Reading 56

Fixed Income · Interest Rate Risk and Return

MODULE 56.1: INTEREST RATE RISK AND RETURN

LOS 56.a

Calculate and interpret the sources of return from investing in a fixed-rate bond.

There are three sources of return from investing in a fixed-rate bond:

Coupon and principal payments
Interest earned on coupon payments that are reinvested over the investor's holding period for the bond
Any capital gain or loss if the bond is sold before maturity

We will assume that a bond makes all of its promised coupon and principal payments on time (i.e., we are not addressing credit risk). Additionally, we assume that the interest rate earned on reinvested coupon payments is the same as the prevailing yield to maturity (YTM) on the bond.

Given the assumptions just listed, we may draw five key results:

An investor who holds a fixed-rate bond to maturity will earn an annualized rate of return equal to the YTM of the bond when purchased, if the YTM (and reinvestment rate) does not change over the life of the bond.
An investor who sells a bond before maturity will earn a rate of return equal to the YTM at purchase if the YTM has not changed since purchase.
If market YTM (the assumed reinvestment rate) increases (decreases) after purchase but before the first coupon date, an investor who holds the bond to maturity will earn a realized return that is higher (lower) than the original YTM at purchase.
If market YTM increases after purchase but before the first coupon date, a bond investor will earn a rate of return lower than the YTM at purchase if the bond is held for a short period.
If market YTM decreases after purchase but before the first coupon date, a bond investor will earn a rate of return lower than the YTM at purchase if the bond is held for a long period.

The time the bond will be held is the investor's investment horizon, which may be shorter than the bond's maturity. A bond investor's horizon yield is the compound annual return earned from the bond over the investment horizon. It is calculated by comparing the purchase price to the end value (coupons + reinvestment income + sale price or principal at maturity).

中文翻譯

固定利率債券投資的三個收益來源：

票息與本金支付
投資人持有期間將票息再投資所賺取的利息（再投資收益）
若於到期前出售，所產生的資本利得或損失

本章假設債券準時支付所有承諾現金流（不考慮信用風險），且再投資利率等於當下的到期殖利率（YTM）。

由此可得五個關鍵結果：

持有至到期且 YTM 不變：實現年化報酬率＝買入時的 YTM。
到期前出售但 YTM 不變：實現報酬率＝買入時的 YTM。
若 YTM 於首次付息前上升（下降），持有至到期者的實現報酬率將高於（低於）原 YTM。
若 YTM 於首次付息前上升，短期持有者的實現報酬率將低於原 YTM。
若 YTM 於首次付息前下降，長期持有者的實現報酬率將低於原 YTM。

投資期限（investment horizon）＝投資人實際持有債券的期間，可能短於到期日；期限收益率（horizon yield）＝該期間內的年化複合報酬率，比較買入價與終值（票息＋再投資利息＋售價／到期本金）即可求得。

Unchanged YTM, Bond Held to Maturity

We illustrate the first key result with a 6% annual-pay 3-year bond purchased at a YTM of 7% and held to maturity.

N = 3; I/Y = 7; PMT = 6; FV = 100; CPT → PV = −97.376

The bond's purchase price is $97.376$. At maturity, coupon income plus reinvestment income equals the future value of an annuity of three $6 coupon payments at the bond's YTM:

$$6(1.07)^2 + 6(1.07) + 6 = \$19.289$$

N = 3; I/Y = 7; PV = 0; PMT = 6; CPT → FV = −19.289

Reinvestment income alone is $19.289 - 3(6) = \$1.289$.

Adding the principal repayment of $100 to $19.289, the investor's compound annual return over the 3-year holding period is:

$$\left(\frac{119.289}{97.376}\right)^{1/3} - 1 = 7\%$$

Demonstrating the first result: a fixed-rate bond with no default and a reinvestment rate equal to the YTM, held to maturity, earns a rate of return equal to the YTM at purchase.

中文翻譯 — YTM 不變，持有至到期

以一張 3 年期、6% 年付息債券、買入時 YTM＝7% 為例：

買入價＝97.376（PV 計算）
票息＋再投資利息（FV 年金）＝$6(1.07)^2+6(1.07)+6=19.289$
單純再投資利息＝19.289 − 3×6 ＝ 1.289
3 年複合報酬率＝(119.289 / 97.376)^(1/3) − 1 ＝ 7%

結論：YTM 與再投資率均不變，持有至到期實現報酬率＝買入時 YTM。

Unchanged YTM, Bond Sold Before Maturity

If the YTM remains unchanged, the value of a bond moves toward par by the maturity date. Between purchase and maturity, the value of a bond at the same YTM as when it was purchased is its carrying value, which reflects amortization of the discount or premium since purchase.

教授提醒

Carrying value is a price along a bond's constant-yield price trajectory — i.e., the value of the bond at a given time after purchase assuming the original YTM has not changed. It is called "carrying value" because it is often shown on the balance sheet when a bond is held to maturity. This is not the same as the market value of the bond if its yield has changed.

Capital gains or losses at sale are measured relative to this carrying value.

Example: Capital gain or loss on a bond

An investor purchases a 20-year bond with a 5% semiannual coupon and a YTM of 6%. Five years later, the investor sells the bond for a price of 91.40. Determine whether the investor realizes a capital gain or loss, and calculate its amount.

Answer:

Any capital gain or loss is based on the bond's carrying value at the time of sale, when it has 15 years (30 semiannual periods) to maturity. The carrying value uses the original YTM:

N = 30; I/Y = 3; PMT = 2.5; FV = 100; CPT → PV = −90.20

Because the selling price of 91.40 exceeds the carrying value of 90.20, the investor realizes a capital gain of $91.40 - 90.20 = 1.20$ per 100 of face value.

Bonds held to maturity have no capital gain or loss. Bonds sold before maturity at the same YTM as at purchase also have no capital gain or loss. Returning to our 3-year 6% bond, an investor with a 2-year horizon experiences:

Price at end of Year 2 (YTM = 7%):

N = 1; I/Y = 7; FV = 100; PMT = 6; CPT → PV = −99.065

$106 / 1.07 = 99.065$, which is the carrying value.

Coupon interest and reinvestment income for two years:

$$6(1.07) + 6 = \$12.420$$

N = 2; I/Y = 7; PV = 0; PMT = 6; CPT → FV = −12.420

Annual compound return over the two-year holding period:

$$\left(\frac{12.420 + 99.065}{97.376}\right)^{1/2} - 1 = 7\%$$

This demonstrates the second key result: for a bond investor with a horizon less than the bond's term to maturity, the annual return equals the YTM at purchase if the bond is sold at that YTM and all coupons are reinvested at the original YTM.

中文翻譯 — YTM 不變，到期前出售

若 YTM 維持不變，債券價格會沿著「等殖利率價格軌跡」逐漸趨近面值，這條軌跡上的點稱為帳面價值（carrying value），反映折／溢價的攤銷。資本利得／損失以售價對帳面價值衡量。

教授提醒：帳面價值是「假設殖利率不變」的價值，常用於財務報告中對持至到期債券的列示，不等於 YTM 已變動後的市價。

例題：20 年期、5% 半年付息、買入 YTM＝6% 的債券，5 年後以 91.40 售出。剩餘 15 年（30 期半年），以原 YTM 計算帳面價值＝90.20，故實現資本利得＝91.40 − 90.20 ＝ 1.20。

回到 3 年 6% 例子，投資期限 2 年、YTM 不變：

第 2 年末售價（剩 1 年）＝106/1.07＝99.065（即帳面價值，無資本利得／損失）
2 年票息＋再投資＝6(1.07)+6＝12.420
年化報酬＝[(12.420+99.065)/97.376]^(1/2) − 1 ＝ 7%

結論：投資期限短於到期且 YTM 不變時，年化報酬仍等於買入時 YTM。

Changed YTM, Bond Held to Maturity

If rates rise (fall) before the first coupon date, an investor who holds a bond to maturity will earn a rate of return greater (less) than the YTM at purchase. The intuition: the YTM is also the reinvestment rate; an increase (decrease) raises (lowers) reinvestment income, raising (lowering) the realized return.

Case A: YTM rises to 8% just after purchase. For the same 3-year 6% bond purchased at 97.376:

Coupons and reinvestment interest:

$$6(1.08)^2 + 6(1.08) + 6 = \$19.478$$

N = 3; I/Y = 8; PV = 0; PMT = 6; CPT → FV = −19.478

Annual compound holding period return:

$$\left(\frac{119.478}{97.376}\right)^{1/3} - 1 = 7.06\%$$

which is greater than the 7% YTM at purchase.

Case B: YTM falls to 6% just after purchase.

Coupons and reinvestment interest:

$$6(1.06)^2 + 6(1.06) + 6 = \$19.102$$

N = 3; I/Y = 6; PV = 0; PMT = 6; CPT → FV = −19.102

Annual compound holding period return:

$$\left(\frac{119.102}{97.376}\right)^{1/3} - 1 \approx 6.94\%$$

which is less than the 7% YTM at purchase. In both cases the realized return lies between the YTM at purchase and the new (assumed reinvestment) YTM.

中文翻譯 — YTM 變動，持有至到期

於首次付息前 YTM 上升（下降）→ 再投資利率同步變動 → 持至到期者報酬率高於（低於）買入 YTM。

YTM 升至 8%：票息＋再投資＝19.478；3 年報酬＝(119.478/97.376)^(1/3) − 1 ≈ 7.06%（>7%）
YTM 降至 6%：票息＋再投資＝19.102；3 年報酬 ≈ 6.94%（<7%）

實現報酬率介於原 YTM 與新 YTM 之間。

Changed YTM, Bond Sold Before Maturity

Consider the same 3-year 6% bond purchased at 97.376 by an investor with a one-year investment horizon.

Case A: YTM rises from 7% to 8% just after purchase. Bond price just after the first coupon (with 2 years remaining at YTM = 8%):

N = 2; I/Y = 8; FV = 100; PMT = 6; CPT → PV = −96.433

With one $6 coupon received and no reinvestment income, the holding period return is:

$$\frac{6 + 96.433}{97.376} - 1 = 5.19\%$$

which is less than the 7% YTM at purchase.

Case B: YTM falls to 6% just after purchase. Bond price just after the first coupon (YTM = 6%):

N = 2; I/Y = 6; FV = 100; PMT = 6; CPT → PV = −100.00

Holding period return:

$$\frac{6 + 100.00}{97.376} - 1 = 8.86\%$$

which is greater than the YTM at purchase.

The intuition rests on the tradeoff between two risks:

Price risk — uncertainty about a bond's price due to uncertainty about market YTM at the time of sale.
Reinvestment risk — uncertainty about the total of coupon and reinvestment income due to uncertainty about future reinvestment rates.

For a bond held to maturity, par value is received regardless of rate changes, so there is no price risk — only reinvestment risk. For a short horizon, there is little reinvestment risk (few coupons reinvested) and large price risk (must sell at the prevailing market price). To summarize:

Short investment horizon: price risk > reinvestment risk
Long investment horizon: reinvestment risk > price risk

中文翻譯 — YTM 變動，到期前出售

同一張 3 年 6% 債券（買入價 97.376），持有 1 年後出售：

YTM 升至 8%：1 年後賣價＝96.433；報酬＝(6+96.433)/97.376 − 1 ≈ 5.19%（<7%）
YTM 降至 6%：1 年後賣價＝100.00；報酬＝(6+100)/97.376 − 1 ≈ 8.86%（>7%）

關鍵直覺＝價格風險 vs. 再投資風險：

持至到期：只有再投資風險（無價格風險）→ 利率上升者報酬增加
短期持有：以價格風險為主（再投資風險小）→ 利率上升者反而虧損

結論：短期持有→價格風險＞再投資風險；長期持有→再投資風險＞價格風險。

LOS 56.b

Describe the relationships among a bond's holding period return, its Macaulay duration, and the investment horizon.

LOS 56.c

Define, calculate, and interpret Macaulay duration.

Is there an investment horizon at which price risk and reinvestment risk are in balance? Yes — as the next example shows.

Example: Investment horizon yields

Consider a 5-year, 11% annual-coupon bond priced at 86.59 to yield 15% to maturity. Calculate the horizon yield for an investment horizon of 4 years, assuming the YTM:

(a) Falls to 14% before the first coupon date; (b) Rises to 16% before the first coupon date.

Answer (a) — YTM falls to 14%, sale after 4 years:

Sale price (1 year remaining at 14%):

N = 1; PMT = 11; FV = 100; I/Y = 14; CPT → PV = −97.368

Coupons and interest on reinvested coupons (4 years at 14%):

N = 4; PMT = 11; PV = 0; I/Y = 14; CPT → FV = −54.133

Horizon return:

$$\left(\frac{97.368 + 54.133}{86.59}\right)^{1/4} - 1 = 15.0\%$$

Answer (b) — YTM rises to 16%, sale after 4 years:

Sale price (1 year remaining at 16%):

N = 1; PMT = 11; FV = 100; I/Y = 16; CPT → PV = −95.690

Coupons and interest on reinvested coupons (4 years at 16%):

N = 4; PMT = 11; PV = 0; I/Y = 16; CPT → FV = −55.731

Horizon return:

$$\left(\frac{95.690 + 55.731}{86.59}\right)^{1/4} - 1 = 15.0\%$$

For a 4-year horizon, the horizon return equals the original 15% YTM regardless of the YTM change.

For this particular bond, an investment horizon of 4 years is neither short enough to face net price risk nor long enough to face net reinvestment risk. If YTM falls, the loss on reinvestment income is exactly offset by the gain in sale price; if YTM rises, the gain on reinvestment income is exactly offset by the price loss.

中文翻譯 — 期限與兩種風險的平衡

是否存在某個「平衡點」，使價格風險與再投資風險互相抵消？答案是肯定的。

例題：5 年期、11% 年付息債券、價格 86.59、YTM＝15%；投資期限 4 年。

(a) YTM 降至 14%：售價＝97.368；4 年票息＋再投資＝54.133；報酬＝[(97.368+54.133)/86.59]^(1/4) − 1 ＝ 15.0%
(b) YTM 升至 16%：售價＝95.690；4 年票息＋再投資＝55.731；報酬同樣為 15.0%

當投資期限＝4 年時，無論 YTM 變動方向為何，報酬率仍等於原 15%。利率下降造成的再投資損失剛好被售價上漲抵消；反之亦然。

How can we find this "sweet spot" horizon? It is the bond's Macaulay duration — the weighted average of the number of years until each promised cash flow is paid, where weights are the PV of each cash flow as a percentage of the bond's full price.

Macaulay duration of the 5-year 11% bond at YTM = 15%

t	Cash Flow $C_t$	PV at 15%	Weight $w_t = PV_t / 86.59$
1	11	$11/1.15 = 9.565$	$9.565 / 86.59 = 0.1105$
2	11	$11/1.15^2 = 8.318$	$8.318 / 86.59 = 0.0961$
3	11	$11/1.15^3 = 7.233$	$7.233 / 86.59 = 0.0835$
4	11	$11/1.15^4 = 6.289$	$6.289 / 86.59 = 0.0726$
5	111	$111/1.15^5 = 55.187$	$55.187 / 86.59 = 0.6373$
Total		86.59	1.0000

The PVs sum to 86.59 (the full bond price) and the weights sum to 1. Macaulay duration:

$$D_{\text{Mac}} = 0.1105(1) + 0.0961(2) + 0.0835(3) + 0.0726(4) + 0.6373(5) = 4.03 \text{ years}$$

Interpretation: at an investment horizon of 4 years, this bond still earns its original YTM even if the YTM changes immediately after purchase.

教授提醒

For a semiannual-pay bond, the same procedure produces a Macaulay duration measured in semiannual periods; divide by 2 to express it in years.

中文翻譯 — Macaulay Duration 計算

「平衡點」即 Macaulay Duration（麥考利存續期間）：以各期現金流現值佔債券總現值的比重為權重，對「年數」做加權平均。

5 年 11% 債券（YTM＝15%）的計算如表所示，加總得：

$D_\text{Mac} = 0.1105(1)+0.0961(2)+0.0835(3)+0.0726(4)+0.6373(5) = 4.03$ 年

解讀：投資期限 4 年時，即便 YTM 立刻變動，仍能實現原 YTM。

教授提醒：半年付息債券計算結果是「半年期數」，除以 2 得到年數。

Duration Gap

The difference between a bond's Macaulay duration and the bondholder's investment horizon is the duration gap:

$$\text{duration gap} = \text{Macaulay duration} - \text{investment horizon}$$

Positive duration gap (Macaulay > horizon) → exposes the investor to price risk from rising interest rates.
Negative duration gap (Macaulay < horizon) → exposes the investor to reinvestment risk from falling interest rates.

中文翻譯 — 存續期間缺口

存續期間缺口（duration gap）＝ Macaulay duration − 投資期限

正缺口（Mac > 期限）：暴露於利率上升的價格風險
負缺口（Mac < 期限）：暴露於利率下降的再投資風險

Module Quiz 56.1

1. The largest component of returns for a 7-year zero-coupon bond yielding 8% and held to maturity is:

A. capital gains.
B. interest income.
C. reinvestment income.

B — The increase in value of a zero-coupon bond over its life is interest income. A zero-coupon bond has no reinvestment risk over its life. A bond held to maturity has no capital gain or loss. (LOS 56.a)

2. An investor buys a 10-year bond with a 6.5% annual coupon and a YTM of 6%. Before the first coupon payment is made, the YTM for the bond decreases to 5.5%. Assuming coupon payments are reinvested at the YTM, the investor's return when the bond is held to maturity is:

A. less than 6.0%.
B. equal to 6.0%.
C. greater than 6.0%.

A — The investment horizon is maturity, so the investor faces reinvestment risk and zero price risk. The decrease in YTM to 5.5% lowers reinvestment income over the life of the bond, so the investor earns less than 6%, the YTM at purchase. (LOS 56.a)

3. Assuming coupon interest is reinvested at a bond's YTM, what is the interest portion of an 18-year, $1,000 par, 5% annual coupon bond's return if it is purchased at par and held to maturity?

A. $576.95.
B. $1,406.62.
C. $1,476.95.

B — The interest portion is the sum of coupon payments and reinvestment income over the holding period:

N = 18; PMT = 50; PV = 0; I/Y = 5%; CPT → FV = −1,406.62

(LOS 56.a)

4. An investor buys a 15-year, £800,000 zero-coupon bond with an annual YTM of 7.3%. If she sells the bond after three years for £346,333, she will have:

A. a capital gain.
B. a capital loss.
C. neither a capital gain nor a capital loss.

A — The price after three years that would generate neither gain nor loss is the bond's carrying value at the original YTM of 7.3%: $800{,}000 / 1.073^{12} = 343{,}473.57$. The sale price of £346,333 exceeds the carrying value, so she realizes a capital gain. (LOS 56.a)

5. An investor with an investment horizon of six years buys a bond with a Macaulay duration of seven years. This investment has:

A. no duration gap.
B. a positive duration gap.
C. a negative duration gap.

B — Duration gap = Macaulay duration − investment horizon. With Macaulay duration (7) greater than the horizon (6), the investment has a positive duration gap. (LOS 56.b)

6. The Macaulay duration (in years) of a 2-year semiannual-pay 7% coupon bond yielding 5% is closest to:

A. 0.38.
B. 1.90.
C. 3.81.

B — Discount semiannual coupons of 3.5 (and 103.5 at maturity) at I/Y = 2.5%:

PV1 = 3.5/1.025 = 3.415; w1 = 3.415/103.762 = 0.0329
PV2 = 3.5/1.025² = 3.331; w2 = 3.331/103.762 = 0.0321
PV3 = 3.5/1.025³ = 3.250; w3 = 3.250/103.762 = 0.0313
PV4 = 103.5/1.025⁴ = 93.766; w4 = 93.766/103.762 = 0.9037
Sum of PVs = 103.762; weights sum to 1

$D_\text{Mac}^\text{(semi)} = 0.0329(1) + 0.0321(2) + 0.0313(3) + 0.9037(4) = 3.806$ semiannual periods. Annualized: $3.806 / 2 = 1.90$ years. (LOS 56.c)

Key Concepts — Reading 56

LOS 56.a

Sources of return from a bond investment include:

Coupon and principal payments
Reinvestment of coupon payments
Capital gain or loss if the bond is sold before maturity

Changes in YTM produce price risk (uncertainty about a bond's price) and reinvestment risk (uncertainty about income from reinvested coupons). An increase (decrease) in YTM decreases (increases) a bond's price but increases (decreases) its reinvestment income.

LOS 56.b

Over a short investment horizon, a change in YTM affects price more than it affects reinvestment income.

Over a long investment horizon, a change in YTM affects reinvestment income more than it affects price.

Macaulay duration may be interpreted as the investment horizon for which a bond's price risk and reinvestment risk offset each other:

$$\text{duration gap} = \text{Macaulay duration} - \text{investment horizon}$$

LOS 56.c

Macaulay duration is the weighted average of the number of years until each of the bond's promised cash flows is to be paid, where the weights are the PV of each cash flow as a percentage of the bond's full value.

中文翻譯 — 重點整理

【LOS 56.a】債券投資三大收益來源：票息與本金、票息再投資利息、到期前出售之資本利得／損失。YTM 變動產生兩類風險：價格風險（影響賣價）與再投資風險（影響再投資收益）。YTM 上升 → 價格下跌但再投資收益上升；YTM 下降 → 價格上升但再投資收益下降。

【LOS 56.b】短期持有者：價格風險主導；長期持有者：再投資風險主導。Macaulay duration 即兩種風險互相抵消的投資期限。存續期間缺口＝Macaulay duration − 投資期限。

【LOS 56.c】Macaulay duration＝以各期現金流現值佔債券總現值之比重為權重的「年數加權平均」；半年付息者的計算結果除以 2 即為年化值。