4. Fomula-Valuation and Risk Models_frm p1b4笔记:valuation and risk models

1. Bond Price

1.1 Quotes

1.1.1 Treasury Bills

Treasury bills are instruments issued by government to finance its short-term funding needs. They last one year or less.

• The bid quote(买价) gives the price at which a market maker is prepared to buy the Treasury bill.

• The ask quote(卖价) (or the offer quote) gives the price at which a market maker is prepared to sell the Treasury bill.

• The mid-market price is the average of bid and ask prices.

• The quoted price ($Q$) and the cash price ($C$)
  $$C=100-\frac{n}{360}\times Q$$

  • where $n$ is the number of calendar days until the maturity of the Treasury bill, whose face value is $USD100$.
  • The quote $Q$ is the interest earned over a 360-day period as a percentage of the face value.

1.1.2 Treasury Bonds

A Treasury bond lasts more than one year. Bonds with a maturity between one and ten years are sometimes referred to as Treasury notes.

“32nds” quotation convention

Example: The bond quoted as 83-5.
$$(83+\frac{5}{32})\%facevalue$$

The quoted price ($Q$) and the cash price ($C$)

• Cash price(Dirty price) = Quoted price(Clean) + accrued interest
• Accrued interest(AI): is the interest earned between the most recent coupon date and the settlement date.
  

Day-count conventions:

• Actual/actual: most commonly for government bonds
• 30/360: most commonly for corporate and municipal bonds

1.2 The Law of One Price and Arbitrage

1.3 STRIPS

STRIPS is an acronym Separate Trading of Registered Interest and Principal of Securities.

STRIPS are created by investment dealers when a coupon-bearing bond is delivered to the Treasury and exchanged for its principal and coupon components.

• C-STRIPS(or TINTs, INTs): The securities created from the coupon payments.
• P-STRIPS(or TPs, Ps): The securities created from principal payments.

1.4 Interest Rates

1.4.1 Equivalent Rate(等价利率)

Convert $R_1$ (compounded $m_1$ times per annum) to the equivalent rate $R_2$(compounded $m_2$ times per annum):
$$(1+\frac{R_1}{m_1}{)}^{m_1}=(1+\frac{R_2}{m_2}{)}^{m_2}$$
Convert $R_m$ (compounded $m$ times per annum) to the equivalent rate $R_c$ (continuously compounded rate).

$$e^{R_c\ast m}=(1+\frac{R_m}{m}{)}^m$$

1.4.2 Spot Rate

1.4.2.1 Bond Valuation Based on Spot Rates

The spot rate is the interest rate earned when cash is received at just one future time. It is also referred to as the zero-coupon interest rate, or just the “zero”.

Use a sequence of spot rates that correspond to the cash flow dates to calculate the bond price(The law of one price).
$$P=\sum _{t=1}^n\frac{C}{(1+Z_t{)}^t}+\frac{FV}{(1+Z_n{)}^n}$$

$Z_n$: spot rate for period $n$.

1.4.2.2 Determine Spot Rates (Zero Rates)

Bootstrap: instruments lasting longer than one year usually make regular payments prior to maturity.
One way of calculating the zero-coupon rates implied by these instruments is by working forward and fitting the zero-coupon rates to progressively longer maturity instruments.

1.4.3 Discount Factor

Spot rate give the same information as discount factors.

Suppose the discount factor for $t$ years is $d(t)$ and that the $t-year$ spot rate is $z(t)$ with semi-annual compound
$$d(t)=\frac{1}{(1+\frac{z(t)}{2}{)}^{2t}}$$

1.4.3.1 Discount factor from T-bills(零息债券)

1.4.3.2 Discount factor from coupon-bearing bonds(付息债券)

1.4.3.3 Perpetuity(永续年金)

Perpetuity is a security that pays coupons forever(票息除以折现率)

$$Perpetuity=\frac{C}{y}$$

1.4.3 Forward Rate

1.4.3.1 Calculating Forward Rate

Forward rates: the future spot rates implied by today’s spot rates.

Implied forward rates(IFR, forward yields): a break-even reinvestment rate that are calculated from spot rates.

When rates are expressed with continuous compounding, the forward rate for the period between time $T_1$ and $T_2$ is
$$F=\frac{R_2{T}_2-R_1{T}_1}{T_2-T_1}$$

1.4.3.2 Bond Valuation Based on Forward Rates

Forward rates can be used to value a bond in the same manner as spot rates because they are interconnected.

Discount bond cash flows one period by one period with forward rates.

1.4.4 Par Rate

Par rate is the coupon rate which bond is priced at par value.

For an asset with a par amount of one unit that makes semiannual payments and matures in $T$ years

$$\frac{p}{2}\sum _{t=1}^{2t}d(\frac{t}{2})+d(T)=1$$
$$A(T)=\sum _{t=1}^{2t}d(\frac{t}{2})$$

• $p$: the par rate，$A(T)$: the annual factor

1.5 YTM

Yield to maturity is a single discount rate which if applied to all the bond’s cash flows, would make the cash flow’s present value equal to the bond’s market price.

When YTM expressed with semi-annual compounding, the bond’s market price is :

$$P=\frac{c/2}{1+y/2}+\frac{c/2}{(1+y/2{)}^2}+\cdots +\frac{c/2+100}{(1+y/2{)}^{2T}}$$

2. Duration 久期

2.1 Macaulay Duration

2.2 Modified Duration

$$\text{Modefied duration}=\frac{\Delta P/P}{\Delta y}$$

2.3 Effective Duration

$$\text{Effective duration}=\frac{V_{-}-V_{+}}{2\times V_0\times \Delta y}$$

注意和convexity 公式对比

2.4 DV01

$$DV01=-\frac{\Delta P}{10,000\times \Delta y}$$

$$DV01=\text{Duration}\times \text{Bond Value}\times 0.0001$$

$$HR=\frac{DV01(\text{initial position})}{DV01(\text{hedging instrument})}$$

$DV01$更多用于对冲

3. Convexity

3.1 Approximate Convexity

$$\text{convexity}=\frac{V_{-}+V_{+}-2V_0}{(\Delta y{)}^2{V}_0}$$

注意和effective duration 公式对比

3.2 Duration and Convexity Estimate

$$\Delta P=-D^{\ast }\times P\times \Delta y+0.5\times C\times P\times (\Delta y{)}^2$$

第一项有符号，第二项有0.5

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4. VaR

4.1 VaR Calculation

$$VaR(X\%)=|E(R)-Z_{X\%}\times \sigma |$$

$$VaR(X\%{)}_{\text{dollar}}=|E(R)-Z_{X\%}\times \sigma |\times \text{Asset Value}$$

Z X % = { 1.28 , when  X = 10 1.65 , when  X = 5 2.33 when  X = 1   Z_{X\%} =

$$\begin{cases} 1.28, & \text{when $X=10$} \\ 1.65, & \text{when $X=5$}\\ 2.33 & \text{when $X=1$ } \end{cases}$$ ZX%​=⎩⎪⎨⎪⎧​1.28,1.65,2.33​when X=10when X=5when X=1 ​

Assuming return is normally distributed and $E(R)=0$, $VaR(X\%)=|Z_{X\%}\times \sigma |$

The VaR for an investment opportunity is a function of two parameters:

The time horizon: VaR increases when the holding period is longer. Square root rule can be used.

$${\sigma }_{J-periods}={\sigma }_{1-period}\times \sqrt{J}\to VaR(X\%{)}_{J-periods}=VaR(X\%{)}_{1-period}\times \sqrt{J}$$

• Assuming the volatility is mean reverting.
• If today’s volatility is higher than the long-term mean, using square root may overstate the volatility / $\text{VaR}$.
• If today’s volatility is lower than the long-term mean, using square root rule may understate the volatility / $\text{VaR}$.

The confidence level: VaR increases when the degree of confidence increases.

$$VaR(New\%)=VaR(Old\%)\times (Z_{New\%}/Z_{Old\%})$$

4.2 VaR Estimation

4.2.1 Linear and Non-linear Derivatives

Linear derivatives has a linear relationship between underlying risk factors(底层风险因子) and derivative instruments. The transmission parameter($\delta$) needs to be constant(e.g. forward, futures) 远期的 $\delta$ 为$1$
$$\Delta P_{\text{Linear Derivative}}=\delta \times \Delta S_{\text{Risk Factor}}$$

$${\text{VaR}}_{\text{Linear Derivative}}=\delta \times {\text{VaR}}_{\text{Risk Factor}}$$

Nonlinear derivatives has a changing relationship between underlying risk factors and derivative instruments depending on the state of underlying asset. The transmission parameter($\delta$) is inconstant(e.g. option, bond).
$$\Delta P_{\text{Non-linear Derivative}}=\delta \times \Delta S_{\text{Risk Factor}}$$

$${\text{VaR}}_{\text{Non-linear Derivative}}=\delta \times {\text{VaR}}_{\text{Risk Factor}}$$

4.2.2 Delta Approximation

Delta normal approach(Delta approximation) is based on the risk factor’s deltas of a portfolio and assumes normal distributions for the risk factors.

For standalone non-linear derivatives:

• Step 1: Calculate the $\text{VaR}$ of the underlying risk factor.
• Step 2: Use delta with respect to the underlying to transmit the risk factor $\text{VaR}$ to the nonlinear derivatives $\text{VaR}$.

$${\text{VaR}}_{\text{option}}=|\Delta |\times {\text{VaR}}_{\text{stock}}$$

$${\text{VaR}}_{\text{bond}}=|-D\times P|\times {\text{VaR}}_{\text{yield}}$$

对于call option, $\Delta =N(d_1)$, 对于put option, $\Delta =1-N(d_1)$

When a call call option is at the money , $\Delta$ is 0.5

For a portfolio:

• Step 1: Calculate individual risk factors’ mean and standard deviation and their correlation between each other.
• Step 2: Calculate the means and volatility of the portfolio
• Step3: Calculate the portfolio’s $\text{VaR}$ assuming normality of value change of the portfolio.

$$\text{VaR}(X\%)=|{\mu }_{\text{portfolio}}-Z_{X\%}\times {\sigma }_{\text{portfolio}}|$$

${\sigma }_{\text{portfolio}}^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2{\rho }_{1,2}$ 用资产价值作为替代$w_1$，$w_2$作为权重

4.2.3 Delta-Gamma Approximation

Delta-Gamma approximation: the change in the derivative value is approximated by slope and curvature. The first derivative is delta linear approximation and second derivative is the gamma correction.

$${\text{VaR}}_{\text{bond}}=|-D\times P|\times {\text{VaR}}_{\text{yield}}-\frac{1}{2}\times C\times P\times {\text{VaR}}_{\text{yield}}^2$$

$${\text{VaR}}_{\text{option}}=|\Delta |\times {\text{VaR}}_{\text{stock}}-\frac{1}{2}\times \Gamma \times {\text{VaR}}_{\text{stock}}^2$$

注意公式是减去 gamma correction
The result under delta-gamma approach should be less than the delta-approximation

4.3 Measuring Volatility

4.3.1 Equally Weighted Standard Deviation

In risk management, the volatility of an asset is the standard deviation of its return in one day.

$${\sigma }_n^2=\frac{1}{m-1}\sum _{i=1}^m(r_{n-i}-\overline{r}{)}^2$$

• $r_i=(S_i-S_{i-1})/S_{i-1}$
• $\overline{r}=1/m{\sum }_{i=1}^m{r}_{n-i}$

Replace $m-1$ with $m$ (reasonable when $m$ is big) and make $\overline{r}=0$ (reasonable as the mean of daily return approximates $0$)
$${\sigma }_n^2=\frac{1}{m}\sum _{i=1}^m(r_{n-i}{)}^2$$

4.3.2 Exponentially weighted moving average(EWMA)

Exponentially weighted moving average(EWMA) gives more weight to more recent information and places exponentially declining weights on distant information(rather than using the equal weight).

$$\omega +\omega \lambda +\omega {\lambda }^2+\omega {\lambda }^3+\cdots +\omega {\lambda }^{k-1}=100\%\to \omega =1-\lambda$$

• $\lambda (0<\lambda <1)$ is introduced as a smoothing parameter.
• $k$: days of data.
• $\omega$: the weight applied to the most recent return.

$${\sigma }_n^2=(1-\lambda )r_{n-1}^2+\lambda {\sigma }_{n-1}^2$$

• Relatively little data needs to be stored
• High $\lambda$ respond relatively slowly to new information provided by the daily percentage changes.
• RiskMetric found that $\lambda =0.94$ to be a good choice.

EWMA can also be used to update correlation: assuming the return for asset $X$ and $Y$ are $x_n$ and $y_n$, for consistency we should use the same value of $\lambda$ for updating both variance rates using and covariances using
$${\rho }_n=\frac{co{v}_n}{{\sigma }_{x,n}\times {\sigma }_{y,n}}$$

The updated covariance between the asset $X$ and $Y$ is
$$co{v}_n=\lambda co{v}_{n-1}+(1-\lambda )x_{n-1}\times y_{n-1}$$

4.3.3 GARCH

The GARCH model can be regarded as an extension of EWMA.
In GARCH(1,1), we not only give some weight to the most recent variance rate and the latest squared return, but also give some weight to a long run average variance rate.

$${\sigma }_n^2=\gamma V_L+\alpha r_{n-1}^2+\beta {\sigma }_{n-1}^2$$

• Since weights must sum to $1$, so $\gamma +\alpha +\beta =1$
• The EWMA model is a particular case of the GARCH(1,1) model where $\alpha =1-\lambda$, $\beta =\lambda$
• Mean reversion: the $V_L$ term provides a ''pull" toward the long-run average mean. We can prove that:
  $$E({\sigma }_{n+t}^2)=V_L+(\alpha +\beta {)}^t({\sigma }_n^2-V_L)$$
  • When $t$ is large, the $E({\sigma }_{n+t}^2)$ approximates to $V_L$
  • $\alpha +\beta$ is called the Persistence Level(持续水平), which defines the speed at which shocks to the variance revert to their long-run value. The bigger the persistence level, the slower the variance revert to their long-run value.

Setting $\omega =\gamma V_L$, the GARCH(1,1) model is

$${\sigma }_n^2=\omega +\alpha r_{n-1}^2+\beta {\sigma }_{n-1}^2$$

$$V_L=\omega /(1-\alpha -\beta ),\alpha +\beta <1$$

5. Unexpected Loss

$$UL=EA\times \sqrt{PD\times {\sigma }_{LR}^2+L{R}^2\times {\sigma }_{PD}^2}$$

注意根号下的“非对称”

6. Risk Contribution

$${RC}_1={UL}_1\frac{{UL}_1+({\rho }_{1,2}\times {UL}_2)}{{UL}_p}$$

$${RC}_2={UL}_2\frac{{UL}_2+({\rho }_{1,2}\times {UL}_1)}{{UL}_p}$$

7. One-Factor Model

$$U_i=a_{i}F+\sqrt{1-a_i^2}{Z}_i$$

$a_i$ 介于 $-1$ 和 $1$ 之间