Reading 58

Fixed Income · Yield-Based Bond Convexity and Portfolio Properties

MODULE 58.1: YIELD-BASED BOND CONVEXITY AND PORTFOLIO PROPERTIES

LOS 58.a

Calculate and interpret convexity and describe the convexity adjustment.

Modified duration is a linear approximation of the price-yield relationship. Because the true relationship is convex, duration-based price estimates become increasingly inaccurate as the size of the yield change grows. The estimates are too low for a yield decrease and too high (in absolute value) for a yield increase.

Convexity measures the curvature of the price-yield relation. The more curved the relation, the larger the convexity adjustment needed to refine a duration-only estimate.

One way to calculate convexity is by considering each of a bond's cash flows separately. The convexity of a single cash flow at period $t$ is:

$$\text{convexity of cash flow at period } t = \frac{t(t+1)}{(1+r)^{2}}$$

where $r$ is the periodic yield of the cash flow ($\text{YTM} / \text{periodicity}$).

The convexity of a coupon-paying bond is the weighted average convexity of its individual cash flows, using the present value of each cash flow as the weight (the same weighting used for Macaulay duration).

中文翻譯

修正存續期間（modified duration）只是價格-殖利率關係的線性近似。但實際關係是凸性的，因此殖利率變動越大，僅以存續期間估計的價格誤差越大。對於殖利率下降，存續期間會低估價格上漲；對於殖利率上升，則會高估價格下跌的幅度。

凸性（Convexity）衡量價格-殖利率關係曲線的彎曲程度。曲線越彎，需要的凸性調整就越大。

單一現金流在第 $t$ 期的凸性為：

$$\text{凸性} = \frac{t(t+1)}{(1+r)^{2}}$$

其中 $r$ 為週期殖利率（YTM ÷ 每年付息次數）。

付息債券的凸性為各現金流凸性的加權平均，權重採用各現金流現值（與 Macaulay duration 同一套權重）。

Example: Convexity by Cash Flow

A 5-year, 11% annual coupon bond is priced at 86.59 to yield 15% YTM. Convexity for the Time 1 coupon is $(1\times 2)/1.15^{2}=1.512$; for the Time 2 coupon $(2\times 3)/1.15^{2}=4.537$. The full table is:

$t$	$C_t$	PV at 15%	Weight $w_t$	$t(t+1)/(1+r)^2$
1	11	9.565	0.1105	1.512
2	11	8.318	0.0961	4.537
3	11	7.233	0.0835	9.074
4	11	6.289	0.0726	15.123
5	111	55.187	0.6373	22.684
Total		86.59	1.0000

Bond Convexity

$0.1105(1.512) + 0.0961(4.537) + 0.0835(9.074) + 0.0726(15.123) + 0.6373(22.684) \approx 16.915$

教授提醒

For bonds with non-annual coupons, divide the convexity result by the periodicity squared to annualize. Example: a semiannual bond's convexity is annualized by dividing by $2^2=4$.

中文翻譯

例題：用現金流逐筆計算凸性。一張 5 年期、年付 11% 票息的債券，價格 86.59，殖利率 15%。第 1 期票息凸性 $=(1\times 2)/1.15^{2}=1.512$；第 2 期 $=(2\times 3)/1.15^{2}=4.537$；其餘類推。

各現金流以現值占比作為權重，加權後得到 債券凸性 ≈ 16.915。

教授提醒：非年付債券要把凸性除以「每年付息次數的平方」進行年化。半年付一次的債券，凸性除以 $2^{2}=4$。

Just as we approximated modified duration, we can also approximate convexity:

$$\text{approximate convexity} = \frac{V_{-} + V_{+} - 2V_{0}}{(\Delta\text{YTM})^{2}\, V_{0}}$$

where:

$V_{-}$ = price of the bond if YTM is decreased by $\Delta\text{YTM}$
$V_{+}$ = price of the bond if YTM is increased by $\Delta\text{YTM}$
$V_{0}$ = current full price of the bond

Example: Calculating Approximate Convexity

For the same 5-year, 11% bond priced at $V_{0} = 86.59138$ at 15% YTM: a +50 bp shift gives $V_{+} = 85.09217$, and a –50 bp shift gives $V_{-} = 88.12721$. Compute the approximate convexity.

Answer

$$\text{approx. convexity} = \frac{88.12721 + 85.09217 - 2(86.59138)}{(0.005)^{2}(86.59138)} \approx 16.92$$

Essentially the same as the by-cash-flow value of 16.915.

A bond's convexity is influenced by the same factors that drive duration. Longer maturity, lower coupon, or lower YTM all increase convexity, and vice versa. For two bonds with equal duration, the bond whose cash flows are more dispersed over time has greater convexity.

中文翻譯

類似於 duration 的估算，凸性也有近似公式：

$$\text{approximate convexity} = \frac{V_{-} + V_{+} - 2V_{0}}{(\Delta\text{YTM})^{2}\, V_{0}}$$

其中 $V_{-}$、$V_{+}$ 分別為殖利率下降／上升 $\Delta\text{YTM}$ 後的價格，$V_{0}$ 為目前價格。

例題：5 年期、11% 票息債券，$V_{0}=86.59138$；±50 bp 後 $V_{+}=85.09217$，$V_{-}=88.12721$。代入公式得 ≈ 16.92，與逐筆現金流計算 16.915 幾乎一致。

影響 duration 的因素也以相同方向影響凸性：到期日越長、票面利率越低、YTM 越低 → 凸性越大。在 duration 相同的情況下，現金流分布越分散，凸性越大。

LOS 58.b

Calculate the percentage price change of a bond for a specified change in yield, given the bond's duration and convexity.

Combining duration (the first-order effect) with convexity (the second-order effect) yields a substantially better estimate of the bond's percentage price change, especially for large yield moves:

$$\%\Delta\text{full price} = -\,\text{ModDur}\,(\Delta\text{YTM}) + \tfrac{1}{2}\,\text{Convexity}\,(\Delta\text{YTM})^{2}$$

Example: Estimating Price Changes With Duration and Convexity

The 5-year, 11% bond above has modified duration 3.50 and convexity 16.9. Estimate the new price for a 50 bp decrease in yield (from $V_{0}=86.59138$).

Answer

Duration effect: $-3.50 \times -0.005 = +1.75\%$
Convexity effect: $\tfrac{1}{2} \times 16.9 \times (-0.005)^{2} = 0.0002113 \approx +0.0211\%$
Total expected $\%\Delta P = 1.7711\%$
Estimated new price $= 86.59138 \times 1.017711 \approx 88.125$

Analogous to money duration (MoneyDur), the money convexity (MoneyCon) of a bond position is expressed in currency units:

$$\text{MoneyCon} = \text{annual convexity} \times \text{full price of bond position}$$

The currency change in a bond's price is then:

$$\Delta\text{full price} = -\,\text{MoneyDur}\,(\Delta\text{YTM}) + \tfrac{1}{2}\,\text{MoneyCon}\,(\Delta\text{YTM})^{2}$$

Example: Money Duration and Money Convexity

For a $\text{\$}10$ million par position in the same bond (price 86.59138 per 100 par; ModDur 3.50; convexity 16.9), find the money duration, money convexity, and the new market value for a 50 bp decrease in YTM.

Answer

Market value $= 0.8659138 \times 10{,}000{,}000 = \$8{,}659{,}138$
MoneyDur $= 3.50 \times \$8{,}659{,}138 = \$30{,}306{,}983$
MoneyCon $= 16.9 \times \$8{,}659{,}138 = \$146{,}339{,}432$
Duration effect: $-(\$30{,}306{,}983 \times -0.005) = +\$151{,}534.92$
Convexity effect: $\tfrac{1}{2} \times \$146{,}339{,}432 \times (-0.005)^{2} = +\$1{,}829.25$
Total $\Delta P = +\$153{,}364.17$
New value $\approx \$8{,}659{,}138 + \$153{,}364.17 = \$8{,}812{,}502$ (consistent with 88.125 in the previous example)

The convexity adjustment is always positive for bonds with positive convexity (option-free bonds), regardless of the direction of the yield change:

For a yield decrease, duration alone understates the price increase → convexity adjustment adds to it.
For a yield increase, duration alone overstates the price decrease → convexity adjustment reduces the loss.

中文翻譯

同時納入存續期間（一階）與凸性（二階）後，可得到更準確的價格變動率估計，特別是殖利率變動較大時：

$$\%\Delta P = -\,\text{ModDur}\,(\Delta\text{YTM}) + \tfrac{1}{2}\,\text{Convexity}\,(\Delta\text{YTM})^{2}$$

例題（百分比）：ModDur=3.50、凸性=16.9，殖利率下降 50 bp。Duration 效果 $=+1.75\%$；凸性效果 $=+0.0211\%$；合計 $+1.7711\%$，估計新價約 $88.125$。

Money Duration / Money Convexity：以貨幣單位表示。$\text{MoneyCon} = \text{年化凸性} \times \text{部位全額價值}$。價格變動：

$$\Delta P = -\,\text{MoneyDur}\,(\Delta\text{YTM}) + \tfrac{1}{2}\,\text{MoneyCon}\,(\Delta\text{YTM})^{2}$$

例題（金額）：$\text{\$}10$ 百萬面額部位，市值 $\$8{,}659{,}138$。MoneyDur $=\$30{,}306{,}983$；MoneyCon $=\$146{,}339{,}432$。50 bp 下降後 Duration 效果 $+\$151{,}535$，凸性效果 $+\$1{,}829$，合計 $+\$153{,}364$；新市值 $\approx \$8{,}812{,}502$，與百分比法 88.125 一致。

凸性調整恆為正：無嵌入選擇權的債券具正凸性，無論殖利率上升或下降，凸性調整皆為正值——降息時放大漲幅、升息時縮小跌幅。

LOS 58.c

Calculate portfolio duration and convexity and explain the limitations of these measures.

There are two approaches to estimating portfolio duration and convexity:

Aggregate cash-flow approach: compute a single duration/convexity from the portfolio's pooled cash flows. Theoretically correct but harder to implement.
Weighted-average approach: compute each bond's duration/convexity, then weight by each bond's share of total portfolio market value. Used most often in practice:

$$\text{portfolio duration} = w_{1}D_{1} + w_{2}D_{2} + \dots + w_{N}D_{N}$$

where $w_{i}$ = full price of bond $i$ ÷ total portfolio value, $D_{i}$ = duration of bond $i$, and $N$ = number of bonds. The same weighting formula is used for portfolio convexity.

教授提醒 — Limitation

The weighted-average approach assumes the YTM of every bond in the portfolio changes by the same amount—a parallel shift of the yield curve. Real-world curve moves are rarely parallel: steepening, flattening, and twist are common. Under non-parallel shifts, weighted-average duration/convexity will not give the true percentage change in portfolio value.

中文翻譯

計算投資組合的存續期間與凸性有兩種方法：

彙總現金流法：把組合中所有債券的現金流彙總後，計算單一的 duration / convexity。理論上正確，但實務上較難實施。
加權平均法：分別算出每一債券的 duration / convexity，再以各自市值占比加權平均。實務上最常用：

$$\text{組合 duration} = w_{1}D_{1} + w_{2}D_{2} + \dots + w_{N}D_{N}$$

其中 $w_{i}$ 為第 $i$ 檔債券全額價值占組合總值之比例。凸性的加權公式相同。

侷限：加權平均法假設組合中所有債券的 YTM 變動幅度相同，亦即殖利率曲線發生平行移動。但實務上曲線變動很少是純平行的，陡峭化、平坦化、扭轉均屬常見。非平行變動時，加權平均的 duration / convexity 並不能精確反映組合價值變化。

📝 Module Quiz 58.1

1. The annualized convexity of a 2-year, annual-pay, 10% coupon bond trading at par is closest to:

A. 1.65.
B. 4.66.
C. 4.96.

B — At par, $r=10\%$. Convexity at $t=1$: $(1\cdot 2)/1.10^{2}=1.653$; at $t=2$: $(2\cdot 3)/1.10^{2}=4.959$. Weights: $w_1 = 9.091/100 = 0.09091$, $w_2 = 90.909/100 = 0.90909$. Bond convexity $= 0.09091(1.653) + 0.90909(4.959) \approx 4.66$. (LOS 58.a)

2. A bond is trading at 104.4518. If its yield increases by 10 bp, the price falls to 103.9954; if its yield decreases by 10 bp, the price rises to 104.9108. The approximate convexity is closest to:

A. 12.45.
B. 24.89.
C. 49.78.

B — $\dfrac{104.9108 + 103.9954 - 2(104.4518)}{(0.001)^{2}(104.4518)} \approx 24.89$. (LOS 58.a)

3. The annualized convexity of a 2-year, semiannual-pay, 10% coupon bond trading at par is closest to:

A. 4.12.
B. 4.66.
C. 16.47.

A — At par with $r=5\%$ per period. Cash-flow convexities: $1.8141, 5.4422, 10.8844, 18.1406$ with weights $0.0476, 0.0454, 0.0432, 0.8638$. Periodic convexity $\approx 16.47$. Annualized: $16.47/2^{2} = 4.12$. (LOS 58.a)

4. A bond has a convexity of 114.6. The convexity effect on price for a 110 bp decrease in yield is closest to:

A. −1.673%.
B. +0.693%.
C. +1.673%.

B — Convexity effect $= \tfrac{1}{2}\times 114.6 \times (-0.011)^{2} = 0.00693 = 0.693\%$. The convexity adjustment is positive regardless of the direction of the yield change. (LOS 58.b)

5. Portfolio duration based on weighted-average durations of constituents assumes:

A. yields change uniformly across all maturities.
B. the portfolio does not include bonds with embedded options.
C. the portfolio's internal rate of return equals its cash flow yield.

A — The weighted-average method assumes a parallel shift of the yield curve, i.e., the discount rate at every maturity changes by the same amount. (LOS 58.c)

Key Concepts — Reading 58

LOS 58.a

Convexity measures the curvature of the price-yield relation. Convexity of a single cash flow at period $t$:

$$\frac{t(t+1)}{(1+r)^{2}}$$

The convexity of a coupon bond is the weighted average of its cash-flow convexities (PV weights). For non-annual coupons, divide by periodicity squared to annualize. Approximate convexity:

$$\text{approx. convexity} = \frac{V_{-} + V_{+} - 2V_{0}}{(\Delta\text{YTM})^{2}\, V_{0}}$$

LOS 58.b

Percentage price change including convexity:

$$\%\Delta P = -\,\text{ModDur}\,(\Delta\text{YTM}) + \tfrac{1}{2}\,\text{Convexity}\,(\Delta\text{YTM})^{2}$$

Money convexity (currency units): $\text{MoneyCon} = \text{annual convexity} \times \text{full price of position}$.

$$\Delta P = -\,\text{MoneyDur}\,(\Delta\text{YTM}) + \tfrac{1}{2}\,\text{MoneyCon}\,(\Delta\text{YTM})^{2}$$

The convexity adjustment is positive for both yield increases and decreases (positive-convexity bonds).

LOS 58.c

Two methods for portfolio duration and convexity:

Single measure based on the portfolio's aggregate cash flows (theoretically correct).
Weighted average of the individual bonds' durations / convexities, weighted by market-value share — used most often in practice but assumes a parallel shift of the yield curve.

中文翻譯 — 重點整理

【LOS 58.a】凸性衡量價格-殖利率曲線的彎曲度。單一現金流凸性 $=t(t+1)/(1+r)^{2}$；債券凸性為各現金流凸性的 PV 加權平均。非年付債券需除以「每年付息次數平方」以年化。近似凸性公式 $=(V_{-}+V_{+}-2V_{0})/[(\Delta YTM)^{2} V_{0}]$。

【LOS 58.b】含凸性的價格變動率：$\%\Delta P = -\text{ModDur}\cdot \Delta YTM + \tfrac{1}{2}\text{Convexity}\cdot (\Delta YTM)^{2}$。Money Convexity 以貨幣單位表示，$\text{MoneyCon}=\text{年凸性}\times \text{部位全額價值}$。對於正凸性債券，凸性調整無論升降息皆為正值。

【LOS 58.c】計算組合 duration / convexity 的兩種方法：①以組合彙總現金流計算（理論正確）；②以各債券市值占比作加權平均（實務常用，但假設殖利率曲線平行移動）。