Free SOA Exam FM (Financial Mathematics) Practice Questions

The **term** of an annuity is the duration of time from when payments begin to when they end, typically measured by the number of payment periods.

(A) describes the interest rate, not the term. (B) describes the total undiscounted payments. (D) is the PV, not the term. (E) is the FV, not the term.

The answer is (C).

Statement (D) is NOT correct. For standard fixed-coupon bonds with fixed cash flows, convexity is always **positive**. This is because the second derivative of the price with respect to yield is positive:
$$C = \frac{1}{P} \frac{d^2P}{dy^2} > 0$$

Positive convexity means the price-yield curve is convex (bows upward), which benefits the bondholder: prices rise more than duration predicts when yields fall, and fall less when yields rise.

Negative convexity occurs for callable bonds or mortgage-backed securities where cash flows change with interest rates.

(B) Correct -- convexity measures the second-order (curvature) effect.
(C) Correct -- with positive convexity, the duration approximation always understates the true price for parallel yield shifts in either direction.
(A) Correct -- the second-order term $\frac{1}{2}C \cdot P \cdot (\Delta y)^2$ improves accuracy.
(E) Correct -- zero-coupon bonds concentrate all cash flow at maturity, producing higher convexity for the same duration.

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

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

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

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

Choice C is correct.
Choice A is incorrect because it uses an incorrect approximation for the exponential.
Choice B is incorrect because it confuses $d^{(6)}$ with the nominal interest $i^{(6)}$.
Choice D is incorrect because it fails to convert from nominal interest to nominal discount.
Choice E is incorrect because it uses $i^{(6)}$ directly without the discount conversion.