Finally finished.
| High quality questions |
concept |
18TIPS, Fisher Effect and Proof of Real Rate of Return 27Using CDS to examine the difference between credit risk and interest rate risk |
| calculate |
The opening price of 9YTM may not be 1,000, which is easily miscalculated. 12Cash flow, interest accrual period and discount rate should change simultaneously 15 days of interest accrual, and the US market 1/32 quotation method |
|
| Lightning protection question |
Chinese mistranslation |
21 The number in the Chinese version is copied incorrectly, the redemption price in the material should be changed to 1100 Question 28 is mistranslated, causing students to mistakenly think it is a multiple-choice question. It is actually asking you to discuss the changes in YTM under three situations. |
| The questions are not rigorous |
Question 24 is as loose as Tsinghua’s circulating bonds last year. You can use the 30/360 accounting method and get by with the number of months. In addition, the 101:04 quotation method corresponds to the decimal method 101.125%, which has been changed in the 12th edition of the English version. 31Fifth question: The tax law is too complicated and not suitable for postgraduate entrance examinations. CFA question 2, the calculation of RCR secretly replaces semi-annual interest payments with annual interest payments CFA question 4, there is a problem with the bond price. The English version is that the price is 77 yuan, but the answer is calculated as 770 yuan, which is very confusing. Other CFA questions are of little practical significance and the quality is not as good as Bodie’s own questions. |
Base
1
Catastrophe bonds are bonds issued in response to terrorist attacks or emergencies that prevent a company from holding major events. Normally, the coupon rate is higher than that of ordinary bonds; but after a disaster, the company does not have to pay subsequent cash flows. .
Eurobonds are bonds issued in a foreign country that are denominated in their own currency.
A zero-coupon bond is a bond that has no coupon income during the holding period and receives a one-time face value income upon maturity. Bonds are issued at a discount, with all proceeds derived entirely from capital gains.
Samurai bonds are yen-denominated bonds issued in Japan by non-Japanese persons.
Junk bonds are bonds rated below B, also known as speculative-grade bonds, and are bonds with a greater risk of default. It was used in leveraged buyouts and hostile takeovers in the last century.
A convertible bond is one in which the holder can convert the bond into stock at a certain percentage. The conversion price and conversion ratio have been agreed upon, and are generally much higher than the current stock price; during conversion, the company does not issue new shares, but directly converts liabilities into equity and capital reserves. Once converted, the holder will no longer receive subsequent coupon and principal income. The coupon rate of convertible bonds is generally lower than that of ordinary bonds.
g?
h?
i?
j?
Redeemable bonds are bonds that the company can repurchase at an agreed redemption price after certain conditions are met. It is generally triggered when interest rates fall and bond prices rise. At this time, the value is only the redemption price, which is lower than the pure bond value. For the company, it is equivalent to holding a call option. Therefore, the price of callable bonds is lower than that of ordinary bonds.
Puttable bonds are bonds whose holders can recover their principal early if the price falls below a certain level. The holder is equivalent to owning a long position in a put option.
//Focus on a few unpopular products that cannot be answered to answer the multiple-choice questions:
Amortization bonds: The principal can be amortized over several repayments to reduce the pressure of principal repayment at maturity.
Equipment Covenant Bond: A bond secured by a mortgage on company-owned equipment
Discount bonds initially issued: discount bonds, issued at a price below the face value
Indexed Bond: A bond whose payment is determined by anchoring it to a price index
2 Callable bonds have a higher yield to maturity. Because when prices rise in the future, the company is likely to exercise its options, causing losses to investors. This part of the risk will be added to the required rate of return, reflected in a higher cost of capital than ordinary bonds.
intermediate
3Zero-coupon bonds have no coupon income during the period, and all income comes entirely from capital gains.
4. When market interest rates rise, according to the bond pricing formula, the present value of all cash flows will become lower, thus making the price lower. Its economic significance is: investors can invest in other products at higher interest rates, but the current fixed-income model can provide returns that are lower than market returns. Therefore, bond prices would have to fall for yields to increase in line with other products.
5The default is semi-annual payment.
24/970=0.0247
//The answer is to pay in one year. Just know it. This question is not answered well.
6
For three-month Treasury bills,
97645*(1+r)=100000
Solve (1+r)=100000/97645=1.024118
Annualized is 1.024118^4-1=10.00%
//Question: What is the actual annual interest rate in question 6 and the actual annual rate of return in question 7?
//Answer: The application of EAR and annualized HPR in investment science.
//Q: Three-month interest rate, x4 simple interest, or annual interest rate based on compound interest?
//Answer: Effective interest rate, of course EAR, requires exponentiation operation
For bonds issued at par, the interest rate during the interest period is equal to the coupon interest rate of 5%.
Therefore, the annual conversion is 1.05^2-1=10.25%
Considering the annual real interest rate, coupon bonds are higher.
7
According to the meaning of the question, for bonds issued at par, the interest rate during the interest accrual period is equal to the coupon rate during the interest accrual period of 4%.
Therefore YTM=1.04^2-1=0.0816
If the interest payment is changed to once a year and the issue is still at par, the coupon rate should be YTM8.16%
8
When the discount rate remains unchanged, since the yield to maturity is lower than the coupon rate, this is a premium bond. Within a year, the bond's price will fall.
Considering the periodic function image of discrete interest payments, the bond value will gradually move closer to the face value before interest payments.
9
To calculate the yield to maturity, only the known current price, coupon and par value models are considered.
令953.10=80*(P/A,YTM,3)+1000*(P/F,YTM,3)
The solution is YTM=9.8820%
To calculate the realized compound rate of return, everything needs to be unified to the final value for calculation.
| period |
0 |
1 |
2 |
3 |
| Current price |
-953.10 |
|||
| coupon |
80 |
80 |
80 |
|
| face value |
1000 |
|||
| Coupon reinvestment |
80*1.1=88 |
80*1.1*1.12=98.56 80*1.12=89.6 |
||
| end value |
1268.16 |
We can get the cubic equation of one variable 953.1*(1+RCR)^3=1268.16
Solve to get RCR=(1268.16/953.1)^(1/3)-1=9.987%
//Error point: not all bonds are parity bonds
You can get the cubic equation of one variable1000*(1+RCR)^3=1268.16
解得RCR=1.26816^(1/3)-1=8.2409%
10
1) According to the meaning of the question, three bonds are valued at a debt capital cost of 8%.
PV1=1000*1.08^(-10)=463.19 US dollars
The coupon rate of the second one is equal to the discount rate, so the bond is issued at par with a price of $1,000.
PV3=100*(1-1.08^(-10))/0.08+1000*1.08^(-10)=1134.20 USD
11
a
950=40*(P/A,r,40)+1000(P/A,r,40)
The solution is r=4.2626%
Equivalent annual YTM=8.5252%
Actual annual YTM=1.042626^2-1=8.7069%
b
Issue at par, r=4%
Equivalent annual YTM=8%
Actual annual YTM=8.16%
c
1050=40*(P/A,r,40)+1000(P/A,r,40)
Solve to get r=3.7565%
Equivalent annual YTM=7.513%
Actual annual YTM=1.037565^2-1=7.654%
12
//Note two points: First, for annual cash flow, APR=EAR, there is no difference between equivalent and effective
Second, the cash flow, discount rate and interest accrual period change at the same time, you cannot change just one parameter!
a
950=80*(P/A,r,20)+1000(P/A,r,20)
The solution is r=8.529%
b
Issue at par, r=8%
c
1050=80*(P/A,r,20)+1000(P/A,r,20)
The solution is r=7.509%
//Error-prone tips: cash flow, discount rate and interest period must be unified! If one changes, the rest must change!
a
950=80*(P/A,r,20)+1000(P/A,r,20)
解得r=4.3804%
Equal YearYTM=8.7608%
Real YearYTM=8.9527%
b
Plain line,r=8%
YTM=8%
YTM=8%
c
1050=80*(P/A,r,20)+1000(P/A,r,20)
解得r=3.4636%
equal yearsYTM=6.9272%
Real YearYTM=1.034636^2-1=7.0472%
The data is lower than the previous question because the frequency of cash flow payments has become lower and more of the present value depends on the uncertain present value of the future, so the expected return is lower.
13
Use a calculator to solve equations.
4.688%
3.526%
7.177%
385.54
463.19
11.9
14
Solution: According to the meaning of the question, this is a premium bond. The proof is as follows:
According to the flattened bond pricing model, B0=50*(P/A,4%,6)+1000*(P/F,4%,6)=1052.42
After the next period's coupon is ex-rights, the revaluation result is B1=50*(P/A,4%,5)+1000*(P/F,4%,5)=1044.52
Defined by HPR, the 6-month expected return is
E(HPR)=(50+1044.52-1052.42)/1052.42=4.00%
15
//Tip: First, count the number of days, excluding the last interest payment date, but including today. The tail is not the beginning.
Second, 1/32 is added to the percentage, but it actually has to be divided by 100.
The accrued interest is 35*(15/182)=2.8846
Full price=1000*(100+2/32)/100+2.88=1003.505
全价=1000*(1+2/32)+2.88=1065.38
16
According to the bond theorem, this is a discount bond. The yield to maturity is higher than the current yield because YTM includes potential capital gains. Only discount bonds can continue to appreciate to their face value over time.
17
According to the bond theorem, a high yield to maturity means a coupon rate below 9%.
18
The data for inflation-protected bonds are as follows:
Defined by HPR, the nominal rate of return in the second year is
HPR2=(42.02+1020*0.03)/1020=0.0712
Due to the Fisher effect, the actual rate of return in the second year is
r=(1+0.0712)/(1+0.03)-1=0.04
Defined by HPR, the nominal rate of return in year 3 is
HPR3=(42.02+1050.6*0.01)/1050.6=0.0500
According to the Fisher effect, the actual rate of return in the third year is
r=(1+0.0500)/(1+0.01)-1=0.04
It can be proved that:
The coupon and face value of inflation-protected bonds increase with the rate of inflation, resulting in a higher nominal yield. Taking into account the Fisher effect, the actual yield remains the same as the originally promised coupon income.
19
| period |
0 |
1 |
2 |
19 |
20 |
| Expected price when YTM remains unchanged |
214.55 |
231.71 |
250.25 |
925.93 |
1000 |
| taxable interest income |
17.16 |
18.54 |
74..07 |
20
Solution: For discount bonds, YTM is higher than the coupon rate.
800=40*(P/A,YTM,10)+1000*(P/F,YTM,10)
Solve to get YTM=0.0682
Based on this calculation, the price after ex-dividend next year is
40*(P/A,YTM,9)+1000*(P/F,YTM,9)=814.86
Therefore, the taxable income is 814.86-800+40=54.86
//taxable income≠incurred interest
So, the taxable income is14.86
21
//The number in the question is wrong, the redemption price is 1100.
1) First use YTM to calculate the bond price.
V0=40*(1-1.035^(-60))/0.035+1000*1.035^(-60)=1124.72
Consider redeeming it after 5 years, you can get
1124.72=40*(P/A,YTC,10)+1100*(P/F,YTC,10)
The semi-annual redemption yield is 3.37%
Therefore, the redemption yield is 6.74%
2)
1124.72=40*(P/A,YTC,10)+1050*(P/F,YTC,10)
The solution is half-year YTC=2.98%
Therefore, the total YTC=5.96%
3)
1124.72=40*(P/A,YTC,4)+1100*(P/F,YTC,4)
The solution is half-year YTC = 3.03%
Therefore, the total YTC=6.06%
It shows that the earlier you redeem, the lower the YTC will be.
22
Calculate Stated yield to maturity using original contract data
900=140*(P/A,SYTM,10)+1000*(P/F,SYTM,10)
The solution is SYTM=16.07%
Calculate expected yield to maturity using new contract data
900=70*(P/A,EYTM,10)+1000*(P/F,EYTM,10)
Solve to get EYTM=8.53%
23
Solution: According to the meaning of the question, bonds are issued at par, YTM=C=10%
Create a table of expected cash flows as follows
| period |
0 |
1 |
2 |
||
| selling price |
-1000 |
||||
| coupon |
100 |
100 |
|||
| principal |
1000 |
||||
| 1 period coupon reinvestment |
100*1.08 |
100*1.1 |
100*1.12 |
||
| end value |
1208 |
1210 |
1212 |
||
| RCR |
9.91% |
10% |
10.09% |
24
The list of interest accrual days is as follows:
| January |
16 |
| February |
28 |
| 3 |
31 |
| 4 |
15 |
| 5 |
31 |
| 6 |
30 |
| 7 |
15 |
| TOTAL |
181 |
Accrued interest 50*90/181=24.86
//Tip: 101:04, the following colon is the American 1/32 quotation method, (101+4/32)/100=101%+(4/32)%=101.125%
Starting from the 12th edition, Bodie's textbooks have been completely changed to the international decimal point quotation method.
Transaction price=24.86+1011.25=1036.11
//Q: Can I take it directly for 182 days?
//Answer: Barely ok, but not in practice. However, this question can be passed by using the 30/360 accounting method and directly mentioning the number of months. It’s the same as Tsinghua’s 2022 bond question.
25
First, the ABC issuance period is shorter, the waiting time to recover the face value is shorter, and the risk is lower.
Second, ABC is guaranteed, so default rates are lower regardless of the original credit rating.
Third, ABC bonds are not redeemable, so they are more conducive to protecting the rights of holders.
Fourth, the low coupon rate is because the value already includes the value of the holder's long call option on the bond.
//There is also a fifth point in the standard answer: large-amount issuance has stronger liquidity
26
//Note: Please note that this question is out of scope and examines the liquidity of the swap market that was not mentioned.
Consider the probability distribution function of the expiration price.
If there is a default and the price falls, both CDS and short selling can be profitable.
However, if there is no default and the price rises, the CDS will be invalid, but the short-selling transaction will have pressure to close the position and suffer losses.
To sum up, choose a
27A
Credit default swaps transfer the risk of default, not the risk of rising or falling interest rates.
//Q: Where does interest rate risk come from?
//A: Bond price and RCR yield part.
28
Interest coverage ratio = EBIT/interest ↑, then short-term solvency ↑, risk ↓, yield to maturity ↓
Equity ratio = B/S↑, then long-term solvency ↓, risk ↑, yield to maturity ↑
Current ratio = current assets/current liabilities ↑, then short-term solvency ↑, risk ↓, yield to maturity ↓
//The Chinese version of the question is mistranslated and is not a multiple-choice question at all. If it is a multiple-choice question, the reference answers are as follows:
The equity ratio and current ratio are both stocks, and the interest coverage ratio is a flow, which is directly related to debt solvency and will affect the relationship between expected YTM and committed YTM.
29
1) The coupon cash flow of floating interest rate will change with the interest rate, thus stabilizing the present value. The coupon cash flow of a fixed interest rate will not change, so when market interest rates fluctuate greatly, the price will also change greatly.
2)//The question is about the sources of risks of floating rate bonds, which are all related to time series and probability theory models. Just look at the subscript to answer.
First, the interest rate spread relative to government bonds may widen, thereby increasing the opportunity cost for investors.
Second, the company's credit risk may increase, causing its solvency to be questioned when interest rates rise.
Third, even floating interest rates do not change at any time. Contracts generally change coupons once a year, and the risk of price fluctuations during the period cannot be eliminated.
3)//Questions 3-4 compare the redemption probabilities of callable bonds under fixed interest rates and floating interest rates. The essence is binary function analysis.
For bonds to be redeemed, interest rates must fall and bond prices must rise. For fixed cash flows, redemptions may increase as the discount rate falls. However, if cash flows are also changing, it will be difficult for the price to significantly exceed the redemption price.
Therefore, fixed-rate bonds must be issued at a discount, corresponding to higher YTM compensation risk.
5) If issued at par, the coupon rate should be set at 9.9% of the pre-tax debt capital cost
6)未来现金流不确定，是多个随机变量的函数，无法用黎曼微积分计算。更为准确的指标是yield to recoupon
30
1)推测题意是年付息。平价债券YTM=C=8.75%
对折价债券，
580=40*(P/A,YTM,20)+1000*(P/F,YTM,20)
解得YTM=8.4087%
如果不考虑赎回条款，平价债券的YTM应该低于折价债券。这里YTM高于折价债券，原因是：
首先，从公司金融学来看，利率下跌同样水平，平价债券更容易触发赎回条款，给投资人带来损失。
其次，从投资学来看，同样触发赎回条件，平价债券的资本利得和资本利得率明显低于折价债券，因此需要用更高的stated/promised YTM补偿。
2)HPR=CY+CG
如果利率波动极大，价格涨幅会很高。但是，平价债券会被赎回，从而有价格上限。因此，投资者的最佳策略是购买折价债券。
3)赎回条款利于发行人、不利投资者。折价使可赎债的被赎回概率大幅下降，因为在极端情况下，也尚需时日才有可能触发赎回。
高级题
31
//中文版的数字又抄错了，YTM是8%，不改的话白算半个小时……
1)到期收益率高于票面利率，该债券为折价债券。
P0=50*(P/A,8%,20)+1000*(P/F,8%,20)=705.46
P1=50*(P/A,7%,19)+1000*(P/F,7%,19)=793.29
故HPR=(50+793.29-705.46)/705.46=32.83%
2)//从2-3对应代数法。也可以用财务学的表格法算账。
一年后constant YTM price=P'1=50*(P/A,8%,19)+1000*(P/F,8%,19)==711.89
implicit interest = 711.89-705.46=6.43
根据OID原始发行折旧税法，
应缴利息所得税=(50+6.43)*0.4=22.57
又∵应缴资本利得税=(793.29-711.89)*0.3=24.42
∴总纳税额22.57+24.42=46.99元
3)税后HPR=(50+(793.29-705.46)-46.99)/705.46=12.88%
//法二：会计表格法
根据OID原始发行折旧法，对该投资者现金流建模如下
| 科目 |
现金流 |
适用税率 |
税后现金流 |
| 利息收入 |
50 |
0.4 |
30 |
| 隐含利息 |
711.89-705.46=6.43 |
0.4 |
3.86 |
| 资本利得 |
793.29-711.89=81.4 |
0.3 |
56.98 |
| 总计收益 |
|
|
90.84 |
税后HPR=90.84/705.46=12.88%
数学证明很简单，把税前税后现金流剥离，把应该交的税单独考察，实际就是代数法的思路。
关键在于，算HPR，资本利得减去的是真实期初成本；算资本利得税，减的是YTM不变的情况下的隐含价格。
4)以7%的YTM计算，P2=50*(P/A,7%,18)+1000*(P/F,7%,18)=798.82
如果用实现的复合收益率代替到期收益率，列现金流如下
| 时期 |
0 |
1 |
2 |
| 息票收入 |
50 |
50 |
|
| 债券变现 |
798.82 |
||
| 利息再投资 |
50*1.03=51.5 |
||
| 终值 |
900.32 |
解一元二次方程：
705.46*(1+RCR)^2=900.32
得RCR=12.79%
CFA考题
1
asinking fund是为了保障本金的偿付，要求发行人定期赎回一定数量债券的条款。赎回价格是市场价格和面值的较低者。
bsinking fund会要求将债券发行拆分为不同到期日的债券，因此预期久期会缩短。由于偿债基金的赎回价格是市值和面值的较低者，因此有可能最终的面值和利息收益比承诺的低。
//不改变面值和利息，只改变支付时间。和股利无关论的思路一致。
c偿债基金虽然降低了最终的收益，但是降低了发行人的还本压力，因此信用风险更低；偿债基金的赎回条款，保证了在利率上行的时候，可以及时从发行人处收回本金以作其他投资。
2
当期收益率为35/960=3.65%→7.3%
缺点在于，没有考虑货币的时间价值和利息偿付的风险。
到期收益率：960=35*(P/A,r,10)+1000*(P/F,r,10)=4%→8%
缺点在于：假设期间YTM不发生变化，息票收入可以按YTM一直再投资
三年后的债券价格是
35*(P/A,4%,4)+1000*(P/F,4%,4)=981.85
普通年金终值系数为
(1.03^6-1)/0.03=6.4684
现金流列表如下
| 时期 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
| 息票收入 |
35 |
35 |
35 |
35 |
35 |
35 |
|
| 变现所得 |
981.85 |
||||||
| 息票再投资年金终值 |
226.39 |
||||||
| 终值 |
1208.24 |
设960*(1+r)^6=1208.24
解得r=3.91%
所以年实现综合收益率为7.82%，说明由于再投资利率低，RCR低于YTM。
实现复合收益率的缺点是，假设用当前的YTM估计未来价格，实际是假设YTM不变
三种指标的缺点
| CY |
YTM |
RCR |
| 没考虑资本利得，HPR不完整 没考虑再投资风险 |
直接用到期时间代替了持有期 而且假设YTM不变 |
有几个指标很难估值： 再投资率，到期YTM，持有期 |
//提示：题目不严谨，RCR默认偷换成了年付息的平价债券……
3
//补充条件：半年付息
由题意，这两个都是新债。
1)市场利率下跌100BP后
S的价格为3*(P/A,2.5%,20)+100*(P/F,2.5%,20)=107.80
C的价格为3.1*(P/A,2.6%,20)+100*(P/F,2.6%,20)=106.18
但C是可赎债，因此价格不会超过赎回价格102
2)如果选择可赎债，投资者预期利率会上升，价格会下跌，不会触发赎回条款。
3)如果利率波动加剧，特别是利率下行的时候，债券价格会暴涨，B债券触发赎回可能性增强，因而价格下跌。
//标答：投资学视角：可赎债价值=纯债券价值+期权价值，因此波动性增大的时候，期权价值↑
6
//第一问问的是HPR和YTM 的关系，和前面比较RCR HPR YTM CY CG的题目思路一样
D
C
C
B