Exam FM · practice
Exam FM — Interest-Rate Swaps Practice Flashcards
Thirty original SOA/CAS Exam FM-style multiple-choice problems on interest-rate swaps, spanning swap mechanics, deriving swap rates from spot rates and discount factors, forward-rate weighting, deferred swaps, post-inception market value, net interest payments, and amortizing/accreting notionals, each with a fully worked solution.
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- mechanicsWhich of the following statements about a standard ("plain vanilla") interest-rate swap is **TRUE**? (A) The notional amount is exchanged between the two parties at maturity. (B) The fixed-rate payer benefits when market interest rates subsequently fall. (C) The swap rate is chosen so the present values of the fixed and floating legs are equal at inception. (D) At inception the swap has positive value to the floating-rate payer. (E) The floating-rate payments are fixed and known at inception.**Answer: (C).** The swap rate $R$ is set so that $\text{PV}(\text{fixed leg})=\text{PV}(\text{floating leg})$, giving the swap **zero value to both parties at inception** — neither pays the other to enter. That makes (C) correct and (D) false. (A) is false: the notional is never exchanged; it only scales the interest cash flows. (B) is false: the fixed-rate *payer* has locked in a rate and benefits when rates **rise** (the floating receipts grow). (E) is false: the floating leg **resets** each period to the prevailing reference rate, so those payments are uncertain at inception.
- swap-rateLet $P_t$ be the price today of a zero-coupon bond paying $\$1$ at time $t$. The discount factors are $P_1=0.9615$, $P_2=0.9246$, $P_3=0.8890$. Calculate the 3-year level swap rate. (A) $3.70\%$ (B) $3.85\%$ (C) $4.00\%$ (D) $4.16\%$ (E) $11.10\%$**Answer: (C).** The level $n$-period swap rate is $R=\dfrac{1-P_n}{\sum_{t=1}^{n}P_t}$. Numerator: $1-P_3=1-0.8890=0.1110$. Denominator: $\sum P_t=0.9615+0.9246+0.8890=2.7751$. $R=\dfrac{0.1110}{2.7751}=0.040000\approx 4.00\%.$ (E) $11.10\%$ is the trap of dividing $1-P_3$ by $P_3$ alone instead of by the full annuity $\sum P_t$; (D) divides by $P_2+P_3$ only.
- swap-rateAnnual effective spot rates are $s_1=2.5\%$, $s_2=3.0\%$, $s_3=3.5\%$. Calculate the 3-year level swap rate. (A) $3.00\%$ (B) $3.30\%$ (C) $3.47\%$ (D) $3.50\%$ (E) $4.01\%$**Answer: (C).** Discount factors: $P_1=1.025^{-1}=0.975610$, $P_2=1.03^{-2}=0.942596$, $P_3=1.035^{-3}=0.901943$. $\sum P_t=0.975610+0.942596+0.901943=2.820149$. $R=\dfrac{1-P_3}{\sum P_t}=\dfrac{1-0.901943}{2.820149}=\dfrac{0.098057}{2.820149}\approx 0.034770.$ So $R\approx 3.47\%$. (D) $3.50\%$ is the trap of just reading off $s_3$; the swap rate is a discount-weighted blend of the forwards and lands slightly below $s_3$ on this upward curve. (B) is the simple average of the spots.
- forward-ratesAnnual effective spot rates are $s_1=4\%$ and $s_2=5\%$. Calculate the implied one-year forward rate $f_2$ (the rate applying to the second year). (A) $4.50\%$ (B) $5.00\%$ (C) $5.50\%$ (D) $6.01\%$ (E) $9.00\%$**Answer: (D).** The no-arbitrage relation is $(1+s_2)^2=(1+s_1)(1+f_2)$, so $f_2=\dfrac{(1+s_2)^2}{1+s_1}-1=\dfrac{1.05^2}{1.04}-1=\dfrac{1.1025}{1.04}-1=1.060096-1=0.060096.$ So $f_2\approx 6.01\%$. (A) is the simple average of $s_1,s_2$; (B) just reads off $s_2$; (E) is the crude error $2s_2-s_1$. On an upward-sloping curve the forward exceeds $s_2$.
- net-paymentsOn a swap with notional $\$2{,}000{,}000$ and fixed swap rate $4.25\%$, the floating reference rate for the upcoming period resets at $4.80\%$. Calculate the net payment for this period and identify its direction, from the perspective of the fixed-rate payer. (A) Fixed payer pays $\$11{,}000$ (B) Fixed payer receives $\$11{,}000$ (C) Fixed payer pays $\$85{,}000$ (D) Fixed payer receives $\$96{,}000$ (E) Fixed payer receives $\$181{,}000$**Answer: (B).** Net to the fixed payer $=N(r_t-R)$. $=2{,}000{,}000\,(0.0480-0.0425)=2{,}000{,}000\,(0.0055)=\$11{,}000.$ The net is **positive**, so the fixed payer **receives** $\$11{,}000$ (the floating reset exceeded the locked fixed rate). (A) flips the sign. (C) uses $N\cdot R$ alone, (D) uses $N\cdot r_t$ alone, and (E) adds the two gross legs instead of netting — all common errors from forgetting that only the net difference changes hands.
- market-valueA company enters a 3-year swap to pay fixed at $R=3.80\%$ on a notional of $\$5{,}000{,}000$. The current discount factors are $P_1=0.970874$, $P_2=0.924556$, $P_3=0.863838$. Calculate the market value of the swap to this fixed-rate payer. (A) $-\$156{,}551$ (B) $-\$78{,}000$ (C) $\$0$ (D) $\$78{,}000$ (E) $\$156{,}551$**Answer: (E).** Value to the fixed payer $=N\big[(1-P_3)-R\sum P_t\big]$. $\sum P_t=0.970874+0.924556+0.863838=2.759268$. PV(floating received) $=N(1-P_3)=5{,}000{,}000(1-0.863838)=5{,}000{,}000(0.136162)=\$680{,}810$. PV(fixed paid) $=N\cdot R\cdot\sum P_t=5{,}000{,}000(0.038)(2.759268)=\$524{,}261$. $V_{\text{payer}}=680{,}810-524{,}261=\$156{,}549\approx\$156{,}551$ (rounding of the discount factors). The current par rate is $R_{\text{mkt}}=\frac{0.136162}{2.759268}=4.935\%>3.80\%$, so the fixed payer locked in a cheap rate and the swap is an **asset** ($V>0$). (A) flips the sign; (C) wrongly assumes the swap is still at par.
- deferred-swapA deferred swap is agreed today but its exchanges occur only at the ends of years $2$, $3$, and $4$ (no exchange in year $1$). The discount factors are $P_1=0.9709$, $P_2=0.9426$, $P_3=0.9151$, $P_4=0.8885$. Calculate the deferred swap rate. (A) $2.88\%$ (B) $3.00\%$ (C) $3.18\%$ (D) $3.85\%$ (E) $4.00\%$**Answer: (B).** For a swap deferred past year $a$ with exchanges over $t=a+1,\dots,b$, the rate is $R=\dfrac{P_a-P_b}{\sum_{t=a+1}^{b}P_t}$. Here $a=1$, $b=4$. Numerator: $P_1-P_4=0.9709-0.8885=0.0824$. Denominator: $P_2+P_3+P_4=0.9426+0.9151+0.8885=2.7462$. $R=\dfrac{0.0824}{2.7462}=0.030005\approx 3.00\%$. (E) $4.00\%$ uses $1-P_4$ in the numerator (the non-deferred floating PV), and (A) keeps $P_1$ in the denominator — both classic deferred-swap traps.
- forward-ratesThe swap rate equals a weighted average of the implied forward rates. Which weighting is correct for a level-notional swap? (A) Equal weights: $R=\frac{1}{n}\sum_{t=1}^{n} f_t$. (B) Weighted by the notional only: $R=\frac{\sum Q_t f_t}{\sum Q_t}$. (C) Weighted by the forward rates themselves. (D) Weighted by the discount factors: $R=\frac{\sum_{t=1}^{n} f_t P_t}{\sum_{t=1}^{n} P_t}$. (E) Weighted by time to maturity: $R=\frac{\sum t\, f_t}{\sum t}$.**Answer: (D).** Setting $\text{PV(fixed)}=\text{PV(floating)}$ gives $R\sum P_t=\sum f_t P_t$, hence $R=\dfrac{\sum_{t=1}^{n} f_t P_t}{\sum_{t=1}^{n} P_t}.$ Each forward is weighted by its own discount factor $P_t$. (A) the simple average is the single most common error — it overweights distant, more heavily discounted periods and overstates $R$ on an upward curve. (B) applies only to a varying-notional swap and still needs the $P_t$ factor: $R=\frac{\sum Q_t f_t P_t}{\sum Q_t P_t}$.