Reading 72

Derivatives · Pricing and Valuation of Interest Rates and Other Swaps

MODULE 72.1: SWAP VALUATION

LOS 72.a

Describe how swap contracts are similar to but different from a series of forward contracts.

In a simple interest-rate swap, one party pays a floating rate and the other pays a fixed rate on a notional principal amount. Consider a 1-year swap with quarterly payments, one party paying a fixed rate and the other a floating rate equal to a 90-day market reference rate (MRR). At each payment date the difference between the swap fixed rate and the MRR is paid to the party that owes the least, that is, a net payment is made from one party to the other.

We can separate these payments into a known payment and three unknown payments that are equivalent to the payments on three forward rate agreements (FRAs). Let $\text{MRR}_n$ represent the floating-rate payment (based on the 90-day MRR) owed at the end of quarter $n$ and $F$ be the fixed payment owed at the end of each quarter. We can represent the swap payment to be received by the fixed-rate payer at the end of period $n$ as $\text{MRR}_n - F$. We can replicate each of these payments to (or from) the fixed-rate payer in the swap with a forward contract — specifically, a long position in an FRA with a contract rate equal to the swap fixed rate and a settlement value based on the 90-day MRR.

中文翻譯

在最簡單的利率交換（interest-rate swap）中，一方就名目本金支付浮動利率，另一方支付固定利率。以一年期、每季結算的交換為例：一方付固定，一方付浮動利率（等於 90 天市場參考利率 MRR）。每個結算日就「固定利率付款」與「MRR 付款」之差結算，由欠較多的一方淨額支付給另一方。

這些付款可拆分為「一筆已知付款」加上「三筆未知付款」，後者相當於三份 FRA（遠期利率協議）的結算。設 $\text{MRR}_n$ 為第 $n$ 季末根據 90 天 MRR 應付的浮動利息，$F$ 為每季末固定利息。固定利率支付方在第 $n$ 期末實收金額即為 $\text{MRR}_n - F$。每一筆此類現金流，皆可用「合約利率等於交換固定利率」的 FRA 多頭部位來複製。

We illustrate this separation below for a 1-year fixed-for-floating swap with a fixed rate of $F$ and floating-rate payments for period $n$ of $\text{MRR}_n$. Note that if the fixed rate and MRR are quoted as annual rates, the payments will be $(\text{MRR}_n - F)$ times one-fourth of the notional principal.

First (90d):$\text{MRR}_1 - F$ — known at time zero because the payment 90 days from now is based on the 90-day MRR observed at time zero and the swap fixed rate $F$, both of which are known at the initiation of the swap.
Second (180d):Equivalent to a long position in an FRA with contract rate $F$ that settles in 180 days and pays $\text{MRR}_2 - F$.
Third (270d):Equivalent to a long position in an FRA with contract rate $F$ that settles in 270 days and pays $\text{MRR}_3 - F$.
Fourth (360d):Equivalent to a long position in an FRA with contract rate $F$ that settles in 360 days and pays $\text{MRR}_4 - F$.

教授提醒

A forward on a 90-day MRR that settles 90 days from now actually pays the present value of $(\text{MRR} - F)$ at the settlement date — so the FRAs in our example pay on days 90, 180, and 270. However, the amounts paid are economically equivalent to the differences between fixed and floating payments due on days 180, 270, and 360, which is the framing used above.

中文翻譯

以「固定收／浮動付」一年期、每季結算的交換為例（固定利率 $F$，第 $n$ 期浮動利率 $\text{MRR}_n$，皆以年利率報價，故實際每期付款 = $(\text{MRR}_n - F) \times \tfrac{1}{4} \times$ 名目本金）：

第 1 期（90 天後）：$\text{MRR}_1 - F$，此筆在期初即已確定（因為當下的 90 天 MRR 與固定利率 $F$ 皆已知）。
第 2 期（180 天後）：等同於「合約利率 = $F$、180 天後結算、給付 $\text{MRR}_2 - F$」的 FRA 多頭。
第 3 期（270 天後）：等同於 270 天後結算、給付 $\text{MRR}_3 - F$ 的 FRA 多頭。
第 4 期（360 天後）：等同於 360 天後結算、給付 $\text{MRR}_4 - F$ 的 FRA 多頭。

教授提醒：以 90 天 MRR 為標的的 FRA，雖在結算日（如第 90、180、270 天）即支付當期 $(\text{MRR} - F)$ 之現值，但其經濟金額等同於利息實際應付日（第 180、270、360 天）的差額，故上述拆解仍然成立。

Therefore, we can describe an interest-rate swap as equivalent to a series of FRAs, each with a forward contract rate equal to the swap fixed rate. However, there is one important difference. Because the forward contract rates are all equal in the FRAs that are equivalent to the swap, these would not be zero-value forward contracts at the initiation of the swap.

Recall that forward contracts are based on a contract rate for which the value of the forward at initiation is zero. There is no reason to suspect that the swap fixed rate results in a zero-value forward contract for each of the future dates. Instead, a swap is most likely to consist of some forwards with positive values and some with negative values; the sum of their values equals zero at initiation.

Finding the swap fixed rate that gives the swap a zero value at initiation — known as the par swap rate — is not difficult if we follow the principle of no-arbitrage pricing. The fixed-rate payer in a swap can replicate that derivative position by borrowing at a fixed rate and lending the proceeds at a variable (floating) rate. For the swap in our example, borrowing at fixed rate $F$ and lending the proceeds at the 90-day MRR will produce the same cash flows as the swap. At each date, the payment due on the fixed-rate loan is $F$ and the interest received on lending at the floating rate is $\text{MRR}_n$.

中文翻譯

因此利率交換可視為「一系列 FRA 的組合」，且每份 FRA 的合約利率都等於交換固定利率 $F$。但有一個重要差異：因為這些 FRA 的合約利率全部相同，並不會在交換期初時剛好都是零值合約。

標準遠期合約以「初始時合約價值為零」的合約利率定價；交換的固定利率沒有理由剛好讓每一筆未來付款都是零值遠期。實際上，交換內部會有些 FRA 為正值、有些為負值，但其總值在期初恰好相加為零。

要找到讓交換期初總值為零的固定利率（即平價交換利率，par swap rate），可運用無套利定價：固定利率支付方的部位，可由「以固定利率 $F$ 借款，以浮動利率 MRR 放款」來複製，現金流完全等同。每期需付固定利息 $F$，並收到浮動利息 $\text{MRR}_n$。

LOS 72.b

Contrast the value and price of swaps.

As with FRAs, the price of a swap is the fixed rate of interest specified in the swap contract (the par swap rate), and the value depends on how expected future floating rates change over time. At initiation, a swap has zero value because the present value of the fixed-rate payments equals the present value of the expected floating-rate payments.

We can solve for the no-arbitrage fixed rate — the par swap rate — from the following equality:

$$\frac{\text{MRR}_1}{1+S_1} + \frac{\text{MRR}_2}{(1+S_2)^2} + \frac{\text{MRR}_3}{(1+S_3)^3} + \frac{\text{MRR}_4}{(1+S_4)^4} \;=\; \frac{F}{1+S_1} + \frac{F}{(1+S_2)^2} + \frac{F}{(1+S_3)^3} + \frac{F}{(1+S_4)^4}$$

where $S_1$ through $S_4$ are the current effective spot rates for 90, 180, 270, and 360 days; $\text{MRR}_1$ through $\text{MRR}_4$ are the forward 90-day rates implied by the spot rates; and $F$ is the fixed-rate payment.

Given the current spot rates ($S_1$ to $S_4$), we calculate the implied (expected) forward rates ($\text{MRR}$s), then solve for the $F$ that gives the swap zero value.

An increase in expected future 90-day rates produces an increase in the value of the fixed-rate payer position; a decrease in expected rates decreases the value of that position. At any point in time, the value of the fixed-rate payer side of a swap equals:

$$V_{\text{fixed-payer}} \;=\; \text{PV(expected future floating-rate payments)} \;-\; \text{PV(future fixed-rate payments)}$$

This calculation uses the spot rates and implied future 90-day rates at that point in time and can be used for any required mark-to-market payments.

中文翻譯

與 FRA 相同：交換的「價格」是合約中指定的固定利率（即 par swap rate）；「價值」則隨對未來浮動利率預期的改變而變動。期初時交換價值為零，因為固定利息的現值 = 預期浮動利息的現值。

從以下等式可解出無套利的固定利率（par swap rate）：

$\displaystyle \sum_{n=1}^{4}\frac{\text{MRR}_n}{(1+S_n)^n} = \sum_{n=1}^{4}\frac{F}{(1+S_n)^n}$

$S_1\sim S_4$：當前 90、180、270、360 天的有效即期利率。
$\text{MRR}_1\sim\text{MRR}_4$：由即期利率推導的隱含 90 天遠期利率。
$F$：每期固定利率付款。

已知 $S_n$ 即可求得隱含遠期利率，再代回式中解 $F$，即得使交換期初價值為零的固定利率。

價值變動方向：未來 90 天利率預期上升→ 固定利率支付方部位價值上升；預期下降則部位價值下降。任何時點，固定利率支付方部位的價值 = 預期未來浮動利息現值 − 未來固定利息現值；可用於依市值結算（mark-to-market）。

📝 Module Quiz 72.1

1. Which of the following is most similar to the floating-rate receiver position in a fixed-for-floating interest-rate swap?

A. Buying a fixed-rate bond and a floating-rate note.
B. Buying a floating-rate note and issuing a fixed-rate bond.
C. Issuing a floating-rate note and buying a fixed-rate bond.

B — The floating-rate receiver (i.e., fixed-rate payer) in a fixed-for-floating interest-rate swap has a position similar to issuing a fixed-coupon bond and buying a floating-rate note. (LOS 72.a)

2. The price of a fixed-for-floating interest-rate swap:

A. is specified in the swap contract.
B. is paid at initiation by the floating-rate receiver.
C. may increase or decrease during the life of the swap contract.

A — The price of a fixed-for-floating interest-rate swap is defined as the fixed rate specified in the swap contract. Typically a swap is priced such that it has a value of zero at initiation and neither party pays the other to enter the swap. (LOS 72.b)

Key Concepts — Reading 72

LOS 72.a

In a simple interest-rate swap, one party pays a floating rate and the other pays a fixed rate on a notional principal amount. The first payment is known at initiation; the remaining payments are unknown. The unknown payments are equivalent to the payments on FRAs. The par swap rate is the fixed rate at which the sum of the present values of these FRAs equals zero.

LOS 72.b

The price of a swap is the fixed rate of interest specified in the swap contract. The value depends on how expected future floating rates change over time. An increase in expected future short-term rates increases the value of the fixed-rate payer position; a decrease in expected future rates decreases the value of the fixed-rate payer position.

中文翻譯 — 重點整理

【LOS 72.a】單純利率交換中，一方付浮動、一方付固定（同一名目本金）。第 1 期付款於期初即已確定；後續各期相當於一系列 FRA。par swap rate 即為使這些 FRA 現值總和為零的固定利率。

【LOS 72.b】交換的價格 = 合約固定利率；價值取決於對未來浮動利率預期的變動。預期未來短期利率上升→ 固定利率支付方部位升值；反之則貶值。