公司理财：利率和债券估值

债券估值方法
债券估值有着重要的作用，既可以为投资者提供投资参考，又能为发行者提供有可靠的报酬。

债券估值涉及市场利率、风险、特殊情况以及财政政策的计算，主要有三种估值方法，即久期合计法、折现法和实施估值法。

久期合计法是最容易懂的方法，从原理上来说，它从债券本金、利息总和以及债券存续期内的税收信息出发，将所有应得款项进行累计，得出该债券的总值。

这种方法十分方便，但适合于波动性比较小的中小投资者。

折现法是另一种经常被投资者采用的债券估值方式，它以债券到期时的价值为基础，计算债券目前的价值。

折现法的优势在于能够反应市场的变动，它考虑到了投资者的预期回报以及债券的拥有者所处的非同一投资期。

最后，实行估值法也是一种常用的债券估值方法，其基本原理是，计算投资者的潜在现金流，折算为当前价值，最后根据市场变动进行调整。

这种方式在预测宏观经济状况、财政政策等市场未来发展方面有各种利弊。

债券估值是一项负责任的金融任务，不仅涉及到投资者的投资参考，还与财政政策密切相关。

详尽的债券估值方法可以有助于投资者更好地了解财务市场，并选择合适的投资机会。

一、单选题1、关于股利稳定增长模型，下列表述错误的是（）。

A.每股股票的预期股利越高，股票价值越大B.股利增长率越大，股票价值越大C.股利增长率为一常数，且折现率大于股利增长率D.必要收益率越小，股票价值越小正确答案：D2、一个投资人持有ABC公司的股票，他的投资必要报酬率为15%。

预计ABC公司未来3年股利分别为0.5元、0.7元、1元。

在此以后转为正常增长，增长率为8%。

则该公司股票的内在价值为（）元。

A.10.08B.12.2C.11.77D.12.08正确答案：C3、某企业于2017年6月1日以10万元购得面值为1000元的新发行债券100张，票面利率为8%，2年期，每年支付一次利息，则2017年6月1日该债券到期收益率为（）。

A.16%B.8%C.12%D.10%正确答案：B4、投资者李某计划进行债券投资，选择了同一资本市场上的A和B 两种债券。

两种债券的面值、票面利率相同，A债券将于一年后到期，B债券到期时间还有半年。

已知票面利率均小于市场利率。

下列有关两债券价值的说法中正确的是（）。

（为便于分析，假定两债券利息连续支付）A.债券A的价值较高B.两只债券的价值相同C.两只债券的价值不同，但不能判断其高低D.债券B的价值较高正确答案：D5、ABC公司以平价购买刚发行的面值为1000元，5年期，每半年支付利息40元的债券，该债券按年计算的有效到期收益率为（）。

A.7.84%B.4%C.8%D.8.16%正确答案：D6、埃米特公司发行了面值1000元的零息债券。

如果适当的贴现率为4%，且债券在8年内到期，下列最接近债券的正确价格的值是（）。

A.1032.00元B.730.69元C.1000.00元D.968.00元正确答案：B7、Alpha普通股的股息收益率为5.2%。

该公司刚刚支付了2.10美元的股息。

明年的股息为2.30美元。

预计股息增长率将保持在当前水平不变。

Alpha股票的回报率是（）。

债券估值三种模型计算公式1. 利息法债券估值模型中最常用的方法是利息法，它基于债券的利息收益来计算现值。

利息法可以通过以下公式计算：PV = C1/(1+r)^1 + C2/(1+r)^2 + … + Cn/(1+r)^n +FV/(1+r)^n其中，PV是当前债券的现值，C是每年付息的现金流，FV是债券到期后的回收价值，r是市场利率。

这个公式的原理是假设债券发行人将每年所支付的利息以及回收本金持续投资在市场上，因此每个现金流必须通过现值化来反映其贡献。

这个方法的一个优点是它比较简单，同时考虑了利息和回收本金，但它也有一些明显的缺点。

例如，它不考虑通货膨胀和信用风险。

2. 折现法折现法是另一种常用的债券估值方法。

这个方法假设投资者采用市场上当前的无风险利率来贴现债券的预期现金流。

折现法可以通过以下公式计算：PV = (C1/1+R) + (C2/ (1+r)^2) + ... + (Cn/(1+r)^n) +FV/(1+r)^n其中，PV是当前债券的现值，C是每年付息的现金流，FV是债券到期后的回收价值，r是市场利率，R是无风险收益率。

这个方法的主要优点是它更好地体现了现金流在市场上的时间价值和利率水平。

折现法从概念上类似于银行互贷，但将利率转化为股息和回收金额，使其更适用于债券评估。

3. 市场价值法市场价值法假设市场上其他类似的债券可以提供与当前债券相似的风险和收益，因此它的估值可以从市场上类似的债券价格得出。

这个方法可以通过以下公式计算：PV = M * P其中，PV是当前债券的现值，M是债券面值，P是市场上债券的价格。

尽管这个方法相对来说较简单，但它有一个重要的缺点：如果市场上找不到类似的债券，它就会失去效力。

此外，如果市场上债券价格波动，那么这个方法的应用也将受到限制。

第八章：利率和债券估值1. a. P = $1,000/(1 + .05/2)⌒20 = $610.27b. P = $1,000/(1 + .10/2)⌒20 = $376.89c. P = $1,000/(1 + .15/2)⌒20 = $235.412.a. P = $35({1 – [1/(1 + .035)]⌒50 } / .035) + $1,000[1 / (1 + .035)⌒50]= $1,000.00When the YTM and the coupon rate are equal, the bond will sell at par.b. P = $35({1 – [1/(1 + .045)]⌒50 } / .045) + $1,000[1 / (1 + .045)⌒50]= $802.38When the YTM is greater than the coupon rate, the bond will sell at a discount.c. P = $35({1 – [1/(1 + .025)]⌒50 } / .025) + $1,000[1 / (1 + .025)⌒50]= $1,283.62When the YTM is less than the coupon rate, the bond will sell at a premium.3. P = $1,050 = $39(PVIFAR%,20) + $1,000(PVIFR%,20) R = 3.547%YTM = 2 *3.547% = 7.09%4. P = $1,175 = C(PVIFA3.8%,27) + $1,000(PVIF3.8%,27) C = $48.48年收益：2 × $48.48 = $96.96则票面利率：Coupon rate = $96.96 / $1,000 = .09696 or 9.70%5. P = €84({1 – [1/(1 + .076)]⌒15 } / .076) + €1,000[1 / (1 + .076)⌒15] = €1,070.186. P = ¥87,000 = ¥5,400(PVIFAR%,21) + ¥100,000(PVIFR%,21) R = 6.56%7. 近似利率为：R = r + h= .05 –.039 =.011 or 1.10%根据公式(1 + R) = (1 + r)(1 + h)→(1 + .05) = (1 + r)(1 + .039)实际利率= [(1 + .05) / (1 + .039)] – 1 = .0106 or 1.06%8. (1 + R) = (1 + r)(1 + h)→R = (1 + .025)(1 + .047) – 1 = .0732 or 7.32%9. (1 + R) = (1 + r)(1 + h)→h = [(1 + .17) / (1 + .11)] – 1 = .0541 or 5.41%10. (1 + R) = (1 + r)(1 + h)→r = [(1 + .141) / (1.068)] – 1 = .0684 or 6.84%11. The coupon rate is 6.125%. The bid price is:买入价= 119:19 = 119 19/32 = 119.59375%⨯ $1,000 = $1,195.9375The previous day‘s ask price is found by:pr evious day‘s ask price = Today‘s asked price – Change = 119 21/32 – (–17/32) = 120 6/32 前一天的卖出价= 120.1875% ⨯ $1,000 = $1,201.87512.premium bond当前收益率= Annual coupon payment / Asked price = $75/$1,347.1875 = .0557 or 5.57% The YTM is located under the ―Asked yield‖column, so the YTM is 4.4817%.Bid-Ask spread = 134:23 – 134:22 = 1/3213.P = C(PVIFAR%,t) + $1,000(PVIFR%,t)票面利率为9%：P0 = $45(PVIFA3.5%,26) + $1,000(PVIF3.5%,26) = $1,168.90P1 = $45(PVIFA3.5%,24) + $1,000(PVIF3.5%,24) = $1,160.58P3 = $45(PVIFA3.5%,20) + $1,000(PVIF3.5%,20) = $1,142.12P8 = $45(PVIFA3.5%,10) + $1,000(PVIF3.5%,10) = $1,083.17P12 = $45(PVIFA3.5%,2) + $1,000(PVIF3.5%,2) = $1,019.00P13 = $1,000票面利率为7%：P0 = $35(PVIFA4.5%,26) + $1,000(PVIF4.5%,26) = $848.53P1 = $35(PVIFA4.5%,24) + $1,000(PVIF4.5%,24) = $855.05P3 = $35(PVIFA4.5%,20) + $1,000(PVIF4.5%,20) = $869.92P8 = $35(PVIFA4.5%,10) + $1,000(PVIF4.5%,10) = $920.87P12 = $35(PVIFA4.5%,2) + $1,000(PVIF4.5%,2) = $981.27P13 = $1,00014.PLaurel = $40(PVIFA5%,4) + $1,000(PVIF5%,4) = $964.54PHardy = $40(PVIFA5%,30) + $1,000(PVIF5%,30) = $846.28Percentage change in price = (New price -Original price) / Original price△PLaurel% = ($964.54 -1,000) / $1,000 = -0.0355 or -3.55%△PHardy% = ($846.28 -1,000) / $1,000 = -0.1537 or -15.37%If the YTM suddenly falls to 6 percentPLaurel = $40(PVIFA3%,4) + $1,000(PVIF3%,4) = $1,037.17PHardy = $40(PVIFA3%,30) + $1,000(PVIF3%,30) = $1,196.00△PLaurel% = ($1,037.17 -1,000) / $1,000 = +0.0372 or 3.72%△PHardy% = ($1,196.002 -1,000) / $1,000 = +0.1960 or 19.60%15. Initially, at a YTM of 10 percent, the prices of the two bonds are:P Faulk = $30(PVIFA5%,16) + $1,000(PVIF5%,16) = $783.24P Gonas = $70(PVIFA5%,16) + $1,000(PVIF5%,16) = $1,216.76If the YTM rises from 10 percent to 12 percent:P Faulk = $30(PVIFA6%,16) + $1,000(PVIF6%,16) = $696.82P Gonas = $70(PVIFA6%,16) + $1,000(PVIF6%,16) = $1,101.06Percentage change in price = (New price – Original price) / Original price△PFaulk% = ($696.82 -783.24) / $783.24 = -0.1103 or -11.03%△PGonas% = ($1,101.06 -1,216.76) / $1,216.76 = -0.0951 or -9.51%If the YTM declines from 10 percent to 8 percent:PFaulk = $30(PVIFA4%,16) + $1,000(PVIF4%,16) = $883.48PGonas = $70(PVIFA4%,16) + $1,000(PVIF4%,16) = $1,349.57△PFaulk% = ($883.48 -783.24) / $783.24 = +0.1280 or 12.80%△PGonas% = ($1,349.57 -1,216.76) / $1,216.76 = +0.1092 or 10.92%16.P0 = $960 = $37(PVIFAR%,18) + $1,000(PVIFR%,18) R = 4.016% YTM = 2 *4.016% = 8.03%Current yield = Annual coupon payment / Price = $74 / $960 = .0771 or 7.71% Effective annual yield = (1 + 0.04016)⌒2 – 1 = .0819 or 8.19%17.P = $1,063 = $50(PVIFA R%,40) + $1,000(PVIF R%,40) R = 4.650% YTM = 2 *4.650% = 9.30%18.Accrued interest = $84/2 × 4/6 = $28Clean price = Dirty price – Accrued interest = $1,090 – 28 = $1,06219.Accrued interest = $72/2 × 2/6 = $12.00Dirty price = Clean price + Accrued interest = $904 + 12 = $916.0020.Current yield = .0842 = $90/P0→P0 = $90/.0842 = $1,068.88P = $1,068.88 = $90{[(1 – (1/1.0781)⌒t ] / .0781} + $1,000/1.0781⌒t $1,068.88 (1.0781)⌒t = $1,152.37 (1.0781)⌒t – 1,152.37 + 1,000t = log 1.8251 / log 1.0781 = 8.0004 ≈8 years21.P = $871.55 = $41.25(PVIFA R%,20) + $1,000(PVIF R%,20) R = 5.171% YTM = 2 *5.171% = 10.34%Current yield = $82.50 / $871.55 = .0947 or 9.47%22.略23.P: P0 = $90(PVIFA7%,5) + $1,000(PVIF7%,5) = $1,082.00P1 = $90(PVIFA7%,4) + $1,000(PVIF7%,4) = $1,067.74Current yield = $90 / $1,082.00 = .0832 or 8.32%Capital gains yield = (New price – Original price) / Original priceCapital gains yield = ($1,067.74 – 1,082.00) / $1,082.00 = –0.0132 or –1.32%D: P0 = $50(PVIFA7%,5) + $1,000(PVIF7%,5) = $918.00P1 = $50(PVIFA7%,4) + $1,000(PVIF7%,4) = $932.26Current yield = $50 / $918.00 = 0.0545 or 5.45%Capital gains yield = ($932.26 – 918.00) / $918.00 = 0.0155 or 1.55%24. a.P0 = $1,140 = $90(PVIFA R%,10) + $1,000(PVIF R%,10) R = YTM = 7.01%b.P2 = $90(PVIFA6.01%,8) + $1,000(PVIF6.01%,8) = $1,185.87P0 = $1,140 = $90(PVIFA R%,2) + $1,185.87(PVIF R%,2)R = HPY = 9.81%The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped by 1 percent; bond prices rise when yields fall.25.PM = $800(PVIFA4%,16)(PVIF4%,12)+$1,000(PVIFA4%,12)(PVIF4%,28)+ $20,000(PVIF4%,40) PM = $13,117.88Notice that for the coupon payments of $800, we found the PV A for the coupon payments, and then discounted the lump sum back to todayBond N is a zero coupon bond with a $20,000 par value; therefore, the price of the bond is the PV of the par, or:PN = $20,000(PVIF4%,40) = $4,165.7826.(1 + R) = (1 + r)(1 + h)1 + .107 = (1 + r)(1 + .035)→r = .0696 or 6.96%EAR = {[1 + (APR / m)]⌒m }– 1APR = m[(1 + EAR)⌒1/m – 1] = 52[(1 + .0696)⌒1/52 – 1] = .0673 or 6.73%Weekly rate = APR / 52= .0673 / 52= .0013 or 0.13%PVA = C({1 – [1/(1 + r)]⌒t } / r)= $8({1 – [1/(1 + .0013)]30(52)} / .0013)= $5,359.6427.Stock account:(1 + R) = (1 + r)(1 + h) →1 + .12 = (1 + r)(1 + .04) →r = .0769 or 7.69%APR = m[(1 + EAR)1/⌒1/m– 1]= 12[(1 + .0769)⌒1/12– 1]= .0743 or 7.43%Monthly rate = APR / 12= .0743 / 12= .0062 or 0.62%Bond account:(1 + R) = (1 + r)(1 + h)→1 + .07 = (1 + r)(1 + .04)→r = .0288 or 2.88%APR = m[(1 + EAR)⌒1/m– 1]= 12[(1 + .0288)⌒1/12– 1]= .0285 or 2.85%Monthly rate = APR / 12= .0285 / 12= .0024 or 0.24%Stock account:FVA = C {(1 + r )⌒t– 1] / r}= $800{[(1 + .0062)360 – 1] / .0062]}= $1,063,761.75Bond account:FVA = C {(1 + r )⌒t– 1] / r}= $400{[(1 + .0024)360 – 1] / .0024]}= $227,089.04Account value = $1,063,761.75 + 227,089.04= $1,290,850.79(1 + R) = (1 + r)(1 + h)→1 + .08 = (1 + r)(1 + .04) →r = .0385 or 3.85%APR = m[(1 + EAR)1/m– 1]= 12[(1 + .0385)1/12– 1]= .0378 or 3.78%Monthly rate = APR / 12= .0378 / 12= .0031 or 0.31%PVA = C({1 – [1/(1 + r)]t } / r )$1,290,850.79 = C({1 – [1/(1 + .0031)]⌒300 } / .0031)C = $6,657.74FV = PV(1 + r)⌒t= $6,657.74(1 + .04)(30 + 25)= $57,565.30。