Cook's distance

D_{i} = \frac{r _{i}^{2}}{p + 1} \cdot \frac{h _{ii}}{1 - h _{ii}}

— r = studentized residual, h = leverage, p = number of predictors; D > 1 is influential

High-leverage cutoff

h_{ii} > \frac{2 ( p + 1 )}{n}

— h = leverage (hat matrix diagonal), p = number of predictors, n = sample size

Tweedie variance function

V (μ) = ϕ μ^{p}

— φ = dispersion, μ = mean, p = power parameter (p=0 normal, p=1 Poisson, 1<p<2 compound Poisson-gamma, p=2 gamma)

GLM link function and linear predictor

g (μ) = η = X β

— g = link function, μ = mean response, η = linear predictor, X = design matrix, β = coefficient vector

GLM mean and variance from b(theta)

E [Y] = b^{'} (θ) = μ, Var (Y) = b^{''} (θ) a (ϕ)

— θ = natural parameter, φ = dispersion, μ = mean, b'' a(φ) = V(μ)·a(φ)

Exponential family density form

f (y; θ, ϕ) = exp {[y θ - b (θ)] / a (ϕ) + c (y, ϕ)}

— y = data, θ = natural parameter, φ = dispersion, a/b/c = family-specific functions

Exponential family density (GLM random component)

f (y; θ, ϕ) = exp {[y θ - b (θ)] / a (ϕ) + c (y, ϕ)}

— θ = canonical parameter, φ = dispersion, b(θ) sets mean and variance

OLS coefficient estimator (closed form)

β^= (X^{T} X)^{- 1} X^{T} Y

— X = design matrix of predictors, Y = response vector,

β^

= least-squares coefficient vector

GLM variance structure

Var (Y) = ϕ V (μ)

— φ = dispersion parameter, V(μ) = variance function (μ for Poisson, μ(1−μ) for binomial, μ² for gamma)

GLM deviance (goodness-of-fit)

D = 2 [ℓ (saturated) - ℓ (μ^)]

— ℓ = log-likelihood, saturated = perfect-fit model,

μ^

= fitted means; analog of RSS

Box-Cox transformation

Y^{(λ)} = (Y^{λ} - 1) / λ

— Y = positive response, λ = power parameter; λ=0 gives log transform, λ=1 gives no transform

Marginal effect of a predictor with an interaction term

\partial η / \partial X_{1} = β_{1} + β_{12} X_{2}

— β_1 = main effect of X_1, β_12 = interaction coefficient, X_2 = moderator value

t-statistic for a regression coefficient

t = β^_{j} / S E (β^_{j})

— β̂_j = estimated coefficient, SE = standard error, compared to t-distribution with n-p-1 degrees of freedom

Likelihood ratio test statistic

Λ = 2 (lo g L_{F} - lo g L_{R}) \sim χ_{p - q}^{2}

— L_F = full model log-likelihood, L_R = reduced model log-likelihood, p-q = parameters dropped

K-nearest neighbors classification probability

Pr (Y = j ∣ X = x_{0}) = \frac{1}{K} \sum_{x_{i} \in N_{K} (x_{0})} I (y_{i} = j)

— j = class, K = neighbors, I(·) = indicator function, N_K(x₀) = K nearest training points

Ridge regression effective degrees of freedom

df (λ) = t r [X (X^{T} X + λ I)^{- 1} X^{T}]

— X = design matrix, λ = ridge tuning parameter, I = identity matrix, tr = matrix trace

K-nearest neighbors regression prediction

f^(x_{0}) = \frac{1}{K} \sum_{x_{i} \in N_{K} (x_{0})} y_{i}

— x₀ = query point, K = neighborhood size, N_K(x₀) = K nearest training points, y_i = neighbor responses

Ridge regression closed-form estimator

β^^{r i d g e} = (X^{T} X + λ I)^{- 1} X^{T} y

— X = design matrix, y = response vector, λ = ridge tuning parameter, I = identity matrix

Small log-coefficient percent-change approximation

e^{β^_{j}} - 1 \approx β^_{j}

for

∣ β^_{j} ∣ < 0.2

— β_j = coefficient on log(Y); approximate percent change in Y per unit increase in X_j is 100β_j%

Effective slope with interaction term

slope_{X_{1}} = β^_{1} + β^_{3} X_{2}

— β_1 = main effect on X_1, β_3 = coefficient on X_1 X_2 interaction, X_2 = value of interacting variable

Log-log elasticity interpretation

%Δ Y \approx β^_{j} \cdot %Δ X_{j}

— β_j = coefficient when both X_j and Y are log-transformed; β_j is the elasticity of Y with respect to X_j

Log-response multiplicative effect of a predictor

E [Y ∣ X_{j} + 1] = E [Y ∣ X_{j}] \cdot e^{β^_{j}}

— β_j = coefficient in log(Y) model, X_j = predictor, e^{β_j} = multiplicative factor on Y

Confidence interval for the mean response in simple linear regression

y^_{0} \pm t_{n - 2, α /2} \cdot s \frac{1}{n} + \frac{( x _{0} - x ˉ ) ^{2}}{S _{xx}}

— s = residual SE, n = sample size, S_xx = Σ(x_i-x̄)²

Prediction standard error from confidence standard error

S E_{p r e d} = S E (y^_{0})^{2} + s^{2}

— SE(ŷ₀) = CI standard error of the fit, s = residual standard error

Prediction interval for a new observation in simple linear regression

y^_{0} \pm t_{n - 2, α /2} \cdot s 1 + \frac{1}{n} + \frac{( x _{0} - x ˉ ) ^{2}}{S _{xx}}

— extra +1 inside radical captures irreducible noise σ²

Variance of the fitted mean in multiple regression

Var (y^_{0}) = σ^{2} x_{0}^{⊤} (X^{⊤} X)^{- 1} x_{0}

— x_0 = predictor vector at new point, X = design matrix, σ² = error variance

Time Series Models 17 items

AR(1) model

X_{t} = ϕ X_{t - 1} + ε_{t}, ε_{t} \sim W N (0, σ^{2})

Stationary iff

∣ ϕ ∣ < 1

MA(1) model

X_{t} = ε_{t} + θ ε_{t - 1}, ε_{t} \sim W N (0, σ^{2})

Always stationary

ARMA(1,1) model

X_{t} = ϕ X_{t - 1} + ε_{t} + θ ε_{t - 1}

Stationary iff

∣ ϕ ∣ < 1

ACF of AR(1)

ρ (h) = ϕ^{h}, h = 0, 1, 2, \dots

Decays geometrically; PACF cuts off after lag 1

Ljung-Box test statistic

Q=n(n+2)∑k=1m​n−kρ^​k2​​
Tests H0​: first m autocorrelations are zero
Distributed χ2(m) under H0​

Random walk with drift h-step point forecast

Y^_{T + h} = Y_{T} + h δ

—

Y_{T}

= last observed value,

δ

= drift per period, h = forecast horizon

h-step prediction interval for a time series forecast

Y^_{T + h} \pm z_{α /2} Var (e_{T + h})

—

Y^_{T + h}

= point forecast,

z_{α /2}

= normal quantile (1.96 for 95%),

Var (e_{T + h})

= h-step forecast error variance

AR(1) h-step forecast error variance

Var (e_{T + h}) = σ^{2} \cdot \frac{1 - ϕ ^{2 h}}{1 - ϕ ^{2}}

—

σ^{2}

= white noise variance,

ϕ

= AR(1) coefficient (

∣ ϕ ∣ < 1

), h = forecast horizon

AR(1) long-run (unconditional) forecast variance ceiling

Var_{\infty} = \frac{σ ^{2}}{1 - ϕ ^{2}}

—

σ^{2}

= white noise variance,

ϕ

= AR(1) coefficient with

∣ ϕ ∣ < 1

Simple exponential smoothing one-step forecast

Y^_{t + 1} = α Y_{t} + (1 - α) Y^_{t}

— α ∈ (0,1) = smoothing constant, Y_t = current observation, Ŷ_t = previous smoothed forecast

ARCH(q) conditional variance

σ_{t}^{2} = ω + α_{1} ε_{t - 1}^{2} + \dots + α_{q} ε_{t - q}^{2}

— ω > 0 = constant, α_i ≥ 0 = ARCH weights, ε_{t-i} = past residuals

Long-run mean of a stationary AR(1) process

μ = \frac{c}{1 - ϕ _{1}}

— c = intercept, φ₁ = AR(1) coefficient with |φ₁| < 1 for stationarity

GARCH(1,1) conditional variance

σ_{t}^{2} = ω + α_{1} ε_{t - 1}^{2} + β_{1} σ_{t - 1}^{2}

— ω = constant, α₁ = ARCH term, β₁ = GARCH lagged-variance term, ε_{t-1} = prior residual

Random walk variance

Var (Y_{t}) = t σ^{2}

where

Y_{t} = Y_{0} + \sum_{i = 1}^{t} ε_{i}

— Y_0 = starting value, σ² = white noise variance, t = time index; variance grows linearly so series is non-stationary

White noise 95 percent significance band for sample ACF

ρ^_{k} \in \pm \frac{1.96}{T}

— T = sample size; sample autocorrelations inside this band are consistent with zero at the 5% level

Autocorrelation function at lag k

ρ_{k} = \frac{γ _{k}}{γ _{0}} = \frac{Cov ( Y _{t} , Y _{t - k} )}{Var ( Y _{t} )}

— γ_k = autocovariance at lag k, γ_0 = variance, ρ_0 = 1, |ρ_k| ≤ 1

Sample autocorrelation at lag k

ρ^_{k} = \frac{\sum _{t = k + 1}^{T} ( Y _{t} - Y ˉ ) ( Y _{t - k} - Y ˉ )}{\sum _{t = 1}^{T} ( Y _{t} - Y ˉ ) ^{2}}

— Ȳ = sample mean, T = sample size, k = lag

Decision Trees 14 items

Gini impurity

G = \sum_{k = 1}^{K} p^_{k} (1 - p^_{k}) = 1 - \sum_{k = 1}^{K} p^_{k}^{2}

p^_{k}

=fraction of class

k

in node

Entropy (node impurity)

H = - \sum_{k = 1}^{K} p^_{k} lo g p^_{k}

Bagging (bootstrap aggregation)

Train B trees on bootstrap samples; aggregate predictions
f^​(x)=B1​∑b=1B​f^​b​(x)
Reduces variance without increasing bias

Out-of-bag mean squared error

OOB MSE = \frac{1}{n} \sum_{i = 1}^{n} (y_{i} - y^_{i}^{OOB})^{2}

—

y_{i}

= actual response,

y^_{i}^{OOB}

= average of trees not trained on i

Out-of-bag observation fraction

(1 - 1/ n)^{n} \to e^{- 1} \approx 0.368

— n = training rows, fraction of rows omitted from any given bootstrap sample

Boosting ensemble update rule

f^(x) \leftarrow f^(x) + λ f^^{b} (x)

—

f^

= current ensemble,

f^^{b}

= new tree fit to residuals,

λ

= shrinkage/learning rate

Random forest default predictors per split

m = ⌊ p ⌋

(classification);

m = ⌊ p /3 ⌋

(regression) — m = predictors sampled at each split, p = total predictors

Piecewise-constant decision tree prediction function

Y^= \sum_{m = 1}^{M} c_{m} \cdot 1 {X \in R_{m}}

— M = number of leaves, R_m = m-th rectangular region, c_m = constant prediction in leaf m, 1{·} = indicator function

Number of unique binary splits for an unordered categorical predictor

N = 2^{q - 1} - 1

— q = number of unordered levels of the categorical predictor

Cost-complexity criterion for a classification tree

C_{α} (T) = \sum_{m = 1}^{∣ T ∣} \sum_{i \in R_{m}} L (y_{i}, y^_{R_{m}}) + α ∣ T ∣

— L = Gini or entropy impurity, |T| = number of terminal nodes, α ≥ 0 = complexity penalty

Recursive binary splitting objective for a regression tree

min_{j, s} [\sum_{i \in R_{1} (j, s)} (y_{i} - y ˉ_{R_{1}})^{2} + \sum_{i \in R_{2} (j, s)} (y_{i} - y ˉ_{R_{2}})^{2}]

— j = predictor, s = cutpoint, R_1,R_2 = half-planes

Cost-complexity pruning criterion for a regression tree

C_{α} (T) = \sum_{m = 1}^{∣ T ∣} \sum_{i : x_{i} \in R_{m}} (y_{i} - y^_{R_{m}})^{2} + α ∣ T ∣

— |T| = number of leaves, α = complexity penalty, R_m = leaf region

Classification tree leaf prediction

C^_{R_{j}} = ar g max_{k} p^_{R_{j}, k}

— R_j = leaf region, k = class index, p̂_{R_j,k} = training proportion of class k in leaf

Regression tree leaf prediction

y^_{R_{j}} = \frac{1}{∣ R _{j} ∣} \sum_{i : x_{i} \in R_{j}} y_{i}

— R_j = leaf region, |R_j| = number of training observations in leaf, y_i = response

Unsupervised Learning Techniques 16 items

PCA — proportion of variance explained

PVE_{k} = \frac{λ _{k}}{\sum _{j = 1}^{p} λ _{j}}

λ_{k}

k

th eigenvalue of covariance (or correlation) matrix

K-means objective function

min_{C_{1}, \dots, C_{K}} \sum_{k = 1}^{K} \sum_{i \in C_{k}} ∥ x_{i} - x ˉ_{k} ∥^{2}

x ˉ_{k}

=centroid of cluster

k

Average linkage distance between clusters

d (A, B) = \frac{1}{∣ A ∣∣ B ∣} \sum_{a \in A} \sum_{b \in B} ∥ a - b ∥

— A, B = clusters, |A|, |B| = cluster sizes, ||·|| = Euclidean distance

Complete linkage distance between clusters

d (A, B) = max_{a \in A, b \in B} ∥ a - b ∥

— A, B = clusters, a, b = observations in each cluster, ||·|| = Euclidean distance

Single linkage distance between clusters

d (A, B) = min_{a \in A, b \in B} ∥ a - b ∥

— A, B = clusters, a, b = observations in each cluster, ||·|| = Euclidean distance

K-means centroid update

μ_{k} = \frac{1}{∣ C _{k} ∣} \sum_{i \in C_{k}} x_{i}

— μ_k = centroid of cluster k, C_k = set of points in cluster k, |C_k| = cluster size, x_i = observation

First principal component as a linear combination

Z_{1} = ϕ_{11} X_{1} + ϕ_{21} X_{2} + \dots + ϕ_{p 1} X_{p}, \sum_{j} ϕ_{j 1}^{2} = 1

— Z₁ = PC1 score, φⱼ₁ = loading of Xⱼ on PC1, Xⱼ = centered predictor

Cumulative proportion of variance explained through component M

Cum PVE_{M} = \sum_{m = 1}^{M} λ_{m} / \sum_{j = 1}^{p} λ_{j}

— λₘ = m-th eigenvalue, M = components retained, p = total predictors

PCA loading vector as eigenvector of the covariance matrix

S ϕ_{m} = λ_{m} ϕ_{m}

— S = p×p sample covariance matrix, φₘ = unit eigenvector (loading vector for PC m), λₘ = eigenvalue = Var(Zₘ)

Principal component score for observation i on component m

z_{im} = ϕ_{1 m} x_{i 1} + ϕ_{2 m} x_{i 2} + \dots + ϕ_{p m} x_{i p}

— zᵢₘ = score, φⱼₘ = loading of Xⱼ on PC m, xᵢⱼ = centered value of Xⱼ for obs i

Total within-cluster sum of squares (pairwise form)

W (k) = \sum_{c = 1}^{k} \frac{1}{2∣ C _{c} ∣} \sum_{i, j \in C_{c}} ∥ x_{i} - x_{j} ∥^{2}

— k = number of clusters, C_c = cluster c, |C_c| = its size, x_i = observation vectors

Silhouette coefficient for an observation

s (i) = \frac{b ( i ) - a ( i )}{m a x { a ( i ) , b ( i )}}

— a(i) = mean within-cluster distance, b(i) = mean distance to nearest other cluster; s(i) in [-1, 1]

One-standard-error rule for the gap statistic

Gap (k) \geq Gap (k + 1) - s_{k + 1}

— choose smallest k satisfying this; s_{k+1} = standard error of Gap(k+1) across B reference samples

Gap statistic for choosing number of clusters

Gap (k) = E^{*} [lo g W^{*} (k)] - lo g W (k)

— W(k) = observed within-cluster SS, W*(k) = SS on uniform reference data, E* = expectation over B reference samples

Principal component score

Z_{k} = ϕ_{1 k} X_{1} + ϕ_{2 k} X_{2} + \dots + ϕ_{p k} X_{p}

— Z_k = kth component, φ_jk = loading of variable j on PC k, X_j = standardized predictor

Loading vector unit-length constraint

\sum_{j = 1}^{p} ϕ_{j k}^{2} = 1

— φ_jk = loading of variable j on PC k, p = number of predictors

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