Pragya the best FRM revision course!

FRM 2017 Part 1

Book 4 – Valuation and Risk Models

Pragya the best FRM revision course!

VOLATILITY IN VAR

Reading: Quantifying Volatility in VaR Models (Chapter 2, Linda Allen, Jacob Boudoukh and Anthony Saunders,

Understanding Market, Credit and Operational Risk (Oxford: Blackwell Publishing, 2004))

1. Deviation from Normal Returns:

a. Fat Tails: A fat-tailed distribution is characterized by having more probability weight (observations) in its

tails relative to the normal distribution.

b. Skewed: A skewed distribution in our case refers to the empirical fact that declines in asset prices are

more severe than increases. This is in contrast to the symmetry that is built into the normal distribution

c. Unstable: Unstable parameter values are the result of varying market conditions, and their effect, for

example, on volatility

d. Unconditional return: On any given day we assume the same distribution exists, regardless of market

and economic conditions

e. Conditional returns: Different market conditions may cause the mean and variance of the return

distribution to change over time.

2. Regime-switching: A regime switching volatility model assumes different market regimes exist with high or low

volatility. The conditional distribution of returns are always normal with a constant mean but either have high

volatility or low volatility.

3. VaR methods:

a. Historical:

i. Parametric (Delta Normal Method): [E(R) − Zσσ] ×Vp

ii. Non-parametric (Historical Simulation)

iii. Hybrid

b. Implied Volatility: Uses derivative pricing models and current derivative prices in order to impute an

implied volatility without having to resort to historical data

4. Return Aggregation: Under the assumption that asset returns are jointly normal, the return on a portfolio is also

normally distributed. Using the variance–covariance matrix of asset returns we can calculate portfolio volatility

and VaR

5. Implied Volatility as future of Volatility: Take the at-the-money (ATM) implied volatility from puts and calls and

extrapolate an average implied, over different maturity periods.

a. The most important reservation stems from the fact that implied volatility is model dependent

6. Time Conversion: VARyear = VARDaily ×√days at the same significance level

VAR AT WORK

Reading: Putting VaR to Work (Chapter 3, Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market,

Credit and Operational Risk (Oxford: Blackwell Publishing, 2004))

1. Linear Derivative: A derivative is defined as linear when the relationship between an underlying and the

derivative is linear in nature (E.g. Futures). The Delta (Change in price of future to change in price of underlying)

must be constant for all levels of the factor, but not necessary equal to 1.

2. Non-linear Derivative: Primary example is an Option. The price change in option for at-the-money options are

different than for out-of or in0the-money options.

3. VaR for Linear Derivatives: VaR for linear derivative is given as: VARderivative = δ ×VARunderlying

4. Option Delta-Gamma: Bond Duration and Convexity is the same as Option Delta and Gamma. Both

approximations are based on Taylor series. The series is not useful for callable bonds.

5. Correlations during Crisis: In times of crisis, the correlations increase substantially and strategies that rely on

low correlations fall apart. Diversifications benefits also fail. A simulation using Monte Carlo is not capable of

predicting scenarios during the time of crisis as the correlation between factors changes during crisis

6. WCS (Worst Case Scenario): It extends the VaR by estimating the extent of loss given an unfavorable event. In

contrast to VaR, WCS focuses on the distribution of the loss during the worst trading period.

MEASURES OF FINANCIAL RISK

Reading: Measures of Financial Risk (Chapter 2, Kevin Dowd, Measuring Market Risk, 2nd Edition(West Sussex, England:

John Wiley & Sons, 2005))

1. Mean Variance framework: It assumes that return distributions are elliptical distributions (like Normal)

a) Use of standard deviation as a risk measurement quantity is not appropriate for non-normal distributions

2. Value at Risk: VaR is an estimate of loss that can occur with a given confidence interval. Recall that Delta Normal

VaR is μ – Zσ which requires a confidence interval (a limitation of VaR). VaR also increases with increase in

holding period (another limitation). It also does not tell the investor the amount of actual loss expected

3. Coherent Risk Measures:

a) Monotonicity: A portfolio with greater returns will have less risk

b) Subadditivity: The risk of a portfolio is at most equal to the risk of assets within the portfolio

c) Positive homogeneity: Size of a portfolio will affect the size of its risk

d) Translation Invariance: Risk of a portfolio is dependent upon the assets of the portfolio

4. Expected Shortfall: Its the mean percent loss among the returns falling below q-percentile. It gives us the

magnitude of loss expected.

a) VaR does not satisfy the property of subadditivity and hence is not a coherent measure of risk

b) ES satisfies all coherent properties

5. Spectral Risk Measures: A more general risk measure is the risk spectrum. It measures the weighted averages of

return quantiles from the loss distribution

a) Weights are set equal in ES for all quantiles below q

b) For Var, weight is 1 for q - quantile and 0 for everything else

BINOMIAL TREES

Reading: Binomial Trees (Chapter 12, Hull, Options, Futures, and Other Derivatives, 8th Edition))

1. Binomial Trees: A useful and very popular technique for pricing an option involves constructing a binomial tree.

This is a diagram representing different possible paths that might be followed by the stock price over the life of

an option. The underlying assumption is that the stock price follows arandom walk. In each time step, it has a

certain probability of moving up by a certain percentage amount and a certain probability of moving down by a

certain percentage amount. In the limit, as the time step becomes smaller, this model is the same as the Black–

Scholes–Merton model.

2. Steps in valuing an Option:

a. U = eσ√t and d = 1

u

b. p =ert−d

u−d

c. C = e−rt[(Fu×p) + (1 − p)Fd]

d. C = e−rt[(Fu×p2) + (1 − p)2Fd + p(1 − p)Fud]

3. As the t becomes small, the binomial model becomes the Black-Scholes Model

4. Variations:

a. If there is a dividend yield, formula for p at ert changes to e(r-q)t

b. If there is a currency, formula for p at ert changes to e(rdomestic−rforeign)t

c. If there is a futures, which is costless to enter and risk-neutral, then ert is replaced by 1

Calculate u and d. U is the Up-move and d is the down move

P is the probability of up move. This is done for the first step only

Call price C, Fu is the payoff from the up move and Fd is the payoff

from the down move

Fud is the payoff at middle step. This is for a

two-step binomial tree

BLACK SCHOLES MERTON

Reading: The Black-Scholes-Merton Mode (Chapter 14, Hull, Options, Futures, and Other Derivatives, 8th Edition))

1. Lognormal Property of Stock Prices: It assumes that percentage changes in stock prices are normally distributed

over a short period of time. A variable that has a lognormal distribution can take any value between zero and

infinity. the continuously compounded rate of return per annum is normally distributed with mean µ - (σ2/2) and

standard deviation σ/√t

2. Assumptions of BSM:

a. The stock price follows the process developed with µ and σ constant.

b. The short selling of securities with full use of proceeds is permitted.

c. There are no transactions costs or taxes. All securities are perfectly divisible.

d. There are no dividends during the life of the derivative.

e. There are no riskless arbitrage opportunities.

f. Security trading is continuous.

g. The risk-free rate of interest, r, is constant and the same for all maturities

3. The BSM pricing formula:

The variables c and p are the European call and European put price, S0 is the stock price at time zero, K is the

strike price, r is the continuously compounded risk-free rate,σ is the stock price volatility, and T is the time

to maturity of the option.

GREEK LETTERS

Reading: The Greek Letters (Chapter 18, Hull, Options, Futures, and Other Derivatives, 8th Edition))

1. Naked Option: A naked position occurs when one party sells a call option or purchases a put option without

owning the underlying asset.

2. Greek Letters:

Letter Symbol Comparison

Delta Δ Change in option price vs. change in underlying security

Gamma Γ Change in Delta vs. change in underlying security

Theta θ Change in option price vs. change in maturity

Vega Change in option price vs. change in volatility

Rho ρ Change in option price vs. change in interest rate

3. European Call and Put:

a. Delta of an European call option is N(d1) from the BSM model

b. Delta of an European put is N(d1) – 1

4. Delta-Neutral Hedging:

a. No. of Options = (No. of Shares/ Delta of Option).

b. The delta hedging works only for small changes in the underlying security.

c. Portfolio Delta is the weighted average of all the Delta of each security

5. Gamma Neutral Hedging: Gamma neutrality provides protection against larger movements in this stock price

between hedge rebalancing. We can make a portfolio Delta, Gamma and Vega Neutral.

BOND PRICES, DISCOUNT FACTORS

Reading: Prices, Discount Factors, and Arbitrage (Chapter 1, Bruce Tuckman, Fixed Income Securities, 3rd Edition

(Hoboken, NJ: John Wiley & Sons, 2011))

1. Law of One Price: Absent confounding factors (e.g., liquidity, special financing rates, taxes, credit risk), two

securities (or portfolios of securities) with exactly the same cash flows should sell for the same price.

2. STRIPS: Zero coupon bond issued by US Treasury are called STRIPS (separate trading of registered interest and

principal securities). They are created when a coupon bond is presented to the treasury. The bond is stripped

into two components 1) Principal(P-STRIPS) and 2) Coupon(C-STRIPS)

a. STRIPS have more sensitivity to interest rates than coupon bonds

b. Longer term tend to trade cheap and shorter term tend to trade rich

3. Replicating Bond Portfolio:

a. B = F1 y% + F2 (y+x)% + F3 (x+y+z)%

b. Set all initial F1 and F2 to zero ad solve for F3 first. Then solve backwards.

SPOT, FORWARD AND PAR RATES

Reading: Spot, Forward and Par Rates (Chapter 2, Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ:

John Wiley & Sons, 2011))

1. Discount Rates from Swap Rates:

a. Swap rates represent bond coupon payments and swap notional amount represents the bonds par

value.

b. [ 100 + r ] x Discount Rate = 100 for 6 month

c. [ r ]Discount6 Month + [ 100 + r]Discount1Year = 100; we insert 6 month rate as calculated above

2. STRIPS Price: STRIPS Price can be used to calculate Discount factors. Discount Factor = (STRIPS Price/100) for

a given maturity. Also, to calculate SPOT rate, we can use the calculator functions.

3. Forward Rates: Forward Rates are rates that will apply starting at a future date.

a. Spot1 year2 = Spot6 month

2 ×Forward Rate6 month

4. Par Rate: The rate at which the current value of bond is same as par value. A swap rate is the same as par rate.

5. Bond Prices and Maturity:

a. Bond prices tend to increase with maturity when coupon rates are above relevant forward rates

b. If short term rates are more than forward rates, shorter investment rolled over will outperform longer

investments

6. Flattening and Steepening:

a. A normal yield curve is where forward rates are higher than short term rates, so the curve has positive

slope.

b. A flat curve has all interest rates at similar levels

c. An inverted yield curve is where long term rates are smaller than short term rates

d. Flattening of yield curve means spread between short and long term rates has narrowed

e. Steepening of yield curve means spreads have increased

RETURNS, SPREADS AND YIELDS

Reading: Returns, Spreads and Yields (Chapter 3, Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John

Wiley & Sons, 2011))

1. Definitions:

a. Gross realized return for a bond is its total return over life i.e. (End- Beginning)/Beginning

b. Net realized return is its gross realized return – per period financing cost. To calculate realized return for

a bond over multiple periods, we must keep track of rates at which coupon are reinvested.

2. Bond Spread: Difference between bond market price and bond price according to the term structure of interest

rates. Used to determine if bond is trading cheap or rich relatively

3. Yield to Maturity: It is the discount rate that will equate all the future cash flows of the bond to its current

market price. It can also be viewed as realized return assuming that all coupons are reinvested at YTM. If rates

are in BEY(Bond Equivalent Yield), then divide it by two to get semiannual rate.

4. Perpetuity: Cash flows are received indefinitely and there is no principal payment as the perpetuity does not

end. It is valued as PVPerpetuity =Coupon

YTM

5. Carry-Roll-Down: It account for price changes due to interest rate movements from the original term structure

to an expected structure.

ONE FACTOR RISK METRICS

Reading: One-Factor Risk Metrics and Hedges (Chapter 4, Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken,

NJ: John Wiley & Sons, 2011))

1. DV01: DV01 means change in fixed income security’s value for every 1 basis point (0.01%) change in interest

rates. It is also known as PVBP i.e. the price value of basis point. DV01 = − [Change in Bond Value

(1000 ×Change in yield)]. The DV01

is preceded by a negative sign because when rates decline, price increases.

2. Hedge Ratio: [DV01 of Instrument/ DV01 of Hedging Instrument] x Value of Position.

3. Duration: A bond price volatility is a function of its coupon, maturity and initial yield. Duration captures all these

variables in a single measure.

a. Macaulay Duration: Bonds interest rate sensitivity based on time in years until promised cash flows

arrive.

b. Modified Duration: It is given as Macaulay Duration/(1+YTM)

c. Effective Duration: (BV−∆y−BV+∆y)

2×BV0×∆y

4. DV01 and Duration: DV01 = Duration x 0.0001 x Bond Value

5. Convexity: Duration is a linear estimate and is good for small changes in interest rates. As rate changes grow

larger, we need to adjust for convexity also. Convexity measures curvature relationship between bond yield and

price.

a. Convexity = (BV−∆y+BV+∆y−2BV0)

BV0×∆y2

b. Change in bond Price % = −[duration × ∆y ×100] + [0.5×convexity×∆y2×100]

c. Negative Convexity: A callable bond has an effective price cap at the call price. Thus curve exhibits

negative convexity

6. Portfolio:

a. Duration is the weighted average of all individual durations

MULTIFACTOR RISK METRICS

Reading: Multi-Factor Risk Metrics and Hedges (Chapter 5, Bruce Tuckman, Fixed Income Securities, 3rd Edition

(Hoboken, NJ: John Wiley & Sons, 2011))

1. Issues with One-factor:

a. Assumes that term structure shifts in parallel fashion

2. Key Rate Exposures:

a. Key Rate makes an assumption that all rates can be determined as a function of a few key rates e.g. US

Treasury 2 years, 5 years, 10 years and 30 years bonds.

b. Calculations:

i. DV01Key rate = (−1

10,000) (

∆BV

∆yk )

ii. DurationKey rate = (−1

BV) (

∆BV

∆yk )

3. Partial 01: It will measure change in value of portfolio from 1 basis point change in fitted rate(Swap Rate Curve)

and subsequent refitting of curve. Swap curves are refitted daily using par rates and short term money market/

future rates.

4. Forward 01: Computed by shifting forward rate curve over several regions of the term structure one region at a

time after the term structure is divided into several regions/ buckets

COUNTRY RISK: DETERMINANTS, MEASURES &

IMPLICATIONS

Reading: Aswath Damodaran, Country Risk: Determinants, Measures and Implications - The 2015 Edition (July 14, 2015)

1. Definitions:

a. Discontinuous Risk: Government policies that change infrequently (like in authoritarian regimes) but can

be difficult to protest

b. Continuous Risk: Government policies change as government changes

2. Country Risks

a. Corruption and Side Costs: Corruption is like an implicit tax on income that reduces ROI

b. Physical Violence

c. Nationalisation/ Expropriation Risk: Some business like Mines are more prone to appropriation

d. Legal Risks: Legal system needs to be effective (Enforcing laws in fair manner) and efficient (dispose

matter quickly)

3. Measures of Country Risk

a. Degree of indebtness: Measured as government debt to GDP

b. Pension/ Social Service commitments

c. Revenue/ Inflows to Government and stability of revenues

d. Political Risk

4. Why local currency defaults happen?

a. Gold Standard: Prior to 1971, currency had to be backed by Gold and thus could not be freely printed

b. Shared Currency: Greece could not print Euros to fulfil its debt obligations

5. Implications of Default:

a. Negative impact on GDP by 0.5% to 2%

b. Long term borrowing costs rise and impacts sovereign ratings

c. Export oriented industries are harmed due to possible trade retaliation

d. Defaults make banking system fragile

e. Increases likelihood of political change

EXTERNAL & INTERNAL RATINGS

Reading: External and Internal Ratings (Chapter 2, Arnaud de Servigny and Olivier Renault, Measuring and Managing

Credit Risk(New York: McGraw-Hill, 2004))

1. External Ratings: They convey information about either a specific instrument, called an issue specific rating or

information about the entity that issued the instrument called the issuer credit rating. They are one-dimensional

and rating scales are uniform

a. S&P uses ratings of AAA, AA, A, BBB, BB, B, CCC, CC, C and D. Anything above BBB is investment grade

and below BB is non-investment grade. D means default

b. Moody’s uses Aaa, Aa, A, Baa, Ba, B, Caa, Ca and C. Ratings of Baa and above are investment grade.

2. Ratings Transition Matrix: It shows the frequency of rating change or default over a given time period.

a. Probability of default increases with time.

b. Ratings are designed to be stable over business cycles.

c. Interpreting ratings may vary depending upon the industry but not much based on geography

3. Internal Ratings: Ratings used by banks for their internal calculations.

a. At-the point approach: Short term credit score which varies through the business cycles

b. Through-the cycle: Score over a longer horizon using more qualitative information. Tends to be more

stable over cycles

c. Pro-cyclical effect of ratings: After economic trough has been reached, bank may downgrade a company

poised for recovery with use of additional credit from bank.

d. For creating an internal transition matrix, it is important to back-test the rating system

4. Internal Rating system bias:

a. Time horizon bias (Using at the point or through the cycle)

b. Homogeneity Bias (non-consistent ratings)

c. Principal-Agent Bias

d. Information Bias(Insufficient information)

e. Criteria Bias (Unstable criteria)

f. Backtesting bias (Incorrect linking of system to default rates)

g. Distribution Bias(Incorrect model)

h. Scale Bias(Ratings are unstable over time)

CAPITAL STRUCTURE IN BANKS

Reading: Capital Structure in Banks (2nd edition, Shroeck, 2002)

1. Definitions:

a. Probability of Default(PD): Likelihood that borrower will default

b. Exposure at Default(EAD): Remaining exposure at default

c. Loss given Default (LGD): Likely percentage loss if a borrower defaults. It is also (1 - Recovery Rate)

d. Expected Loss(EL): Defined as expected deterioration in asset quality and is given as PD x EAD x LGD

e. Unexpected Loss: Variation in the expected loss amount.

i. It is given as UL = EAD × √(PD×σLGD2 ) + (LGD2 ×σPD

2 )2

ii. σPD2 = PD×(1 − PD)

f. Economic Capital: Estimated reserves required for dealing with Unexpected loss

2. Portfolio Expected Loss: Sum of expected losses of each asset is the expected loss of the portfolio. It is simply

given as ELPortfolio = ∑ (PDi×EADi×LGDi)i

3. Portfolio Unexpected Loss: ULPortfolio = √∑ ∑ ULi ×ULj× ρijji2

OPERATIONAL RISK

Reading: Operational Risk (Chapter 20, John Hull, Risk Management and Financial Institutions, 3rd Edition(Boston:

Pearson Prentice Hall, 2012))

1. Regulatory capital: Three types of methods to determine regulatory capital are:

a) Basic Indicator: 15% of banks annual gross income over a 3 year period

b) Standardized approach: Eight business lines with different factors

c) Advanced measurement approach: Banks VaR measure at 99.9% confidence

2. Operational Risk: Basel Committee segregates operational risk into seven types namely Business Practices,

Internal Fraud, External Fraud, Damage to physical assets, process management, System failures, employment

practices.

3. Definitions:

a) Loss Frequency: Number of losses over a specific period of time (Poisson Distribution)

b) Loss Severity: Value of financial loss suffered (Lognormal Distribution)

c) Convolution: Combining frequency and severity (Monte Carlo Simulation)

d) RCSA(Risk and Control Self Assessment): Survey managers directly responsible for Risk Management

e) Power Law: EVT used to evaluate nature of tails of a given distribution

f) Moral Hazard: Presence of insurance causes the company to take more risks

g) Adverse selection: An insurance company cannot identify good or bad firm

STRESS TESTING

Reading: Stress Testing (Chapter 1 and 2, Stress Testing: Approaches, Methods, and Applications, Edited by Akhtar

Siddique and Iftekhar Hasan (London: Risk Books, 2013))

1. Effective governance: Key elements of effective governance and control over stress testing include governance

structure, policies and procedures, documentation, validation and independent review and internal audit

2. Difference between Stress tests/ VaR and EC (Economic Capital)

a) Stress tests look at fewer scenarios as compared to VaR measures

b) Stress tests are conditional whereas VaR and EC tend to be unconditional

3. Expected Loss

a) Expected Loss = PD x LGD x EAD