CHAPTER 4
第 4 章
BOND PRICE VOLATILITY
债券价格波动
ANSWERS TO QUESTIONS FOR CHAPTER 4
第 4 章问题的解答
(Questions are in bold print followed by answers.)
（问题以粗体显示，后跟答案。
1. The price value of a basis point will be the same regardless if the yield is increased or decreased by 1 basis point. However, the price value of 100 basis points (i.e., the change in price for a 100-basis-point change in interest rates) will not be the same if the yield is increased or decreased by 100 basis points. Why?
1. 无论收益率是增加还是减少 1 个基点，一个基点的价格值都是相同的。但是，如果收益率增加或减少 100 个基点，则 100 个基点的价格值（即利率变化 100 个基点的价格变化）将有所不同。为什么？
The convex relationship explains why the price value of a basis point (i.e., the change in price for a 1-basis-point change in interest rates) will be roughly the same regardless if the yield is increased or decreased by 1 basis point, while the price value of 100 basis points will not be the same if the yield is increased or decreased by 100 basis points. More details are below.
凸关系解释了为什么无论收益率增加还是减少 1 个基点，一个基点的价格值（即利率变化 1 个基点的价格变化）都将大致相同，而如果收益率增加或减少 100 个基点，100 个基点的价格值将不相同。更多详细信息如下。
When the price-yield relationship for any option-free bond is graphed, it displays a convex shape. When the price of the option-free bond declines, we can observe that the required yield rises. However, this relationship is not linear. The convex shape of the price-yield relationship generates four properties concerning the price volatility of an option-free bond. First, although the prices of all option-free bonds move in the opposite direction from the change in yield required, the percentage price change is not the same for all bonds. Second, for very small changes in the yield required (like 1 basis point), the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases. Third, for large changes in the required yield (like 100 basis points), the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield. Fourth, for a given large change in basis points, the percentage price increase is greater than the percentage price decrease.
当绘制任何无期权债券的价格-收益率关系时，它会显示凸形。当无期权债券的价格下跌时，我们可以观察到所需的收益率上升。但是，这种关系不是线性的。价格-收益率关系的凸形产生了关于无期权债券价格波动的四个属性。首先，尽管所有无期权债券的价格走势与所需的收益率变化相反，但所有债券的价格变化百分比并不相同。其次，对于所需收益率的非常小的变化（例如 1 个基点），无论所需收益率是增加还是减少，给定债券的价格变化百分比都大致相同。第三，对于所需收益率的大幅变化（如 100 个基点），所需收益率增加的价格百分比变化与所需收益率降低的价格百分比变化不同。第四，对于给定的基点大幅变化，价格上涨百分比大于价格下跌百分比。
2. Calculate the requested measures in parts (a) through (f) for bonds A and B (assume that each bond pays interest semiannually):
2. 计算债券 A 和 B 的 （a） 至 （f） 部分中的所请求度量值（假设每只债券每半年支付一次利息）：
Bond A | Bond B | |
Coupon | 8% | 9% |
Yield to maturity | 8% | 8% |
Maturity (years) | 2 | 5 |
Par | $100.00 | $100.00 |
Price | $100.00 | $104.055 |
(a) What is the price value of a basis point for bonds A and B?
（a） 债券 A 和 B 的基点价格是多少？
For bond A, we get a bond quote of $100.00 for our initial price if we have a 2-year maturity, an 8% coupon rate and an 8% yield. If we change the yield one basis point so the yield is 8.01%, then we have the following variables and values:
对于债券 A，如果我们有 2 年的期限、8% 的票面利率和 8% 的收益率，我们的初始价格将获得 100.00 美元的债券报价。如果我们将收益率更改一个基点，使收益率为 8.01%，则我们有以下变量和值：
C = $40, y = 0.0801 / 2 = 0.04005 and n = 2(2) = 4.
C = 40 美元，y = 0.0801 / 2 = 0.04005 和 n = 2（2） = 4。
Inserting these values into the present value of the coupon payments formula, we get:
将这些值代入息票支付公式的现值中，我们得到：
== $145.179.
== 145.179 美元。
Computing the present value of the par or maturity value of $1,000 gives:
计算 1,000 美元的面值或到期值的现值，得到：
= = $854.640.
= = 854.640 美元。
If we add a basis point to the yield, we get the value of Bond A as: P = $145.179 + $854.640 = $999.819 with a bond quote of $99.9819. For bond A the price value of a basis point is about $100 – $99.9819 = $0.0181 per $100.
如果我们在收益率上加上一个基点，我们得到债券 A 的价值为：P = 145.179 美元 + 854.640 美元 = 999.819 美元，债券报价为 99.9819 美元。对于债券 A，一个基点的价格价值约为 100 美元 – 99.9819 美元 = 每 100 美元 0.0181 美元。
Using the bond valuation formulas as just completed above, the value of bond B with a yield of 8%, a coupon rate of 9%, and a maturity of 5 years is: P = $364.990 + $675.564 = $1,040.554 with a bond quote of $104.0554. If we add a basis point to the yield, we get the value of Bond B as: P = $364.899 + $675.239 = $1,040.139 with a bond quote of $104.0139. For bond B, the price value of a basis point is $104.0554 – $104.0139 = $0.0416 per $100.
使用上面刚刚完成的债券估值公式， 收益率为 8%、票面利率为 9% 且期限为 5 年的债券 B 的价值为：P = 364.990 美元 + 675.564 美元 = 1,040.554 美元，债券报价为 104.0554 美元。如果我们在收益率上加上一个基点，我们得到债券 B 的价值为：P = 364.899 美元 + 675.239 美元 = 1,040.139 美元，债券报价为 104.0139 美元。对于债券 B，一个基点的价格值为 104.0554 美元 – 104.0139 美元 = 每 100 美元 0.0416 美元。
(b) Compute the Macaulay durations for the two bonds.
（b） 计算两个键的麦考利久期。
For bond A with C = $40, n = 4, y = 0.04, P = $1,000 and M = $1,000, we have:
对于 C = 40 美元、n = 4、y = 0.04、P = 1,000 美元和 M = 1,000 美元的债券 A，我们有：
Macaulay duration (half years) = =
麦考利持续时间（半年） = =
= = 3.77509.
= = 3.77509 的。
Macaulay duration (years) = Macaulay duration (half years) / 2 = 3.77509 / 2 = 1.8875.
麦考利持续时间（年）= 麦考利持续时间（半年）/ 2 = 3.77509 / 2 = 1.8875。
For bond B with C = $45, n = 10, y = 0.04, P= $1,040.55 and M = $1,000, we have:
对于 C = 45 美元、n = 10、y = 0.04、P = 1,040.55 美元和 M = 1,000 美元的债券 B，我们有：
Macaulay duration (half years) = =
麦考利持续时间（半年） = =
= = 8.3084.
Macaulay duration (years) = Macaulay duration (half years) / 2 = 8.3084 / 2 = 4.1542.
麦考利持续时间（年）= 麦考利持续时间（半年）/ 2 = 8.3084 / 2 = 4.1542。
(c) Compute the modified duration for the two bonds.
（c） 计算两个键的修改久期。
Taking our answer for the Macaulay duration in years in part (b), we can compute the modified duration for bond A by dividing by 1.04. We have:
以 （b） 部分中以年为单位的麦考利久期的答案，我们可以通过除以 1.04 来计算债券 A 的修改久期。我们有：
modified duration = Macaulay duration / (1+y) = 1.8875 / 1.04 = 1.814948.
修改后的持续时间 = 麦考利持续时间 / （1+y） = 1.8875 / 1.04 = 1.814948。
Taking our answer for the Macaulay duration in years in part (b), we can compute the modified duration for bond B by dividing by 1.04. We have:
以 （b） 部分中以年为单位的麦考利久期的答案，我们可以通过除以 1.04 来计算债券 B 的修改久期。我们有：
modified duration = Macaulay duration / (1+y) = 4.1542 / 1.04 = 3.994417.
修改后的持续时间 = 麦考利持续时间 / （1+y） = 4.1542 / 1.04 = 3.994417。
[NOTE. We could get the same answers for both bonds A and B by computing the modified duration using an alternative formula that does not require the extensive calculations required by the procedure in parts (a) and (b). This shortcut formula is:
[注意。通过使用替代公式计算修改后的久期，我们可以得到键 A 和 B 的相同答案，该公式不需要 （a） 和 （b） 部分中程序所需的大量计算。这个快捷公式是：
modified duration = .
修改的持续时间 = 。
where C is the semiannual coupon payment, y is the semiannual yield, n is the number of semiannual periods, and P is the bond quote in 100’s.
其中 C 是半年度息票支付，y 是半年度收益率，n 是半年度周期数，P 是以 100 为单位的债券报价。
For bond A (expressing numbers in terms of a $100 bond quote), we have: C = $4, y = 0.04, n = 4, and P = $100. Inserting these values in our modified duration formula, we can solve as follows:
对于债券 A（以 100 美元的债券报价表示数字），我们有：C = 4 美元，y = 0.04，n = 4，P = 100 美元。将这些值插入我们修改后的 duration 公式中，我们可以按如下方式求解：
= =
($362.98952 + $0) / $100 = 3.6298952. Converting to annual number by dividing by two gives a modified duration for bond A of 1.814948 which is the same answer shown above.
（362.98952 美元 + 0 美元）/ 100 美元 = 3.6298952。通过除以 2 转换为年度数字，可以得出债券 A 的修改久期 为 1.814948，这与上面显示的答案相同。
For bond B, we have C = $4.5, n = 20, y = 0.04, and P = $104.055. Inserting these values in our modified duration formula, we can solve as follows:
F或债券 B，我们有 C = 4.5 美元，n = 20，y = 0.04，P = 104.055 美元。将这些值插入我们修改后的 duration 公式中，我们可以按如下方式求解：
= =
($912.47578 – $811.9762) / $104.055 = 7.988834. Converting to annual number by dividing by two gives a modified duration for bond B of 3.994417 which is the same answer shown above.]
（912.47578 美元 – 811.9762 美元）/ 104.0 美元55 = 7.988834。通过除以 2 转换为年度数字，得到债券 B 的修改久期 为 3.994417，这与上面显示的答案相同。
(d) Compute the approximate duration for bonds A and B using the shortcut formula by changing yields by 20 basis points and compare your answers with those calculated in part (c).
（d） 使用捷径公式，将收益率改变 20 个基点，计算债券 A 和 B 的大致久期，并将您的答案与 （c） 部分中计算的答案进行比较。
To compute the approximate measure for bond A, which is a 2-year 8% coupon bond trading at 8% with an initial price (P0) of $1,000 (thus, it trades at its par value of $1,000), we proceed as follows.
债券 A 是一只 2 年期 8% 票息债券，交易利率为 8%，初始价格 （P0） 为 1,000 美元（因此，它以 1,000 美元的面值交易），我们进行如下操作。
First, we increase the yield on the bond by 20 basis points from 8% to 8.20%. Thus, ∆y is 0.20% or 0.002 in decimal form. The new price (P+) can be computed using our bond valuation formula. Doing this we get $996.379 with a bond price quote of $99.6379.
首先，我们将债券收益率从 8% 上调 20 个基点至 8.20%。因此，∆y 为 十进制形式的 0.20% 或 0.002。新价格 （P+） 可以使用我们的债券估值公式计算。这样做，我们得到 996.379 美元，债券报价为 99.6379 美元。
Second, we decrease the yield on the bond by 20 basis points from 8% to 7.8%. The new price (P_) can be computed using our bond valuation formula. Doing this we get $1,003.638 with a bond price quote of $100.3638.
其次，我们将债券收益率从 8% 上调 20 个基点至 7.8%。新价格 （P_） 可以使用我们的债券估值公式计算。这样做，我们得到 1,003.638 美元，债券报价为 100.3638 美元。
Third, with the initial price, P0, equal to $100 (when expressed as a bond quote), the duration can be approximated as follows:
第三，初始价格 P0 等于 100 美元（以债券报价表示时），久期可以近似如下：
approximate duration =
大致持续时间 =
where ∆y is the change in yield used to calculate the new prices (in decimal form). What the formula is measuring is the average percentage price change (relative to the initial price) per 20-basis-point change in yield. Inserting in our values, we have:
其中 ∆y 是用于计算新价格的产量变化（十进制形式）。该公式衡量的是收益率每变化 20 个基点的平均价格变化百分比（相对于初始价格）。插入我们的值，我们有：
approximate duration == 1.814948.
大约持续时间 == 1.814948。
This compares with 1.814948 computed in part (c). Thus, the approximate duration measure (to six decimal places for this problem) is the same as the modified duration computed in part (c).
这与 （c） 部分中计算的 1.814948 形成鲜明对比。因此，近似持续时间度量（此问题精确到小数点后六位）与（c）部分中计算的修改后的持续时间相同。
To compute the approximate measure for bond B, which is a 5-year 9% coupon bond trading at 8% with an initial price (P0) of $104.0554, we proceed as follows.
为了计算债券 B 的近似度量，债券 B 是 5 年期 9% 票息债券，交易利率为 8%，初始价格 （P0） 为 104.0554 美元，我们进行如下操作。
First, we increase the yield on the bond by 20 basis points from 8% to 8.20%. Thus, ∆y is 0.002. The new price (P+) can be computed using our bond valuation formula. Doing this we get $1,032.283 with a bond price quote of $103.2283.
首先，我们将债券收益率从 8% 上调 20 个基点至 8.20%。因此，∆y 为 0.002。新价格 （P+） 可以使用我们的债券估值公式计算。这样做，我们得到 1,032.283 美元，债券报价为 103.2283 美元。
Second, we decrease the yield on the bond by 20 basis points from 8% to 7.8%. The new price (P_) can be computed using our bond valuation formula. Doing this we get $1,048.909 with a bond price quote of $104.8909.
其次，我们将债券收益率从 8% 上调 20 个基点至 7.8%。新价格 （P_） 可以使用我们的债券估值公式计算。这样做，我们得到 1,048.909 美元，债券报价为 104.8909 美元。
Third, with the initial price, P0, equal to $104.0554 (when expressed as a bond quote), the duration can be approximated as follows:
第三，初始价格 P0 等于 104.0554 美元（以债券报价表示时），久期可以近似如下：
approximate duration =
大致持续时间 =
where ∆y is the change in yield used to calculate the new prices (in decimal form). What the formula is measuring is the average percentage price change (relative to the initial price) per 20-basis-point change in yield. Inserting in our values, we have:
其中 ∆y 是用于计算新价格的产量变化（十进制形式）。该公式衡量的是收益率每变化 20 个基点的平均价格变化百分比（相对于初始价格）。插入我们的值，我们有：
approximate duration == 3.994507.
大约持续时间 ==3.994507。
This compares with 3.994417 computed in part (c). Thus, the approximate duration measure is virtually the same as the modified duration computed in part (c).
这与 （c） 部分中计算的 3.994417 形成鲜明对比。因此，近似持续时间度量实际上与在 （c） 部分中计算的修改持续时间相同。
Besides the above approximate duration approach, there is another approach that is shorter than the Macaulay duration and modified duration approach. With this approach, we proceed as follows. For bond A, we add 20 basis points and get a yield of 8.20%. We now have C = $40, y = 4.10%, n = 4, and M = $1,000. Before we use this shortcut approach, we first compute P. As given above, we can use our bond valuation formula to get $996.379.
除了上述近似久期方法外，还有另一种比麦考利久期和修正久期方法短的方法。使用这种方法，我们按如下方式进行。对于债券 A，我们增加 20 个基点，收益率为 8.20%。我们现在有 C = 40 美元，y = 4.10%，n = 4，M = 1,000 美元。在我们使用这种捷径方法之前，我们首先计算 P。如上所述，我们可以使用我们的债券估值公式来获得 996.379 美元。
Now we can compute the modified duration for bond A using the below formula:
现在我们可以使用以下公式计算修改后的持续时间 f或键 A：
modified duration = .
修改的持续时间 = 。
Putting in all applicable variables in terms of $100, we have: C = $4, n = 4, y = 0.041 and P = $99.6379. Inserting these values in our modified duration formula, we can solve as follows:
以 100 美元为单位放入所有适用的变量，我们有：C = 4 美元，n = 4，y = 0.041 和 P = 99.6379 美元。将这些值插入我们修改后的 duration 公式中，我们可以按如下方式求解：
= ($353.3031039 + $7.9803601) / $99.6379 =
= （353.30310 美元39 + 7.9 美元 803601） / 99.6379 美元 =
3.6259643. Converting to annual number by dividing by two gives a modified duration for bond A of 1.812982. This compares with 1.814948 computed in part (c). Since both round off to 1.81, the 20 point change in basis does not exercise any noticeable effect on our computation as the difference is only 1.814948 – 1.82982 = 0.001966.
3.6259643。通过除以 2 转换为年度数字，债券 A 的修改久期 为 1.812982。相比之下，在 （c） 部分中计算的是 1.814948。由于两者都四舍五入到 1.81，因此基差的 20 点变化不会对我们的计算产生任何明显影响，因为差异仅为 1.814948 – 1.82982 = 0.001966。
For bond B, we add 20 basis points and get a yield of 8.20%. We now have C = $45, y = 4.10%, n = 10, and M = $1,000. Before we use the modified duration formula, we first compute P. Using our bond valuation formula, we get $1,032.283.
对于债券 B，我们增加 20 个基点，收益率为 8.20%。我们现在有 C = 45 美元，y = 4.10%，n = 10，M = 1,000 美元。在使用修改后的久期公式之前，我们首先计算 P。使用我们的债券估值公式，我们得到 1,032.283 美元。
Now we can compute the modified duration for bond B as above for bond A. Given C = $4.5, n = 10, y = 0.041, and P =$103.2237, and inserting these values into the above formula for modified duration gives:
现在我们可以计算键 A 的修改久期 f或键 B，如上所示。假设 C = 4.5 美元，n = 10，y = 0.041，P =103.2237 美元，并将这些值代入上述公式中，得到：
modified duration = ($885.80511 – $627.0730) / $103.2283 = 7.97357.
修改后的持续时间 = （885.80511 USD – 627.0730 USD） / 103.2 USD， 283 = 7.97357。
Converting to annual number by dividing by two gives a modified duration for bond B of about 3.9868. This compares with 3.9944 computed in part (c). Since both round off to 3.99, the 20-point change in basis does not exercise any noticeable effect on our computation. However, for the two year bond, we only had a difference of about 0.0020, while for the five year bond, we have a difference of about 0.0076. Thus, this shortcut approach gives a wider disagreement for the longer-term bond (bond B).
通过除以 2 转换为年度数字，债券 B 的修改久期约为 3.9868。相比之下，第 （c） 部分计算的值为 3.9944。由于两者都四舍五入到 3.99，因此基数的 20 个点变化不会对我们的计算产生任何明显影响。然而，对于两年期债券，我们只差了大约 0.0020，而对于五年期债券，我们的差额约为 0.0076。因此，这种捷径方法对长期债券（债券 B）产生了更广泛的分歧。
(e) Compute the convexity measure for both bonds A and B.
（e） 计算键 A 和 B 的凸度测度。
In half years, the convexity measure =. Noting that
半年后，凸性测度 =。注意到
= ,
= ，
we can insert this quantity into our convexity measure (half year) formula to get:
我们可以将此量代入我们的凸性度量（半年）公式中，得到：
convexity measure = .
凸度测度 = 。
For bond A, we have a 2-year 8% coupon bond trading at 8% with an initial price (P0) of $1,000 with a bond quote of $100. Expressing numbers in terms of a $100 bond quote, we have: C = $4, y = 0.04, n = 4 and P = $100. Inserting these numbers into our convexity measure formula gives:
对于债券 A，我们有 2 年期 8% 票息债券，交易利率为 8%，初始价格 （P0） 为 1,000 美元，债券报价为 100 美元。以 100 美元的债券报价表示数字，我们有：C = 4 美元，y = 0.04，n = 4 和 P = 100 美元。将这些数字代入我们的凸性测度公式中，得到：
convexity measure (half years) =
凸度测量（半年）=
=
[$125,000(0.14519581) – $16,438.542135 + $0]=
[125,000 美元 （0.14519581） – 16,438.542135 美元 + 0 美元]=
[$18,149.4761 - $16,438.5421 + 0]=1,710.934[0.01] = 17.10934.
[18,149.4761 美元 - 16,438.5421 美元 + 0]=1,710.934[0.01] = 17.10934.
Convexity measure (years) == 17.10934 / 2(2) = 17.10934 / 4 = 4.277335. Dollar convexity measure = convexity measure (years) times P = 4.277335($100) equals about $427.73.
凸度测量（年） == 17.10934 / 2（2） = 17.10934 / 4 = 4.277335。美元凸性度量 = 凸性度量（年） times P = 4.277335（100 美元）等于约 427.73 美元。
[NOTE. We can get the same convexity by proceeding as follows. First, we increase the yield on the bond by 10 basis points from 8% to 8.1%. Thus, ∆y is 0.001. The new price (P+) can be computed using our bond valuation formula. Doing this we get $998.187 with a bond quote of $99.8187. Second, we decrease the yield on the bond by 10 basis points from 8% to 7.9%. The new price (P_) can be computed using our bond valuation formula. Doing this we get $100.1817. Third, with the initial price, P0, equal to $100, the convexity measure of any bond can be approximated using the following formula:
[注意。我们可以通过如下操作来获得相同的凸度。首先，我们将债券收益率从 8% 下调 10 个基点至 8.1%。因此，∆y 为 0.001。新价格 （P+） 可以使用我们的债券估值公式计算。这样做，我们得到 998.187 美元，债券报价为 99.8187 美元。其次，我们将债券收益率从 8% 上调 10 个基点至 7.9%。新价格 （P_） 可以使用我们的债券估值公式计算。这样做，我们得到 100.1817 美元。第三，初始价格 P0 等于 100 美元，任何债券的凸度都可以使用以下公式进行近似计算：
approximate convexity measure = .
近似凸度测度 = 。
Inserting in our values, the approximate convexity measure for bond A is
插入我们的值，键 A 的近似凸度 测度为
approximate convexity measure == 4.277338.
近似凸度测量 == 4.277338。
The approximate convexity measure of 4.277338 is almost identical to the convexity measure of 4.277335 computed above.]
近似的凸度测量值 4.277338 与上面计算的 4.277335 的凸度测量值几乎相同。
For bond B, we have a 5-year 9% coupon bond trading at 8% with an initial price (P0) of $104.055. Expressing numbers in terms of a $100 bond quote, we have: C = $4.5, y = 0.04, n = 10, and P = $104.0554. Inserting these numbers into our convexity measure formula gives:
对于债券 B，我们有 5 年期 9% 票息债券，交易利率为 8%，初始价格 （P0） 为 104.055 美元。以 100 美元的债券报价表示数字，我们有：C = 4.5 美元，y = 0.04，n = 10，P = 104.0554 美元。将这些数字代入我们的凸性测度公式中，得到：
convexity measure = .
凸度测度 = 。
=
[$140,625[0.32443583] – $36,538.9274 + –$1,375.00] =
[140,625 美元[0.32443583] – 36,538.9274 美元 + –1,375.00 美元] =
[$45,623.7888 – $36,538.9274 + –$858.8209] = 8,226.04[0.0096103] = 79.0544.
[45,623.7888 美元 – 36,538.9274 美元 + –858.8209 美元] =8,226.04[0.0096103] = 79.0544。
Convexity measure (years) == 79.0544 / 2(2) =
凸度测量（年） == 79.0544 / 2（2）=
79.0544 / 4 = 19.7636077. Dollar convexity measure = convexity measure (years) times P = 19.7636077($1,040.55) equals about $2,056.51.
79.0544 / 4 = 19.7636077。美元凸度测量 = 凸性测量（年）乘以 P = 19.7636077（1,040.55 美元）等于约 2,056.51 美元。
[NOTE. We can get the same convexity measure by proceeding as follows. First, we increase the yield on the bond by 10 basis points from 8% to 8.1%. Thus, ∆y is 0.001. The new price (P+) can be computed using our bond valuation formula. Doing this we get $1,036.408 with a bond quote of $103.6408. Second, we decrease the yield on the bond by 10 basis points from 8% to 7.9%. The new price (P_) can be computed using our bond valuation formula. Doing this we get $104.4721. Third, with the initial price, P0, equal to $104.0554, the convexity measure of any bond can be approximated using the following formula:
[注意。我们可以通过如下方式获得相同的凸度测度。首先，我们将债券收益率从 8% 下调 10 个基点至 8.1%。因此，∆y 为 0.001。新价格 （P+） 可以使用我们的债券估值公式计算。这样做，我们得到 1,036.408 美元，债券报价为 103.6408 美元。其次，我们将债券收益率从 8% 上调 10 个基点至 7.9%。新价格 （P_） 可以使用我们的债券估值公式计算。这样做，我们得到 104.4721 美元。第三，初始价格 P0 等于 104.0554 美元，任何债券的凸度度量可以使用以下公式近似计算：
approximate convexity measure = .
近似凸度测度 = 。
Inserting in our values, the approximate convexity measure for bond B is
插入我们的值，键 B 的近似凸度 为
approximate convexity measure == 19.7636548.
近似凸度测量 ==19.7636548。
The approximate convexity measure of 19.7636548 is almost identical to the convexity measure of 19.7636077 computed above.]
19.7636548 的近似凸度测度与上面计算的 19.76360 77 的凸度测度几乎相同。
(f) Compute the approximate convexity measure for bonds A and B using the shortcut formula by changing yields by 20 basis points and compare your answers to the convexity measure calculated in part (e).
（f） 使用快捷公式通过将收益率改变 20 个基点来计算债券 A 和 B 的近似凸度测量，并将您的答案与第 （e） 部分计算的凸性测量进行比较。
To compute the approximate convexity measure for bond A, which is a 2-year 8% coupon bond trading at 8% with an initial price (P0) of $100, we proceed as follows.
为了计算债券 A 的近似凸性度量，债券 A 是一只 2 年期 8% 票息债券，交易利率为 8%，初始价格 （P0） 为 100 美元，我们进行如下操作。
First, we increase the yield on the bond by 20 basis points from 8% to 8.2%. Thus, ∆y is 0.002. The new price (P+) can be computed using our bond valuation formula. Doing this we get $99.6379.
首先，我们将债券收益率从 8% 下调 20 个基点至 8.2%。因此，∆y 为 0.002。新价格 （P+） 可以使用我们的债券估值公式计算。这样做，我们得到 99.6379 美元。
Second, we decrease the yield on the bond by 20 basis points from 8% to 7.8%. The new price (P_) can be computed using our bond valuation formula. Doing this we get $1,003.638 with a bond price quote of $100.3638.
其次，我们将债券收益率从 8% 上调 20 个基点至 7.8%。新价格 （P_） 可以使用我们的债券估值公式计算。这样做，我们得到 1,003.638 美元，债券报价为 100.3638 美元。
Third, with the initial price, P0, equal to $100, the convexity measure of any bond can be approximated using the following formula:
第三，初始价格 P0 等于 100 美元，任何债券的凸度都可以使用以下公式进行近似计算：
approximate convexity measure = .
近似凸度测度 = 。
Inserting in our values, the approximate convexity measure is
插入我们的值，近似凸度测度为
approximate convexity measure == 4.2773486.
近似凸度测量 == 4.2773486。
This answer of 4.2773486 for the approximate convexity measure is very similar to that computed in part (e) using the convexity measure where we got 4.2773350. [NOTE. The 4.2773486 for a change of 20 basis points is almost identical to the 4.2773384 that we can compute for a change of 10 basis points.]
近似凸度测度的答案 4.2773486 与在 （e） 部分使用凸度测度计算的答案非常相似，我们得到了 4.2773350。[注意。变化 20 个基点的 4.2773486 与我们计算的 4.2773384 变化 10 个基点几乎相同。
To compute the approximate convexity measure for bond B which is a 5-year 9% coupon bond trading at 8% with an initial price (P0) of $104.0554 (worth $1,040.554 and with a par value = M = $1,000), we proceed as follows.
为了计算债券 B 的近似凸性度量，债券 B 是 5 年期 9% 票息债券，交易利率为 8%，初始价格 （P0） 为 104.0554 美元（价值 1,040.554 美元，面值 = M = 1,000 美元），我们进行如下操作。
First, we increase the yield on the bond by 20 basis points from 8% to 8.2%. Thus, ∆y is 0.001. The new price (P+) can be computed using our bond valuation formula. Doing this we get $103.2283.
首先，我们将债券收益率从 8% 下调 20 个基点至 8.2%。因此，∆y 为 0.001。新价格 （P+） 可以使用我们的债券估值公式计算。这样做，我们得到 103.2283 美元。
Second, we decrease the yield on the bond by 20 basis points from 8% to 7.8%. The new price (P_) can be computed using our bond valuation formula. Doing this we get $1,048.909 with a bond price quote of $104.8909.
其次，我们将债券收益率从 8% 上调 20 个基点至 7.8%。新价格 （P_） 可以使用我们的债券估值公式计算。这样做，我们得到 1,048.909 美元，债券报价为 104.8909 美元。
Third, with the initial price, P0, equal to $104.0554, the convexity measure of any bond can be approximated using the following formula:
第三，初始价格 P0 等于 104.0554 美元，任何债券的凸度度量可以使用以下公式近似计算：
approximate convexity measure = .
近似凸度测度 = 。
Inserting in our values, the approximate convexity measure is:
插入我们的值，近似凸度度量为：
approximate convexity measure = = 19.763824.
近似凸度测量 = = 19.763824。
This answer of 4.2773486 for the approximate convexity measure is very similar to that computed in part (e) using the convexity measure where we got 19.7636077 [NOTE. The 19.7636077 for a change of 20 basis points is almost identical to the 19.7636548 computed for a change of 10 basis points.]
近似凸度测度的答案 4.2773486 与在 （e） 部分使用凸度测度计算的答案非常相似，我们得到 19.7636077 [注.变动 20 个基点的 19.7636077 与变动 10 个基点的计算结果为 19.7636548 几乎相同。
3. Can you tell from the following information which of the following three bonds will have the greatest price volatility, assuming that each is trading to offer the same yield to maturity?
3. 您能否从以下信息中判断出以下三种债券中哪一种的价格波动最大，假设每只债券都以提供相同的到期收益率？
Bond | Coupon Rate (%) | Maturity (years) |
X | 8 | 9 |
Y | 10 | 11 |
Z | 11 | 12 |
The price of a bond will change over time as a result of a change in the perceived credit risk of the issuer. Thus, if one of the three bonds undergoes greater change in credit risk then that bond might be expected to experience more volatility unless other factors dominate. Below we describe these factors: term to maturity, coupon rate, and yield to maturity.
由于发行人感知的信用风险发生变化，债券的价格将随着时间的推移而变化 。因此，如果三种债券中的一种货币的信用风险变化较大，那么除非其他因素占主导地位，否则预计该债券的波动性可能会更大。下面我们描述了这些因素：到期期限、票面利率和到期收益率。
For a given term to maturity and initial yield, the price volatility of a bond will increase as the coupon rate becomes smaller. Thus, ceteris paribus, we would expect bond X to have greater price volatility that bond Y, and bond Y to have greater price volatility than bond Z. However, the differences in coupon rates for bonds Y and Z are not that great. Thus, if just looking at these two bonds, the differences in price volatilities may not be that recognizable.
对于给定的到期期限和初始收益率，债券的价格波动性将随着票面利率变小而增加。因此，ceteris paribus，我们预计债券 X 的价格波动性大于债券 Y，债券 Y 的价格波动性大于债券 Z。然而，债券 Y 和 Z 的票面利率差异并不大。因此，如果只看这两只债券，价格波动的差异可能并不那么明显。
For a given coupon rate and initial yield, longer terms to maturity will produce greater price volatility. Thus, ceteris paribus, bond Z will have more price volatility than bond Y and bond Y will have more price volatility than bond X. However, the differences in maturities for bonds Y and Z are not that great. Thus, if just comparing these two bonds, the differences in price volatilities may not be that identifiable.
对于给定的票面利率和初始收益率，较长的到期期限将产生更大的价格波动。因此，同等价格，债券 Z 的价格波动性大于债券 Y，债券 Y 的价格波动性大于债券 X。然而，债券 Y 和 Z 的到期日差异并不大。因此，如果只比较这两种债券，价格波动的差异可能不是那么明显。
We should note that the expectations of price volatility based upon coupon rates and maturities are the reverse for bonds X and Z. For example, we expect bond X to have the greatest price volatility based upon coupon rate but the lowest based upon maturity. For bond Z, we expect it to have lowest price volatility based upon coupon rate but the greatest based upon maturity. If the coupon rate and the maturity factors or characteristics balance out then it is possible all three bonds will experience price volatilities that are very similar.
我们应该注意到，基于票面利率和到期日的价格波动预期与债券 X 和 Z 相反。例如，我们预计债券 X 的价格波动性（票面利率）最大，但到期时价格波动最低。对于债券 Z，我们预计其票面利率的价格波动最低，但到期日价格波动最大。如果票面利率和到期因素或特征平衡，那么这三种债券的价格波动都可能非常相似。
[NOTE. An implication of the maturity factor is that investors who want to increase a portfolio’s price volatility because they expect interest rates to fall, all other factors being constant, should hold bonds with long maturities in the portfolio. To reduce a portfolio’s price volatility in anticipation of a rise in interest rates, bonds with shorter-term maturities should be held in the portfolio.]
[注意。期限因子的一个含义是，由于预期利率下降而希望增加投资组合价格波动的投资者，所有其他因素保持不变，应该在投资组合中持有长期债券。为了减少预期利率上升的投资组合价格波动，投资组合中应持有期限较短的债券。
Although the yield to maturity is held constant for bonds X, Y, and Z, the yield to maturity can also play a role as a factor impacting a bond’s price volatility. Ceteris paribus, the higher the yield to maturity at which a bond trades, then the lower the price volatility should be.
尽管债券 X、Y 和 Z 的到期收益率保持不变，但到期收益率也可以作为影响债券价格波动的因素发挥作用。Ceteris paribus，债券交易的到期收益率越高，那么价格波动性应该就越低。
In addition to the above factors, we have to keep in mind four important properties concerning the price volatility of an option-free bond that result from the convex shape of the price-yield relationship.
除了上述因素之外，我们还必须牢记关于无期权债券价格波动的四个重要属性，这些属性是由价格-收益率关系的凸形引起的。
First, although the prices of all option-free bonds move in the opposite direction from the change in yield required, the percentage price change (e.g., price volatility) depends on the convexity relationship between price and yield for each of the three bonds. Thus, for bonds X, Y, and Z, we will not expect the same price volatility due to likely differences in convexity. Whether the market yield rises or falls, the bond with the greatest convexity will achieve a higher price. That is, if the required yield rises, the capital loss for this bond will be less while a fall in the required yield will generate greater price appreciation.
首先，尽管所有无期权债券的价格都与所需的收益率变化相反，但价格变化百分比（例如价格波动）取决于三种债券的价格和收益率之间的凸关系。因此，对于债券 X、Y 和 Z，由于凸性可能存在差异，我们不会预期相同的价格波动。无论市场收益率是上升还是下降，凸性最大的债券都会获得更高的价格。也就是说，如果所需收益率上升，该债券的资本损失将减少，而所需收益率的下降将产生更大的价格升值。
Second, for very small changes in the yield required, the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases. Thus, for bonds X, Y, and Z if the percentage price change is very small, we will not likely detect which bond has the greatest price volatility.
其次，如果所需收益率发生变化或非常小的变化，则无论所需收益率是增加还是减少，给定债券的价格变化百分比都大致相同。因此，对于债券 X、Y 和 Z，如果百分比价格变化非常小，我们不太可能检测到哪种债券的价格波动最大。
Third, for large changes in the required yield, the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield. Thus, whichever bond or bonds change, the price volatility will depend on the direction of the change.
第三，所需产量发生较大变化，所需产量增加的价格百分比变化与所需产量降低的价格百分比变化不同。因此，无论哪种债券发生变化，价格波动都将取决于变化的方向。
Fourth, for a given large change in basis points, the percentage price increase is greater than the percentage price decrease. Thus, whichever bond or bonds change, the price volatility will be relatively greater if there is a percentage price increase as opposed to a decrease. The implication of this fourth property is that if an investor owns a bond, the price appreciation that will be realized (if the required yield decreases) is greater than the capital loss that will be realized if the required yield rises by the same number of basis points. For an investor who is “short” a bond, the reverse is true: the potential capital loss is greater than the potential capital gain if the required yield changes by a given number of basis points.
第四，f或给定的基点变化较大，价格上涨的百分比大于价格下跌的百分比。因此，无论债券或债券发生变化，如果价格上涨百分比而不是下跌，价格波动性将相对较大。这第四个属性的含义是，如果投资者拥有债券，则将实现的价格升值（如果所需收益率下降）大于如果所需收益率上升相同数量的基点将实现的资本损失。对于“做空”债券的投资者来说，情况正好相反：如果所需收益率变化给定数量的基点，则潜在资本损失大于潜在资本收益。
4. Answer the below questions for bonds A and B.
4. 对于债券 A 和 B，请回答以下问题。
Bond A | Bond B | |
Coupon | 8% | 9% |
Yield to maturity | 8% | 8% |
Maturity (years) | 2 | 5 |
Par | $100.00 | $100.00 |
Price | $100.00 | $104.055 |
(a) Calculate the actual price of the bonds for a 100-basis-point increase in interest rates.
（a） 计算利率上调 100 个基点的债券实际价格。
For Bond A, we get a bond quote of $100 for our initial price if we have an 8% coupon rate and an 8% yield. If we change the yield 100 basis point so the yield is 9%, then the value of the bond (P) is the present value of the coupon payments plus the present value of the par value. We have C = $40, y = 4.5%, n = 4, and M = $1,000. Inserting these numbers into our present value of coupon bond formula, we get:
对于债券 A，如果我们的票面利率为 8%，收益率为 8%，我们的初始价格为 100 美元的债券报价。如果我们将收益率改变 100 个基点，使收益率为 9%，那么债券的价值 （P） 就是息票支付的现值加上面值的现值。我们有 C = 40 美元，y = 4.5%，n = 4 和 M = 1,000 美元。将这些数字代入我们的息票债券现值公式中，我们得到：
== $143.501.
== 143.501 美元。
The present value of the par or maturity value of $1,000 is:
1,000 美元的面值或到期价值的现值为：
= = $838.561.
= = 838.561 美元。
Thus, the value of bond A with a yield of 9%, a coupon rate of 8%, and a maturity of 2 years is: P = $143.501 + $838.561 = $982.062. Thus, we get a bond quote of $98.2062.
因此， 收益率为 9%、票面利率为 8% 且期限为 2 年的债券 A 的价值为：P = 143.501 美元 + 838.561 美元 = 982.062 美元。因此，我们得到 98.2062 美元的债券报价。
We already know that bond B will give a bond value of $1,000 and a bond quote of $100 since a change of 100 basis points will make the yield and coupon rate the same, For example, inserting the values of C = $45, y = 4.5%, n = 10, and M = $1,000 into our bond valuation formula gives: P = $356.072 + $643.928 = $1,000.00 with a bond quote of $100.
我们已经知道债券 B 将给出 1,000 美元的债券价值和 100 美元的债券报价，因为 100 个基点的变化将使收益率和票面利率相同，例如，将 C = 45 美元、y = 4.5%、n = 10 和 M = 1,000 美元的值插入我们的债券估值公式中，得到： P = 356.072 美元 + 643.928 美元 = 1,000.00 美元，债券报价为 100 美元。
(b) Using duration, estimate the price of the bonds for a 100-basis-point increase in interest rates.
（b） 使用久期，估计利率上调 100 个基点的债券价格。
To estimate the price of bond A, we begin by first computing the modified duration. We can use an alternative formula that does not require the extensive calculations required by the Macaulay procedure. The formula is:
要估算债券 A 的价格，我们首先要计算修改后的久期。我们可以使用不需要麦考利程序所需的大量计算的替代公式。公式为：
modified duration = .
修改的持续时间 = 。
Putting all applicable variables in terms of $100, we have C = $4, n = 4, y = 0.045, and P = $98.2062. Inserting these values, in the modified duration formula gives:
将所有适用的变量以 100 美元为单位，我们有 C = 4 美元，n = 4，y = 0.045，P = 98.2062 美元。在修改后的 duration 公式中插入这些值会得到：
= =
($1,975.308642[0.161439] + $35.664491) / $98.2062 = ($318.89117 + $35.664491) / $98.2062 = $354.555664 / $98.2062 = 3.6103185 or about 3.61. Converting to annual number by dividing by two gives a modified duration of 1.805159 (before the increase in 100 basis points it was 1.814948).
（1975.308642 美元 [0.161439] + 35.664491 美元） / 98.2062 美元 = （318.89117 美元 + 35.664491 美元） / 98.2062 美元 = 354.555664 美元 / 98.2062 美元 = 3.6103185 或约 3.61。通过除以 2 转换为年度数字，修正久期为 1.805159（在增加 100 个基点之前为 1.814948）。
We next solve for the change in price using the modified duration of 1.805159 and dy = 100 basis points = 0.01. We have:
接下来，我们使用修改后的久期 1.805159 和 dy = 100 个基点 = 0.01 来求解价格变化。我们有：
= (modified duration)(dy) = 1.805159(0.01) = 0.0180515.
= （修改后的持续时间）（dy） = 1.805159（0.01） = 0.0180515。
We can now solve for the new price of bond A as shown below:
我们现在可以求解债券 A 的新价格，如下所示：
= (1 + 0.0180515)$1,000 = (0.9819484)$1,000 = $981.948.
= （1 + 0.0180515）1,000 美元 = （0.9819484）1,000 美元 = 981.948 美元。
This is slightly less than the actual price of $982.062. The difference is $982.062 – $981.948 = $0.114.
这略低于 982.062 美元的实际价格。差额为 982.062 美元 – 981.948 美元 = 0.114 美元。
To estimate the price of bond B, we follow the same procedure just shown for bond A. Using the alternative formula for modified duration that does not require the extensive calculations required by the Macaulay procedure and noting that C = $45, n = 10, y = 0.045, and P = $100, we get: modified duration =
为了估算债券 B 的价格，我们遵循刚才显示的债券 A 的相同程序。使用修改久期的替代公式，该公式不需要麦考利程序所需的大量计算，并注意到 C = 45 美元，n = 10，y = 0.045 和 P = 100 美元，我们得到：修改后的久期 =
= =
($791.27182 + $0) / $100 = 7.912718 or about 7.91 (before the increase in 100 basis points it was 7.988834 or about 7.99). Converting to an annual number by dividing by two gives a modified duration of 3.956359 (before the increase in 100 basis points it was 3.994417). We will now estimate the price of bond B using the modified duration measure. With 100 basis points giving dy = 0.01 and an approximate duration of 3.956359, we have:
（791.27182 美元 + 0 美元）/ 100 美元 = 7.912718 或约 7.91（在增加 100 个基点之前为 7.988834 或约 7.99）。通过除以 2 转换为年度数字，修正久期为 3.956359（在增加 100 个基点之前为 3.994417）。 我们现在将使用修改后的久期指标来估算债券 B 的价格。100 个基点给出 dy = 0.01，大约久期为 3.956359，我们有：
= (modified duration)(dy) = 3.956359(0.01) = 0.0395635.
= （修改后的持续时间）（dy） = 3.956359（0.01） = 0.0395635.
Thus, the new price is = (1 – 0.0395635)$1,040.55 = (0.9604364)$1,040.55 = $999.382. This is slightly less than the actual price of $1,000. The difference is $1,000 – $999.382 = $0.618.
因此，新价格为 = （1 – 0.0395635）1,040.55 美元 = （0.9604364）1,040.55 美元 = 999.382 美元。这比实际价格 1,000 美元略低。差额为 1000 USD – 999.382 USD = 0.618 USD。
(c) Using both duration and convexity measures, estimate the price of the bonds for a 100-basis-point increase in interest rates.
（c） 使用久期和凸性度量，估计利率上调 100 个基点的债券价格。
For bond A, we use the duration and convexity measures as given below.
对于键 A，我们使用 duration 和 convexic 度量，如下所示。
First, we use the duration measure. We add 100 basis points and get a yield of 9%. We now have C = $40, y = 4.5%, n = 4, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $982.062. Prior to that, the price sold at par (P = $1,000) since the coupon rate and yield were then equal. The actual change in price is: ($982.062 – $1,000) = $17.938 and the actual percentage change in price is: $17.938 / $1,000 = 0.017938%.
首先，我们使用 duration 度量。我们增加 100 个基点，收益率为 9%。我们现在有 C = 40 美元，y = 4.5%，n = 4 和 M = 1,000 美元。注意。在第 （a） 部分，我们计算了实际的债券价格，得到 P = 982.062 美元。在此之前，由于票面利率和收益率相等，因此以面值 （P = 1,000） 出售。实际价格变化为： （$982.062 – $1,000） = $17.938，实际价格百分比变化为： $17.938 / $1,000 = 0.017938%.
We will now estimate the price by first approximating the dollar price change. With 100 basis points giving dy = 0.01 and a modified duration computed in part (b) of 1.805159, we have:
现在，我们将首先通过近似美元价格变化来估计价格。100 个基点给出 dy = 0.01，在 （b） 部分计算的修改久期为 1.805159，我们得到：
= (modified duration)(dy) = 1.805159(0.01) = 0.01805159 or about 1.805159%.
= （修改后的持续时间）（dy） = 1.805159（0.01） = 0.01805159 或大约 1.805159%。
This is slightly more negative than the actual percentage decrease in price of 1.7938%. The difference is 1.7938% – (1.805159%) = 1.7938% + 1.805159% = 0.011359%.
这比实际价格下降百分比 1.7938% 略负。差值为 1.7938% – （1.805159%） = 1.7938% + 1.805159% = 0.011359%。
Using the 1.805159% just given by the duration measure, the new price for bond A is:
使用久期度量所给出的 1.805159%，债券 A 的新价格为：
= (1 + 0.018051598)$1,000 = (0.9819484)$1,000 = $981.948.
= （1 + 0.018051598）1,000 美元 = （0.9819484）1,000 美元 = 981.948 美元。
This is slightly less than the actual price of $982.062. The difference is $982.062 – $981.948 = $0.114.
这略低于 982.062 美元的实际价格。差额为 982.062 美元 – 981.948 美元 = 0.114 美元。
Next, we use the convexity measure to see if we can account for the difference of 0.011359%. We have: convexity measure (half years) = =
接下来，我们使用凸性度量来查看我们是否可以解释 0.011359% 的差异。我们有：凸度测量（半年）= =
.
For bond A, we add 100 basis points and get a yield of 9%. We now have C = $40, y = 4.5%, n = 4, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $982.062. Prior to that, the price sold at par (P = $1,000) since the coupon rate and yield were then equal. Expressing numbers in terms of a $100 bond quote, we have: C = $4, y = 0.045, n = 4, and P = $98.2062. Inserting these numbers into our convexity measure formula gives:
对于债券 A，我们增加 100 个基点，收益率为 9%。我们现在有 C = 40 美元，y = 4.5%，n = 4 和 M = 1,000 美元。注意。在第 （a） 部分，我们计算了实际的债券价格，得到 P = 982.062 美元。在此之前，由于票面利率和收益率相等，因此以面值 （P = 1,000） 出售。以 100 美元的债券报价表示数字，我们有：C = 4 美元，y = 0.045，n = 4，P = 98.2062 美元。将这些数字代入我们的凸性测度公式中，得到：
convexity measure (half years) =
凸度测量（半年）=
=
1,662.88 [0.0101827] = 16.9325.
1,662.88 [0.0101827] = 16.9325。
The convexity measure (in years) = = 16.9325 / 2(2) = 16.9325 / 4 = 4.233125.
凸度测量（以 s 年为单位）= =16.9325 / 2（2） = 16.9325 / 4 = 4.233125。
[NOTE. Dollar convexity measure = convexity measure (years) times P = 4.233125($98.2062) = $415.7192.]
[注意。美元凸性度量 = 凸性度量（年）乘以 P = 4.233125（98.2062 美元） = 415.7192 美元。
The percentage price change due to convexity is: convexity measure (dy)2. Inserting in our values, we get: = 0.00021166. Thus, we have 0.021166% increase in price when we adjust for the convexity measure.
由于凸性引起的价格变化百分比为：convexity measure （dy）2。插入我们的值，我们得到：= 0.00021166。因此，当我们调整凸性度量时，我们的价格上涨了 0.021166%。
Adding the duration measure and the convexity measure, we get 1.805159% + 0.021166% = 1.783994%. Recall the actual change in price is: ($982.062 – $1,000) = $17.938 and the actual percentage change in price is: $17.938 / $1,000 = −0.017938 or approximately 1.7938%. Using the 1.783994% resulting from both the duration and convexity measures, we can estimate the new price for bond A. We have:
将久期测度和凸性测度相加，我们得到 1.805159% + 0.021166% = 1.783994%。回想一下，价格的实际变化是： （$982.062 – $1,000） = $17.938，实际价格的百分比变化是： $17.938 / $1,000 = −0.017938 或大约 1.7938%。使用久期和凸性测量得出的 1.783994%，我们可以估计债券 A 的新价格。我们有：
new price is = (1 + 0.01783994)$1,000 = (0.9819484)$1,000 = $982.160.
新价格为 = （1 + 0.01783994）1000 美元 = （0.9819484）1000 美元 = 982.160 美元。
This estimated price for bond A of $982.160 is slightly more than the actual price of $982.062. The difference is $982.160 – $982.062 = $0.098. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of $0.114 to $0.098.
债券 A 的估计价格为 982.160 美元，略高于实际价格 982.062 美元。差额为 982.160 美元 – 982.062 美元 = 0.098 美元。因此，使用凸性测量和久期测量将估计价格从 0.114 美元缩小到 0.098 美元。
For bond B, we can also estimate its price using both the duration and convexity measures as just shown for bond A.
对于债券 B，我们还可以使用刚才显示的债券 A 的久期和凸性度量来估计其价格。
First, we use the duration measure. We add 100 basis points and get a yield of 9%. We now have C = $45, y = 4.5%, n = 10, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $1,000 since the coupon rate and yield were then equal. Prior to that, the price sold at par (P = $1,040.55). The actual change in price is: ($1,000 – $1,040.55) = $40.55 and the actual percentage change in price is: $40.55 / $1,040.55 = −0.0389697 or about −3.896978%.
首先，我们使用 duration 度量。我们增加 100 个基点，收益率为 9%。我们现在有 C = 45 美元，y = 4.5%，n = 10，M = 1,000 美元。注意。n 部分 （a） 我们计算实际债券价格，得到 P = 1,000 美元，因为票面利率和收益率相等。在此之前，价格以面值 （P = 1,040.55 美元） 出售。价格的实际变化为：（1,000 美元 – 1,040.55 美元） = 40.55 美元，价格的实际百分比变化为：40.55 美元 / 1,040.55 美元 = −0.0389697 或约 −3.896978%。
We will now estimate the price by first approximating the dollar price change. With 100 basis points giving dy = 0.01 and a modified duration computed in part (b) of 3.956359, we have:
现在，我们将首先通过近似美元价格变化来估计价格。100 个基点给出 dy = 0.01，在 （b） 部分计算的修改久期为 3.956359，我们得到：
= (modified duration)(dy) = 3.956359(0.01) = 0.0395635 or about 3.95635%.
= （修改后的持续时间）（dy） = 3.956359（0.01） = 0.0395635 或大约 3.95635%。
This is slightly more negative than the actual percentage decrease in price of −3.896978%. The difference is 3.896978 – 3.95635% = 3.896978 + 3.95635% = 0.059382%.
这比实际价格下跌百分比 -3.896978 略负。差值为 3.896978 – 3.95635% = 3.896978 + 3.95635% = 0.059382%。
Using the 3.95635% just given by the duration measure, the new price for bond B is:
使用久期度量所给出的 3.95635%，债券 B 的新价格为：
= (1 – 0.0395635)$1,040.55 = (0.96043641)$1,040.55 = $999.382.
= （1 – 0.0395635）1,040.55 美元 = （0.96043641）1,040.55 美元 = 999.382 美元。
This is slightly less than the actual price of $1,000. The difference is $1,000 – $999.382 = $0.618.
这比实际价格 1,000 美元略低。差额为 1000 USD – 999.382 USD = 0.618 USD。
Next, we use the convexity measure to see if we can account for the difference of 0.0594%. We have:
接下来，我们使用凸性度量来查看我们是否可以解释 0.0594% 的差异。我们有：
convexity measure (half years) = =
凸度测量（半年） = =
.
For bond B, we add 100 basis points and get a yield of 9%. We now have C = $45, y = 4.5%, n =10, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $1,000 since the coupon rate and yield were then equal. Prior to that, the price sold at P = $1,040.55. Expressing numbers in terms of a $100 bond quote, we have: C = $4.5, y = 0.045, n = 10, and P = $100. Inserting these numbers into our convexity measure formula gives:
对于债券 B，我们增加 100 个基点，收益率为 9%。我们现在有 C = 45 美元，y = 4.5%，n = 10，M = 1,000 美元。注意。在第 （a） 部分，我们计算了实际债券价格，得到 P = 1,000 美元，因为票面利率和收益率相等。在此之前，售价为 P = 1,040.55 美元。以 100 美元的债券报价表示数字，我们有：C = 4.5 美元，y = 0.045，n = 10，P = 100 美元。将这些数字代入我们的凸性测度公式中，得到：
convexity measure (half years) =
凸度测量（半年）=
=
7,781.03[0.01000] = 77.8103.
7,781.03[0.01000] = 77.8103。
The convexity measure (in years) = = 77.8103 / 2(2) = 77.8103 / 4 = 19.452564.
凸度测量（以 s 年为单位）= =77.8103 / 2（2）= 77.8103 / 4 = 19.452564。
[NOTE. Dollar convexity measure = convexity measure (years) times P = 19.452564($100) = $1,945.2564.]
[注意。美元凸性度量 = 凸性度量（年）乘以 P = 19.452564（100 美元） = 1,945.2564 美元。
The percentage price change due to convexity is: convexity measure (dy)2. Inserting in our values, we get: = 0.00097463. Thus, we have 0.097463% increase in price when we adjust for the convexity measure.
由于凸性引起的价格变化百分比为：convexity measure （dy）2。插入我们的值，我们得到：= 0.00097463。因此，当我们调整凸性度量时，我们的价格上涨了 0.097463%。
Adding the duration measure and the convexity measure, we get 3.956359% + 0.097263% equals 3.859096%. Recall the actual change in price is: ($1,000 – $1,040.55) = $40.55 and the actual percentage change in price is: $40.55 / $1,040.55 = 0.0389697 or about 3.896978%. Thus, the new price is = (1 – 0.03859096)$1,040.55 = (0.9614091)$1,040.55 = $1,000.394 for bond A. This is about the same as the actual price of $1,000. The difference is $1,000.394 – $1,000 = $0.394. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of −$0.618 to $0.394.
将久期测度和凸性测度相加，我们得到 3.956359% + 0.097263% 等于 3.859096%。回想一下，价格的实际变化是： （$1,000 – $1,040.55） = $40.55，实际价格的百分比变化是： $40.55 / $1,040.55 = 0.03896977 或大约 3.896978%。 因此，债券 A 的新价格为 = （1 – 0.03859096）1,040.55 美元 = （0.9614091）1,040.55 美元 = 1,000.394 美元。这与 1,000 美元的实际价格大致相同。差额为 1000.394 USD – 1000 USD = 0.394 USD。因此，使用凸性指标和久期指标将估计价格从 −0.618 美元缩小到 0.394 美元。
(d) Comment on the accuracy of your results in parts b and c, and state why one approximation is closer to the actual price than the other.
（d） 在 b 部分和 c 部分中评论您的结果的准确性，并说明为什么一个近似值比另一个近似值更接近实际价格。
For bond A, the actual price is $982.062. When we use the duration measure, we get a bond price of $981.948 that is $0.114 less than the actual price. When we use duration and convex measures together, we get a bond price of $982.160. This is slightly more than the actual price of $982.062. The difference is $982.160 – $982.062 = $0.098. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of $0.114 to $0.0981.
对于债券 A，实际价格为 982.062 美元。当我们使用久期度量时，我们得到的债券价格为 981.948 USD，比实际价格低 0.114 USD。当我们同时使用久期和凸度量时，我们得到的债券价格为 982.160 美元。这比实际价格 982.062 美元略高。差额为 982.160 美元 – 982.062 美元 = 0.098 美元。因此，使用凸性度量和久期度量将估计价格从 0.114 美元缩小到 0.0981 美元。
For bond B, the actual price is $1,000. When we use the duration measure, we get a bond price of $999.382 that is $0.618 less than the actual price. When we use duration and convex measures together, we get a bond price of $1,000.394. This is slightly more than the actual price of $1,000. The difference is $1,000.394 – $1,000 = $0.394. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of −$0.618 to $0.394
对于债券 B，实际价格为 1000 美元。当我们使用久期度量时，我们得到的债券价格为 999.382 USD，比实际价格低 0.618 USD。当我们同时使用久期和凸度量时，我们得到的债券价格为 1,000.394 美元。这比实际价格 1,000 美元略高。差额为 1000.394 USD – 1000 USD = 0.394 USD。因此，使用凸性度量和久期度量将估计价格从 −0.618 美元缩小到 0.394 美元
As we see, using the duration and convexity measures together is more accurate. The reason is that adding the convexity measure to our estimate enables us to include the second derivative that corrects for the convexity of the price-yield relationship. More details are offered below.
正如我们所看到的，同时使用持续时间和凸性度量更准确。原因是，将凸性度量添加到我们的估计中使我们能够包括修正价格-收益率关系的凸性的二阶导数。下面提供了更多详细信息。
Duration (modified or dollar) attempts to estimate a convex relationship with a straight line (the tangent line). We can specify a mathematical relationship that provides a better approximation to the price change of the bond if the required yield changes. We do this by using the first two terms of a Taylor series to approximate the price change as follows:
Duration （modified or dollar） 尝试估计与直线 （切线） 的凸关系。我们可以指定一个数学关系，如果所需的收益率发生变化，它可以更好地近似于债券的价格变化。我们通过使用泰勒级数的前两项来近似价格变化，如下所示：
(1).
Dividing both sides of this equation by P to get the percentage price change gives us:
将此方程的两边除以 P 得到价格变化百分比，得到：
(2).
The first term on the right-hand side of equation (1) is equation for the dollar price change based on dollar duration and is our approximation of the price change based on duration. In equation (2), the first term on the right-hand side is the approximate percentage change in price based on modified duration. The second term in equations (1) and (2) includes the second derivative of the price function for computing the value of a bond. It is the second derivative that is used as a proxy measure to correct for the convexity of the price-yield relationship. Market participants refer to the second derivative of bond price function as the dollar convexity measure of the bond. The second derivative divided by price is a measure of the percentage change in the price of the bond due to convexity and is referred to simply as the convexity measure.
等式 （1） 右侧的第一项是基于美元久期的美元价格变化方程，是我们基于久期的价格变化的近似值。在等式 （2） 中，右侧的第一项是基于修改后的久期价格的近似百分比变化。方程 （1） 和 （2） 中的第二项包括用于计算债券价值的价格函数的二阶导数。它是用作代理度量的第二个导数，用于校正价格-收益率关系的凸性。市场参与者将债券价格函数的二阶导数称为债券的美元凸性度量。二阶导数除以价格是衡量债券价格因凸性而发生的百分比变化的指标，简称为凸性指标。
(e) Without working through calculations, indicate whether the duration of the two bonds would be higher or lower if the yield to maturity is 10% rather than 8%.
（e） 在不进行计算的情况下，说明如果到期收益率为 10% 而不是 8%，则两只债券的久期是更高还是更低。
Like term to maturity and coupon rate, the yield to maturity is a factor that influences price volatility. Ceteris paribus, the higher the yield level, the lower the price volatility. The same property holds for modified duration. Thus, a 10% yield to maturity will have both less volatility than an 8% yield to maturity and also a smaller duration.
与到期期限和票面利率一样，到期收益率是影响价格波动的一个因素。Ceteris paribus，收益率水平越高，价格波动性越小。同一属性在修改后的持续时间内保持不变。因此，10% 的到期收益率将具有较低的波动性，而 8% 的到期收益率将具有更短的持续时间。
There is consistency between the properties of bond price volatility and the properties of modified duration. When all other factors are constant, a bond with a longer maturity will have greater price volatility. A property of modified duration is that when all other factors are constant, a bond with a longer maturity will have a greater modified duration. Also, all other factors being constant, a bond with a lower coupon rate will have greater bond price volatility. Also, generally, a bond with a lower coupon rate will have a greater modified duration. Thus, bonds with greater durations will greater price volatilities.
债券价格波动的属性与修改后的久期属性之间存在一致性。当所有其他因素保持不变时，期限较长的债券将具有更大的价格波动。修正久期的一个特性是，当所有其他因素保持不变时，期限较长的债券将具有更长的修正久期。此外，在所有其他因素不变的情况下，票面利率较低的债券将具有更大的债券价格波动性。此外，一般来说，票面利率较低的债券将具有更长的修改久期。因此，久期较长的债券将增加价格波动。
5. State why you would agree or disagree with the following statement: As the duration of a zero-coupon bond is equal to its maturity, the price responsiveness of a zero-coupon bond to yield changes is the same regardless of the level of interest rates.
5. 说明您为什么同意或不同意以下陈述：由于零息债券的久期等于其到期日，因此无论利率水平如何，零息债券对收益率变化的价格反应都是相同的。
As seen in Exhibit 4-3, the price responsiveness of a zero-coupon bond is different as yields change. Like other bonds, zero-coupon bonds have greater price responsiveness for changes at higher levels of maturity as interest rates change. Like other bonds, zero-coupon bonds also have greater price responsiveness for changes at lower levels of interest rates compared to higher levels of interest rates.
如图 4-3 所示，零息债券的价格响应能力随着收益率的变化而变化。与其他债券一样，随着利率的变化，零息债券对更高期限水平的变化具有更大的价格响应能力。与其他债券一样，与 较高利率水平相比，零息债券对较低利率水平的变化也具有更大的价格响应能力。
Except for long-maturity deep-discount bonds, bonds with lower coupon rates will have greater modified and Macaulay durations. Also, for a given yield and maturity, zero-coupon bonds have higher convexity and thus greater price responsiveness to changes in yields.
除长期深度贴现债券外，票面利率较低的债券将具有更长的修改久期和麦考利久期。此外，对于给定的收益率和期限，零息债券具有更高的凸性，因此对收益率变化的价格响应性更强。
6. State why you would agree or disagree with the following statement: When interest rates are low, there will be little difference between the Macaulay duration and modified duration measures.
6. 说明您为什么同意或不同意以下陈述： 当利率较低时，麦考利久期和修改后的久期指标之间几乎没有差异。
The Macaulay duration is equal to the modified duration times one plus the yield. Rearranging this expression gives:
麦考利久期等于修改后的久期乘以 1 加上产量。重新排列此表达式可得到：
modified duration = .
修改的持续时间 = 。
It follows that the modified duration will approach equality with the Macaulay duration as yields approach zero. Thus, if by low interest rates one means rates approaching zero, then one would agree with the statement.
因此，当收益率接近零时，修改后的久期将与麦考利久期接近相等。因此，如果低利率是指利率接近零，那么人们会同意这种说法。
[NOTE. Like term to maturity and coupon rate, the yield to maturity is a factor that will influence price volatility. All other factors constant, the higher the yield level, the lower the price volatility. The same property holds for duration. There is also consistency between the properties of bond price volatility and the properties of modified duration. When all other factors are constant, a bond with a longer maturity will have greater price volatility. A property of modified duration is that when all other factors are constant, a bond with a longer maturity will have a greater modified duration. Also, all other factors being constant, the lower the coupon rate, the greater the bond price volatility. Also, generally, lower coupon rates will render greater modified durations. Thus, bonds with greater modified durations will have greater the price volatilities.]
[注意。与到期期限和票面利率一样，到期收益率也是影响价格波动的一个因素。所有其他因素保持不变，收益率水平越高，价格波动性越低。同一属性的持续时间保持不变。债券价格波动的属性与修改后的久期属性之间也存在一致性。当所有其他因素保持不变时，期限较长的债券将具有更大的价格波动。修正久期的一个特性是，当所有其他因素保持不变时，期限较长的债券将具有更长的修正久期。此外，在所有其他因素不变的情况下，票面利率越低，债券价格波动性就越大。此外，一般来说，较低的票面利率将呈现更大的修改期限。因此，久期修改后的债券将具有更大的价格波动性。
7. State why you would agree or disagree with the following statement: If two bonds have the same dollar duration, yield, and price, their dollar price sensitivity will be the same for a given change in interest rates.
7. 说明您为什么同意或不同意以下陈述：如果两种债券具有相同的美元久期、收益率和价格，则对于给定的利率变化，它们的美元价格敏感性将相同。
If the two bonds have the same dollar duration then their percentage change in price is the same. This implies they will have the same dollar price sensitivity. This possibility is seen from the following equation:
如果两种债券具有相同的美元久期，则它们的价格变化百分比相同。这意味着他们将具有相同的美元价格敏感性。从以下等式中可以看出这种可能性：
= (modified duration)P
= （修正持续时间）P
where the expression on the right-hand side is the estimated dollar duration. By having the same dollar duration, price (P), and yield, we see they can have the same price change (dP) for a given change in yield (dy). Thus, their dollar price change or dollar price sensitivity can be the same.
其中右侧的表达式是估计的美元持续时间。通过具有相同的美元久期、价格 （P） 和收益率，我们看到对于给定的收益率变化 （dy），它们可以具有相同的价格变化 （dP）。因此，它们的美元价格变化或美元价格敏感性可以相同。
There are possible caveats to the above argument that make it possible that the dollar price sensitivity can be different for a given change in interest rates. For example, for an increase in the required yield, the estimated dollar price change is more than the actual price change. The reverse is true for a decrease in the required yield. Thus, to the extent the estimated changes can differ, their percentage changes in price can differ. This makes it possible that their dollar price sensitivity will be different for a given change in interest rates. Also, coupon rates are a factor in determining price behavior. For an option-free bond with a given term to maturity and initial yield, the price volatility of a bond will increase as the coupon rate decreases. Thus, unless coupon rates are actually the same when the dollar duration, yield, and price are the same, then their dollar price sensitivity will not necessarily be the same for a given change in interest rates.
上述论点可能存在一些警告，这使得美元价格敏感性对于给定的利率变化可能会有所不同。例如，对于所需产量的增加，估计的美元价格变化大于实际价格变化。所需产量的降低则相反。因此，如果估计的变化可能不同，它们的价格变化百分比也可能不同。这使得它们的美元价格敏感性可能会因给定的利率变化而不同。此外，票面利率是决定价格行为的一个因素。对于具有给定到期期限和初始收益率的无期权债券，债券的价格波动性将随着票面利率的降低而增加。因此，除非票面利率实际上相同，而美元久期、收益率和价格相同，否则对于给定的利率变化，它们的美元价格敏感性不一定相同。
8. State why you would agree or disagree with the following statement: For a 1-basis point change in yield, the price value of a basis point is equal to the dollar duration.
8. 说明您为什么同意或不同意以下陈述：对于收益率变化 1 个基点，一个基点的价格值等于美元久期。
The validity of the above statement is discussed below.
上述陈述的有效性将在下面讨论。
The price value of a basis point, also referred to as the dollar value of a 01, is the change in the price of the bond if the required yield changes by 1 basis point. For small changes in the required yield, the below equation does a good job in estimating the change in price:
基点的价格值，也称为 01 的美元价值，是所需收益率变化 1 个基点时债券价格的变化。对于所需产量的微小变化，以下等式可以很好地估计价格的变化：
dP = (dollar duration)(dy).
dP= （美元久期）（dy）.
Consider a 6% 25-year bond selling at $70.3570 to yield 9%. The dollar duration is 747.2009. For a 1-basis-point (0.0001) increase in the required yield, the estimated price change per $100 of face value is
考虑以 70.3570 美元的价格出售 6% 的 25 年期债券，收益率为 9%。美元久期为 747.2009。所需收益率增加 1 个基点 （0.0001） 时，每 100 美元面值的估计价格变化为
dP = (dollar duration)(dy) = ($747.2009)(0.0001) = $0.07472.
dP= （美元久期）（dy） = （$747.2009）（0.0001） = $0.07472。
If we change the yield one basis point so the yield is 9.01%, then the value of the bond is: P = $592.378 + $110.445 = $702.824 with a bond quote of $70.2824. The price value of a basis point is about $70.2824 – $70.3570 = $0.07464. The dollar duration for a 1-basis point change gives about the same value as the price value of a basis point as both round off to $0.0747.
如果我们将收益率更改一个基点，使收益率为 9.01%，那么债券的价值为：P = 592.378 美元 + 110.445 美元 = 702.824 美元，债券报价为 70.2824 美元。一个基点的价格值约为 70.2824 美元 – 70.3570 美元 = 0.07464 美元。1 个基点变化的美元久期与一个基点的价格值大致相同，因为两者都四舍五入到 0.0747 美元。
9. The November 26, 1990, issue of BondWeek includes an article, “Van Kampen Merritt Shortens.” The article begins as follows:
9. 1990 年 11 月 26 日的《邦德周刊》刊登了一篇文章，“Van Kampen Merritt Shortens”。文章的开头如下：
“Peter Hegel, first v.p. at Van Kampen Merritt Investment Advisory, is shortening his $3 billion portfolio from 110% of his normal duration of 6½ years to 103-105% because he thinks that in the short run the bond rally is near an end.”
“Van Kampen Merritt Investment Advisory 的首席副总裁 Peter Hegel 正在将他的 30 亿美元投资组合从他正常期限 61/2 年的 110% 缩短到 103-105%，因为他认为在短期内债券涨势已接近尾声。”
Explain Hegel’s strategy and the use of the duration measure in this context.
解释黑格尔的策略以及在这种情况下持续时间度量的使用。
If Hegel thinks the bond rally is over it implies that he thinks bond prices will not go up. This implies the belief that Hegel thinks interest rates will stop falling.
如果黑格尔认为债券反弹已经结束，那就意味着他认为债券价格不会上涨。这意味着相信黑格尔认为利率将停止下降。
If interest rates begin going up then one does not want to lock in longer-term bonds at lower rates. This implies you want your portfolio of bonds to focus more on shorter-term bonds. Thus, you want a portfolio with a shorter duration. A shorter duration will mean not only less sensitivity to interest rates but if interest rates go up then Hegel will later capitalize on this because as bonds in his portfolio mature quicker (than would be achieved with a portfolio with a higher duration) he will be able to buy new bonds and lock in higher rates.
如果利率开始上升，那么人们不想以较低的利率锁定长期债券。这意味着您希望您的债券投资组合更多地关注短期债券。因此，您需要一个持续时间较短的投资组合。较短的期限不仅意味着对利率的敏感性降低，而且如果利率上升，那么黑格尔稍后将利用这一点，因为随着他投资组合中的债券到期更快（比使用更高期限的投资组合实现），他将能够购买新债券并锁定更高的利率。
In brief, Hegel uses the duration measure to optimize the value of his portfolio based upon his belief about how interest rates change.
简而言之，黑格尔根据他对利率如何变化的信念，使用久期度量来优化他的投资组合的价值。
10. Consider the following two Treasury securities:
10. 考虑以下两种国债：
Bond | Price | Modified duration (years) |
A | $100 | 6 |
B | $80 | 7 |
Which bond will have the greater dollar price volatility for a 25-basis-point change in interest rates?
如果利率变化 25 个基点，哪种债券的美元价格波动更大？
The estimated dollar price change can be obtained by using the below equation:
估计的美元价格变化可以通过使用以下公式获得：
dP = (modified duration)P(dy).
dP= （修改后的持续时间）P（dy）.
Inserting in our values for bond A, we have:
插入债券 A 的值，我们有：
dP = (modified duration)P(dy) = (6)$100(0.0025) = $1.50
dP= （修改后的持续时间）P（dy） = （6）$100（0.0025） = $1.50
which is the estimated dollar price change or volatility for a 25-basis-point change. The percentage change in price is:
这是 25 个基点变化的估计美元价格变化或波动性。价格的百分比变化为：
= = 0.0150 or 1.50%.
= = 0.0150 或 1.50%。
Inserting in our values for bond B, we have:
插入键 B 的值，我们有：
dP = (modified duration)P(dy) = (7)$80(0.0025) = $1.40
dP= （修改后的持续时间）P（dy） = （7）$80（0.0025） = $1.40
which is the estimated dollar price change or volatility for a 25-basis-point change. The percentage change in price is:
这是 25 个基点变化的估计美元价格变化或波动性。价格的百分比变化为：
= = 0.0175 or 1.750%.
= = 0.0175 或 1.750%。
Thus, we see that while bond A has a greater estimated dollar price volatility compared to bond B, it has a lower percentage change in price. From an investor’s point of view, every dollar invested in bond B has greater volatility.
因此，我们看到，虽然与债券 B 相比，债券 A 的估计美元价格波动更大，但它的价格变化百分比较低。从投资者的角度来看，投资于债券 B 的每一美元都有更大的波动性。
11. What are the limitations of using duration as a measure of a bond’s price sensitivity to interest-rate changes?
11. 使用久期作为衡量债券价格对利率变化的敏感性的指标有哪些限制？
Below we discuss three limitations of using duration.
下面我们讨论使用 duration 的三个限制。
First, duration measures are only approximations for small changes in yield. They do not capture the effect of the convexity of a bond on its price performance when yields change by more than a small amount. To get improved accuracy, the duration measure should be supplemented with an additional measure to capture the curvature or convexity of a bond. It is important to note that investors can be misled if they rely on duration as the sole measure of the price volatility of a bond.
首先，久期度量只是产量微小变化的近似值。当收益率变化超过少量时，它们没有捕捉到债券的凸性对其价格表现的影响。为了获得更高的精度，持续时间测量应辅以额外的测量来捕获粘合的曲率或凸度。重要的是要注意，如果投资者依赖久期作为衡量债券价格波动的唯一指标，他们可能会被误导。
Second, in the derivation of the relationship between modified duration and bond price volatility, we started with the bond price equation that assumes that all cash flows for the bond are discounted at the same discount rate. In essence we are assuming that the yield curve is flat and all shifts are parallel. This assumption does not always hold. This is very important when we try to use a portfolio’s duration to quantify the responsiveness of a portfolio’s value to a change in interest rates. If a portfolio has bonds with different maturities, the duration measure may not provide a good estimate for unequal changes in interest rates of different maturities.
其次，在推导修正久期与债券价格波动之间的关系时，我们从债券价格方程式开始，该方程式假设债券的所有现金流都以相同的贴现率进行贴现。从本质上讲，我们假设收益率曲线是平坦的，并且所有变化都是平行的。这个假设并不总是成立的。当我们尝试使用投资组合的久期来量化投资组合价值对利率变化的反应性时，这一点非常重要。如果投资组合包含不同期限的债券，久期指标可能无法很好地估计不同期限利率的不相等变化。
Third, we must be careful when applying our duration equations to bonds that are not option-free bonds. When changes in yields result in a change in the expected cash flow for a bond, which is the case for bonds with embedded options, the duration and convexity measures are appropriate only in certain circumstances.
第三，在将久期方程应用于非无期权债券时，我们必须谨慎。当收益率的变化导致债券的预期现金流发生变化时（嵌入期权的债券就是这种情况），久期和凸性指标仅在某些情况下适用。
12. The following excerpt is taken from an article titled “Denver Investment to Make $800 Million Treasury Move,” that appeared in the December 9, 1991, issue of BondWeek, p. 1:
12. 以下摘录自一篇题为“丹佛投资进行 8 亿美元国债搬迁”的文章，该文章发表在 1991 年 12 月 9 日的《债券周刊》上，第 1 页：
“Denver Investment Advisors will swap $800 million of long zero-coupon Treasuries for intermediate Treasuries. . . . The move would shorten the duration of its $2.5 billion fixed-income portfolio. . . .”
“Denver Investment Advisors 将把 8 亿美元的长期零息国债换成中间国债。. . . . 此举将缩短其 25 亿美元固定收益投资组合的久期。. . . .“
Why would the swap described here shorten the duration of the portfolio?
为什么这里描述的掉期会缩短投资组合的久期？
Duration captures the price sensitivity of a fixed-income investment to changes in yields. Thus, lowering the duration should lower the sensitivity. This is desired if one feels interest rates are going to increase in which case the value of your fixed-income investment would decline.
久期反映了固定收益投资对收益率变化的价格敏感性。因此，降低持续时间应该会降低敏感度。如果有人认为利率会上升，那么这是理想的，在这种情况下，您的固定收益投资的价值会下降。
Denver Investment Advisors are swapping $800 million long zero-coupon Treasuries for intermediate Treasuries. As a percentage of its portfolio, the proposed swap involves $800 million ÷ $2.5 billion = 0.32 or 32%. Because the portfolio duration is the weighted average of its individual investments, the swap of $800 million long zero-coupon Treasuries for intermediate Treasuries will lower its duration if the $800 million being swapped actually has a lower duration or price sensitivity.
Denver Investment Advisors 正在将 8 亿美元的多头零息国债换成中间国债。作为其投资组合的百分比，拟议的掉期涉及 8 亿美元÷ 25 亿美元 = 0.32 或 32%。由于投资组合久期是其单个投资的加权平均值，因此，如果被交换的 8 亿美元实际上具有较低的久期或价格敏感性，那么将 8 亿美元的长期零息国债换成中间国债将缩短其久期 。
As seen in Exhibit 4-3, the price responsiveness of a bond is different as yields change. For example, bonds have greater price responsiveness for changes at higher levels of maturity as interest rates change. Furthermore, bonds have greater price responsiveness for changes at lower levels of interest rates. However, the exhibit also shows that bonds have greater percentage price changes especially for longer-term securities. This indicates that swapping its long zero-coupon Treasuries for intermediate Treasuries could have an important impact on lowering Denver Investment’s duration.
如图 4-3 所示，债券的价格响应能力随着收益率的变化而变化。 例如，随着利率的变化，B onds 对更高期限水平的变化具有更强的价格响应能力。此外，债券对较低利率水平下的变化具有更强的价格响应能力。然而，该证据还显示，债券的价格变化百分比更大，尤其是长期证券。这表明将其长期零息国债换成中间国债可能会对缩短 Denver Investment 的久期产生重要影响。
13. You are a portfolio manager who has presented a report to a client. The report indicates the duration of each security in the portfolio. One of the securities has a maturity of 15 years but a duration of 25. The client believes that there is an error in the report because he believes that the duration cannot be greater than the security’s maturity. What would be your response to this client?
13. 您是一名投资组合经理，已向客户提交了一份报告。该报告显示投资组合中每种证券的久期。其中一种证券的期限为 15 年，但期限为 25 年。客户认为报告有错误，因为他认为久期不能超过证券的到期日。您对这位客户有什么反应？
Unfortunately, market participants often confuse the main purpose of duration by constantly referring to it as some measure of the weighted average life of a bond. This is because of the original use of duration by Macaulay where the cash flow for each period divided by the market value formed a weight with the weights adding up to one. If you rely on this interpretation of duration, it will be difficult for you to understand why a security with a maturity of 15 years can have a duration greater than 25 years. For example, consider collateralized mortgage obligation (CMO) bond classes. Certain CMO bond classes have a greater duration than the underlying mortgage loans (because CMO bond classes are leveraged instruments whose price sensitivity or duration are a multiple of the underlying mortgage loans from which they were created). That is, a CMO bond class can have a duration of 25 although the underlying mortgage loans from which the CMO is created can have a maturity of 15 years.
不幸的是，市场参与者经常混淆久期的主要目的，不断将其称为债券加权平均期限的某种衡量标准。这是因为麦考利最初使用久期，其中每个时期的现金流除以市场价值形成一个权重，权重加起来为 1。如果您依赖这种对期限的解释，您将很难理解为什么期限为 15 年的证券可以拥有超过 25 年的期限。例如，考虑抵押贷款义务 （CMO） 债券类别。某些 CMO 债券类别的久期长于标的抵押贷款（因为 CMO 债券类别是杠杆工具，其价格敏感性或久期是创建它们的标的抵押贷款的倍数）。也就是说，CMO 债券类别的期限为 25 年，尽管创建 CMO 的基础抵押贷款的期限为 15 年。
The answer to the puzzle (about duration being greater than maturity) is that duration is the approximate percentage change in price for a small change in interest rates. Thus, a CMO bond class with a duration of 25 does not mean that it has some type of weighted average life of 15 years. Instead, it means that for a 100-basis-point change in yield, that bond’s price will change by roughly 40%. Similarly, we interpret the duration of an option in the same way. A call option can have a duration of 25 when the time to expiration of the option is much less than 25 years. This is confusing to someone who interprets duration as some measure of the life of an option.
这个难题的答案（关于久期大于到期日）是久期是利率微小变化的价格变化的近似百分比变化。因此，久期为 25 的 CMO 债券类别并不意味着它具有 15 年的某种加权平均寿命。相反，这意味着收益率每变化 100 个基点，该债券的价格将变化约 40%。同样，我们以相同的方式解释期权的久期。当期权到期的时间远少于 25 年时，看涨期权的久期可以为 25 年。对于将 duration 解释为选项寿命的某种衡量标准的人来说，这令人困惑。
14. Answer the below questions.
14. 回答以下问题。
(a) Suppose that the spread duration for a fixed-rate bond is 2.5. What is the approximate change in the bond’s price if the spread changes by 50 basis points?
（a） 假设固定利率债券的价差存续期为 2.5。如果利差变化 50 个基点，债券价格的大致变化是多少？
A measure of how a non-Treasury bond’s price will change if the spread sought by the market changes is referred to as spread duration. A spread duration for a fixed-rate security is interpreted as the approximate change in the price of a fixed-rate bond for a 100-basis-point change in the spread. If the change is 2.5% (as given by a duration of 2.5) for 100 basis points then it would be about 1.25% for 50 basis points as shown below in more detail.
如果市场寻求的价差发生变化，非国债价格将如何变化的指标称为价差持续时间。固定利率证券的价差久期被解释为价差变化 100 个基点，固定利率债券价格的近似变化。如果 100 个基点的变化为 2.5%（以 2.5 的久期计算）， 那么 50 个基点的变化约为 1.25%，如下文更详细地显示。
Let us begin by noting that
让我们首先注意到
= (modified duration)(dy).
=（修改后的持续时间）（dy）.
Substituting spread duration for modified duration to approximate the percentage price change for a given change in the yield we get:
用价差久期代替修改后的久期，以近似我们得到的收益率中给定变化的价格变化百分比：
= (spread duration)(dy).
=（价差持续时间）（dy）.
Putting in 2.5 for the spread duration and 0.005 for dy (since the spread changes by 50 basis points), we get:
将价差持续时间输入 2.5，将 dy 输入 0.005（因为价差变化了 50 个基点），我们得到：
= 2.5(0.005) = 0.0125.
= 2.5（0.005）= 0.0125.
Thus, the change in the bond’s price if the spread changes by 50 basis points in percentage terms is 1.25%, which can be interpreted as the approximate percentage change in price for a 50-basis-point change in yield.
因此，如果利差以百分比形式变化 50 个基点，债券价格的变化为 1.25%，这可以解释为收益率变化 50 个基点的价格的近似百分比变化。
(b) What is the spread duration of a Treasury security?
（b） 国债的价差久期是多久？
The spread represents compensation for credit risk. The price of a non-Treasury bond is exposed to a change in the spread that is called credit spread risk. For a Treasury security, there is no credit risk and thus the spread duration for a Treasury security is zero.
利差代表信用风险的补偿。非国债的价格受到利差变化的影响，称为信用利差风险。对于美国国债证券，不存在信用风险，因此美国国债的价差持续时间为零。
15. What is meant by the spread duration for a floating-rate bond?
15. 浮息债券的价差存续期是什么意思？
A floating-rate bond’s price sensitivity will depend on whether the spread that the market wants changes. The spread is reflected in the quoted margin in the coupon reset formula. The quoted margin is fixed over the life of a typical floating-rate bond (or floater). Spread duration is a measure used to estimate the sensitivity of a floater’s price sensitivity to a change in the spread. A spread duration of 1.4 for a floater would mean that if the spread the market requires changes by 100 basis points, the floater’s price will change by about 1.4%.
浮动利率债券的价格敏感性将取决于市场希望的价差是否发生变化。点差反映在优惠券重置公式中的报价保证金中。报价保证金在典型浮动利率债券（或浮动利率债券）的期限内是固定的。价差持续时间是用于估计浮动价格对价差变化的敏感性的指标。浮动股票的价差持续时间为 1.4 意味着如果市场要求的价差变化 100 个基点，浮动货币的价格将变化约 1.4%。
16. Explain why the duration of an inverse floater is a multiple of the duration of the collateral from which the inverse floater is created.
16. 解释为什么反向浮动利率的存续期是创建反向浮动利率的抵押品存续期的倍数。
In general, an inverse floater is created from a fixed-rate security. The security from which the inverse floater is created is called the collateral. From the collateral two bonds are created: a floater and an inverse floater. The two bonds are created such that (i) the total coupon interest paid to the two bonds in each period is less than or equal to the collateral’s coupon interest in each period, and (ii) the total par value of the two bonds is less than or equal to the collateral’s total par value. Equivalently, the floater and inverse floater are structured so that the cash flow from the collateral will be sufficient to satisfy the obligation of the two bonds.
一般来说，反向浮动是从固定利率证券创建的。创建反向浮动的证券称为抵押品。从抵押品中可以创建两种债券：浮动债券和反向浮动债券。这两只债券的设立使得 （） 在每个期间支付给两只债券的总票面利息小于或等于抵押品在每个期间的票面利息，以及 （ii） 两只债券的总面值小于或等于抵押品的总面值。等效地，浮动和反向浮动的结构使抵押品的现金流足以满足两种债券的义务。
The duration of the inverse floater is related to the duration of the collateral and the duration of the floater. Assuming that the duration of the floater is close to zero, it can be shown that the duration of an inverse floater is as follows:
反向浮动的持续时间与抵押品的持续时间和浮动的持续时间有关。 假设 floater 的持续时间接近于零，则可以证明反向 floater 的持续时间如下：
duration of an inverse floater = (1 + L)(duration of collateral) x
反向浮动货币的存续期 = （1 + L）（抵押品存续期） x
where L is the ratio of the par value of the floater to the par value of the inverse floater. For example, if collateral with a par value of $100 million is used to create a floater with a par value of $80 million and an inverse floater with a par value of $20 million, then L = ($80 million / $20 million) = 4.
其中 L 是浮动币的面值与反向浮动币的面值的比率。例如，如果使用面值为 1 亿美元的抵押品创建面值为 8000 万美元的浮动货币和面值为 2000 万美元的反向浮动货币，则 L = （8000 万美元 / 2000 万美元） = 4。
We can illustrate why an inverse’s duration is a multiple of the collateral. Suppose that the par value of the collateral of $50 million is split as follows: $40 million for the floater and $10 million for the inverse floater. Suppose also that the collateral and inverse are trading at par so that the ratio of the prices is 1 and that the duration for the collateral is 8. For a 100-basis-point change in interest rates, the collateral’s price will decline by 8% or 0.08 ($500 million) = $4 million. Assuming that the floater’s price does not change when interest rates increase, the $4 million decline must come from the inverse floater. For the inverse floater to realize a decline in value of $4 million when its value is $10 million, the duration must be 40. That is, a duration of 40 will produce a 40% change in value or 0.04($10 million) = $4 million. Thus, the duration is five times the collateral’s duration of 8. Or equivalently, because L is 4, it is (1 + 4) times the collateral’s duration.
我们可以说明为什么反向交易的久期是抵押品的倍数。假设 5000 万美元抵押品的面值按以下方式分配：浮动债券为 4000 万美元，反向浮动债券为 1000 万美元。还假设抵押品和反向抵押品以面值交易，因此价格比率为 1，抵押品的期限为 8。利率变化 100 个基点，抵押品的价格将下跌 8% 或 0.08（5 亿美元）= 400 万美元。假设浮动货币的价格在利率上升时没有变化，那么 400 万美元的下降一定来自反向浮动货币。要使反向浮动货币在其价值为 1000 万美元时实现 400 万美元的下降，持续时间必须为 40。也就是说，持续时间为 40 将产生 40% 的价值变化，即 0.04（1000 万美元）= 400 万美元。因此，期限是抵押品期限 8 的 5 倍。或者等效地，因为 L 是 4，所以它是抵押品期限的 （1 + 4） 倍。