R55 练习: The Term Structure of Interest Rates

R55 练习: 利率期限结构

考纲范围

Define spot rates and the spot curve, and calculate the price of a bond using spot rates.

Define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates.

Compare the spot curve, par curve, and forward curve.

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Q1.

In term of the relationship between spot rates and YTM, which of the following statements is least likely correct?

A. YTM refers to the single discount rate that can be applied to value a bond. However, if spot rates are used to calculate bond value, cash flows occurring at different time should be discounted at different spots rates.

B. Spot rates are equal to YTM for zero-coupon bonds.

C. YTM is the arithmetic average of relevant spot rates.

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答案：C

解析：YTM是使债券所有现金流现值之和等于债券价格的单一折现率，它是即期利率的加权平均而非算术平均。对于零息债券，spot rate确实等于YTM（因为只有一笔现金流）。

选项	判断	解析
A	✗	正确描述了YTM和spot rate的区别
B	✗	零息债券的spot rate = YTM，正确
C	✓	YTM是spot rates的加权平均，不是算术平均，最不正确

关联：R55: 利率期限结构

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Q2.

Suppose that the spot rates for 1-year, 2-year, and 3-year are 4%, 5%, and 6%, respectively. The price of a three-year bond that makes a 6% annual coupon is:

A. 1002.11

B. 1000.00

C. 1055.50

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答案：A

解析：使用各期即期利率分别折现每笔现金流：

$PV=\frac{60}{(1.04{)}^1}+\frac{60}{(1.05{)}^2}+\frac{1060}{(1.06{)}^3}$

$=57.6923+54.4218+890.00\approx 1002.11$

注意：虽然coupon rate = 3年期spot rate = 6%，但由于短期spot rate较低，债券价格略高于面值。

选项	判断	解析
A	✓	逐笔用对应期限spot rate折现，价格略高于面值
B	✗	仅当YTM = coupon rate时价格等于面值，此处spot rate不同
C	✗	计算有误

关联：R55: 利率期限结构

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Q3.

Regarding the maturity structure of interest rates, which of the followings is least accurate?

A. Spot curve is a series of yields-to-maturity on par bonds.

B. Forward curve describes the relationship between maturity and forward rate.

C. The bonds on a par curve should own similar credit risk.

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答案：A

解析：Spot curve是即期利率与到期期限的关系曲线，不是par bonds的YTM系列。Par curve才是面值债券YTM与期限的关系。Forward curve描述的是远期利率与期限的关系。Par curve上的债券确实应具有相似的信用风险。

选项	判断	解析
A	✓	描述的是par curve而非spot curve，最不准确
B	✗	forward curve确实描述远期利率与期限的关系
C	✗	par curve上的债券应具有相似信用风险，正确

关联：R55: 利率期限结构

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Q4.

The one-year spot rate is 3.8%, the two-year spot rate is 4.3%, and the three-year spot rate is 5.2%. The implied one-year forward rate two years from now is closest to:

A. 7.02%.

B. 6.38%.

C. 4.32%.

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答案：A

解析：利用即期利率求远期利率：

$(1+S_3{)}^3=(1+S_2{)}^2\times (1+f_{2,1})$

$(1.052{)}^3=(1.043{)}^2\times (1+f_{2,1})$

$f_{2,1}=\frac{(1.052{)}^3}{(1.043{)}^2}-1=\frac{1.16410}{1.08785}-1\approx 7.02\%$

选项	判断	解析
A	✓	正确使用公式求得2y1y forward rate ≈ 7.02%
B	✗	计算有误
C	✗	计算有误

关联：R55: 利率期限结构

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Q5.

Use the following zero-coupon bonds’ YTM to calculate the implied forward rate 1y2y with annual compounding.

Maturity	YTM
1-year	2%
2-year	3%
3-year	4%

A. 5.01%

B. 3%

C. 7.07%

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答案：A

解析：1y2y表示从1年后开始的2年期远期利率：

$(1+S_1)\times (1+f_{1y2y}{)}^2=(1+S_3{)}^3$

$(1.02)\times (1+f_{1y2y}{)}^2=(1.04{)}^3$

$(1+f_{1y2y}{)}^2=\frac{1.124864}{1.02}=1.10281$

$f_{1y2y}=(1.10281{)}^{0.5}-1\approx 5.01\%$

选项	判断	解析
A	✓	正确计算1y2y forward rate ≈ 5.01%
B	✗	3%是2年期spot rate
C	✗	可能是2y1y的计算结果

关联：R55: 利率期限结构

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Q6.

Currently, “0y1y”, “1y1y”, “2y1y” and “3y1y” forward rates are 0.80%, 1.12%, 3.94% and 4.28% respectively, all of which are effective annual rates. The 3-year implied spot rate is closest to:

A. 2.03%

B. 1.94%

C. 1.78%

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答案：B

解析：由远期利率求即期利率：

$(1+S_3{)}^3=(1+f_{0y1y})\times (1+f_{1y1y})\times (1+f_{2y1y})$

$=(1.008)\times (1.0112)\times (1.0394)=1.05945$

$S_3=(1.05945{)}^{1/3}-1\approx 1.94\%$

选项	判断	解析
A	✗	计算有误
B	✓	正确使用3个1年期远期利率求3年期spot rate ≈ 1.94%
C	✗	计算有误

关联：R55: 利率期限结构

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Q7.

Assuming “0y1y”, “1y1y”, “2y1y” and “3y1y” forward rates are 0.80%, 1.12%, 3.94% and 4.28% respectively, the fair value of an 8% four-year annual payment bond with a par value of 100 is closest to:

A. 115.55

B. 118.76

C. 121.09

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答案：A

解析：使用远期利率逐期折现每笔现金流：

$PV=\frac{8}{1.008}+\frac{8}{1.008\times 1.0112}+\frac{8}{1.008\times 1.0112\times 1.0394}+\frac{108}{1.008\times 1.0112\times 1.0394\times 1.0428}$

逐步计算后得到约115.55。

选项	判断	解析
A	✓	正确使用远期利率逐期折现 ≈ 115.55
B	✗	计算有误
C	✗	计算有误

关联：R55: 利率期限结构

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Q8.

Christy is now making her decision on investing in a 2% annual coupon payment rate government bond with 3 years remaining to maturity. She observes the following forward curve:

Time Period	Forward Rate
0y1y	1.45%
1y1y	1.62%
2y1y	2.24%
3y1y	3.31%

Suppose the bond is now priced at USD 99.28, should Christy purchase the bond?

A. Yes.

B. No.

C. Cannot determine.

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答案：A

解析：计算债券公允价值：

$PV=\frac{2}{1.0145}+\frac{2}{1.0145\times 1.0162}+\frac{102}{1.0145\times 1.0162\times 1.0224}$

计算得出公允价值 > 99.28，因此市场价格低于公允价值，债券被低估，应该购买。

选项	判断	解析
A	✓	公允价值高于市场价99.28，债券被低估，应购买
B	✗	债券被低估，应该购买
C	✗	可以通过计算公允价值做出判断

关联：R55: 利率期限结构

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Q9.

Assume the following annual forward rates are observed in the market.

Tenor	Forward rates
0y1y	1.8%
1y1y	5.2%
2y1y	7.6%

Which of the following is most likely accurate?

A. The spot rate curve lies above the forward rate curve.

B. The par rate curve lies above the forward rate curve.

C. The spot rate curve lies above the par rate curve.

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答案：C

解析：远期利率递增（1.8% < 5.2% < 7.6%），说明收益率曲线向上倾斜。当收益率曲线向上倾斜时，三条曲线的关系为：par < spot < forward。因此spot rate curve在par rate curve之上。

选项	判断	解析
A	✗	向上倾斜时spot < forward，spot在forward下方
B	✗	向上倾斜时par < forward，par在forward下方
C	✓	向上倾斜时par < spot < forward，spot在par上方

关联：R55: 利率期限结构