Temp_store

LO 5.2: Describe exogenous and endogenous liquidity risk and explain how they m ight be integrated into VaR models.
During times of a financial crisis, market liquidity conditions change, which changes the liquidity horizon of an investment (i.e., the time to unwind a position without materially affecting its price). Two types of liquidity risk are exogenous liquidity and endogenous liquidity. Both types of liquidity are important to measure; however, academic studies suggest that risk valuation models should first account for the impact of endogenous liquidity.
Professors Note: In Book 3, we w ill examine the estimation o f liquidity risk using the exogenous spread approach and the endogenous price approach.
Exogenous liquidity is handled through the calculation of a liquidity-adjusted VaR (LVaR) measure, and represents market-specific, average transaction costs. The LVaR measure incorporates a bid/ask spread by adding liquidity costs to the initial estimate of VaR.
Endogenous liquidity is an adjustment for the price effect of liquidating positions. It depends on trade sizes and is applicable when market orders are large enough to move prices. Endogenous liquidity is the elasticity of prices to trading volumes and is more easily observed in instances of high liquidity risk.
Poor market conditions can cause a flight to quality, which decreases a traders ability to unwind positions in thinly traded assets. Thus, endogenous liquidity risk is most applicable to exotic/complex trading positions and very relevant in high-stress market conditions, however, endogenous liquidity costs will be present in all market conditions.
R i s k M e a s u r e s

LO 5.1: Explain the following lessons on VaR implementation: tim e horizon over which VaR is estimated, the recognition o f time varying volatility in VaR risk factors, and VaR backtesting.
There is no consensus regarding the proper time horizon for risk measurement. The appropriate time horizon depends on the risk measurement purpose (e.g., setting capital limits) as well as portfolio liquidity. Thus, there is not a universally accepted approach for aggregating various VaR measures based on different time horizons.
Time-varying volatility results from volatility fluctuations over time. The effect of time- varying volatility on the accuracy of VaR measures decreases as time horizon increases. However, volatility generated by stochastic (i.e., random) jumps will reduce the accuracy of long-term VaR measures unless there is an adjustment made for stochastic jumps. It is important to recognize time-varying volatility in VaR measures since ignoring it will likely lead to an underestimation of risk. In addition to volatility fluctuations, risk managers should also account for time-varying correlations when making VaR calculations.
To simplify VaR estimation, the financial industry has a tendency to use short time horizons. This approach is computationally attractive for larger portfolios. However, a 10- day VaR time horizon, as suggested by the Basel Committee on Banking Supervision, is not always optimal. It is more preferred to instead allow the risk horizon to vary based on specific investment characteristics. When computing VaR over longer time horizons, a risk manager needs to account for the variation in a portfolios composition over time. Thus, a longer than 10-day time horizon may be necessary for economic capital purposes.
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Historically, VaR backtesting has been used to validate VaR models. However, backtesting is not effective when the number of VaR exceptions is small. In addition, backtesting is less effective over longer time horizons due to portfolio instability. VaR models tend to be more realistic if time-varying volatility is incorporated; however, this approach tends to generate a procyclical VaR measure and produces unstable risk models due to estimation issues.
In t e g r a t i n g L i q u i d i t y R i s k i n t o V a R M o d e l s

LO 4.7: Describe the method o f m apping forwards, forward rate agreements, interest rate swaps, and options.
M a p p i n g A p p r o a c h e s f o r L i n e a r D e r i v a t i v e s
Forward Contracts
The delta-normal method provides accurate estimates of VaR for portfolios and assets that can be expressed as linear combinations of normally distributed risk factors. Once a portfolio, or financial instrument, is expressed as a linear combination of risk factors, a covariance (correlation) matrix can be generated, and VaR can be measured using matrix multiplication.
Forwards are appropriate for the application of the delta-normal method. Their values are a linear combination of a few general risk factors, which have commonly available volatility and correlation data.
The current value of a forward contract is equal to the present value of the difference between the current forward rate, F , and the locked in delivery rate, K, as follows:
Forwardt = (Ft K)e rt
Suppose you wish to compute the diversified VaR of a forward contract that is used to purchase euros with U.S. dollars one year from now. This forward position is analogous to the following three separate risk positions:
1. A short position in a U.S. Treasury bill.
2. A long position in a one-year euro bill.
3. A long position in the euro spot market.
Figure 10 presents the pricing information for the purchase of $100 million euros in exchange for $126.3 million, as well as the correlation matrix between the positions.
Figure 10: Monthly VaR for Forward Contract and Correlation Matrix
Risk Factor EUR spot Long EUR bill Short USD bill EUR forward
Price/Rate 1.2500 0.0170 0.0292 1.2650
VaR% 4.5381 0.1396 0.2121
EUR Spot
1.000 0.115 0.073
1 Yr EUR 0.115 1.000 -0.047
lY rU S 0.073 -0.047 1.000
In this example, we have a long position in a EUR contract worth $122.911 million today and a short position in a one-year U.S. T-bill worth $122.911 today, as illustrated in Figure 11. The fourth column represents the investment present values. The fifth column represents the absolute present value of cash flows multiplied by the VaR percentage from Figure 10.
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Figure 11: Undiversified and Diversified VaR of Forward Contract
Position EUR spot Long EUR bill Short USD bill
Undiversified V aR
PVfactor
CF
0.9777 0.9678
100.0 126.5
122.911 122.911 122.911
D iversified VaR *Note that some rounding has occurred.
/ w 5.578 0.172 0.261 6.010
xA VaR 31.116 0.142 -0.036 31.221 5.588
The undiversified VaR for this position is $6.01 million, and the diversified VaR for this position is $5,588 million. Recall that the diversified VaR is computed using matrix algebra.
The general procedure weve outlined for forwards also applies to other types of financial instruments, such as forward rate agreements and interest rate swaps. As long as an instrument can be expressed as linear combinations of its basic components, the delta- normal VaR may be applied with reasonable accuracy.
Forward Rate Agreements (FRA)
Suppose you have an FRA that locks in an interest rate one year from now. Figure 12 illustrates data related to selling a 6 x 12 FRA on $100 million. This amount is equivalent to borrowing $100 million for a 6-month period (180 days) and investing the proceeds at the 12-month rate (360 days). Assuming that the 360-day spot rate is 4.5% and the 180-day spot rate is 4.1%, the present values of the cash flows are presented in the second column of Figure 12. The present value of the notional $100 million contract is x = $100 / 1.0205 = $97,991 million. This will be invested for a 12-month period. The forward rate is then computed as follows: (1 + F, 9 / 2) = [1.045 / (1 + 0.041 / 2)] = [(1.045 / 1.0205) 1] X 2= 4.80/0.
The sixth column computes the undiversified VaR of $0.62 million at the 95% confidence level using the VaR percentages in the third column multiplied by the absolute value of the present values of cash flows. Matrix algebra is then used to multiply this vector by the correlation matrix presented in columns four and five to compute the diversified VaR of $0,348 million.
Figure 12: Calculating VaR for an FRA
PV(CF), x -97.991 97.991
VaR% 0.1629 0.4696
Position 180 days 360 days
Undiversified V aR
D iversified V aR
Correlations (R) 0.79 1
0.79
1
0.160 0.460 0.620
xAVaR -0.0325 0.1537
0.1212 0.348
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Interest Rate Swaps
Interest rate swaps are commonly used to exchange interest rates from fixed to floating rates or from floating to fixed rates. Thus, an interest rate swap can be broken down into fixed and floating parts. The fixed part is priced with a coupon-paying bond and the floating part is priced as a floating-rate note.
Suppose you want to compute the VaR of a $100 million four-year swap that pays a fixed rate for four years in exchange for a floating-rate payment. The necessary steps to compute the undiversified and diversified VaR amounts are as follows:
Step 1: Begin by creating a present value of cash flows showing the short position of the fixed portion as we agree to pay the fixed interest rates and fixed bond maturity. Then, add the long present value of the variable rate bond at a present value of $ 100 million today.
Step 2: Multiply the vector representing the absolute present values of cash flows by the VaR percentages at the 95% confidence level and sum the values to compute the undiversified VaR amount.
Step 3: Use matrix algebra to multiply the correlation matrix by the absolute values to compute the diversified VaR amount. Again, recall that the diversified VaR is computed using matrix algebra.
M a p p i n g A p p r o a c h e s f o r N o n l i n e a r D e r i v a t i v e s
As mentioned, the delta-normal VaR method is based on linear relationships between variables. Options, however, exhibit nonlinear relationships between movements of the values of the underlying instruments and the values of the options. In many cases, the delta-normal method may still be applied because the value of an option may be expressed linearly as the product of the option delta and the underlying asset.
Unfortunately, the delta-normal VaR cannot be expected to provide an accurate estimate of the true VaR over ranges where deltas are unstable. In other words, over longer periods of time, the delta is not a constant, which makes linear methods inappropriate. Conversely, over short periods of time, such as one day, a linear approximation of the delta is more accurate. However, the accuracy of this approximation is dependent on parameter inputs (i.e., delta increases with the underlying spot price).
For example, assume the strike price of an option is $100 with a volatility of 25%. If we are only concerned about a one-day risk horizon, then the one-day loss could be computed as follows:
aScj%/T = -1 .645 x $ 100 x 0.25 x , = -$2.59
252 V 252
Thus, over a one-day horizon, the worst case scenario at the 95% confidence level is a loss of $2.59, which brings the position down to $97.41. Linear approximations using this method may be reliable for longer maturity options if the risk horizon is very short, such as a one-day time horizon.
Professors Note: Options are usually mapped using a Taylor series approximation and using the delta-gamma method to calculate the option VaR.
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K e y C o n c e p t s
LO 4.1
Value at risk (VaR) mapping involves replacing the current values of a portfolio with risk factor exposures. Portfolio exposures are broken down into general risk factors and mapped onto those factors.
LO 4.2
Specific risk decreases as more risk factors are added to a VaR model.
LO 4.3
Fixed-income risk mapping methods include principal mapping, duration mapping, and cash flow mapping. Principal mapping considers only the principal cash flow at the average life of the portfolio. Duration mapping considers the market value of the portfolio at its duration. Cash flow mapping is the most complex method considering the timing and correlations of all cash flows.
LO 4.4
The primary difference between principal, duration, and cash flow mapping techniques is the consideration of the timing and amount of cash flows.
Undiversified VaR is calculated as:
Undiversified VaR =
x Vj
Diversified VaR is computed using matrix algebra as follows:
Diversified VaR =
= yj(x x V)7R(x x V)
LO 4.5
Stress testing each zero-coupon bond by its VaR is a simpler approach than incorporating correlations; however, this method ceases to be viable if correlations are anything other than 1.
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LO 4.6
A popular use of VaR is to establish a benchmark portfolio and measure VaR of other portfolios in relation to this benchmark. The tracking error VaR is smallest for portfolios most closely matched based on cash flows.
LO 4.7
Delta-normal VaR can be applied to portfolios of many types of instruments as long as the risk factors are linearly related. Application of the delta-normal method with options and other derivatives does not provide accurate VaR measures over long risk horizons in which deltas are unstable.
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C o n c e p t C h e c k e r s
1.
2.
Which of the following methods is not one of the three approaches for mapping a portfolio of fixed-income securities onto risk factors? A. Principal mapping. B. Duration mapping. C. Cash flow mapping. D. Present value mapping.
If portfolio assets are perfectly correlated, portfolio VaR will equal: A. marginal VaR. B. component VaR. C. undiversified VaR. D. diversified VaR.
3.
Which of the following could be considered a general risk factor?
I. Exchange rates. II. Zero-coupon bonds. A. B. C. Both I and II. D. Neither I nor II.
I only. II only.
4.
The VaR percentages at the 93% confidence level for a bond with maturities ranging from one year to five years are as follows:
Maturity
1 2 3 4 5
VAR % 0.4696 0.9868 1.4841 1.9714 2.4261
A bond portfolio consists of a $100 million bond maturing in two years and a $100 million bond maturing in four years. What is the VaR of this bond portfolio using the principal VaR mapping method? A. $1,484 million. B. $1,974 million. C. $2,769 million. D. $2,968 million.
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5. Suppose you are calculating the tracking error VaR for two zero-coupon bonds using a
$ 100 million benchmark bond portfolio with the following maturities and market value weights. Which of the following combinations of two zero-coupon bonds would most likely have the smallest tracking error? month Maturity month 1 1 year 2 year 3 year 4 year 5 year 7 year 10 year 20 year 30 year
Benchmark
1.00 10.00 13.00 24.00 12.00 18.00 9.25 6.50 4.75 1.50
A. 1 year and 7 year. B. 2 year and 4 year. C. 3 year and 3 year. D. 4 year and 7 year.
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C o n c e p t C h e c k e r An s w e r s
1. D
Present value mapping is not one of the approaches.
2. C
If we assume perfect correlation among assets, VaR would be equal to undiversified VaR.
3. A
Exchange rates can be used as general risk factors. Zero-coupon bonds are used to map bond positions but are not considered a risk factor. However, the interest rate on those zeros is a risk factor.
4. D The VaR percentage is 1.4841 for a three-year zero-coupon bond [(2 + 4) / 2 = 3]. We
compute the VaR under the principal method by multiplying the VaR percentage times the market value of the average life of the bond: principal mapping VaR = $200 million x 1.4841% = $2,968 million.
5. C The three-year and five-year cash flows are highest for the benchmark portfolio at $24
million and $ 18 million, respectively. Thus, tracking error VaR will likely be the lowest for the portfolio where the cash flows of the benchmark and zero-coupon bond portfolios are most closely matched.
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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:
M e s s a g e s f r o m t h e A c a d e m ic L it e r a t u r e o n R i s k M e a s u r e m e n t f o r t h e Tr a d in g B o o k
Topic 5
E x a m F o c u s
This topic addresses tools for risk measurement, including value at risk (VaR) and expected shortfall. Specifically, we will examine VaR implementation over different time horizons and VaR adjustments for liquidity costs. This topic also examines academic studies related to integrated risk management and discusses the importance of measuring interactions among risks due to risk diversification. Note that several concepts in this topic, such as liquidity risk, stressed VaR, and capital requirements, will be discussed in more detail in Book 3, which covers operational and integrated risk management and the Basel Accords.
V a l u e a t R i s k (Va R ) Im p l e m e n t a t i o n

LO 4.6: Explain how VaR can be used as a performance benchmark.
It is often convenient to measure VaR relative to a benchmark portfolio. This is what is referred to as benchmarking a portfolio. Portfolios can be constructed that match the risk factors of a benchmark portfolio but have either a higher or a lower VaR. The VaR of the deviation between the two portfolios is referred to as a tracking error VaR. In other words, tracking error VaR is a measure of the difference between the VaR of the target portfolio and the benchmark portfolio.
Suppose you are trying to benchmark the VaR of a $100 million bond portfolio with a duration of 4.77 to a portfolio of two zero-coupon bonds with the same duration at the 95% confidence level. The market value weights of the bonds in the benchmark portfolio and portfolios of two zero-coupon bonds are provided in Figure 7.
Figure 7: Benchmark Portfolio and Zero-Coupon Bond Portfolio Weights
Benchmark
A
B
C
D
E
84.35 month 3year 2 M aturity 1 month 3 month 6 month 1 year 2 year 3 year 4 year 5 year 7 year 9 year 10 year 15 year 20 year 30 year
Total Value
1.00 1.25 2.00 12.50 23.50 17.50 12.00 8.00 6.50 4.50 3.50 3.00 3.25 1.50 100.00
23.00 77.00
55.75
44.25
60.50
39.50
58.10
41.90
100.00
100.00
100.00
100.00
15.65 100.00
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The first step in the benchmarking process is to match the duration with two zero-coupon bonds. Therefore, the weights of the market values of the zero-coupon bonds in Figure 7 are adjusted to match the benchmark portfolio duration of 4.77. Figure 8 illustrates the creation of five two-bond portfolios with a duration of 4.77. The market values of all bonds in the zero-coupon portfolios are adjusted to match the duration of the benchmark portfolio. For example, portfolio A in Figures 7 and 8 is comprised of a four-year zero- coupon bond with a market weight of 23% and a five-year zero-coupon bond with a market weight of 77%. This results in a duration for portfolio A of 4.77, which is equivalent to the benchmark. The other zero-coupon bond portfolios also adjust their weights of the two zero-coupon bonds to match the benchmarks duration.
Figure 8: Matching Duration of Zero-Coupon Bond Portfolios to Benchmark month 3year 2 Time 1 month 3 month 6 month 1 year 2 year 3 year 4 year 5 year 7 year 9 year 10 year 15 year 20 year 30 year Duration
Benchmark
A
B
c
D
0.00 0.00 0.01 0.13 0.47 0.53 0.48 0.40 0.46 0.41 0.35 0.45 0.65 0.45 4.77
0.92 3.85
1.67
3.10
1.21
3.56
0.58
4.19
4.77
4.77
4.77
4.77
E 0.07
4.70 4.77
Figure 9 presents the absolute VaR by multiplying the market value weights of the bonds (presented in Figure 7) by the VaR percentages presented in column 2 of Figure 9. The VaR percentages are for a monthly time horizon. The absolute VaR for the benchmark portfolio is computed as $1.99 million. Notice this is very close to the VaR percentage for the four- year note in Figure 9.
Next, the absolute VaR for the five portfolios each consisting of two zero-coupon bonds is computed by multiplying the VaR percentage times the market value of the zero- coupon bonds. We define the new vector of market value positions for each zero-coupon bond portfolio presented in Figure 7 as x and the vector of market value positions of the benchmark as xQ. Then the relative performance to the benchmark is computed as the tracking error (TE) VaR as follows:
Tracking error VaR = a J ( x x q /V ‘V x x q )
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The tracking error or difference between the VaR for the benchmark and zero-bond portfolios is due to nonparallel shifts in the term structure of interest rates. However, the tracking error of $0.45 million for zero-coupon bond portfolio A and the benchmark is much less than the VaR for the benchmark at $1.99. In this example, the smallest tracking error is for portfolio C. Notice that the benchmark portfolio has the largest market weight in the two-year note. Thus, the cash flows are most closely aligned with portfolio C, which contains a two-year zero-coupon bond. This reduces the tracking error to $0.17 million for that portfolio. Also notice that minimizing the absolute VaR in Figure 9 is not the same as minimizing the tracking error. Portfolio E is a barbell portfolio with the highest tracking error to the index, even though it has the lowest absolute VaR.
Tracking error can be used to compute the variance reduction (similar to R-squared in a regression) as follows:
Variance improvement = 1 (tracking error / benchmark VaR)
Variance improvement for portfolio C relative to the benchmark is computed – (0.17 / 1.99)2 = 99.3% 1 – (0.17 / 1.99)2 = 99.3%
Figure 9: Absolute VaR and Tracking Error Relative to Benchmark Portfolio month 3year 2 Time 1 month 3 month 6 month 1 year 2 year 3 year 4 year 5 year 7 year 9 year 10 year 15 year 20 year 30 year
VaR% Benchmark 0.022 0.065 0.163 0.47 0.987 1.484 1.971 2.426 3.192 3.913 4.25 6.234 8.146 11.119 Absolute VaR Tracking Error VaR
0.00 0.00 0.00 0.06 0.23 0.26 0.24 0.19 0.21 0.18 0.15 0.19 0.26 0.17 1.99
A
B
C
D
E 0.02
0.45 1.87
0.83
1.41
0.60
1.55
0.27
1.78
2.32
2.24
2.14
2.05
1.74 1.76
0.45
0.31
0.17
0.21
0.84
0.00
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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11

LO 4.5: Describe how m apping o f risk factors can support stress testing.
If we assume that there is perfect correlation among maturities of the zeros, the portfolio VaR would be equal to the undiversified VaR (i.e., the sum of the VaRs, as illustrated in column 3 of Figure 5). Instead of calculating the undiversified VaR directly, we could reduce each zero-coupon value by its respective VaR and then revalue the portfolio. The difference between the revalued portfolio and the original portfolio value should be equal to the undiversified VaR. Stressing each zero by its VaR is a simpler approach than incorporating correlations; however, this method ceases to be viable if correlations are anything but perfect (i.e., 1).
Using the same two-bond portfolio from the previous example, we can stress test the VaR measurement, assuming all zeros are perfectly correlated, and derive movements in the value of zero-coupon bonds. Figure 6 illustrates the calculations required to stress test the portfolio. The present value factor for a one-year zero-coupon bond discounted at 3.5% is simply 1 / (1.035) = 0.9662. The VaR percentage movement at the 95% confidence level for a one-year zero-coupon bond is provided in column 5 (0.4696). Thus, there is a 95% probability that a one-year zero-coupon bond will fall to 0.9616 [computed as follows: 0.9662 x (1 – 0.4696 / 100) = 0.9616].
The VaR adjusted present values of zero-coupon bonds are presented in column 7 of Figure 6. The last column simply finds the present value of the portfolios cash flows using the VaR% adjusted present value factors. The sum of these values suggests that the change in portfolio value is $2.67 (computed $200.00 $197.33). Notice that the $2.67 is equivalent to the undiversified VaR previously computed in Figure 5.
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Figure 6: Stress Testing a Portfolio
Portfolio
CF
$108.5
$5 $5 $5 $105
Year 1 2 3 4 5
Spot Rate 3.50% 3.90% 4.19% 4.21% 5.10%
PV(CF) $104.83 $4.63 $4.42 $4.24 $81.88 $200.00
VAR % 0.4696 0.9868 1.4841 1.9714 2.4261
P V
Factor 0.9662 0.9263 0.8841 0.8479 0.7798
VaR Adj. P V Factor 0.9616 0.9172 0.8710 0.8312 0.7609
N ew Zero
Value $104.34 $4.59 $4.36 $4.16 $79.89 $197.33
B e n c h m a r k i n g a P o r t f o l i o

LO 4.4: Summarize how to map a fixed income portfolio into positions o f standard instruments.
To illustrate principal, duration, and cash flow mapping, we will use a two position fixed- income portfolio consisting of a one-year bond and a five-year bond. You will notice in the following examples that the primary difference between these mapping techniques is the consideration of the timing and amount of cash flows.
Suppose a portfolio consists of two par value bonds. One bond is a one-year $100 million bond with a coupon rate of 3.5%. The second bond is a five-year $100 million bond with a coupon rate of 5%. In this example, we will differentiate between the timing and cash flows used to map the VaR for this portfolio using principal mapping, duration mapping, and cash flow mapping. The risk percentages (or VaR percentages) for zero-coupon bonds with maturities ranging from one to five years (at the 95% confidence level) are as follows:
Maturity
1 2 3 4 5
VAR % 0.4696 0.9868 1.4841 1.9714 2.4261
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Principal mapping is the simplest of the three techniques as it only considers the timing of the redemption or maturity payments of the bonds. While this simplifies the process, it ignores all coupon payments for the bonds. The weights in this example are both 50% (i.e., $100 million / $200 million). Thus, the weighted average life of this portfolio for the two bonds is three years [0.50(1) + 0.50(5) = 3].
As Figure 3 illustrates, the principal mapping technique assumes that the total portfolio value of $200 million occurs at the average life of the portfolio, which is three years. Note that the VaR percentage at the 95% confidence level is 1.4841 for a three-year zero-coupon bond. We compute the VaR under the principal method by multiplying the VaR percentage times the market value of the average life of the bond, as follows:
Principal mapping VaR = $200 million x 1.4841% = $2,968 million
Figure 3: Fixed-Income Mapping Techniques
Year 1 2
2.768
3 4 5
CFs fo r
5- Year Bond
$5 $5
$5 $5 $105
CFs fo r
1-Year Bond
$103.5
$0
$0 $0 $0
Spot Rates 3.50% 3.90%
4.19% 4.21% 5.10%
M apping Technique
Principal
D uration
$200
$200
$200
$200
PV(CF) $104.83 $4.63
$4.42 $4.24 $81.88 $200.00
In the last three columns of Figure 3, you can see the differences in the amounts and timing of cash flows for all three methods. To calculate the VaR of this fixed-income portfolio using duration mapping, we simply replace the portfolio with a zero-coupon bond that has the same maturity as the duration of the portfolio. Figure 4 demonstrates the calculation of Macaulay duration for this portfolio. The numerator of the duration calculation is the sum of time, t, multiplied by the present value of cash flows, and the denominator is simply the present value of all cash flows. Duration is then computed as $553.69 million / $200 million = 2.768.
Figure 4: Duration Calculation
Year 1 2 3 4 5
C F for
5- Year Bond
CF fo r
1-Year Bond
$5 $5 $5 $5 $105
$103.5
$0 $0 $0 $0
Spot Rate PV(CF) 3.50% $104.83 3.90% $4.63 $4.42 4.19% 4.21% $4.24 $81.88 5.10% $200.00
t X PV(CF) $104.83 $9.26 $13.26 $16.96 $409.38 $553.69
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The next step is to interpolate the VaR for a zero-coupon bond with a maturity of 2.768 years. Recall that the VaR percentages for two-year and three-year zero-coupon bonds were 0.9868 and 1.4841, respectively.
The VaR of a 2.768 year maturity zero-coupon bond is interpolated as follows:
0.9868 + (1.4841 – 0.9868) x (2.768 – 2 ) = 0.9868 + (0.4973 x 0.768) = 1.3687
We now have the information needed to calculate the VaR for this portfolio using the interpolated VaR percentage for a zero-coupon bond with a 2.768 year maturity:
Duration mapping VaR = $200 million x 1.3687% = $2,737 million
In order to calculate the VaR for this fixed-income portfolio using cash flow mapping, we need to map the present value of the cash flows (i.e., face amount discounted at the spot rate for a given maturity) onto the risk factors for zeros of the same maturities and include the inter-maturity correlations. Figure 3 summarizes the required calculations. The second column of Figure 3 provides the present value of cash flows that were computed in Figure 3. The third column of Figure 5 multiplies the present value of cash flows times the zero- coupon VaR percentages.
Figure 5: Cash Flow Mapping
Year 1 2 3 4 5
X
104.83 4.63 4.42 4.24 81.88
Undiversified VaR
x x V 0.4923 0.0457 0.0656 0.0836 1.9864 2.674
1Y 1
0.894 0.887 0.871 0.861
Correlation Matrix (R) 2F 4Y 0.894 0.871 0.964 0.992
3Y 0.887 0.99 1 1 1 0.99 0.964 0.954
0.992 0.987
1
0.996
5T 0.861 0.954 0.987 0.996
1
xAVaR 1.17 0.115 0.170 0.217 5.168 6.840
If the five zero-coupon bonds were all perfectly correlated, then the undiversified VaR could be calculated as follows:
Diversified VaR
Undiversified VaR = ^^|xj x Vj
N
i= l
In this example, the undiversified VaR is computed as the sum of the third column: 2.674.
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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
The correlation matrix provided in the fourth through eighth columns of Figure 5 provides the inter-maturity correlations for the zero-coupon bonds for all five maturities. The diversified VaR can be computed using matrix algebra as follows:
Diversified VaR = ol^Jx ‘^ ^ x = ^ (x x V )rR(x x V)
Where x is the present value of cash flows vector, Vis the vector of VaR for zero-coupon bond returns and R is the correlation matrix. The last column of Figure 5 summarizes the computations for the matrix algebra. The square root of the sum of this column (6.840) is the diversified VaR using cash flow mapping and is calculated as 2.615.
Notice that in order to calculate portfolio diversified VaR using the cash flow mapping method, we need to incorporate the correlations between the zero-coupon bonds. As you can see, cash flow mapping is the most precise method, but it is also the most complex.
Professors Note: The complex calculations required fo r cash flow mapping would be very time consuming to perform using a fin an cial calculator. Therefore, this calculation it is highly unlikely to show up on the exam.
S t r e s s T e s t i n g

LO 4.3: Differentiate am ong the three methods o f m apping portfolios o f fixed income securities.
After we have selected our general risk factors, we must map our portfolio onto these factors. The three methods of mapping for fixed-income securities are (1) principal mapping, (2) duration mapping, and (3) cash flow mapping.
Principal mapping. This method includes only the risk of repayment of principal amounts. For principal mapping, we consider the average maturity of the portfolio. VaR is calculated using the risk level from the zero-coupon bond that equals the average maturity of the portfolio. This method is the simplest of the three approaches.
Duration mapping. With this method, the risk of the bond is mapped to a zero-coupon bond of the same duration. For duration mapping, we calculate VaR by using the risk level of the zero-coupon bond that equals the duration of the portfolio. Note that it may be difficult to calculate the risk level that exactly matches the duration of the portfolio.
Cash flow mapping. With this method, the risk of the bond is decomposed into the risk of each of the bonds cash flows. Cash flow mapping is the most precise method because we map the present value of the cash flows (i.e., face amount discounted at the spot rate for a given maturity) onto the risk factors for zeros of the same maturities and include the inter- maturity correlations.

LO 4.2: Explain how the m apping process captures general and specific risks.
So how many general risk factors (or primitive risk factors) are appropriate for a given portfolio? In some cases, one or two risk factors may be sufficient. O f course, the more risk factors chosen, the more time consuming the modeling of a portfolio becomes. However, more risk factors could lead to a better approximation of the portfolios risk exposure.
In our choice of general risk factors for use in VaR models, we should be aware that the types and number of risk factors we choose will have an effect on the size of residual or specific risks. Specific risks arise from unsystematic risk or asset-specific risks of various positions in the portfolio. The more precisely we define risk, the smaller the specific risk.
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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
For example, a portfolio of bonds may include bonds of different ratings, terms, and currencies. If we use duration as our only risk factor, there will be a significant amount of variance among the bonds that we referred to as specific risk. If we add a risk factor for credit risk, we could expect that the amount of specific risk would be smaller. If we add another risk factor for currencies, we would expect that the specific risk would be even smaller. Thus, the definition of specific risk is a function of general market risk. 1)] / 2) to evaluate the correlation between each risk factor. To simplify the number of As an example, suppose an equity portfolio consists of 5,000 stocks. Each stock has a market risk component and a firm-specific component. If each stock has a corresponding risk factor, we would need roughly 12.5 million covariance terms (i.e., [5,000 x (5,000
1)] / 2) to evaluate the correlation between each risk factor. To simplify the number of parameters required, we need to understand that diversification will reduce firm-specific components and leave only market risk (i.e., systematic risk or beta risk). We can then map the market risk component of each stock onto a stock index (i.e., changes in equity prices) to greatly reduce the number of parameters needed.
Suppose you have a portfolio of N stocks and map each stock to the market index, which is defined as your primitive risk factor. The risk exposure, /3-, is computed by regressing the return of stock i on the market index return using the following equation:
Ri = oq +(3jRy + j
We can ignore the first term (i.e., the intercept) as it does not relate to risk, and we will also assume that the last term, which is related to specific risk, is not correlated with other stocks or the market portfolio. If the weight of each position in the portfolio is defined as then the portfolio return is defined as follows:
r p = ^ wiR; = y ^ w i(3iRM + y > ii
N
i=i
N
i=i
N
i=i
Aggregating all risk exposures, /T, based on the market weights of each position determines the risk exposure as follows: =X}WiPi p P =X}WiPi
N
i=l
We can then decompose the variance, V, of the portfolio return into two components, which consist of general market risk exposures and specific risk exposures, as follows:
V(Rp) = Pp x V(Rm) + wf x<j2j
N
i= l
General market risk: (3p x V (R m)
Specific risk:
N
i= l
wf x cr^
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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
M a p p i n g A p p r o a c h e s f o r F i x e d -In c o m e P o r t f o l i o s

LO 4.1: Explain the principles underlying VaR m apping, and describe the m apping process.
Value at risk (VaR) mapping involves replacing the current values of a portfolio with risk factor exposures. The first step in the process is to measure all current positions within a portfolio. These positions are then mapped to risk factors by means of factor exposures. Mapping involves finding common risk factors among positions in a given portfolio. If we have a portfolio consisting of a large number of positions, it may be difficult and time consuming to manage the risk of each individual position. Instead, we can evaluate the value of these positions by mapping them onto common risk factors (e.g., changes in interest rates or equity prices). By reducing the number of variables under consideration, we greatly simplify the risk management process.
Mapping can assist a risk manager in evaluating positions whose characteristics may change over time, such as fixed-income securities. Mapping can also provide an effective way to manage risk when there is not sufficient historical data for an investment, such as an initial public offering (IPO). In both cases, evaluating historical prices may not be relevant, so the manager must evaluate those risk factors that are likely to impact the portfolios risk profile.
The principles for VaR risk mapping are summarized as follows:
VaR mapping aggregates risk exposure when it is impractical to consider each position
separately. For example, there may be too many computations needed to measure the risk for each individual position.
VaR mapping simplifies risk exposures into primitive risk factors. For example, a
portfolio may have thousands of positions linked to a specific exchange rate that could be summarized with one aggregate risk factor.
VaR risk measurements can differ from pricing methods where prices cannot be
aggregated. The aggregation of a number of positions to one risk factor is acceptable for risk measurement purposes.
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VaR mapping is useful for measuring changes over time, as with bonds or options. For example, as bonds mature, risk exposure can be mapped to spot yields that reflect the current position.
VaR mapping is useful when historical data is not available.
The first step in the VaR mapping process is to identify common risk factors for different investment positions. Figure 1 illustrates how the market values (MVs) of each position or investment are matched to the common risk factors identified by a risk manager.
Figure 1: Mapping Positions to Risk Factors
Figure 2 illustrates the next step, where the risk manager constructs risk factor distributions and inputs all data into the risk model. In this case, the market value of the first position, MVp is allocated to the risk exposures in the first row, x^ , x12, and x^y The other market value positions are linked to the risk exposures in a similar way. Summing the risk factors in each column then creates a vector consisting of three risk exposures.
Figure 2: Mapping Risk Exposures
Investment Market Value
Risk Factor 1 Risk Factor 2 Risk Factor 3
1 2 3 4 5
MVX m v 2 MV, m v 4 m v 5
*n
*21
*31
*41
*51
*12
*22
*32
*42
*52
*13
*23
*33
*43
*53

LO 3.6: Describe the Basel rules for backtesting.
In the backtesting process, we attempt to strike a balance between the probability of a Type I error (rejecting a model that is correct) and a Type II error (failing to reject a model that is incorrect). Thus, the Basel Committee is primarily concerned with identifying whether exceptions are the result of bad luck (Type I error) or a faulty model (Type II error). The Basel Committee requires that market VaR be calculated at the 99% confidence level and backtested over the past year. At the 99% confidence level, we would expect to have 2.3 exceptions (230 x 0.01) each year, given approximately 250 trading days.
Regulators do not have access to every parameter input of the model and must construct rules that are applicable across institutions. To mitigate the risk that banks willingly commit a Type II error and use a faulty model, the Basel Committee designed the Basel penalty zones presented in Figure 5. The committee established a scale of the number of exceptions and corresponding increases in the capital multiplier, k. Thus, banks are penalized for exceeding four exceptions per year. The multiplier is normally three but can be increased to as much as four, based on the accuracy of the banks VaR model. Increasing k significantly increases the amount of capital a bank must hold and lowers the banks performance measures, like return on equity.
Notice in Figure 5 that there are three zones. The green zone is an acceptable number of exceptions. The yellow zone indicates a penalty zone where the capital multiplier is increased by 0.40 to 1.00. The red zone, where 10 or more exceptions are observed, indicates the strictest penalty with an increase of 1 to the capital multiplier.
Figure 5: Basel Penalty Zones
N um ber o f Exceptions
M ultiplier (k)
Zone
Green
Yellow
0 to 4
5
6
7
8
9
3.00
3.40
3.50
3.65
3.75
3.85
4.00
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Red
10 or more
Topic 3 Cross Reference to GARP Assigned Reading – Jorion, Chapter 6
As shown in Figure 3, the yellow zone is quite broad (five to nine exceptions). The penalty (raising the multiplier from three to four) is automatically required for banks with 10 or more exceptions. However, the penalty for banks with five to nine exceptions is subject to supervisors discretions, based on what type of model error caused the exceptions. The Committee established four categories of causes for exceptions and guidance for supervisors for each category:

The basic integrity o f the model is lacking. Exceptions occurred because of incorrect data or errors in the model programming. The penalty should apply.
Model accuracy needs improvement. The exceptions occurred because the model does not

accurately describe risks. The penalty should apply. Intraday trading activity. The exceptions occurred due to trading activity (VaR is based on static portfolios). The penalty should be considered. Bad luck. The exceptions occurred because market conditions (volatility and correlations among financial instruments) significantly varied from an accepted norm. These exceptions should be expected to occur at least some of the time. No penalty guidance is provided.
Although the yellow zone is broad, an accurate model could produce five or more exceptions 10.8% of the time at the 99% confidence level. So even if a bank has an accurate model, it is subject to punishment 10.8% of the time (using the required 99% confidence level). However, regulators are more concerned about Type II errors, and the increased capital multiplier penalty is enforced using the 97% confidence level. At this level, inaccurate models would not be rejected 12.8% of the time (e.g., those with VaR calculated at the 97% confidence level rather than the required 99% confidence level). While this seems to be only a slight difference, using a 99% confidence level would result in a 1.24 times greater level of required capital, providing a powerful economic incentive for banks to use a lower confidence level. Exemptions may be excluded if they are the result of bad luck that follows from an unexpected change in interest rates, exchange rates, political event, or natural disaster. Bank regulators keep the description of exceptions intentionally vague to allow adjustments during major market disruptions.
Industry analysts have suggested lowering the required VaR confidence level to 93% and compensating by using a greater multiplier. This would result in a greater number of expected exceptions, and variances would be more statistically significant. The one- year exception rate at the 95% level would be 13, and with more than 17 exceptions, the probability of a Type I error would be 12.5% (close to the 10.8% previously noted), but the probability of a Type II error at this level would fall to 7.4% (compared to 12.8% at a 97.5% confidence level). Thus, inaccurate models would fail to be rejected less frequently.
Another way to make variations in the number of exceptions more significant would be to use a longer backtesting period. This approach may not be as practical because the nature of markets, portfolios, and risk changes over time.
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K e y C o n c e p t s
LO 3.1
Backtesting is an important part of VaR model validation. It involves comparing the number of instances where the actual loss exceeds the VaR level (called exceptions) with the number predicted by the model at the chosen level of confidence. The Basel Committee requires banks to backtest internal VaR models and penalizes banks with excessive exceptions in the form of higher capital requirements.
LO 3.2
VaR models are based on static portfolios, while actual portfolio compositions are dynamic and incorporate fees, commissions, and other profit and loss factors. This effect is minimized by backtesting with a relatively short time horizon such as daily holding periods. The backtesting period constitutes a limited sample, and a challenge for risk managers is to find an acceptable level of exceptions.
LO 3.3
The failure rate of a model backtest is the number of exceptions divided by the number of observations: N / T. The Basel Committee requires backtesting at the 99% confidence level over the past year (230 business days). At this level, we would expect 230 x 0.01, or 2.5 exceptions.
LO 3.4
In using backtesting to accept or reject a VaR model, we must balance the probabilities of two types of errors: a Type I error is rejecting an accurate model, and a Type II error is failing to reject an inaccurate model. A log-likelihood ratio is used as a test for the validity of VaR models.
LO 3.5
Unconditional coverage testing does not evaluate the timing of exceptions, while conditional coverage tests review the number and timing of exceptions for independence. Current market or trading portfolio conditions may require changes to the VaR model.
LO 3.6
The Basel Committee penalizes financial institutions when the number of exceptions exceeds four. The corresponding penalties incrementally increase the capital requirement multiplier for the financial institution from three to four as the number of exceptions increase.
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C o n c e p t C h e c k e r s
1.
2.
3.
4.
5.
In backtesting a value at risk (VaR) model that was constructed using a 97.3% confidence level over a 232-day period, how many exceptions are forecasted? A. 2.5. B. 3.7. C. 6.3. D. 12.6.
Unconditional testing does not reflect the: A. size of the portfolio. B. number of exceptions. C. confidence level chosen. D. timing of the exceptions.
Which of the following statements regarding verification of a VaR model by examining its failure rates is false? A. The frequency of exceptions should correspond to the confidence level used for
the model.
B. According to Kupiec (1995), we should reject the hypothesis that the model is
correct if the log-likelihood ratio (LR) > 3.84.
C. Backtesting VaR models with a higher probability of exceptions is difficult
because the number of exceptions is not high enough to provide meaningful information.
D. The range for the number of exceptions must strike a balance between the
chances of rejecting an accurate model (a Type I error) and the chances of failing to reject an inaccurate model (a Type II error).
The Basel Committee has established four categories of causes for exceptions. Which of the following does not apply to one of those categories? A. The sample is small. B. Intraday trading activity. C. Model accuracy needs improvement. D. The basic integrity of the model is lacking.
A risk manager is backtesting a sample at the 95% confidence level to see if a VaR model needs to be recalibrated. He is using 252 daily returns for the sample and discovered 17 exceptions. What is the 2;-score for this sample when conducting VaR model verification? A. 0.62. B. 1.27. C. 1.64. D 2.86.
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C o n c e p t C h e c k e r An s w e r s
1. C
(1 – 0.975) X 252 = 6.3
2. D Unconditional testing does not capture the timing of exceptions.
3. C Backtesting VaR models with a lower probability of exceptions is difficult because the number
of exceptions is not high enough to provide meaningful information.
4. A Causes include the following: bad luck, intraday trading activity, model accuracy needs
improvement, and the basic integrity of the model is lacking.
5. B The z-score is calculated using x = 17, p = 0.05, c = 0.95, and N = 252, as follows:
1 7 -0 .0 5 (2 5 2 ) _ 1 7 -1 2 .6 _
4.4
_ ^ ^
~ ^0.05(0.95)252 ~ V ll.9 7 ~~ 3.4598 ~~
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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:
VaR M a p p i n g
E x a m F o c u s
Topic 4
This topic introduces the concept of mapping a portfolio and shows how the risk of a complex, multi-asset portfolio can be separated into risk factors. For the exam, be able to explain the mapping process for several types of portfolios, including fixed-income portfolios and portfolios consisting of linear and nonlinear derivatives. Also, be able to describe how the mapping process simplifies risk management for large portfolios. Finally, be able to distinguish between general and specific risk factors, and understand the various inputs required for calculating undiversified and diversified value at risk (VaR).
T h e M a p p i n g P r o c e s s