LO 4.7: Describe the method o f m apping forwards, forward rate agreements,

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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Topic 4 C ross Reference to G A R P A ssigned R eading – Jorion , C h apter 11
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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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
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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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
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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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
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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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
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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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
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

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