{
  "deckName": "Exam FM — Interest-Rate Swaps",
  "examCode": "Exam FM",
  "cards": [
    {
      "front": "In plain terms, what is an **interest-rate swap**?",
      "back": "An interest-rate swap is an agreement between two parties to exchange streams of interest payments on a stated **notional amount** over a set term. Typically one party pays a **fixed** rate and receives a **floating** rate; the other does the reverse. The notional itself is never exchanged — it only scales the interest cash flows.",
      "tag": "mechanics"
    },
    {
      "front": "What is the **notional amount** in a swap, and is it actually exchanged between the parties?",
      "back": "The notional amount is the principal figure used only to **compute** each interest payment. It is **not** exchanged at any point — neither at inception nor at maturity. Only the periodic interest amounts (and in practice just the *net* difference) change hands.",
      "tag": "mechanics"
    },
    {
      "front": "Identify the **fixed leg** and the **floating leg** of a swap, and who bears interest-rate risk on each.",
      "back": "The **fixed leg** pays a constant rate (the swap rate $R$) each period on the notional, so its cash flows are known at inception. The **floating leg** pays a market reference rate that **resets each period**, so its future payments are uncertain. The fixed-rate *payer* benefits if rates rise; the fixed-rate *receiver* benefits if rates fall.",
      "tag": "mechanics"
    },
    {
      "front": "What does it mean to say a swap has **zero value at inception**, and why is that the case?",
      "back": "At inception the present values of the two legs are equal, so the net market value is $0$ to both parties — neither pays the other to enter. The fixed rate $R$ (the **swap rate**) is chosen precisely so that $\\text{PV}(\\text{fixed leg})=\\text{PV}(\\text{floating leg})$, making the deal fair at the start.",
      "tag": "mechanics"
    },
    {
      "front": "Let $P_t$ denote the price today of a zero-coupon bond paying $1$ at time $t$ (the discount factor implied by the spot curve). State the formula for the level $n$-period **swap rate** $R$.",
      "back": "$R=\\dfrac{1-P_{n}}{\\sum_{t=1}^{n}P_{t}}$.\n\nThe numerator $1-P_n$ is the present value of the floating leg per unit notional; the denominator is the PV of an annuity of $1$ per period. Setting fixed-leg PV equal to floating-leg PV and solving gives this $R$.",
      "tag": "swap-rate"
    },
    {
      "front": "Why does the floating leg of a standard swap have present value $1-P_{n}$ per unit notional (where $P_n$ is the time-$n$ discount factor)?",
      "back": "A floating-rate note that pays the per-period market rate and returns principal at time $n$ is always worth its face value $1$ today. Stripping off the final principal payment, worth $P_n$ today, leaves just the floating **interest** stream, worth $1-P_n$. That is exactly the floating leg of the swap.",
      "tag": "swap-rate"
    },
    {
      "front": "Spot rates are $s_1=3\\%$, $s_2=3.5\\%$, $s_3=4\\%$ (annual effective). Find the **3-year swap rate**.",
      "back": "Discount factors: $P_1=1.03^{-1}=0.970874$, $P_2=1.035^{-2}=0.933511$, $P_3=1.04^{-3}=0.888996$.\n\nSum $\\sum P_t=2.793381$.\n\nSwap rate $R=\\dfrac{1-P_3}{\\sum P_t}=\\dfrac{1-0.888996}{2.793381}=\\dfrac{0.111004}{2.793381}\\approx0.039738$.\n\nSo $R\\approx 3.97\\%$.",
      "tag": "swap-rate"
    },
    {
      "front": "How is the swap rate $R$ related to the **forward rates** $f_t$ embedded in the spot curve?",
      "back": "$R$ is a **discount-factor-weighted average** of the one-period forward rates: $R=\\dfrac{\\sum_{t=1}^{n}f_{t}\\,P_{t}}{\\sum_{t=1}^{n}P_{t}}$, where $f_t$ is the forward rate for period $t$ and $P_t$ the time-$t$ discount factor. It is **not** a simple arithmetic average of the $f_t$ — weighting by $P_t$ is essential.",
      "tag": "forward-rates"
    },
    {
      "front": "Define the one-period forward rate $f_t$ in terms of spot rates, then compute $f_2$ given $s_1=3\\%$ and $s_2=3.5\\%$.",
      "back": "The no-arbitrage forward rate satisfies $(1+s_t)^{t}=(1+s_{t-1})^{t-1}(1+f_t)$, so $f_t=\\dfrac{(1+s_t)^{t}}{(1+s_{t-1})^{t-1}}-1$.\n\nHere $f_2=\\dfrac{(1.035)^{2}}{1.03}-1=\\dfrac{1.071225}{1.03}-1\\approx0.040024$, i.e. about $4.00\\%$.",
      "tag": "forward-rates"
    },
    {
      "front": "Using forward rates $f_1=3.00\\%$, $f_2=4.00\\%$, $f_3=5.01\\%$ with discount factors $P_1=0.970874$, $P_2=0.933511$, $P_3=0.888996$, confirm the 3-year swap rate via the forward-weighting formula.",
      "back": "Numerator $\\sum f_t P_t = (0.03)(0.970874)+(0.040024)(0.933511)+(0.050073)(0.888996)$\n$=0.029126+0.037363+0.044515=0.111004$.\n\nDenominator $\\sum P_t=2.793381$.\n\n$R=\\dfrac{0.111004}{2.793381}\\approx0.039738\\approx3.97\\%$ — matching the $\\frac{1-P_n}{\\sum P_t}$ result, as it must.",
      "tag": "forward-rates"
    },
    {
      "front": "What is a **deferred swap**, and how does deferral change the swap-rate formula?",
      "back": "A deferred swap is agreed today but its exchanges begin in a **future** period rather than the next one. If exchanges run over periods $a+1,\\dots,b$, the swap rate is $R=\\dfrac{P_{a}-P_{b}}{\\sum_{t=a+1}^{b}P_{t}}$. The numerator is the floating-leg PV over only that window; the denominator sums discount factors only over the active periods.",
      "tag": "deferred-swap"
    },
    {
      "front": "Discount factors are $P_1=0.9709$, $P_2=0.9426$, $P_3=0.9151$, $P_4=0.8885$. Find the rate on a swap **deferred one year** that exchanges at the ends of years $2$, $3$, and $4$.",
      "back": "Active periods are $t=2,3,4$, deferred past year $1$ (so $a=1$, $b=4$).\n\n$R=\\dfrac{P_1-P_4}{P_2+P_3+P_4}=\\dfrac{0.9709-0.8885}{0.9426+0.9151+0.8885}=\\dfrac{0.0824}{2.7462}\\approx0.030005$.\n\nSo the deferred swap rate is about $3.00\\%$.",
      "tag": "deferred-swap"
    },
    {
      "front": "After inception, how do you compute the **market value of an existing swap** to the fixed-rate payer once rates have moved?",
      "back": "Value the two legs on the **current** curve and net them: $V_{\\text{payer}}=\\text{PV}(\\text{floating received})-\\text{PV}(\\text{fixed paid})$. Using current discount factors $P_t$, this equals $V_{\\text{payer}}=N\\big[(1-P_n)-R\\sum_{t}P_t\\big]$, where $N$ is notional and $R$ the original contracted fixed rate.",
      "tag": "market-value"
    },
    {
      "front": "Show that the market value of a swap to the fixed payer can be written as $V=N\\,(R_{\\text{mkt}}-R)\\sum_t P_t$, where $R_{\\text{mkt}}$ is the current par swap rate.",
      "back": "Start from $V_{\\text{payer}}=N[(1-P_n)-R\\sum P_t]$. Since the current par rate satisfies $R_{\\text{mkt}}=\\dfrac{1-P_n}{\\sum P_t}$, we have $1-P_n=R_{\\text{mkt}}\\sum P_t$. Substituting: $V_{\\text{payer}}=N\\,[R_{\\text{mkt}}\\sum P_t-R\\sum P_t]=N\\,(R_{\\text{mkt}}-R)\\sum_t P_t$. The fixed payer gains when current rates exceed the rate they locked in.",
      "tag": "market-value"
    },
    {
      "front": "A party pays fixed at $R=4.5\\%$ on a 3-year swap, notional $\\$1{,}000{,}000$. Rates fall and current discount factors are $P_1=0.97$, $P_2=0.94$, $P_3=0.90$. Find the swap's market value to this fixed payer.",
      "back": "$\\sum P_t=0.97+0.94+0.90=2.81$.\n\nPV(floating received) $=N(1-P_3)=1{,}000{,}000(1-0.90)=\\$100{,}000$.\n\nPV(fixed paid) $=N\\cdot R\\cdot\\sum P_t=1{,}000{,}000(0.045)(2.81)=\\$126{,}450$.\n\n$V_{\\text{payer}}=100{,}000-126{,}450=-\\$26{,}450$.\n\nRates fell, so the fixed payer (locked at $4.5\\%$) is **out of the money** by about $\\$26{,}450$.",
      "tag": "market-value"
    },
    {
      "front": "In the previous example, verify the result using the current par swap rate $R_{\\text{mkt}}$ and the formula $V=N\\,(R_{\\text{mkt}}-R)\\sum P_t$.",
      "back": "Current par rate $R_{\\text{mkt}}=\\dfrac{1-P_3}{\\sum P_t}=\\dfrac{0.10}{2.81}=0.035587$ (about $3.56\\%$).\n\n$V_{\\text{payer}}=N(R_{\\text{mkt}}-R)\\sum P_t=1{,}000{,}000\\,(0.035587-0.045)(2.81)$\n$=1{,}000{,}000(-0.009413)(2.81)\\approx-\\$26{,}450$ — the same answer.",
      "tag": "market-value"
    },
    {
      "front": "How are **net interest payments** on a swap determined each period, and what changes hands?",
      "back": "Each period the fixed payer owes $N\\cdot R$ and the floating payer owes $N\\cdot r_t$ (the reset reference rate for that period). Only the **net** difference is exchanged. Net to the fixed payer $=N(r_t-R)$: positive (a receipt) when the floating reset exceeds the fixed rate, negative (a payment) when it falls short.",
      "tag": "net-payments"
    },
    {
      "front": "On a swap with notional $\\$500{,}000$ and fixed rate $4\\%$, the floating reference for the upcoming period resets at $3.6\\%$. Find the net payment and state who pays whom.",
      "back": "Fixed obligation $=500{,}000(0.04)=\\$20{,}000$.\n\nFloating obligation $=500{,}000(0.036)=\\$18{,}000$.\n\nNet $=N(r_t-R)=500{,}000(0.036-0.04)=-\\$2{,}000$.\n\nThe net is negative to the fixed payer, so the **fixed-rate payer pays $\\$2{,}000$** to the counterparty this period.",
      "tag": "net-payments"
    },
    {
      "front": "Why do the **net** swap payments generally differ from period to period, even though the fixed leg is constant?",
      "back": "The fixed payment $N\\cdot R$ is the same each period, but the floating payment $N\\cdot r_t$ changes as the reference rate **resets** to the prevailing one-period rate (well-approximated by the implied forward rates). Since the net is $N(r_t-R)$, it varies as $r_t$ moves above or below the fixed swap rate.",
      "tag": "net-payments"
    },
    {
      "front": "How are the expected (implied) **floating payments** of a swap projected for valuation purposes?",
      "back": "Each future floating reset is projected using the **implied one-period forward rate** $f_t$ from today's spot curve. The expected floating payment for period $t$ is $N\\cdot f_t$. Discounting these at the corresponding $P_t$ reproduces the floating-leg PV of $N(1-P_n)$ — consistent with no-arbitrage pricing.",
      "tag": "forward-rates"
    },
    {
      "front": "What is an **amortizing swap** versus an **accreting swap**?",
      "back": "Both have a notional that **changes** over the term. In an **amortizing** swap the notional $Q_t$ **decreases** period to period (mirroring a paydown schedule, e.g. a swap hedging an amortizing loan). In an **accreting** swap the notional **increases** over time (e.g. matching a drawdown schedule). A swap whose notional varies arbitrarily is a *varying-notional* swap.",
      "tag": "varying-notional"
    },
    {
      "front": "For a swap with a period-$t$ notional $Q_t$, state the **varying-notional swap rate** formula in terms of forward rates $f_t$ and discount factors $P_t$.",
      "back": "$R=\\dfrac{\\sum_{t=1}^{n}Q_{t}\\,f_{t}\\,P_{t}}{\\sum_{t=1}^{n}Q_{t}\\,P_{t}}$.\n\nThe fixed leg pays $Q_t R$ each period; the floating leg's expected payment is $Q_t f_t$. Setting their present values equal and solving for $R$ gives this notional-weighted, discount-factor-weighted average of the forward rates.",
      "tag": "varying-notional"
    },
    {
      "front": "An amortizing swap has notionals $Q_1=\\$1{,}000{,}000$, $Q_2=\\$700{,}000$, $Q_3=\\$400{,}000$. Forward rates are $f_1=3.00\\%$, $f_2=4.00\\%$, $f_3=5.01\\%$ and discount factors $P_1=0.970874$, $P_2=0.933511$, $P_3=0.888996$. Find the swap rate.",
      "back": "Numerator $\\sum Q_t f_t P_t$:\n$1{,}000{,}000(0.03)(0.970874)=29{,}126$\n$700{,}000(0.040024)(0.933511)=26{,}154$\n$400{,}000(0.050073)(0.888996)=17{,}806$\nSum $=73{,}086$.\n\nDenominator $\\sum Q_t P_t$:\n$970{,}874+653{,}458+355{,}598=1{,}979{,}930$.\n\n$R=\\dfrac{73{,}086}{1{,}979{,}930}\\approx0.036913\\approx3.69\\%$.",
      "tag": "varying-notional"
    },
    {
      "front": "Why does an amortizing swap (notional declining over time) typically have a **lower** fixed rate than a level swap on the same upward-sloping curve?",
      "back": "On an upward-sloping curve the later forward rates $f_t$ are the **highest**, but an amortizing swap assigns them the **smallest** notionals $Q_t$. The notional-weighted average $R=\\frac{\\sum Q_t f_t P_t}{\\sum Q_t P_t}$ therefore puts less weight on the high late-period forwards, pulling the swap rate **below** the level-notional swap rate.",
      "tag": "varying-notional"
    },
    {
      "front": "From the fixed-rate payer's perspective, summarize the **two equivalent ways** to value any swap (level or varying notional) at a point in time.",
      "back": "**(1) Leg-by-leg:** $V=\\text{PV}(\\text{floating leg})-\\text{PV}(\\text{fixed leg})=\\sum_t Q_t f_t P_t-\\sum_t Q_t R\\,P_t$.\n\n**(2) Net-cash-flow:** discount each period's net $Q_t(f_t-R)$ at its own factor: $V=\\sum_t Q_t(f_t-R)P_t$.\n\nThe two are algebraically identical; at inception both give $0$ by the choice of $R$.",
      "tag": "market-value"
    },
    {
      "front": "A common sign-error trap: when valuing a swap, why must you weight each period's net cash flow by its **own** discount factor $P_t$ rather than using an average rate?",
      "back": "The net cash flows $Q_t(f_t-R)$ occur at different future dates and the curve is generally not flat, so each must be discounted at the factor $P_t$ for **its** maturity. Collapsing to a single average yield ignores the term structure and mis-times the cash flows, giving a wrong value — especially for steep curves or varying notionals.",
      "tag": "market-value"
    },
    {
      "front": "A swap is currently **off-market** (its contracted fixed rate differs from today's par rate). What does that imply about its value, and how is it reflected in pricing?",
      "back": "An off-market swap has **non-zero** value: $V_{\\text{payer}}=N(R_{\\text{mkt}}-R)\\sum_t P_t$. If the contracted $R$ is below the current par rate $R_{\\text{mkt}}$, the fixed payer's swap is an **asset** ($V>0$); if $R$ exceeds $R_{\\text{mkt}}$, it is a **liability** ($V<0$). One party may make an upfront payment equal to $|V|$ to enter such a swap fairly.",
      "tag": "market-value"
    },
    {
      "front": "Common pitfall: a candidate computes a 3-year swap rate as the simple average $\\tfrac{1}{3}(f_1+f_2+f_3)$ of the forward rates. Why is this wrong, and what is the correct approach?",
      "back": "The swap rate is a **discount-factor-weighted** average, $R=\\frac{\\sum f_t P_t}{\\sum P_t}$, not the unweighted mean. Because later forwards are discounted more heavily (smaller $P_t$), a simple average overweights distant periods on an upward-sloping curve and gives a rate that is too high. Always weight by the $P_t$ — equivalently, just use $R=\\frac{1-P_n}{\\sum P_t}$.",
      "tag": "swap-rate"
    }
  ]
}