SCHWESERNOTES* FOR THE HRM1 EXAM FRM 2013 Book Part I * A7 i Mi y Valuation and Risk Models KAPLAN SCHWESER FRM PART I BOOK 4: VALUATION AND RISK MODELS READING ASSIGNMENTS AND AIM STATEMENTS VALUATION AND RISK MODELS VaR Methods 12 34: Measures of Financial Risk 35: Quantifying Volatility in VaR Models 23 36: Putting VaR to Work 37: Binomial Trees 38: The Black-Scholes-Merton Model 39: The Greek Letters 40: Prices, Discount Factors, and Arbitrage 41: Spot, Forward, and Par Rates 42: Returns, Spreads, and Yields 43: One-Factor Risk Metrics and Hedges 44: Multi-Factor Risk Metrics and Hedges 45: Empirical Approaches to Risk Metrics and Hedges 46: Country Risk Models 47: External and Internal Ratings 48: Loan Portfolios and Expected Loss 49: Unexpected Loss 50: Operational Risk 51 : Stress Testing 52: Principles for Sound Stress Testing Practices and Supervision ' 35 56 70 87 105 125 141 159 175 192 205- 216 223 233 ’ 243 249 262 271 SELF-TEST: VALUATION AND RISK MODELS 284 PAST FRM EXAM QUESTIONS 291 FORMULAS 3i7 APPENDIX 322 INDEX 325 ©2013 Kaplan, Inc Page FRM PART I BOOK 4: VALUATION AND RISK MODELS ©2013 Kaplan, Inc., d.b.a Kaplan Schweser All rights reserved.' Printed in the United States of America ISBN: 978-1-4277-4473-9 / 1-4277-4473-4 PPN: 3200-3232 Required Disclaimer: GARP® does not endorse, promote, review, or warrant the accuracy of the products or services offered by Kaplan Schweser of FRM® related information, nor does it endorse any pass rates claimed by the provider Further, GARP® is not responsible for any fees or costs paid by the user to Kaplan Schweser, nor is GARP® responsible for any fees or costs of any person or entity providing any services to Kaplan Schweser FRM®, GARP®, and Global Association of Risk Professionals™ are trademarks owned by the Global Association of Risk Professionals, Inc GARP FRM Practice Exam Questions are reprinted with permission Copyright 2012, Global Association of Risk Professionals All rights reserved These materials may not be copied without written permission from the author The unauthorized duplication of these notes is a violation of global copyright laws Your assistance in pursuing potential violators of this law is greatly appreciated Disclaimer: The SchweserNotes should be used in conjunction with the original readings as set forth by GARP® The information contained in these books is based on the original readings and is believed to be accurate However, their accuracy cannot be guaranteed nor is any warranty conveyed as to your ultimate exam success Page ©2013 Kaplan, Inc READING ASSIGNMENTS AND AIM STATEMENTS The following material is a review of the Valuation and Risk Models principles designed to address the AIM statements setforth by the GlobalAssociation of Risk Professionals READING ASSIGNMENTS Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, England: John Wiley & Sons, 2005) 34 “Measures of Financial Risk,” Chapter (page 23) Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach 35- “Quantifying Volatility in VaR Models,” Chapter (page 35) 36 “Putting VaR to Work,” Chapter (page 56) John Hull, Options, Futures, and Other Derivatives, 8th Edition (New York: Pearson Prentice Hall, 2012) 37 “Binomial Trees,” Chapter 12 (page 70) 38 “The Black-Scholes-Merton Model,” Chapter 14 (page 87) (page 105) 39 “The Greek Letters,” Chapter 18 Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011) 40 “Prices, Discount Factors, and Arbitrage,” Chapter (page 125) 41 “Spot, Forward, and Par Rates,” Chapter (page 141) 42 “Returns, Spreads and Yields,” Chapter (page 159) 43 “One-Factor Risk Metrics and Hedges,” Chapter (page 175) 44 “Multi-Factor Risk Metrics and Hedges,” Chapter (page 192) 45 “Empirical Approaches to Risk Metrics and Hedges,” Chapter (page 205) Caouette, Altman, Narayanan, and Nimmo, Managing Credit Risk, 2nd Edition (New York: John Wiley & Sons, 2008) 46 “Country Risk Models,” Chapter 23 ©2013 Kaplan, Inc (page 216) Page Book Reading Assignments and ATM Statements Arnaud de Servigny and Olivier Renault, Measuring and Managing Credit Risk (New York: McGraw-Hill, 2004) 47 “External and Internal Ratings,” Chapter (page 223) Michael Ong, Internal Credit Risk Models: Capital Allocation and Performance Measurement (London: Risk Books, 2003) 48 “Loan Portfolios and Expected Loss,” Chapter (page 233) 49 “Unexpected Loss,” Chapter (page 243) John Hull, Risk Management and Financial Institutions, 2nd Edition (Boston: Pearson Prentice Hall, 2010) 50 Operational Risk,” Chapter 18 (page 249) Philippe Jorion, Value-at-Risk: The New Benchmarkfor Managing Financial Risk, 3rd Edition (New York: McGraw Hill, 2007) 51 “Stress Testing,” Chapter 14 (page 262) 52 “Principles for Sound Stress Testing Practices and Supervision” (Basel Committee on Banking Supervision Publication, Jan 2009) (page 271) Page €>2013 Kaplan, Inc Book Reading Assignments and AIM Statements AIM STATEMENTS 34 Measures of Financial Risk Candidates, after completing chis reading, should be able to: Describe the mean-variance framework and the efficient frontier, (page 23) Explain the limitations of the mean-variance framework with respect to assumptions about the return distributions, (page 25) Define the Value-at-Risk (VaR) measure of risk, describe assumptions about return distributions and holding period, and explain the limitations of VaR (page 26) Define the properties of a coherent risk measure and explain the meaning of each property, (page 27) Explain why VaR is not a coherent risk measure, (page 28) Explain and calculate expected shortfall (ES), and compare and contrast VaR and ES (page 28) Describe spectral risk measures, and explain how VaR and ES are special cases of spectral risk measures, (page 29) Describe how the results of scenario analysis can be interpreted as coherent risk measures, (page 29) 56 35 Quantifying Volatility in VaR Models Candidates, after completing this reading, should be able to: Explain how asset return distributions tend to deviate from the normal distribution (page 35) Explain potential reasons for the existence of frit tails in a return distribution and describe the implications fat tails have on analysis of return distributions, (page 35) Distinguish between conditional and unconditional distributions, (page 35) Describe the implications of regime switching on quantifying volatility, (page 37) Explain the various approaches for estimating VaR (page 38) ' Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: • Historical standard deviation Exponential smoothing GARCH approach Historic simulation Multivariate density estimation Hybrid methods (page 38) Explain the process of return aggregation in the context of volatility forecasting methods, (page-48) Describe implied volatility as a predictor of future volatility and its shortcomings (page 48) Explain long horizon volatility/VaR and the process of mean reversion according to an AR(l) model, (page 49) • • • • • • • 36 Putting VaR to Work Candidates, after completing this reading, should be able to: Explain and give examples of linear and non-linear derivatives, (page 56) Explain bow to calculate VaR for linear derivatives, (page 58) Describe the delta-normal approach to calculating VaR for non-linear derivatives (page 58) ©2013 Kaplan, Inc Page Book Reading Assignments ami AIM Statements Describe the limitations of the delta-normal method, (page 58) Explain the full revaluation method for computing VaR (page 62) Compare delta-normal and full revaluation approaches, (page 62) Explain structural Monte Carlo, stress testing and scenario analysis methods for computing VaR, identifying strengths and weaknesses of each approach, (page 62) Describe the implications of correlation breakdown for scenario analysis, (page 62) Describe worst-case scenario (WCS) analysis and compare WCS to VaR (page 65) 37 Binomial Trees Candidates, after completing this reading, should be able to: Calculate the value of a European call or put option using the one-step and twostep binomial model, (page 70) Calculate the value of an American call or put option using a two-step binomial model, (page 78) Describe how volatility is captured in the binomial model, (page 77) Describe how the binomial model value converges as time periods are added (page 80) Explain how the binomial model can be altered to price options on: stocks with dividends, stock indices, currencies, and futures, (page 77) 38 The Black-Scholes-Merton Model • • Candidates, after completing this reading, should be able to: Explain the lognormal property of stock prices, the distribution of rates of return, and the calculation of expected return, (page 87) Compute the realized return and historical volatility of a stock, (page 87) List and describe the assumptions underlying the Black-Scholes-Merton option pricing model, (page 90) Compute the value of a European option using the Black-Scholes-Merton model on a non dividend paying stock, (page 91) 5- Identify rhe complications involving the valuation of warrants, (page 97) 6.- Define implied volatilities and describe how to compute implied volatilities from market prices of options using the Black-Scholes-Merton model, (page 97) Explain how dividends affect the early decision for American call and put options (page 96) Compute the value of a European option using the BlackjScholes-Merton model on a dividend paying stock, (page 93) Use Blacks Approximation to compute the value of an American call option on a dividend-paying stock, (page 96) 39 The Greek Letters Candidates, after completing this reading, should be able to: Describe and assess the risks associated with naked and covered option positions (page 105) Explain how' naked and covered option positions generate a stop loss trading strategy, (page 106) Describe delta hedging for an option, forward, and futures contracts, (page 106) Compute delta for an option, (page 106) Describe the dynamic aspects of delta hedging, (page 109) Define the delta of a portfolio, (page 112) Define and describe theta, gamma, vega, and rho for option positions, (page 113) Page ©2013 Kaplan, Inc Back Reading Assignments and AJM Statements Explain how to implement and maintain a gamma neutral position, (page 113) Describe the relationship between delta, theta, and gamma, (page 113) 10 Describe how hedging activities take place in practice, and describe how scenario analysis can be used to formulate expected gains and losses with option positions (page 119) 11 Describe how portfolio insurance can be created through option instruments and stock index futures, (page 120) 40 Prices, Discount Factors, and Arbitrage Candidates, after completing this reading, should be able to: Define discount factor and use a discount function to compute present and future values, (page 128) Define the “law of one price,” explain it using an arbitrage argument, and describe how it can be applied to bond pricing, (page 130) Identify the components of a U.S Treasury coupon bond, and compare and contrasr the structure to Treasury STRIPS, including the difference between P-STRIPS and C-STRIPS (page 132) Construct a replicating portfolio using multiple fixed income securities to match the cash flows of a given fixed income security, (page 133) Identify arbitrage opportunities for fixed income securities with certain cash flows (page 130) Differentiate between “clean” and “dirty” bond pricing and explain the implications of accrued interest with respect to bond pricing, (page 134) Describe the common day-count conventions used in bond pricing, (page 134) 41 Spot, Forward, and Par Rates Candidates, after completing this reading, should be able to: Calculate and describe the impact of different compounding frequencies on -a bonds value, (page 141) Calculate discount factors given interest rate swap rates, (page 142) Compute spot rates given discount factors, (page 144) Define and interpret the forward rate, and compute forward rates given spot rates (page 146) Define par rate and describe the equation for the par rate of a bond, (page 148) Interpret the relationship between spot, forward and par rates, (page 149) Assess the impact of maturity on the price of a bond and the returns generated by bonds, (page 151) Define the “flattening” and “steepening” of rate curves and construct a hypothetical trade ro reflect expectations that a curve will flatten or steepen, (page 151) ' 42 Returns, Spreads, and Yields Candidates, after completing this reading, should be able to: Distinguish between gross and net realized returns, and calculate the realized return for a bond over a holding period including reinvestments, (page 59) Define and interpret the spread of a bond, and explain how a spread is derived from a bond price and a term structure of rates, (page 161) Define, interpret, and apply a bond’s yield-to-maturity (YTM) to bond pricing (page 161) Compute a bond’s YTM given a bond structure and price, (page 161) Calculate the price of an annuity and a perpetuity, (page 165) ©2013 Kaplan, Inc Page Book Reading Assignments and AIM Statements Explain the relationship berween spot rates and YTM (page 166) Define the coupon effect and explain the relationship between coupon rate, YTM, and bond prices, (page 167) S Explain the decomposition oi P&L for a bond into separate factors including carry roll-down, rate change and spread change effects, (page 168) Identify the most common assumptions in carry roll-down scenarios, including realized forwards, unchanged term structure, and unchanged yields, (page 169) 43 One-Factor Risk Metrics and Hedges Candidates, after completing this reading, should be able to: Describe an interest rate factor and identify common examples of interest rate factors, (page 175) Define and compute the DV01 of a fixed income security given a change in yield and the resulting change in price, (page 176) Calculate the face amount of bonds required to hedge an option position given the DV01 of each, (page 176) Define," compute, and interpret the effective duration of a fixed income security given a change in yield and the resulting change in price, (page 178) and contrast DV01 and effective duration as measures of price sensitivity Compare 80) (page Define, compute, and interpret the convexity of a fixed income security given a change in yield and the resulting change in price, (page 181) Explain the process of calculating the effective duration and convexity of a portfolio of fixed income securities, (page 183) Explain the impact of negative convexity on the hedging of fixed income securities (page 184) Construct a barbell portfolio to match the cost and duration of a given bullet investment, and explain the advantages and disadvantages of bullet versus barbell portfolios.- (page 185) 44 Multi-Factor Risk Metrics and Hedges _ Page Candidates, after completing this reading, should be able to: Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques, (page 192) Define key rate exposures and know the characteristics of key rate exposure factors including partial ‘01s and forward-bucket ‘01s (page 193) Describe key-rate shift analysis, (page 193) Define, calculate, and interpret key rate ‘01 and key rate duration, (page 194) Describe the key rate exposure technique in multi-factor hedging applications and summarize its advantages and disadvantages, (page 195) Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile, (page 195) Describe the relationship between key rates, partial ‘01s and forward-bucket ‘01s, 7and calculate the forward-bucket ‘01 for a shift in rates in one or more buckets (page 197) Construct an appropriate hedge for a position across its entire range of forward bucket exposures, (page 198) Explain how key rare and multi-factor analysis may be applied in estimating portfolio volatility, (page 199) ©2013 Kaplan, Inc Book Reading Assignments and AIM Statements 45 Empirical Approaches to Risk Metrics and Hedges Candidates, after completing this reading, should be able to: I Explain the drawbacks to using a DVO1-neutral hedge for a bond position (page 205) Describe a regression hedge and explain how it improves on a standard DV01neutral hedge, (page 206) 3- Calculate the regression hedge adjustment factor, beta, (page 207) Calculate the face value of an offsetting position needed to carry out a regression hedge, (page 207) Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge, (page 208) Compare and contrast between level and change regressions, (page 209) 7- Describe principal component analysis and explain how it is applied in constructing a hedging portfolio, (page 209) 46 Country Risk Models Candidates, after completing this reading, should be able to: Define and differentiate between country risk and transfer risk and describe some of the factors that might lead to each, (page 21 6) Describe country risk in a historical context, (page 216) Identify and describe some of the major risk factors that are relevant for sovereign risk analysis, (page 217) Compare and contrast corporate and sovereign historical default rate patterns (page 218) Explain approaches for and challenges in assessing country risk, (page 218) Describe how country risk ratings are used in lending and investment decisions (page 219) Describe some of the challenges in country risk analysis, (page 219) 47 External and Internal Ratings Candidates, after completing this reading, should be able to: Describe external raring scales, the rating process, and the link between ratings and default, (page 223) Describe the impact of time horizon, economic cycle, industry, and geography on external ratings, (page 225) Review the results and explanation of the impact of ratings changes on bond and stock prices, (page 226) Compare external and internal ratings approaches, (page 226) Explain and compare the through-the-cycle and at-the-point internal ratings approaches, (page 227) Define and explain a ratings transition matrix and its elements, (page 224) Describe the process for and issues with building, calibrating and backtesting an internal rating system, (page 227) Identify and describe the biases that may affect a rating system, (page 228) ©2013 Kaplan, Inc Page Book Past FRM Exam Answers Question from the 2011 FRM Practice Exam 45 C BBB loans have an 88.21% chance of being upgraded in one year A Incorrect The chance of BBB loans being upgraded over year is 4.08% (0.02 + 0.21 + 3.85) B Incorrect The chance of BB loans staying at the same rate over year is 75.73% C Correct 88.21 % represents the chance of BBB loans staying at BBB or being upgraded over year D Incorrect The chance of BB loans being downgraded over year is 5.72% (0.04 + 0.08 + 0.33 + 5.27) (See Topic 47) Questionfrom the 2011 FRM Practice Exam 46 A AAA’ loans have 0% chance of ever defaulting AAA loans can default eventually, through consecutive downgrading, even though they are calculated to not default in one year AA — — > AA is 86.65% A * A is 86.96% BBB -* AAA/AA/A (sum) = 4.08% (See Topic 47) Question from the 2009 FRM Practice Exam 47 C 1.375% The easiest way to determine the answer would be to make this a square matrix including default in initial state Then self-multiplying the matrix three times yields three-period transition matrix We can also manually the calculation; After year there is a 0% chance of default and 5% chance of being in state B, After year there is 95% * 5% + 80% * 5% chance of being in state B and 5% * 10% chance of defaul t After year there is a (95% * 5% + 80% * 5%) * 10% additional chance of default A Incorrect Only considers the third year transition from B to default B Incorrect Only considers the second year transition from B to default D Incorrect Mistakenly doubles the second year transition from B to default (See Topic 47) Question from the 2009 FRM Practice Exam '48 C 9.62% Total Number of A’ rated issuances = 52 Probability of A’ rated issues downgraded to BBB (P,) = / 52 = 0.0385 Probability of A’ rated issues downgraded to Default (P2) = / 52 = 0.0577 Probability of A’ rated issues to be downgraded in one year = P, + P2 = 0.0962 = 9.62% A Incorrect Is the number of upgrades from A (2 + 5) / 52 = 3.46% B Incorrect Is the number of downgrades to A / 51 + / 42 = 3.92% + 9.52% = 13.44% D Incorrect Is the number of downgrades to BBB / 52 = 3.85% (See Topic 47) Page 314 ©2013 Kaplan, Inc Book Fast f-RJVi txam Answers Question from the 2008 FRIvI Practice Exam 49 D A sound interna] system uses at-the-point-in-time scoring for small-to-medium-sized companies and private firms and through-the-cycle scoring for large firms Explanation: The approaches are not compatible or directly comparable, and using the two approaches for different firms can yield highly inconsistent and misleading results (See Topic 47) Question from the 2008 FRM Practice Exam 50 C 18.5% Explanation: To answer this question the test taker must understand transition matrices The easiest way to determine the answer would be to make this a square matrix including default in initial state Then self multiplying the matrix to get the two year transition matrix We can also manually the calculation: After year there is a 10% chance of default and 80% chance of still being B and a 10% chance oi being an A In year there is a 10% chance of default if B rated (or 80% * 10% = 8%) and a 5% chance of default if A rated (10% * 5% = 0.5%) The total probability is therefore 18.5% Answer ‘a* assumes just one year Answer ‘b' ignores the probability of default after becoming A rated Answer ‘d’ simply adds two 10% (inappropriately does the second year probability) (See Topic 47) Question from the 2011 FRM Practice Exam 51 B USD 176,400 The risky portion of the asset value at the horizon is Outstanding + (Commitment— Outstanding) * Drawdown Given Default = USD 20,000,000 + (USD 100,000,000 - USD 20,000,000) * 0.80 = USD 84,000,000 This is the adjusted exposure on default (AE) The expected loss EL = AE * EDF * LGD, or USD 84,000,000 * 0.0035 * 0.6 = USD 176,400 This is the amount that the bank should set aside as a loss reserve (See Topic 48) Question from the 2009 FRM Practice Exam 52 C USD 6.0 million Expected Loss equals exposure multiplied by the risk of default and by the recovery rate, or E{L) = Exposure * PD * (1 - Recovery Rate) E(L) = 200 million USD x 5% x 60% = million USD Correlation amongst issuers does not matter for computing expected losses A Incorrect Incorrectly set E(L) = 200 * 0.05 * 0.4 = 4.0 B Incorrect Incorrectly set E(L) = 200 * 0.05 * 0.5 = 5.0 D Incorrect Incorrectly set E(L) = 200 * 0.05 * 0.8 = 8.0 (See Topic 48) ©2013 Kaplan, Inc Page 315 Book Past FRM Exam Answers Question from the 2008 FRM practice exam 53 A It is used to evaluate the potential impact on portfolio values of events or movements in a set of financial variables A Correct It describes stress testing’ B Incorrect It is not about ‘stress testing’ C Incorrect As B is incorrect D Incorrect As A is correct (See Topic 51) Page 316 ©2013 Kaplan, Inc unlikely, although plausible, FORMULAS Valuation anti Risk Models VaR Methods VaR (X%) = zX%a where: VaR (X%) 7jx% = the X% probability value at risk = the critical z-value based on the normal distribution and the selected X% probability the = standard deviation of daily returns on a percentage basis a VaR (X%)dollar basis = VaR (X%)decimal basis x asset value = (zÿ%a) x asset value VaR(X%)J-djv- = V,R(X%)WI/7T VaR = |Rp-(Z)(n)]Vp Topic 35 GARCH(U): = a+brt2_u+corÿ.1 Topic 36 — Taylor Series approximation (order two): f(x) = f(x0) + ff(xg)(x XQ) +— fw(x0)(x — Xg)2 Topic 37 risk-neutral valuation: U = size of the up-move factor = e°ÿ D = size of the down-move factor = e — = — TTU = eaft U ert D iru = probability of an up move Ttd = probability of a down move =ÿ U-D ©2013 Kaplan, Inc Page 317 Book Formulas Topic 38 expected value: E(S-p) = Sÿeÿ' Black-Scholes-Merton Option Pricing Model: c0 = [S0 x N(d] )] - |x x e“RfxT x N(d2)J Po = {xxe-RfxT X[l-N(d2)]}-{S0 x[l N(dj ))} where: = d, d2 = b(x)+!sf+(°-5x°2)l xT ax 7T dj — continuously compounded returns: u; = In —— lSi-l S; put-call parity: j Po — c0 s0+(xxe-R;*T] c0 — Po + S0 ~ Xe R* xT or ~ Topic 39 delta = A = —dc portfolio delta = Ap = i=J gamma: r=®S ds2 relationship among delta, theta, and gamma: rll = + rSA + 0.5cr2S2r Wga rho Page 318 = dc — —dcOx ©2013 Kaplan, Inc Book kormuias Topic 40 accrued interest: AI = number of days from last coupon to the settlement date number of days in coupon period where: c = coupon payment — clean price = dirty price accrued interest Topic 41 —— iKc i spot rate: z(t) = FVn = PV0 x + d(t) -1 — m where: r m n = annual rate = number of compounding periods per year = number of years l HPR: r = m 'EY* par rate, Gp ~-X njxn PVoJ -1 ]£d|ÿJ + d(T) = Topic 42 PV of a perpetuity = —C realized return: Rt_ijt = BVt+Ct— BVt_! BVÿ ©2013 Kaplan, Inc Page 319 Book Formulas bond price: p~ -c, (! + >•)' (1 + y)* C3 ; + + (1 -t- y) ' (l + yf where: P = the price of the security Ck = the annual cash flow in year k N - term to maturity in years y = the annual yield or YTM on the security Topic 43 DV01 = - ABV 10,000x Ay where: ABV = change in bond value Ay = change in yield modified duration — modified duration = effective duration Macaulay duration (l + periodic market yield ) ABV BV Ay BV_Ay -BV+Ay 2xBV0 xAy where: • = estimated price if yield decreases by a given amount, Ay BY_A>, ' BV+.£y = estimated price if yield increases by a given amount, Ay BV0 = initial observed bond price Ay change in required yield, in decimal form DV01 = duration x 0.0001 x bond value convexity BV_Ay +BV+Ay-2xBV0 BV0 x Ay2 percentage price change a duration effect + convexity effect = [—duration X Ay X 100] + X convexity X (Ay)2 X 10ol K duration of portfolio = wj X Dj j=! Page 320 ©2013 Kaplan, Inc Book Formulas Topic 44 key rate ‘01: key rate DV01k duration: —— Dk = ABV 10,000 Ayk ABV BV Topic 45 face value of offsetting position: Ayk FR = FN x DV01N' X0 DV01R change regression: Ayt = a + (3Ax( + As( where: Ayt = yt-yt_, = - xt-i Axt x level regression: y = a + 0x( + E( Topic 48 expected loss = exposure x loss given default x probabili ty of default expected loss = ELH = AE x LGD x EDF adjusted exposure = OS + a x COMy Topic 49 unexpected loss: UL = AE X JEDF X a2LGD + LGD2 x a2F.Dl ©2013 Kaplan, Inc Page 321 USING THE CUMULATIVE Z-TABLE Probability Example Assume that the annual earnings per share (EPS) for a large sample of firms is normally distributed with a mean of $5.00 and a standard deviation of $1.50 What is the approximate probability of an observed EPS value falling between $3.00 and $7.25? If EPS = x = $7.25, then z = (x - p)/o = ($7.25 - S5.00)/$l 50 =+1.50 If EPS = x = $3.00, then z = (x- p)/cr = ($3.00 - $5.00)/$1.50 = -1.33 For 7,-value of 1.50: Use the row headed 1.5 and the column headed to find the value 0.9332 This represents the1 area under the curve to the left of the critical value 1.50 For z- value of -1.33: Use the row headed and the column headed to find the value 0.9082 This represents the area under the curve to the left of the critical value +1 33 The area to the left of -1.33 is 0.9082 = 0.0918 - The area between these critical values is 0.9332 - 0.0918 = 0.8414, or 84.14% Hypothesis Testing - One-Tailed Test Example A sample of a stocks returns on 36 non-consecutive days results in a mean return of 2.0% Assume the population standard deviation is 20.0% Can we say with 95% confidence that the mean return is greater than 0%? H(): p < 0.0%, HA: p > 0.0% The test statistic = z-statistic = x-M-O o I Vn = (2.0 - 0.0) / (20.0 / 6) = 0.60 The significance level = 1.0 - 0.95 = 0.05, or 5% " • Since this is a one-tailed test with an alpha of 0.05, we need to find the value 0.95 in the cumulative e-table The closest value is 0.9505, with a corresponding critical z-value of 1.65 Since the test statistic is less than the critical value, we fail to reject HQ - Hypothesis Testing Two-Tailed Test Example Using the same assumptions as before, suppose that the analyst now w'ants to determine if he can say with 99% confidence that the stocks return is not equal to 0.0% Hq: p = 0.0%, Ha: p 0.0% The test statistic (z-value) = (2.0 - 0.0) / (20.0 / 6) = 0.60 The significance level = 1.0 - 0.99 = 0.01, or 1% Since this is a two-tailed test with an alpha of 0.01, there is a 0.005 rejection region in both tails Thus, we need to find die value 0.995 (1.0- 0.005) in the table The closest value is 0.9951, which corresponds to a critical z-value of 2.58 Since the test statistic is less than the critical value, we fail to reject Hfl and conclude that the stocks recurn equals 0.0% Page 322 ©2013 Kaplan, Inc p(/j ' CUMULATIVE Z-TABLE P(Z < z) = N(z) for z > P(Z < -z) = - N(z) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.5000 0.5040 0.5080 0.5120 0.5517 0.5160 0.5557 0.5199 0.5239 0.5279 0.5319 0.5596 0.5636 0.5910 0.5948 0.6331 0.6700 0.5987 0.6368 0.6026 0.5675 0.6064 0.6103 0.5359 0.5753 0.6141 0.6736 0.6772 0.7054 0.7088 0.7123 0.7389 0.7422 0.7704 0.7734 0.7454 0.7764 0.1 0.5398 0.2 0.5793 0.5438 0.5832 0.5478 0.5871 0.3 0.6179 0.6217 0.4 0.6554 0.6591 0.6255 0.6628 0.5 0.6915 0.6 0.7257 0.6985 0.7324 0.7 0.7580 0.6950 0.7291 0.761 0.7642 0.7019 0.7357 0.7673 0.6293 0.6664 0.6406 0.5714 0.6443 0.6808 0.6480 0.6844 0.6517 0.6879 0.7157 0.7486 0.7190 0.7224 0.7517 0.7794 0.7823 0.7549 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.8106 0.8133 0.8186 0.8212 0.8238 0.8023 0.8289 0.8078 0.8159 0.7995 0.8264 0.8051 0.9 0.8315 0.8-340 0.8365 0.8389 0.8413 0.8643 0.8849 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8599 0.8621 0.8665 0.8686 0.8708 0.8729 0.8770 0.8810 0.8830 0.8869 0.8888 0.8907 0.8925 0.8749 0.8944 0.8577 0.8790 0.8980 0.9147 0.8997 0.9015 0.9115 0.9265 0.8962 0.9131 0.9279 0.9292 0.9162 0.9306 0.9319 0.9394 0.9406 0.9418 0.9429 1.1 1.2 13 0.9032 0.9049 0.9066 0.9082 0.9099 1.4 0.9192 0.9207 0.9222 0,9236 0.9251 1.5 0.9332 0.9357 0.9474 0.937 0.9382 0.9495 0.9591 1.6 0.9452 1.7 1.8 0.9554 0.9641 0.9345 0.9463 0.9564 0.9649 1.9 0.9713 0.9772 2.1 0.9821 0.9861 2.2 2.3 2.4 0.9893 0.9918 0.9484 0.9515 0.9525 0.9608 0.9686 0.9750 0.9616 0.9693 0.9535 0.9625 0.9699 0.9756 0.9761 0.9441 0.9545 0.9633 0.9706 0.9767 0.9808 0.9812 0.9817 0.9656 0.9582 0.9664 0,9671 0.9505 0.9599 0.9678 0.9719 0.9726 0.9732 0.9738 0.9744 0.9778 0.9826 0.9864 0.9896 0.9920 0.9783 0.983 0.9788 0.9793 0.9803 0.9834 0.9838 0.9798 0.9842 0,9846 0.985 0.9854 0.9868 6.9871 0.9875 0.9878 0.9881 0.9884 0.9898 0.9901 0.9904 0.9906 0.9925 0.9927 0.9929 0.9909 0.9931 0.9911 0.9922 0.9887 0.9913 0.9934 0.9857 0.989 0.9916 0.9936 0.9943 0.9957 0.9968 0.9977 0.9945 0.9946 0.9959 0.9960 0.9970 0.9948 0.9961 0.9949 0.9962 0.9972 0.9979 0.9985 0.9951 0.9963 0.9964 0.9973 0.9980 0.9986 0.9981 0.9986 0.9989 0.9990 0.9990 0.9573 0.9938 0.994 2.7 0.9953 0.9965 0.9955 0.9966 2.8 0.9974 0.9975 0.9941 0.9956 0.9967 0.9976 2.9 0.9981 0.9982 0.9982 0.9983 0.9969 0.9977 0.9984 0,9984 0.9979 0.9983 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0,9989 2.5 2.6 0.9177 0.9978 ©2013 Kaplan, Inc 0.9971 0.9932 0.9952 0.9974 Page 323 — ALTERNATIVE Z-TABLE P(Z < z) = N(z) for z P(Z < -z) = - N(z) z 0.01 y\L = 0.02 0.03 0.04 0.05 0.06 0.07 P(0.u s Z £ 0.08 0.09 0.0279 0.0319 0.0359 0.0675 0.1064 0.07)4 0.0753 0.1103 0.114) 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.1 0.0398 0.0438 0.0478 0.0517 0.0636 0.0793 0.0832 0.0871 0.0910 0.0557 0.0948 0.0596 0.2 0.0987 0.1026 0.3 0.1179 0.1217 0.1255 0.1368 0.1406 0.1443 0.1480 0.1517 0.1554 0.1591 0.1628 0.1293 0.1664 0.1331 0.4 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.2486 0.2794 0.2517 0.2549 0.2823 0.2852 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.7 0.2580 0.261 0.2642 0.2673 0.2704 0.2764 0.8 0.2881 0.2910 0.2939 0.2967 0.3051 0.307S 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.2995 0.3264 0.2734 0.3023 0.3289 0.3315 0.3340 0.3356 0.3389 1.0 0.3413 0.3461 0.3485 0.3508 0.353! 0.3599 0.3621 0.3643 0.3849 0.3686 0,3708 0.3729 0.3554 0.3770 0.3577 1.1 0.3438 0.3665 0.3869 0.3790 0.3810 0.3830 0.3888 0.3907 0.3925 0.3962 0.3980 1.4 0.4032 0.4192 0.4049 0.4207 0.4015 0.4177 1.5 0.4332 1.6 0.4452 0.4554 0.4345 0.4463 1.2 1.3 0.4066 0.4082 0.4099 0.3749 0.3944 0.4115 0.4131 0.4147 0.3997 0.4162 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 0.4357 0.4474 0.4573 0.4370 0.4394 0.4505 0.4406 0.4418 0.444 0.4515 0.4525 0.4429 0.4535 0.4599 0.4608 0.4616 0.4625 0,4656 0.4582 0.4664 0.4382 0.4495 0.4591 0.4671 0.4678 0.4686 0.4726 0.4732 0.4738 0.4744 0.4750 0.4693 0.4756 0.4699 0.4761 0.4788 0.4793 0.4838 0.4798 0.4842 0.4878 0.4803 0.4808 0.4846 0.4850 0.4881 0.4S84 0.4901 0.4925 0.4927 0.4906 0.4929 0.4909 0.4931 0.4911 0.4920 0.4783 0.4830 0.4868 0.4898 0.4922 0.4812 0.4854 0.4887 0.4913 0.4934 0.4940 0.494! 0.4943 0.4945 0.4957 0.4959 0.4969 0.4948 0.4961 0.4971 0.4972 0.4979 0.4985 0.4979 0.4951 0.4963 0.4973 0.4980 0.4985 0.4986 0.4952 0.4964 0.4974 0.4981 0.4986 0,4989 0.4989 0.4990 0.4990 1.8 0.4641 1.9 0.4713 0.4564 0.4649 0.4719 2.0 0.4772 0.4778 0.4826 0.4864 1.7 Page 324 0.00 > / HU 2.1 0,4821 2.2 0.4861 2.3 2.4 0.4893 0.4918 2.5 2.6 0.4896 2.7 0.4939 0.4953 0.4965 2.8 0.4974 0.4955 0.4966 0.4975 2.9 0.4981 0.4982 0.4956 0.4967 0.4976 0.4982 3.0 0.4987 0.4987 0.4987 0.4484 0.4834 0.4871 0.4875 0.4904 0.4977 0.4983 0.4977 0.4984 0.4946 0.4960 0.4970 0.4978 0.4984 0.4988 0.4988 0.4989 0.4968 ©20.13 Kaplan, Inc 0.4932 0.4949 0.4962 0.4545 0.4633 0.4706 0.4767 0.4817 0.4857 0.4890 0.4916 0.4936 INDEX A D accrued interest 34 adjusted exposure 235, 243 advanced measurement approach 250 adverse selection 257 Altman Z-score 219 annuity 165 annuity factor 49 arbitrage opportunity 131 at-the-point approach 227 day-count convention 135 default frequency 243 delta 59, 73, 106 delta-gamma approximation 62 delta hedging 106 delta neutral 107 deka-neucrai portfolio 109 delta-normal VaR 8, 62 Derivatives Policy Group 263 dirty price 136 discount bond 67 discount factors 128,142 distribution bias 228 dividend yield 77 dollar value of a basis point 76 duration 178 DV01 176 DVO -neutral hedge 205 B backtesting 50 backtesting bias 228 barbell strategy 185 basic indicator approach 250 basis 277 basis risk 277 beta factors 250 binomial model 70 Black's approximation 96 Black-Scholes-Merton model 57, 87 bond equivalent yield (BEY) 163 bond valuation 125 bootstrapping 146 bullet strategy 185 C callable debt 184 carry-roll-down component 169 causal relationships 254 clean price 136 coherent risk measures 27 collateral 238 commitments 234 conditional distribution 36 conditional scenario method 265 conditional VaR 28 contagion 63, 216 contingent risk 279 convexity 181 country risk 216 coupon effect 68 covenants 235 covered position 105 criteria bias 228 C-STRIPS 132 E economic capital 245, 273 effective duration 179 - efficient frontier 23’ event-driven scenario 263 expected loss 233, 243' expected shortfall 28 expected tail loss (ETL) '28 F factor push method 265 forward-bucket '01s 193 forward rates 146 full valuation methods 16, 62 funding liquidity risk 279 G gamma 115 gamma-neutral 116 GARCH model 42 gross realized return 59 H hedge adjustment factor 207 hedge ratio 72, 176 hedging 72 ©2013 Kaplan, Inc Page 325 Book Index historical-based approaches 38 hisroricai scenarios 265 historical simulation method 20, 43, 64 historical volatility 98 holding period return 142 homogeneity bias 228 hybrid approach 39, 45 implied volatility 39, 98 information bias 228 interest rate factors 175 International Country Risk Guide 219 investment grade 223 K key rate ’01 194 key rate duration 95 key rate exposures 193 key rate shift 193 key risk indicators 255 L law of one price 130 legal risk 249 linear derivative 58 linear- methods 16 local delta 59 lognormal distribution 252 loss frequency 252 loss.given default 243 loss severity 252 • M Macaulay duration 78 market portfolio 24 mean reversion 49 mean-variance framework 23 modified duration 179 monoline insurers 278 tnonotonicicy 27 Monte Carlo simulation 21,252 moral hazard 256 multidimensional scenario analysis 264 multivariate density estimation 47 N naked position 105 negative convexity 184 net realized return 60 non-investment grade 223 nonparallel shift 152 Page 326 o operational risk 249 outstandings 234 P I issuer credit rating 223 nonparametric approach 39 normal distribution 23 parallel shift 152 parametric approaches lor VaR 39 par rate 48 partial ‘01s 193 par value 67 perfect hedge 72 perpetuity 165 pipeline risk 278 Poisson distribution 252 political risk 21? portfolio-driven scenario 263 portfolio insurance 120 positive homogeneity 27 premium bond 167 price-yield curve 60, 127 principal/agenr bias 228 principal components analysis 209 prospective scenarios 265 P-STRIPS 132 pull to par 67 put-call parity 92 R rate changes component 169 ratings process 223 ratings scale 223 realized forward scenario 169 realized return 89, 159, 163 realized yield 164 reconstitute 132 regime-switching volatility model 37 regression hedge 206 reinvestment risk 160,164 reputational risk 249 return decomposi tion 68 rho 118 risk and control self assessment 254 risk governance 272 RiskMetrics 39 risk spectrum 29 _ s scale bias 228 scenario analysis 29, 120,254,263 scorecard approach 255 ©2013 Kaplan, Inc Book Index securitization 278 seniority 238 sensitivity analysis 266, 275 sovereign risk SPAN system 263 spot rate 44 spot rate curve 144 spread V value at risk 26, 38, 273 vega 1 w warrants 97 case scenario 65 wrong-way risk 278 worst spread change component 69 standardized approach 250 stop-loss strategy 06 strategic risk 249 stress testing 64, 262, 271 STRIPS 132 structured Monte Carlo (SMC) approach 62 subadditivity 27 swap rates 142 Y yield curve 151 yield curve butterfly shifts 153 yield curve risk 192 yield curve twists 152 yield to maturity 161 T Taylor Series approximation 59 theta 11 through-the-cycle approach 227 rime horizon bias 228 total price appreciation 168 transfer risk 216 translation invariance 27 u unchanged term structure scenario 170 unchanged yields scenario 70 unconditional distribution 36 unexpected loss 234, 243 unidimensional scenario analysis 264 usage given default 236 ©2013 Kaplan, Inc, Page 327 Notes - ... 87 10 5 12 5 14 1 15 9 17 5 19 2 205- 216 223 233 ’ 243 249 262 2 71 SELF-TEST: VALUATION AND RISK MODELS 2 84 PAST FRM EXAM QUESTIONS 2 91 FORMULAS 3i7 APPENDIX 322 INDEX 325 ©2 013 Kaplan, Inc Page FRM. .. FRM PART I BOOK 4: VALUATION AND RISK MODELS ©2 013 Kaplan, Inc., d.b.a Kaplan Schweser All rights reserved.' Printed in the United States of America ISBN: 978 -1- 42 77 -44 73-9 / 1- 42 77 -44 73 -4 PPN:... if|§fi Page 14 m ©2 013 Kaplan, Inc VaR Methods Answer: VaR = [E(R) - zo] x portfolio value [0.0 018 8 - 1. 65(0. 012 5)] x $10 0,000,000 ' -=-0. 018 745 x $10 0,000,000 » - $1, 8 74, 500 , - 7v 4" 1IHi| WpWWj