FRM Level 1 Part 4 Flashcards
chapter 1: fundamental fund and valuation
what the bond’s features are?
- issuer: government/sovereign or corporate
- face value = par value = principle (通常为100或1000)
- maturity:
a. money market securities (T <= 1 year) e.g. T-bills
b. capital market securities (T > 1 year) e.g. T-notes (1-10 years), T-bond (> 10 years)
c. perpetual bonds (永续债券) P = coupon / r - coupon(rate: a coupon rate is the yield paid by fixed- income security; Frequency: annual and semiannual(需要转为年化的))
- currency denomination (币种)
- quotation
1. T-bills discount rate: price = (1 - discount rate x t / T)*FV 2. T-notes and T-bonds 32nds： 90-5 表示 90+5/32, 即每100的面值对应90+5/32的限制
- full price (daily price) = flat price (clean price) + AI
4. AI = t*PMT/T Government Bond (actual/actual) corporate bond (30/360) Money market (Actual/360) t为上一次付息日距离现在的时间，T为两个付息日之间的时间间隔
- bond pricing
1. 计算付息节点的bond price (dirty price) 五因素 P = the sum of (Ct/(1 + r)^t + FV/(1 + r)^n) [t=1 to n] FV, PV: inflow 为正，outflow 为负 N，I/Y，PMT： 注意frequency
2. 计算两个付息日之间的price 算出AI 五因素法算出题目要求的时间点上一时期的daily price 算出来的daily price 复利到题目要求的时间点 如果要求Clean price 就减去AI
根据计算Coupon和discount rate
coupon rate < discount rate —— discount bond
coupon rate = discount rate —— par bond
coupon rate > discount rate —— premium bond
- bond price with different rate
different interest rate
spot rate/zero rate
definition: the yield to maturity on a zero coupon bond
实际应用：Treasury STIRPS
Classification:
1. coupon or interest STIRPS are called C-STIRPS—–short-term (long-term) C-STRIPS often trade rich (cheap)
- principle STRPS are called P-STRPS——P-STRPS可能会继承原债券的特性
advantage: 可以构建任何组合的现金流
swap rate
a swap rate is the rate of the fixed leg of a swap as determined by its particular market and the parties involved
forward rate
the interest rate on a bond or money market instrument traded in a forward market
par rate
the coupon rate which bond is price at par value
yield to maturity
the internal rate of return on the cash flow
- discount factor
definition: 把不同期限的即期利率折现的式子看成一个整体，即折现因子
the relationship between interest rate and discount rate
1. spot rate
2. swap rate —– 在0时刻， PV 固定 = PV 浮
3. par rate
转换步骤：
a. 使用boostrappping 的方法，以半年复利为例，先用半年的平价债券的par rate，通过折现到0时刻的式子求出半年的折现因子D（0.5）
b. 用D（0.5）和一年期的平价债券的par rate, 通过折现到0时刻的的式子求出一年的折现因子D（1）
c. 重复上述步骤就可以求出更长的折现因子
- the relationship between different interest rate
- forward rate and spot rate
离散：(1+Z)^A x (1 + IFR)^(B-A) = (1+Z)^B
IFR(A,B-A) is a forward rate that starts in period A and ends in B
连续：Rf = (R2T2 - R1T1)/(T2-T1)
- Forward rate, spot rate, and par rate
总结：
Spot curve increase:
forward > spot > par
spot curve increase:
forward < spot < par
- YTM and spot rate
YTM is a kind of average of all the spot rates
p = CF1/(1 + YTM) + … + CFn/(1 + YTM)^n
=CF1/(1 + Z1) + … + CFn/(1 + Zn)^n
coupons effect : fair price bond with the same maturity but different coupons have different YTM
spot curve increase:
coupon rate 越大，YTM 越小
spot curve decrease:
coupon rate 越大，YTM越小 yield > spot > forward
4. spot rates, YTM and forward rate upward-slopping term spot curve increase: forwspot curve decrease: yield > spot > forward ard>spot >yield
- bond pricing
- spot rate
p = sum(C/(1 + Z)^t) + FV/(1 + Zn) - forward rate
p = CF/(1 + F0,1) +CF2/(1 + F0,1)(1 + F1,2) + …+ CFn/(1 + F0,1)(1 + F1,2)(…)(1 + Fn-1, N)
the impact of maturity on bond price (if term of rate unchange)
coupon rate > forward rate, 则 maturity increase， bond price increase
coupon rate < forward rate, 则 maturity decrease， bond price decrease
3. YTM p = sum(PMT/(1 + YTM)^i) + F/(1 +YTM)^n YTM 与 bond price 反相关： coupon rate > YTM: at premium coupon rate = YTM at par coupon rate < YTM at discount "put to par" effect : if no default and the yield keep constant, t increase, bond price tend to be the par value
- the spread of the bond
definition: spread refers to the difference between two prices, rates or yields
the market price of any securities can be thought of as its value computed using some item structure of interest rate plus a spread
p = CF1/(1 + Z1 + spread) +…+ CFn/(1 + Zn + spread)^n
- bond replication
law of one price: identical future cash flows should have
arbitrage: a type of transaction undertaken when two assets produce identical result but sell for different prices
用两个债券复制第三个债券
- 匹配现金流
- 市值权重相加等于1，列出第二个方程
- 解方程并根据权重计算出第三个债券的价值。判断是否能够套利
- curve shift
- parallel shift (平行移动)
all the points in the curve move the same number of basic points - non-parallel ship (非平行移动)
different points in the curve move the different number of basic points
- others
- annuity 在一段时间内按照一定频率支付固定金额
- perpetuity
perpetuity = C/y
a security that pays coupons forever - gross realized return
R = (P[t+1] + c - P[t])/P[t] - gross realized net return
GRNR = gross realized return - financial cost - total price appreciation
carry roll-down: the price changes due to the passage of time but with no change in the spread:
the realized forward scenario (forward rates = future rates)
the unchanged term structure scenario (term structure keep constant)
the unchanged yields scenario (bond’s yield keep constant)
rate change: the price changes due to rates changing
spread change: the prices changes due to the bond’s individual spread
chapter 3: one factor risk metric and hedges
1. duration
1.1. identification: the sensitivity of bond’s full price to changes in benchmark interest rates
1.2 classification page 5 Macaulay duration(Dmac): the weighted average time to receipt of the bond's promised payments 它是用来对债券进行具体的数值分析，以衡量其价格对利率(或收益率)变动的敏感程度的一个指标。
方法讲解
将债券未来各部分现金流入量的到期时间分别加权后再汇总，权重是各个现金流入量的现值，然后用这个加权的总到期时间除以所有的现金流量现值之和(即债券的价格)，得出的就是麦考利持续期的数值。这个数值，表面上看是该债券收益的一种平均到期时间，而奇妙之处在于，它又是债券价格对收益率变化敏感性的比例系数。要知道利率(收益率)变动时债券价格的反应，只要用麦考利持续期数值来乘以收益率变化量就可以了。假定某种债券的麦考利持续期是10，该债券的收益率在瞬间要从9%升至9．10%，那么收益率的变化是0．10，10乘以0．10，得1，这个数字就是该债券价格变动的百分比数值。也就是说，当某债券的收益率可能要上升10个基点(0．1个百分点)的时候，如果债券的麦考利持续期是10，那么它的价格将下降1%。上面提到过，债券的收益率与其价格总是反方向运动的，所以上述计算过程列成公式时必须加上一个负号。
公式：y = 收益率 P = sum(Ct/(1 + y)^t) Dmac = sum(t x Ct/(1 +y)^t)/P
小结
由此可见，债券的麦考利持续期越大，它的价格对收益率变动的敏感性就越强。
modified duration(Dmod): provides an linear estimate of the percentage price change for a bond given a change in yield 修正持久期是衡量价格对收益率变化的敏感度的指标。在市场利率水平发生一定幅度波动时，修正持久期越大的债券，价格波动越大(按百分比计)。
对于给定的到期收益率的微小变动，债券价格的相对变动与其麦考利久期成比例。当然，这种比例关系只是一种近似的比例关系，它的成立是以债券的到期收益率很小为前提的。为了更精确地描述债券价格对于到期收益率变动的灵敏性，又引入了修正久期模型（Modified Duration Model）。修正久期被定义为：
Dmad = Dmac/(1 + y)
公式推导：将债券价格公式看作P与1+y之间的函数，同时求导，在同除P; 债券价格变化的百分比恰好等于修正久期与债券到期收益率变化的乘积。因此，修正久期可以用来测度债券在利率变化时的风险暴露程度。
从这个式子可以看出，对于给定的到期收益率的微小变动，债券价格的相对变动与修正久期之间存在着严格的比例关系。所以说修正久期是在考虑了收益率项y的基础上对 Macaulay久期进行的修正，是债券价格对于利率变动灵敏性的更加精确的度量。
修正久期大抵抗利率上升风险弱，抵抗利率下降风险能力强；久期小抵抗利率上升风险能力强，抵抗利率下降风险能力弱。
当我们判断当前的利率水平存在上升可能，就可以集中投资于短期品种、缩短债券久期；而当我们判断当前的利率水平有可能下降，则拉长债券久期、加大长期债券的投资，这就可以帮助我们在债市的上涨中获得更高的溢价。
Money/dollar duration: the dollar duration measures the dollar change in a bond’s value to a change in the market interest rate
DVO1: the money change in the full price of a bond when it yield changes by one basis point (0.01%)
effective duration: a measure of the price change in units of currency given a change in yield
- duration hedge
price change of underlying asset + price change of hedging instrument = 0
Hedge ratio = - DVO1(underlyingAsset)/DVO1(HedgingInstrument) - property
maturity 越大，duration越大
coupon rate越大，duration越小
YTM 越大，duration 越小
- duration calculation
page 6 四种不能用来计算含权债券的方法 Macaulay duration modified duration money/dollar duration DVO1 一种含不含权都能用 effective duration
effective duration
有效持久期是具有嵌入式期权的债券的期限计算。 该持久期的衡量方法考虑到预期现金流量会随利率变化而波动的事实，因此，它是一种风险度量。 如果具有嵌入式期权的债券的行为类似于无期权债券，则可以使用修改的期限来估计有效期限。
Effective duration is a duration calculation for bonds that have embedded options.
Cash flows are uncertain in bonds with embedded options, making it difficult to know the rate of return.
The impact on cash flows as interest rates change is measured by effective duration.
Effective duration calculates the expected price decline of a bond when interest rates rise by 1%.
Effective Duration Calculation
The formula for effective duration contains four variables. They are:
P(0) = the bond’s original price per $100 worth of par value.
P(1) = the price of the bond if the yield were to decrease by Y percent.
P(2) = the price of the bond if the yield were to increase by Y percent.
Y = the estimated change in yield used to calculate P(1) and P(2).
The complete formula for effective duration is:
Effective duration = (P(1) - P(2)) / (2 x P(0) x Y)
As an example, assume that an investor purchases a bond for 100% par and that the bond is currently yielding 6%. Using a 10 basis-point change in yield (0.1%), it is calculated that with a yield decrease of that amount, the bond is priced at $101. It is also found that by increasing the yield by 10 basis points, the bond’s price is expected to be $99.25. Given this information, the effective duration would be calculated as:
Effective duration = ($101 - $99.25) / (2 x $100 x 0.001) = $1.75 / $0.20 = 8.75
The effective duration of 8.75 means that if there were to be a change in yield of 100 basis points, or 1%, then the bond’s price would be expected to change by 8.75%. This is an approximation. The estimate can be made more accurate by factoring in the bond’s effective convexity.
DV01
DV01 or Dollar Value of 1 basis point measures the interest rate risk of bond or portfolio of bonds by estimating the price change in dollar terms in response to change in yield by a single basis point ( One percent comprise of 100 basis points). DV01 is also known as Dollar Duration of a Bond and is the foundation of all Fixed Income instruments risk analysis and is used in abundance by Risk Managers and Bond Dealers.
In other words, where Duration is basically the ratio of the percentage change in the price of a security to a change in yield in percent, DV01 helps to interpret the same in Dollar terms, thereby enabling relevant stakeholders to understand the price impact of change in yields.
For instance, suppose a Bond has a Modified Duration of 5 and Market Value of the Bond as on date is $1.0 million, the DV01 is calculated as Modified Duration multiplied by Market Value of the Bond multiplied by 0.0001 i.e. 5 * $1 million* 0.0001= $500. Thus the bond will change by $500 for a one-point change in basis point in yield.
Dollar Duration or DV01 can also be calculated if one is aware of the Bonds Duration, current yield, and change in yield.
The calculation of Dollar Value of one basis point aka DV01 is very simple and there are multiple ways to calculate it. One of the most common formulas used to calculate DV01 is as follows:
DV01 Formula = – (ΔBV/10000 * Δy)
- convexity
page 6
1. identification
a measure of the curvature (non-linear relationship ) in the relationship between bond yield and price
凸度是衡量债券价格和债券收益率之间关系的曲率或曲线的程度。 凸度说明了债券的期限如何随利率的变化而变化。 投资组合经理将使用凸度作为风险管理工具，以衡量和管理投资组合的利率风险敞口。
- calculation
convexity = 1/P X d2P/dy2
price change using both duration and convexity - 负凸性 (callable bond) vs 正凸性 (puttable bond)
正凸性：涨多跌少
负凸性：涨少跌多
4.性质和组合
maturity 越大 convexity 越大
coupon rate 越大 convexity 越小
YTM 越大 convexity 越小
for bonds with same duration, has the greater dispersion of cash has the greater convexity
- portfolio duration and convexity
1. formula duration sum(wi x Di) convexity sum(wi x Ci) 2. classification barbell 现金流集中在两端 ballet 现金流集中在中间 如果两个组合有Same duration, barbell 的凸性更大
