Reading 59

Fixed Income · Curve-Based and Empirical Fixed-Income Risk Measures

MODULE 59.1: CURVE-BASED AND EMPIRICAL FIXED-INCOME RISK MEASURES

LOS 59.a

Explain why effective duration and effective convexity are the most appropriate measures of interest rate risk for bonds with embedded options.

LOS 59.b

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

So far, all of our duration measures have been calculated using the YTM and prices of straight (option-free) bonds. This is straightforward because both the future cash flows and their timing are known. This is not the case with bonds that have embedded options, such as callable bonds, putable bonds, or a mortgage-backed security (MBS). Embedded options may bring about the early termination of a bond, either at the choice of the investor (for a putable bond) or at the choice of the issuer or underlying borrowers (for callable bonds and MBSs).

教授提醒

A callable bond gives the issuer the right to buy the bond back before maturity, and is equivalent to a straight (option-free) bond plus a short call option position.

A putable bond gives the investor the right to sell the bond back to the issuer before maturity, and is equivalent to a straight bond plus a long put option position.

MBSs resemble callable bonds because mortgage borrowers have the right to prepay their loans.

中文翻譯

到目前為止，我們所學的所有存續期間衡量都是用直接（無期權）債券的YTM與價格計算。因為這類債券的未來現金流與其發生時點都是已知，計算很直接。但對於含嵌入選擇權的債券（可贖回債券callable、可賣回債券putable、不動產抵押貸款證券MBS）情況就不同──嵌入選擇權可能導致債券提前終止，可能由投資人（putable）或由發行人／借款人（callable、MBS）行使。

教授提醒：

可贖回債券＝直接債券＋空頭買權（發行人有權於到期前買回）。
可賣回債券＝直接債券＋多頭賣權（投資人有權於到期前賣回）。
MBS類似可贖回債券，因為房貸借款人擁有提前清償權。

Because bonds with embedded options have uncertain future cash flows and redemption dates, they do not have a single well-defined yield. The yield depends on whether the embedded option is exercised (e.g., a callable bond has a YTM and a yield to each call date).

Therefore, analyzing interest rate risk for bonds with embedded options is based on shifts in the benchmark curve (e.g., government par rates), rather than on changes in the bond's own yield. This measure of price sensitivity is called effective duration (EffDur).

Calculating effective duration is the same as calculating approximate modified duration, but we replace the change in YTM with $\Delta\text{Curve}$, the change in the benchmark yield curve (used with a bond pricing model to generate $V_-$ and $V_+$):

$$\text{EffDur} = \dfrac{V_- - V_+}{2\,V_0\,\Delta\text{Curve}}$$

Another difference between effective duration and the methods discussed earlier is that effective duration separates the effects of changes in benchmark yields from changes in the spread for credit and liquidity risk. Modified duration makes no such distinction. Effective duration reflects only the sensitivity of the bond's value to changes in the benchmark yield curve, and assumes all else (including spreads) remains the same.

中文翻譯

含嵌入選擇權的債券，因未來現金流及到期日不確定，沒有唯一明確的收益率（callable 既有 YTM 也有 yield-to-call）。因此，這類債券的利率風險分析必須改以基準曲線（例如政府公債 par rate）的變動為基礎，而非以該債券自身收益率的變動為基礎，這種敏感度衡量稱為有效存續期間（EffDur）。

計算方式與「近似修正存續期間」相同，但把 ΔYTM 替換為基準曲線變動 $\Delta\text{Curve}$（再用定價模型推算 $V_-$、$V_+$）：

$\text{EffDur} = \dfrac{V_- - V_+}{2\,V_0\,\Delta\text{Curve}}$

另外，有效存續期間會把基準殖利率變動與信用／流動性利差變動的影響分開；修正存續期間則不區分。EffDur 只衡量基準曲線變動對債券價值的影響，並假設其他條件（含利差）不變。

When calculating convexity for bonds with embedded options, we use an analogous measure, effective convexity (EffCon), also based on changes in the benchmark curve rather than YTM:

$$\text{EffCon} = \dfrac{V_- + V_+ - 2V_0}{(\Delta\text{Curve})^2 \cdot V_0}$$

While the convexity of any option-free bond is positive, a callable bond can exhibit negative convexity. At low yields the call option becomes more valuable and the call price puts an effective ceiling on increases in bond value (see Figure 59.1). For a bond with negative convexity, the price increase from a decrease in YTM is smaller than the price decrease from an equal-sized increase in YTM. Hence the duration of a callable bond is less than that of an equivalent option-free bond at low yields.

中文翻譯

計算含嵌入選擇權債券的凸性時，使用對應的有效凸性（EffCon），同樣以基準曲線變動為基礎：

$\text{EffCon} = \dfrac{V_- + V_+ - 2V_0}{(\Delta\text{Curve})^2 \cdot V_0}$

無期權債券的凸性必為正；但可贖回債券在低殖利率時會呈現負凸性。低殖利率下，買權價值升高，贖回價對債券價格形成上限（圖 59.1）。負凸性意味著「殖利率下降帶來的漲幅」小於「等量殖利率上升帶來的跌幅」，因此低殖利率時可贖回債券的存續期間比同條件無期權債券短。

Figure 59.1: Price-Yield Function — Callable vs. Option-Free Bond (Call Price = 102)

[圖表描述] 同一張價格–殖利率圖上繪出兩條曲線：

無期權債券：標準的凸性向上之曲線（隨殖利率下降，價格無上限地上升）。
可贖回債券：在高殖利率區間與無期權債券幾乎重疊；當殖利率下降至接近贖回觸發水準時，價格被「贖回價（102）」壓平，曲線出現向下凹的負凸性區段。

兩條曲線在低殖利率時相距最大（買權價值＝兩條曲線之差）。

A putable bond always has positive convexity. At higher yields the put becomes more valuable, so the value of the putable bond decreases less than that of an option-free bond as yield rises (see Figure 59.2). This means the duration of a putable bond is less than that of an equivalent option-free bond at high yields.

Figure 59.2: Price-Yield Curves — Option-Free vs. Putable Bond

[圖表描述] 同一張價格–殖利率圖：

無期權債券：標準凸性向上曲線。
可賣回債券：低殖利率時與無期權債券重疊；殖利率上升時，價格因「賣回價」形成下限而下跌幅度受限，但曲線仍維持正凸性（不會出現負凸性）。

兩條曲線在高殖利率時差距最大（賣權價值）。

中文翻譯

圖 59.1（贖回價＝102）：可贖回債券在低殖利率區間，因贖回價形成價格上限，曲線變平甚至向下凹（負凸性）；高殖利率區間與無期權債券近乎重合。

可賣回債券始終為正凸性。高殖利率時賣權價值升高，曲線下跌速度因賣回價形成的下限而放緩（圖 59.2）。因此高殖利率時可賣回債券的存續期間，比同條件無期權債券短。

For an option-free bond, small differences can be observed between modified duration (with respect to $\Delta\text{YTM}$) and effective duration (with respect to $\Delta\text{Curve}$). It might seem natural to assume that a shift in the government par yield curve would automatically flow through into a similar change in risky bond yields. However, in a non-flat yield curve environment, a shift in the benchmark par yield curve generates a nonparallel shift in the government spot curve. If credit spreads above government spot rates remain the same, the change in the risky bond's yield will be slightly different from the original par-curve shift, causing ModDur to differ slightly from EffDur.

教授提醒

Bond yields are a weighted average of the spot rates that apply to the bond's individual cash flows. Shifting the par curve necessarily changes the spot curve, but not all spot rates have equal weight in any one par yield, so the spot curve undergoes a nonparallel shift. Consequently, yields on risky bonds (assumed to keep a constant spread over the spot curve) may not move exactly in line with the original $\Delta\text{Curve}$.

For the exam: for option-free bonds, ModDur and EffDur are not exactly the same for a given $\Delta\text{Curve}$ unless the yield curve is flat.

We can estimate the expected price change for a bond using EffDur and EffCon, analogously to ModDur and convexity:

$$\dfrac{\Delta P}{P} \approx -\text{EffDur} \times \Delta\text{Curve} + \tfrac{1}{2}\,\text{EffCon} \times (\Delta\text{Curve})^2$$

Unlike modified duration and convexity, effective duration and convexity do not necessarily provide better estimates of bond prices for smaller changes in yield. For bonds with embedded options, considerations beyond the level of government rates determine whether the option will be exercised (e.g., the level of credit spreads on a corporate bond, or the principal outstanding on a mortgage).

中文翻譯

即使是無期權債券，修正存續期間（對 ΔYTM）與有效存續期間（對 ΔCurve）也會有微小差異。直覺上以為政府 par 曲線的平移會等量傳導到風險債券殖利率，但在非平坦曲線環境下，par 曲線的平移會造成非平行的即期（spot）曲線變動；若假設信用利差固定，則風險債券殖利率的變動會與原始 par 曲線變動略有不同，使 ModDur 與 EffDur 出現微小差異。

教授提醒：債券殖利率是其各期現金流對應 spot 利率的加權平均；不同到期 spot 對某一 par yield 的權重不同，因此 par 曲線移動會導致 spot 曲線的非平行移動。考試重點：無期權債券下，除非殖利率曲線完全平坦，否則 ModDur 與 EffDur 對相同 ΔCurve 並不完全相等。

價格變動估計式：

$\dfrac{\Delta P}{P} \approx -\text{EffDur}\cdot\Delta\text{Curve} + \tfrac{1}{2}\,\text{EffCon}\cdot(\Delta\text{Curve})^2$

另需注意：對含嵌入選擇權債券而言，EffDur／EffCon 對較小殖利率變動未必更精準，因為選擇權是否被執行還取決於政府利率以外的因素（例如公司債的信用利差、抵押貸款的本金餘額等）。

LOS 59.c

Define key rate duration and describe its use to measure price sensitivity of fixed-income instruments to benchmark yield curve changes.

Effective duration is an adequate measure of bond price risk only for parallel shifts in the benchmark yield curve. The impact of nonparallel shifts can be measured using key rate duration. A key rate duration (also called a partial duration) is the sensitivity of the value of a bond or portfolio to a change in the benchmark yield at a specific maturity, holding other yields constant. The sum of a bond's key rate durations equals its effective duration.

Key rate duration is particularly useful for measuring shaping risk—the effect of a nonparallel shift in the yield curve on a bond portfolio. We use the key rate duration for each maturity to compute the effect on the portfolio of a yield change at that maturity; the overall effect is the sum of the individual effects.

The key rate duration of a cash flow in a portfolio equals the cash flow's modified duration multiplied by its weight in the portfolio.

中文翻譯

LOS 59.c：EffDur 僅適用於基準曲線的平行移動。對非平行移動，需用關鍵利率存續期間（key rate duration，又稱 partial duration）──衡量在「其他到期點殖利率不變、僅某一特定到期點殖利率變動」下，債券或投組價值的敏感度。各關鍵利率存續期間之和＝該券的有效存續期間。

關鍵利率存續期間特別適合衡量形狀風險（shaping risk），即殖利率曲線非平行變動對投組的影響：以各到期點的關鍵利率存續期間，分別計算該到期點殖利率變動對投組的影響，再加總。

投組中某一現金流的關鍵利率存續期間＝該現金流的修正存續期間 × 其在投組中的權重。

Example: Key rate duration

A portfolio has equally weighted investments in a 5-year zero-coupon bond yielding 5% and a 10-year zero-coupon bond yielding 6% (annual coupon basis). What is the performance of the portfolio if 5-year yields increase by 50 bps and 10-year yields decrease by 25 bps?

Answer:

Recall that for a single cash flow, $\text{ModDur} = \dfrac{\text{Macaulay Dur}}{1+y}$, and Macaulay duration of a single cash flow equals its maturity.

5-year cash flow: $\text{ModDur} = \dfrac{5}{1.05} = 4.762$. Key rate duration $= 4.762 \times 0.5 = 2.381$. Impact of +50 bps $= -2.381 \times 0.0050 = -0.0119 = -1.19\%$.

10-year cash flow: $\text{ModDur} = \dfrac{10}{1.06} = 9.434$. Key rate duration $= 9.434 \times 0.5 = 4.717$. Impact of −25 bps $= -4.717 \times (-0.0025) = 0.0118 = +1.18\%$.

Total: $-1.19\% + 1.18\% = \mathbf{-0.01\%}$ — portfolio value remains roughly unchanged in response to this nonparallel shift.

中文翻譯

例題：關鍵利率存續期間。投組以等權重持有 5 年期零息債（YTM 5%）與 10 年期零息債（YTM 6%）。若 5 年殖利率上升 50bps、10 年殖利率下降 25bps，投組績效為何？

解答：單期現金流 Macaulay Dur ＝期限；ModDur ＝ Mac／(1+y)。

5 年：ModDur ＝ 5/1.05 ＝ 4.762；KRD ＝ 4.762 × 0.5 ＝ 2.381；影響＝ −2.381 × 0.005 ＝ −1.19%。
10 年：ModDur ＝ 10/1.06 ＝ 9.434；KRD ＝ 9.434 × 0.5 ＝ 4.717；影響＝ −4.717 × (−0.0025) ＝ +1.18%。

合計：−1.19% ＋ 1.18% ＝ −0.01%。對此非平行變動，投組價值幾乎不變。

LOS 59.d

Describe the difference between empirical duration and analytical duration.

The duration measures we have introduced so far—based on mathematical analysis—are often referred to as analytical durations. A different approach is to estimate empirical durations using the actual observed historical relationship between benchmark yield changes and bond price changes.

When we estimate corporate bond durations based on a shift in the benchmark (government) yield curve, we implicitly assume that the credit spread for the corporate bond remains unchanged—i.e., changes in the benchmark yield curve and the bond's yield spread are uncorrelated. When this assumption is not justified, empirical-duration estimates—based on actual observed price–yield behavior—may be more appropriate.

An example is a "flight to quality": in a market-uncertainty episode, investor demand shifts sharply toward low-credit-risk bonds. Government yields fall while credit spreads widen at the same time. Government bond prices rise, but corporate bond prices rise less—or possibly fall. For a corporate bond portfolio, an empirical-duration estimate accounting for this effect would be lower (less price response to a decrease in benchmark yields) than an analytical-duration estimate would indicate. An analytical estimate is still appropriate for portfolios of government securities; an empirically-derived estimate is more appropriate for credit-risky corporate bond portfolios.

中文翻譯

LOS 59.d：前述以數學模型計算的存續期間統稱為解析型存續期間（analytical duration）；另一種方式是從歷史價格與殖利率資料的實際關係，迴歸估計經驗型存續期間（empirical duration）。

用基準殖利率曲線變動估算公司債存續期間時，隱含假設「公司債信用利差不變」，亦即基準曲線變動與信用利差變動無相關。若此假設不成立，採用以實際觀察數據估算的經驗型存續期間更合適。

典型情境：「品質飛升（flight to quality）」。市場不確定性升高時，資金湧向低信用風險債券，政府債殖利率下降但信用利差同時擴大。政府債價格上漲，公司債價格漲幅較小、甚至下跌。考慮此效應的經驗型存續期間，會低於解析型估計值（亦即公司債對基準殖利率下降的價格反應較小）。對政府債組合，解析型估計仍合適；對信用風險公司債組合，經驗型估計更合適。

Module Quiz 59.1

1. Effective duration is more appropriate than modified duration for estimating interest rate risk for bonds with embedded options because these bonds:

A. tend to have greater credit risk than option-free bonds.
B. exhibit high convexity that makes modified duration less accurate.
C. have uncertain cash flows that depend on the path of interest rate changes.

C — Because bonds with embedded options have cash flows that are uncertain and depend on future interest rates, effective duration must be used. (LOS 59.a)

2. When an embedded option has significant value, relative to an equivalent option-free bond, the effective duration of a bond with an embedded option will most likely be:

A. lower.
B. the same.
C. higher.

A — Embedded options decrease the duration of the bond relative to an equivalent option-free bond when the option has significant value. (LOS 59.b)

3. Which of the following bonds is most likely to exhibit negative convexity?

A. Callable bonds in a low-yield environment.
B. Callable bonds in a high-yield environment.
C. Putable bonds in a high-yield environment.

A — Negative convexity is exhibited only by callable bonds in a low-yield environment, where the call price creates a ceiling on the price of the callable bond and slows the rate at which prices rise as yields fall. Putable bonds never exhibit negative convexity. (LOS 59.b)

4. A bond portfolio manager who wants to estimate the sensitivity of the portfolio's value to changes in the 5-year yield only should use a(n):

A. key rate duration.
B. Macaulay duration.
C. effective duration.

A — Key rate duration refers to the sensitivity of a bond or portfolio value to a change in one specific maturity yield. (LOS 59.c)

5. Assume that a bond has an effective duration of 10.5 and a convexity of 97.3. Using both of these measures, the estimated percentage change in price for this bond, in response to a decline in the yield curve of 200 basis points, is closest to:

A. 19.05%.
B. 22.95%.
C. 24.89%.

B — Total estimated price change ＝ duration effect ＋ convexity effect ＝ {[−10.5 × (−0.02)] ＋ [½ × 97.3 × (−0.02)²]} × 100 ＝ 21.0% ＋ 1.95% ＝ 22.95%. (LOS 59.b)

Key Concepts — Reading 59

LOS 59.a

Bonds with embedded options have uncertain cash flows, so they do not have a single well-defined yield. Effective duration and effective convexity must therefore be calculated with respect to shifts in the benchmark curve rather than the bond's yield.

Effective duration is a linear estimate of the percentage change in a bond's price for a 1% change in the benchmark yield curve:

$\text{EffDur} = \dfrac{V_- - V_+}{2\,V_0\,\Delta\text{Curve}}$ ， $\text{EffCon} = \dfrac{V_- + V_+ - 2V_0}{(\Delta\text{Curve})^2 \cdot V_0}$

Callable bonds and MBS may exhibit negative convexity at low yields.

LOS 59.b

Expected price change for a bond given $\Delta\text{Curve}$:

$\dfrac{\Delta P}{P} \approx -\text{EffDur}\cdot\Delta\text{Curve} + \tfrac{1}{2}\,\text{EffCon}\cdot(\Delta\text{Curve})^2$

LOS 59.c

Key rate duration measures the price sensitivity of a bond or portfolio to a change in yield at a specific maturity, with all other yields held constant. Key rate durations sum to effective duration and can be used to estimate the impact of nonparallel (shaping) shifts in the curve.

LOS 59.d

Macaulay, modified, and effective duration are analytical measures. Empirical duration is estimated from historical data. Empirical duration may be lower than analytical duration when the analytical assumptions break down—for example, for credit-risky bonds in a flight-to-quality scenario.

中文翻譯 — 重點整理

【LOS 59.a】含嵌入選擇權的債券現金流不確定，無單一明確殖利率，須以基準曲線變動為基礎計算 EffDur／EffCon。可贖回債券與 MBS 在低殖利率時可能出現負凸性。

$\text{EffDur} = \dfrac{V_- - V_+}{2\,V_0\,\Delta\text{Curve}}$； $\text{EffCon} = \dfrac{V_- + V_+ - 2V_0}{(\Delta\text{Curve})^2 \cdot V_0}$

【LOS 59.b】價格變動估計式：$\dfrac{\Delta P}{P} \approx -\text{EffDur}\cdot\Delta\text{Curve} + \tfrac{1}{2}\,\text{EffCon}\cdot(\Delta\text{Curve})^2$。

【LOS 59.c】關鍵利率存續期間衡量「僅某一到期點殖利率變動」對債券或投組價格的敏感度，其和等於有效存續期間，可用於估計非平行（形狀）變動的影響。

【LOS 59.d】Macaulay／修正／有效存續期間屬解析型；經驗型存續期間由歷史資料迴歸估算。當解析假設不成立時（如信用債在 flight-to-quality 環境），經驗型估計常低於解析型。