Reading 25 - Risk Management: VaR

VAR in terms of minimum loss

% chance that portfolio will lose that amount/% in the given time period

Earnings at Risk (EAR)

analogous to CFAR only from an accounting earnings standpoint CFAR and EAR are often used to add validity to VAR calculations

3. Maximum loss limit

is the maximum allowable loss each unit or PM is given individual max loss limits pros: - with this method total max loss will never exceed firm's capital cons: - possibility of all units exceeding their max loss limits simultaneously

Other tools and actions to used in conjunction with VaR

- VAR projections should be continually back-tested to compare actual results across multiple time periods - Incremental VAR (IVAR) - Cash Flow at Risk (CFAR) - Earnings at Risk (EAR) - Tail Value at Risk (TVAR) - Credit VAR - Stress Testing

Issues to consider with VaR

- VaR time period should relate to the nature of situation > traditional stock/bond portfolio will focus on longer monthly/quarterly VaR while highly leveraged derivatives portfolio will focus on shorter daily VaR - probability % selected will affect VaR (typically 1 or 5% - 1% is more conservative) > prob of 1% will produce greater VaR risk estimates vs a 5% probability - examine the left tail to see "how much worse can the loss be?"

1. Nominal position limits (notional position limits or monetary position limits)

- defines the amount to be allocated to each unit or PM - management allocates capital where they feel it will produce the highest risk adjusted returns pro: - easy to understand Cons: - individual may artificially exceed limit imposed by using other assets that can replicate a given position (not a good capital allocation method from a risk control standpoint)

Advantages of VaR vs other risk measures

- has become the industry standard for risk measurement and is required by many regulators - aggregates all risk into one single, easy to understand number - can be used in capital allocation

Limitations of VaR

- some methods (Monte Carlo) are difficult and expensive - different computation methods can generate different estimates of VaR - can generate false sense of security (output only as good as inputs) - one sided and focuses on the left side of the tail and ignores any upside

2. VAR-based position limits - alternative or supplement to nominal position limit

- to achieve desired overall risk exposure (measured by VAR), capital is allocated according to VAR pros: - overall VAR is the sum of individual unit VARs con: - does not measure correlation of different positions (leading to overestimating firm VAR and misallocating capital)

Stress Testing

- used as a complement to VAR and may reveal outcomes not reflected in a typical VAR - typically an extreme scenario

VAR will be larger in 1% or 5% probability?

1%. A 1% probability is equivalent to 99 confidence level and this requires 2.33 standard deviation movement in the direction of lower returns.

Measuring and allocating capital can be done with

1. Nominal position limits (notional position limits or monetary position limits) 2. VAR-based position limits 3. Maximum loss limit 4. Internal capital requirements 5. regulatory capital requirements 5. Behavioral conflicts

Stressing models can be done as

1. factor push analysis - simple stress test where the analyst pushes factors to the most disadvantageous combination of possible circumstances and measures the resulting impact on the portfolio 2. maximum loss optimization - uses more sophisticated mathematical and computer modeling to find this worst combination of factors 3. worst-case scenario - the worst case that the analyst thinks is likely to occur

7 Risk Factors to be modeled in scenario analysis (stylized) recommended by the Derivatives Policy Group -

1. parallel yield curve shifting by +- 100 bps 2. changes in steepness of yield curves (curve twisting) by +- 25 bps 3.each of the 4 combos of #1 and #2 4. changes in yield volatilities by +- 20% from current levels 5. changes in the value of equity indices by +- 10% 6. changes in the value of key currencies (relative to USD) moving by +- 6% and other currencies by +- 20% 7. changes in swap spreads by +- 20 bps

Methods for computing VaR

Analytical (variance-covariance) Historical Monte Carlo

5. Behavioral conflicts

ERM system must recognize the potential for incentive conflicts between management (the one that allocates the risk) and PMs (those who make investment decisions

Historical VaR - accumulate a number of past daily returns, rank the returns from highest to lowest, and identify the lowest 5% or 1% of returns > the HIGHEST of the lowest 5% or 1% is the 1 day, 5% or 1% VaR

Pros: - easy to calculate and understand - does not assume returns distributions - can be applied to different time periods Cons: - assumes that the pattern od historical returns will repeat in the future

Can total portfolio VAR be calculated from aggregating all the individual VARs?

No simply aggregating individual position VAR will not be enough to calculate portfolio VAR. Correlations of the positions must be taken into account.

Analytical (variance-covariance) VaR -based on normal distribution and one tailed confidence intervals

Pros: - easy to calculate and understood - allows modeling the correlations of risks - can be applied to shorter or longer time periods Cons: - assumes normal distribution of returns - many assets experience leptokurtosis (fat tails) and having that tends to underestimate the loss and its associated probability of extreme returns - difficult to estimate SD in very large portfolios

$100M portfolio has VaR of 1.37% at 5% probability over one week

Portfolio could lose at least 1.37% of its value or $1.37M over one week. There's a 5% chance the loss will be greater than 1.37% and 95% that it will be less then 1.37%

Monte Carlo VaR - uses computer software to generate hundreds or thousands of possible outcomes from the distributions of inputs specified by the user - user can specify the type of distributions

Pros: - can incorporate any assumptions regarding return patterns, correlations, and other factors the analyst believes are relevant Cons: - output is only as good as the input

Types of scenario analysis

Stylized scenarios - analyst changes one or more risk factors to measure the effect on the portfolio - instead of manager selecting risk factors, some stylized scenarios are more like industry standards Actual extreme events - analyst measures impact of major past events on portfolio value Hypothetical events - extreme events that might occur but have not previously occurred

Tail Value at Risk (TVAR)

VAR + average outcomes in the tail gives additional insight if VAR is exceeded VAR is $1M and TVAR $2.7M, 5% losses > $1M and average lost over $1M is another $1.7M

Value at risk = for time period < 1 yr divide annual return by 12 (monthly), 52 (weekly) divide annual SD with square root of 12 (monthly), 52 (weekly) For a short one day VaR assume return = 0

[expected portfolio return - z value*sd] * port value z value that corresponds to the desired level of significance 5% level = 1.65 1% level = 2.33

Incremental VAR (IVAR)

effect of an individual item on overall risk of the portfolio calculated by measuring portfolio VAR before and after an additional asset class or asset or another change in the portfolio

5. regulatory capital requirements

firms are required to incorporate this if they are covered by the regulation

increase in correlation will result in higher or lower VAR?

high correlation will increase SD which would result in higher calculated VAR (greater losses)

When VaR is a gain (VaR is typically a loss)

it may mean that the portfolio is more conservative

higher expected returns will result in higher or lower VAR?

lower calculated VAR (smaller losses)

Cash Flow at Risk (CFAR)

measures the risk of the company's cash flows interpreted like VAR but substitutes value with cash flow minimum cash flow loss at a given probability

VAR is larger over a month or day??

month. VAR is larger for longer time periods because the possibility of losses is larger

Credit VAR (credit risk or default VAR)

projects risk due to credit events VAR was initially developed as a measure of market risk but have been extended to handle credit risk (probability that a counterparty will not pay what it owes)

4. Internal capital requirements

specify the level of capital requirements management feels is appropriate for the firm - historically specified using ratio to capital to assets firms subject to regulatory capital requirements (banks and securities firms) that might overrule internal capital requirements if higher

VaR - single aggregate risk measure

states at some probability (often 1-5%) the minimum expected loss during a specified time period loss that would be exceeded with a given probability over a specified time period can be a % or nominal amount

VAR and stress testing in setting capital requirements

unusual events are not typically captured by VAR so stress testing (scenario analysis) is used to estimate their effect on the value of a firm or portfolio stress testing helps determine whether the firm has sufficient capital to withstand unusual events to determine the allocation of capital across business units or portfolio managers, sr. management must determine the allocation in a way that maximizes potential returns without placing the viability of the firm in jeopardy - measure expected return and potential losses of individual units or PMs and ensure that the amount of capital at risk never exceeds total firm capital

Scenario analysis

used to measure the effect on the portfolio of simultaneous movements in one or several factors user defines events and compares the value of portfolios before and after the events Cons: -inability to accurately measure by products of major factor movements or include effects of simultaneous movements in risk factors - model output will only be as good as inputs