R57 练习: Yield-Based Bond Duration Measures

R57 练习: 基于收益率的久期度量

考纲范围

Define, calculate, and interpret modified duration, money duration, and the price value of a basis point (PVBP).

Explain how a bond’s maturity, coupon, and yield level affect its interest rate risk.

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Q1.

An analyst determines the present value of a 3-year bond under different yield scenarios. The results are presented in the following table. The bond’s approximate modified duration is closest to:

Yield Assumption	Present Value of the Bond
8.5%	USD 95.21
9%	USD 94.75
9.5%	USD 94.42

A. 0.83.

B. 1.67

C. 8.34.

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答案：A

解析：近似修正久期公式：

$ApproxModDur=\frac{P{V}_{-}-P{V}_{+}}{2\times \Delta y\times P{V}_0}=\frac{95.21-94.42}{2\times 0.005\times 94.75}=\frac{0.79}{0.9475}=0.8339\approx 0.83$

选项	判断	解析
A	✓	(95.21 - 94.42) / (2 × 0.005 × 94.75) ≈ 0.83
B	✗	可能将分母中的2遗漏
C	✗	可能将delta y用错

关联：R57: 基于收益率的久期度量

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Q2.

A 3-year bond with semi-annual coupon of 5% is currently priced at 101.3886. If the current yield to maturity is 4.5% and the interest rate changes by 50bps, what is the approximate modified duration of this bond?

A. 2.7371.

B. 2.7624.

C. 5.5256.

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答案：B

解析：先求两个新价格（YTM ± 50bps），然后使用近似公式。N=6, PMT=2.5, FV=100。

当YTM=5%时：I/Y=2.5%, PV- = ? 当YTM=4%时：I/Y=2%, PV+ = ?

然后代入公式：ApproxModDur = (PV- - PV+) / (2 × 0.005 × 101.3886) ≈ 2.7624

选项	判断	解析
A	✗	计算有细微差异
B	✓	正确计算近似修正久期 ≈ 2.7624
C	✗	可能未使用正确的delta y

关联：R57: 基于收益率的久期度量

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Q3.

Given a bond has a modified duration of 6.5 and is currently priced at 1,000, how would the bond’s value change if its yield to maturity decreases by 20 basis points?

A. Decrease by 13.

B. Increase by 13.

C. Increase by 130.

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答案：B

解析：

$\Delta P\approx -ModDur\times \Delta y\times P=-6.5\times (-0.002)\times 1000=13$

YTM下降20bps，债券价格上升约13。

选项	判断	解析
A	✗	利率下降时价格应上升
B	✓	-6.5 × (-0.002) × 1000 = +13
C	✗	可能将bps换算错误（用了200bps而非20bps）

关联：R57: 基于收益率的久期度量

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Q4.

An option-free bond is now sold at 100, and the modified duration is 6. If the yield decreases by 100bps, the approximate price change in units of the currency for this bond is?

A. 0.06.

B. 6.

C. -6.

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答案：B

解析：Money duration = Modified duration × Price = 6 × 100 = 600。

$\Delta P=-ModDur\times \Delta y\times P=-6\times (-0.01)\times 100=6$

利率下降100bps，价格上升6个货币单位。

选项	判断	解析
A	✗	可能仅计算了百分比变化而非金额变化
B	✓	-6 × (-0.01) × 100 = +6
C	✗	利率下降时价格应上升（正值）

关联：R57: 基于收益率的久期度量

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Q5.

The table below shows the information about a newly issued bond. Price is per 100 of par value.

Item	Bond
Time to maturity	3 years
Coupon	8%, annually
Price	95.026296

The estimated price value of a basis point is closest to:

A. 0.024

B. 0.048

C. 0.194

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答案：A

解析：PVBP是收益率变动1个基点时债券价格的变动。需先求YTM，然后计算YTM ± 1bp时的价格差。

$PVBP=\frac{|P{V}_{-1bp}-P{V}_{+1bp}|}{2}$

或简化为：PVBP ≈ ModDur × 0.0001 × Price

选项	判断	解析
A	✓	PVBP ≈ 0.024 per 100 of par value
B	✗	可能计算了2个基点的变化而未除以2
C	✗	计算有误

关联：R57: 基于收益率的久期度量

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Q6.

Which of the following statements correctly describes Macaulay duration?

A. The maturity of bond negatively related with Macaulay duration.

B. The Macaulay duration of zero-coupon bond is larger than its time to maturity.

C. The duration of bond is inversely related to its YTM.

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答案：C

解析：久期与YTM反向相关：YTM越高，远期现金流的现值权重越小，久期越短。期限与Macaulay duration正相关（而非负相关）。零息债券的Macaulay duration等于其到期期限（而非大于）。

选项	判断	解析
A	✗	maturity与MacDur正相关，期限越长久期越大
B	✗	零息债券的MacDur恰好等于time to maturity
C	✓	YTM越高，久期越短，反向相关

关联：R57: 基于收益率的久期度量

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Q7.

An analyst forecasts that the market will experience a rapid rise in interest rate volatility. Which of the following option-free bonds will face more interest rate risk?

A. 4% coupon bond with 20 years to maturity.

B. 4% coupon bond with 10 years to maturity.

C. 10% coupon bond with 10 years to maturity.

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答案：A

解析：利率风险（久期）越大的债券面临的利率风险越高。久期与期限正相关、与票息率负相关、与YTM负相关。4%票息/20年期限的债券具有最长期限和最低票息，因此久期最大，利率风险最高。

选项	判断	解析
A	✓	低票息+长期限 = 最大久期 = 最高利率风险
B	✗	虽票息低但期限短于A
C	✗	高票息降低了久期

关联：R57: 基于收益率的久期度量

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Q8.

Assume market discount rates decrease by the same amount. Which of the following statements related to bond characteristics is most likely correct?

A. For two bonds with the same coupon rate, the price of the longer-term bond always increases more than that of the shorter-term bond.

B. Among the three zero-coupon bonds, the one with the longest maturity would always have the greatest price volatility compared with the other two bonds.

C. Holding other factors constant, a lower-coupon bond has a smaller percentage change than a higher-coupon bond.

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答案：B

解析：零息债券的Macaulay duration = 到期期限，因此最长期限的零息债券久期最大，价格波动性一定最大。对于付息债券，长期限并非”总是”意味着更大波动（极端情况可能反转）。低票息债券的价格变化百分比更大而非更小。

选项	判断	解析
A	✗	对付息债券不一定”always”成立
B	✓	零息债券中最长期限的必定波动最大
C	✗	低票息债券变化百分比更大，而非更小

关联：R57: 基于收益率的久期度量