# LO 4.6: Explain how VaR can be used as a performance benchmark.

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 2018 Kaplan, Inc. Page 45 Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11 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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2018 Kaplan, Inc.
Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
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
2018 Kaplan, Inc.
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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11