固定收益系列课程之二：固定收益市场 100分答案

6
9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
9.2188% 9.2700% 9.3194% 9.3672% 9.4132% 9.4575% 9.5000% 9.5408% 9.5799% 9.6172% 9.6528% 9.6866% 9.7188% 9.7491% 9.7778% 9.8047% 9.8299% 9.8533% 9.8750% 9.8950% 9.9132% 9.9297% 9.9444%
折现因子 1 0.9619 0.9245 0.8880 0.8523 0.8175 0.7836 0.7506 0.7186 0.6876 0.6575 0.6285 0.6004 0.5733 0.5472 0.5220 0.4978 0.4745 0.4522 0.4308 0.4102
现金流量 0 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80 0 80
9、当期的平价到期收益曲线 1 如下：
1
平价收益率是指证券价格等于面值时的到期收益率 3
到期日 1 2 3 4 5
平价收益率 10% 15% 20% 23% 25%
假设平价到期收益率的单位 为年，按年复利计息。 利用上面提供的信息，计算以下债券在零期的价格，该债券获得的现金流如下： C1 =10元, C2 = 10元, C3 = 110元，其中Ct是在第t期获得的现金收入。
r3 = 21.56%
债券的价值为
10 10 110 + + = 77.84 2 1.1 (1.1539) (1.2156) 3

固定收益证券试题及答案一、单项选择题（每题2分，共20分）1. 固定收益证券的主要风险不包括以下哪一项？A. 利率风险B. 信用风险C. 流动性风险D. 汇率风险答案：D2. 以下哪个不是固定收益证券的特点？A. 收益固定B. 投资期限长C. 风险较低D. 价格波动大答案：D3. 债券的票面利率与市场利率的关系是：A. 总是相等的B. 总是不等的C. 有时相等，有时不等D. 以上都不对答案：C4. 如果市场利率上升，而债券的票面利率保持不变，那么债券的：A. 价格上升B. 价格下降C. 价格不变D. 与市场利率无关答案：B5. 以下哪个不是固定收益证券的种类？A. 政府债券B. 企业债券C. 股票D. 金融债券答案：C6. 债券的到期收益率是指：A. 债券的票面利率B. 债券的当前市场价格C. 投资者持有到期的年化收益率D. 债券的发行价格答案：C7. 以下哪个因素不会影响固定收益证券的收益率？A. 发行主体的信用等级B. 债券的期限C. 市场利率水平D. 投资者的风险偏好答案：D8. 债券的久期是指：A. 债券的到期时间B. 债券的加权平均到期时间C. 债券的票面金额D. 债券的发行时间答案：B9. 以下哪个不是影响债券价格的因素？A. 债券的票面利率B. 债券的信用等级C. 债券的发行量D. 市场利率的变化答案：C10. 以下哪个是固定收益证券投资的主要目的？A. 资本增值B. 获得稳定的现金流C. 参与公司决策D. 投机取利答案：B二、多项选择题（每题3分，共15分）11. 固定收益证券的收益来源主要包括哪些？（ACD）A. 利息收入B. 股票升值C. 资本利得D. 再投资收益12. 以下哪些因素会影响固定收益证券的信用风险？（ABD）A. 发行主体的财务状况B. 经济环境的变化C. 投资者的个人偏好D. 法律和政策环境13. 固定收益证券的流动性通常与以下哪些因素有关？（ACD）A. 债券的发行量B. 债券的票面利率C. 市场交易的活跃度D. 债券的到期时间14. 以下哪些措施可以降低固定收益证券的投资风险？（ABD）A. 分散投资B. 选择信用等级较高的债券C. 增加投资金额D. 关注市场利率变动15. 固定收益证券的久期与以下哪些因素有关？（ABC）A. 债券的现金流时间B. 每笔现金流的金额C. 每笔现金流的现值D. 债券的票面利率三、判断题（每题1分，共10分）16. 固定收益证券的风险总是低于股票。

固定收益证券-课后习题与答案.第1章固定收益证券概述1．固定收益证券与债券之间是什么关系？解答：债券是固定收益证券的一种，固定收益证券涵盖权益类证券和债券类产品，通俗的讲，只要一种金融产品的未来现金流可预期，我们就可以将其简单的归为固定收益产品。

2．举例说明，当一只附息债券进入最后一个票息周期后，会否演变成一个零息债券？解答：可视为类同于一个零息债券。

3．为什么说一个正常的附息债券可以分拆为若干个零息债券？并给出论证的理由。

解答：在不存在债券违约风险或违约风险可控的前提下，可以将附息债券不同时间点的票面利息视为零息债券。

4．为什么说国债收益率是一国重要的基础利率之一。

解答：一是国债的违约率较低；二是国债产品的流动性在债券类产品中最好；三是国债利率能在一定程度上反映国家货币政策的走向，是衡量一国金融市场资金成本的重要参照。

5．假如面值为100元的债券票面利息的计算公式为：1年期银行定期存款利率某2+50个基点－1年期国债利率，且利率上限为5%，利率下限为4%，利率每年重订一次。

如果以后5年，每年利率重订日的1年期银行存款利率和国债利率如表1.4所示，计算各期债券的票面利息额。

表1.41年期定期存款利率和国债利率重订日第1次第2次第3次第4次第5次1年期银行存款利率（%）1.52.84.15.46.71年期国债利率（%）2.53.04.55.87.0债券的息票利率4%4%4.7%5%5%解答：第1次重订日计算的债券的票面利率为：1.5%某2+0.5%-2.5%=1%，由于该票面利率低于设定的利率下限，所以票面利率按利率下限4%支付。

此时，该债券在1年期末的票面利息额为100某4%=4元第2次重订日计算的债券的票面利率为：2.8%某2+0.5%-3%=3.1%，由于该票面利率低于设定的利率下限，所以票面利率仍按利率下限4%支付。

此时，该债券在2年期末的票面利息额为100某4%=4元第3次重订日计算的债券的票面利率为：4.1%某2+0.5%-4.5%=4.2%，由于该票面利率介于设定的利率下限和利率上限之间，所以票面利率按4.7%支付。

23、假设货币市场期限为3个月、6个月和9个月 的债券的实际季度收益率分别为0.75%、1.5%和 2%，再假设该市场上存在期限为3个月和9个月 的两种贴现国债，面值都是100元。如果投资者 的投资期限是3个月，并假定收益率曲线在未来3
个月里不会变化。请问该投资者应选择哪一种债 券投资？
3、一张期限为10年的等额摊还债券，每年等 额偿还的金额为100元；另有一张永久债券， 每年支付利息为50元。如果市场利率为8%， 试比较它们价格的大小。
4、若市场上有下表所示的两个债券，并假设 市场利率的波动率是10%，构建一个二期的利率 二叉树。
市场债券品种假设
品种 A
到期期限 息票利率 折现率 当前价格
12、考虑票面金额1000元、票面利率为8%、 期限为5年的每年付息一次的债券，现有两种 情况：到期收益率为7%时，上升1个百分点 所引起的债券价格变化率为多少? 到期收益 率为8%时，上升1个百分点所引起的债券价 格变化率为多少?哪种情况下债券价格变化率 大？
13、某投资者购买了10张面值为100元，票 面利率为6%、每年付息一次的债券，债券刚 付息，持有3年，获得3年末的利息后出售。 期间获得的利息可以再投资，假设再投资收 益率为4.5%。每份债券购买价为103元，出 售价为107元。求该投资者的总收益率。
14、某一次还本付息债券，面值100元，票面 利率3.5%，期限3年，2011年12月10日到期。 债券交易的全价为99.40元，结算日为2009年9 月15日，试计算其到期收益率。
15、假设有3个不同期限债券，它们的数据
见下表，其中第一个为零息债券，后两个是附
息债券，且都是每年付息一次。试给出1年期
6、设某债券与上题B债券条件相同，但 为可回售债券，持有人有权在发行后的 第一年末以99.50元的价格向发行人回售， 利率二叉树与上题亦相同，试计算该债 券的价格。

固定收益证券习题答案北大第二章计算题答案1、假定到期收益率曲线是水平的，都是5%。

一个债券票面利率为6%，每年支付一次利息，期限3年。

如果到期收益率曲线平行上升一个百分点，请计算债券价格的变化。

债券原价格P=__++=102.7223 1.051.051.05新价格P=__++=1001.061.0621.063债券价格下降2.72元2、假定某债券面值为100元，期限为3年，票面利率为年6%，一年支付两次利息。

请计算债券本息累积到第3年年底的总价值（再投资收益率为半年3%）。

=3(1+3%)5+(1+3%)4+(1+3%)3+(1+3%)2+(1+3%)1+(1+3%)0+100 =119.43、假定某债券面值为100元，期限为3年，票面利率为年6%，一年支付两次利息。

投资者购买价格为103元，请计算在再投资收益率为年4%的情况下投资者的年收益率。

[]总收益__=3 (1+2%)+(1+2%)+(1+2%)+(1+2%)+(1+2%)+(1+2%) +100=118.92收益率13y=(118.92/103) 1=4.91%4、一个投资者按85元的价格购买了面值为100元的两年期零息债券。

投资者预计这两年的通货膨胀率将分别为4%和5%。

请计算该投资者购买这张债券的真实到期收益率。

因为2（（（=1+i真）*1+4%）*1+5%）10085所以i真=3.8%5、一个债券期限为5年，票面利率为5%，面值为100元，一年支付利息两次，目前的价格为105元。

求该债券的到期收益率和年有效收益率。

约当收益率=2*1.94%=3.88% 年有效收益率=3.92%(1+1.94%)2 1=3.92%北大6、一个20年期限的债券，面值100元，现在价格110元，票面利率6%，一年两次付息。

5年后可以按面值回购，计算该债券的到期收益率和至第一回购日的到期收益率。

到期收益率（b.e.b.）=2.6%*2=5.2% 至第一回购日的到期收益率=1.89%*2=3.98%7、某投资者购买一张债券，面值为1000元，价格为1100元。

第1章固定收益证券概述1．固定收益证券与债券之间是什么关系？解答：债券是固定收益证券的一种，固定收益证券涵盖权益类证券和债券类产品，通俗的讲，只要一种金融产品的未来现金流可预期，我们就可以将其简单的归为固定收益产品。

2．举例说明，当一只附息债券进入最后一个票息周期后，会否演变成一个零息债券？解答：可视为类同于一个零息债券。

3．为什么说一个正常的附息债券可以分拆为若干个零息债券？并给出论证的理由。

解答：在不存在债券违约风险或违约风险可控的前提下，可以将附息债券不同时间点的票面利息视为零息债券。

4．为什么说国债收益率是一国重要的基础利率之一。

解答：一是国债的违约率较低；二是国债产品的流动性在债券类产品中最好；三是国债利率能在一定程度上反映国家货币政策的走向，是衡量一国金融市场资金成本的重要参照。

5．假如面值为100元的债券票面利息的计算公式为：1年期银行定期存款利率×2+50个基点－1年期国债利率，且利率上限为5%，利率下限为4%，利率每年重订一次。

如果以后5年，每年利率重订日的1年期银行存款利率和国债利率如表1.4所示，计算各期债券的票面利息额。

表1.4 1年期定期存款利率和国债利率解答：第1次重订日计算的债券的票面利率为：1.5%×2+0.5%-2.5%=1%，由于该票面利率低于设定的利率下限，所以票面利率按利率下限4%支付。

此时，该债券在1年期末的票面利息额为100×4%=4元第2次重订日计算的债券的票面利率为：2.8%×2+0.5%-3%=3.1%，由于该票面利率低于设定的利率下限，所以票面利率仍按利率下限4%支付。

此时，该债券在2年期末的票面利息额为100×4%=4元第3次重订日计算的债券的票面利率为：4.1%×2+0.5%-4.5%=4.2%，由于该票面利率介于设定的利率下限和利率上限之间，所以票面利率按4.7%支付。

此时，该债券在3年期末的票面利息额为100×4.2%=4.2元第4次重订日计算的债券的票面利率为：5.4%×2+0.5%-5.8%=5.5%，由于该票面利率高于设定的利率上限，所以票面利率按利率上限5%支付。

固定收益证券试题及部分答案班级序号：学号：姓名：成绩：1）Explain why you agree or disagree with the following statement: “The price of a floater will always trade at its par value.”Answer:I disagree with the statement: “The price of a floater will always trade at its par value.”First, the coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some spread or margin. For example, the coupon rate of a floater can reset at the rate on a three-month Treasury bill (the reference rate) plus 50 basis points (the spread). Next, the price of a floater depends on two factors: (1) the spread over the reference rate and (2) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price of a floater will trade close to its par value as long as (1) the spread above the reference rate that the market requires is unchanged and (2) neither the cap nor the floor is reached. However, if the market requires a larger (smaller) spread, the price of a floater will trade below (above) par. If the coupon rate is restricted from changing to the reference rate plus the spread because of the cap, then the price of a floater will trade below par.2）A portfolio manager is considering buying two bonds. Bond A matures in three years and has a coupon rate of 10% payable semiannually. Bond B, of the same credit quality, matures in 10 years and has a coupon rate of 12% payable semiannually. Both bonds are priced at par.(a) Suppose that the portfolio manager plans to hold the bond that is purchased for three years. Which would be the best bond for the portfolio manager to purchase?Answer:The shorter term bond will pay a lower coupon rate but it will likely cost less for a given market rate.Since the bonds are of equal risk in terms of creit quality (The maturity premium for the longer term bond should be greater),the question when comparing the two bond investments is:What investment will be expecte to give the highest cash flow per dollar invested?In other words,which investment will be expected to give the highest effective annual rate of return.In general,holding the longer term bond should compensate the investor in the form of a maturity premium and a higher expected return.However,as seen in the discussion below,the actual realized return for either investment is not known with certainty.To begin with,an investor who purchases a bond can expect to receive a dollar return from(i)the periodic coupon interest payments made be the issuer,(ii)ancapital gainwhen the bond matures,is called,or is sold;and (iii)interest income generated from reinvestment of the periodic cash flows.The last component of the potential dollar return is referred to as reinvestment income.For a standard bond(our situation)that makes only coupon payments and no periodic principal payments prior to the maturity date,the interim cash flows are simply the coupon payments.Consequently,for such bonds the reinvestment income is simply interest earned from reinvesting the coupon interest payments.For these bonds,the third component of the potential source of dollar return is referred to as the interest-on-interest components.If we are going to coupute a potential yield to make a decision,we should be aware of the fact that any measure of a bond’s potential yield should take into consideration each of the three components described above.The current yield considers only the coupon interest payments.No consideration is given to any capital gain or interest on interest.The yield to maturity takes into account coupon interest and any capitalgain.It also considers the interest-on-interest component.Additionally,implicit in the yield-to-maturity computation is the assumption that the coupon payments can be reinvested at the computed yield to maturity.The yield to maturity is a promised yield and will be realized only if the bond is held to maturity and the coupon interest payments are reinvested at the yield to maturity.If the bond is not held to maturity and the couponpayments are reinvested at the yield to maturity,then the actual yield realized by an investor can be greater than or less than the yield to maturity.Given the facts that(i)one bond,if bought,will not be held to maturity,and(ii)the coupon interest payments will be reinvested at an unknown rate,we cannot determine which bond might give the highest actual realized rate.Thus,we cannot compare them based upon this criterion.However,if the portfolio manager is risk inverse in sense that she or he doesn’t want to buy a longer term bond,which will likel have more variability in its return,then the manager might prefer the shorter term bond(bondA) of thres years.This bond also matures when the manager wants to cash in the bond.Thus,the manager would not have to worry about any potential capital loss in selling the longer term bond(bondB).The manager would know with certainty what the cash flows are.IfThese cash flows are spent when received,the manager would know exactly how much money could be spent at certain points in time.Finally,a manager can try to project the total return performance of a bond on the basis of the panned investment horizon and expectations concerning reinvestment rates and future market yields.This ermits the portfolio manager to evaluate thich of several potential bonds considered for acquisition will perform best over the planned investment horizon.As we just rgued,this cannot be done using the yield to maturity as a measure of relative .coming total returnto assess performance over some investment horizon is called horizon analysis.When a total return is calculated oven an investment horizon,it is referred to as a horizon return.The horizon analysis framwor enabled the portfolio manager to analyze the performance of a bond under different interest-rate scenarios for reinvestment rates and future market yields.Only by investigating multiple scenarios can the portfolio manager see how sensitive the bond’s performance will be to each scenario.This can help the manager choosebetween the two bond choices.(b) Suppose that the portfolio manager plans to hold the bond that is purchased for six years instead of three years. In this case, which would be the best bond for the portfolio manager to purchase?Answer:Similear to our discussion in part(a),we do not know which investment would give the highest actual relized return in six years when we consider reinvesting all cash flows.If the manager buys a three-year bond,then there would be the additional uncertainty of now knowing what three-year bond rates would be in three years.The purchase of the ten-year bond would be held longer than previously(six years compared to three years)and render coupon payments for a six-year period that are known.If these cash flows are spent when received,the manager will know exactly how much money could be spent at certain points in timeNot knowing which bond investment would give thehighest realized return,the portfolio manager would choose the bond that fits the firm’s goals in terms of maturity.3）Answer the below questions for bonds A and B.Bond A Bond BCoupon 8% 9%rYield to maturity 8% 8%Maturity (years) 2 5Par $100.00 $100.00Price $100.00 $104.055(a) Calculate the actual price of the bonds for a 100-basis-point increase ininterest rates.Answer: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:41111(1)(10.045)$40$143.5010.045nr P C r ????--???? ===????????????????The present value of the par or maturity value of $1,000 is:4$1,000$838.561(1)(1.045)n M r == 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. 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 couponrate the same, For example, inserting 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. 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(b) Using duration, estimate the price of the bonds for a 100-basis-point increase in interest rates.Answer: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:211(100/)1(1)(1)n n Cn C y y y y Modified Duration P ??-- ?? ??=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: 212451(100/)$414($100$4/0.045)11(1)(1)0.045(1.045)(1.045)98.2062n n C n C y y y y Modified Duration P ????--- - ???? ????===($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). We next solve for the change in price using the modified duration of 1.805159 and dy = 100 basis points = 0.01. We have: ()() 1.805159(0.01)0.0180515dP Modified Duration dy P=-=-=- We can now solve for the new price of bond A as shown below:(1)(10.0180515)$1,000$981.948dP P P=-= This is slightly less than the actual price of $982.062. The difference is $982.062 –$981.948 = $0.114. 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:21210111(100/)$4.5110($100$4.5/0.045)11(1)(1)0.045(1.045)(1.045)$100n n C n C y y y y Modified Duration P ????--- - ???? ????== ($791.27182 $0) / $100 = 7.912718 or about 7.91 (before the increase in 100 basis points it was7.988834 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 durationmeasure. With 100 basis points giving dy = 0.01 and an approximate duration of 3.956359, we have:()() 3.956359(0.01)0.0395635dP Modified Duration dy P=-=-=-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.(c) Using both duration and convexity measures, estimate the price of the bonds for a 100-basis-point increase in interest rates. Answer:For bond A, we use the duration and convexity measures as given below. 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 bondprice 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%. 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:()() 1.805159(0.01)0.01805159dP Modified Duration dy P=-=-=- This is slightly more negative than the actual percentage decrease in price of1.7938%. The difference is 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:(1)(10.01805159)$1,000$981.948dP P P=-= This is slightly less than the actual price of $982.062. The difference is $982.062 –$981.948 = $0.114. Next, we use the convexity measure to see if we can account for the difference of 0.011359%. We have: convexity measure(half years) =2232121212(1)(100/)11(1)(1)(1)n n n d P C Cn n n C y dy P y y y y y P ???? -??=-- ?????? ?????? 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:convexity measure (half years) = 342562$412($4)44(5)(100$4/0.045)1116.93250.045(1.045)0.045(1.045)(1.045)$ 98.2062y ????-=??-- =???????????? 2216.9325() 4.2331252Convexity Measure inm period per year TheConvexity Measure in years m === 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:Pr (1)(10.01783994)$1,000(0.9819484)$1,000$982.160dP New ice P P= = -== 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:()() 3.056359(0.01)0.0395635dP Modified Duration dy P=-=-=- This is slightly more negative than the actual percentage decrease in price of -3.896978%. The difference is (-3.896978%)-(-3.95635%)=0.059382%Using the -3.95635%just given by the duration measure, the new price for Bond B is: (1)(10.0395635)$1,040.55$999.382dP P P=-=This is slightly less than the actual price of $1,000. This difference is $1,000-$999.382=$0.618We use the convexity measure to see if we can account for the difference of 00594%. We have:2232121212(1)(100/)1()1(1)(1)(1)n n n d P C Cn n n C y Convexity Measure half years dy P y y y y y P ???? -??==-- ?????? ?????? For Bond B, 100 basis points are added 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:310211122($4.5)12($4.5)410(11)(100$4.5/0.045)1()1(0.045)(1.045)(0.045)(1.04 5)(1.045)$100Convexity Measure half years ????-??=-- ???????????? 7,781.03[0.01000]77.8103==The convexity measure (in years)=2277.810319.4525642convexitymeasureinm period per year m == Note. DollarConvexity Measure=Convexity Measure (years) times P=19.452564($100)=$1,945.2564.The percentage price change due to convexity is 21()2dP convexity measure dy P = Inserting in the values, we get 21(77.8103)(0.01)0.000974632dP P == Thus, we have 0.097463% increase in price when we adjust for convexity measure.Adding the duration measure and convexity measure, we get -3.9563659% 0.097263% equals -3.859096%. Recall the actual change in price is ($1,000-$1,040.55)=-$40.55 and the actual new price is(1)(10.03859096)$1,040.55(0.9614091)$1,040.55$1,000.394dP P P-=-== 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.(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.Answer: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. 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 $0.394As 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. Moredetails are offered below. Duration (modified or dollar) attempts to estimatea 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:2221()(1)2dP d P dP dy dy error dy dy = Dividing both sides of this equation by P to get the percentage price change gives us: 22211()(2)2dP dP d P error dy dy P dy P dy P =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 tosimply as the convexity measure.(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%.Answer: 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. 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.4）Suppose a client observes the following two benchmark spreads for two bonds:Bond issue U rated A: 150 basis pointsBond issue V rated BBB: 135 basis pointsYour client is confused because he thought the lower-rated bond (bond V) should offer a higher benchmark spread than the higher-rated bond (bond U). Explain why the benchmark spread may be lower for bond U.5）The bid and ask yields for a Treasury bill were quoted by a dealer as 5.91% and 5.89%, respectively. Shouldn’t the bid yield be less than the ask yield, because the bid yield indicates how much the dealer is willing to pay and the ask yield is what the dealer is willing to sell the Treasury bill for?Answer:The higher bid means a lower price. So the dealer is willing to pay less than would be paid for the lower ask price. We illustrate this below. Given the yield on a bank discount basis (Yd), the price of a Treasury bill is found by first solving the formula for the dollar discount (D), as follows:()()360d t D Y F = The price is then Price = F-DFor the 100-day Treasury bill with a face value (F) of $100,000, if the yield on a bank discount basis (Yd) is quoted as 5.91%, D is equal to:100()()0.0591($100,000)()$1,641.67360360d t D Y F ===Therefore, price = $100,000 –$1,641.67 = $98,358.33. For the 100-day Treasury bill with a face value (F) of $100,000, if the yield on a bank discount basis (Yd) is quoted as 5.89%, D is equal to:100()()0.0589($100,000)()$1,636.11360360d t D Y F === Therefore, price is: P = F –D = $100,000 –$1,636.11 = $98,363.89.Thus, the higher bid quote of 5.91% (compared to lower ask quote 5.89%) gives a lower selling price of $98,358.33 (compared to $98,363.89). The 0.02% higher yield translates into a selling price that is $5.56 lower. In general, the quoted yield on a bank discount basis is not a meaningful measure of the return from holding a Treasury bill, for two reasons. First, the measure is based on a face-value investment rather than on the actual dollar amount invested. Second, the yield is annualized according to a 360-day rather than a 365-day year, making it difficult to compare Treasury bill yields with Treasury notes and bonds, which pay interest on a 365-day basis. The use of 360 days for a year is a money market convention for some money market instruments, however. Despite its shortcomings as a measure of return, this is the method that dealers have adopted to quote Treasury bills. Many dealer quote sheets, and some reporting services, provide two other yield measures that attempt tomake the quoted yield comparable to that for a coupon bond and other money market instruments.6）What is the difference between a cash-out refinancing and a rate-and-term refinancing?Answer:When a lender is evaluating an application from a borrower who is refinancing, the loan-to-value ratio (LTV) is dependent upon the requested amount of the new loan and the market value of the property as determined byan appraisal. When the loan amount requested exceeds the original loan amount, the transaction is referred to as a cash-out-refinancing. If instead, there is financing where the loan balance remains unchanged, the transaction is said to be a rate-and-term refinancing or no-cash refinancing. That is, the purpose of refinancing the loan is to either obtain a better note rate or change the term of the loan.7）Describe the cash flow of a mortgage pass-through security.Answer:The cash flow of a mortgage pass-through security depends on the cash flow of the underlying mortgage.The cash flow consists of monthly mortgage payments representing interest,the scheduled repayment of principal,and any prepayments. Payments are made to security holders each month.Neither theamount nor the timing,however,of the cash flow from the pool of mortgages is identical to that of the cash flow passed through to investors.The monthly cash flow for a pass-through is less than the monthly cash flow of the underlying mortgages by an amount equal to servicing and other fees.The other fees are those charged by the issuer or guarantor of the pass-through for guaranteeing the issue.The coupon rage on a pass-through,called the pass-through coupon rate,is less than the mortgage rage on the underlying pool of mortgage loans by an amount equal to the servicing and guaranteeing feesThe timing of the cash flow,like the amount of the cash flow,is also different.The monthly mortgage payment is due from each mortgagor on the first day of each month,but there is a delay in passing through the corresponding monthly cash flow to the securityholders.The length of the delay varies by the type of pass-through security. Because of prepayments,the cash flow of a pass-through is also not known with certainty.8）Explain the effect on the average lives of sequential-pay structures of including an accrual tranche in a CMO structure.。

固定收益证券作业及答案1.三年后收到的100元现在的价值是多少？分别考虑复利20%、复利100%、复利0%、复利20%（半年计息）、复利20%（季计息）和复利20%（连续计息）的情况。

2.以连续复利方式计息，分别计算复利4%、复利20%（年计息）、复利20%（季计息）和复利100%的利率。

3.考虑以下问题：a。

___在交易日92年9月16日给出了票面利率为91/8's在92年12月31日到期，92年9月17日结算的政府债券，其标价为买入价101：23，卖出价101：25.求该债券的买入和卖出的收益率。

b。

在同一交易日，___对同时在92年12月31日到期和在92年9月17日结算的T-bill报出的买入和卖出折现率分别是2.88%和2.86%。

是否存在套利机会？（“买入”和“卖出”是从交易者的角度出发，你是以“买入价”卖出，以“卖出价”买入）4.在交易日92年9月16日，以10-26的价格买入了一张面值为2000万美元、到期日为2021年11月15日的STRIPs （零息债券）。

求该债券的到期收益率。

5.今天是1994年10月10日，星期一，是交易日。

以下是三种债券的相关信息：发行机构票面利率到期日到期收益___ 10% 8.00% 星期二，1/31/95费城（市政） 9% 7.00% 星期一，12/2/95___（机构） 8.50% 8% 星期五，7/28/95这三种债券的面值均为100美元，每半年付息一次。

注意到上表中最后一列是到期收益，它反映了给定到期日、某种特定债券的标准惯例。

在计算日期时，不考虑闰年，同时也要忽略假期。

回答以下问题时，需要写清楚计算过程，不能只是用计算器计算价格。

a。

计算___发行的国债的报价，假定该国债按照标准结算方式结算。

b。

计算费城发行的城市债券的报价，假定该债券的标准结算期为三天。

c。

计算___发行的机构债券的报价，假定该债券按照标准结算方式结算。

本题需要根据给定的到期收益曲线来计算固定付息债券的全价，以及在曲线上下移动100个基点时的全价。

（1）
V ( t )+ = 2.5 × e −5.3%×0.5 + 2.5 × e −5.3%×1 + 2.5 × e −5.3%×1.5 + 102.5 × e −5.3%×2 = 99.31( 元 )
到期收益率上升 50 个基点，债券价格下降 0.95 元。 （2）
V ( t )− = 2.5 × e −4.3%×0.5 + 2.5 × e −4.3%×1 + 2.5 × e −4.3%×1.5 + 102.5 × e −4.3%×2 = 101.24 ( 元 )
FV = 1000 × (1 + 6% ) = 5975.32 ( 元)
i i =1 5
4. 若某剩余期限为 5 年的零息票国债，当前市场价格为 82 元，到期支付本金 100 元，请根据该债券计算连续复利形式下 5 年期即期利率。 由 82 × e
R (0,5)×5
= 100 ，求得连续复利形式下 5 年期即期利率 R ( 0,5 ) = 3.97%
Copyright © 2011 Chen, Rong & Zheng, Zhenlong 4
《固定收益证券》参考答案 陈蓉 郑振龙 北京大学出版社
到期收益率下降 50 个基点。 由式（2.11）该债券的价格
V ( t ) = 2.5 × e −4.8%×0.5 + 2.5 × e −4.8%×1 + 2.5 × e −4.8%×1.5 + 102.5 × e −4.8%×2 = 100.26 ( 元 )
4% + 0.25% = 0.0213A 2 假设该债券的利差保持在 0.25%不变，则其合理的贴现率应为 A×

) r
(.5)
⎞
⎛
f (1) ⎞
⎜⎜⎝1 +
2 ⎟⎟⎠ = ⎜⎜⎝1+
2
⎟⎟⎠
⎜1+ ⎝
2
⎟ ⎠
⎛ ⎜⎜⎝1 +
Байду номын сангаас
) r
(1.5)
⎞3
2 ⎟⎟⎠
⎛ = ⎜⎜⎝1+
) r
(1)
2
⎞2 ⎟⎟⎠
⎛ ⎜1 ⎝
+
f
(1.5) ⎞
2
⎟ ⎠
to
get
) r
(.5)
=
.25%
;
) r
(1)
=
.3250%
;
) r
(1.5)
.5 [.998752 + .996758] + 99.6758 = 100.17468
2
Chapter 3
3.1 The price of the 3/4s of May 31, 2012 was 99.961 as of May 31, 2010. Calculate its
price using the discount factors in Table 2.3. Is the bond trading cheap or rich to those
of .5%. If over the subsequent six months the term structure remains unchanged, will
the price of the .5% bond increase, decrease, or stay the same? Try to answer the

3、一张期限为10年的等额摊还债券，每年等 额偿还的金额为100元；另有一张永久债券， 每年支付利息为50元。如果市场利率为8%， 试比较它们价格的大小。
4、若市场上有下表所示的两个债券，并假设 市场利率的波动率是10%，构建一个二期的利率 二叉树。 市场债券品种假设
品种 A 到期期限 息票利率 （年） （%） 1 3.50 折现率 （%） 3.50 当前价格 （元） 100.00
15、假设有3个不同期限债券，它们的数据 见下表，其中第一个为零息债券，后两个是附 息债券，且都是每年付息一次。试给出1年期 到3年期的即期收益率。 三个不同期限债券的数据
期限（年） 1 2 3 面值（元） 100 100 100 息票利率（%） 市场价格（元） 0 95.60 5.42 102.38 6.78 105.56
18、试计算面值为100元，到期收益率为5%， 期限为5年的贴现债券的久期和修正久期。
19、有一债券，面值100元，期限20年，息 票利率8%，每年付息一次，到期收益率8%， 价格是100元。当市场利率上升10个基点时， 市场价格是99.0254元；当市场利率下降10个 基点时，市场价格是100.9892元。求该债券 的有效久期。
20、假设有一个债券，面值100元，期限3年， 票面利率5%，每年付息一次，市场利率4%， 试计算其凸度。
21、有一债券面值是100元，初始到期收益率 为8%，修正久期是7.95年，凸度是84.60， 债券价格是84.9278元。当收益率下降100个 基点时，试计算用修正久期预测的债券价格 和考虑凸度调整后的债券价格。
9、有一附息债券，一年付息一次，期限5年， 票面金额为1000元，票面利率5.2%。某投资 者在该债券发行时以998元的发行价购入，持 满3年即以1002.20元的价格卖出。请计算该 投资者的持有期收益率是多少（可用简化公 式）？当期收益率有一企业债券，面值100元，期限3年， 票面利率4%，到期一次还本付息，利息所得 税税率为20%，请计算持有该债券到期的税 后复利到期收益率。

固定收益证券 第一章作业 1 解：()22.746%)51/(100031.751%101/100063=+==+=p p 半年复利：年复利：85.740)365/%101/(100074.741)12/%101/(100065.743%)5.21/(100010953612=+==+==+=p p p 天复利：月复利：季复利：82.740/1000)/%101/(10003.03==+=e n p n 连续复利：2解：0885.02000)2/110000905.02000)1(1000168=⇒=+=⇒=+r r r r （半年复利：年复利：0866.02)/1(0867.02365/10870.0212/10876.024/1829209632=⇒=+=⇒=+=⇒=+=⇒=+r n r r r r r r r n 连续利率：）天利率：（）月利率：（）季复利：（3解：1823.02.1ln %201%200392.004.1ln %41%,4==⇒=+==⇒=+r e r e r r ，年计息：复利年计息：复利1984.0)12/%201ln(12/%201%201952.005.1ln 4/%201%20121244=+=⇒=+==⇒=+r e r e r r ）（，月计息：复利）（，季计息：复利4解：应计利息净价全价全价应计利息净价+=⨯++=⨯⨯==+=1841051)2/09125.01(100184/792/09125.010071875.10132/231012x解得： 029911851.0=x 故买入收益率为0.029911851同理 卖出收益率为0.027782327 5解：答：（1）设逆浮动利率债券的利率公式为X 5000*9%=3000*（LIBOR+3%）+2000X 解得X=18%-1.5LIBOR逆浮动利率债券的利率确定公式为18%-1.5 month LIBOR （2）因为0<LIBOR<=12%浮动利率债券的利率=1month LIBOR+3%，故顶为15%，底为3%逆浮动利率债券的利率=18%-1.5monthLIBOR ，故顶为18%，底为0 6 解：5000/100*98.25 =4912.5元所以投资者购买该债券需支付的金额 4912.5元 7 解：浮动利率= LIBOR+1.25%=6.5%+1.25%=7.725%半年支付的利率水平=7.725%/2=3.875%第二章作业1. 设债券的面值是P ，到期收益率为5%时的债券价格为1P ，到期收益率为6%时的债券价格为2P .则123223126%6%6%(15%)(15%)(15%)6%6%6%(16%)(16%)(16%)0.027P P P PP P P P P P P P P +=++++++=+++++-= 即价格下降2.72% 2.54321003%(13%)1003%(13%)1003%(13%)1003%(13%)1003%(13%)1003%100119.4FV =⨯⨯++⨯⨯++⨯⨯++⨯⨯++⨯⨯++⨯+=3、假定某债券面值为100元，期限为3年，票面利率为年6%，一年支付2次利息，投资者购买价格为103元，请计算在再投资收益率为年4%的情况下投资者的年收益率。

一．名词解释一般责任债券：指地方政府以其信用承诺支付本息而发行的债券。

一般责任债券并无特定的还款来源，地方政府必须利用税收来偿还本息。

金融债券：银行等金融机构作为筹资主体为筹措资金而面向个人发行的一种有价证券，是表明债务、债权关系的一种凭证。

债券按法定发行手续，承诺按约定利率定期支付利息并到期偿还本金。

它属于银行等金融机构的主动负债.资产抵押债券：是以资产（通常是房地产）的组合作为抵押担保而发行的债券,是以特定“资产池(Asset Pool）”所产生的可预期的稳定现金流为支撑，在资本市场上发行的债券工具。

零息债券：零息债券是指以贴现方式发行,不附息票，而于到期日时按面值一次性支付本利的债券.零息债券发行时按低于票面金额的价格发行，而在兑付时按照票面金额兑付，其利息隐含在发行价格和兑付价格之间。

零息债券的最大特点是避免了投资者所获得利息的再投资风险。

连续复利：如果利率是按年复利,那么投资终值为:A(1+R)N ，如果利率对应一年复利m 次.则投资终值为：:A(1+R/m)Nm,当m趋于无穷大时所对应的利率称为连续复利，在连续复利下，可以证明数量为A的资金投资N年时，投资终值为Aern收益债券：指规定无论利息的支付或是本金的偿还均只能自债券发行公司的所得或利润中拔出的公司债券。

公司若无盈余则累积至有盈余年度始发放,这种债券大多于公司改组或重整时才发生，一般不公开发行.这种债券的利息并不固定，发期有无利润和利润大小而定，如无利润则不付息。

因此，这种债券与优先股类似.所不同的是优先股无到期日，而它需到期归还本金。

扬基债券：在美国债券市场上发行的外国债券，即美国以外的政府、金融机构、工商企业和国际组织在美国国内市场发行的、以美元为计值货币的债券。

即期利率：借贷交易达成后立即贷款形成的利率称为即期利率.即期利率是指债券票面所标明的利率或购买债券时所获得的折价收益与债券当前价格的比率。

它是某一给定时点上无息证券的到期收益率。

第一章概览1、如何理解固定收益证券？固定收益证券是指其发行人未来向其持有人支付的货币流量将符合某种合约规定的证券。

（1）代表了其持有人的一种权利；（2）发行人未来要支付一定的货币流量；（3）发行人未来支付的货币流量要符合某种合约约定；（4）发行人未来支付的货币流量并不一定是固定不变的。

2、固定收益证券有哪些种类？（1）债务证券类；（2）股权证券类的优先股；（3）资产证券化类。

3、债券的特征体现在哪些方面？（1）偿还性；（2）收益性；（3）安全性；（4）流动性。

4、说明债券与股票的联系与区别。

（1）联系：债券与股票都属于有价证券、债券与股票都是筹措资金的手段、债券和股票的收益率互相影响；（2）区别：债券与股票的权利体现不同的关系、债券与股票的发行目的不完全相同、债券与股票的期限有着不同的属性、债券与股票的债息或股息收益情况存在差异、债券与股票的风险水平不同。

5、说明债券的完整过程。

6、债券可以如何分类？第二章债券发行和偿还1、说明债券发行方式的种类。

（1）直接发行是指债券发行人不依靠中介机构，自己直接向债券认购人出售债券。

间接发行是指债券发行人依靠中介机构的帮助，由中介机构来代理完成债券发行事宜。

（2）公开募集是指债券向全社会公开发行，一般不具体指定债券发行对象或者对债券发行对象做过多的限制。

私下募集是指债券不向全社会公开发行，而是只允许特定类别的投资者承购债券。

（3）竞价发行是指由代表债券认购人方面的竞价人通过一定的竞价程序来决定债券发行价格的方式。

定价发行是指由债券发行人单方面确定债券发行价格、而后由债券认购人认购的方式。

2、债券主要的发行方式有哪些？（1）承购包销法，是指债券发行人与债券承销人签订承购包销合同，由债券承销人负责其所承受的债券发行的销售任务。

（2）招标发行法，是指债券发行人先不规定债券发行价格（或其他某一债券发行条件），由投标人直接竞价，然后发行人根据投标所产生的结果来发行债券。

1.一份20年期的债券，票息率是9%，收益率是6%，债券价格是134.41. 如果收益率增加10个基点，价格减少至132.99. 当收益率减少10个基点，价格增加至135.85. 计算债券的修正久期。

解：D*=-(dp/p)/dy=[(135.85-132.99)/134.41]/[0.001*2]=10.632.一份7年的零息债券年收益率是6.75%，6年的灵犀债券年收益率是5.87%。

计算六年后的一年期远期利率。

解：(1+R7)7 = (1+R6)6(1+F6,7), F6,7=12.19%3.普通年利率是8%，半年付息一次，那么其对应的每年付息一次的普通复利是多少。

解：(1+y s/2)2=(1+y), y = 0.08164.支付固定利息的债券，价格是102.9，1年后到期，票息率是8%。

假设利息是每半年支付一次，计算债券的到期收益率。

解：4/(1+y/2)+104/(1+y/2)2 = 102.9, y=5%5.一份两年的债券，票息率是6%每半年支付一次，收益率是5.2%。

如果麦考利久期是1.92年，其修正久期是多少。

解：D*=D/(1+y)=1.92/(1+0.052/2)=1.876.对于一份10年的par 债券（收益率=票息率），票面价值是100，其修正久期是7，凸性是50. 计算收益率增加10个基点对于债券价格的影响。

解：ΔP=-D*PΔy+(1/2)CP(Δy)2=-(7*100)(0.001)+(1/2)(50*100)(0.001)2=-0.69757.假设三年不含权债券的票面价值是1000，年票息率是10%每年付息一次，到期收益率是5%，计算其修正久期。

解：根据计算公式，久期是2.75，修正久期是2.62.8.假设一个6×9的远期利率协议（FRA），合同中的执行利率是6.35%，本金是1千万美元。

如果结算利率是6.85%，计算持有者当前的损失。

日期按照30/360计算。