Part 1 Full Length Interactive Mock Exam 2

The classic risk management process affirms the job of a risk manager to include four activities: identifying risks; analyzing and measuring risks; assessing the impact of risk events; and managing risks. This process culminates in the series of decisions as to how to handle identified risks. Which of the following is (TRUE as) a common activity of the risk manager?

Below are a set of innovations over ten steps (from initial t = 0 to t = 10) and the first innovation ε(1) = 0.26. The innovations are random Gaussian white noise, ε(t) ~ N(0, σ^2 = 1).

Consider two time series models. The AR(1) model has an intercept, δ, of 0.70, and an AR parameter, φ, of 0.50. The MA(1) model has a mean, μ, of 0.70, and a weight of 0.50. Which of the following is nearest the value at t = 4; i.e., which are the missing values inside the red rectangle?

Consider the price of an asset that begins and $30.00 and ends, after 20 periods, lower at $8.55. Also highlighted are its maximum ($39.23) and minimum price ($6.79) during this 20-period life:

Among the following choices, which lookback option has the HIGHEST payoff if its life matches the 20-period interval shown?

Consider a large $20.0 million portfolio of 100 loans. In its general form, the portfolio’s unexpected loss is given by:[1]

However, each loan in this portfolio has approximately the same characteristics and size; i.e., the size of each is about $200,000. For modeling purposes, we can set the pairwise correlation coefficient to be constant ρ(i,j) = 0.160 for all i ≠ j. These assumptions greatly simplify the calculation of the portfolio’s unexpected loss and each loan’s contribution to portfolio risk. In this situation, which of the following statements is TRUE?

[1] Gerhard Schroeck, Risk Management and Value Creation in Financial Institutions, (New York: Wiley, 2002

A big part of a risk manager’s job is to identify her firm’s risk factors. Each of the following statements about risk factors is true EXCEPT which is false?

Below are displayed 15 pairwise (X,Y) trials. The simple regression line based on all 15 observations is given by Y1 = 0.488 + 0.425*X. We consider the possibility that the 12th Trial, given by point (X = 2.50, Y = -3.00) might be an outlier. If this point is removed, then the regression based on the remaining 14 observations is given by Y2 = 0.761 + 0.574*X. These results are displayed, including selected summary statistics.

According to Cook’s distance, is the 12th Trial an outlier?

Assume the current price of a stock is $30.00 and imagine that we can only trade the following four options at two strike prices:

• At a strike price of $28.00, we can employ either a call or a put, where c(K=28.00) = $3.98 and p(K=28.00) = $1.46
• At a strike price of $32.00, we can employ either a call or a put, where c(K=32.00) = $2.05 and p(K=32.00) = $3.46

Each of these prices is approximately accurate for a six-month option when the volatility is 31.2% (but these details are not necessary to answer the question). If we want to implement a bull spread, how could we do that?

Finlux International seeks to build an internal rating system for its considerable credit portfolio and assigns the project to a team including Alice, Bob, Chris, and Denise. In order to cast a wide net for ideas, each of the team members builds a mini-prototype:

I. Alice (A) developed an internal migration matrix based on a sample taken during a recession such that (related) her probabilities are not Markovian, yet to retrieve 5-year cumulative default probabilities she raises the matrix to the fifth power (ie., 5-year cumulative matrix = M^5) a calculation that might be valid if her probabilities were Markovian​
II. Bob (B), in order to maximize the universe of rated bonds, combines ratings for all of the major agencies across industries, countries, asset classes​
III Chris (C) uses a method of polling the salespeople who originate the loans on the theory that these are the people with the best “on the ground” knowledge of credit risk​
IV. Denise (D) mixed at-the-point-in-time and through-the-cycle ratings because she was unaware of which methodological assumption applied to each sourced dataset​

Which of the following accurately matches each team member to the bias that afflicts their approach?

One of the risk management building blocks is enterprise risk management (ERM). Which of the following is TRUE as a feature or implication of ERM?

Sally is a portfolio manager at an investment management firm. She wants to test her primary equity portfolio’s reaction to the factors in the Fama-French three-factor model. She collected excess returns (i.e., net of the risk-free rate) over the last eight years, so that the sample size, n = 96 months. The response (aka, explained, dependent) variable is the portfolio’s excess return. The three explanatory variables are the market factor (MKT), the size factor (SMB), and the value factor (HML). The size factor captures the excess return of small capitalization stocks (SMB = “small minus big”) and the value factor captures the excess returns of value stocks (HML = “high book-to-market minus low book-to-market”)’. Sally’s regression results are displayed below.

Which of the following descriptions of her portfolio is the most accurate?

Consider an at-the-money (ATM) stock option with a strike price of $50.00 and six months’ time to expiration; i.e., S(0) = K = $50.00 and T = 0.5 years. Now imagine the following four variations (I., II., III. and IV) on this option:

I. It is a European CALL option on a non-dividend-paying stock while the risk-free rate is 3.0%

II. It is a European CALL option on a stock that pays 1.60% dividend yield (D = $0.40) while the risk-free rate is 3.0%

III. It is a European PUT option on a stock that pays 1.60% dividend yield (D = $0.40) while the risk-free rate is 3.0%

IV. It is a European PUT option on a stock that pays 1.60% dividend yield (D = $0.40) while the risk-free rate is ZERO!

For the three variations where the stock pays a continuous 1.60% dividend, the equivalent present value (over the life of the option) is given by the lump sum, D = $0.40. For those interested, although it is beyond the scope of this question, this translation is given by the following: the PV of dividend, D = -S(0)*[exp(-q*T)-1]; in this case, D = $50.00*[exp(-0.0160*0.5)-1] = $0.3980.

Each of the above options has a different minimum value (aka, lower bound). However, among the four, which has the LOWEST minimum value?

As of the twentieth century (i.e., beginning 1900 and afterward), each of the following is TRUE about the consequences of sovereign default EXCEPT which is false?

According to GARP, “a recent trend among corporations is to use a board-approved risk appetite to guide management and (potentially) to inform investors.” Which of the following statements is TRUE about the firm’s risk appetite?

Shown below is the autocorrelation function (ACF) for a time series object that contains the total quarterly beer production in Australia (in megalitres) from 1956:Q1 to 2010:Q2 (source: https://cran.r-project.org/web/packages/fpp2/index.html).

About this ACF and its implications, each of the following statements is true EXCEPT which statement is false?

A non-dividend paying stock is currently trading at a price $35.00 when its volatility is 30.0% and the riskfree rate is 3.0%. Consider a chooser option with a strike price of $30.00 that gives the holder the right to choose (a call or put option) in three months and the chosen option, at that point in time, will have a remaining time to maturity of nine months; i.e., T1 = +0.25 years and T2 = +1.0 years. The price of this chooser is $7.710. Which of the following changes, ceteris paribus, will INCREASE the value of this chooser?

The exhibit below combines Tuckman’s Tables 5.6 and 5.7. The situation starts with the underlying trade: a 5×10 payer swaption struck at 4.044% (which gives the buyer the right to buy a fixed rate of 4.044% on a 10-year EUR swap in five years such that the underlying security in this option is a 10-year swap). This initial swaption trade is highlighted in light blue below. Additionally, the forward-bucket exposures of four swaps are shown in the upper panel. The lower panel highlights three different hedges (discussed in Tuckman) and their respective implied net positions:

Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)

About these hedges, which of the following statements is TRUE?

The global financial crisis (GFC) of 2007 to 2009 engendered regulatory responses to corporate risk governance. Below are summarized ten key developments. The first Sarbanes-Oxley (SOX) occurred prior to the GFC but is listed for context. The others are grouped naturally into three responses: Basel III and BCBS, Dodd-Frank, and the European response.

I. Prior to the GFC, Sarbanes-Oxley (SOX) required that the CEO and CFO affirm the accuracy of financial disclosures.​​

Basel III and BCBS:​

II. Basel III (BIII) was a direct response to the GFC. BIII limited core Tier 1 capital to common equity and retained earnings. BIII also imposed new ratios for short-term liquidity (i.e., LCR) and long-term liquidity (i.e., NSFR).​

III. Basel III designed a macroprudential overlay that included a 3.0% leverage ratio; countercyclical capital buffer (CCCB; aka, CCyB); and total loss-absorbing capital (TLAC) standards applicable to G-SIBs.​

IV. The Basel III framework was revised again in 2016 with the Fundamental Review of the Trading Book (FRTB; aka, part of Basel IV) which included enhanced disclosure requirements.​

V. The Basel Committee on Banking Supervision (BCBS) issued Corporate Governance Principles for Banks which–in addition to identifying the importance of an independent risk management function–defines roles for the board, board risk committees, senior management, CROs and internal auditors​​

Dodd-Frank:​

VI. The 2010 Dodd-Frank Act strengthened the regulatory reach of the Fed; ended too-big-too-fail (TBTF); launched overhaul of derivatives markets; introduced the Volcker Rule; created the Consumer Financial Protection Bureau (CFPB).​

VII. The Dodd-Frank Act also instituted a new approach to scenario analysis and stress testing that included: a top-down approach with macroeconomic scenarios unfolding over several quarters; a focus on the effects of macroeconomic downturns on a series of risk types, including credit risk, liquidity risk, market risk, and operational risk; an approach that is computationally demanding, because risk drivers are not stationary, as well as realistic, allowing for active management of the portfolios; a stress testing framework that is fully incorporated into a bank’s business, capital, and liquidity planning processes; and an approach that not only looks at each bank in isolation but across all institutions. This allows for the collection of systemic information showing how a major common scenario would affect the largest banks collectively.​​

The European response:​

VIII.For banks in Europe, the Supervisory Review and Evaluation Process (SREP) introduced three new principles to banking supervision: (i) A forward-looking emphasis on the sustainability of each bank’s business model, including during conditions of stress; (ii) An assessment methodology based on best practices within the banking industry, and (iii) An expectation that every bank will ultimately operate under the same standards.​

IX. The two key components of SREP are (i) the internal capital adequacy assessment process (ICAAP) and (ii) the internal liquidity adequacy assessment process (ILAAP). The ICAAP incorporates scenario analysis and stress testing; it outlines how stress testing supports capital planning. The ILAAP incorporates potential losses from asset liquidations and increased funding costs during stressful periods.​

X. European banks with assets of EUR 30 billion and above must run European Banking Authority (EBA) stress tests. These stress tests are run at the consolidated banking group level (insurance activities are excluded). Two supervisory macroeconomic scenarios covering a three-year period are provided by the regulator: a baseline scenario and an adverse scenario​

In regard to the above list of regulatory responses to the GFC, which of the following statements is TRUE?

Debra is an analyst at a governmental agency. Her boss asked her to investigate whether the Phillips curve applies during high-inflation regimes. To answer the question, Debra collected data from the FRED database at the St. Louis Fed (https://fred.stlouisfed.org/). The Phillips curve describes an inverse relationship between unemployment rates and inflation rates; https://en.wikipedia.org/wiki/Phillips_curve. Debra collected monthly data and she regressed the inflation rate against the unemployment rate (conditional on high-inflation regimes simply for narrative purposes). Her independent variable is the unemployment rate (FRED code: UNRATE) and here, the dependent variable is the Inflation rate (CPIAUCSL). The units are percentages not decimals; e.g., the dataset includes the month of January in 1982 when the unemployment rate was 8.90 and the inflation rate was 6.38. Her regression results are presented here.

Debra wants to know if an inverse relationship is observed. Which of the following statements about the regression is TRUE?

[This is tedious and difficult. Inspired by the final LO above and Hull’s[1] EOC Question 6.11] Today it is December 31, 2018. The cheapest-to-deliver bond in an August 2019 Treasury bond futures contract is a 9.0% coupon bond, and delivery is expected to be made on August 28, 2019. Coupon payments on the bond are made on April 2 and October 2 each year. In this case, therefore, as of settlement today (December 31, 2018) there were 90 days since the last coupon and there will be 92 days until the next coupon. Delivery will be in 240 days (and subsequent coupon date 35 days after delivery, or 275 days from today). The term structure is flat, and the rate of interest with continuous compounding is 3.0% per annum. The conversion factor for the bond is 1.380. The current quoted bond price (for this bond which is assumed to be the cheapest to deliver) is $115.00. What is the quoted futures price for the contract?

[1] John C. Hull, Options, Futures, and Other Derivatives, 10th Edition (New York: Pearson Prentice Hall, 2017). Example 6.2 from Hull, but spreadsheets handcrafted by David Harper.

Suzanne the Risk Analyst is building an interest rate term structure and she is evaluating various candidate models. Her first candidate is Tuckman’s Model 1[1] (aka, normally distributed rates and no drift) which has the advantage of extreme simplicity and is specified by (Tuckman 9.1)27: dr σ*dw. Her colleague Peter observes this is a single-factor model: the model’s only factor is the short-term interest rate. Among the following, which is probably the strongest criticism against this model as a single-factor model?

[1] Bruce Tuckman’s Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)

The Acme Investment Trading Company perceives the credit risk of a certain public retailer is mispriced by the market. Acme is considering buying or selling a credit default swap (CDS) for the purpose of speculating on this view with respect to the retailer’s credit profile. In comparison to buying or shorting the retailer’s cash bond, Acme has already identified an advantage to the CDS: it has better liquidity. On the other hand, which of the following is a disadvantage of the CDS?

Mary belies that the average net promoter score (NPS) in financial services, as a population, is at least 50. Her one-sided null hypothesis is H0: μ(NPS) ≤ 50 and her alternative hypothesis is HA: μ(NPS) > 50. Among a collected sample of 40 firms, her staff observes a sample average NPS of 53.60 with a standard deviation of 9.0. Her staff informs her that the test statistic is 2.53 and the two-sided p-value is 1.556% per the Excel function T.DIST.2T(2.530, 39) = 0.015563. Their report also includes these 95.0% critical t-values: T.INV(0.95, 39) = 1.685 and T.INV.2T(0.050, 39) = 2.023; as expected, these values are slightly higher than, respectively, the critical Z-values of 1.645 and 2.33. Each of the following is true EXCEPT which is false?

It is August and Sally is a fund manager with $50.0 million invested in government bonds who is worried that interest rates are expected to be volatile over the next quarter (note: this question is inspired by Hull’s EOC Problem 6.18). She decides to use the December Treasury bond (“T-bond”) futures contract to hedge the value of the portfolio. The current futures price is 108-00 or $108.00. Because each contract is for the delivery of $100,000 face value of bonds, the futures contract price is therefore $108,000.00. Suppose the modified duration of the bond portfolio in three months will be 13.0 years. The cheapest-to-deliver (CTD) in the T-bond contract is anticipated to be a bond with 18.0 years to maturity that pays a 5.0% semi-annual coupon; at maturity, the duration of this CTD bond is expected to be about 12.0 years.

However, the manager does not want to completely neutralize duration. Rather, she wants to REDUCE the portfolio’s duration by 7.0 years, from 13.0 years to 6.0 years. About how many T-bond futures contracts should she trade to achieve this reduction in duration of the net portfolio?

The exhibit below modifies Tuckman’s Table 4.2[1] and shows the prices on May 28, 2019 for two instruments: 10-year U.S. note futures contracts, TYU0, and call options with a strike of 120 on the same futures contracts, TYU0C 120.

What is the effective duration of, respectively, the futures, TYU0, and the options, TYU0C 120?

[1] Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011. Modified table from Tuckman; spreadsheet handcrafted by David Harper.

Consider the market portfolio plus three other portfolios (A, B, C) with the following features while the risk-free rate is 3.0%:

Please note the returns are expected gross returns; e.g., the market’s expected return is 9.0% so that its expected excess return is 6.0%. If we assume the capital asset pricing model (CAPM) is valid then which of the following statements is TRUE?

Below is the probability mass function (pmf) of a binomial random variable where the probability of success over 15 trials is 10.0%; i.e., p = 0.10 and n = 15. What is the interquartile range (IQR) of this probability distribution?

Consider an asset with a current price of $120.00 and volatility of 16.0% while the risk-free rate is 3.0%. A regular (aka, vanilla) but deeply out-of-the-money (OTM) one-year European put option on the stock with a strike price of $100.00 has a price of $0.740; i.e., p(S = $120.00, K = $100.00, σ = 0.160, Rf = 0.030, T = 1.0 year) = $0.740.

Now consider the modification of this regular put option into a gap put option with the addition of a trigger price, denoted K2. In this case, the price of the gap put option is given by p(S = $120.00, K1 = $100.00, K2 = trigger price, σ = 0.160, Rf = 0.030, T = 1.0 year). Each of the following statements about this gap option is true EXCEPT which is false?

As shown in the exhibit below in a format similar to Tuckman’s table 3.2[1], the price of a bond as of 5/28/2019 is $113.335. This bond pays a semi-annual 10.0% coupon and has maturity of 1.5 years; that is, it matures on 11/30/2020. Currently, the six-month forward rates are 0.60%, 1.00% and 1.50% (see green row).

If we make a scenario assumption of UNCHANGED TERM STRUCTURE, which of the following is nearest to the carry roll-down over the next six months?

[1] Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011. Spreadsheet handcrafted by David Harper.

Let Rm(P) denote the monthly return of the portfolio and Rm(B) denote the monthly return of its benchmark. Over a three-year measurement period, the following statistics are calculated:

• The average monthly return for, respectively, the portfolio and the benchmark was Rm(P) = 8.50% and Rm(B) = 6.90%; therefore, on average, the portfolio outperformed its benchmark by +1.60%.
• The monthly standard deviation of the difference between the portfolio’s and benchmark’s return, σ[Rm(P) – Rm(B)] = 11.80%
• A regression of the portfolio’s excess return against the benchmark’s excess return produced the sample regression function, ERm(P) = -0.0140 + 1.35 * ERm(B); therefore, the regression intercept (aka, alpha) is -1.40%
• The standard error of the regression (SER), which approximates the volatility of alpha, σ(α), is 11.0%

Although the periodicity is monthly, an information ratio is generally annualized. In regard to annualized information ratios, which of the following statements is accurate?

Below is the joint distribution of analyst ratings (i.e., negative, neutral, or positive) and stock returns (-5, zero, or +5 in percentage terms).

We are interested in the conditional distribution of stock returns given a negative analyst rating. For example, the expected stock return conditional on a negative analyst rating is [-5.0 * (20.0%/25.0%)] + [0 * (5.0%/25.0%)] + [+5 * (0%/25.0%)]= -4.0. The variance of the stock return conditional on a negative analyst rating is [-5.0 – (-4.0)]^2 * 80% + [0 – (-4.0)]^2 * 20% + [+5.0 – (-4.0)]^2 * 0% = 4.0. What is the skew of the stock return conditional on a negative analyst rating?

You are trying to model the theoretical forward price of silver using the cost of carry model. Your model is informed by the following four propositions. Each is correct except which is FALSE?

Let the two-year term structure of zero rates include the following four spot rates: 1.0% @ 0.5 years, 2.0% @ 1.0, 3.0% @ 1.5, and 4.0% @ 2.0 years. Using these discount rates, the price of a two-year $100.00 face value bond with 4.0% coupon rate is $100.10 (see blue cell) as shown in the exhibit below, where $100.10 is the sum of four discounted cash flows:

Assume the term structure above (i.e., 1.0% @ 0.5 years, 2.0% @ 1.0, 3.0% @ 1.5, and 4.0% @ 2.0 years) remains valid, but a different bond trades at a price of only $95.12. Which of the following is nearest to this bond’s spread?

Each of the following statements is true about the arbitrage pricing theory (APT) model EXCEPT which is false?

Let X = µ + σ*T be the generalized student’s t distribution where X is a linear transformation of the classic student’s t distribution, as represented by T ~ t(df) or sometimes T ~ t(v). Consequently, X is a generalized random variable denoted by X ~ Gen. t(v)(µ, σ^2). If the generalized student’s t distribution happens to be standardized, then it could be represented as X ~ Gen. t(v)(0, 1). If X is a generalized student’s t distribution with µ = 2.0 and σ = 3.0 with degrees of freedom, v = 8, then each of the following is true EXCEPT which is false?

Consider the following upward-sloping but smooth zero rate curve:

There are various theories that attempt to explain the factors that determine the shape of the zero rate curve. If the above zero curve is observed, each of the following theories is plausible EXCEPT which of the following theories is the LEAST LIKELY to be true, if only because it does not comport with the observed zero curve?

Below is a graph of swap rates (represented by the solid green line) and the corresponding, implied six-month forward rates (represented by the dotted orange line). Specifically, the six-month, 1.0-year, 1.5-year, 2.0-year and 2.5-year swap rates are, respectively, 0.70%, 1.20%, 1.50%, 1.50% and 2.0%. The six-month forward rates implied by this swap rate curve are: f(0, 0.5) = s(0.5) = 0.7000%; f(0.5, 1.0) = 1.7043%; f(1.0, 1.5) = 2.1089%; f(1.5, 2.0) = 1.5000%; f(2.0, 2.5) = 4.0674%.

Consider the fixed side of a 2.5-year swap which pays the swap rate of 2.0%. Because 2.0% is the 2.5-year par rate, the present value of 100 face amount of the fixed side of the 2.5-year swap is 100. Consider the price of the swap over the following year, in two six-month steps:

• After six months, as the swap ages (toward maturity) from 2.5-year to a 2.0-year swap (but with the same fixed rate, of course)
• After another six months, as the swap ages from a 2.0-year swap to a 1.5-year swap

If we assume an unchanged term structure as the swap matures from a 2.5-year swap to a 1.5-year swap, what happens to the price of the fixed side?

Relative to the other principles, compliance rates have been stronger for the Risk Reporting Practices outlined in BCBS 239. In regard to effective risk reporting, each of the following is true EXCEPT which is false?

Let Z be a random variable that is a linear function of random variables X and Y, where Z = 3*X + 7*Y? If the standard deviation of X and Y, respectively, are σ(X) = 4.0 and σ(Y) = 5.0 and the correlation between X and Y is ρ(X,Y) = 0.50, then what is the standard deviation of Z, σ(Z)?

A three-year bond $1,000.00 face value bond pays a 10.0% semi-annual coupon and has a semi-annual (aka, bond equivalent) yield of 14.0%. Its price is therefore $904.67. The chart below also shows cash flows as proportional weights:

We can use modified duration to estimate the price impact of a small change in yield. Which of the following is NEAREST to a duration-based (i.e., linearly approximate) estimate of the bond’s price change given a 26 basis point (0.26%) drop (shock down) to the yield?

Assume the following discount function (note that a discount function is a series of discount factors):

Which of the following is nearest to the implied six-month forward rate starting in two years, f(2.0, 2.5)? Assume rates are expressed as per annum with semi-annual compounding.

According to GARP, each of the following is one of the top ten benefits of enterprise risk management (ERM) EXCEPT which is NOT a benefit of ERM?

Among a set of filtered stocks, a stock screener assigns stocks to one of three style categories: value, quality, or momentum. At the end of each month, the stock’s performance is compared to the S&P such that it either beats or does not beat the index The prior beliefs (aka, unconditional probabilities) are the following: Pr(Style = Value) = 15.0%, Pr(Style = Quality) = 30.0%, and Pr(Style = Momentum) = 55.0%. The stock screener also knows that a Moment stock is more likely than a Quality stock, and much more likely than a Value stock, to beat the index; specifically, the screener knows the following conditional probabilities:

• Pr(Beat | Value) = 40.0%
• Pr(Beat | Quality) = 60.0%
• Pr(Beat | Momentum ) = 80.0%

If we observe that a stock beats the index, what is the probability it is a momentum stock; i.e.., what is Pr(Momentum | Beat)?

Bonus question: if we observe the stock beats the index two months in a row, what is the probability it is a momentum stock; i.e., what is Pr(Momentum | Two consecutive Beats)?

Consider the steep spot (aka, zero) rate curve illustrated below: 3.0% at 0.5 years, 4.0% at 1.0 year, 4.6% at 1.5 years and 5.0% at 2.0 years. Each of these zero rates is per annum with continuous compounding.

Which of the following is nearest to the theoretical price of a two-year $100.00 face value bond that pays an 8.0% semi-annual coupon (4.0% coupon every six months)?

In regard to Tuckman’s discussion of the components and structure of U.S. Treasury STRIPS, which of the following statements is TRUE?

Scenario analysis is an ascendant tool. After the global financial crisis (GFC), regulators insisted that systemically important banks demonstrate their ability to withstand adverse and severely adverse scenarios. Meanwhile, GARP explains, “Scenario analysis, along with stress and sensitivity testing, have risen to become the preeminent risk identification tools for many enterprise risk management (ERM) programs. This is a result of the weaknesses in probabilistic risk metrics (e.g., VaR) that were revealed by the global financial crisis of 2007–2008.”[1]

Which of the following statements is TRUE about scenario analysis?

[1] 2020 FRM Part I: Foundations of Risk Management, 10th Ed. Pearson Learning Solutions, 10/2019. VitalBook file.

Barbara observes a sample with the following statistics: mean of X and standard deviation of Y. She thinks the true (aka, population) mean is Z. She does NOT know the population’s variance, nor does she even know the population’s distribution, including she cannot assume it is normal. If she wants to conduct a hypothesis test of the observed sample mean, each of the following statements is true EXCEPT which is incorrect?

Sally the portfolio manager oversees a $9.0 million large-cap equities portfolio with a beta of 1.30. She has decided the portfolio’s beta is too high and calibrates a new target beta of 0.70. If she employs S&P 500 index futures contracts to reduce the beta when the index futures price is 2,400 (the contract size is $250 * S&P 500 index per the specification), then which is nearest to the trade?

On Monday, a bank’s position (the underlying portfolio) on the dollar-euro exchange rate has an initial position delta of 11,300 and a position gamma of -25,000. The exchange rate is EURUSD $0.860; i.e., per the currency priority convention, Euro is the base currency. By the end of the week, on Friday, the exchange rate had jumped by +$0.050 to $0.910. The bank entered two trades:

• On Monday, while the exchange rate is EURUSD $0.860, the bank’s first trade neutralized delta; aka, makes the net position “delta neutral”
• On Friday, after the exchange rate had increased to EURUSD $0.910, the bank’s second trade re-established the net position’s delta neutrality

Which of the following were the bank’s trades? Please note: this question is inspired by Hull’s EOC Problem 19.22[1].

[1] John C. Hull, Risk Management and Financial Institutions, 4th Ed. (Hoboken, NJ: John Wiley & Sons, 2015).

Three famous financial disasters are Continental Illinois (whose failure in 1984 was the largest bank failure in US history prior to the global financial crisis), Lehman Brothers (who was the fourth-largest U.S. investment bank before filing for bankruptcy in 2008), and Northern Rock (who was a fast-growing British mortgage bank, who in 2008 was forced into public ownership as an alternative to insolvency). What did Continental Illinois, Lehman Brothers, and Northern Rock have in common?

An analyst at your firm has developed a new trading strategy called AlphaGen. She claims there is a 70.0% probability on any given day that the strategy offers true alpha and will generate a profit. After observing ten (10) days of performance, in fact the strategy was profitable only four days. Assume there are only two possible states of the world: either the analyst is correct and the strategy offers true alpha; or the strategy does not offer true alpha and the profit/loss outcome is equally likely to be a gain or loss on any given day. Your prior assumption was that the two states of the world are equally likely: there is s 50.0% probability the strategy offers true alpha, and a 50% probability it does not. These assumptions are illustrated below; e.g., if the strategy does generate alpha, then P[Profit | Alpha] = 70.0%, but if the strategy does not generate alpha, then P[Profit | No Alpha] = 50.0%.

We can also represent the prior probabilities as follows: P[p = 0.70] = 50.0% and P[p = 0.50] = 50.0%. Additionally, because you are an FRM candidate, you are already able to compute the following binomial probabilities:

• Prob[Exactly 4 profits out of 10 days | True Alpha] = binomial[4 successes, 10 trials, p = 0.70, false = p.m.f.] = 0.70^4*0.30^6*C(10,4) = 3.68%
• Prob[Exactly 4 profits out of 10 days | No Alpha] = binomial[4 successes, 10 trials, p = 0.50, false = p.m.f.] = 0.50^4*0.50^6*C(10,4) = 20.51%.

Given the evidence that the strategy was profitable on four days out of ten, which is nearest to the (posterior) probability that the analyst is correct, and the strategy generates true alpha? (note: this question is inspired by Miller’s EOC question 6.4)[1].

[1] Michael Miller, Mathematics and Statistics for Financial Risk Management, 2nd Edition (Hoboken, NJ: John Wiley & Sons, 2013)

Barbara is a value investor who bought 1,000 shares of Apple (ticker: AAPL) in 2014 when the share price was $95.00 and she considered them under-valued. In hindsight, her view was correct as the shares currently (as of mid-2017) trade at $140.00. She now thinks the shares are slightly over-valued. However, she does not want to sell them unless there is a market crash. This is because she believes the shares are likely to trade in a range and may even gain modestly in the future. However, she also believes there is something like a 10.0% probability of a technology sector crash (a possibility enabled by the low interest rate regime). If the technology sector does crash, she fears the AAPL shares could plummet. If the AAPL shares drop, Barbara does want to sell, however, she does not want to sell in a panic at fire-sale prices.

Specifically, if the shares were to quickly lose more than 13.0% of their current value, Barbara will be eager to sell them. However, she also wants to ensure that she realizes a minimum holding period return (HPR) of 30.0%; and to this HPR dividends have already contributed 6.0%. Therefore, she only wants to sell if the price appreciation (from her $95.00 cost basis) is at least +24.0%. She justifies this conditional view on a belief that if the shares plunge too far such that she cannot realize her HPR threshold, the market will have overreacted. In this case of an over-reaction, she believes it will be better to avoid selling in a panic, and instead, she will be better off to await an eventual recovery. Which of the following orders is most consistent with her strategy?

A market maker takes a short position in 1,000 European call options on copper futures. The options mature in three months, and the futures contract underlying the option matures in four months. The current four-month futures price is $3.30 per pound, the exercise price of the options is $3.00, the risk-free interest rate is 4.0% per annum, and the volatility of silver futures prices is 20.0% per annum. Which is nearest to the position delta? (note: this question is inspired by Hull’s EOC Question 19.10)[1]

[1] John C. Hull, Risk Management and Financial Institutions, 4th Ed. (Hoboken, NJ: John Wiley & Sons, 2015).

In early 2012, J.P. Morgan Chase (JPM) lost billions on trades executed by the London Whale, the notorious nickname for Bruno Iksil, who assumed massive exposures (masquerading as hedges) in a large credit derivative portfolio. Which of the following BEST summarizes the root cause of the debacle?

Your colleague Peter is selecting probability distributions in order to perform several Monte Carlo simulations. Each of the following choices appears to be logical or sensible, EXCEPT which choice prima facie appears be a mistake?

PlanetZim Financial Bank just entered a position in a derivatives contract. Which of the following features of the derivative position is MOST likely to indicate the trade is a case of SPECULATION, in contrast to a case of a hedge, arbitrage, or market-making?

Mark the Risk Analyst is evaluating an at-the-money (ATM) call option on a non-dividend-paying stock when the stock’s price is $30.00; i.e., S(0) = K = $30.00. For this option, he has the table shown below. The table displays option prices generated by the Black-Scholes-Merton option pricing model according to selected input variations of volatility and maturity. In other words, the table displays different values of c = BSM[S(0) = $30.00, K = $30.00, σ, Rf = 0.030, T].

Mark’s option actually expires in eight months; T = 8/12. Further, he observes that the option’s traded price (aka, observed market price) is $3.00. In this case, which of the following is nearest to the call option’s implied volatility?

The Volkswagen emissions scandal concerned over ten million cars during the years 2009 to 2015. The scandal unfolded with significant financial repercussions and massive reputational damage to the company. GARP writes that “the damage to Volkswagen, the world’s biggest carmaker, was significant … Its reputation, particularly in the important US market, took a severe hit. The reputational effect extended beyond the company itself as German government officials expressed concerns that the value of the imprimatur ‘Made in Germany’ would be diminished because of Volkswagen’s actions.” Which of the following most accurately summarizes the Volkswagen emissions case study?

Portfolio manager Peter manages a large portfolio with 100 component positions. He is interested in analyzing the non-trivial cross moments in the portfolio (trivial cross-moments are the position’s coskew/cokurtosis with itself, which is simply the position’s standard skew or kurtosis, so these are analogous to the diagonal of a covariance matrix which is mere variances. Each of the following statements is true EXCEPT which is inaccurate?

The Investment Committee at your endowment just analyzed the historical performance of its asset allocation to hedge funds, which was 20.0% of the fund. It has determined that net of fees these hedge funds did not outperform the S&P 500 on a risk-adjusted basis. Consequently, the Committee wants to re-allocate this portion to a fund that tracks the S&P 500 index; and the Committee is comfortable mirroring the index with minimum tracking error. An outside consultant proposes an exchange-traded fund (ETF) such as the “Spider” (ticker SPY), but some members want to compare the ETF to an open-ended or closed-ended mutual fund that tracks the S&P 500.

In addition to highlighting the fact that the expense ratios tend to be lower for ETFs than mutual funds, the consultant offers the following arguments in favor of an ETF:

I. In contrast to an open-ended mutual fund, the advantage of the SPDR ETF can be traded at any time, can be shorted, and does not have to be partially liquidated to accommodate redemptions

II. In contrast to a closed-ended mutual fund whose price tends to trade at a discount to its fair market value, there is never any appreciable difference between the traded price of the SPDR EFT and its fair market value.

Which of the consultant’s argument(s) is (are) TRUE?

A stock has a current price of $71.00 and follows geometric Brownian motion. It offers an expected return of 14.0% with volatility of 23.0% per annum. Which is nearest to the probability that a deeply out-of-the-money (OTM) European call option on the stock with a maturity of one year (T = 1.0 year) and an exercise price of $100.00 will be exercised?

According to GARP, “a wave of uncertainty over the valuation of asset-backed structured products exacerbated the [global financial] crisis” as counterparty risk suddenly became priced expensively by counterparties. Consequently, the valuation of these illiquid assets became problematic due to their inherent opacity (until mid-2007 counterparty risk was a factor that effectively had NOT even been priced by the market!). Coincidentally, with these VALUATION and TRANSPARENCY issues, which of the following is TRUE about the ensuing liquidity crunch?

Below are the joint probabilities for a cumulative bivariate normal distribution with a correlation parameter, ρ, of 0.30.

If V(1) and V(2) are each variables characterized by a uniform distribution, which is nearest to the joint probability Pr[V(1) < 0.050, V(2) < 0.050] under a Gaussian copula model?

A defined benefit pension fund is 50.0% invested in equities and 50.0% invested in bonds. If we assume the simplest possible balance sheet, which is MOST LIKELY to be the net effect of a scenario where equities are approximately flat, but interest increase by 100 basis points? Please note this is inspired by Hull’s EOC Question 3.18[1], so it makes simplifying assumptions such as (i) the rate increase is a parallel shift of both short- and long-term interest rates, (ii) durations are not managed, and (iii) the fund is not hedged.

[1] Source for Hull’s 3.18: John C. Hull, Risk Management and Financial Institutions, 5th edition (Hoboken, New Jersey: John Wiley & Sons, 2018).

Peter initially values a one-year European put option on a non-dividend-paying (q = 0%) stock with the following assumptions:

• One-year put option maturity with twelve steps in the binomial model (one for each month): Δt = 1/12
• The strike price is equal to the stock’s current price of $40.00; i.e., at-the-money
• Volatility of stock, σ = 25.0% per annum
• Riskfree rate is 3.0% per annum with continuous compounding, r = 3.0%

Once he performs the twelve-step valuation, he decides to re-price the same option but with only one difference: he will increase the number of steps in the binomial tree to 50 (one for each week) so that each time step Δt = 1/50. In regard to the switch from twelve steps to 50 steps (i.e., Δt = 1/12 → Δt = 1/50), each of the following statements is true EXCEPT which is false?

Peter is a Risk Analyst (and certified FRM) tasked by his firm to evaluate a half dozen vendors and make a recommendation to purchase one of their risk management software packages. The firm has a specific need, and the Risk Committee of the board has given the list of six vendors to Peter because they are each reputable vendors in the software category. Upon seeing the list, Peter realizes that he is friends with one of the CEOs (among the six vendors) because they both volunteer their time to a local charity and, due to this shared interest, they frequently socialize together.

According to GARP’s Code of Conduct, should Peter take any action with respect to a potential conflict of interest?

The data and plot below show a bivariate normal sample distribution. The first two columns, z(1) and z(2), display random standard normal variables, N(0,1). Because X(1) is a (non-standard) random normal variable with mean, µ, of 2.0 and standard deviation, σ, of 4.0, it is given by X(1) = 2.0 + 4.0 * z(1). On the other hand, X(2) is correlated with X(1) according to the selected correlation parameter, ρ(X1, X2), of 0.80.

What is the missing third realization (“???”) of X(2)?

Below is an extract (selected rows) from a mortality table:

Which is nearest to the probability of a man aged 80 years old dying in the second year (between ages 81 and 82)?

In the event of certain operational loss, the severity of the loss ranges from three to 18. As luck would have it, the loss distribution is similar to the roll of three dice.

For example, in the event of a loss occurrence, the probability of a loss of either three (the minimum loss) or 18 (the maximum loss) is given by approximately (1/6)^3 = 0.4630%; the probability of a loss of four or 17 is 3/6^3 = 1.389%. Which are nearest, respectively, to the 95.0% value at risk (VaR) and 95.0% expected shortfall (ES) for the severity of this loss?

Gorton says, “The recent crisis is often described as being the worst global crisis since the Great Depression, and the evidence supports this label.”[1] The financial crisis included two main panic periods: August 2007 and September‐October 2008. Gorton’s literature review compares the crisis to the long history of previous and numerous financial crises. This comparison produced two interesting features: on the one hand, the authors found a crucial SIMILARITY to historical predicates but, on the other hand, they found a novel DIFFERENCE from previous crises. Which of the following best summarizes, respectively, the feature common (aka, similarity) to previous crises and the novel difference observed in the recent crisis?

[1] Gary Gorton and Andrew Metrick, 2012. “Getting Up to Speed on the Financial Crisis: A One-Weekend-Reader’s Guide”, Journal of Economic Literature 50:1, 128—150.

You have assumed a single-index model and regressed the returns for two stocks, stock (A) and stock (B), against the index in separate univariate regressions. This produced the following two linear functions, such that β(A) is 0.80 and β(B) is 1.40:

Additionally, the volatility of the index, σ(M), is 20.0%, the volatility of stock A, σ(A), is 32.0% and the volatility of stock B, σ(B), is 40.0%. Which of the following is the implied correlation between the two stocks?

You are meeting with your FRM study group when one of the members of the group says they are a bit unclear on the definition of the term “short hedge.” The following conversation ensues:

I. Albert says, “It’s simple, if a company owns an asset but wants to hedge its plan to sell the asset at the future spot price, a short hedge is appropriate”​

II. Barbara says, “Yes, Albert that is true, but if the company plans to sell the commodity in the future at a predetermined price, then a long hedge is appropriate!”​

III. Chris says, “Barbara is correct because a short hedge is simply a hedge where a short futures position is taken.”​

IV. Donald says, “Exactly true, Chris. And that means that a short hedge can also be a cross-hedge; i.e., these terms are not mutually exclusive.”​

V. Erin says, “And I would like to add that the company does not need to own the asset in order to conduct a short hedge.”​

VI. Fred says, “And I would like to add that a short hedge implies negative basis, just as a long hedge implies positive basis.”​

Which of the statements is (are) accurate?

Patricia the Portfolio Manager owns an option portfolio that contains both long and short positions in European call and put options. The position gamma of the portfolio is negative, specifically it is -19,500. Recall that position gamma is the product of option quantity and each option’s percentage gamma. For example, a long position in 10,000 call options that each have a percentage (per option) gamma of 0.050 has a position gamma of +500; on the other hand, a short position in the same number of options has a position gamma of -500. Position gamma can be summed across the option portfolio.

Patricia initially estimates her option portfolio’s value at risk (VaR) based on a delta approximation. Her analysis quotes VaR in Loss(+)/Profit(-) format; aka, L/P units. In this way, losses and VaR are expressed as positive values. However, she realizes that such an estimate omits the portfolio’s negative position gamma, so she re-computes the portfolio’s VaR by using a DELTA-GAMMA approximation. How does the revised estimate compare to the first delta-only estimate?

Suppose that three factors have been identified for the U.S. economy:

• Expected inflation rate (IR) is +2.00%
• Expected 10-year Treasury yield (T-NOTE) is 2.40%
• Expected growth in productivity (PROD) is +3.00%

A stock with an expected return of 9.0% has the following betas with respect to these factors: β(IR) = +1.50, β(T-NOTE) = -1.20 and β(PROD) = 0.70. In turns out that that economy’s actual factor performance is given by the following set of results:

• Actual inflation rate (IR) is + 2.60%
• Actual 10-year Treasury yield (T-NOTE) is 3.00%
• Actual growth in productivity (PROD) +2.00%

What is the revised estimate of the stock’s expected rate of return (note: this is a variation on Bodie’s Problem 10.1)?[1]

[1] Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition (New York: McGraw-Hill, 2013)

You would like to describe an account that begins at TIME(0) = $100.00 and compounds continuously at 9.0% per annum. What is a function that characterizes the value of this account, A(t), over time according to such a continuous and constant growth trend?

The current price of a technology index is 3,000 and its yield, q, is 2.0% per annum with continuous compounding; i.e., about 2.010% per annum with semi-annual compounding. The risk-free rate is 3.0% per annum with continuous compounding. The discount rate for the index can be determined by the capital asset pricing model (CAPM) where its beta is 1.80 and the market’s expected return is 9.0%; i.e., the market’s expected excess return is 6.0%. Which is nearest to the index’s expected future spot price in 10 months, E[S(+0.833)]?

Which of the following statements is TRUE about the role of validation in stress testing governance?

Assume a sock’s returns are accurately characterized by the Fama-French three-factor model. The stock’s own factor betas and the risk model’s risk premiums are given below:

What is the firm’s size/value style characteristics and what is the firm’s required rate of return?

What is the crucial difference between bootstrapping and Monte Carlo simulation?

The price of a 72-day Treasury bill is quoted as 7.00. Which is nearest to the continuously compounded return (on an actual/365 basis) that an investor will earn on the Treasury bill for the 72-day period? (note: inspired by Hull’s EOC Problem 6.8)[1]

[1] John C. Hull, Options, Futures, and Other Derivatives, 9th Edition (New York: Pearson Prentice Hall, 2014)

Linda Allen[1] introduces the following value at risk (VaR) estimation approaches:

• Historical simulation (HS)
• GARCH (1,1)
• Hybrid
• Multivariate density estimation (MDE)
• Historical standard deviation (STDEV)
• Adaptive volatility (AV)

Consider these six summary descriptions:

I. The simplest parametric approach whose weakness is sensitivity to window length and extreme observations

II. The most convenient and prominent non-parametric approach whose weakness is inefficient use of data

III. An interpretation of the exponentially weighted moving average (EWMA) that gives the risk manger a rule that can used to adapt prior beliefs about volatility in the face of news

IV. A parametric approach that assumes conditional returns are normal but unconditional tails are heavy; and that returns are not correlated but conditional variance is mean-reverting

V. An approach that weights past squared returns not by time but instead according to the difference between current and past states of the world

VI. An approach that modifies historical simulation by assigning exponentially declining weights to past data such that recent (distant) returns are assigned more (less) weight

Which sequence below correctly matches the VaR estimation approach with its summary description?

[1] Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach (Oxford: Blackwell Publishing, 2004)

According to Ben Bernanke, among the following trigger/vulnerability pairs, which were the primary TRIGGER and (among) the primary VULNERABILITIES, respectively, that led to the financial crisis?

During a recent workweek, Peter recorded the number of dropped calls to the company’s support line. For the tiny sample of five days (n = 5), the average number of dropped calls was 9.60 per day. Peter used Excel’s VAR.P() to retrieve the variance and the result was 1.840. However, his colleague Mary pointed out that this is a biased estimate of the unknown population variance. What is the unbiased estimate of the variance?

A $100.0 million interest rate swap has a remaining life of 15 months. Under the terms of the swap, six-month LIBOR is exchanged for 3.60% per annum (compounded semiannually). Six-month LIBOR forward rates for all maturities are 3.00% (with semiannual compounding). Two Three months ago, the six-month LIBOR rate was 2.90% (this assumption is shown in purple cell below). OIS rates for all maturities are 2.80% with continuous compounding.

Which is nearest to the current value of the swap to the counterparty who is paying the floating rate? (Inspired by Hull’s EOC Problem 7.2, 10th Edition)[1]

[1] Hull, Options, Futures, and Other Derivatives, 10th Edition (New York: Pearson, 2017).

Rebecca has determined that her equity portfolio’s 25-day 95.0% confident absolute value at risk (aVaR) is given by -µ*Δt + σ*α*sqrt(Δt) = -12,000 + 208,000 = $196,000. She subsequently decides that she wants to translate this into a 10-day 99.0% confident aVaR. If the returns are i.i.d. and normally distributed, which of the following is nearest to the translated VaR?

Among the Basel Committee’s Principles for effective risk data aggregation and risk reporting, Principle 2 is “Data architecture and IT infrastructure: A bank should design, build and maintain data architecture and IT infrastructure which fully supports its risk data aggregation capabilities and risk reporting practices not only in normal times but also during times of stress or crisis, while still meeting the other Principles.”[1] The principle includes paragraph 33, where two terms have been replaced with “[keyword #1]” and “[keyword #2]”:

“A bank should establish integrated data [keyword #1 here] and architecture across the banking group, which includes information on the characteristics of the data–i.e., [keyword#2 inserts here]–as well as use of single identifiers and/or unified naming conventions for data including legal entities, counterparties, customers and accounts.” The first keyword, [keyword #1], refers to the categorization or classifications of data; for example, market risk and credit risk are categories of risk. The second keyword, [keyword #2], refers to information about the data.”[1]

Which terms correctly replace, respectively, [keyword #1] and [keyword #2]?

[1] “Principles for Sound Stress Testing Practices and Supervision” (Basel Committee on Banking Supervision Publication, May 2009)

Peter Parker at Betalab Bank emailed a survey to the bank’s customers. The survey included a question that asked them to rank their customer satisfaction on a scale from one to 10. She received 51 responses, and she considers that a random sample (n = 51). Among this sample, the average satisfaction score (on a scale of one to 10) is 8.50 with a sample standard deviation of 1.90. Betalab’s CEO is Mary-Jane, FRM, and she hopes that the bank’s average customer satisfaction is at least 9.0. Mary-Jane holds the FRM designation, so she understands that acceptance of the null is more accurately a failure to reject the null, but she is a practical person. Her null hypothesis is that the population’s average customer satisfaction is at least 9.0 (i.e., H0: μ ≥ 9.0 and H1: μ < 9.0). Peter shares his sample findings with five of his colleagues, and each colleague gives different input, as follows:

I. Albert says the test statistic is (8.5 – 9.0) ÷ [1.90 / SQRT(51)] = -1.88, or |-1.88| = 1.88​

II. Betty says that if the sample size were doubled, ceteris paribus (i.e., same sample mean and sample standard deviation), the test statistic will increase about +41%​

III. Chris says that (for n = 51) Mary should use a one-sided test, and with one-sided 95.0% confidence (aka, 5.0% significance) she should reject the null​

IV. Derek says that (for n = 51) Mary can accept (aka, fail to reject) the CEO’s null hypothesis with 95.0% confidence but only if she artificially switches to a two-sided hypothesis (i.e., H0: μ = 9.0 and H1: μ ≠ 9.0). Notice how this insincerely “games” the test: the setup here calls for a one-sided test.​

V. Erin agrees with Chris and says that Mary should use a one-sided test per the CEO’s one-sided hypothesis but notes that Peter can accept (aka, fail to reject) the null with one-sided 99.0% confidence (aka, 1.0% significance)​

Which of the five statements is (are) correct?

As she analyzes a stock, Barbara contemplates a straddle because she wants to express her view that there will be a big change in the stock price, either dramatically up or down, but she is uncertain as to the direction. In this way, she wants an option combination strategy that is “long volatility.” However, the straddle is more expensive than she anticipated. Which of the following will allow her to generally express her “long volatility” view (giving her the potential for a large, or even uncapped, payoff in the event of a dramatic price move) but with a cost that is REDUCED in comparison to the straddle?

Shown below is a bank’s portfolio of wholesale exposures (this is Siddique and Hasan’s Table 2.2):[1]

The bank needs to estimate the sensitivities of this portfolio’s losses to changes in two macro variables: gross domestic product (GDP) growth and unemployment. If the bank conducts a stress test, each of the following is likely–or at least reasonable–to be included in the bank’s stress test assumptions EXCEPT which is the LEAST likely to be an assumption in the stress test?

[1] Akhtar Siddique and Iftekhar Hasan, Stress Testing: Approaches, Methods, and Applications (London: Risk Books, 2013)

“Agency costs” is a term used to describe the costs in a situation where the interests of two parties are not perfectly aligned. In the events leading up to the Global Financial Crisis (CFC; aka, credit crisis), which of the following was a source of agency cost?

Barbara just received a dataset. She runs the dataset through five different regression models, where each regression model employs a different estimator for the slope. The properties of the slope estimators include the following:

• Estimator A is biased but has the smallest variance (3.7)
• Estimator B is linear and biased but has a small variance (4.6)
• Estimator C is linear and unbiased but has a medium variance (9.5)
• Estimator D is nonlinear and biased but is consistent and has a large variance (11.8)
• Estimator E is nonlinear and unbiased but has the largest variance (14.1)

Which of these estimators is BLUE?

Patricia wants to hedge her portfolio of exotic options. The portfolio consists mostly of barrier options. She is comparing a classic delta-hedge to a static options replication; the static option replication entails shorting a portfolio that replicates certain boundary conditions. Each of the following is a good argument in favor of a static option replication, for the purpose of hedging her portfolio, EXCEPT which is WEAKEST argument?

Assume we follow Tuckman’s[1] illustration and estimate the key rate exposures (i.e., KR01s and key rate durations) for a zero-coupon bond with 30 years to maturity; e.g., C-STRIP. The key interest rate is a par yield. The analysis selects four key rates: 2-year, 5-year, 10-year, and 30-year. Further, as Tuckman explains about the rule for rates between neighbors, “each of the four shapes is called a key-rate shift. Each key rate affects par yields from the term of the previous key rate (or zero) to the term of the next key rate (or the last term). For example, the 10-year key rate affects par yields of terms 5 to 30 years only. Furthermore, the impact of each key rate is normalized to be one basis point at its own maturity and then assumed to decline linearly, reaching zero at the terms of the adjacent key rates. For the two-year shift at terms of less than 2 years and for the 30-year shift at terms greater than 30 years, however, the assumed change is constant at one basis point.”[1]

About this key rate exposure technique, each of the following statements is true EXCEPT which is false?

[1] Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)

Each of the following is true about the re-empowered role of the Chief Risk Officer (CRO) EXCEPT which is false?

Over the previous six months, two assets generated the monthly returns displayed below under columns X(i) and Y(i). The third and fourth columns display the squared difference-from-average-returns that inform the univariate variance; the fifth column shows the associated cross-product that informs their covariance.

Because this is a very small sample, we want to retrieve unbiased sample moments. What are, respectively, the unbiased sample variances of the two variables and their sample covariance?

The price of a dividend-paying stock is $44.00 while the riskfree rate is 3.0%. Consider a European call option and a European put option with identical strike prices, K = $40.00, and identical times to expiration of nine months, T = 0.75 years. The call has a price of $8.95 and the put has a price of $5.36. What is the present value of the dividends expected during the life of the option?

Consider the following one-year rating transition (aka, migration) matrix illustrated below:

About this migration matrix, each of the following statements is true EXCEPT which is false?

Crouhy writes that “understanding [the difference between expected loss and unexpected loss] is the key to understanding modern risk management concepts such as economic capital attribution and risk-adjusted pricing.”[1]

Which of the following statements is TRUE about unexpected loss (UL)?

[1] Michel Crouhy, Dan Galai, and Robert Mark, The Essentials of Risk Management, 2nd Edition (New York: McGraw-Hill, 2014)

A stock (X1) has three possible returns: -5%, 0, or +5%. The analyst rating (X2) can be negative (denoted by X2= -1), neutral (X2 = 0), or positive (X2 = +1). The probability matrix is displayed below (inside the square). Six joint probabilities are given, but three are missing; for example, the joint probability of a negative analyst rating and a negative stock return, P(X1 = -5% ∩ X2 = -1) = 14.0%.

The bottom row (outside the square) displays the unconditional (aka, marginal) probabilities for the stock; for example, the unconditional Pr(X1 = -5%) = 20.0%. What is the unconditional (aka, marginal) probability that the analyst rating is positive, Pr(X2 = +1)?

The spot price of wheat is $5.00 per bushel while the risk-free rate is 3.0% per annum with continuous compounding. The cost to store wheat is 12.00% per annum as a proportion of the spot price. The traded (observed) price of a nine-month wheat futures contract, F(0, 0.75), is $5.430. Among the following choices, which of the following scenarios is the most likely?

Peter works for an ESG fund that evaluates political risk. Among the components of political risk, Peter’s fund assigns the greatest weight to corruption. If corruption is the most important criteria, into which of the following countries is Peter’s firm MOST LIKELY to invest; i.e., which is least corrupt?