Chapter 7
第七章
The Risk and Term Structure of Interest Rates
利率風險與期限結構

Conceptual and Analytical Problems
概念與分析問題

Consider a firm that issued a large quantity of commercial paper in the period leading to a financial crisis. (LO1)
考慮一家在金融危機前導期間發行大量商業紙券的公司。（LO1）

How would you expect the credit rating of the commercial paper to evolve as the crisis unfolds?
您預期商業紙券的信用評等在危機發展過程中如何變化？

Would you alter your prediction if, rather than commercial paper, the firm was instead issuing asset-backed commercial paper?
您會改變您的預測嗎？如果公司不是發行商業紙，而是發行資產支持商業紙呢？

Answer:
回答：

Commercial paper is generally issued without collateral, so that its rating will depend on the creditworthiness of the issuer, which faces downgrade risk in a crisis
商業本票通常無抵押發行，因此其評級將取決於發行者之信用狀況，在危機中面臨降級風險.

If the commercial paper is asset-backed, the probability of a downgrade would depend on the creditworthiness of the collateral
如果商業本票是資產支持型，降級的可能性將取決於抵押品的信用狀況.

Suppose that a major foreign government defaults on its debt. What, if anything, will happen to the position and slope of the U.S. yield curve? (LO1)
假設一個主要外國政府違約其債務。如果有的話，美國利差曲線的位置和斜率將會發生什麼變化？（LO1）

Answer: If Treasury debt is a substitute for the foreign government debt, then U.S. yield curve would shift downward, reflecting a flight to quality. If the holders of the foreign government debt were concentrated at the short end of the maturity spectrum, then the U.S. yield curve would shift down and become more steeply sloped; if the holders were concentrated at the long end of the maturity spectrum, then the U.S. yield curve would shift down and become flatter.
答：如果國庫債務是外國政府債務的替代品，則美國利差曲線將向下移動，反映對質量的避難。如果外國政府債務持有者集中在短期到期範圍，則美國利差曲線將向下移動並變得更加陡峭；如果持有者集中在長期到期範圍，則美國利差曲線將向下移動並變得更加平坦。

What was the connection between house price movements, the growth in subprime mortgages, and securities backed by these mortgages—on the one hand—and on the other hand—the difficulties encountered by some financial institutions during the 2007-2009 financial crisis? (LO1)
房屋價格波動、次級貸款的增長以及由這些貸款作為保險的證券——一方——與另一方面——在 2007-2009 年金融危機中一些金融機構遇到的困難之間有何聯繫？（LO1）

Answer: The viability of many subprime mortgages—and in particular ARMs—depended on being able to refinance the loan before the interest rate reset to a higher level. When house prices began to fall in 2006, pushing home values below the loan amount for many of these mortgages, refinancing was no longer an option. Unable to pay the higher interest rates, borrowers began to default on these subprime mortgages at an increasing rate. This, in turn, led to significant falls in the prices of securities backed by subprime mortgages, causing difficulties for institutions that had sizable holdings of these securities.
答案：許多次級貸款的可行性——特別是可調利率貸款——取決於在利率重置為更高水平之前能夠重新貸款。當房價於 2006 年開始下跌，將許多這些貸款的房價推低至貸款金額以下時，重新貸款不再是一個選擇。無法支付更高的利率，借款人開始以越來越高的比率違約這些次級貸款。這，轉而導致了由次級貸款擔保的證券價格的顯著下跌，為持有這些證券規模較大的機構帶來困難。

Suppose that the interest rate on one-year bonds is currently 4 percent and is expected to be 5 percent in one year and 6 percent in two years. Using the expectations hypothesis, compute the yields on two- and three-year bonds and plot the yield curve. (LO3)
假設一年期債券的利率目前為 4%，預計一年後為 5%，兩年後為 6%。使用預期假說，計算兩年期和三年期債券的收益率並繪製收益率曲線。（LO3）

Answer:
回答：

 Yield on one-year bond = 4%
一年期债券收益率 = 4%

 Yield on two-year bond = (4% + 5%)/2 = 4.5%
兩年期債券收益率 = (4% + 5%)/2 = 4.5%

 Yield on three-year bond = (4% + 5% + 6%)/3 = 5%
三年債券收益率 = (4% + 5% + 6%)/3 = 5%

*According to the liquidity premium theory, if the yield on both one-and two-year bonds are the same, would you expect the one-year yield in one-year’s time to be higher, lower or the same? Explain your answer. (LO3)
根據流動性溢價理論，如果一年期和兩年期債券的收益率相同，您預期一年後的一年期收益率會更高、更低還是相同？請解釋您的答案。（LO3）

Answer: According to the liquidity premium theory, the two-year yield (i_2,t) is an average of this year’s and next year’s one-year yields (i_1,t + i^e _{1, t +1}) divided by 2 plus a risk premium (rp) to compensate for the inflation and interest-rate risk associated with the longer maturity.
答：根據流動性溢價理論，兩年期收益率（i _{2, t} ）是今年和明年一年期收益率（i _{1, t} + i ^e _{1, t +1} ）的平均值，除以 2，並加上風險溢價（rp）以彌補與較長到期日相關的通脹和利率風險。

i_2,t = rp + (i_1,t + i^e _{1, t +1})/2

As we can see from the formula, if the current one-and two-year yields are the same and there is a risk premium included in the two-year yield, then next year’s one-year yield must be lower than this year’s.
從公式中我們可以看出，如果當前的一年和兩年期收益相同，且兩年期收益中包含風險溢價，那麼明年的一年期收益必須低於今年。

You have $1,000 to invest over an investment horizon of three years. The bond market offers various options. You can buy (i) a sequence of three one-year bonds; (ii) a three-year bond; or (iii) a two-year bond followed by a one-year bond. The current yield curve tells you that the one-year, two-year, and three-year yields to maturity are 3.5 percent, 4.0 percent, and 4.5 percent, respectively. You expect that one-year interest rates will be 4 percent next year and 5 percent the year after that. Assuming annual compounding, compute the return on each of the three investments, and discuss which one you would choose. (LO3)
您有 1,000 美元可以投資，投資期限為三年。債券市場提供各種選擇。您可以購買（i）三個一年期的債券；（ii）一個三年期的債券；或（iii）一個兩年期債券後跟一個一年期債券。當前的收益率曲線告訴您，一年期、兩年期和三年期的到期收益率分別為 3.5％、4.0％和 4.5％。您預期明年的一年期利率將為 4％，後年為 5％。假設年複利，計算三種投資的回報，並討論您會選擇哪一種。（LO3）

Answer:
回答：

Expected return for (i) = (1.035) × (1.04) × (1.05) – 1 = 13.02%
預期回報率(i) = (1.035) × (1.04) × (1.05) – 1 = 13.02%

Expected return for (ii) = (1.045)³ - 1 = 14.12%
預期回報率（ii）=（1.045） ³ - 1 = 14.12%

Expected return for (iii) = (1.04)² × (1.05) – 1 = 13.57%
預期回報率(iii) = (1.04) ² × (1.05) – 1 = 13.57%

The second and third options have higher expected returns than the first, but both options involve investing in longer-term bonds (three-year and two-year bonds, respectively). Long-term bonds have higher inflation risk and interest-rate risk; investors require compensation for this additional risk, which is why longer-term bonds generally have higher yields than would be suggested by the expectations hypothesis. In selecting an investment strategy, it is important to take these additional risks into account. An investor’s investment horizon is also important; to reduce interest rate risk, someone with a short-term horizon would be better off choosing the first option, while someone with a three-year horizon should probably choose the second option.
第二和第三選項的預期回報率比第一選項高，但這兩個選項都涉及投資於長期債券（分別為三年期和兩年期債券）。長期債券具有更高的通脹風險和利率風險；投資者需要為這個額外風險獲得補償，這也是為什麼長期債券的收益率通常比預期假說所建議的更高。在選擇投資策略時，考慮這些額外風險非常重要。投資者的投資時間範圍也很重要；為了減少利率風險，短期時間範圍的人選擇第一選項會更好，而三年時間範圍的人可能應該選擇第二選項。

Suppose that the yield curve shows that the one-year bond yield is 3 percent, the two-year yield is 4 percent, and the three-year yield is 5 percent. Assume that the risk premium on the one-year bond is zero, the risk premium on the two-year bond is 1 percent, and the risk premium on the three-year bond is 2 percent. (LO3)
假設收益曲線顯示一年期債券收益為 3%，兩年期收益為 4%，三年期收益為 5%。假設一年期債券的風險溢價為零，兩年期債券的風險溢價為 1%，三年期債券的風險溢價為 2%。（LO3）

What are the expected one-year interest rates next year and the following year?
明年和後年的預期一年期利率是多少？

If the risk premiums were all zero, as in the expectations hypothesis, what would the slope of the yield curve be?
如果風險溢價都為零，如同預期假說中所述，那麼利差曲線的斜率會是什麼樣子？

Answer:
回答：

With the one-year yield at 3 percent (or i_1t = 0.03), the risk premium on the two-year bond at 1 percent (0.01), and with the two-year yield at 4 percent (or i_2t = 0.04), the implied one-year rate next year will solve:
一年收益為 3%（或 i _{1 t} = 0.03），兩年期債券風險溢價為 1%（0.01），以及兩年期收益為 4%（或 i _2t = 0.04），明年預期的年利率將解決：

i_2t = 0.01 + (0.03 + i_{1,t +1}) / 2 = 0.04,
i _{2 t} = 0.01 + (0.03 + i _{1, t +1} ) / 2 = 0.04

which implies that the one-year interest rate next year will be 0.03 or 3 percent
哪意味著明年的一年期利率將為 0.03 或 3%.

Using this result, with a risk premium on the three-year bond of 0.02, we have:
使用此結果，在三年期債券的風險溢價為 0.02 的情況下，我們得到：

i_3t = 0.02 + (0.03 + 0.03 + i_{1,t +2}) / 3 = 0.05

which implies that the one-year rate in two years will also be 0.03 or 3 percent
這意味著兩年後的一年期利率也將是 0.03 或 3%.

As the solutions in part (a) show, the current and prospective one-year rates are all 3 percent, so the expectations hypothesis yield curve would be flat.
當部分（a）中的解決方案顯示，當前和預期的年利率都是 3%，因此預期假設的收益率曲線將會平坦。

*If inflation and interest rates become more volatile, what would you expect to see happen to the slope of the yield curve? (LO3)
如果通脹和利率變得更加波動，您預期會看到收益曲線的斜率發生什麼變化？（LO3）

Answer: Investors are likely to demand a higher risk premium in the face of increased volatility. There is more uncertainty regarding the real return on investments and the price for which you could sell a bond before maturity. Assuming the uncertainty rises the longer the term to maturity, you should expect the yield curve to become steeper as investors demand a larger premium to compensate for inflation risk and interest rate risk.
投資者可能會在面對波動性增加的情況下要求更高的風險溢價。對於投資的實際回報和您在到期前賣出債券的價格的不確定性更高。假設不確定性隨著到期日的延長而上升，您應該預期收益率曲線會變得更加陡峭，因為投資者要求更高的溢價來補償通脹風險和利率風險。

Suppose your local government, threatened with bankruptcy, decided to tax the interest income on its own bonds as part of an effort to rectify serious budgetary woes. What would you expect to see happen to the yields on these bonds? (LO2)
假設您的當地政府面臨破產威脅，決定對其自身債券的利息收入徵稅，作為改善嚴重預算困境的一部分。您預期這些債券的收益率會發生什麼變化？（LO2）

Answer: You would expect the yields to rise to compensate investors for the loss of the tax-exempt status. These yields also would have to compensate investors for the heightened probability of default by the local government.
回答：您會預期收穫率上升以彌補投資者因稅收豁免狀態的失去。這些收穫率還必須彌補投資者對地方政府提高的違約風險。

*Suppose the yields on tax-exempt local government bonds in Problem 9 initially were below the Treasury yields of the same maturity. If the tax-exempt status were then removed from the local government bonds, would you expect their yield spreads versus treasuries to narrow, to disappear, or to change sign? Explain your answer. (LO1, LO2)
假設問題 9 中稅免的地方政府債券的收益率最初低於相同到期日的國庫券收益率。如果當時從地方政府債券中移除稅免狀態，您預期它們與國庫券的利差會縮小、消失還是變化符號？請解釋您的答案。（LO1，LO2）

Answer: We can attribute the lower yields on the local government bonds versus Treasury issues to their tax-exempt status, as investors would view the federal government as being at least as creditworthy as the local government. Given the serious budgetary woes of the local government, investors likely would regard these bonds as riskier than Treasury issues. The local bond yield also would have to compensate investors for the new tax obligation, so the spread would change sign.
我們可以將地方政府債券與國庫券之間較低的收益率歸因於其免稅狀態，因為投資者會認為聯邦政府至少與地方政府一樣值得信賴。考慮到地方政府的嚴重預算危機，投資者可能會認為這些債券比國庫券風險更高。地方債券收益率還必須補償投資者新的稅收義務，因此利差將變化符號。

Suppose the risk premium on U.S. corporate bonds increases. How would the change affect your forecast of future economic activity, and why? (LO4)
假設美國企業債的風險溢價增加。這個變化將如何影響您對未來經濟活動的預測，以及為什麼？（LO4）

Answer: An increasing risk premium can be a sign of an impending recession, so you would be more likely to forecast an economic downturn. During cyclical downturns, private companies have a more difficult time repaying their debt, while the U.S. Treasury typically is not affected, increasing the risk premium on company debt
答案：隨著風險溢價的上升，可能是一個即將來臨的經濟衰退的徵兆，因此你更有可能預測經濟衰退。在循環性衰退期間，私人公司還債更加困難，而美國國庫通常不受影響，增加公司債的風險溢價.

If regulations restricting institutional investors to investment grade bonds were lifted, what do you think would happen to the spreads between yields on investment grade and speculative grade bonds? (LO1)
如果取消限制机构投资者投资投资级债券的规定，您认为投资级和投机级债券之间的利差会发生什么变化？（LO1）

Answer: If institutional investors were willing to hold speculative-grade bonds in the absence of a legal restriction, the spread between speculative and investment-grade yields would narrow. Institutional demand for investment grade bonds would shift to the left as these investors switch to higher-yielding speculative bonds. Prices of investment grade bonds would decline and the yields would rise. In addition, demand for speculative grade bonds would shift to the right, increasing prices and lowering the yields. As a result, the yield spread between speculative grade and investment grade bonds would narrow.
如果機構投資者願意在沒有法律限制的情況下持有投機級債券，則投機級與投資級利差的會縮窄。機構對投資級債券的需求會向左移動，因為這些投資者轉向更高收益的投機級債券。投資級債券的價格將下跌，收益率將上升。此外，對投機級債券的需求將向右移動，提高價格並降低收益率。結果，投機級與投資級債券之間的利差將縮窄。

Consider a struggling emerging market economy where, in contrast to developed economies, the perceived risk associated with holding sovereign bonds is affected by the state of the economy. Suppose vast quantities of valuable minerals were unexpectedly discovered on government-owned land. How might the government’s bond rating be affected? Using the model of demand and supply for bonds, what would you expect to happen to the yields on that country’s government bonds? (LO1, LO4)
考慮一個困難的新興市場經濟體，與發達經濟體相比，持有國家債券相關的感知風險受到經濟狀況的影響。假設在政府所有的土地上意外發現大量有價礦產。政府的債券評等可能會受到什麼影響？使用債券的供需模型，你預期該國政府債券的收益率會發生什麼變化？（LO1，LO4）

Answer: The ratings of the bonds would likely be upgraded, as the outlook for the economy would improve and the reduction in the perceived riskiness of the bonds would shift the demand curve to the right. Revenues from the minerals could also mitigate the government’s need to borrow, shifting the supply of bonds to the left. Bond prices would increase and yields would fall.
答：債券的評級可能會提升，因為經濟前景將改善，對債券風險感知的降低將將需求曲線向右移動。礦產品的收入也可能減緩政府借貸的需求，將債券供給向左移動。債券價格將上升，收益率將下降。

Select the circumstance in which the impact on government bond yields of a new source of revenue (such as a natural resource discovery) would be largest. Explain your choice. (LO1, LO4)
選擇新收入來源（如自然資源發現）對政府債券收益率影響最大的情況。解釋你的選擇。（LO1，LO4）

Before the discovery, the government was heavily indebted with a crippling debt-service burden or
在發現之前，政府負債累累，負擔沉重的債務服務負擔或

Before the discovery, the government had a very low debt burden.
在發現之前，政府負債負擔相當低。

Answer: The impact would be larger in the case of a heavily indebted government, especially if the debt was denominated in another currency, such as U.S. dollars. These factors make the government more vulnerable to global interest rate and currency movements. The benefit of an additional source of revenue through taxing the mineral discoveries would be greater for a government at risk of defaulting due to high debt servicing costs than in a country where this risk was not an issue.
答：在債務沉重的政府情況下，影響會更大，特別是如果債務以另一種貨幣計價，如美元。這些因素使政府對全球利率和貨幣波動更加脆弱。對於因高債務服務成本而面臨違約風險的政府來說，通過對礦產發現徵稅來增加收入來源的益處，將比在這種風險不是問題的國家更大。

The misrating of mortgage-backed securities by rating agencies contributed to the financial crisis of 2007-2009. List some recommendations you would make to avoid such mistakes in the future. (LO1)
評級機構對抵押担保證券的誤評導致了 2007-2009 年的金融危機。列出一些您會提出的建議，以避免未來犯下這樣的錯誤。（LO1）

Answer: In the run-up to the 2007-2009 crisis, the absence of data capturing a period of falling house prices at a national level caused models to underestimate the default risk of the mortgages underlying the mortgage-backed securities. Running tests to see what outcomes these models would predict in extreme circumstances (stress testing) may help avoid such underestimations in the future. Publishing data on the accuracy of various bond rating firms in anticipating bond defaults also could encourage more reliable ratings. Similarly, encouraging professional asset managers to develop their own risk assessments would diminish the influence of credit ratings in portfolio selection. Finally, changing the relationship between the rating agencies and the securities issuers could reduce potential conflicts of interest. These conflicts can arise from payments by the bond issuers to the credit rating agencies in return for having their bonds rated
回答：在 2007-2009 年危机前夕，缺乏捕捉全国房价下跌期间的数据导致模型低估了抵押贷款支持证券背后的抵押贷款违约风险。对模型进行测试以查看在极端情况下这些模型会预测出什么结果（压力测试）可能有助于避免未来的低估。发布关于各种债券评级公司在预测债券违约方面的准确性的数据也可能鼓励更可靠的评级。同样，鼓励专业资产管理人开发自己的风险评估将减少信用评级在投资组合选择中的影响。最后，改变评级机构与证券发行人之间的关系可能减少潜在的潜在利益冲突，这些冲突可能源于债券发行人向信用评级机构支付费用以获得其债券评级。.

How do you think the abolition of investor protection laws would affect the risk spread between corporate and government bonds? (LO1)
您認為廢除投資者保護法會如何影響企業債和政府債之間的風險傳播？（LO1）

Answer: These laws were likely to be much more important in protecting purchasers of corporate bonds rather than purchasers of government bonds. Their abolition would raise the relative riskiness of corporate bonds, widening the spread between corporate and government bond yields.
這些法律可能對保護公司債購買者的重要性遠大於政府債購買者。其廢除將提高公司債的相對風險性，擴大公司債與政府債收益率的差距。

You and a friend are reading The Wall Street Journal and notice that the Treasury yield curve is slightly upward sloping. Your friend comments that all looks well for the economy but you are concerned that the economy is heading for trouble. Assuming you are both believers in the liquidity premium theory, what might account for your difference of opinion? (LO3, LO4)
你和一位朋友正在讀《華爾街日報》，並注意到國庫券收益率曲線略微上揚。你的朋友評論說經濟看起來一切正常，但你擔心經濟正走向困境。假設你們都是流動性溢價理論的信仰者，你的觀點差異可能是由什麼造成的？（LO3，LO4）

Answer: The difference in opinion could reflect different views on the size of the risk premium. If the risk premium is large enough, a slightly upward-sloping yield curve could mean that interest rates are expected to fall, indicating a weak economy. However, if the risk premium is small, the slight upward slope could reflect expectations that interest rates will rise and that the economy is expected to be healthy.
答案：意見差異可能反映了對風險溢價大小的不同觀點。如果風險溢價足夠大，略微上升的利息曲線可能意味著預期利率將下降，這表明經濟疲軟。然而，如果風險溢價較小，略微上升的斜率可能反映了預期利率將上升，並且預期經濟將健康。

Do you think the term spread was an effective predictor of the recession that started in December 2007? Why or why not? (LO4)
您認為「擴散」這個術語是有效預測 2007 年 12 月開始的經濟衰退的指標嗎？為什麼或為什麼不呢？（LO4）

Answer: An inverted yield curve (negative term spread) is often a sign that the economy is about to go into recession. Looking at the term spread (10-year yield minus the three-month interest rate) in Figure 7.7, we see that the term spread did indeed turn negative in late 2006/early 2007. On the other hand, the severity of the recession was not predicted by the yield curve.
答案：倒掛的收益曲線（負利差）通常是經濟將進入衰退的徵兆。觀察圖 7.7 中的利差（十年收益率減去三個月利率），我們看到利差在 2006 年底/2007 年初確實變為負值。另一方面，衰退的嚴重程度並未由收益曲線預測。

*Given the data in the accompanying table, would you say that this economy is heading for a boom or for a recession? Explain your choice. (LO4)
給予附表中的數據，您會說這個經濟是走向繁榮還是衰退？請解釋您的選擇。（LO4）

Answer: The information in both the term structure and the risk structure point to a healthy economy. The term spread (the gap between the 10-year Treasury bond yield and the three-month Treasury bill rate) is positive and widening. This tells us that the yield curve is upward sloping and getting more steeply upward sloping. This implies that interest rates are expected to continue to rise in the future—a sign that the economy is expected to do well.
答案：兩者之間的期限結構和風險結構所提供的信息都指向一個健康的经济。期限利差（十年國庫券收益率與三月國庫券利率之間的差距）為正值並擴大。這告訴我們，收益率曲線呈上升趋势並且越來越陡峭。這意味著預期未來利率將繼續上升——這是經濟預期表現良好的跡象。

The risk spread (the gap between the Treasury and corporate 10-year bonds) is narrowing. This is a sign of a healthy economy as people do not require such a high-risk premium on corporate bonds.
風險擴散（國庫債券與企業 10 年期債券之間的差距）正在縮小。這是經濟健康的跡象，因為人們不需要在企業債券上支付如此高的風險溢價。

Suppose recent regulatory reforms relating to credit rating agencies are perceived to improve the reliability and accuracy of credit ratings of corporate bond issues. Imagine further that you manage a corporation interested in issuing new bonds, in addition to past issues by the firm that already trade in the market. Identify one way in which your firm might lose and one way in which it might gain from these reforms. Explain your answer. (LO1)
假設近期與信評機構相關的監管改革被認為能夠提高企業債券發行的信評可靠性及準確性。想像一下，如果你管理一家對發行新債券感興趣的公司，除了該公司已在市場上交易的過去發行外。確定你公司可能因這些改革而損失的一種方式，以及可能獲益的一種方式。解釋你的答案。（LO1）

Answer: If, prior to the reforms, your bond issues enjoyed inflated credit ratings, you may receive a lower rating and therefore face higher borrowing costs if the reforms are successful in bringing about more accurate ratings.
回答：若在改革前，您的债券发行享有膨胀的信用评级，若改革成功使评级更加准确，您可能会获得较低的评级，因此面临更高的借贷成本。

Core Principle 3 states that information is the basis for decisions. If, prior to the reforms, lack of investor confidence in credit ratings made it difficult or impossible to issue bonds, the reforms may improve market access. Put differently, if the reforms restore investor confidence in credit ratings, your corporation may be better able to issue new bonds at a reasonable price.
核心原則 3 指出，資訊是決策的基礎。如果改革前，投資者對信用評級的信心不足使得發行債券變得困難或不可能，改革可能會改善市場進入。換句話說，如果改革恢復了投資者對信用評級的信心，您的公司可能更能以合理的價格發行新債券。