Articles by kenli

LO 1.6: Evaluate estimators o f risk measures by estimating their standard errors.
Sound risk management practice reminds us that estimators are only as useful as their precision. That is, estimators that are less precise (i.e., have large standard errors and wide confidence intervals) will have limited practical value. Therefore, it is best practice to also compute the standard error for all coherent risk measures.
Professors Note: The process o f estimating standard errors for estimators o f coherent risk measures is quite complex, so your focus should be on interpretation o f this concept.
First, lets start with a sample size of n and arbitrary bin width of h around quantile, q. Bin width is just the width of the intervals, sometimes called bins, in a histogram. Computing standard error is done by realizing that the square root of the variance of the quantile is equal to the standard error of the quantile. After finding the standard error, a confidence interval for a risk measure such as VaR can be constructed as follows:
[q + se(q)X za ] > VaR > [q se(q)X za ] 2018 Kaplan, Inc.
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Example: Estimating standard errors
Construct a 90% confidence interval for 3% VaR (the 93% quantile) drawn from a standard normal distribution. Assume bin width = 0.1 and that the sample size is equal to 500.
Answer:
The quantile value, q, corresponds to the 5% VaR which occurs at 1.65 for the standard normal distribution. The confidence interval takes the following form:
[1.65 + 1.65 x se(q)] > VaR > [1.65 1.65 x se(q)] Professors Note: Recall that a confidence interval is a two-tailed test (unlike VaR), so a 90% confidence level will have 5% in each tail. Given that this is equivalent to the 5% significance level o f VaR, the critical values o f 1.65 will he the same in both cases.
Since bin width is 0.1, q is in the range 1.65 0.1/2 = [1.7, 1.6]. Note that the left tail probability, p , is the area to the left o f1.7 for a standard normal distribution.
Next, calculate the probability mass between [1.7, 1.6], represented 2&f(q). From the standard normal table, the probability of a loss greater than 1.7 is 0.045 (left tail). Similarly, the probability of a loss less than 1.6 (right tail) is 0.945. Collectively, f(q) = 1 -0 .0 4 5 -0 .9 4 5 = 0.01
The standard error of the quantile is derived from the variance approximation of q and is equal to:
yp(l ~ p ) / n
f(q)
Now we are ready to substitute in the variance approximation to calculate the confidence interval for VaR:
1.65 + 1.65V 4 –
0.01
= 3.18 > VaR > 0 .1 2
‘ 500 > V aR >
, 1.65 1.65
, , c V0.045(l 0.045) / 500
0.01
Lets return to the variance approximation and perform some basic comparative statistics. What happens if we increase the sample size holding all other factors constant? Intuitively, the larger the sample size the smaller the standard error and the narrower the confidence interval.
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2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Now suppose we increase the bin size, h, holding all else constant. This will increase the probability mass/fg’) and reducep , the probability in the left tail. The standard error will decrease and the confidence interval will again narrow.
Lastly, suppose that p increases indicating that tail probabilities are more likely. Intuitively, the estimator becomes less precise and standard errors increase, which widens the confidence interval. Note that the expression p(l p) will be maximized at p = 0.3.
The above analysis was based on one quantile of the loss distribution. Just as the previous section generalized the expected shortfall to the coherent risk measure, we can do the same for the standard error computation. Thankfully, this complex process is not the focus of the LO.
Quantile-Quantile Plots

LO 1.3: Estimate risk measures by estimating quantiles.
A more general risk measure than either VaR or ES is known as a coherent risk measure. A coherent risk measure is a weighted average of the quantiles of the loss distribution where the weights are user-specific based on individual risk aversion. ES (as well as VaR) is a special case of a coherent risk measure. When modeling the ES case, the weighting function is set to [1 / (1 confidence level)] for all tail losses. All other quantiles will have a weight of zero.
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Under expected shortfall estimation, the tail region is divided into equal probability slices and then multiplied by the corresponding quantiles. Under the more general coherent risk measure, the entire distribution is divided into equal probability slices weighted by the more general risk aversion (weighting) function.
This procedure is illustrated for n = 10. First, the entire return distribution is divided into nine (i.e., n 1) equal probability mass slices at 10%, 20%, …, 90% (i.e., loss quantiles). Each breakpoint corresponds to a different quantile. For example, the 10% quantile (confidence level = 10%) relates to 1.2816, the 20% quantile (confidence level = 20%) relates to 0.8416, and the 90% quantile (confidence level = 90%) relates to 1.2816. Next, each quantile is weighted by the specific risk aversion function and then averaged to arrive at the value of the coherent risk measure.
This coherent risk measure is more sensitive to the choice of n than expected shortfall, but will converge to the risk measures true value for a sufficiently large number of observations. The intuition is that as n increases, the quantiles will be further into the tails where more extreme values of the distribution are located.

LO 1.4: Define coherent risk measures.

LO 1.2: Estimate VaR using a parametric approach for both normal and lognormal return distributions.
In contrast to the historical simulation method, the parametric approach (e.g., the delta- normal approach) explicitly assumes a distribution for the underlying observations. For this LO, we will analyze two cases: (1) VaR for returns that follow a normal distribution, and (2) VaR for returns that follow a lognormal distribution.
Norm al VaR
Intuitively, the VaR for a given confidence level denotes the point that separates the tail losses from the remaining distribution. The VaR cutoff will be in the left tail of the returns distribution. Flence, the calculated value at risk is negative, but is typically reported as a positive value since the negative amount is implied (i.e., it is the value that is at risk). In equation form, the VaR at significance level a is:
VaR(a% ) = pp/L + CTP/L x za
where p and a denote the mean and standard deviation of the profit/loss distribution and 2: denotes the critical value (i.e., quantile) of the standard normal. In practice, the population parameters (i and a are not likely known, in which case the researcher will use the sample mean and standard deviation.
Example: Computing VaR (normal distribution)
Assume that the profit/loss distribution for XYZ is normally distributed with an annual mean of $ 13 million and a standard deviation of $ 10 million. Calculate the VaR at the 93% and 99% confidence levels using a parametric approach.
Answer:
VaR(5%) = -$15 million + $10 million x 1.65 = $1.5 million. Therefore, XYZ expects to lose at most $1.5 million over the next year with 95% confidence. Equivalently, XYZ expects to lose more than $1.5 million with a 5% probability.
VaR(l%) = -$15 million + $10 million x 2.33 = $8.3 million. Note that the VaR (at 99% confidence) is greater than the VaR (at 95% confidence) as follows from the definition of value at risk.
Now suppose that the data you are using is arithmetic return data rather than profit/loss data. The arithmetic returns follow a normal distribution as well. As you would expect, because of the relationship between prices, profits/losses, and returns, the corresponding VaR is very similar in format:
VaR(a%) = (pr + CTr x za ) x Pt-1
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Example: Computing VaR (arithmetic returns)
A portfolio has a beginning period value of $100. The arithmetic returns follow a normal distribution with a mean of 10% and a standard deviation of 20%. Calculate VaR at both the 95% and 99% confidence levels.
Answer:
VaR(l%) = (-10% + 2.33 x 20%) x 100 = $36.6
Lognormal VaR
The lognormal distribution is right-skewed with positive outliers and bounded below by zero. As a result, the lognormal distribution is commonly used to counter the possibility of negative asset prices (P ). Technically, if we assume that geometric returns follow a normal distribution (jiR, crR), then the natural logarithm of asset prices follows a normal distribution and P follows a lognormal distribution. After some algebraic manipulation, we can derive the following expression for lognormal VaR:
VaR(a%) = Pt1 x (l – e^R -^R *^)
Example: Computing VaR (lognormal distribution)
A diversified portfolio exhibits a normally distributed geometric return with mean and standard deviation of 10% and 20%, respectively. Calculate the 5% and 1% lognormal VaR assuming the beginning period portfolio value is $100.
Answer:
Lognormal VaR(5%) = 100 x (1 exp[0.1 0.2 x 1.65])
= 100 x (1 – exp[-0.23]) = $20.55
Lognormal VaR(l%) = 100 x (1 exp[0.1 0.2 x 2.33])
= 100 x (1 exp[0.366]) = $30.65
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Note that the calculation of lognormal VaR (geometric returns) and normal VaR (arithmetic returns) will be similar when we are dealing with short-time periods and practical return estimates.
E x p e c t e d S h o r t f a l l

LO 1.1: Estimate VaR using a historical simulation approach.
Estimating VaR with a historical simulation approach is by far the simplest and most straightforward VaR method. To make this calculation, you simply order return observations from largest to smallest. The observation that follows the threshold loss level denotes the VaR limit. We are essentially searching for the observation that separates the tail from the body of the distribution. More generally, the observation that determines VaR for n observations at the (1 a) confidence level would be: (a x n) + 1.
Professors Note: Recall that the confidence level, (1 a), is typically a large value (e.g., 95% ) whereas the significance level, usually denoted as a , is much smaller (e.g., 5%).
To illustrate this VaR method, assume you have gathered 1,000 monthly returns for a security and produced the distribution shown in Figure 1. You decide that you want to compute the monthly VaR for this security at a confidence level of 93%. A ta95% confidence level, the lower tail displays the lowest 3% of the underlying distributions returns. For this distribution, the value associated with a 95% confidence level is a return of
15.5%. If you have $1,000,000 invested in this security, the one-month VaR is $155,000 (-15.5% x $1,000,000).
Figure 1: Histogram of Monthly Returns – u C a j 3 crv U – i
70 –
60 –
50 –
40 –
30 –
20-
10-
0
/
5%
ProbabilityJ of Loss ———-

\
6
/
N ‘8 V
4
/
& 4 ”
q
/
/
b
.v \
Monthly Return
Q ni r i ‘i i \o o
lo
\O
4 V5
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2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Example: Identifying the VaR limit
Identify the ordered observation in a sample of 1,000 data points that corresponds to VaR at a 93% confidence level.
Answer:
Since VaR is to be estimated at 93% confidence, this means that 5% (i.e., 50) of the ordered observations would fall in the tail of the distribution. Therefore, the 51st ordered loss observation would separate the 5% of largest losses from the remaining 95% of returns.
Professors Note: VaR is the quantile that separates the tail from the body of the distribution. With 1,000 observations at a 95% confidence level, there is a certain level o f arbitrariness in how the ordered observations relate to VaR. In other words, should VaR be the 50th observation (i.e., a x n), the 51st observation [i.e., (a x n) + 1], or some combination o f these observations? In this example, using the 51st observation was the approximation for VaR, and the method used in the assigned reading. However, on past FRM exams, VaR using the historical simulation method has been calculated as just: (a x n), in this case, as the 50th observation.
Example: Computing VaR
A long history of profit/loss data closely approximates a standard normal distribution (mean equals zero; standard deviation equals one). Estimate the 5% VaR using the historical simulation approach.
Answer:
The VaR limit will be at the observation that separates the tail loss with area equal to 5% from the remainder of the distribution. Since the distribution is closely approximated by the standard normal distribution, the VaR is 1.65 (5% critical value from the stable). Recall that since VaR is a one-tailed test, the entire significance level of 5% is in the left tail of the returns distribution.
>From a practical perspective, the historical simulation approach is sensible only if you expect future performance to follow the same return generating process as in the past. Furthermore, this approach is unable to adjust for changing economic conditions or abrupt shifts in parameter values.
2018 Kaplan, Inc.
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Pa r a m e t r i c E s t i m a t i o n A p p r o a c h e s

LO 1.1: Estimate VaR using a historical simulation approach.
Estimating VaR with a historical simulation approach is by far the simplest and most straightforward VaR method. To make this calculation, you simply order return observations from largest to smallest. The observation that follows the threshold loss level denotes the VaR limit. We are essentially searching for the observation that separates the tail from the body of the distribution. More generally, the observation that determines VaR for n observations at the (1 a) confidence level would be: (a x n) + 1.
Professors Note: Recall that the confidence level, (1 a), is typically a large value (e.g., 95% ) whereas the significance level, usually denoted as a , is much smaller (e.g., 5%).
To illustrate this VaR method, assume you have gathered 1,000 monthly returns for a security and produced the distribution shown in Figure 1. You decide that you want to compute the monthly VaR for this security at a confidence level of 93%. A ta95% confidence level, the lower tail displays the lowest 3% of the underlying distributions returns. For this distribution, the value associated with a 95% confidence level is a return of
15.5%. If you have $1,000,000 invested in this security, the one-month VaR is $155,000 (-15.5% x $1,000,000).
Figure 1: Histogram of Monthly Returns – u C a j 3 crv U – i
70 –
60 –
50 –
40 –
30 –
20-
10-
0
/
5%
ProbabilityJ of Loss ———-

\
6
/
N ‘8 V
4
/
& 4 ”
q
/
/
b
.v \
Monthly Return
Q ni r i ‘i i \o o
lo
\O
4 V5
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2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Example: Identifying the VaR limit
Identify the ordered observation in a sample of 1,000 data points that corresponds to VaR at a 93% confidence level.
Answer:
Since VaR is to be estimated at 93% confidence, this means that 5% (i.e., 50) of the ordered observations would fall in the tail of the distribution. Therefore, the 51st ordered loss observation would separate the 5% of largest losses from the remaining 95% of returns.
Professors Note: VaR is the quantile that separates the tail from the body of the distribution. With 1,000 observations at a 95% confidence level, there is a certain level o f arbitrariness in how the ordered observations relate to VaR. In other words, should VaR be the 50th observation (i.e., a x n), the 51st observation [i.e., (a x n) + 1], or some combination o f these observations? In this example, using the 51st observation was the approximation for VaR, and the method used in the assigned reading. However, on past FRM exams, VaR using the historical simulation method has been calculated as just: (a x n), in this case, as the 50th observation.
Example: Computing VaR
A long history of profit/loss data closely approximates a standard normal distribution (mean equals zero; standard deviation equals one). Estimate the 5% VaR using the historical simulation approach.
Answer:
The VaR limit will be at the observation that separates the tail loss with area equal to 5% from the remainder of the distribution. Since the distribution is closely approximated by the standard normal distribution, the VaR is 1.65 (5% critical value from the stable). Recall that since VaR is a one-tailed test, the entire significance level of 5% is in the left tail of the returns distribution.
>From a practical perspective, the historical simulation approach is sensible only if you expect future performance to follow the same return generating process as in the past. Furthermore, this approach is unable to adjust for changing economic conditions or abrupt shifts in parameter values.
2018 Kaplan, Inc.
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Pa r a m e t r i c E s t i m a t i o n A p p r o a c h e s

LO 16.9: Explain the impact of a single asset price jump on a volatility smile.
Price jumps can occur for a number of reasons. One reason may be the expectation of a significant news event that causes the underlying asset to move either up or down by a large amount. This would cause the underlying distribution to become bimodal, but with the same expected return and standard deviation as a unimodal, or standard, price-change distribution.
Implied volatility is affected by price jumps and the probabilities assumed for either a large up or down movement. The usual result, however, is that at-the-money options tend to have a higher implied volatility than either out-of-the-money or in-the-money options. Away-from-the-money options exhibit a lower implied volatility than at-the-money options. Instead of a volatility smile, price jumps would generate a volatility frown, as in Figure 3.
Figure 3: Volatility Smile (Frown) W ith Price Jump Implied volatility
Strike price
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Topic 16 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
K e y C o n c e p t s
LO 16.1 When option traders allow implied volatility to depend on strike price, patterns of implied volatility resemble volatility smiles.
LO 16.2 Put-call parity indicates that the deviation between market prices and Black-Scholes-Merton prices will be equivalent for calls and puts. Hence, implied volatility will be the same for calls and puts.
LO 16.3 Currency traders believe there is a greater chance of extreme price movements than predicted by a lognormal distribution. Equity traders believe the probability of large down movements in price is greater than large up movements in price, as compared with a lognormal distribution.
LO 16.4 The volatility pattern used by traders to price currency options generates implied volatilities that are higher for deep in-the-money and deep out-of-the-money options, as compared to the implied volatility for at-the-money options.
LO 16.5 The volatility smile exhibited by equity options is more of a smirk, with implied volatility higher for low strike prices. This has been attributed to leverage and crashophobia effects.
LO 16.6 Alternative methods to studying volatility patterns include: replacing strike price with strike price divided by stock price, replacing strike price with strike price divided by the forward price for the underlying asset, and replacing strike price with option delta.
LO 16.7 Volatility term structures and volatility surfaces are used by traders to judge consistency in model-generated option prices.
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Topic 16 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
LO 16.8 Volatility smiles that are not flat require the use of implied volatility functions or trees to correctly calculate option Greeks.
LO 16.9 Price jumps may generate volatility frowns instead of smiles.
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Topic 16 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
C o n c e p t C h e c k e r s
1.
2.
3.
4.
3.
The market price deviations for puts and calls from Black-Scholes-Merton prices indicate: A. equivalent put and call implied volatility. B. equivalent put and call moneyness. C. unequal put and call implied volatility. D. unequal put and call moneyness.
An empirical distribution that exhibits a fatter right tail than that of a lognormal distribution would indicate: A. equal implied volatilities across low and high strike prices. B. greater implied volatilities for low strike prices. C. greater implied volatilities for high strike prices. D. higher implied volatilities for mid-range strike prices.
the same across maturities for given strike prices. the same for short time periods. The sticky strike rule assumes that implied volatility is: A. B. C. the same across strike prices for given maturities. D. different across strike prices for given maturities.
Compared to at-the-money currency options, out-of-the-money currency options exhibit which of the following volatility traits? A. Lower implied volatility. B. A frown. C. A smirk. D. Higher implied volatility.
Which of the following regarding equity option volatility is true? A. There is higher implied price volatility for away-from-the-money equity options. B. Crashophobia suggests actual equity volatility increases when stock prices
decline.
C. Compared to the lognormal distribution, traders believe the probability of large
down movements in price is similar to large up movements.
D. Increasing leverage at lower equity prices suggests increasing volatility.
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Topic 16 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
C o n c e p t C h e c k e r A n s w e r s
1. A Put-call parity indicates that the implied volatility of a call and put will be equal for the same
strike price and time to expiration.
2. C An empirical distribution with a fat right tail generates a higher implied volatility for higher
strike prices due to the increased probability of observing high underlying asset prices. The pricing indication is that in-the-money calls and out-of-the-money puts would be expensive.
3. B The sticky strike rule, when applied to calculating option sensitivity measures, assumes
implied volatility is the same over short time periods.
4. D Away-from-the-money currency options have greater implied volatility than at-the-money
options. This pattern results in a volatility smile.
5. D There is higher implied price volatility for low strike price equity options. Crashophobia is based on the idea that large price declines are more likely than assumed in Black-Scholes- Merton prices, not that volatility increases when prices decline. Compared to the lognormal distribution, traders believe the probability of large down movements in price is higher than large up movements. Increasing leverage at lower equity prices suggests increasing volatility.
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Se l f -Te s t : M a r k e t Ri s k M e a su r e me n t a n d Ma n a g e me n t
10 Questions: 30 Minutes
1.
2.
3.
4.
An analyst for Z Corporation is determining the value at risk (VaR) for the corporations profit/loss distribution that is assumed to be normally distributed. The profit/loss distribution has an annual mean of $3 million and a standard deviation of $3.5 million. Using a parametric approach, what is the VaR with a 99% confidence level? A. $0,775 million. B. $3,155 million. C. $5,775 million. D. $8,155 million.
The Basel Committee requires backtesting of actual losses to VaR calculations. How many exceptions would need to occur in a 250-day trading period for the capital multiplier to increase from three to four? two to five. A. B. five to seven. C. seven to nine. D. ten or more.
The top-down approach to risk aggregation assumes that a banks portfolio can be cleanly subdivided according to market, credit, and operational risk measures. In contrast, a bottom-up approach attempts to account for interactions among various risk factors. In order to assess which approach is more appropriate, academic studies evaluate the ratio of integrated risks to separate risks. Regarding studies of top-down and bottom-up approaches, which of the following statements is incorrect? A. Top-down studies suggest that risk diversification is present. B. Bottom-up studies sometimes calculate the ratio of integrated risks to separate
C. Bottom-up studies suggest that risk diversification should be questioned. D. Top-down studies calculate the ratio of integrated risks to separate risks to be
risks to be less than one.
greater than one.
Commercial Bank Z has a $3 million loan to company A and a $3 million loan to company B. Companies A and B each have a 5% and 4% default probability, respectively. The default correlation between companies A and B is 0.6. What is the expected loss (EL) for the commercial bank under the worst case scenario? a. b. c. d.
$83,700. $133,900. $165,600. $233,800.
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5.
6.
7.
8.
Book 1 Self-Test: Market Risk Measurement and Management
A risk manager should always pay careful attention to the limitations and advantages of applying financial models such as the value at risk (VaR) and Black-Scholes- Merton (BSM) option pricing model. Which of the following statements regarding financial models is correct? a. Financial models should always be calibrated using most recent market data
because it is more likely to be accurate in extrapolating trends.
b. When applying the VaR model, empirical studies imply asset returns closely
follow the normal distribution.
c. The Black-Scholes-Merton option pricing model is a good example of
the advantage of using financial models because the model eliminates all mathematical inconsistences that can occur with human judgment.
d. A good example of a limitation of a financial model is the assumption of
constant volatility when applying the Black-Scholes-Merton (BSM) option pricing model.
Assume that a trader wishes to set up a hedge such that he sells $100,000 of a Treasury bond and buys TIPS as a hedge. Using a historical yield regression framework, assume the DV01 on the T-bond is 0.072, the DV01 on the TIPS is 0.051, and the hedge adjustment factor (regression beta coefficient) is 1.2. What is the face value of the offsetting TIPS position needed to carry out this regression hedge? A. $138,462. B. $169,412. C. $268,499. D. $280,067.
A constant maturity Treasury (CMT) swap pays ($1,000,000 / 2) x (y^-p 9%) every six months. There is a 70% probability of an increase in the 6-month spot rate and a 60% probability of an increase in the 1-year spot rate. The rate change in all cases is 0.50% per period, and the initial yCMT is 9%. What is the value of this CMT swap? A. $2,325. B. $2,229. C. $2,429. D. $905.
Suppose the market expects that the current 1-year rate for zero-coupon bonds with a face value of $1 will remain at 5%, but the 1-year rate in one year will be 3%. What is the 2-year spot rate for zero-coupon bonds? A. 3.995%. B. 4.088%. C. 4.005%. D. 4.115%.
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Book 1 Self-Test: Market Risk Measurement and Management
9.
10.
An analyst is modeling spot rate changes using short rate term structure models. The current short-term interest rate is 5% with a volatility of 80bps. After one month passes the realization of dw, a normally distributed random variable with mean 0 and standard deviation Vdt, is -0.5. Assume a constant interest rate drift, X, of 0.36%. What should the analyst compute as the new spot rate? A. 5.37%. B. 4.63%. C. 5.76%. D. 4.24%.
Which of the following statements is incorrect regarding volatility smiles? A. Currency options exhibit volatility smiles because the at-the-money options have
higher implied volatility than away-from-the-money options.
B. Volatility frowns result when jumps occur in asset prices. C. Equity options exhibit a volatility smirk because low strike price options have
greater implied volatility.
D. Relative to currency traders, it appears that equity traders expectations of
extreme price movements are more asymmetric.
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Se l f -Te st A n s w e r s: M a r k e t Ri s k M e a su r e me n t a n d Ma n a g e me n t
1. B The population mean and standard deviations are unknown; therefore, the standard normal
z-value of 2.33 is used for a 99% confidence level.
VaR(l%) = -5.0 million + ($3.5 million)(2.33) = -5.0 million + 8.155 million = 3.155 million (See Topic 1)
2. D Ten or more backtesting violations require the institution to use a capital multiplier of four.
(See Topic 3)
3. D Top-down studies calculate this ratio to be less than one, which suggests that risk
diversification is present and ignored by the separate approach. Bottom-up studies also often calculate this ratio to be less than one; however, this research has not been conclusive, and has recently found evidence of risk compounding, which produces a ratio greater than one. Thus, bottom-up studies suggests that risk diversification should be questioned. (See Topic 5)
4. C The default probability of company A is 5%. Thus, the standard deviation for company A is:
V0.05(l 0.05) – 0.2179
Company B has a default probability of 4% and, therefore, will have a standard deviation of 0.1960. We can now calculate the expected loss under the worst case scenario where both companies A and B are in default. Assuming that the default correlation between A and B is 0.6, the joint probability of default is:
P(AB) = 0.6^0.05(0.95) x 0.04(0.96) + 0.05 x 0.04 = 0.6V0.001824 + 0.002 = 0.0276
Thus, the expected loss for the commercial bank is $165,600 (= 0.0276 x $6,000,000). (See Topic 6)
5. D The Black-Scholes-Merton (BSM) option pricing model assumes strike prices have a
constant volatility. However, numerous empirical studies find higher volatility for out-of- the-money options and a volatility skew in equity markets. Thus, this is a good example of a limitation of financial models. The choice of time period used to calibrate the parameter inputs for the model can have a big impact on the results. Risk managers used volatility and correlation estimates from pre-crisis periods during the recent financial crisis, and this resulted in significantly underestimating the risk for financial models. All financial models should be stress tested using scenarios of extreme economic conditions. VaR models often assume asset returns have a normal distribution. However, empirical studies find higher kurtosis in return distributions. High kurtosis implies a distribution with fatter tails than the normal distribution. Thus, the normal distribution is not the best assumption for the underlying distribution. Financial models contain mathematical inconsistencies. For example, in applying the BSM option pricing model for up-and-out calls and puts and down-and-out calls and puts, there are rare cases where the inputs make the model insensitive to changes in implied volatility and option maturity. (See Topic 8)
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Book 1 Self-Test Answers: Market Risk Measurement and Management
6 . B Defining JF^ and P* as the face amounts of the real and nominal bonds, respectively, and
their corresponding DVO Is as DV01R and DV01R, a DV01 hedge is adjusted by the hedge adjustment factor, or beta, as follows:
F r = FN x
DV01 N DV01 R x(3 0.072 10.051 J
(cid:31)R F = 1 0 0 ,0 0 0 x
x 1.2 = 169,412
(See Topic 10)
7 . A The payoff in each period is ($1,000,000 / 2) x (yCMT – 9 % ). For example, the 1-year payoff of $5,000 in the figure below is calculated as ($1,000,000 / 2) x (10% 9% ) = $5,000. The other numbers in the year one cells are calculated similarly.
In six months, the payoff if interest rates increase to 9.50% is ($1,000,000 / 2 ) x (9.5% 9.0%) = $2,500. Note that the price in this cell equals the present value of the probability weighted 1 -year values plus the 6-month payoff:
months, U
($5,000×0.6)+ ($0x0.4)
+ 0.095 1
+ $2,500 = $5,363.96
The other cell value in six months is calculated similarly and results in a loss of $4,418.47.
The value of the CMT swap today is the present value of the probability weighted 6-month values:
($5,363.96 x 0.7) + ( – $4,418.47 x 0.3)
+ 0.09 1
$2,324.62
1 0 % yCMT Price = $5,000
Tc mt
8 .5 %
Price = -$ 4 ,4 18
7 c m t _ 8 %
Price = -$5,000
Today
6 months year 1 year
Thus the correct response is A. The other answers are incorrect because they do not correctly discount the future values or omit the 6-month payoff from the 6-month values.
(See Topic 11)
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2017 Kaplan, Inc.
Book 1 Self-Test Answers: Market Risk Measurement and Management
8. A The 2-year spot rate is computed as follows:
f (2 ) = 2/(1 .05)(1.03) – 1 = 3.995%
(See Topic 12)
9. B This short rate process has an annualized drift of 0.36%, so it requires the use of Model 2
(with constant drift). The change in the spot rate is computed as:
dr = Xdt + crdw
dr = (0.36% / 12) + (0.8% x -0.5) = -0.37% = -3 7 basis points
Since the initial short-term rate was 5% and d r is 0.37%, the new spot rate in one month is:
5% – 0.37% = 4.63% (See Topic 13)
10. A Currency options exhibit volatility smiles because the at-the-money options have lower
implied volatility than away-from-the-money options.
Equity traders believe that the probability of large price decreases is greater than the probability of large price increases. Currency traders beliefs about volatility are more symmetric as there is no large skew in the distribution of expected currency values (i.e., there is a greater chance of large price movements in either direction).
(See Topic 16)
2017 Kaplan, Inc.
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Market Risk Measurement and Management
F o r m u l a s
Topic 1
profit/loss data: P/Lt = Pt + Dt Pt l
arithmetic return: rt
geometric return: Rt =
+ Dt ~ Pt-l
P t-1
P . + D . P -l
J
delta-normal VaR: VaR(a%) = (p,r + crr X za ) x Pt_1
lognormal VaR: VaR(a%) = Pt_1 x |l e^R CTRXZa j
standard error of a quantile: se (q) =
V p(l p)/n
f(q)
Topic 2 age-weighted historical simulation: w(i) = XM (1-X )
1-X ”
Topic 3
model accuracy test:
x pT
V p(l p)T
unconditional coverage test statistic:
LR = -2ln[(l – p)T-NpN] + 2ln{[l – (N/T)]t “n (N/T)n}
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2017 Kaplan, Inc.
Topic 6 portfolio mean return: pp = wxpx + wYpY
Book 1 Formulas
portfolio standard deviation: Op = V W X X + w y a y + 2 w x w y c o v x y 2 2 j 2 2
2 2
n ( X t -|j,x )(Yt – PY) t1 covariance: cov^y —-

n 1
covXY correlation: PXY = a x o Y
realized correlation: Prealized 2n n E ft,;
>)
correlation swap payoff: notional amount x (preajizej pfixeci)
joint probability of default: P(AB) = Pa b ^PDa CI PDa ) x PDb (1 PDB) + PDA X PDg
Topic 7 mean reversion rate: St – S j = a(p, S j)
autocorrelation: AC(pt,pt
) _ cov(Pt>Pt-l) a(pt)xa(P[_,)
t_1
Topic 8
correlation w ith expectation values: PXY =
E(XY) – E(X)E(Y)
Ve (X2)-(E (X ))2 x Ve (Y2)-(E (Y ))2
Spearmans rank correlation: Ps 1 ~ i = l
n(n2 -1 )
n
6 E d ?
Kendall s r: r n(n 1) / 2
2017 Kaplan, Inc.
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Book 1 Formulas
Topic 12
2- year spot rate: r (2) = %J(l + tj )(l + r2) 1
3- year spot rate: r (3) = %J(l + q ) (l + r2) (l + r3) 1
Jensens inequality: E
1
(l + r)
1
Topic 13
Model 1:
dr = crdw
where: dr = change in interest rates over small time interval, dt dt = small time interval (measured in years) a = annual basis-point volatility of rate changes dw = normally distributed random variable with mean 0 and standard deviation \fdt
Model 2: dr = Xdt + crdw
Vasicek model:
dr = k(9 – r)dt + crdw
where: k = a parameter that measures the speed of reversion adjustment 9 = long-run value of the short-term rate assuming risk neutrality r = current interest rate level
long-run value of short-term rate: X A
9 q -| k
where: the long-run true rate of interest = the long-run true rate of interest
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2017 Kaplan, Inc.
Book 1 Formulas
Topic 14
Model 3:
dr = \(t)dt + cre-atdw
where: cr = volatility at t = 0, which decreases exponentially to 0 for a > 0
CIR model: dr = k(0 r)dt + cr Vr dw
Model 4: dr = ardt + crdw
Topic 16
put-call parity: c – p = S – PV(X)
2017 Kaplan, Inc.
Page 205
U si n g t h e C u mu l a t i v e Z-Ta b l e
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)/a = ($7.25 – $5.00)/$ 1.50 – +1.50
If EPS = x = $3.00, then z = (x – p)/a = ($3.00 – $5.00)/$1.50 – -1.33
Forz-value o f 1.50: Use the row headed 1.5 and the column headed 0 to find the value 0.9332. This represents the area under the curve to the left of the critical value 1.50.
Forz-value o f 1.33: Use the row headed 1.3 and the column headed 3 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 of1.33 is 1 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%? (2.0 – 0.0) / (20.0 / 6) = 0.60. Hq: p < 0.0%, Ha : p > 0.0%. The test statistic = ^-statistic = = (2.0 – 0.0) / (20.0 / 6) = 0.60.
x – p o
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 z-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 wants 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 the 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 HQ and conclude that the stocks return equals 0.0%.
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2017 Kaplan, Inc.
C u mu l a t i v e -Z-Ta b l e P(Z < z) = N(z) for z > 0 P(Z < -z) = 1 - N(z) z 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 0 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.994 0.9955 0.9966 0.9975 0.9982 0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.983 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.937 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.985 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.989 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 3 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 2017 Kaplan, Inc. Page 207 A l t e r n a t iv e Z-Ta b l e P(Z < z) = N(z) for z > 0 P(Z < -z) = 1 - N(z) z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 0.00 0.0000 0.0398 0.0793 0.1179 0.1554 0.1915 0.2257 0.2580 0.2881 0.3159 0.3413 0.3643 0.3849 0.4032 0.4192 0.4332 0.4452 0.4554 0.4641 0.4713 0.4772 0.4821 0.4861 0.4893 0.4918 0.4939 0.4953 0.4965 0.4974 0.4981 0.01 0.0040 0.0438 0.0832 0.1217 0.1591 0.1950 0.2291 0.2611 0.2910 0.3186 0.3438 0.3665 0.3869 0.4049 0.4207 0.4345 0.4463 0.4564 0.4649 0.4719 0.4778 0.4826 0.4864 0.4896 0.4920 0.4940 0.4955 0.4966 0.4975 0.4982 0.02 0.0080 0.0478 0.0871 0.1255 0.1628 0.1985 0.2324 0.2642 0.2939 0.3212 0.3461 0.3686 0.3888 0.4066 0.4222 0.4357 0.4474 0.4573 0.4656 0.4726 0.4783 0.4830 0.4868 0.4898 0.4922 0.4941 0.4956 0.4967 0.4976 0.4982 0.03 0.0120 0.0517 0.0910 0.1293 0.1664 0.2019 0.2357 0.2673 0.2967 0.3238 0.3485 0.3708 0.3907 0.4082 0.4236 0.4370 0.4484 0.4582 0.4664 0.4732 0.4788 0.4834 0.4871 0.4901 0.4925 0.4943 0.4957 0.4968 0.4977 0.4983 0.04 0.0160 0.0557 0.0948 0.1331 0.1700 0.2054 0.2389 0.2704 0.2995 0.3264 0.3508 0.3729 0.3925 0.4099 0.4251 0.4382 0.4495 0.4591 0.4671 0.4738 0.4793 0.4838 0.4875 0.4904 0.4927 0.4945 0.4959 0.4969 0.4977 0.4984 0.05 0.0199 0.0596 0.0987 0.1368 0.1736 0.2088 0.2422 0.2734 0.3023 0.3289 0.3531 0.3749 0.3944 0.4115 0.4265 0.4394 0.4505 0.4599 0.4678 0.4744 0.4798 0.4842 0.4878 0.4906 0.4929 0.4946 0.4960 0.4970 0.4978 0.4984 0.06 0.0239 0.0636 0.1026 0.1406 0.1772 0.2123 0.2454 0.2764 0.3051 0.3315 0.3554 0.3770 0.3962 0.4131 0.4279 0.4406 0.4515 0.4608 0.4686 0.4750 0.4803 0.4846 0.4881 0.4909 0.4931 0.4948 0.4961 0.4971 0.4979 0.4985 0.07 0.0279 0.0675 0.1064 0.1443 0.1808 0.2157 0.2486 0.2794 0.3078 0.3340 0.3577 0.3790 0.3980 0.4147 0.4292 0.4418 0.4525 0.4616 0.4693 0.4756 0.4808 0.4850 0.4884 0.4911 0.4932 0.4949 0.4962 0.4972 0.4979 0.4985 0.08 0.0319 0.0714 0.1103 0.1480 0.1844 0.2190 0.2517 0.2823 0.3106 0.3356 0.3599 0.3810 0.3997 0.4162 0.4306 0.4429 0.4535 0.4625 0.4699 0.4761 0.4812 0.4854 0.4887 0.4913 0.4934 0.4951 0.4963 0.4973 0.4980 0.4986 0.09 0.0359 0.0753 0.1141 0.1517 0.1879 0.2224 0.2549 0.2852 0.3133 0.3389 0.3621 0.3830 0.4015 0.4177 0.4319 0.4441 0.4545 0.4633 0.4706 0.4767 0.4817 0.4857 0.4890 0.4916 0.4936 0.4952 0.4964 0.4974 0.4981 0.4986 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 Page 208 2017 Kaplan, Inc. St u d e n t s T-Di s t r i b u t i o n df df i 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 OO 0.100 0.20 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 1.282 Level of Significance for One-Tailed Test 0.050 0.025 0.01 0.005 0.0005 Level of Significance for Two-Tailed Test 0.10 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645 0.05 12.706 4.303 3.182 2.776 2.571 2 .447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 1.960 0.02 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.358 2.326 0.01 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 2.576 0.001 636.619 31.599 12.294 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373 3.291 2017 Kaplan, Inc. Page 209 aussian copula 100 Gaussian default time copula 101 generalized extreme value distribution 80 general risk factors 36 G G edge adjustment factor 111 historical scenarios 46 historical simulation approach 2 Ho-Lee model 156 H h ntegrated risk measurement 46 interest rate drift 123 interest rate expectations 137 interest rate tree 120 interest rate volatility 139 I i ensens inequality 141 Johnson SB distribution 80 J J endalls t 91 K K everage 189 LIBOR-OIS spread 179 lognormal model 170 lognormal VaR 5 London Interbank Offered Rate (LIBOR) 178 L l mbedded options 131 endogenous liquidity 45 exception 25 exogenous liquidity 45 expected shortfall 6, 46, 64 exposures 34 E e ailure rate 26 filtered historical simulation 19 F f ge-weigh ted historical simulation 17 arbitrage-free models 156 autocorrelation 79 A a acktesting 25 backward induction 121 balance sheet management 48 Basel penalty zones 28 best-fit distributions 80 binomial interest rate model 120 Black-Karasinski model 172 bootstrap historical simulation 15 bottom-up approach 47 B b allable bonds 131 Cholesky decomposition 103 cleaned returns 25 coherent risk measure 6 compartmentalized approach 47 concentration ratio 66 concentration risk 66 conditional coverage 29 constant drift 155 constant maturity Treasury swap 127 convexity effect 141 copula function 99 correlation coefficient 55,88 correlation copula 99 correlation options 56 correlation risk 52 correlation swap 58 correlation trading strategies 56 correlation-weighted historical simulation 18 covariance 55 Cox-Ingersoll-Ross (CIR) model 169 crashophobia 189 credit default swaps 53 cyclical feedback loop 48 c c efault correlation 64 default time 103 DVOl-neutral hedge 109 dynamic financial correlations 53 D d In d e x Page 210 2017 Kaplan, Inc. Book 1 Index Sharpe ratio 35 Spearmans rank correlation 89 specific risks 36 spectral risk measures 46 standard deviation 54 state-dependent volatility 126 static financial correlations 53 statistical correlation measures 88 sticky delta rule 191 sticky strike rule 190 stressed VaR 46 stress testing 46, 87 surrogate density function 16 systemic risk 65 ime-dependent volatility 167 time-varying volatility 44 top-down approach 47 tracking error VaR 39 true probabilities 123 Type I error 26 Type II error 26 u unconditional coverage 29 undiversified VaR 38 unified approach 47 T t alue at risk 1, 44, 59 variance-covariance method 60 Vasicek model 157 volatility skew 187 volatility smiles 187 volatility surface 190 volatility term structure 190 volatility-weighted historical simulation 18 w wrone-way risk 53 6 7 V v 2017 Kaplan, Inc. Page 211 mean 54 mean reversion 77 mean reversion rate 77 mean-reverting process 157 mechanical-search stress tests 46 migration risk 64 Model 1 152 Model 2 155 Model 3 168 Model 4 170 model risk 35 modified duration 35 egative convexity 131 nonmonotonous 53 nonrecombining trees 126 N n IS rate 178 OIS zero curve 179 option-adjusted spread 129 ordinal risk measures 92 overnight indexed swap 178 o O earson correlation 55, 88 position-based method 35 predefined scenarios 46 price jumps 191 principal components analysis 113 putable bonds 132 put-call parity 186 p P uantile 4 quantile-quantile plot 9 quanto option 57 Q q _ i R i recombining tree 126 regression hedge 110 return-based analysis 35 risk-averse investor 145 risk diversification 47 risk engine 34 risk factors 34 risk-free rate 177 risk-neutral investor 145 risk-neutral pricing 123 risk-neutral probabilities 123 risk premium 145 Required Disclaimers: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by Kaplan Schweser. CFA Institute, CFA, and Chartered Financial Analyst are trademarks owned by CFA Institute. Certified Financial Planner Board of Standards Inc. owns the certification marks CFP, CERTIFIED FINANCIAL PLANNER M, and federally registered CFP (with flame design) in the U.S., which it awards to individuals who successfully complete initial and ongoing certification requirements. Kaplan University does not certify individuals to use the CFP, CERTIFIED FINANCIAL PLANNER1, and CFP (with flame design) certification marks. CFP certification is granted only by Certified Financial Planner Board of Standards Inc. to those persons who, in addition to completing an educational requirement such as this CFP Board-Registered Program, have met its ethics, experience, and examination requirements. Schweser and Kaplan University are review course providers for the CFP Certification Examination administered by Certified Financial Planner Board of Standards Inc. 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CALAA is not responsible for any fees or costs paid by the user to Kaplan Schweser nor is CALAA responsible for any fees or costs of any person or entity providing any services to Kaplan Schweser. CAIA, CALA Association, Chartered Alternative Investment Analyst, and Chartered Alternative Investment Analyst Association^ are service marks and trademarks owned by CHARTERED ALTERNATIVE INVESTMENT ANALYST ASSOCIATION, INC., a Massachusetts non-profit corporation with its principal place of business at Amherst, Massachusetts, and are used by permission. 2018 SchweserNotes Part Market Risk Measurement and Management eBook 1 K A P L A N ' ) S C H W E S E R Getting Started FRM Exam Part II Welcome As the VP of Advanced Designations at Kaplan Schweser, I am pleased to have the opportunity to help you prepare for the 2018 FRM Exam. Getting an early start on your study program is important for you to sufficiently prepare, practice, and perform on exam day. Proper planning will allow you to set aside enough time to master the learning objectives in the Part II curriculum. Now that you've received your SchweserNotes, here's how to get started: Step 1: Access Your Online Tools Visit www.schweser.com/frm and log in to your online account using the button located in the top navigation bar. After logging in, select the appropriate part and proceed to the dashboard where you can access your online products. Step 2: Create a Study Plan Create a study plan with the Schweser Study Calendar (located on the Schweser dashboard). Then view the Candidate Resource Library on-demand videos for an introduction to core concepts. Step 3: Prepare and Practice Read your SchweserNotes Our clear, concise study notes will help you prepare for the exam. At the end of each reading, you can answer the Concept Checker questions for better understanding of the curriculum. Attend a Weekly Class Attend our Live Online Weekly Class or review the on-demand archives as often as you like. Our expert faculty will guide you through the FRM curriculum with a structured approach to help you prepare for the exam. (See our instruction packages to the right. Visit www.schweser.com/frm to order.) Practice with SchweserPro QBank Maximize your retention of important concepts and practice answering exam-style questions in the SchweserPro QBank and taking several Practice Exams. Use Schweser's QuickSheet for continuous review on the go. (Visit www.schweser.com/frm to order.) Step 4: Final Review A few weeks before the exam, make use of our Online Review Workshop Package. Review key curriculum concepts in every topic, perform by working through demonstration problems, and practice your exam techniques with our 8-hour live Online Review Workshop. Use Schweser's Secret Sauce for convenient study on the go. Step 5: Perform As part of our Online Review Workshop Package, take a Schweser Mock Exam to ensure you are ready to perform on the actual FRM Exam. Put your skills and knowledge to the test and gain confidence before the exam.. Again, thank you fortrusting Kaplan Schweser with your FRM Exam preparation! Sincerely, Derek Burkett, CFA, FRM, CAIA VP, Advanced Designations, Kaplan Schweser The Kaplan Way Acquire new knowledge through demonstration and examples. Practice < ?xl-----bs Apply new knowledge through simulation and practice. Evaluate mastery of new knowledge and identify achieved outcomes. PremiumPlus Package FRM Instruction Packages: PremiumPlus Package Premium Instruction Package Live Instruction*: Remember to join our Live Online Weekly Class. Register online today at www.schweser.com/frm. *Dates, tim es, and instructors subject to change MKT-005736 Contact us for questions about your study package, upgrading your package, purchasing additional study materials, or for additional information: www.schweser.com/frm Toll-Free: 888.325.5072 International: +1 608.779.8397 FRM Pa r t II B o o k 1: M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t W e l c o m e t o t h e 2018 S c h w e s e r N o t e s R e a d i n g A s s i g n m e n t s a n d L e a r n i n g O b j e c t i v e s M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t 1: Estimating Market Risk Measures: An Introduction and Overview 2: Non-parametric Approaches 3: Backtesting VaR 4: VaR Mapping 5: Messages from the Academic Literature on Risk Measurement for the Trading Book 6: Some Correlation Basics: Properties, Motivation, Terminology 7: Empirical Properties of Correlation: How Do Correlations Behave in the Real World? 8: Statistical Correlation Models Can We Apply Them to Finance? 9: Financial Correlation ModelingBottom-Up Approaches 10: Empirical Approaches to Risk Metrics and Hedging 11: The Science of Term Structure Models 12: The Evolution of Short Rates and the Shape of the Term Structure 13: The Art of Term Structure Models: Drift 14: The Art of Term Structure Models: Volatility and Distribution 15: Volatility Smiles S e l f -Te s t : M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t F o r m u l a s A p p e n d i x In d e x v viii 1 15 25 38 56 64 8 8 98 111 121 132 149 164 179 189 199 205 209 213 2018 Kaplan, Inc. Page iii FRM 2018 PART II BOOK 1: MARKET RISK MEASUREMENT AND MANAGEMENT 2018 Kaplan, Inc. All rights reserved. Published in 2018 by Kaplan, Inc. Printed in the United States of America. ISBN: 978-1-4754-7029-1 Required Disclaimer: GARP does not endorse, promote, review, or warrant the accuracy of the products or services offered by Kaplan 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, nor is GARP responsible for any fees or costs of any person or entity providing any services to Kaplan. FRM, GARP, and Global Association of Risk Professionals are trademarks owned by the Global Association of Risk Professionals, Inc. 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 iv 2018 Kaplan, Inc. We l c o m e t o t h e 2018 S c h w e s e r N o t e s Thank you for trusting Kaplan Schweser to help you reach your career and educational goals. We are very pleased to be able to help you prepare for the FRM Part II exam. In this introduction, I want to explain the resources included with the SchweserNotes, suggest how you can best use Schweser materials to prepare for the exam, and direct you toward other educational resources you will find helpful as you study for the exam. Besides the SchweserNotes themselves, there are many online educational resources available at Schweser.com. Just log in using the individual username and password you received when you purchased the SchweserNotes. Schw eserN otes The SchweserNotes consist of four volumes that include complete coverage of all FRM assigned topics and learning objectives (LOs), Concept Checkers (multiple-choice questions for every topic), and Self-Test questions to help you master the material and check your retention of key concepts. Online Practice Questions To retain what you learn, it is important that you quiz yourself often. We offer an online version of the SchweserPro QBank, which contains hundreds of Part II practice questions and explanations. Quizzes are available for each topic or across multiple topics. Build your own exams by specifying the topics and the number of questions. Practice Exams Schweser offers two full 4-hour practice exams. These exams are important tools for gaining the speed and skills you will need to pass the exam. The Practice Exams book contains answers with full explanations for self-grading and evaluation. Schweser Study Calendar Use your Online Access to tell us when you will start and what days of the week you can study. The online Schweser Study Calendar will create a study plan just for you, breaking the curriculum into daily and weekly tasks to keep you on track and help you monitor your study progress. 2018 Kaplan, Inc. Page v Book 1 Welcome to the 2018 SchweserNotes The FRM Part II exam is a formidable challenge (covering 79 assigned readings and almost 500 learning objectives), and you must devote considerable time and effort to be properly prepared. There are no shortcuts! You must learn the material, know the terminology and techniques, understand the concepts, and be able to answer 80 multiple choice questions quickly and (at least 70%) correctly. A good estimate of the study time required on average is 250 hours, but some candidates will need more or less time, depending on their individual backgrounds and experience. To help you really master this material and be well-prepared for the FRM exam, we offer several other educational resources, including: Online Weekly Class Our Online Weekly Class is offered each week, beginning in February for the May exam and August for the November exam. This online class brings the personal attention of a classroom into your home or office with 30 hours of real-time instruction, led by David McMeekin, CFA, GALA, FRM. The class offers in-depth coverage of difficult concepts, instant feedback during lecture and Q&A sessions, and discussion of sample exam questions. Archived classes are available for viewing at any time throughout the season. Candidates enrolled in the Online Weekly Class also have full access to supplemental on- demand video instruction in the Candidate Resource Library and an e-mail address link for sending questions to the instructor at any time. Late-Season Review Late-season review and exam practice can make all the difference. Our Review Package helps you evaluate your exam readiness with products specifically designed for late-season studying. This Review Package includes the Online Review Workshop (8-hour live and archived online review of essential curriculum topics), the Schweser Mock Exam (one 4-hour exam), and Schweser s Secret Sauce (concise summary of the FRM curriculum). Page vi 2018 Kaplan, Inc. Book 1 Welcome to the 2018 SchweserNotes Part II Exam Weights In preparing for the exam, pay attention to the weights assigned to each knowledge domain within the curriculum. The Part II exam weights are as follows: Book Knowledge Domains Exam Weight Exam Questions 1 2 3 4 4 Market Risk Measurement and Management Credit Risk Measurement and Management Operational and Integrated Risk Management Risk Management and Investment Management Current Issues in Financial Markets 25% 25% 25% 15% 10% 20 20 20 12 8 H ow to Succeed There are no shortcuts to studying for this exam. Expect the Global Association of Risk Professionals (GARP) to test you in a way that will reveal how well you know the Part II curriculum. You should begin studying early and stick to your study plan. You should first read the SchweserNotes and complete the Concept Checkers for each topic. At the end of each book, you should answer the provided Self-Test questions to understand how concepts may be tested on the exam. You should finish the overall curriculum at least two weeks before the FRM exam. This will allow sufficient time for Practice Exams and further review of those topics you have not yet mastered. I would like to take this opportunity to thank the content developers, editors, and graphic designers who worked countless hours to create the 2018 FRM SchweserNotes. I would especially like to thank Derek Burkett, CFA, FRM, CAIA; Adam Stueber, CAIA; Craig Prochaska, CFA; Kent Westlund, CFA, FRM; Kurt Schuldes, CFA, CAIA; Tim Greive, CFA, CAIA; Jeff Bahr, Andy Bauer, Allie Bottcher, Katherine Bourgeois, Alyssa Brunner, Lindsey Casto, Laura Goetzinger, Ryan Henry, Hannah Kelley, Alissa Knop, Genevieve Kretschmer, Gretchen Panzer, Jessica Pearse, Ashley Sinclair, Ben Strong, and Debbie White for their contributions. Best regards, /Uc Snut& Eric Smith, CFA, FRM Content Manager Kaplan Schweser 2018 Kaplan, Inc. Page vii R e a d i n g A s s i g n m e n t s a n d L e a r n i n g O b j e c t i v e s The following material is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by the Global Association of Risk Professionals. R e a d i n g A s s i g n m e n t s Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, U.K.: John Wiley & Sons, 2005). 1. Estimating Market Risk Measures: An Introduction and Overview, Chapter 3 (page 1) 2. Non-parametric Approaches, Chapter 4 (page 15) Philippe Jorion, Value- at-Risk: The New Benchmark for Managing Financial Risk, 3rd Edition (New York, NY: McGraw Hill, 2007). 3. Backtesting VaR, Chapter 6 4. VaR Mapping, Chapter 11 (page 25) (page 38) 5. Messages from the Academic Literature on Risk Measurement for the Trading Book, Basel Committee on Banking Supervision, Working Paper No. 19, Jan 2011. (page 56) Gunter Meissner, Correlation Risk Modeling and Management (New York, NY: John Wiley & Sons, 2014). 6. Some Correlation Basics: Properties, Motivation, Terminology, Chapter 1 (page 64) 7. 8. 9. Empirical Properties of Correlation: How Do Correlations Behave in the Real World?, Chapter 2 Statistical Correlation Models Can We Apply Them to Finance?, Chapter 3 (page 88) (page 98) Financial Correlation Modeling Bottom-Up Approaches, Chapter 4, Sections 4.3.0 (intro), 4.3.1, and 4.3.2 only (page 111) Bruce Tuckman and .Angel Serrat, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011). 10. Empirical Approaches to Risk Metrics and Hedging, Chapter 6 11. The Science of Term Structure Models, Chapter 7 12. The Evolution of Short Rates and the Shape of the Term Structure, Chapter 8 (page 121) (page 132) (page 149) Page viii 2018 Kaplan, Inc. Book 1 Reading Assignments and Learning Objectives 13. The Art of Term Structure Models: Drift, Chapter 9 (page 164) 14. The Art of Term Structure Models: Volatility and Distribution, Chapter 10 (page 179) John C. Hull, Options, Futures, and Other Derivatives, 10th Edition (New York, NY: Pearson, 2017). 13. Volatility Smiles, Chapter 20 (page 189) 2018 Kaplan, Inc. Page ix Book 1 Reading Assignments and Learning Objectives L e a r n i n g O b j e c t i v e s 1. Estimating Market Risk Measures: An Introduction and Overview After completing this reading, you should be able to: 1. Estimate VaR using a historical simulation approach, (page 2) 2. Estimate VaR using a parametric approach for both normal and lognormal return distributions, (page 4) 3. Estimate the expected shortfall given P/L or return data, (page 6) 4. Define coherent risk measures, (page 6) 3. Estimate risk measures by estimating quantiles, (page 6) 6. Evaluate estimators of risk measures by estimating their standard errors, (page 7) 7. Interpret Q Q plots to identify the characteristics of a distribution, (page 9) 2. Non-parametric Approaches After completing this reading, you should be able to: 1. Apply the bootstrap historical simulation approach to estimate coherent risk measures, (page 13) 2. Describe historical simulation using non-parametric density estimation, (page 16) 3. Compare and contrast the age-weighted, the volatility-weigh ted, the correlation- 4. weighted, and the filtered historical simulation approaches, (page 17) Identify advantages and disadvantages of non-parametric estimation methods. (page 19) 3. Backtesting VaR After completing this reading, you should be able to: 1. Define backtesting and exceptions and explain the importance of backtesting VaR models, (page 25) 2. Explain the significant difficulties in backtesting a VaR model, (page 26) 3. Verify a model based on exceptions or failure rates, (page 26) 4. Define and identify Type I and Type II errors, (page 28) 5. Explain the need to consider conditional coverage in the backtesting framework, (page 32) 6. Describe the Basel rules for backtesting, (page 33) 4. VaR Mapping After completing this reading, you should be able to: 1. Explain the principles underlying VaR mapping, and describe the mapping process, (page 38) 2. Explain how the mapping process captures general and specific risks, (page 39) 3. Differentiate among the three methods of mapping portfolios of fixed income securities, (page 41) 4. Summarize how to map a fixed income portfolio into positions of standard instruments, (page 41) 5. Describe how mapping of risk factors can support stress testing, (page 44) 6. Explain how VaR can be used as a performance benchmark, (page 45) 7. Describe the method of mapping forwards, forward rate agreements, interest rate swaps, and options, (page 48) Page x 2018 Kaplan, Inc. Book 1 Reading Assignments and Learning Objectives 5. Messages from the Academic Literature on Risk Measurement for the Trading Book After completing this reading, you should be able to: 1. Explain the following lessons on VaR implementation: time horizon over which VaR is estimated, the recognition of time varying volatility in VaR risk factors, and VaR backtesting, (page 56) 2. Describe exogenous and endogenous liquidity risk and explain how they might be integrated into VaR models, (page 57) 3. Compare VaR, expected shortfall, and other relevant risk measures, (page 57) 4. Compare unified and compartmentalized risk measurement, (page 58) 5. Compare the results of research on top-down and bottom-up risk aggregation methods, (page 59) 6. Describe the relationship between leverage, market value of asset, and VaR within an active balance sheet management framework, (page 60) 6. Some Correlation Basics: Properties, Motivation, Terminology After completing this reading, you should be able to: 1. Describe financial correlation risk and the areas in which it appears in finance. (page 64) 2. Explain how correlation contributed to the global financial crisis of 2007 to 2009. (page 74) 3. Describe the structure, uses, and payoffs of a correlation swap, (page 70) 4. Estimate the impact of different correlations between assets in the trading book on the VaR capital charge, (page 71) 5. Explain the role of correlation risk in market risk and credit risk, (page 76) 6. Relate correlation risk to systemic and concentration risk, (page 76) 7. Empirical Properties of Correlation: How Do Correlations Behave in the Real World? After completing this reading, you should be able to: 1. Describe how equity correlations and correlation volatilities behave throughout various economic states, (page 88) 2. Calculate a mean reversion rate using standard regression and calculate the 3. corresponding autocorrelation, (page 89) Identify the best-fit distribution for equity, bond, and default correlations. (page 92) 8. Statistical Correlation Models Can We Apply Them to Finance? After completing this reading, you should be able to: 1. Evaluate the limitations of financial modeling with respect to the model itself, calibration of the model, and the models output, (page 98) 2. Assess the Pearson correlation approach, Spearmans rank correlation, and Kendalls , and evaluate their limitations and usefulness in finance, (page 100) t 9. Financial Correlation Modeling Bottom-Up Approaches After completing this reading, you should be able to: 1. Explain the purpose of copula functions and the translation of the copula equation. (page 111) 2. Describe the Gaussian copula and explain how to use it to derive the joint probability of default of two assets, (page 112) 3. Summarize the process of finding the default time of an asset correlated to all other assets in a portfolio using the Gaussian copula, (page 115) 2018 Kaplan, Inc. Page xi Book 1 Reading Assignments and Learning Objectives 10. Empirical Approaches to Risk Metrics and Hedging After completing this reading, you should be able to: 1. Explain the drawbacks to using a DV01-neutral hedge for a bond position. (page 121) 2. Describe a regression hedge and explain how it can improve a standard DV01- neutral hedge, (page 122) 3. Calculate the regression hedge adjustment factor, beta, (page 123) 4. Calculate the face value of an offsetting position needed to carry out a regression hedge, (page 123) 3. Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge, (page 124) 6. Compare and contrast level and change regressions, (page 123) 7. Describe principal component analysis and explain how it is applied to constructing a hedging portfolio, (page 125) 11. The Science of Term Structure Models After completing this reading, you should be able to: 1. Calculate the expected discounted value of a zero-coupon security using a binomial tree, (page 132) 2. Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios, (page 132) 3. Define risk-neutral pricing and apply it to option pricing, (page 135) 4. Distinguish between true and risk-neutral probabilities, and apply this difference to interest rate drift, (page 135) 5. Explain how the principles of arbitrage pricing of derivatives on fixed income securities can be extended over multiple periods, (page 136) 6. Define option-adjusted spread (OAS) and apply it to security pricing, (page 141) 7. Describe the rationale behind the use of recombining trees in option pricing. (page 138) 8. Calculate the value of a constant maturity Treasury swap, given an interest rate tree and the risk-neutral probabilities, (page 139) 9. Evaluate the advantages and disadvantages of reducing the size of the time steps on the pricing of derivatives on fixed income securities, (page 142) 10. Evaluate the appropriateness of the Black-Scholes-Merton model when valuing derivatives on fixed income securities, (page 142) 11. Describe the impact of embedded options on the value of fixed income securities, (page 143) 12. The Evolution of Short Rates and the Shape of the Term Structure After completing this reading, you should be able to: 1. Explain the role of interest rate expectations in determining the shape of the term structure, (page 149) 2. Apply a risk-neutral interest rate tree to assess the effect of volatility on the shape of the term structure, (page 151) 3. Estimate the convexity effect using Jensens inequality, (page 153) 4. Evaluate the impact of changes in maturity, yield, and volatility on the convexity of a security, (page 153) 5. Calculate the price and return of a zero coupon bond incorporating a risk premium, (page 157) Page xii 2018 Kaplan, Inc. Book 1 Reading Assignments and Learning Objectives 13. The Art of Term Structure Models: Drift After completing this reading, you should be able to: 1. Construct and describe the effectiveness of a short term interest rate tree assuming normally distributed rates, both with and without drift, (page 164) 2. Calculate the short-term rate change and standard deviation of the rate change using a model with normally distributed rates and no drift, (page 163) 3. Describe methods for addressing the possibility of negative short-term rates in term structure models, (page 166) 4. Construct a short-term rate tree under the Ho-Lee Model with time-dependent drift, (page 168) 3. Describe uses and benefits of the arbitrage-free models and assess the issue of fitting models to market prices, (page 168) 6. Describe the process of constructing a simple and recombining tree for a short-term rate under the Vasicek Model with mean reversion, (page 169) 7. Calculate the Vasicek Model rate change, standard deviation of the rate change, expected rate in T years, and half life, (page 172) 8. Describe the effectiveness of the Vasicek Model, (page 173) 14. The Art of Term Structure Models: Volatility and Distribution After completing this reading, you should be able to: 1. Describe the short-term rate process under a model with time-dependent volatility, (page 179) 2. Calculate the short-term rate change and determine the behavior of the standard deviation of the rate change using a model with time dependent volatility. (page 179) 3. Assess the efficacy of time-dependent volatility models, (page 180) 4. Describe the short-term rate process under the Cox-Ingersoll-Ross (CIR) and lognormal models, (page 181) 5. Calculate the short-term rate change and describe the basis point volatility using the CIR and lognormal models, (page 181) 6. Describe lognormal models with deterministic drift and mean reversion, (page 183) 15. Volatility Smiles After completing this reading, you should be able to: 1. Define volatility smile and volatility skew, (page 190) 2. Explain the implications of put-call parity on the implied volatility of call and put options, (page 189) 3. Compare the shape of the volatility smile (or skew) to the shape of the implied distribution of the underlying asset price and to the pricing of options on the underlying asset, (page 190) 4. Describe characteristics of foreign exchange rate distributions and their implications on option prices and implied volatility, (page 191) 5. Describe the volatility smile for equity options and foreign currency options and provide possible explanations for its shape, (page 191) 6. Describe alternative ways of characterizing the volatility smile, (page 192) 7. Describe volatility term structures and volatility surfaces and how they may be used to price options, (page 193) 8. Explain the impact of the volatility smile on the calculation of the Greeks. (page 193) 9. Explain the impact of a single asset price jump on a volatility smile, (page 194) 2018 Kaplan, Inc. Page xiii The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in: E s t im a t in g M a r k e t R i s k M e a s u r e s : A n In t r o d u c t i o n a n d O v e r v ie w Topic 1 E x a m F o c u s In this topic, the focus is on the estimation of market risk measures, such as value at risk (VaR). VaR identifies the probability that losses will be greater than a pre-specified threshold level. For the exam, be prepared to evaluate and calculate VaR using historical simulation and parametric models (both normal and lognormal return distributions). One drawback to VaR is that it does not estimate losses in the tail of the returns distribution. Expected shortfall (ES) does, however, estimate the loss in the tail (i.e., after the VaR threshold has been breached) by averaging loss levels at different confidence levels. Coherent risk measures incorporate personal risk aversion across the entire distribution and are more general than expected shortfall. Quantile-quantile (QQ) plots are used to visually inspect if an empirical distribution matches a theoretical distribution. E s t i m a t i n g R e t u r n s To better understand the material in this topic, it is helpful to recall the computations of arithmetic and geometric returns. Note that the convention when computing these returns (as well as VaR) is to quote return losses as positive values. For example, if a portfolio is expected to decrease in value by $ 1 million, we use the terminology expected loss is $1 million rather than expected profit is -$1 million. Profit/loss data: Change in value of asset/portfolio, P at the end of period t plus any interim payments, D . P/Lt = Pt + D t - P Arithmetic return data: Assumption is that interim payments do not earn a return (i.e., no reinvestment). Hence, this approach is not appropriate for long investment horizons. Pt +Dt ~Pt-i Pt-1 Geometric return data: Assumption is that interim payments are continuously reinvested. Note that this approach ensures that asset price can never be negative. 2018 Kaplan, Inc. Page 1 Topic 1 Cross Reference to GARP Assigned Reading - Dowd, Chapter 3 H i s t o r i c a l S i m u l a t i o n A p p r o a c h 2018 SchweserNotes Part Market Risk Measurement and Management eBook 1 K A P L A N ' ) S C H W E S E R Getting Started FRM Exam Part II Welcome As the VP of Advanced Designations at Kaplan Schweser, I am pleased to have the opportunity to help you prepare for the 2018 FRM Exam. Getting an early start on your study program is important for you to sufficiently prepare, practice, and perform on exam day. Proper planning will allow you to set aside enough time to master the learning objectives in the Part II curriculum. Now that you've received your SchweserNotes, here's how to get started: Step 1: Access Your Online Tools Visit www.schweser.com/frm and log in to your online account using the button located in the top navigation bar. After logging in, select the appropriate part and proceed to the dashboard where you can access your online products. Step 2: Create a Study Plan Create a study plan with the Schweser Study Calendar (located on the Schweser dashboard). Then view the Candidate Resource Library on-demand videos for an introduction to core concepts. Step 3: Prepare and Practice Read your SchweserNotes Our clear, concise study notes will help you prepare for the exam. At the end of each reading, you can answer the Concept Checker questions for better understanding of the curriculum. Attend a Weekly Class Attend our Live Online Weekly Class or review the on-demand archives as often as you like. Our expert faculty will guide you through the FRM curriculum with a structured approach to help you prepare for the exam. (See our instruction packages to the right. Visit www.schweser.com/frm to order.) Practice with SchweserPro QBank Maximize your retention of important concepts and practice answering exam-style questions in the SchweserPro QBank and taking several Practice Exams. Use Schweser's QuickSheet for continuous review on the go. (Visit www.schweser.com/frm to order.) Step 4: Final Review A few weeks before the exam, make use of our Online Review Workshop Package. Review key curriculum concepts in every topic, perform by working through demonstration problems, and practice your exam techniques with our 8-hour live Online Review Workshop. Use Schweser's Secret Sauce for convenient study on the go. Step 5: Perform As part of our Online Review Workshop Package, take a Schweser Mock Exam to ensure you are ready to perform on the actual FRM Exam. Put your skills and knowledge to the test and gain confidence before the exam.. Again, thank you fortrusting Kaplan Schweser with your FRM Exam preparation! Sincerely, Derek Burkett, CFA, FRM, CAIA VP, Advanced Designations, Kaplan Schweser The Kaplan Way Acquire new knowledge through demonstration and examples. Practice < ?xl-----bs Apply new knowledge through simulation and practice. Evaluate mastery of new knowledge and identify achieved outcomes. PremiumPlus Package FRM Instruction Packages: PremiumPlus Package Premium Instruction Package Live Instruction*: Remember to join our Live Online Weekly Class. Register online today at www.schweser.com/frm. *Dates, tim es, and instructors subject to change MKT-005736 Contact us for questions about your study package, upgrading your package, purchasing additional study materials, or for additional information: www.schweser.com/frm Toll-Free: 888.325.5072 International: +1 608.779.8397 FRM Pa r t II B o o k 1: M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t W e l c o m e t o t h e 2018 S c h w e s e r N o t e s R e a d i n g A s s i g n m e n t s a n d L e a r n i n g O b j e c t i v e s M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t 1: Estimating Market Risk Measures: An Introduction and Overview 2: Non-parametric Approaches 3: Backtesting VaR 4: VaR Mapping 5: Messages from the Academic Literature on Risk Measurement for the Trading Book 6: Some Correlation Basics: Properties, Motivation, Terminology 7: Empirical Properties of Correlation: How Do Correlations Behave in the Real World? 8: Statistical Correlation Models Can We Apply Them to Finance? 9: Financial Correlation ModelingBottom-Up Approaches 10: Empirical Approaches to Risk Metrics and Hedging 11: The Science of Term Structure Models 12: The Evolution of Short Rates and the Shape of the Term Structure 13: The Art of Term Structure Models: Drift 14: The Art of Term Structure Models: Volatility and Distribution 15: Volatility Smiles S e l f -Te s t : M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t F o r m u l a s A p p e n d i x In d e x v viii 1 15 25 38 56 64 8 8 98 111 121 132 149 164 179 189 199 205 209 213 2018 Kaplan, Inc. Page iii FRM 2018 PART II BOOK 1: MARKET RISK MEASUREMENT AND MANAGEMENT 2018 Kaplan, Inc. All rights reserved. Published in 2018 by Kaplan, Inc. Printed in the United States of America. ISBN: 978-1-4754-7029-1 Required Disclaimer: GARP does not endorse, promote, review, or warrant the accuracy of the products or services offered by Kaplan 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, nor is GARP responsible for any fees or costs of any person or entity providing any services to Kaplan. FRM, GARP, and Global Association of Risk Professionals are trademarks owned by the Global Association of Risk Professionals, Inc. 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 iv 2018 Kaplan, Inc. We l c o m e t o t h e 2018 S c h w e s e r N o t e s Thank you for trusting Kaplan Schweser to help you reach your career and educational goals. We are very pleased to be able to help you prepare for the FRM Part II exam. In this introduction, I want to explain the resources included with the SchweserNotes, suggest how you can best use Schweser materials to prepare for the exam, and direct you toward other educational resources you will find helpful as you study for the exam. Besides the SchweserNotes themselves, there are many online educational resources available at Schweser.com. Just log in using the individual username and password you received when you purchased the SchweserNotes. Schw eserN otes The SchweserNotes consist of four volumes that include complete coverage of all FRM assigned topics and learning objectives (LOs), Concept Checkers (multiple-choice questions for every topic), and Self-Test questions to help you master the material and check your retention of key concepts. Online Practice Questions To retain what you learn, it is important that you quiz yourself often. We offer an online version of the SchweserPro QBank, which contains hundreds of Part II practice questions and explanations. Quizzes are available for each topic or across multiple topics. Build your own exams by specifying the topics and the number of questions. Practice Exams Schweser offers two full 4-hour practice exams. These exams are important tools for gaining the speed and skills you will need to pass the exam. The Practice Exams book contains answers with full explanations for self-grading and evaluation. Schweser Study Calendar Use your Online Access to tell us when you will start and what days of the week you can study. The online Schweser Study Calendar will create a study plan just for you, breaking the curriculum into daily and weekly tasks to keep you on track and help you monitor your study progress. 2018 Kaplan, Inc. Page v Book 1 Welcome to the 2018 SchweserNotes The FRM Part II exam is a formidable challenge (covering 79 assigned readings and almost 500 learning objectives), and you must devote considerable time and effort to be properly prepared. There are no shortcuts! You must learn the material, know the terminology and techniques, understand the concepts, and be able to answer 80 multiple choice questions quickly and (at least 70%) correctly. A good estimate of the study time required on average is 250 hours, but some candidates will need more or less time, depending on their individual backgrounds and experience. To help you really master this material and be well-prepared for the FRM exam, we offer several other educational resources, including: Online Weekly Class Our Online Weekly Class is offered each week, beginning in February for the May exam and August for the November exam. This online class brings the personal attention of a classroom into your home or office with 30 hours of real-time instruction, led by David McMeekin, CFA, GALA, FRM. The class offers in-depth coverage of difficult concepts, instant feedback during lecture and Q&A sessions, and discussion of sample exam questions. Archived classes are available for viewing at any time throughout the season. Candidates enrolled in the Online Weekly Class also have full access to supplemental on- demand video instruction in the Candidate Resource Library and an e-mail address link for sending questions to the instructor at any time. Late-Season Review Late-season review and exam practice can make all the difference. Our Review Package helps you evaluate your exam readiness with products specifically designed for late-season studying. This Review Package includes the Online Review Workshop (8-hour live and archived online review of essential curriculum topics), the Schweser Mock Exam (one 4-hour exam), and Schweser s Secret Sauce (concise summary of the FRM curriculum). Page vi 2018 Kaplan, Inc. Book 1 Welcome to the 2018 SchweserNotes Part II Exam Weights In preparing for the exam, pay attention to the weights assigned to each knowledge domain within the curriculum. The Part II exam weights are as follows: Book Knowledge Domains Exam Weight Exam Questions 1 2 3 4 4 Market Risk Measurement and Management Credit Risk Measurement and Management Operational and Integrated Risk Management Risk Management and Investment Management Current Issues in Financial Markets 25% 25% 25% 15% 10% 20 20 20 12 8 H ow to Succeed There are no shortcuts to studying for this exam. Expect the Global Association of Risk Professionals (GARP) to test you in a way that will reveal how well you know the Part II curriculum. You should begin studying early and stick to your study plan. You should first read the SchweserNotes and complete the Concept Checkers for each topic. At the end of each book, you should answer the provided Self-Test questions to understand how concepts may be tested on the exam. You should finish the overall curriculum at least two weeks before the FRM exam. This will allow sufficient time for Practice Exams and further review of those topics you have not yet mastered. I would like to take this opportunity to thank the content developers, editors, and graphic designers who worked countless hours to create the 2018 FRM SchweserNotes. I would especially like to thank Derek Burkett, CFA, FRM, CAIA; Adam Stueber, CAIA; Craig Prochaska, CFA; Kent Westlund, CFA, FRM; Kurt Schuldes, CFA, CAIA; Tim Greive, CFA, CAIA; Jeff Bahr, Andy Bauer, Allie Bottcher, Katherine Bourgeois, Alyssa Brunner, Lindsey Casto, Laura Goetzinger, Ryan Henry, Hannah Kelley, Alissa Knop, Genevieve Kretschmer, Gretchen Panzer, Jessica Pearse, Ashley Sinclair, Ben Strong, and Debbie White for their contributions. Best regards, /Uc Snut& Eric Smith, CFA, FRM Content Manager Kaplan Schweser 2018 Kaplan, Inc. Page vii R e a d i n g A s s i g n m e n t s a n d L e a r n i n g O b j e c t i v e s The following material is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by the Global Association of Risk Professionals. R e a d i n g A s s i g n m e n t s Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, U.K.: John Wiley & Sons, 2005). 1. Estimating Market Risk Measures: An Introduction and Overview, Chapter 3 (page 1) 2. Non-parametric Approaches, Chapter 4 (page 15) Philippe Jorion, Value- at-Risk: The New Benchmark for Managing Financial Risk, 3rd Edition (New York, NY: McGraw Hill, 2007). 3. Backtesting VaR, Chapter 6 4. VaR Mapping, Chapter 11 (page 25) (page 38) 5. Messages from the Academic Literature on Risk Measurement for the Trading Book, Basel Committee on Banking Supervision, Working Paper No. 19, Jan 2011. (page 56) Gunter Meissner, Correlation Risk Modeling and Management (New York, NY: John Wiley & Sons, 2014). 6. Some Correlation Basics: Properties, Motivation, Terminology, Chapter 1 (page 64) 7. 8. 9. Empirical Properties of Correlation: How Do Correlations Behave in the Real World?, Chapter 2 Statistical Correlation Models Can We Apply Them to Finance?, Chapter 3 (page 88) (page 98) Financial Correlation Modeling Bottom-Up Approaches, Chapter 4, Sections 4.3.0 (intro), 4.3.1, and 4.3.2 only (page 111) Bruce Tuckman and .Angel Serrat, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011). 10. Empirical Approaches to Risk Metrics and Hedging, Chapter 6 11. The Science of Term Structure Models, Chapter 7 12. The Evolution of Short Rates and the Shape of the Term Structure, Chapter 8 (page 121) (page 132) (page 149) Page viii 2018 Kaplan, Inc. Book 1 Reading Assignments and Learning Objectives 13. The Art of Term Structure Models: Drift, Chapter 9 (page 164) 14. The Art of Term Structure Models: Volatility and Distribution, Chapter 10 (page 179) John C. Hull, Options, Futures, and Other Derivatives, 10th Edition (New York, NY: Pearson, 2017). 13. Volatility Smiles, Chapter 20 (page 189) 2018 Kaplan, Inc. Page ix Book 1 Reading Assignments and Learning Objectives L e a r n i n g O b j e c t i v e s 1. Estimating Market Risk Measures: An Introduction and Overview After completing this reading, you should be able to: 1. Estimate VaR using a historical simulation approach, (page 2) 2. Estimate VaR using a parametric approach for both normal and lognormal return distributions, (page 4) 3. Estimate the expected shortfall given P/L or return data, (page 6) 4. Define coherent risk measures, (page 6) 3. Estimate risk measures by estimating quantiles, (page 6) 6. Evaluate estimators of risk measures by estimating their standard errors, (page 7) 7. Interpret Q Q plots to identify the characteristics of a distribution, (page 9) 2. Non-parametric Approaches After completing this reading, you should be able to: 1. Apply the bootstrap historical simulation approach to estimate coherent risk measures, (page 13) 2. Describe historical simulation using non-parametric density estimation, (page 16) 3. Compare and contrast the age-weighted, the volatility-weigh ted, the correlation- 4. weighted, and the filtered historical simulation approaches, (page 17) Identify advantages and disadvantages of non-parametric estimation methods. (page 19) 3. Backtesting VaR After completing this reading, you should be able to: 1. Define backtesting and exceptions and explain the importance of backtesting VaR models, (page 25) 2. Explain the significant difficulties in backtesting a VaR model, (page 26) 3. Verify a model based on exceptions or failure rates, (page 26) 4. Define and identify Type I and Type II errors, (page 28) 5. Explain the need to consider conditional coverage in the backtesting framework, (page 32) 6. Describe the Basel rules for backtesting, (page 33) 4. VaR Mapping After completing this reading, you should be able to: 1. Explain the principles underlying VaR mapping, and describe the mapping process, (page 38) 2. Explain how the mapping process captures general and specific risks, (page 39) 3. Differentiate among the three methods of mapping portfolios of fixed income securities, (page 41) 4. Summarize how to map a fixed income portfolio into positions of standard instruments, (page 41) 5. Describe how mapping of risk factors can support stress testing, (page 44) 6. Explain how VaR can be used as a performance benchmark, (page 45) 7. Describe the method of mapping forwards, forward rate agreements, interest rate swaps, and options, (page 48) Page x 2018 Kaplan, Inc. Book 1 Reading Assignments and Learning Objectives 5. Messages from the Academic Literature on Risk Measurement for the Trading Book After completing this reading, you should be able to: 1. Explain the following lessons on VaR implementation: time horizon over which VaR is estimated, the recognition of time varying volatility in VaR risk factors, and VaR backtesting, (page 56) 2. Describe exogenous and endogenous liquidity risk and explain how they might be integrated into VaR models, (page 57) 3. Compare VaR, expected shortfall, and other relevant risk measures, (page 57) 4. Compare unified and compartmentalized risk measurement, (page 58) 5. Compare the results of research on top-down and bottom-up risk aggregation methods, (page 59) 6. Describe the relationship between leverage, market value of asset, and VaR within an active balance sheet management framework, (page 60) 6. Some Correlation Basics: Properties, Motivation, Terminology After completing this reading, you should be able to: 1. Describe financial correlation risk and the areas in which it appears in finance. (page 64) 2. Explain how correlation contributed to the global financial crisis of 2007 to 2009. (page 74) 3. Describe the structure, uses, and payoffs of a correlation swap, (page 70) 4. Estimate the impact of different correlations between assets in the trading book on the VaR capital charge, (page 71) 5. Explain the role of correlation risk in market risk and credit risk, (page 76) 6. Relate correlation risk to systemic and concentration risk, (page 76) 7. Empirical Properties of Correlation: How Do Correlations Behave in the Real World? After completing this reading, you should be able to: 1. Describe how equity correlations and correlation volatilities behave throughout various economic states, (page 88) 2. Calculate a mean reversion rate using standard regression and calculate the 3. corresponding autocorrelation, (page 89) Identify the best-fit distribution for equity, bond, and default correlations. (page 92) 8. Statistical Correlation Models Can We Apply Them to Finance? After completing this reading, you should be able to: 1. Evaluate the limitations of financial modeling with respect to the model itself, calibration of the model, and the models output, (page 98) 2. Assess the Pearson correlation approach, Spearmans rank correlation, and Kendalls , and evaluate their limitations and usefulness in finance, (page 100) t 9. Financial Correlation Modeling Bottom-Up Approaches After completing this reading, you should be able to: 1. Explain the purpose of copula functions and the translation of the copula equation. (page 111) 2. Describe the Gaussian copula and explain how to use it to derive the joint probability of default of two assets, (page 112) 3. Summarize the process of finding the default time of an asset correlated to all other assets in a portfolio using the Gaussian copula, (page 115) 2018 Kaplan, Inc. Page xi Book 1 Reading Assignments and Learning Objectives 10. Empirical Approaches to Risk Metrics and Hedging After completing this reading, you should be able to: 1. Explain the drawbacks to using a DV01-neutral hedge for a bond position. (page 121) 2. Describe a regression hedge and explain how it can improve a standard DV01- neutral hedge, (page 122) 3. Calculate the regression hedge adjustment factor, beta, (page 123) 4. Calculate the face value of an offsetting position needed to carry out a regression hedge, (page 123) 3. Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge, (page 124) 6. Compare and contrast level and change regressions, (page 123) 7. Describe principal component analysis and explain how it is applied to constructing a hedging portfolio, (page 125) 11. The Science of Term Structure Models After completing this reading, you should be able to: 1. Calculate the expected discounted value of a zero-coupon security using a binomial tree, (page 132) 2. Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios, (page 132) 3. Define risk-neutral pricing and apply it to option pricing, (page 135) 4. Distinguish between true and risk-neutral probabilities, and apply this difference to interest rate drift, (page 135) 5. Explain how the principles of arbitrage pricing of derivatives on fixed income securities can be extended over multiple periods, (page 136) 6. Define option-adjusted spread (OAS) and apply it to security pricing, (page 141) 7. Describe the rationale behind the use of recombining trees in option pricing. (page 138) 8. Calculate the value of a constant maturity Treasury swap, given an interest rate tree and the risk-neutral probabilities, (page 139) 9. Evaluate the advantages and disadvantages of reducing the size of the time steps on the pricing of derivatives on fixed income securities, (page 142) 10. Evaluate the appropriateness of the Black-Scholes-Merton model when valuing derivatives on fixed income securities, (page 142) 11. Describe the impact of embedded options on the value of fixed income securities, (page 143) 12. The Evolution of Short Rates and the Shape of the Term Structure After completing this reading, you should be able to: 1. Explain the role of interest rate expectations in determining the shape of the term structure, (page 149) 2. Apply a risk-neutral interest rate tree to assess the effect of volatility on the shape of the term structure, (page 151) 3. Estimate the convexity effect using Jensens inequality, (page 153) 4. Evaluate the impact of changes in maturity, yield, and volatility on the convexity of a security, (page 153) 5. Calculate the price and return of a zero coupon bond incorporating a risk premium, (page 157) Page xii 2018 Kaplan, Inc. Book 1 Reading Assignments and Learning Objectives 13. The Art of Term Structure Models: Drift After completing this reading, you should be able to: 1. Construct and describe the effectiveness of a short term interest rate tree assuming normally distributed rates, both with and without drift, (page 164) 2. Calculate the short-term rate change and standard deviation of the rate change using a model with normally distributed rates and no drift, (page 163) 3. Describe methods for addressing the possibility of negative short-term rates in term structure models, (page 166) 4. Construct a short-term rate tree under the Ho-Lee Model with time-dependent drift, (page 168) 3. Describe uses and benefits of the arbitrage-free models and assess the issue of fitting models to market prices, (page 168) 6. Describe the process of constructing a simple and recombining tree for a short-term rate under the Vasicek Model with mean reversion, (page 169) 7. Calculate the Vasicek Model rate change, standard deviation of the rate change, expected rate in T years, and half life, (page 172) 8. Describe the effectiveness of the Vasicek Model, (page 173) 14. The Art of Term Structure Models: Volatility and Distribution After completing this reading, you should be able to: 1. Describe the short-term rate process under a model with time-dependent volatility, (page 179) 2. Calculate the short-term rate change and determine the behavior of the standard deviation of the rate change using a model with time dependent volatility. (page 179) 3. Assess the efficacy of time-dependent volatility models, (page 180) 4. Describe the short-term rate process under the Cox-Ingersoll-Ross (CIR) and lognormal models, (page 181) 5. Calculate the short-term rate change and describe the basis point volatility using the CIR and lognormal models, (page 181) 6. Describe lognormal models with deterministic drift and mean reversion, (page 183) 15. Volatility Smiles After completing this reading, you should be able to: 1. Define volatility smile and volatility skew, (page 190) 2. Explain the implications of put-call parity on the implied volatility of call and put options, (page 189) 3. Compare the shape of the volatility smile (or skew) to the shape of the implied distribution of the underlying asset price and to the pricing of options on the underlying asset, (page 190) 4. Describe characteristics of foreign exchange rate distributions and their implications on option prices and implied volatility, (page 191) 5. Describe the volatility smile for equity options and foreign currency options and provide possible explanations for its shape, (page 191) 6. Describe alternative ways of characterizing the volatility smile, (page 192) 7. Describe volatility term structures and volatility surfaces and how they may be used to price options, (page 193) 8. Explain the impact of the volatility smile on the calculation of the Greeks. (page 193) 9. Explain the impact of a single asset price jump on a volatility smile, (page 194) 2018 Kaplan, Inc. Page xiii The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in: E s t im a t in g M a r k e t R i s k M e a s u r e s : A n In t r o d u c t i o n a n d O v e r v ie w Topic 1 E x a m F o c u s In this topic, the focus is on the estimation of market risk measures, such as value at risk (VaR). VaR identifies the probability that losses will be greater than a pre-specified threshold level. For the exam, be prepared to evaluate and calculate VaR using historical simulation and parametric models (both normal and lognormal return distributions). One drawback to VaR is that it does not estimate losses in the tail of the returns distribution. Expected shortfall (ES) does, however, estimate the loss in the tail (i.e., after the VaR threshold has been breached) by averaging loss levels at different confidence levels. Coherent risk measures incorporate personal risk aversion across the entire distribution and are more general than expected shortfall. Quantile-quantile (QQ) plots are used to visually inspect if an empirical distribution matches a theoretical distribution. E s t i m a t i n g R e t u r n s To better understand the material in this topic, it is helpful to recall the computations of arithmetic and geometric returns. Note that the convention when computing these returns (as well as VaR) is to quote return losses as positive values. For example, if a portfolio is expected to decrease in value by $ 1 million, we use the terminology expected loss is $1 million rather than expected profit is -$1 million. Profit/loss data: Change in value of asset/portfolio, P at the end of period t plus any interim payments, D . P/Lt = Pt + D t - P Arithmetic return data: Assumption is that interim payments do not earn a return (i.e., no reinvestment). Hence, this approach is not appropriate for long investment horizons. Pt +Dt ~Pt-i Pt-1 Geometric return data: Assumption is that interim payments are continuously reinvested. Note that this approach ensures that asset price can never be negative. 2018 Kaplan, Inc. Page 1 Topic 1 Cross Reference to GARP Assigned Reading - Dowd, Chapter 3 H i s t o r i c a l S i m u l a t i o n A p p r o a c h