After completing this reading you should be able to:
 Explain and give examples of linear and nonlinear derivatives.
 Describe and calculate VaR for linear derivatives.
 Describe and explain the historical simulation approach for computing VaR and ES.
 Describe the deltanormal approach for calculating VaR for nonlinear derivatives.
 Describe the limitations of the deltanormal method.
 Explain the full revaluation method for computing VaR.
 Compare deltanormal and full revaluation approaches for computing VaR.
 Explain structured Monte Carlo, stress testing, and scenario analysis methods for computing VaR, and identify strengths and weaknesses of each approach.
 Describe the implications of correlation breakdown for scenario analysis.
 Describe worstcase scenario (WCS) analysis and compare WCS to VaR.
Linear vs. Nonlinear Derivatives
A linear derivative is one whose value is directly related to the market price of the underlying variable. What does that mean?
If the underlying makes a move, the value of the derivative moves with a nearly identical margin. In fact, there is a 1:1 relationship between the derivative and the underlying – explaining why linear derivatives are said to be “deltaone” products. However, the delta itself need not always be equal to 1. Examples of linear derivatives include futures and forwards.
A nonlinear derivative is one whose value/payoff changes with time and space.
Space, in this case, refers to the location of the strike/exercise price with respect to the spot/current price. The payoff varies with the value of the underlying but also exhibits some nonlinear relationship with other variables, including interest rates, dividends, or even volatility. Nonlinear derivatives are generally referred to as options.
For nonlinear derivatives, delta is not constant. Rather, it keeps on changing with the change in the underlying asset. Examples include the Vanilla European option, Vanilla American option, Bermudan option, etc.
VaR for Linear Derivatives
In general terms, the VaR of a linear derivative can be expressed as:
$$ { VaR }_{ linear\quad derivative }=\Delta \times { VaR }_{ underlying\quad factor } $$
\(\Delta\) represents the sensitivity of the derivative’s price to the price of the underlying asset. It’s usually expressed as a percentage.
Example 1
Suppose the permitted lot size of S&P 500 futures contracts is 300 and multiples thereof. What’s the VaR of an S&P 500 futures contract?
$$ { VaR }_{ S\& P\quad 500\quad futures\quad c. }=300\times { VaR }_{ S\& P\quad 500\quad index } $$
Example 2
A 1point increase in the S&P CNX Nifty index increases the value of a Nifty futures contract by $100. What’s the VaR of the Nifty futures contract?
$$ { VaR }_{ Nifty\quad futures\quad c. }=$100\times { VaR }_{ S\& P\quad CNX\quad Nifty\quad index } $$
Deltanormal VaR
The Deltanormal VaR calculation approach involves the delta approximation for nonlinear derivatives. The linear approximation is often used in conjunction with a normality assumption for the distribution.
In fact, the Deltanormal VaR is calculated via the formula above. i.e.,
$$ VaR = \Delta \times { VaR }_{ underlying \quad factor } $$
$$ \Delta =\frac { percentage\quad change\quad in\quad call\quad value }{ percentage\quad change\quad in\quad stock\quad price } =\frac { \Delta C\% }{ \Delta S\% } $$
The method has several disadvantages, chief among them being that:
 It’s computationally easy but quite inaccurate compared to other VaR measurement methods. Put more precisely, it may underestimate the occurrence of extreme losses because of its reliance on the normal distribution.

This method is accurate for small moves of the underlying, but quite inaccurate for large moves. As we can see from the following graph, the slope of the green line is completely different from the slope of the blue line:
For large changes in a nonlinear derivative, we must therefore use the delta + gamma approximation, or full revaluation.
Full Revaluation VaR
Under the full revaluation approach, the VaR of a portfolio is established by fully repricing the portfolio under a set of scenarios over a period of time. It’s hugely popular because it generates reliable VaR estimates. It’s the preferred method for options, especially in the presence of large movements of risk factors.
Full revaluation has several advantages over the deltanormal approach.
 It accounts for nonlinear relationships
 It accounts for extreme observations
However, computations can be particularly burdensome, for instance when repricing complex MBSs, swaptions, or exotic options.
The Structured Monte Carlo Approach to VaR
The structured Monte Carlo Approach relies on thousands of valuation simulations (possible outcomes) for the underlying portfolio or asset, assuming that returns are normally distributed. The simulated outcomes are then used to come up with the VaR estimate for the portfolio.
The approach assumes that returns are normally distributed with a mean \(\mu \) and variance \({ \sigma }^{ 2 }\).
Strength:
 By assuming an underlying distribution, the SMC approach successfully addresses multiple factors. For example, as long as mean and variance have been specified, one can generate, say, 1000 simulations and then estimate the probability of a particular event occurring.
 As the number of simulations increases, the deviation of forecasts from the true mean decreases.
Weakness:
 It’s not 100% accurate, and sometimes increasing the number of simulations does not result in increased accuracy/reliability.
Stress Testing
In finance, contagion is the spread of an economic shock in an economy or region, so that price movements in one market are gradually but increasingly noticed in other markets. During a contagion, both volatility and correlations increase, rendering diversification less effective as a risk mitigation strategy. Stress testing is an attempt to model the contagion effect that would occur in the event of a major crisis.
Some of the historical events that have been used to stress test by various firms include the Mexican crisis of 1994, the Gulf war of 1990, and the near collapse of LTCM in 1998. The underlying question among analysts while stress testing is quite simple: If a similar event occurred right now, how would it affect the firm’s current position?
The other approach to stress testing may not necessarily turn to historical events. It can also be done using predetermined stress scenarios. For example, the firm could seek to find out the effect on its current positions if interest rates short up by, say, 300bps.
Strength:
 Assumptions regarding the underlying distribution are not needed.
Weakness:
 Stress testing based on history is obviously limited to a few past events, and therefore there’s a cap on the usefulness of such a strategy. In fact, a constant problem of historical events is that obtaining accurate data on the exact happenings can be a tough ask. Furthermore, as is the case with all past events, there can be no guarantees that what happened back then will happen in the future.
The Historical Simulation Approach
Unlike stress testing, the historical simulation approach does not limit itself to specific events in the past (major crises). Instead, all past data is used to single out specific instances when extreme losses were recorded in a particular asset class. For example, a firm could seek to identify the ten worst weeks for the bond market and then attempt to evaluate how current bond holdings would be affected if events from those past weeks were to happen again.
The overriding goal of the historical simulation is to identify extreme changes in valuation, movements of the underlying factors notwithstanding.
Strengths:
 It’s relatively simple to implement. All we need to do is keep a record of past price changes
 Distributions need not be normal
 Securities under evaluation can be nonlinear.
Weakness:
 It’s limited to actual historical data
Scenario Analysis Methods
Scenario analysis is basically stress testing without reliance on past data. It’s an attempt to establish how a predetermined set of scenarios would affect current positions. For example, how would an increase in interest rates of 3% affect current floating rate bonds? How would a substantial decrease in tax rates affect equity holders?
Strength:
 It’s not limited to historical events
Weakness:
 The total risk of most asset classes is usually a combination of different risk sources, some of which are actually correlated such that an increase in one leads to a decrease/increase in the other. As such, narrowing of possible events to a specific source of risk, such as a rise in interest rates, normally results in loss estimates that are rough estimates at best. To more accurately model classspecific risk, you have to look at total risk, not specific “causal” factors.
Worstcase Scenario Analysis
Worstcase scenario analysis focuses on extreme losses at the tail end of the distribution. First, firms assume that an unfavorable event is certain to occur. They then attempt to establish the worst possible outcomes that could come out of it.
WCS analysis dissects the tail further so as to establish the range of worst case losses that could be incurred. For example, within the lowest 5% of returns, we can construct a “secondary” distribution that specifies the 1% WCS return.
WCS analysis complements the VaR, and here’s how. Recall that the VaR specifies the minimum loss for a given percentage, but it stops short of establishing the severity of losses in the tail. WCS analysis goes a step further to more precisely describe the distribution of extreme losses.
Questions
Question 1
Which of the following statements about stress testing are correct?

 A weakness of stress testing is that it’s highly subjective
 Stress testing can complement VaR estimation by helping managers
establish crucial vulnerabilities in a portfolio
 The larger the number of test scenarios, the better the understanding of the risk exposure of a portfolio
 Stress testing allows users to include scenarios that did not occur in the lookback horizon of the VAR data but are nonetheless possible.
 All of the above
 I, II, and IV
 I, II, and III
 Only II and IV
The correct answer is B.
In stress testing, the fewer the scenarios under consideration, the easier it is to understand and interpret results. Too many scenarios make it difficult to interpret the risk exposure.
Question 2
Robert Myer, FRM, manages \($200\quad million\) of a distressed bond portfolio. He wishes to conduct stress tests to establish how the portfolio could be affected by various possible outcomes. The portfolio has an annualized return of \(15\%\), with an annualized return volatility of \(30\%\). Over the past three years, there were several days when the daily value change of the portfolio was more than three standard deviations.
Suppose the portfolio suffered a foursigma daily event. Estimate the change in value for this portfolio:
 $12.6 million
 $1.5 million
 $1.89 million
 $10 million
The correct answer is A.
First, you should convert the annual volatility into a daily volatility.
$$ { \sigma }_{ daily }={ \sigma }_{ annual }\times \frac { 1 }{ \sqrt { 252 } } $$
$$ =0.25\times \frac { 1 }{ \sqrt { 252 } } =0.01575 $$
$$ Daily\quad change=investment\quad value \times daily\quad volatility \times number\quad of\quad sigmas $$
$$ =$200\quad million\times 0.01575\times 4=$12.6\quad million $$
The annualized return is irrelevant for these calculations.
Exam tip: Sigma = volatility = standard deviation of returns
Question 3
A risk manager wishes to calculate the VaR for a Nikkei futures contract using the historical simulation approach. The current price of the contract is \(955\) and the multiplier is \(250\). For the last \(301\quad days\), the following return data have been recorded:
Returns: 7.8%, 7.0%, 6.2%, 5.2%, 4.6%, 3.2%, 2.0%,……….3.8%, 4.2%, 4.8%, 5.1%, 6.3%, 6.8%, 7.0%
What is the VaR of the position at 99% using the historical simulation methodology?
 $12,415
 $16,713
 $18,623
 $14,803
The correct answer is D.
The 99% return among 301 observations would be the third worst observation among the returns \(\left( 0.99\times 301=297 \right) \)
Among the returns given above, the third worst return is \(6.2\%\).
A is incorrect. This answer incorrectly uses the fourth worst observation as the 99% return among 301 observations.
B is incorrect. This answer incorrectly uses the second worst observation as the 99% return among 301 observations.
C is incorrect. This answer incorrectly uses the worst observation as the 99% return among 301 observations.
Question 4
Bank \(X\) and Bank \(Y\) are two competing investment banks that are calculating the 1day 99% VaR for an atthemoney call on a nondividendpaying stock with the following information:
 Current stock price: USD 100
 Estimated annual stock return volatility: 20%
 Current BlackScholesMerton option value: USD 4.80
 Option delta: 0.7
To compute VaR, Bank \(X\) uses the linear approximation method, while Bank \(Y\) uses a Monte Carlo simulation method for full revaluation. Which bank will estimate a higher value for the 1day 99% VaR?
 Bank \(X\)
 Bank \(Y\).
 Both will have the same VaR estimate.
 Insufficient information to determine.
The option’s return function is convex with respect to the value of the underlying; therefore the linear approximation method will always underestimate the true value of the option for any potential change in price. Therefore the VaR will always be higher under the linear approximation method than a full revaluation conducted by Monte Carlo simulation analysis. The difference is the bias resulting from the linear approximation, and this bias increases in size with the change in the option price and with the holding period.