R58 练习: Yield-Based Bond Convexity and Portfolio Properties

R58 练习: 基于收益率的凸性与组合特性

考纲范围

Calculate and interpret convexity and describe the convexity adjustment.

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

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

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Q1.

A callable bond is currently priced at 97.33 per 100 of par value. Based on the current yield curve, when the yield curve changes parallelly upward or downward by 30 bps, the bond price will be 96.37 and 98.89, respectively. The bond’s approximate effective convexity is closest to:

A. 342.48

B. 684.95.

C. 4.32

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答案：B

解析：近似有效凸性公式：

$ApproxConvexity=\frac{P{V}_{-}+P{V}_{+}-2\times P{V}_0}{(\Delta y{)}^2\times P{V}_0}=\frac{98.89+96.37-2\times 97.33}{(0.003{)}^2\times 97.33}=\frac{0.60}{0.000876}\approx 684.95$

选项	判断	解析
A	✗	可能在分母中多乘了2
B	✓	正确使用凸性公式计算 ≈ 684.95
C	✗	计算方法完全错误

关联：R58: 基于收益率的凸性与组合特性

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Q2.

Convexity is the second-order effect and indicates the change in the modified duration given a change in the yield-to-maturity. Holding all other things constant, convexity is greater when:

A. the bond has a shorter time-to-maturity.

B. the bond offers a lower coupon rate.

C. the bond has a higher yield-to-maturity.

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答案：B

解析：凸性与久期的影响因素相同方向：期限越长、票息率越低、YTM越低、现金流离散程度越大，凸性越大。低票息率意味着更大的凸性。

选项	判断	解析
A	✗	更短的期限意味着更小的凸性
B	✓	更低的票息率意味着更大的凸性
C	✗	更高的YTM意味着更小的凸性

关联：R58: 基于收益率的凸性与组合特性

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Q3.

A specific bond has a modified duration of 2.3754 and a convexity of 55.3257. If the YTM of this bond increases by 100 bps, the expected percentage price change is closest to:

A. -2.0988%.

B. -2.3755%.

C. -4.3442%.

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答案：A

解析：

$\%\Delta P\approx -ModDur\times \Delta y+\frac{1}{2}\times Convexity\times (\Delta y{)}^2$

$=-2.3754\times 0.01+\frac{1}{2}\times 55.3257\times (0.01{)}^2$

$=-0.023754+0.002766=-0.020988=-2.0988\%$

选项	判断	解析
A	✓	正确计算久期效应 + 凸性调整项 = -2.0988%
B	✗	仅考虑了久期效应，未加凸性调整
C	✗	计算有误

关联：R58: 基于收益率的凸性与组合特性

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Q4.

A specific bond has a modified duration of 3.8257 and a convexity of 89.5739. If the YTM of this bond decreases by 100 bps, which is closest to the expected percentage price change?

A. 4.2736%

B. 3.8366%

C. 2.5467%

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答案：A

解析：

$\%\Delta P\approx -3.8257\times (-0.01)+\frac{1}{2}\times 89.5739\times (0.01{)}^2$

$=0.038257+0.004479=0.042736=4.2736\%$

凸性调整项总是正的（无论利率上升或下降），体现了正凸性的”涨多跌少”特征。

选项	判断	解析
A	✓	久期效应 + 凸性调整 = 3.8257% + 0.4479% = 4.2736%
B	✗	仅考虑了久期效应
C	✗	计算有误

关联：R58: 基于收益率的凸性与组合特性

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Q5.

A fund manager has three bonds in his portfolio as shown in the table below:

Bond	Maturity	Duration	Market Price	Par Value (Total)
A	3Y	2.5	102.01	4 million
B	5Y	4.2	99.46	2 million
C	7Y	6.3	98.37	1 million

By using the weighted average of individual bond durations, the portfolio duration is closest to:

A. 3.5094.

B. 4.1219.

C. 4.3106.

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答案：A

解析：组合久期以市值(market value)为权重计算：

MV_A = 102.01/100 × 4M = 4.0804M, MV_B = 99.46/100 × 2M = 1.9892M, MV_C = 98.37/100 × 1M = 0.9837M

Total MV = 7.0533M

$D_p=\frac{4.0804}{7.0533}\times 2.5+\frac{1.9892}{7.0533}\times 4.2+\frac{0.9837}{7.0533}\times 6.3\approx 3.5094$

选项	判断	解析
A	✓	以market value为权重的加权平均久期 ≈ 3.5094
B	✗	可能使用了par value作为权重
C	✗	可能使用了等权重

关联：R58: 基于收益率的凸性与组合特性

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Q6.

What is the limitation of using the weighted mean of individual bond durations as the calculation of portfolio duration?

A. It cannot be used when the yield curve becomes steeper.

B. It can only be used for option-free bond.

C. Its calculation is very complicated.

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答案：A

解析：组合久期（加权平均法）的主要局限在于它假设收益率曲线平行移动。当收益率曲线形状发生变化（如变陡或变平）时，该方法不再准确。它不仅适用于无权债券，计算也并不复杂。

选项	判断	解析
A	✓	假设收益率曲线平行移动，曲线变陡时不适用
B	✗	该方法不限于option-free bond
C	✗	加权平均计算并不复杂

关联：R58: 基于收益率的凸性与组合特性