Reading 54
MODULE 54.1: YIELD AND YIELD SPREAD MEASURES FOR FLOATING-RATE INSTRUMENTS
Calculate and interpret yield spread measures for floating-rate instruments.
Floating-Rate Note Yields. The values of floating-rate notes (FRNs) are more stable than those of fixed-rate debt of similar maturity because the coupon rate is reset periodically based on a variable market reference rate (MRR). The coupon rate on an FRN consists of a relatively risk-free (usually interbank) MRR plus a fixed margin based on the credit risk of the issuer (at issuance) relative to that of the MRR. The next-period coupon is set using the current MRR for the reset period, and the payment at the end of the period is based on this rate. For this reason, interest is said to be paid in arrears.
If the FRN issuer has more (less) credit risk than the financial institutions from which the MRR is derived, a margin is added to (subtracted from) the MRR. Liquidity and tax treatment also affect the margin.
- Quoted margin (QM): the fixed margin above the MRR actually paid in the coupon.
- Required margin / Discount margin (DM): the margin required to price the FRN at par.
FRNs are usually issued at par with QM = DM at issuance. If credit quality remains unchanged, QM continues to equal DM and the FRN will trade at par on its coupon reset dates.
- If issuer credit quality decreases after issuance → investors demand higher DM → DM > QM → FRN trades at a discount.
- If issuer credit quality improves after issuance → DM < QM → FRN trades at a premium.
A simplified way to value an FRN on a reset date: use the current MRR plus QM to estimate future cash flows, and discount these cash flows at MRR + DM. More complex models produce better estimates of value.
浮動利率票據（FRN）殖利率：FRN 的價值比同到期日的固定利率債務更穩定，因為其票息會週期性地根據可變的市場參考利率（MRR）重設。FRN 的票息率＝相對無風險的（通常是銀行同業）MRR ＋ 固定的信用利差（反映發行人在發行時相對於 MRR 報價金融機構的信用風險）。下一期的票息利率以當期 MRR 設定，期末才支付，因此稱為「後付利息」。
若發行人信用風險高於（低於）MRR 的報價金融機構，則須在 MRR 上加（減）一個利差。流動性與稅負亦會影響此利差。
- 報價利差（QM）：實際在票息中支付、加在 MRR 上的固定利差。
- 所需利差／貼現利差（DM）：使 FRN 平價交易所需要的利差。
FRN 通常以平價發行，發行時 QM = DM。若信用品質維持不變，重設日仍會以平價交易。
- 發行後信用品質惡化 → 投資人要求更高 DM → DM > QM → FRN 折價交易。
- 發行後信用品質改善 → DM < QM → FRN 溢價交易。
教授提醒：這與固定利率債券中票息與殖利率的關係類似。固定票息債券若投資人要求殖利率高於票息，票息「不足」，債券折價交易。FRN 中票息為 MRR + QM，投資人要求殖利率為 MRR + DM；當 MRR + DM > MRR + QM，QM 不足，FRN 即低於面值交易。
重設日 FRN 的簡化估值方法：以「當期 MRR + QM」估計未來現金流，再以「MRR + DM」貼現。更複雜的模型可得到更精確的估值。
A $100,000 FRN with a semiannual coupon pays a 180-day MRR plus a quoted margin of 120 bps. On a reset date with five years remaining to maturity, the 180-day MRR is quoted as 3.0% (annualized), and the discount margin (based on the issuer's current credit rating) is 1.5% (annualized). Estimate the value of the FRN.
- Current annualized coupon rate: 3.0% + 1.2% = 4.2%. Next semiannual coupon = 4.2% / 2 = 2.1% of face value.
- Required return per period (MRR + DM as 180-day rate): 4.5% / 2 = 2.25%.
- Using face value 100, 10 coupons of 2.1, and discount rate 2.25% per period:
N = 10; I/Y = 2.25; FV = 100; PMT = 2.1; CPT → PV = −98.67. - Value ≈ 98.67% of face value, or $98,670.
例題：FRN 估值
一張面額 $100,000、半年付息的 FRN，票息為 180 日 MRR 加 120 bps 報價利差。在某重設日，距到期尚有 5 年；當期 180 日 MRR 為年化 3.0%，依目前信用評等的貼現利差為年化 1.5%。試估值。
解答：
- 當期年化票息率＝3.0% + 1.2% = 4.2%，下一期半年票息＝4.2% ÷ 2 = 2.1%。
- 每期所需報酬率（MRR + DM 的半年率）＝4.5% ÷ 2 = 2.25%。
- 以面值 100、10 期票息 2.1、每期貼現率 2.25% 計算：N=10、I/Y=2.25、FV=100、PMT=2.1，求得 PV ≈ −98.67。
- 估值約為面值的 98.67%，即 $98,670。
Calculate and interpret yield measures for money market instruments.
For money market securities (debt maturing in a year or less), yields are quoted using various conventions:
- Some quotes use a 360-day year, others a 365-day year.
- Add-on yields = interest earned on the amount paid/deposited today.
- Discount yields = annualized current discount from the face value received at maturity.
- Bank CDs, repos, and market reference rates → typically quoted as annualized add-on rates.
- U.S. T-bills and commercial paper → quoted as annualized discounts from face value, based on a 360-day year.
The relation between an annualized add-on yield based on a 365-day year (bond equivalent yield, BEY) and the holding period yield (HPY):
$$\text{quoted add-on yield} = \text{HPY} \times \dfrac{365}{\text{days to maturity}}$$
Example: A 100-day bank CD with an add-on yield (annualized) of 1.5% based on a 365-day year. HPY = 1.5% × 100/365 = 0.41%. A $1,000 CD pays $1,004.10 in 100 days.
The relation between a quoted discount and the actual unannualized discount based on a 360-day year:
$$\text{quoted discount yield} = \text{actual discount} \times \dfrac{360}{\text{days to maturity}}$$
Example: A 180-day U.S. T-bill quoted at 2.2% (annualized) discount yield based on a 360-day year. Actual discount from face = 180/360 × 2.2% = 1.1%. A $1,000 T-bill is priced at (1 − 0.011) × 1,000 = $989. HPY = 1,000/989 − 1 = 1.11%, slightly higher than the discount of 1.1%.
An analyst should be able to convert a yield from one basis to another to compare money market securities quoted differently.
對於貨幣市場證券（到期日在一年以內的債務），殖利率的報價慣例多樣：
- 有的依 360 日計算，有的依 365 日計算。
- 附加殖利率（Add-on）＝今日存入金額所賺得的利息。
- 貼現殖利率（Discount）＝以到期面值為基礎的年化折讓率。
- 銀行存單（CD）、附買回（repo）、MRR → 通常以年化附加利率報價。
- 美國國庫券（T-bill）、商業本票 → 以面值的年化折讓率（360 日）報價。
債券等值殖利率（BEY）是以 365 日為基礎的年化附加殖利率：
quoted add-on yield = HPY × 365 / 到期天數
範例：100 日銀行 CD，年化附加利率 1.5%（365 日）。HPY = 1.5% × 100/365 = 0.41%；$1,000 的 CD 在 100 日後支付 $1,004.10。
貼現報價與實際未年化折讓的關係（360 日基礎）：
quoted discount yield = 實際折讓 × 360 / 到期天數
範例：180 日 T-bill 報價年化貼現率 2.2%（360 日）。實際折讓＝180/360 × 2.2% = 1.1%；$1,000 售價＝(1 − 0.011) × 1,000 = $989。HPY = 1,000/989 − 1 = 1.11%，略高於折讓率 1.1%。
分析師應能在不同報價基礎間轉換，以比較不同慣例下的貨幣市場證券。
- A $1,000 90-day T-bill is priced with an annualized discount of 1.2%. Calculate its market price and its annualized add-on yield based on a 365-day year.
- A $1 million negotiable CD with 120 days to maturity is quoted with an add-on yield of 1.4% based on a 365-day year. Calculate the payment at maturity and its bond equivalent yield.
- A bank deposit for 100 days is quoted with an add-on yield of 1.5% based on a 360-day year. Calculate the BEY and the yield on a semiannual bond basis.
(1) Discount from face = 1.2% × 90/360 × 1,000 = $3; price = 1,000 − 3 = $997.
Add-on yield for 90 days = 3 / 997 = 0.3009%. Annualized (365-day) = 365/90 × 0.3009% = 1.2203%. This is the BEY for a money market security.
(2) Add-on interest for 120 days = 120/365 × 1.4% = 0.4603%.
Maturity payment = $1,000,000 × (1 + 0.004603) = $1,004,603.
The quoted yield 1.4% is already the BEY (an annualized add-on yield based on 365 days).
(3) The 1.5% yield is annualized on a 360-day basis. Convert to a 365-day BEY:
BEY = (365/360) × 1.5% = 1.5208%.
To compare with the YTM of a semiannual-pay bond: convert the holding-period return to an effective semiannual yield, then double it.
100-day HPR = 1.5% × 100/360 = 0.4167%.
Effective annual yield = $1.004167^{365/100} - 1$ = 1.5294%.
Equivalent semiannual yield = $1.015294^{1/2} - 1$ = 0.7618%.
Annual yield on a semiannual bond basis = 2 × 0.7618% = 1.5236%.
Because the periodicity of the money market security (365/100) is greater than the periodicity 2 of a semiannual-pay bond, the simple annual rate (1.5%) is less than the yield on a semiannual bond basis.
例題：貨幣市場殖利率
- $1,000 的 90 日 T-bill，年化貼現率 1.2%。計算市價及 365 日基礎的年化附加殖利率。
- $1,000,000 可轉讓 CD，120 日到期，年化附加殖利率 1.4%（365 日）。計算到期支付金額與 BEY。
- 100 日銀行存款，年化附加殖利率 1.5%（360 日）。計算 BEY 及半年期債券基礎的年化殖利率。
解答：
(1) 折讓＝1.2% × 90/360 × 1,000 = $3；市價＝1,000 − 3 = $997。90 日附加殖利率＝3/997 = 0.3009%；年化（365 日）＝365/90 × 0.3009% = 1.2203%。此即貨幣市場證券的 BEY。
(2) 120 日附加利息＝120/365 × 1.4% = 0.4603%；到期支付＝$1,000,000 × 1.004603 = $1,004,603。1.4% 的報價已是 BEY（365 日基礎的年化附加殖利率）。
(3) 1.5% 為 360 日年化基礎，轉成 365 日 BEY：(365/360) × 1.5% = 1.5208%。
欲與半年付息債券的 YTM 比較：先將 HPR 換算為等值半年期殖利率再乘 2。
100 日 HPR ＝ 1.5% × 100/360 = 0.4167%；EAY = 1.004167^(365/100) − 1 ≈ 1.5294%；等值半年率 = 1.015294^(1/2) − 1 ≈ 0.7618%；半年期債券基礎年化 = 2 × 0.7618% ≈ 1.5236%。
因為貨幣市場證券的計息頻率（365/100）大於半年付息債券的 2，所以其簡單年化利率 1.5% 低於半年期債券基礎的年化殖利率。
- A. equal to par value.
- B. less than par value.
- C. greater than par value.
- A. Add-on yield based on a 365-day year.
- B. Discount yield based on a 360-day year.
- C. Discount yield based on a 365-day year.
- A. A 90-day Treasury bill quoted with a discount of 1% on a 360-day basis.
- B. A 183-day commercial paper quoted with a discount of 1% on a 365-day basis.
- C. A 91-day certificate of deposit offering an add-on rate of 1% on a 365-day basis.
90-day T-bill: discount = 1% × 90/360 = 0.25% → price = 99.75 → HPR = 100/99.75 − 1 = 0.2506% → BEY = 0.2506% × 365/90 = 1.016%.
183-day CP: discount = 1% × 183/365 = 0.5014% → price = 99.4986 → HPR = 100/99.4986 − 1 = 0.5039% → BEY = 0.5039% × 365/183 = 1.005%.
91-day CD: already quoted as BEY = 1.000%. (LOS 54.b)
Floating-rate notes (FRNs) pay a coupon equal to a fixed quoted margin (QM) over a market reference rate (MRR). The required margin (or discount margin, DM) is the extra return over MRR demanded by investors due to credit and liquidity risk. At issuance, FRNs usually have QM = DM, so the FRN is issued at par.
- When credit conditions deteriorate and DM > QM, the FRN trades below par.
- When credit conditions improve and DM < QM, the FRN trades above par.
For money market instruments, yields may be quoted on a discount basis or an add-on basis, and they may use a 360-day or 365-day year. A bond equivalent yield is an add-on yield based on a 365-day year.
【LOS 54.a】FRN 的票息＝固定報價利差（QM）＋ 市場參考利率（MRR）。所需利差／貼現利差（DM）是投資人因發行人信用與流動性風險所要求超過 MRR 的額外報酬。發行時 QM = DM，故 FRN 通常以平價發行。
- 信用品質惡化、DM > QM → FRN 低於面值交易。
- 信用品質改善、DM < QM → FRN 高於面值交易。
【LOS 54.b】貨幣市場工具的殖利率報價可能採貼現基礎或附加基礎，並使用 360 日或 365 日計息基礎。債券等值殖利率（BEY）是以 365 日為基礎的年化附加殖利率。