1 Reinsurance Solutions Knowledge. Experience. Performance. THE POWER OF INSIGHT. ® Value at Risk...

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Reinsurance Solutions

Knowledge. Experience. Performance.

THE POWER OF INSIGHT.®

Value at RiskMay 18, 2006

James E. Backus, FSA

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Value-at-Risk

Brief history Basic VAR Variations and extensions of VAR Practical considerations Advantages of VAR Limitations of VAR

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Brief History

Just sort of gradually entered the financial vocabulary, sort of like finalization and parenting

But it is a natural concept NYSE implemented capital requirements based on the

concept as far back as the 1920s SEC in early 1980s adopted crude measure to assess

capital adequacy of broker-deals The first use of the specific terms "Value at Risk" was in

the 1993 Group of 30 report on derivatives best practices JP Morgan launched its RiskMetrics service in 1994 as a

means for its clients to implement the VAR concept

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Brief History (continued)

BIS Basel Committee on Banking Supervisions' April 1995 proposed revisions to its initial rules

• Capital requirements were to cover market risk in addition to credit risk

• Market risk was defined as three times 2-week/10 day VAR at 99% confidence

• Could use "standard model" from regulators or use proprietary model with regulator's approval

Eventually 1996 amendment, implemented in 1998, basically retained this approach

Basel II, which goes into effect in December 2006, retains same concepts and adds operational risk

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Brief History (continued)

So stockbrokers believe in VAR And bank regulators believe in VAR And they talk about VAR to our stakeholders Like other financial technique, theory is running ahead of

practice and both are still evolving

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Basic VAR

Definitions Explicit features Implicit assumptions Principal advantages of VAR Examples of VAR statements

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Basic Definition

“A forecast of a given percentile, usually in the lower tail, of the distribution of returns on a portfolio over some period; similar in principle to an estimate of the expected return on a portfolio, which is a forecast of the 50th percentile.” (www.gloriamundi.org)

Should be median, not expected return Maximum expected loss on an investment portfolio in a

given interval of time with a given confidence level First application environment was measurement of

market value risk on asset trading portfolio

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Explicit Features

Time period Confidence level Unit of currency Form of distribution

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Implicit Assumptions

Static portfolio Static distribution of returns

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Principal Advantages of VAR for Managing Risk

Easy to explain and communicate Single measurement captures good information Easier to implement and manage than many other

practices Widespread acceptance Regulatory encouragement

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Examples of VAR Statements

Risk Management Policy:• Our 95% one-month VAR will not exceed $10 million

Measurements in Compliance with Policy:• Probability that we will lose more than $10 million over the

next month is 3%

or

• Maximum amount we can lose over the next month at a 95% confidence level is $8 million

in other words

• Our VAR is $8 million

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Competitors for Risk Management

Asset allocation corridors Beta Volatility Greeks Duration and convexity Retention limits Credit quality limits CTE (conditional tail expectation) VAR

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Choosing the Right Words

We keep about 70% in stocks and 30% in bonds, but our portfolio manager can deviate as much as +/- 10% at any one time.

The beta of our portfolio is never allowed to exceed 1.25. We keep volatility under 35%. We never allow gamma to exceed 1, delta to exceed 15,

vega to exceed 0.25, or tau to be more than 5. Our portfolio duration is maintained between 5.5 and 6.5

years, and we never allow convexity to be less than zero.

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Choosing the Right Words (continued)

Our retention is $250,000; we reinsure any excess over this limit

Our credit quality limits are 5% BB, 2% B, 0.5% C, and 0.1% all else; fallen angels must be disposed of in less than 30 days

We arrange our portfolio so that if we lose money over the next year, we probably won't lose more than $10 million

We arrange our portfolio so that there is less than a 5% chance of losing more than $10 million over the next year

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Difficult Questions

Summarize your risks in a single measurement, using anything but VAR or CTE.

Compare your level of risks to those of your three closest competitors using anything but VAR or CTE.

Determine how much capital you should hold, and explain why, using anything but VAR or CTE.

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Three Important Differences

Liabilities Time intervals Non-market risks

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Liabilities

Asset portfolio VAR is not relevant. Solution is to include liabilities as negative assets. Pension managers may focus on unfunded liability or

another value that takes into account both assets and liabilities.

Choose a simplified valuation method for the liabilities:• Judgment call on a mathematical form

• Value fixed benefits at cost of funds

• Risk-neutral valuation by simulation

• Must fit with portfolio valuation technique

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Time Intervals

Frequent valuation of pension and insurance liabilities is not feasible, but an appropriate VAR measure can be developed using a longer time period, annually or perhaps quarterly

The required reaction time gives the basic VAR time period parameter, e.g., a 10-day VAR measure can be used if the organization can react within 10 days to reduce risks to an acceptable level

It should also drive governance, e.g., VAR must be reported at least that often, and portfolios must be adjusted at least that often

Typically, VAR is reported more frequently than the measurement interval. For example, 10-day VAR may be reported at the close of business each day

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Time Intervals (continued)

It should also drive data collection Basic relationship is that volatility is adjusted by the

square root of the time interval• Annual volatility is monthly volatility times the square root

of 12.

This doesn't work very well in practice Economic time series exhibit momentum

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Non-Market Risks

Cost of living/wage inflation/price inflation Longevity and mortality Participation Sponsor reliability Nonmarketable assets (equipment, sponsor equity,

restricted stock, intangibles) If a risk can be simulated, it can be included in the VAR

framework.

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Practical Considerations

Number of variables Distribution function Correlations Non-stationarity

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Lots of Variables

Sample portfolio:• 200 government bonds

• 50 publicly traded corporate bonds

• 20 private placements without market value histories

• 25 equity positions

• 5 real estate investments

• Pension liability

Build a distribution function for each one? Solution is to simulate using fewer variables.

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What’s the Right Distribution?

Convenient distributions are normal, lognormal, or actual historical changes

Historical data is backwards looking. Exactly what historical period are we likely to repeat going forward?

The tail is the most interesting part, and the normal and lognormal have small tails

Fatter tails can be modeled using a mix of normal distributions, stochastic volatility, or the Stable Paretian distribution

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Correlation Is Key

Correlation is has a very significant impact on VAR:• A + B, A – B, or (A2 + B2)0.5

Find the 300x300 return correlation matrix, then apply? Historical data is too sparse for credibility. Modeling typically uses a parameterized security

valuation approach (such as CAPM) to handle correlations implicitly.

BIS/Basel approach assumes perfect correlation among asset classes, producing values inappropriate for management.

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Non-Stationary Distributions

How much can change over the rebalancing interval? Longer interval and more volatile assumptions reduce

value of simplified answers. Solution is to use simulation for some assumptions, e.g.,

future volatilities. Long intervals, frequent trading require simulation

modeling of well-defined portfolio management strategy.

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Simple Numeric Example

S&P 500 Index fund with zero tracking error Current value $1000 5-year historical data of monthly returns has mean 0.87%

and standard deviation of 5.16% Assuming a normal distribution, a sample return can be

calculated as

$1,000 * (0.87% + *5.18%)

where is from a standard normal distribution

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Simple Numeric Example (continued)

A 90% confidence level is equivalent to 1.28 standard deviations below the mean,

The 10th percentile of the monthly gain is a result of

$1000 * (0.0087-1.28*0.0516). = -$57.35 If our risk management policy is to limit our one-month

90% VAR to $100, we have succeeded

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Potential Problems with VAR

Not additive, but manageable Comparability depends on parameters Not sufficient by itself

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Not Additive, but Manageable

Another difficult question: how to assess combined risks. Range of possibilities:

• A + B, A – B, or (A2 + B2)0.5

Not useful for a precise answer but may lead to a reasonable understanding.

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Comparability Depends on Parameters

Original 90% 1-month VAR was $57.35 (S&P Index) 99% 1-month VAR with the same data would have given

-$1000 * (0.0087-2.33*0.0516). = $111.41 90% 1-year VAR with the same data would have given

-$1000 * (0.0087-2.33*0.0516). = $220.47 Using the five prior years of S&P data would have given

-$1000 * (0.0099-1.28*0.0250). = $22.15 The higher the confidence level, the greater the volatility

of the percentile statistics, and the greater the impact of small differences in assumptions

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Not Sufficient by Itself

An additional control in a comprehensive framework, not the only control

Just a model

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Conclusions

VAR can be appropriate for managing all types of risks if they can be realistically simulated.

The exact implementation of VAR must take into account the context in which it is applied

• Pension plans are more concerned with net funding levels than with the supporting assets viewed in isolation.

• Daily valuation of pension liabilities is not necessary or useful.

• Daily rebalancing of pension fund assets is not necessary or useful.

• Plan sponsors can usefully include VAR as one element of a risk management plan.

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