Free SOA Exam P Insurance Applications Practice Questions (2026)

Sample Questions

Question 1 Easy

Losses are lognormal with

ln (X) \sim N (7, 4)

. Calculate

E [X]

Solution

B is correct.

E [X] = e^{μ + σ^{2} /2} = e^{7 + 2} = e^{9}

Question 2 Medium

N ~ Poisson(10), each claim Cᵢ iid Exp(mean 500), independent of N. S = ΣCᵢ. Calculate Var(S).

Solution

B is correct.

Compound Poisson:

Var (S) = λ \cdot E [C^{2}]

E [C^{2}] = Var (C) + (E [C])^{2} = 250, 000 + 250, 000 = 500, 000

Var (S) = 10 (500, 000) = 5, 000, 000

Question 3 Hard

X ~ Uniform(0, 500). Policy: deductible 100, max payment 300. Calculate Var(payment).

Solution

A is correct.

Y = min(max(X−100,0), 300). Cap at X=400.

E [Y]

= (

\frac{( 1 )}{500}

)(

1^{2}

/2) +

\frac{( 300 )}{5}

= 90 + 60 = 150.

E [Y^{2}]

= (

\frac{( 1 )}{500}

)(

1^{3}

/3) +

1^{2}

/5 = 18000 + 18000 = 36000. Var = 36000 − 22500 = 13500.

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