Free SOA Exam FM (Financial Mathematics) General Cash Flows, Portfolios, and Asset-Liability Management Practice Questions

A spot rate $s_t$ is the annual effective yield on a zero-coupon investment from time 0 to time $t$. It represents the rate at which a single cash flow at time $t$ is discounted to the present.

Choice (B) describes a holding-period return, not a spot rate. Choice (C) is partially correct in that $f_{0,1} = s_1$, but this is not the general definition of a spot rate. Choice (D) is not the definition of a spot rate (though there is a relationship between spot and forward rates). Choice (E) describes the par yield, not the spot rate.

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.

Cash flows: 80 at times 1 through 4, and 1080 at time 5. Yield $i = 6\%$.

Compute present values:
$PV_1 = 80/1.06 = 75.472$
$PV_2 = 80/1.06^2 = 71.200$
$PV_3 = 80/1.06^3 = 67.170$
$PV_4 = 80/1.06^4 = 63.368$
$PV_5 = 1080/1.06^5 = 807.072$

Price $P = 1084.28$.

Convexity formula: $C = \frac{\sum t(t+1) \cdot PV_t}{(1+i)^2 \times P}$.

Compute $\sum t(t+1) \cdot PV_t$:
$1(2)(75.472) = 150.94$
$2(3)(71.200) = 427.20$
$3(4)(67.170) = 806.04$
$4(5)(63.368) = 1267.36$
$5(6)(807.072) = 24212.16$

Sum $= 26863.70$.

$C = \frac{26863.70}{(1.06)^2 \times 1084.28} = \frac{26863.70}{1.1236 \times 1084.28} = \frac{26863.70}{1218.30} = 22.05$.

Choice D (24.78) results from omitting the $(1+i)^2$ factor in the denominator.

Choice A (18.19) results from using $t^2$ instead of $t(t+1)$ in the numerator.