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Reading 55

Fixed Income · The Term Structure of Interest Rates: Spot, Par, and Forward Curves

MODULE 55.1: SPOT, PAR, AND FORWARD CURVES

LOS 55.a

Define spot rates and the spot curve, and calculate the price of a bond using spot rates.

A bond's yield to maturity assumes that every cash flow is discounted at a single rate. In practice, the appropriate discount rate depends on the timing of the cash flow. The discount rate that applies to a single payment to be received at a specific point in the future is called a spot rate. Because spot rates can be observed from the prices of zero-coupon bonds, they are sometimes referred to as zero-coupon rates or simply zero rates.

To value a bond using spot rates, discount each individual cash flow by the spot rate corresponding to its maturity, then sum the present values. The general no-arbitrage pricing equation is:

\[ P = \frac{CPN_1}{(1+S_1)^1} + \frac{CPN_2}{(1+S_2)^2} + \cdots + \frac{CPN_n + FV_n}{(1+S_n)^n} \]

Example
Valuing a bond using spot rates

Given the following spot rates, calculate the value of a 3-year, 5% annual-coupon bond (par = 100):

  • 1-year spot: 3%
  • 2-year spot: 4%
  • 3-year spot: 5%
Answer

\[ P = \frac{5}{(1.03)^1} + \frac{5}{(1.04)^2} + \frac{105}{(1.05)^3} = 100.180 \]

The price obtained from spot rates is the no-arbitrage price; any other price would create an arbitrage opportunity. Because the price is slightly above par, the YTM must be slightly below the 5% coupon. Solving for YTM (N=3, PMT=5, FV=100, PV=−100.180) gives I/Y ≈ 4.93%.

教授提醒
Think of each spot rate as the return earned by a single future cash flow. The bond's overall YTM is essentially a weighted average of these spot rates, weighted by the size and timing of the cash flows. In the example, most of the dollars are received in year 3, so the YTM is heavily pulled toward the 5% three-year spot rate, ending up just below 5%.

The spot curve displays spot rates against maturity for a particular type of issuer (e.g., U.S. Treasuries). It is the foundational curve from which par rates and forward rates can be derived.

中文翻譯

即期利率（Spot rate）是指對未來某一單一時點現金流所適用的折現率。由於可從零息債券價格直接觀察出來，故又稱為「零息利率」或「zero rate」。YTM 假設所有現金流用同一個利率折現，但實際上不同到期期間應使用不同的折現率。

使用即期利率為債券估價時，將每筆現金流以其對應到期年限的即期利率分別折現後再加總，即為債券的無套利價格（no-arbitrage price）：

\[ P = \frac{CPN_1}{(1+S_1)} + \frac{CPN_2}{(1+S_2)^2} + \cdots + \frac{CPN_n + FV_n}{(1+S_n)^n} \]

例題：3 年期 5% 年付息債券，1/2/3 年即期利率為 3%、4%、5%，價格 = 5/1.03 + 5/1.04² + 105/1.05³ = 100.180；對應 YTM ≈ 4.93%（略低於 5% 票面利率，因為價格略高於面額）。

教授提醒：YTM 可以視為各期即期利率的加權平均，權重由現金流大小及時點決定。本例中大部分現金流發生在第 3 年（含本金），故 YTM 被第 3 年的 5% 即期利率拉得很近，最終略低於 5%。

即期收益曲線（Spot curve）就是把各到期年限的即期利率對到期日繪成的曲線，是推導 par 利率與 forward 利率的基礎。

LOS 55.b

Define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates.

Par yields are the coupon rates that, given a particular spot curve, would cause a hypothetical bond at each maturity to be priced exactly at par. Equivalently, a par yield is the YTM of a hypothetical par-priced bond at that maturity. For a 3-year annual-pay bond, the par coupon PMT must satisfy:

\[ \frac{PMT}{(1+S_1)} + \frac{PMT}{(1+S_2)^2} + \frac{PMT + 100}{(1+S_3)^3} = 100 \]

Example
Computing the 3-year par rate

With spot rates of 1%, 2%, and 3%, find PMT such that:

\[ \frac{PMT}{1.01} + \frac{PMT}{(1.02)^2} + \frac{PMT+100}{(1.03)^3} = 100 \]

Answer

Solving gives PMT ≈ 2.96, so the 3-year par rate ≈ 2.96%.

教授提醒
On the exam, if a par-rate problem appears in multiple choice, plug the middle answer into the equation and check whether the resulting bond price equals 100. If the price comes out below 100, the higher choice is correct; if above 100, the lower choice is correct. This is much faster than solving algebraically.
中文翻譯

Par yield（票面利率／平價收益率）：在給定即期利率曲線下，使該到期年限的假設債券剛好以面額交易（即價格 = 100）的票面利率，也可視為該到期年限「平價債券」的 YTM。對 3 年期年付息債券：

\[ \frac{PMT}{1+S_1} + \frac{PMT}{(1+S_2)^2} + \frac{PMT+100}{(1+S_3)^3} = 100 \]

例題：S₁=1%、S₂=2%、S₃=3%，求得 PMT ≈ 2.96，故 3 年 par 利率 ≈ 2.96%。

教授提醒：考試遇到此題型時，把選項中的中間值代入公式，看價格是否等於 100。若小於 100，正確答案是較大的；若大於 100，正確答案是較小的──比代數求解快很多。

A forward rate is a borrowing or lending rate locked in today for a loan that will start at some specified date in the future. The standard CFA notation uses two numbers separated by a letter (y for years, m for months): the first denotes when the loan begins, the second denotes its length. Examples:

  • 1y1y — a 1-year rate, one year from now
  • 2y1y — a 1-year rate, two years from now
  • 3y2y — a 2-year rate, three years from now

The fundamental no-arbitrage link between spot rates and forward rates is that investing for N periods at the N-period spot rate must give the same outcome as rolling forward through one-period rates each year. For three years:

\[ (1+S_3)^3 = (1+S_1)(1+1y1y)(1+2y1y) \]

so

\[ S_3 = \big[(1+S_1)(1+1y1y)(1+2y1y)\big]^{1/3} - 1 \]

which is simply the geometric mean of the implied one-period returns. More generally, the N-period spot rate is the geometric mean of the forward rates that apply to each one-period segment between today and N.

Example
Computing a spot rate from forward rates

Given S₁ = 2%, 1y1y = 3%, 2y1y = 4%, what is the 3-year spot rate?

Answer

\[ S_3 = \big[(1.02)(1.03)(1.04)\big]^{1/3} - 1 = 2.997\% \]

Interpretation: $1 compounded at 2.997% for three years grows to the same amount as $1 earning 2%, then 3%, then 4% in successive years.

教授提醒
A simple arithmetic average of the one-period forward rates gives a quick approximation of the multi-period spot rate. In the previous example, (2 + 3 + 4)/3 = 3%, very close to the exact 2.997%. This is fast enough for sanity checks during the exam.
中文翻譯

Forward rate（遠期利率）：今天就鎖定、但實際借貸期間是未來某段時間的利率。CFA 慣用的記法 「AyBy」：第一個數字是「幾年後開始」，第二個是「貸款多久」。例：

  • 1y1y = 1 年後的 1 年期利率
  • 2y1y = 2 年後的 1 年期利率
  • 3y2y = 3 年後的 2 年期利率

核心無套利關係：以 N 年期即期利率投資 N 年，與每年用一年期利率滾動 N 次，最終結果應該相同。3 年的版本：

\[ (1+S_3)^3 = (1+S_1)(1+1y1y)(1+2y1y) \]

所以 N 年期即期利率＝這 N 段一年期遠期利率的幾何平均。

例題：S₁=2%、1y1y=3%、2y1y=4%，則 S₃ = [(1.02)(1.03)(1.04)]^(1/3) − 1 = 2.997%。意義：$1 連續三年分別以 2%、3%、4% 複利所得的終值，等於以 2.997% 三年複利的終值。

教授提醒：用算術平均近似可快速估算──(2+3+4)/3 = 3%，與精確值 2.997% 非常接近，可作為考場上的合理性檢查。

The same no-arbitrage relationship can be inverted to recover forward rates from observed spot rates. The two-period building block is:

\[ (1+S_2)^2 = (1+S_1)(1+1y1y) \]

Example
Computing 1y1y from spot rates

If S₁ = 4% and S₂ = 8%, find the 1-year forward rate one year from now.

Answer

\[ 1y1y = \frac{(1+S_2)^2}{(1+S_1)} - 1 = \frac{(1.08)^2}{1.04} - 1 = \frac{1.1664}{1.04} - 1 = 12.154\% \]

Investors are willing to accept 4% on the 1-year bond today (rather than 8% on the 2-year bond) only because they expect to earn 12.154% on a 1-year bond one year from today — a rate they can lock in today as a forward.

Extending one more period, since \( (1+S_3)^3 = (1+S_2)^2(1+2y1y) \):

\[ 2y1y = \frac{(1+S_3)^3}{(1+S_2)^2} - 1 \]

Example
Computing 2y1y from spot rates

If S₁ = 4%, S₂ = 8%, S₃ = 12%, find 2y1y.

Answer

\[ 2y1y = \frac{(1.12)^3}{(1.08)^2} - 1 = 20.45\% \]

Check: \( S_3 = [(1.04)(1.12154)(1.2045)]^{1/3} - 1 \approx 12.00\% \). ✓

教授提醒
Simple-average shortcuts also work for forward rates from spot rates. If S₁ = 4% and S₂ = 8%, then "two years × 8% = 16%" minus "one year × 4% = 4%" leaves about 12% for the second-year forward (actual 12.154%). For multi-period forwards: with S₂ = 6% and S₄ = 8%, the 2y2y is approximately (4×8 − 2×6)/2 = 10% (actual 10.04%). These approximations work well when rates are small and serve as a great sanity check.

The same formula extends to multi-period forward rates. For example, given a 4-year spot rate S₄ and a 2-year spot rate S₂:

\[ 2y2y = \left[\frac{(1+S_4)^4}{(1+S_2)^2}\right]^{1/2} - 1 \]

中文翻譯

反過來，也可以用即期利率推算遠期利率。兩期版本：

\[ (1+S_2)^2 = (1+S_1)(1+1y1y) \quad \Rightarrow \quad 1y1y = \frac{(1+S_2)^2}{1+S_1} - 1 \]

例題（1y1y）：S₁=4%、S₂=8%，1y1y = (1.08)²/1.04 − 1 = 12.154%。意義：投資者今天願意接受 4% 的 1 年期利率（而非 8% 的 2 年期），是因為預期 1 年後的 1 年期利率會是 12.154%，且此遠期利率可在今日鎖定。

三期版本：\( (1+S_3)^3 = (1+S_2)^2(1+2y1y) \)，故 2y1y = (1+S₃)³/(1+S₂)² − 1。

例題（2y1y）：S₂=8%、S₃=12%，2y1y = (1.12)³/(1.08)² − 1 = 20.45%。

教授提醒：近似法──以 S₁=4%、S₂=8% 為例，2 年×8%=16%，扣掉 1 年×4%=4%，剩約 12%（實際 12.154%）；多期遠期同理：S₂=6%、S₄=8% 下，2y2y ≈ (4×8 − 2×6)/2 = 10%（實際 10.04%）。利率較小時近似結果都很接近。

多期版本：例如 2y2y = [(1+S₄)⁴/(1+S₂)²]^(1/2) − 1。

A bond can also be valued directly from forward rates. For a 3-year annual-pay bond:

\[ P = \frac{CPN_1}{(1+S_1)} + \frac{CPN_2}{(1+S_1)(1+1y1y)} + \frac{CPN_3 + FV}{(1+S_1)(1+1y1y)(1+2y1y)} \]

Example
Bond value using forward rates

S₁ = 4%, 1y1y = 5%, 2y1y = 6%. Value a 3-year annual-pay bond with a 5% coupon and $1,000 par.

Answer

\[ P = \frac{50}{1.04} + \frac{50}{(1.04)(1.05)} + \frac{1{,}050}{(1.04)(1.05)(1.06)} \approx \$1{,}000.98 \]

教授提醒
Notice how this formula mirrors the spot-rate pricing formula — the forward-rate discount factors are mathematically equivalent to spot-rate discount factors because of the no-arbitrage relationship. For a semiannual-pay bond, use semiannual discount rates and count the number of semiannual periods (rather than years).
中文翻譯

債券亦可直接以遠期利率折現估價。3 年期年付息債券：

\[ P = \frac{CPN_1}{1+S_1} + \frac{CPN_2}{(1+S_1)(1+1y1y)} + \frac{CPN_3 + FV}{(1+S_1)(1+1y1y)(1+2y1y)} \]

例題：S₁=4%、1y1y=5%、2y1y=6%，5% 年付息、面額 $1,000、3 年期：

\[ P = \frac{50}{1.04} + \frac{50}{(1.04)(1.05)} + \frac{1{,}050}{(1.04)(1.05)(1.06)} \approx \$1{,}000.98 \]

教授提醒：這個公式與用 spot 折現得到的價格一致──遠期利率的折現因子與即期利率的折現因子在無套利條件下完全等價。半年付息債券則需以半年折現率與半年期數計算。

LOS 55.c

Compare the spot curve, par curve, and forward curve.

The spot curve (also called the zero curve or strip curve) plots spot rates against maturity. For U.S. Treasuries, it is typically built from the prices of zero-coupon stripped Treasuries. A spot curve that rises with maturity is called a normal yield curve; one that declines is inverted. Treasury spot rates are usually quoted on a semiannual bond basis so they are directly comparable to coupon-bond YTMs.

A coupon bond yield curve shows the YTMs of actively traded coupon bonds (e.g., U.S. Treasuries) at various maturities, with intermediate maturities estimated by linear interpolation. Two practical issues arise in building one directly from market prices:

  • Liquidity differences — only "on-the-run" (most recently issued) bonds trade actively, but there may be too few of them to span the curve.
  • Tax distortions — interest income and capital gains/losses on premium or discount bonds may be taxed differently, distorting yields.

The par curve is built to sidestep these issues: it shows hypothetical coupon rates at each maturity that would price a bond exactly at par given the underlying spot curve. Because it is derived from spot rates, it is internally consistent across maturities and avoids tax/liquidity distortion.

The forward curve plots forward rates for future periods (typically 1-year forward rates for each future year, quoted on a semiannual bond basis).

The three curves are tightly linked:

  • Each spot rate is the geometric mean of the one-period forward rates over its maturity.
  • Each par yield is essentially a weighted average of the spot rates that apply to that bond's cash flows, with the heaviest weight on the longest spot rate (where the par payment occurs).

Therefore, in an upward-sloping environment: forward > spot > par, all rising but forwards rising fastest. In a downward-sloping (inverted) environment: forward < spot < par, all falling but forwards falling fastest. An inverted curve is sometimes interpreted as a market expectation of falling future interest rates. When forward rates are constant across maturities, all three curves are flat and equal — the flat yield curve case.

教授提醒
Sanity-check with simple averages. Suppose S₁ = 1% and 1y1y = 3% (rising forwards). The 2-year spot is approximately (1+3)/2 = 2% — rising too, but more slowly than forwards. The 1-year par yield equals the 1-year spot (1%, since there's only one cash flow). The 2-year par yield is a weighted average of the 1-year and 2-year spots, but mostly weighted toward 2% (where the par payment occurs), so it is just a touch below 2%. The takeaway: forwards drive spots, which in turn drive par yields.
中文翻譯

Spot curve（即期曲線；亦稱 zero / strip curve）：以即期利率對到期日繪製。美國公債通常以剝離公債（stripped Treasury）的價格建構，並以半年期 bond basis 報價，可直接與付息債券 YTM 比較。曲線上揚為「正常」，下凹為「反轉」。

Coupon bond yield curve（付息債券收益曲線）：以實際活躍交易付息債券的 YTM 對到期日繪製，缺漏的到期日以線性插補。實務上有兩個問題：①流動性──只有 on-the-run（最近發行）債券交易熱絡，數量可能不足；②稅負──利息收入與買賣價差課稅方式不同，溢價或折價債券會扭曲 YTM。

Par curve（平價曲線）：為避免上述問題，從 spot curve 推導而得，顯示每個到期日「能讓債券剛好以面額交易」的假設票面利率。此曲線在到期日間具一致性，且不受流動性與稅負扭曲。

Forward curve（遠期曲線）：未來各期的遠期利率（通常為各未來年度的 1 年期利率），以半年期 bond basis 報價。

三條曲線的關係：

  • 每個即期利率＝對應期間內每一年遠期利率的幾何平均。
  • 每個 par yield＝對應期間各即期利率的加權平均，最大權重落在最後一年（因有面額還本）。

因此：

  • 正常（上揚）曲線：forward > spot > par，三者皆上揚但 forward 最陡。
  • 反轉（下凹）曲線：forward < spot < par，三者皆下降但 forward 最陡。反轉曲線常被解讀為市場預期未來利率將下跌。
  • 平坦曲線：當所有遠期利率相同，三條曲線重疊。

教授提醒：用算術平均做 sanity check。設 S₁=1%、1y1y=3%（forward 上揚），則 2 年 spot ≈ (1+3)/2 = 2%，比 forward 上揚得慢；1 年 par yield＝1% spot（因為只有一筆現金流）；2 年 par yield 是 1% spot 與 2% spot 的加權平均（重心在 2%），故略低於 2%。重點：forward 推動 spot，spot 推動 par yield。

Module Quiz 55.1
1. If spot rates are 3.2% for one year, 3.4% for two years, and 3.5% for three years, the price of a $100,000 face value, 3-year, annual-pay bond with a coupon rate of 4% is closest to:
  • A. $101,420.
  • B. $101,790.
  • C. $108,230.
A — \( P = \frac{4{,}000}{1.032} + \frac{4{,}000}{(1.034)^2} + \frac{104{,}000}{(1.035)^3} = \$101{,}419.28 \). (LOS 55.a)
2. A market rate of discount for a single payment to be made in the future is a:
  • A. spot rate.
  • B. simple yield.
  • C. forward rate.
A — A spot rate is the discount rate for a single future payment. Simple yield accounts for coupon interest plus straight-line amortization of any premium or discount. A forward rate applies to a loan beginning in the future, not to a single payment received in the future. (LOS 55.a)
3. Which of the following yield curves is least likely to consist of observed yields in the market?
  • A. Forward yield curve.
  • B. Par bond yield curve.
  • C. Coupon bond yield curve.
B — Par bond yield curves are theoretical: they show the coupon rates that would price hypothetical bonds at par at each maturity. Coupon bond yields and forward rates can both be observed from market transactions. (LOS 55.b)
4. The 4-year spot rate is 9.45%, and the 3-year spot rate is 9.85%. What is the 1-year forward rate three years from today?
  • A. 8.258%.
  • B. 9.850%.
  • C. 11.059%.
A — \( (1.0945)^4 = (1.0985)^3 (1+3y1y) \), so \( 3y1y = \frac{(1.0945)^4}{(1.0985)^3} - 1 = 8.258\% \). Approximation check: 4(9.45) − 3(9.85) = 8.25%. (LOS 55.b)
5. Given the following spot and forward rates:
  • Current 1-year spot rate is 5.5%.
  • 1-year forward rate one year from today is 7.63%.
  • 1-year forward rate two years from today is 12.18%.
  • 1-year forward rate three years from today is 15.5%.
The value of a 4-year, 10% annual-pay, $1,000 par-value bond is closest to:
  • A. $870.
  • B. $996.
  • C. $1,009.
C — \( P = \frac{100}{1.055} + \frac{100}{(1.055)(1.0763)} + \frac{100}{(1.055)(1.0763)(1.1218)} + \frac{1{,}100}{(1.055)(1.0763)(1.1218)(1.155)} \approx \$1{,}009.03 \). (LOS 55.b)
6. The 1-year spot rate is 1% and the 1y1y rate is 3%. Which of the following statements is most accurate?
  • A. The 2-year spot is just below 2%, and the 2-year par yield is just below the 2-year spot rate.
  • B. The 2-year spot is just below 2%, and the 2-year par yield is just above the 2-year spot rate.
  • C. The 2-year spot is just below 4%, and the 2-year par yield is just below the 2-year spot rate.
A — The 2-year spot is the geometric mean of the 1% and 3% periodic rates, so it is approximately 2% (slightly below the 2% arithmetic average). With forwards rising, par yields also rise but lag the spot curve, so the 2-year par yield is just below the 2-year spot rate. (LOS 55.c)
Key Concepts — Reading 55
LOS 55.a

Spot rates are market discount rates for single payments to be made in the future. The no-arbitrage price of a bond is the sum of each cash flow discounted at its own spot rate:

\[ P = \frac{CPN_1}{(1+S_1)} + \frac{CPN_2}{(1+S_2)^2} + \cdots + \frac{CPN_n + FV_n}{(1+S_n)^n} \]

LOS 55.b

The par yield at each maturity is the coupon rate that would price a hypothetical bond at par given the spot curve. It can be interpreted as a weighted average of the spot rates applied to each cash flow.

Forward rates are rates locked in today for short-term loans starting in the future. The N-period spot rate is the geometric mean of the forward rates spanning that period; the same identity inverted lets you back out a forward rate from two spot rates. To value a bond directly from forwards, discount each cash flow by the product of one plus each one-period forward rate up to that point:

\[ P = \frac{CPN_1}{(1+S_1)} + \frac{CPN_2}{(1+S_1)(1+1y1y)} + \frac{CPN_3 + FV}{(1+S_1)(1+1y1y)(1+2y1y)} \]

LOS 55.c

The spot curve plots spot rates by maturity (often built from zero-coupon/stripped Treasury prices). The par curve shows the coupon rates that would price bonds at par at each maturity. The forward curve plots short-term rates for future periods.

  • Upward-sloping (normal): forward > spot > par
  • Downward-sloping (inverted): forward < spot < par
  • Flat: forward = spot = par across all maturities
中文翻譯 — 重點整理

【LOS 55.a】Spot rate 是針對單一未來現金流的市場折現率。債券無套利價格＝把每筆現金流以其對應期數的 spot rate 折現後加總：

\[ P = \frac{CPN_1}{1+S_1} + \frac{CPN_2}{(1+S_2)^2} + \cdots + \frac{CPN_n + FV_n}{(1+S_n)^n} \]

【LOS 55.b】Par yield＝在給定 spot curve 下，使債券剛好以面額交易的票面利率，可視為各期 spot rate 的加權平均（最後一期權重最大）。Forward rate＝今天就鎖定、未來才開始的短期借貸利率。N 年 spot rate＝該段期間每年 forward rate 的幾何平均；反之，知道兩個 spot rate 即可解出對應的 forward rate。用 forward rate 為債券估價時，將每筆現金流以從現在到該時點所累乘的「1 + forward rate」折現：

\[ P = \frac{CPN_1}{1+S_1} + \frac{CPN_2}{(1+S_1)(1+1y1y)} + \frac{CPN_3 + FV}{(1+S_1)(1+1y1y)(1+2y1y)} \]

【LOS 55.c】Spot curve 通常由零息／stripped Treasury 價格推導；par curve 顯示各到期日「使債券以面額交易」的票面利率；forward curve 顯示未來各期的短期利率。

  • 正常（上揚）：forward > spot > par
  • 反轉（下凹）：forward < spot < par
  • 平坦：所有到期日下三者相等
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