Free SOA Exam FM (Financial Mathematics) Time Value of Money Practice Questions

Time value of money is the foundational topic on SOA Exam FM, covering interest rate conversions, present and future value calculations, discount factors, and force of interest. The SOA syllabus allocates 10-15% of Exam FM to interest theory fundamentals (SOA).

145 Questions
62 Easy
56 Medium
27 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
Calculate the present value of $50,000 due in 12 years at an effective annual interest rate of 3%.
Solution
B is correct.

The present value is:

PV=FV⋅vn=FV(1+i)n=50,000(1.03)12PV = FV \cdot v^n = \frac{FV}{(1+i)^n} = \frac{50{,}000}{(1.03)^{12}}

Compute (1.03)12(1.03)^{12} by repeated squaring:

(1.03)2=1.0609(1.03)^2 = 1.0609

(1.03)4=(1.0609)2=1.12550881(1.03)^4 = (1.0609)^2 = 1.12550881

(1.03)8=(1.12550881)2=1.26677008(1.03)^8 = (1.12550881)^2 = 1.26677008

(1.03)12=(1.03)8×(1.03)4=1.26677008×1.12550881=1.42576089(1.03)^{12} = (1.03)^8 \times (1.03)^4 = 1.26677008 \times 1.12550881 = 1.42576089

PV=50,0001.42576089=35,069PV = \frac{50{,}000}{1.42576089} = 35{,}069

The present value is $35,069.
Question 2 Medium
A nominal interest rate of 6% compounded semiannually is given. Determine the equivalent force of interest δ\delta.
Solution
A is correct.

Step 1: Find the effective annual interest rate.

1+i=(1+0.062)2=(1.03)2=1.06091 + i = \left(1 + \frac{0.06}{2}\right)^2 = (1.03)^2 = 1.0609

Step 2: Find the force of interest.

δ=ln⁡(1+i)=ln⁡(1.0609)\delta = \ln(1 + i) = \ln(1.0609)

Alternatively: δ=2ln⁡(1.03)=2×0.029559=0.059118\delta = 2\ln(1.03) = 2 \times 0.029559 = 0.059118.

So δ≈0.05912\delta \approx 0.05912.
Question 3 Hard
Given i(4)=10%i^{(4)} = 10\%, calculate d(6)d^{(6)}.
Solution

Choice C is correct.

Convert i(4)i^{(4)} to the force of interest δ\delta.

δ=4ln⁡(1+i(4)4)=4ln⁡(1.025)=4×0.024693=0.098771\delta = 4 \ln\left(1 + \frac{i^{(4)}}{4}\right) = 4 \ln(1.025) = 4 \times 0.024693 = 0.098771

Convert δ\delta to d(6)d^{(6)}:
d(6)=6(1−e−δ/6)=6(1−e−0.016462)d^{(6)} = 6\left(1 - e^{-\delta/6}\right) = 6\left(1 - e^{-0.016462}\right)

e−0.016462=0.98367e^{-0.016462} = 0.98367
d(6)=6×0.01633=0.09796≈9.80%d^{(6)} = 6 \times 0.01633 = 0.09796 \approx 9.80\%
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