Exam FM — Interest-Rate Swaps Flashcards

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.

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.

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.

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.

$R=\dfrac{1-P_{n}}{\sum_{t=1}^{n}P_{t}}$. The 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 $R$.

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.

Discount factors: $P_1=1.03^{-1}=0.970874$, $P_2=1.035^{-2}=0.933511$, $P_3=1.04^{-3}=0.888996$. Sum $\sum P_t=2.793381$. Swap rate $R=\dfrac{1-P_3}{\sum P_t}=\dfrac{1-0.888996}{2.793381}=\dfrac{0.111004}{2.793381}\approx0.039738$. So $R\approx 3.97\%$.

$R$ 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.

The no-arbitrage forward rate satisfies $(1+s_t)^{t}=(1+s_{t-1})^{t-1}(1+f_t)$, so $f_t=\dfrac{(1+s_t)^{t}}{(1+s_{t-1})^{t-1}}-1$. Here $f_2=\dfrac{(1.035)^{2}}{1.03}-1=\dfrac{1.071225}{1.03}-1\approx0.040024$, i.e. about $4.00\%$.

Numerator $\sum f_t P_t = (0.03)(0.970874)+(0.040024)(0.933511)+(0.050073)(0.888996)$ $=0.029126+0.037363+0.044515=0.111004$. Denominator $\sum P_t=2.793381$. $R=\dfrac{0.111004}{2.793381}\approx0.039738\approx3.97\%$ — matching the $\frac{1-P_n}{\sum P_t}$ result, as it must.

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.

Active periods are $t=2,3,4$, deferred past year $1$ (so $a=1$, $b=4$). $R=\dfrac{P_1-P_4}{P_2+P_3+P_4}=\dfrac{0.9709-0.8885}{0.9426+0.9151+0.8885}=\dfrac{0.0824}{2.7462}\approx0.030005$. So the deferred swap rate is about $3.00\%$.

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.

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.

$\sum P_t=0.97+0.94+0.90=2.81$. PV(floating received) $=N(1-P_3)=1{,}000{,}000(1-0.90)=\$100{,}000$. PV(fixed paid) $=N\cdot R\cdot\sum P_t=1{,}000{,}000(0.045)(2.81)=\$126{,}450$. $V_{\text{payer}}=100{,}000-126{,}450=-\$26{,}450$. Rates fell, so the fixed payer (locked at $4.5\%$) is **out of the money** by about $\$26{,}450$.

Current par rate $R_{\text{mkt}}=\dfrac{1-P_3}{\sum P_t}=\dfrac{0.10}{2.81}=0.035587$ (about $3.56\%$). $V_{\text{payer}}=N(R_{\text{mkt}}-R)\sum P_t=1{,}000{,}000\,(0.035587-0.045)(2.81)$ $=1{,}000{,}000(-0.009413)(2.81)\approx-\$26{,}450$ — the same answer.

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.

Fixed obligation $=500{,}000(0.04)=\$20{,}000$. Floating obligation $=500{,}000(0.036)=\$18{,}000$. Net $=N(r_t-R)=500{,}000(0.036-0.04)=-\$2{,}000$. The net is negative to the fixed payer, so the **fixed-rate payer pays $\$2{,}000$** to the counterparty this period.

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.

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.

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.

$R=\dfrac{\sum_{t=1}^{n}Q_{t}\,f_{t}\,P_{t}}{\sum_{t=1}^{n}Q_{t}\,P_{t}}$. The 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.

Numerator $\sum Q_t f_t P_t$: $1{,}000{,}000(0.03)(0.970874)=29{,}126$ $700{,}000(0.040024)(0.933511)=26{,}154$ $400{,}000(0.050073)(0.888996)=17{,}806$ Sum $=73{,}086$. Denominator $\sum Q_t P_t$: $970{,}874+653{,}458+355{,}598=1{,}979{,}930$. $R=\dfrac{73{,}086}{1{,}979{,}930}\approx0.036913\approx3.69\%$.

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.

**(1) Leg-by-leg:** $V=\text{PV}(\text{floating leg})-\text{PV}(\text{fixed leg})=\sum_t Q_t f_t P_t-\sum_t Q_t R\,P_t$. **(2) Net-cash-flow:** discount each period's net $Q_t(f_t-R)$ at its own factor: $V=\sum_t Q_t(f_t-R)P_t$. The two are algebraically identical; at inception both give $0$ by the choice of $R$.

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.

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.

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 $R=\frac{1-P_n}{\sum P_t}$.