Reading 57

Fixed Income · Yield-Based Bond Duration Measures and Properties

MODULE 57.1: YIELD-BASED BOND DURATION MEASURES AND PROPERTIES

LOS 57.a

Define, calculate, and interpret modified duration, money duration, and the price value of a basis point (PVBP).

Modified duration (ModDur) is calculated as Macaulay duration (MacDur) divided by one plus the bond's periodic rate of return (YTM divided by periodicity).

For an annual-pay bond, the general form is:

$$\text{ModDur} = \dfrac{\text{MacDur}}{1 + \text{YTM}}$$

For a semiannual-pay bond with a YTM quoted on a semiannual bond basis:

$$\text{ModDur}_{\text{SEMI}} = \dfrac{\text{MacDur}_{\text{SEMI}}}{1 + \text{YTM}/2}$$

Example: Modified Duration

A 5-year, 11% annual-coupon bond priced at 86.59 to yield 15% to maturity has a Macaulay duration of 4.03. Calculate the modified duration of this bond.

Answer

Because it is an annual-coupon bond (periodicity = 1):

$$\text{ModDur} = \dfrac{4.03}{1.15} = 3.50$$

Modified duration provides an estimate of the percentage change in a bond's price for a 1% change in YTM:

$$\%\Delta P \approx -\text{ModDur} \times \Delta\text{YTM}$$

Based on a ModDur of 3.50, a 0.5% increase in YTM should cause the bond price to fall by approximately $3.50 \times 0.5\% = 1.75\%$. The estimated price is $86.59 \times (1 - 0.0175) = 85.075$, close to the directly calculated value at YTM = 15.5% of 85.092:

N = 5; I/Y = 15.5; FV = 100; PMT = 11; CPT → PV = −85.092

For a semiannual-pay bond, $\text{ModDur}_{\text{SEMI}}$ can be annualized by dividing by two before being applied to a 1% change in the bond's annualized YTM.

中文翻譯

修正存續期間（ModDur）＝ Macaulay 存續期間 ÷（1 + 每期報酬率），其中每期報酬率＝ YTM ÷ 付息頻率。

年付息債券：ModDur = MacDur ÷ (1 + YTM)
半年付息債券：ModDur_SEMI = MacDur_SEMI ÷ (1 + YTM/2)

例：5 年期、年付息 11%、價格 86.59、YTM 15%、MacDur 4.03 的債券：ModDur = 4.03 ÷ 1.15 = 3.50。

ModDur 用來估計 YTM 變動 1% 時債券價格的百分比變動：%ΔP ≈ −ModDur × ΔYTM。例如 YTM 上升 0.5%，價格估計下跌 3.50 × 0.5% = 1.75%，從 86.59 跌到約 85.075，與直接重算得到的 85.092 相近。半年付息債券將 ModDur_SEMI 除以 2 後再用於年化 YTM 1% 變動的估計。

Approximate Modified Duration

We can approximate modified duration directly using bond values calculated for an increase in YTM and a decrease in YTM of the same size. Let:

$V_-$ = the bond price when YTM is decreased by $\Delta\text{YTM}$
$V_+$ = the bond price when YTM is increased by $\Delta\text{YTM}$
$V_0$ = the current price of the bond

$$\text{Approximate ModDur} = \dfrac{V_- - V_+}{2 \times V_0 \times \Delta\text{YTM}}$$

The numerator $V_- - V_+$ is divided by 2 to take the average of the magnitudes of the price increase and decrease. $V_0$ in the denominator converts that average dollar change into a percentage, and the $\Delta\text{YTM}$ term scales the duration measure to a 1% change in yield. Note: $\Delta\text{YTM}$ must be entered as a decimal (e.g., 0.005 for 50 bps), not as a whole percentage.

Example: Calculating Approximate ModDur

Consider a 5-year, 11% annual-coupon bond priced at 86.59 to yield 15% to maturity. If its YTM rises by 50 bps, the price falls to 85.092. If its YTM falls by 50 bps, the price rises to 88.127. Calculate the approximate ModDur.

Answer

$$\text{Approx. ModDur} = \dfrac{88.127 - 85.092}{2 \times 86.59 \times 0.005} = 3.505$$

The approximate price change for a 1% change in YTM is therefore 3.505%. This result is very close to the ModDur of 3.50 calculated earlier.

Modified duration is a linear estimate of the relationship between a bond's price and YTM, whereas the actual price-yield relationship is convex—not a straight line. ModDur therefore provides good estimates for small yield changes, but increasingly poor estimates for larger yield changes as the curvature of the price-yield curve becomes more pronounced. Figure 57.1 illustrates this idea.

Figure 57.1: Approximate ModDur as a Linear (Tangent) Estimate of a Convex Price-Yield Curve

Linear ModDur lies below the convex curve at every other yield level, so ModDur over-estimates the price drop when YTM rises and under-estimates the price gain when YTM falls. The error grows with $|\Delta\text{YTM}|$.

中文翻譯

近似修正存續期間：先把 YTM 上升、下降同樣幅度後重算債券價格，再代入公式：

近似 ModDur = (V₋ − V₊) ÷ (2 × V₀ × ΔYTM)

V₋：YTM 下降 ΔYTM 時的債券價格
V₊：YTM 上升 ΔYTM 時的債券價格
V₀：當前價格

分子除以 2 是取漲跌幅度的平均；除以 V₀ 把美元變動換算為百分比；除以 ΔYTM 把存續期間調為「對應 1% 殖利率變動」的尺度。注意 ΔYTM 須以小數代入（50 bps 寫成 0.005）。

例：5 年期 11% 年付息債券，當前價 86.59、YTM 15%；YTM 升 50 bps 跌至 85.092、降 50 bps 漲至 88.127。近似 ModDur = (88.127 − 85.092) / (2 × 86.59 × 0.005) = 3.505，與前例 3.50 相近。

ModDur 是價格-殖利率關係的線性估計，而實際關係是凸的。因此 ModDur 對小幅 YTM 變動估計準確，對大幅變動誤差會放大（見圖 57.1）。線性切線位於凸曲線下方：YTM 上升時 ModDur 高估跌幅、下降時低估漲幅。

Money Duration

The money duration of a bond position (also called dollar duration) is expressed in currency units:

$$\text{Money Duration} = \text{annual ModDur} \times \text{full price of bond position}$$

Multiplying money duration by a given change in YTM (expressed as a decimal) provides an estimate of the change in the bond position's value for that change in YTM.

Example: Money Duration

1. Calculate the money duration on a coupon date of a $2,000,000 par-value bond that has a ModDur of 7.42 and a full price of 101.32, expressed both for the whole bond and per $100 of face value.
2. What will be the impact on the value of the bond of a 25 bp increase in its YTM?

Answer

1. Money duration for the whole bond:

$$7.42 \times \$2{,}000{,}000 \times 101.32\% = \$15{,}035{,}888$$

Per $100 of par value:

$$7.42 \times 101.32 = \$751.79$$

(Equivalently, $\$15{,}035{,}888 \div (\$2{,}000{,}000/\$100) = \$751.79$.)

2. Estimated change in value:

$$\$15{,}035{,}888 \times 0.0025 = \$37{,}589.72$$

The bond value decreases by $\$37{,}589.72$.

中文翻譯

美元存續期間（Money Duration / Dollar Duration）以貨幣單位表示：

Money Duration = 年化 ModDur × 債券持倉的完整價格（full price）

把 Money Duration 乘以 YTM 變動（小數），即得到該變動下持倉價值的估計變動金額。

例：面額 $2,000,000、ModDur 7.42、full price 101.32 的債券：

整檔債券的 Money Duration = 7.42 × $2,000,000 × 101.32% = $15,035,888
每 $100 面額的 Money Duration = 7.42 × 101.32 = $751.79

YTM 上升 25 bps 時，價值估計下跌 = $15,035,888 × 0.0025 = $37,589.72。

Price Value of a Basis Point (PVBP)

The price value of a basis point (PVBP) is the money change in the full price of a bond when its YTM changes by one basis point (0.01%). It can be computed directly as the average of the price decrease when YTM rises 1 bp and the price increase when YTM falls 1 bp:

$$\text{PVBP} = \dfrac{V_- - V_+}{2} \times \text{par value} \times 0.01$$

Equivalently, $\text{PVBP} = \text{Money Duration} \times 0.0001$.

Example: Calculating PVBP

A newly issued, 20-year, 6% annual-pay straight bond is priced at 101.39. Calculate the PVBP for this bond, assuming a par value of $1,000,000.

Answer

First, find the YTM:

N = 20; PV = −101.39; PMT = 6; FV = 100; CPT → I/Y = 5.88

Now find prices at YTMs of 5.89% and 5.87%:

I/Y = 5.89; CPT → PV = −101.273 ($V_+$)
I/Y = 5.87; CPT → PV = −101.507 ($V_-$)

$$\text{PVBP (per \$100 par)} = \dfrac{101.507 - 101.273}{2} = 0.117$$

For the $1,000,000 par-value bond, each 1 bp change in YTM changes the bond's price by $0.117 \times \$1{,}000{,}000 \times 0.01 = \$1{,}170$.

中文翻譯

PVBP（每基點價格變動）＝ YTM 變動 1 bp（0.01%）時，債券完整價格的貨幣變動。直接計算法是取「YTM 上升 1 bp 跌幅」和「YTM 下降 1 bp 漲幅」的平均：

PVBP = (V₋ − V₊) / 2 × 面額 × 0.01；亦可寫成 PVBP = Money Duration × 0.0001。

例：新發行 20 年期、年息 6%、價格 101.39 的直線債券，面額 $1,000,000。先解 YTM 得 5.88%；再以 5.89%、5.87% 重算價格分別為 101.273（V₊）、101.507（V₋）。

每 $100 面額的 PVBP = (101.507 − 101.273) / 2 = 0.117；對 $1,000,000 面額，每 1 bp 殖利率變動使債券價格變動 0.117 × $1,000,000 × 0.01 = $1,170。

LOS 57.b

Explain how a bond's maturity, coupon, and yield level affect its interest rate risk.

Maturity. Other things equal, bonds with longer maturity usually have higher interest rate risk. The present values of payments made further in the future are more sensitive to changes in the discount rate than are the present values of nearer-term payments.

教授提醒

We say usually because for a discount coupon bond, an increase in maturity can actually decrease Macaulay duration over a range of long maturities. For a discount bond, duration first rises with longer maturity and then falls, eventually approaching the duration of a perpetuity, $(1 + \text{YTM}) / \text{YTM}$.

Between coupon dates, if the YTM remains constant, Macaulay duration decreases smoothly with the passage of time and then jumps up slightly at each coupon payment date as the time to the next coupon resets to a full coupon period.

Coupon rate. Other things equal, a higher coupon rate decreases a bond's interest rate risk. For a given maturity and YTM, a zero-coupon bond has greater duration than a coupon bond. A higher coupon means more of the bond's value comes from payments received sooner, so price is less sensitive to changes in yield.

Floating-rate notes (FRNs). For an FRN whose coupons reset periodically to a market reference rate (MRR), a rise in interest rates also raises the coupon, limiting price risk. Macaulay duration of an FRN is approximately the time remaining to the next coupon reset date.

Yield level. Other things equal, an increase (decrease) in YTM decreases (increases) a bond's interest rate risk. Looking at the convex price-yield curve, the slope is steeper at lower yields and flatter at higher yields, so price is more sensitive to a given yield change when yields are lower.

教授提醒 — Quick Recap

Holding all else equal:

↑ Maturity ⇒ ↑ Duration (with the discount-bond caveat above)
↑ Coupon ⇒ ↓ Duration
↑ YTM ⇒ ↓ Duration
Time passing between coupons ⇒ Duration drifts down, then bumps up at the next coupon date

中文翻譯

到期日：其他條件不變時，到期日越長通常利率風險越大，因為較遠期現金流的現值對折現率較敏感。但折價債券例外——其 Macaulay duration 隨到期日先上升後下降，最終逼近永續債的 (1 + YTM) / YTM。在兩個付息日之間若 YTM 不變，Macaulay duration 會隨時間逐漸下降，每次發放息票後又微幅上跳（因到下一次配息又回到完整一期）。

票面利率：其他條件不變時，票面利率越高利率風險越低。相同到期日與 YTM 下，零息債券的 duration 大於附息債券。票面利率高代表更多價值來自早期現金流，對殖利率變動較不敏感。

浮動利率債券（FRN）：票息會週期性重設為市場參考利率，利率上升時票息也上升，價格風險很低。FRN 的 Macaulay duration 近似為「距離下一次重設日的剩餘時間」。

殖利率水準：其他條件不變時，YTM 越高利率風險越低（duration 越小）。從凸的價格-殖利率曲線看，低殖利率區斜率較陡，價格對殖利率變動較敏感。

速記：到期日↑ → duration↑（折價債券例外）；票息↑ → duration↓；YTM↑ → duration↓；付息期間時間流逝 → duration 緩降，每逢配息日小幅上跳。

📝 Module Quiz 57.1

1. A 14% annual-pay coupon bond has six years to maturity. The bond is currently trading at par. Using a 25 basis point change in yield, the approximate modified duration of the bond is closest to:

A. 0.392.
B. 3.888.
C. 3.970.

B — Trading at par means YTM = 14%. With Δy = 0.0025:
$V_-$: N=6, PMT=14, FV=100, I/Y=13.75 → PV = −100.979
$V_+$: I/Y=14.25 → PV = −99.035
$V_0 = 100.000$.
Approx. ModDur = (100.979 − 99.035) / (2 × 100 × 0.0025) = 3.888. (LOS 57.a)

2. The current price of a $1,000, 7-year, 5.5% semiannual coupon bond is $1,029.23. The bond's price value of a basis point is closest to:

A. $0.05.
B. $0.60.
C. $5.74.

B — First find YTM: PV = −1,029.23, FV = 1,000, PMT = 27.5, N = 14 → I/Y ≈ 2.500 (semiannual), or 5.00% annual.
At 5.01%: I/Y = 2.505 → PV = $1,028.63.
At 4.99%: I/Y = 2.495 → PV = $1,029.82.
PVBP = ($1,029.82 − $1,028.63) / 2 = $0.595. (LOS 57.a)

3. The modified duration of a bond is 7.87. The approximate percentage change in price using duration only for a yield decrease of 110 basis points is closest to:

A. −8.657%.
B. +7.155%.
C. +8.657%.

C — %ΔP ≈ −ModDur × ΔYTM = −7.87 × (−1.10%) = +8.657%. (LOS 57.a)

4. All else equal, which of the following bonds is likely to have the highest price risk?

A. A 10-year maturity semiannual-pay floating-rate note.
B. A 2-year zero-coupon bond.
C. A 2-year 10% semiannual-pay bond.

B — The FRN's price risk is very low because coupons reset to market rates and the price returns toward par at each reset. Among the two 2-year bonds, lower coupons mean greater price risk, so the zero-coupon bond has the highest. (LOS 57.b)

Key Concepts — Reading 57

LOS 57.a

Modified duration is a linear estimate of the percentage change in a bond's price that would result from a 1% change in YTM:

$$\text{ModDur} = \dfrac{\text{MacDur}}{1 + \text{periodic return}}$$

$$\%\Delta P \approx -\text{ModDur} \times \Delta\text{YTM}$$

ModDur can be approximated by repricing the bond at slightly higher and lower yields:

$$\text{Approx. ModDur} = \dfrac{V_- - V_+}{2 \times V_0 \times \Delta\text{YTM}}$$

Money duration is stated in currency units (sometimes per 100 of bond value):

$$\text{Money Duration} = \text{annual ModDur} \times \text{full price of bond position}$$

PVBP is the change in a bond's value, in currency units, for a 1 bp change in YTM:

$$\text{PVBP} = \dfrac{V_- - V_+}{2} \times \text{par value} \times 0.01 = \text{Money Duration} \times 0.0001$$

LOS 57.b

Holding other factors constant:

Duration increases when maturity increases (with a discount-bond exception over very long maturities).
Duration decreases when the coupon rate increases.
Duration decreases when YTM increases.
Duration decreases as time passes between coupon dates, but increases slightly on coupon payment dates.
For floating-rate notes, Macaulay duration ≈ time to the next coupon reset; price risk is therefore low.

中文翻譯 — 重點整理

【LOS 57.a】

修正存續期間 ModDur = MacDur ÷ (1 + 每期報酬率)；%ΔP ≈ −ModDur × ΔYTM
近似 ModDur = (V₋ − V₊) ÷ (2 × V₀ × ΔYTM)
美元存續期間 Money Duration = 年化 ModDur × 債券持倉完整價格
PVBP = (V₋ − V₊) / 2 × 面額 × 0.01；亦可寫為 Money Duration × 0.0001

【LOS 57.b】其他條件不變時：

到期日↑ → duration↑（折價債券在長到期區間例外）
票息↑ → duration↓
YTM↑ → duration↓
付息期間時間流逝 → duration 緩降；配息當日小幅上跳
FRN 的 Macaulay duration ≈ 距下次重設日的時間，價格風險很低