topic 1: pricing of fixed income securities

FRM P1-估值与风险模型Https:www.zbgedu.com

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Topic 1: Pricing of Fixed Income Securities

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Bond Price

Ø Pricing bond with a single discount rate: Bond price is the present value of the

promised cash flows.

• Market discount rate: Discount rate used in the PV calculation, the required

rate of return by investors given the risk of the investment in the bond.

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Discount Factors

Ø The discount factor for a particular term gives the value today, or the

present value of one unit of currency to be received at the end of that term.

The discount function is expressed as d(t), where t denotes time in years.

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Treasury STRIPS

Ø STRIPS: Zero-coupon bonds issued by the Treasury are call STRIPS.

Ø The coupon bond is “stripped” into two components: principal and

coupon (P-STRIPS and C-STRIPS, respectively).

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Constructing a Replication Portfolio

Ø Example: The table below provides the market pricing data for 3-

year U.S. Treasury bonds, paying coupons semiannually.

Assuming we use 3.5% and 9.0% bonds to replicate the 5.0% bond.

Please identify whether the arbitrage opportunity exists or not.

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Ø Cash flows for the three bonds

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Ø Thus, the fair price of the replication portfolio would be:

72.73% * 99.34 + 27.27% * 106.52 = 101.3 < 102.3 of 5.0%

bond, there is a arbitrage opportunity exists.

Ø Arbitrage strategy:

• Buy 72.73% of face value (per $100) 3.5% bond and

27.27% of face value (per $100) 9.0% bond.

• Sell same amount of face value in 5.0% bond.

• The arbitrage profit would be $1 per $100 face value.

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Clean Price and Dirty Price

Ø Constant-yield price trajectory: Illustrates the change of bond price

over time if the yield keep constant.

Ø “Pull to par" effect: If no default and the yield keep constant, bond

price approaches par value as its time-to-maturity approaches zero.

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Ø Three commonly used day count conventions

• U.S Treasury bonds uses actual/actual

• U.S corporate and municipal bonds use 30/360

• U.S money market instruments (Treasury bills) use actual/36010

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Example: Consider a 5% semiannual coupon bond with par value of

100 and required yield of 4.8%. The bond will matures on 15

February 2024 and coupon are made on 15 February and 15 August

of each year. The bond is to be priced for settlement on 14 May 2015,

and that date is 88 days into the 181-day period. What should be the

price paid and what should be the price quoted?

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Compounding

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Ø Effective annual rate (EAR, 有效年利率): The rate by which a unit

of currency will grow in a year with interest on interest included.

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Yield to Maturity

Ø Yield to maturity (YTM): Internal rate of return of on the cash flow.

Ø Three critical assumption for YTM:

• The investor hold the bond until maturity.

• The issuer makes full and timely coupon and principal payments.

• The investor is able to reinvest coupon payments at YTM.

Ø Perpetuity: A perpetuity bond is a bond that pays coupons forever.

The price of a perpetuity is simply the coupon divided by the yield.

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Yield to Maturity

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Reinvestment Risk

Example: A bond portfolio manager invests $20 million in a bond issued at

par that matures in 30 years, and which promises to pay an annual interest

rate of 9%. The interest is paid once per year, and the payments are

reinvested at an annual interest rate of 8%. The first payment is one year

from today. What is the annually realized yield on this investment?

A. 8.185%

B. 8.285%

C. 8.385%

D. 8.415%

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Reinvestment Risk

Answer: C

Step 1-compute value of the coupon payments: N=30; I/Y=8; PV=0;

PMT=$1,800,000 (20 million×0.09); CPT→FV= $203,909,780

Step 2-total Future Value =$20000000(principle)+ $203,909,780

(value of the coupon payments)=$223,909,780

Step 3-annual yield: 8.385%

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Spot Rates

Ø Pricing bonds with spot rates: Use a sequence of spot rates that

correspond to the cash flow dates to calculate the bond price.

Ø Spot rates/zero rates: Yields-to-maturity on zero-coupon bonds

maturing at the date of each cash flow.

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nt

t nt=1 t n

n

Coupon Face valuePri

:

ce =

Spot

+ (1 +

rate foZ ) (1 + r peri n

Z )Z od

Spot Rates

Ø Example: Suppose that the one-year spot rate is 2%, the two-year spot rate

is 3%, and the three-year spot rate is 4%. Then, the price of a three-year

bond with par value of 100 that makes a 5% annual coupon payment is:

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Relationship between Spot Rates and YTM

Ø Example: A bond with $100 par value pays 5% coupon annually

for 4 years. The spot rates are as follows: Year 1: 4.0%; Year 2:

4.5%; Year 3: 5.0%; Year 4: 5.5%; Calculate the price of the

bond using spot rates and determine in the YTM for the bond.

Ø Answer:

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Coupon Effect

Ø For an upward-sloping yield curve, the yield to

maturity declines as the coupon rate increases.

Ø For an downward-sloping yield curve, the yield

to maturity increases as the coupon rate increases.

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Spot Rates

Ø Spot rate curve can be derived from either a series

of STRIPS prices, or the comparable discount factor.

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Forward Rates

Ø Forward rates are the interest rate on a bond or money

market instrument traded in a forward market.

Ø Relationship between forward rates and spot rates

• Implied forward rates (IFR): A break-even

reinvestment rate that are calculated from spot rates.

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A B - A BA A,B - A B(1 + z ) (1 + IFR ) = (1 + z )

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Relationship between Forward Rates and Spot Rates

Ø Spot rate curve can be used to compute the forward rate curve.

Ø Forward rate curve reflects the relation between forward rates and

times-to-maturity. A series of forward rates, each having same tenor.

Ø Example: Compute the 6-month forward rate in six months, given

the following spot rates.

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Maturity effect

Ø If the term structure of rates remains completely unchanged over period:

• Bond prices will increase over the period when coupon rates are

above the relevant forward rates.

• Bond prices will decrease over the period when coupon rates are

below the relevant forward rates.

Ø Example: The table shows present values of 100 face amount of the

bond with 1.44% coupon rate.

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Par Rates

Ø Par rate is the coupon rate at which the present value

of a bond equals its face value (par rate = coupon rate).

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Relationship between Spot, Forward Rates and YTM Curves

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Topic 2: Interest Rate Risk of Fixed Income Securities

Interest Rate Factor

Ø One-factor approach: All rate changes are driven by one interest rate factor.

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Macaulay Duration

Ø Macaulay duration: Weighted average time to receipt of the bond’s

promised payments, where the weights are the shares of full price

that correspond to each of the bond’s promised future payments.

• For zero coupon bond, MacDur = maturity

• For coupon bond, MacDur < maturity

• For perpetuity bond, MacDur = (1 + r)/r

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Macaulay Duration

Ø Example: Two-year bond of par value $1000, coupon rate 10%,

discount rate 10%, semi-annual, calculate the Macaulay duration.

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Modified Duration

Ø Modified duration requires a simple adjustment to Macaulay Duration.

Ø Modified duration provides an linear estimate of the percentage price

change for a bond given a change in its YTM.

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0

PMacDurPModDur = =

r 1 + rr: Periodic market yield

P -PApproximate ModDur =

2 ( Yield) P

P -P ModDur r

Money Duration/Dollar Duration

Ø Money duration/dollar duration: A measure of the price

change in units of currency given a change in its YTM.

Ø Modified duration provides an linear estimate of the

percentage price change for a bond given a change in its YTM.

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P

Dollar duration (DD) = = P ModDurr

P -P ModDur r -DD r

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Effective Duration

Ø Macaulay, modified and money duration is based on the expected

cash flows for an option-free bond, they are not an appropriate

estimate of the price sensitivity of bonds with embedded options.

Ø Effective duration: Sensitivity of bond's price to change in a

benchmark yield curve (Parallel shift). Used for bonds with

embedded option due to uncertain future cash flow.

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DV01

Ø Price value of a basis point (PVBP, DV01): The money change in full

price of a bond when its YTM changes by one basis point (0.01%).

Ø DV01 hedging: The purpose of a hedge is to construct a portfolio,

which consists of the initial position and the hedge position, that will

not change in value for a small change in yield.

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DV01 -0.01% P ModDur -0.01% DD

DV01(Initial position per $100)Hedge ratio = DV01(Hedging instrument per $100)

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DV01 hedging

Example: Assume a $100 million of 10-year semiannual coupon bond

has a DV01 of 0.735, and a 5-year semiannual coupon bond will be

used as the hedging instrument. The 5-year bond has a DV01 of 0.652.

Please calculate the face amount of the 5-year semiannual coupon

bond required to hedge the 10-year semiannual coupon bond.

Face amount required = $100 million * 0.735/0.652 = $112.73 million

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Convexity

Ø Duration is a good approximation of price changes for relatively

small changes in interest rates. As rate changes grow larger, the

curvature of the bond price-yield relationship is important.

Ø Convexity: The “second-order” effect of the price-yield curve.

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Convexity

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Convexity

Ø Duration provides an linear estimate of the percentage price

change for a bond given a change in its YTM.

Ø Duration and convexity provide linear and non-linear estimate of

the percentage price change for a bond given a change in its YTM.

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P -P ModDur r -DD r

2

2

1P -P ModDur r + P Convexity r

21

-DD r + DC r2

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Callable bond

Ø Lower effective duration, especially when interest

rates are falling, due to shorter expected life.

Ø Upside price appreciation in response to decreasing

yields is limited, it has negative convexity.

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Putable bond

Ø Lower effective duration, especially when interest

rates are rising, due to shorter expected life.

Ø Downside price appreciation in response to

increasing yields is limited, it has more convexity.

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Properties of Bond Duration and Convexity

Ø The factors that lead to greater convexity are the same as for duration:

• Longer time-to-maturity

• Lower coupon rate

• Lower yield-to-maturity

• Lower coupon frequency

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Barbell vs. Bullet Portfolio

Ø A barbell strategy is the strategy that managers only invest in bonds

with short and long maturities.

Ø A bullet strategy is the strategy that managers only invest in the

intermediate maturity range.

Ø These two strategies will have the same duration and different

convexity. The advantages and disadvantages of a barbell versus a

bullet portfolio are dependent on the investment manager’s view on

interest rates. Manager believing that rates will be particularly volatile

will prefer the barbell portfolio while a manager believing that rates

will not be particularly volatile will prefer the bullet portfolio.42

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Major Weakness of Single-Factor Approaches

Ø A major weakness of single-factor approaches is the assumption

that movements in the entire term structure can be described by one

interest rate factor. It is widely recognized that rates in different

regions of the term structure are far from perfectly correlated.

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Key Rate Exposures and Shifts

Ø Key Rate Shifts

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Key Rate’ 01s and Key Rate Duration

Ø Key rate‘ 01s: Which is the key rate equivalent of DV01.

Ø Key rate duration: Which is the key rate equivalent of modified durations.

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Example

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Principal Component Analysis

Ø A statistical technique known as principal components analysis can be used to

understand term structure movements in historical data. This technique looks at the

daily movements in rates of various maturities and identifies certain factors. These

factors are term structure movements with the property that:

• The daily term structure movements observed are a linear combination of the

factors (e.g., an observed movement might consist of five units of the first

factor, two units of the second factor, and one unit of the third factor).

• The factors are uncorrelated.

• The first two or three factors account for most of the observed daily movements.

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Ø Because there are eight rates and eight factors, the change on

any particular day can be calculated as a linear combination of

the factors. The change in the jth rate has the form:

• fij: factor loading

• ai: factor score

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Ø The importance of a factor is measured by

the standard deviation of its factor scores.

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Estimating Portfolio Volatility

Ø Banks do not attempt to make all the KR01s zero, but they are required to

calculate risk measures (VaR or expected shortfall) using their KR01

exposures in conjunction with standard deviations of (and correlations

between) the ten rates specified by regulators.

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53

Topic 3: Option Pricing

Properties of Stock Options

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One-step Binomial Model

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Two-period Binomial Model for American Option

Ø Example: A non-dividend-paying stock is currently trading at

72, a put option on this stock has a exercise price of 75 and a

maturity of 2 years. Suppose the interest rate is 3%, U = 1.356

and D = 0.541, πU = 0.6 and πD = 0.4. Calculate the put option

value if it is European-style and American-style respectively.

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Ø Answer: If it is European style

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Ø Answer: If it is American style

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Assumptions underlying the BSM model

Ø The options are European-style.

Ø The underlying asset price follows a geometric Brownian

motion, and moves smoothly from value to value.

Ø Continuously compounded risk-free interest rate is known and

constant; borrowing and lending at risk-free rate is allowed.

Ø Volatility of the underlying asset return is constant and known.

Ø The underlying asset has no cash flow.

Ø The market is frictionless: No transaction costs, no taxes, no

regulatory constraints; No-arbitrage opportunities in the market;

The underlying asset is highly liquid, and continuous trading is

available; Short selling of the underlying asset is permitted.62

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BSM Model-European Options Without Dividends

Ø Formulas for BSM model:

• N(d1): Delta of call

• N(d2): Risk-neutral probability that a call option will be exercised at expiration

• N(-d2): Risk-neutral probability that a put option will be exercised at expiration

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BSM Model-European Options With Dividends

Ø Continuously compounded rate of dividends:

Ø Dollar amount of dividends:

Ø Effects on options: Dividend payment will

increase put values and decrease call values.64

cf

cf

-R T-qT0 0 1 2

-R T -qT0 2 0 1

C = S e N(d ) - Xe N(d )

p = Xe 1 - N(d ) - S e 1 - N(d )

cf

cf

-R T0 0 dividends 1 2

-R T0 2 0 dividends 1

C = (S - PV )N(d ) - Xe N(d )

p = Xe 1 - N(d ) - (S - PV ) 1 - N(d )

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Delta

Ø Delta (Δ): Ratio of change of the option price

with respect to the price of the underlying assets.

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Option

Underlying assets

PDelta =

P

Delta

Ø When t → T, delta is unstable.

Ø Most sensitive to changes of underlying assets when at the money.

66

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Delta

Ø Delta for call option and put option (Non-dividend paying stock):

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call 1

put call 1

Delta = N(d )Delta = Delta - 1 = N(d ) - 1

Delta-Hedging

Ø Delta is also the hedge ratio.

Ø Delta-neutral portfolio: Combine the underlying assets

with the options so that the value of the portfolio does not

change with variation of the price of underlying assets.

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stock

option

N = Delta

N

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Delta-Hedging

Ø Example: We have a short put options on 10,000 shares of stock. Deltaput = –0.419.

Assume each stock option contract is for one share of stock. Calculate the

numbers of stock needed to delta hedge assuming the hedging instrument is stock.

Ø Answer:

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stock put short putN = -N Delta 10000

Sell 419

0.41

0 sh

9 4190

ares of stock

Practice

The current stock price of a company is USD 80. A risk manager is monitoring call

and put options on the stock with exercise prices of USD 50 and 5 days to maturity.

Which of these scenarios is most likely to occur if the stock price falls by USD 1?

A. Scenario A

B. Scenario B

C. Scenario C

D. Scenario D70

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Gamma

Ø Like duration in fixed income securities, delta is a linear estimation for

option and delta-neutral strategy only holds for very small changes in

the value of the underlying asset. So it requires frequently rebalance

hedge ratio (dynamic hedging). If hedge ratio is never adjusted, it is

referred to as static hedging, also known as a hedge-and-forget strategy.

Ø Gamma measures the stability of delta. If gamma is higher, rebalancing

hedge ratio more frequently is required. On the other hand, when

gamma is lower, rebalancing hedge ratio is required less frequently.

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Gamma

Ø Gamma (Γ): Rate of change of the option’s delta with respect to

the price of the underlying asset, and measures the non-linear

estimation of the option price function not captured by delta.

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DeltaGamma =

S

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Gamma

Ø Gamma is the same for call and put options.

Ø Gamma is largest when the option is at-the-money. If the option

is deep in- or out-of-the-money, gamma approaches zero.

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Long Call Short Call Long Put Short PutΓ > 0 Γ < 0 Γ > 0 Γ < 0

Gamma

74

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Gamma

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Gamma-Hedging

Ø Delta neutral: Hedging underlying assets with small changes.

Ø Gamma neutral: Hedging underlying asset with larger changes.

Ø Create delta and gamma neutral:

• Step one: Gamma-neutral by non-linearly instruments such as options and bonds.

• Step two: Delta-neutral by linearly instruments such as stocks and forwards.

76

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Gamma-Hedging

Example: Assume an existing short option position is

delta-neutral but has a gamma of -5,400. If there exists a

traded option with a delta of 0.4 and a gamma of 1.2.

How to create a delta-neutral and gamma-neutral position.

Answer:

1. Gamma-neutral position: Buy 5400/1.2 = 4500 options

2. Delta-neutral position: Sold 4500×0.4 = 1800 shares

of the underlying position.

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Vega

Ø Vega (Λ): Rate of change of the value of the option with respect to the volatility

of the underlying asset. Vega for call option is equal to Vega for put option.

Ø Vega is largest when the option is at-the-money. If the option is deep in- or out-

of-the-money, gamma approaches zero.

78

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Vega

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Theta

Ø Theta (θ): The sensitivity of the option price against time passage (t).

Ø As time passed, most options become less valuable, so theta is usually

negative for long option. With exception to deep in-the-money put

option. Short-term at the money option has a greatest negative theta.

80

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Rho

Ø Rho (ρ): The sensitivity of option price against the risk-free rate.

Ø The equity options are not as sensitive to changes in the risk free rate as

they are to changes in the other variables.

Ø In the money calls and puts are more sensitive to changes in rates than

out-of-the-money options.

Ø For call options, the Rho is positive. For put options, the Rho is negative.

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Rho

82

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Summary of Greek Letters

83

84

Topic 4: Market Risk

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Value at Risk (VaR)

Ø VaR: The maximum loss over a target horizon and for a given confidence level.

Ø Non-parametric approach: Historical simulation

Ø Example: You have accumulated 100 daily returns for your $100M portfolio. After

ranking the returns from highest to lowest, you identify the lowest six daily returns:

-0.0011, -0.0019, -0.0025, -0.0034, -0.0096, -0.0101

Ø VaRdaily(95%) = 0.19% ($190000)

85

Value at Risk (VaR)

Ø Parametric approach: Assume that asset returns follow normal distribution.

Ø Percentage VaR: VaR(X%) = E(R) - zX%×σ

Ø Dollar VaR: VaR(X%)dollar = [E(R) - zX%σ]×asset value

86

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VaR Conversion: Time Horizon

Ø Convert VaR to other time bases (Square Root Rule)

• VaR(X%)J-days = VaR(X%)1-days × (J)0.5

• VaR(95%)weekly = VaR(95%)daily × (5)0.5

• VaR(95%)monthly = VaR(95%)daily × (20)0.5

• VaR(95%)semiannual = VaR(95%)daily × (125)0.5

• VaR(95%)annual = VaR(95%)daily × (250)0.5

87

VaR Conversion: Confidence Level

Ø Convert VaR to different confidence level

VaR(X2%) = VaR(X1%) × zX2%/zX1%

88

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Value at Risk (VaR)

Ø Disadvantages of VaR

• Subject to model risk and implementation risk.

• Do not give us indication of how worse it could be if tail event occurs.

• Not sub-additive.

89

Coherent Risk Measures

Ø Monotonicity: A random cash flow or future value R1 that is always greater than

R2 should have a lower risk.

Ø Subadditivity: The portfolio’s risk should not be greater than the sum of its parts.

Ø Positive Homogeneity: The risk of a position is proportional to its scale or size.

Ø Translation Invariance: Risk is dependent on the assets within portfolio.

90

1 2 1 2R R , then (R ) (R )

1 2 1 2(R R ) (R ) (R )

0, ( R) (R)

(c R) (R) c

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Expected Shortfall (ES)

Ø Expected Shortfall (ES): It is the average of the worst 100×(1-α)% of losses.

Ø ES is a better risk measure than VaR because:

• The ES tells us what to expect in bad states-it gives an idea of how bad

bad might be-while the VaR tells us nothing other than to expect a loss

higher than the VaR itself.

• The ES always satisfies subadditivity, while the VaR does not.

91

Conditional and Unconditional Distribution

Ø Unconditional distribution: On any given day we assume the same distribution

exists, regardless of market and economic conditions, which means the mean

and the standard deviation of asset return on any given day would be unchanged.

Ø Conditional distribution: Information available to market participants about the

distribution of asset returns at any given point in time which may be different

than on other days, which causes the mean and the standard deviation of asset

return at any given day would most likely be dynamic.

92

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Regime-Switching Volatility Model

Ø Regime-switching volatility model

93

Evaluate VaR Estimation Approaches

94

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Parametric Approaches for Conditional Volatility

Ø Historical standard deviation approach assumes asset returns are

equally weighted. Each day, the forecast is updated by adding the

most recent day and dropping the furthest day. In a simple moving

average, all weights on past returns are equal and set to (1/N).

95


96

Ø Exponentially weighted moving average (EWMA) model

• We assign a higher weight on more recent data, and the weights

decline exponentially to zero as the data becomes older.

• The lower value of λ, the lower weight for older data.2 2 2n n-1 n-1

2 2 2n-2 n-2 n-1

mi-1 2 m 2

n-i n-mi=1

σ = λσ + (1 - λ)u

= λ[λσ + (1 - λ)u ] + (1 - λ)u

= (1 - λ) λ u + λ σ

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Example

97

Using a daily RiskMetrics EWMA model with a decay factor

λ = 0.95 to develop a forecast of the conditional variance,

which weight will be applied to the return that is four days old?

A. 0.000

B. 0.043

C. 0.048

D. 0.950 Answer: Bi -1 3(1 - λ)λ 0.05 0.95 0.043


98

Ø Generalized autoregressive conditional heteroskedasticity (GARCH) model

• In GARCH (1, 1), we assign some weight (γ) to the long-run average variance rate (VL).

• EWMA model is a particular case of the GARCH (1, 1) model (γ = 0, α = 1 - �, β = �).

• The GARCH (l, l) model incorporates mean reversion whereas EWMA model does not.

• The GARCH (l, l) is theoretically more appealing than the EWMA model.

L

2 2 2n L n-1 n-1

2 2 2n n-1

L

n-1

γ + α + β = 1 (γ 0,σ = γV + αu + βσ

σ = + αu +

α + β 1)

ω = γV ωωV =

1 - α β

σ

-

β

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99

Ø Mean reversion of the GARCH model

• When the current volatility is above/below the long-term volatility, the GARCH

(1, 1) model estimates a downward/upward-sloping volatility term structure.


100

Ø Mean reversion of the GARCH model

• Persistence: α + β. GARCH (1, 1) model is unstable if persistence > 1. Persistence of one

implies no mean reversion. Persistence of less than one implies “reversion to the mean”.

A lower persistence implies greater reversion of speed to the long-run mean (higher γ).

long-run average variance rate

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Example

101

Which of the following GARCH models will take the shortest time to

revert to long-run value?

A. σ2t = 0.05 + 0.03u2

t-1 + 0.96σ2t-1

B. σ2t = 0.03 + 0.02u2

t-1 + 0.95σ2t-1

C. σ2t = 0.02 + 0.01u2

t-1 + 0.97σ2t-1

D. σ2t = 0.01 + 0.01u2

t-1 + 0.98σ2t-1

Answer: B

Updating Covariance

102

Ø Updating a covariance estimate in the EWMA model

Ø Updating a covariance estimate in the GARCH (1, 1) model

• This model gives some weight to a long-run average covariance (CovL).

n n-1 n-1 n-1

n-1

n-1

Cov = λCov + 1 - λ x yx : The percentage change for x on day n-1y : The percentage change for y on day

( )

n-1

n n-1 n-1 n-1

L L

Cov = ω + αx y + βCovωω = γCov Cov =

1 - α - β

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Non-Parametric Approach

Ø Historical simulation method: Using historical data directly to estimate VaR.

• No parameter estimates and no distribution assumptions are required, the

only thing we need to determine up front is the look back window. Once the

window length is determined, we order returns in descending order, and go

directly to the tail of this ordered vector.

Ø Multivariate density estimation (MDE) allows for weights to vary based on how

relevant the data is to the current market environment, regardless of the timing

of the most relevant data. MDE is very flexible in introducing dependence on

economic variables (called state variables or conditioning variables).

103

Implied Volatility-based Approach

Ø Advantages:• Forward-looking predictive nature.• Implied volatility reacts immediately to changing market conditions.

Ø Disadvantages:• It is model dependent.• The BSM model assumes volatility is constant over the life of the option,

but in reality, volatility tends to change over time. At a given point in time, options on the same underlying asset may trade different implied volatilities. For example, the smile effect-deep out of the money and deep in the money options trade at a higher vol than at the money options.

• Empirical results indicate implied volatility is on average greater than realized volatility.

104

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Long Horizon Volatility and VaR

Ø The effects of mean reversion on long horizon volatility estimation

105

VaR of Linear & Nonlinear Derivatives

Ø Delta approximation

Ø Delta-gamma approximation

106

Fixed income security: VaR( P) = -D P VaR( r)

Option: VaR( f) = VaR( S)

2

2

1Fixed income security: VaR( P) = -D P VaR( r) - C P VaR( r)

21

Option: VaR( f) = VaR( S) - VaR( S)2

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Example

An at-the-money European call option on the DJ EURO STOXX 50 index with a

strike of 2200 and maturing in 1 year is trading at EUR 350, where contract value

is determined by EUR 10 per index point. The risk-free rate is 3% per year, and

the daily volatility of the index is 2.05%. If we assume that the expected return

on the DJ EURO STOXX 50 is 0%, the 99% 1-day VaR of a short position on a

single call option calculated using the delta-normal approach is closest to:

A. EUR 8

B. EUR 53

C. EUR 84

D. EUR 525107

Monte Carlo Simulation Approach

Ø Simulating thousands of returns for underlying assets

based on the assumption of distribution. VaR is

calculated from the simulated outcomes.

• Advantage: Measure risk factors by assuming various

underlying distribution; Able to generate correlated

scenarios based on a statistical distribution.

• Disadvantage: Fail to produce an accurate forecast of

future volatility or correlation and increase number of

simulations will not improve the forecast; Model risk.

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Scenario Analysis (Stress Testing)

Ø Generating scenarios in simulation and claiming that their distribution is

relevant going forward is as problematic as estimating past volatility and

using it as a forecast for future volatility. Generating a larger number of

simulations cannot remedy the problem. Scenario analysis may offer an

alternative that explicitly considers future events.

• Advantage: No distribution assumption; Can models multiple risk factors;

Can specifically focus on the tails (extreme losses); Complements VaR.

• Disadvantage: Highly subjective.

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Worst-Case Scenario (WCS)

Ø In contrast to VaR, WCS focuses on the distribution of the loss during

the worst trading period. The key point is that a worst period will

occur with probability one. An expected loss which is far greater than

the VaR is determined from the worst case distribution analysis.

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111

Topic 5: Credit Risk

Expected Loss

Ø Probability of default (PD), also referred to as expected default frequency (EDF).

Ø Exposure amount (EA), also referred to as exposure at default (EAD).

Ø Loss rate (LR), also referred to as loss given default (LGD). Recovery rate (RR) +

Loss given default (LGD) = 1.

Ø Expected loss (EL) = PD * LGD * EAD

Ø A prudent bank should set aside a certain amount of money (loan loss reserves) to

cover these losses that occur during the normal course of their credit business.

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Unexpected Loss

Ø Unexpected Losses (UL): Risk arises from the unexpected variation in

loss level. It is the standard deviation of credit losses, that is, the standard

deviation of actual credit losses around the expected loss average (EL).

Ø If the default probability follows binomial distribution:

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Economic Capital for Credit Risk

Ø Economic capital is an estimate of the overall capital reserve needed

to guarantee the solvency of a bank for a given confidence level.

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Models for quantifying credit risk

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Example

Suppose a bank has a portfolio with100,000 loans, and each

loan is USD 1 million and has a 1% probability of default in

a year. The recovery rate is 40%. In this case (using USD

million as the unit of measurement) n = 100,000, ρ = 0.1.

Answer:

σi2 (Variance of credit loss from the ith loan) = (PD - PD2) *

(EAD * LGD)2 = (1% - 1%2) * (1 * 60%)2 = 0.003564.

116

σ 1 + (n - 1)ρ 0.0597 1 + 99999 0.1α = = = 0.0189n EAD 100000 1

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Portfolio Credit Risk

Ø The expected loss of a portfolio of credits is straightforward

to calculate because EL is linear and additive.

Ø When measuring unexpected loss at the portfolio level, we

need to consider the effects of diversification because – as

always in portfolio theory – only the contribution of an asset

to the overall portfolio risk matters in a portfolio context.

117

Models for quantifying credit risk (The Gaussian Copula Model)

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Example

119

Risk Allocation

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Risk Allocation

121

Through-the-Cycle and At-the-point-in-time Approach

Ø At-the-point-in-time approach

• Assesses the credit quality of a firm over the coming months (generally 1 year).

• Lag

• Procyclical

Ø Through-the-Cycle Approach

• Capture the creditworthiness of a firm over a longer time horizon.

• Asset classes rated with through-the-cycle tools would be penalized during

growth periods compared with asset classes rated with at-the-point-in-time

tools, and vice versa in recessions.

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Credit Rating Transition Matrix

Example: Given the following ratings transition matrix, calculate

the two-period cumulative probability of default for a 'B' credit.

A. 2.0%

B. 2.5%

C. 4.0%

D. 4.5%123

Answer: D

Conditional and Unconditional Default Probabilities

Ø Table below shows the cumulative average default rates. It shows

the probability that an issuer with a certain rating will default

within one year, within two years, within three years, and so on.

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Conditional and Unconditional Default Probabilities

Ø Calculate the probability of a B-rated bond defaulting during the

fifth year 18.32% − 15.87 % = 2.45% we will refer to them as

unconditional default probabilities.

Ø If the firm survives to the end of year 4, what is the probability

that it will default during year 5? (conditional default probability)

• The probability it will survive to the end of the fourth year:

100% − 15.87% = 84.13%.

• The probability that it will default during the fifth year,

conditional on no earlier default is: 0.0245/0.8413 = 2.91%.

125

Exponential Distribution

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Exponential Distribution

127

Sovereign Default Spread

Ø The credit spread for sovereign debt in a specific currency is the excess

interest paid over the risk-free rate in that currency. There is a strong

correlation between credit spreads and ratings, but credit spreads can

provide extra information on the ability of a country to repay its debt. One

reason for this is that credit spreads are more granular than ratings.

Ø Credit spreads also have the advantage of being able to adjust more quickly

to new information than ratings. However, they are also more volatile.

Ø One source of credit spread data is the credit default swap market. They are

like insurance contracts in that they provide a payoff to the holder if the

country defaults within a certain period (usually five years).128

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129

Topic 6: Operational Risk

Operational Risk Capital Measurement Approaches

Ø Basic Indicator Approach (BIA)

• Operational risk capital is based on 15% of bank's annual

positive gross income (GI) over a three-year period.

130

BIAAvgORC = 15% GI

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Ø Standardized Approach (SA)

• Bank‘s activities are divided into 8 business lines. Operational

capital charge is obtained by multiplying the annual positive

gross income (GI) over a three-year period by the beta factor.

131

1-8 1-8SA 3 Years

Max[ (GI β ), 0 ]ORC =

3


Ø Example of Standardized approach (TSA)

• If the bank has the following revenue results from the past three years

for its two lines of business, corporate finance and retail banking:

132

2.4 + 0.6Capital = = 1.52

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Ø Advanced Measurement Approach (AMA)

• The AMA allows a bank to design its own model for

calculating operational risk capital. The Basel Committee

recognized that they were allowing significant flexibility

for the design of the AMA capital model.

• The model must hold capital for one year 99.9% VaR.

• The elements must be included in the model: internal loss

data, external loss data, scenario analysis and business

environment internal control factors.

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Ø Distribution of operational losses

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Ø Operational risk losses can be classified along two dimensions:

• Loss frequency: Number of losses over a specific time period.

Loss frequency is often modeled with Poisson distribution.

• Loss severity: Value of financial loss suffered. Loss severity

is often modeled with a lognormal distribution.

Ø Loss frequency and loss severity are combined in an effort to

simulate an expected loss distribution. The best technique to

accomplish this simulation is to use Monte Carlo simulation.

Ø Having created the loss distribution, the desired percentile value

can be measured directly.135

Data of AMA

Ø There are two types of operational risk losses: high frequency low-

severity losses (HFLS) and low frequency high-severity losses (LFHS).

Ø A bank should focus its attention on LFHS. These are what create the

tail of the loss distribution. LFHS occur infrequently. Even if good

records have been kept, internal data are liable to be inadequate, and

must be supplemented with external data and scenario analysis.

Ø Both internal and external historical data will be used to estimate the

loss severity and they must be adjusted for inflation. In addition, a

scale adjustment should be made to external data.

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Data of AMA

Ø Two sources of external data:

• Data consortia, which are companies that facilitate the

sharing of data between banks.

• Data vendors, who are in the business of collecting publicly

available data in a systematic way.

Ø Data from data vendors can potentially be biased because only

large losses are usually reported. If the data from a vendor is

used in a direct way to determine the loss severity distribution,

the distribution is likely to be biased toward large losses.

137

Data of AMA

Ø The aim is to generate scenarios covering the full range of possible LFHS.

These scenarios might come from bank's own and other banks’ experience,

the work of consultants, and the risk management group in conjunction with

senior management or business unit managers.

Ø Advantage of generating scenarios is that they include losses that the

financial institution has never experienced, but could incur. The scenario

analysis approach leads to management thinking actively and creatively

about potential adverse events. In some cases, strategies for responding to an

event so as to minimize its probability and severity are likely to be developed.

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Example

An operational risk analyst is attempting to analyze a bank's loss severity

distribution. However, historical data on operational risk losses are limited.

Which of the following is the best way to address this issue?

A. Generate additional data using Monte Carlo simulation and merge it

with the bank's operational losses.

B. Estimate the parameters of a Poisson distribution to model the loss

severity of operational losses.

C. Estimate relevant probabilities using expected loss information that is

published by credit rating agencies.

D. Merge external data from other banks with the bank's internal data

after making appropriate scale adjustments.139

Reducing Operational Risk

Ø Risk control and self assessment (RCSA)

• It is one way in which financial institutions try to understand operational

risks while creating an awareness of operational risk among employees.

• The key term here is self assessment. Line managers and their staff, not

operational risk professionals, are asked to identify risk exposures. The

risks considered should include not just losses that have occurred in the

past , but those that could occur in the future.

• The assessment process should be repeated periodically.

• The RCSA process may lead to improvements reducing the frequency of

losses, the severity of losses, or both.140

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Stress Testing Versus VaR and ES

Ø One disadvantage of VaR and ES is that they are usually backward-

looking. Stress is designed to be forward-looking. Unlike VaR and ES,

stress testing does not provide a probability distribution for losses.

Ø The backward-looking VaR/ ES analysis looks at a wide range of

scenarios (some good for the organization and some bad ) that reflect

history. On the other hand, stress testing looks at a relatively small

number of scenarios (all bad for the organization).

Ø The VaR/ES approach often has a short time horizon (perhaps only one

day), whereas stress testing usually looks at a much longer period.

141

Stressed Measures

Ø Stressed VaR and Stressed ES: The data used to develop scenarios

was from an immediately preceding period. However, bank

regulators have moved to using what are termed stressed VaR and

stressed expected shortfall for determining bank capital requirements.

These are measures based on data from a one-year period that would

be particularly stressful for a bank 's current portfolio.

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Reverse Stress Testing

Ø Reverse stress testing takes the opposite approach. It asks the question: “What

combination of circumstances could lead to the failure of the financial institution?”

• One reverse stress-testing approach involves the use of historical scenarios.

Under this approach, a financial institution would look at a series of adverse

scenarios from the past and determine how much worse each scenario would

have to be for the financial institution to fail.

• One approach is to define a handful of key factors (e.g., GDP growth rate,

unemployment rate, equity price movements, and interest rates changes) and

construct a model relating all other relevant variables to them.

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Internal Audit

Ø It should ensure that stress tests are carried out by

employees with appropriate qualifications, that

documentation is satisfactory, and that the models

and procedures are independently validated.

Ø It assesses the practices used across the whole

financial institution to ensure they are consistent.

Ø It will be able to find ways in which governance,

controls, and responsibilities can be improved. It can

then provide advice to senior management and the

board on changes it considers to be desirable.144

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topic 1: pricing of fixed income securities

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