As a fictional example, a bank's interest rate position has key rate sensitivities of +1 million yen at 1 year, +1 million yen at 5 years and −2 million yen at 10 years. A key rate sensitivity is the change in market value when only the rate for that maturity rises by 1bp. One day, the 1-year rate is unchanged, the 5-year rate rises 4bp and the 10-year rate rises 8bp. Using the key rate sensitivities as an approximation, what is the change in the position's market value on that day?
Quiz: Measuring Risk
Last reviewed: September 26, 2026
This is a draft, shown for review before publication.
6 questions to check your understanding of the chapter “Measuring Risk”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.
Question 1
Answer and explanation
Answer: C
The change in market value is −12 million yen. For each maturity, you multiply the sensitivity by the move in that maturity's rate (in bp) and add up the results. The 1-year point gives +1 million × 0 = 0 yen. The 5-year point gives +1 million × 4 = +4 million yen. The 10-year point gives −2 million × 8 = −16 million yen. The total is 0 + 4 − 16 = −12 million yen, a loss of 12 million yen.
0 yen is the value you get by looking only at DV01. The sum of the key rate sensitivities is roughly equal to DV01, and this position's DV01 is 1 + 1 − 2 = 0, so DV01 suggests there is no profit or loss. DV01 does not show differences in how the rate for each maturity moves. On this day longer-maturity rates rose more, the slope of the yield curve changed, and so there was a loss. The figure of −16 million yen counts only the 10-year point and leaves out the +4 million yen from the 5-year point. The figure of +12 million yen gets the direction of the profit or loss the wrong way round.
If rates moved the other way on a given day, with the 5-year rate falling 4bp and the 10-year rate falling 8bp, the same calculation gives a change in market value of +12 million yen, a gain of 12 million yen.
Learn more: Sensitivities: DV01, Vega and Key Rate Sensitivities
Question 2
A bank sets limits on its interest rate positions not only on the total DV01 across all maturities but also on each key rate sensitivity. Which position do the limits on key rate sensitivities make it possible to stop?
Answer and explanation
Answer: B
The limits make it possible to stop a position whose total DV01 is small but whose profit and loss is concentrated in the rate for a single maturity. Positions at different maturities offset each other when all rates move by the same amount, so the total DV01 looks small. With limits on each key rate sensitivity, you can stop a position that is concentrated in one maturity even when the total is small. Setting limits on vega split into a grid of maturities and strikes follows the same idea.
Key rate sensitivities also do not show how strongly risk factors are correlated. Adding up sensitivities to different risk factors (drivers of a position's value) does not take into account the correlation between the rates for different maturities (the correlation between risk factors). Correlation enters through methods such as the variance-covariance method for VaR (value at risk), which combines the sensitivities with the size of the risk factor moves and their correlations.
Key rate sensitivities also do not show profit and loss growing with large moves. A sensitivity is a first-order approximation for small moves and cannot capture the curvature in the price (convexity). The profit and loss from large moves is found by revaluing positions in stress testing.
Learn more: Sensitivities: DV01, Vega and Key Rate Sensitivities
Question 3
As a fictional example, a position has a VaR of 30 million yen with a 1-day holding period and a 99% confidence level. Which statement correctly describes what this VaR means?
Answer and explanation
Answer: D
This VaR means that the probability of tomorrow's loss exceeding 30 million yen is estimated at 1%. VaR is found by setting a holding period (the period over which the loss is measured) and a confidence level (the probability that the loss stays within VaR). On the other 99% of days, the loss stays within 30 million yen or there is a profit. You would expect the loss to exceed 30 million yen on about 1 in every 100 business days.
VaR is not an upper limit on losses. VaR does not show the size of the losses larger than 30 million yen that occur with a 1% probability. The average of the large losses that occur with a 1% probability, that is, the average loss when the loss exceeds VaR, is shown by expected shortfall, which is a different measure from VaR. Reading VaR as a 1% probability of any loss confuses the days on which a loss occurs with the days on which the loss exceeds VaR. Even with a VaR of 30 million yen, losses smaller than 30 million yen also occur on the other 99% of days.
Learn more: Value at Risk (VaR)
Question 4
As a fictional example, you hold 300 million yen of one asset (daily volatility 1.0%) and 500 million yen of a second asset (daily volatility 0.8%), and the correlation between the two assets is 0. Assume that profit and loss follows a normal distribution with a mean of 0, and that the factor for a 99% confidence level is 2.326. What is the VaR with a 1-day holding period and a 99% confidence level, using the variance-covariance method?
Answer and explanation
Answer: A
The VaR is about 11.63 million yen. Working in millions of yen, the holdings are 300 and 500. The standard deviation of profit and loss on the first asset is 300 × 1.0% = 3 million yen. For the second asset, it is 500 × 0.8% = 4 million yen. The variance of the total profit and loss is 3 × 3 + 4 × 4 + 2 × 0 × 3 × 4 = 25. Its square root, 5 million yen, is the standard deviation of the total profit and loss. VaR is this multiplied by the factor: 2.326 × 5 million = 11.63 million yen.
About 16.28 million yen is the sum of the VaRs for each asset. The first asset's VaR is 2.326 × 3 million, or about 6.98 million yen, and the second asset's is 2.326 × 4 million, or about 9.30 million yen, which add up to about 16.28 million yen. This is the same as the total VaR when the correlation is 1. When the correlation is less than 1, losses on the two assets do not necessarily occur on the same day, so the total VaR is smaller than the sum of the VaRs for each asset. The figure of 5 million yen is the standard deviation of profit and loss left as it is, without multiplying by the confidence-level factor.
Because profit and loss is treated as normally distributed with a mean of 0, there is also the same 1% probability of a profit larger than 11.63 million yen.
Learn more: Value at Risk (VaR)
Question 5
As a fictional example, the 1-day profit and loss of a position is calculated under 200 equally likely scenarios. The 6 largest losses are 50, 25, 20, 18, 17 and 15 million yen. The VaR at a 97.5% confidence level is the fifth-largest loss, 17 million yen. What is the expected shortfall at the same confidence level?
Answer and explanation
Answer: C
The expected shortfall is 26 million yen. Since 2.5% of 200 scenarios is 5, you average the 5 largest losses, including VaR. The sum of these 5 is 50 + 25 + 20 + 18 + 17 = 130 million yen. Dividing this by 5 gives 26 million yen.
17 million yen is VaR itself. VaR does not reflect that there is a scenario with a loss of 50 million yen, but expected shortfall brings this tail loss into the measure as an average. About 24.17 million yen is the average of 6 losses, including the sixth-largest loss of 15 million yen (145 ÷ 6). You average only the 5 losses set by the confidence level. 28.25 million yen is the average of the top 4, leaving out the VaR of 17 million yen (113 ÷ 4). Although expected shortfall is described as "the average loss when the loss exceeds VaR," when you calculate it from a sample of profit and loss you average the 5 scenarios that make up the largest 2.5% of losses. The VaR of 17 million yen is one of those 5 scenarios, so it is included in the average.
Learn more: Expected Shortfall
Question 6
Bank market risk rules include the internal models approach, under which a bank calculates its amount of risk with its own models. Under the Fundamental Review of the Trading Book (FRTB), which measure did the internal models approach adopt as its amount of risk?
Answer and explanation
Answer: B
The FRTB adopted expected shortfall at a 97.5% confidence level, calculated with data from a stress period. Because VaR does not capture tail losses, the Basel Committee on Banking Supervision adopted expected shortfall for the internal models approach under the FRTB. The earlier dual framework of VaR and stressed VaR was merged into a single expected shortfall that uses stress-period data. Japan's capital adequacy notice for banks also requires banks that use the internal models approach to calculate expected shortfall daily at a one-tailed 97.5% confidence level.
99% VaR with a holding period of at least 10 business days is the measure first used under the amendment published in January 1996, which brought market risk into capital requirements. The combination of 99% VaR and stressed VaR is the Basel 2.5 framework published in July 2009. Stressed VaR is VaR calculated with data from a continuous one-year stress period, at a one-tailed 99% confidence level. Even for expected shortfall, the FRTB's confidence level is 97.5%, not 99%. If profit and loss is normally distributed, expected shortfall at 97.5% is about the same size as VaR at 99%.
Learn more: Expected Shortfall