Allianz Case Study Risk Management

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Thinking about Risk

Presented on April 18, 2006

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What is Risk?

Markowitzs key insight: risk can be proxied byvolatility. But in general, risk is not simply thevolatility of cash return

Example: comparing 1 month bills to 10 year zerosWhat is risky depends on your economiccircumstances, in particular on your liabilities

Two steps for dealing with this issue: (1) find the

minimum variance portfolio, which depends on yourliabilities; then (2) calculate portfolio risk as thevolatility around this minimum variance portfolio


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Min. Variance Portfolio forIndividual Investors

The minimum risk portfolio is heavy in TIPS, whichmatch the implicit liability of your future consumptionstream

Volatility of return, calculated around this TIPS-heavy

portfolio, is risk

There are different minimum risk portfolios for differentindividuals, so you and I may differ about which of twoportfolios is the riskier

The relative income hypothesis and Keeping up withthe Joneses risk


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Min. Variance Portfolio forDefined Benefit Pension Plans

For a Pension Fund the usual objective is tomaximize return/risk -- Information Ratio -- relativeto plan liabilities

Usually, liabilities are nominal cash payments

Risk is the volatility of the plans surplus.

The minimum risk portfolio defeases yourliabilities. Or, more accurately, it comes as close to

defeasing it as it can. In practice it is unlikely a trulyzero risk portfolio can be found.

Nominal bonds work, and TIPS work if you haveCOLA


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Min. Variance Portfolio for FundManagers

Positions are managed relative to a benchmark

Risk is fluctuations in return compared to thebenchmarks return

To avoid risk completely, hold the benchmark portfolio,assuming it is investable

You still have business risk if you retreat to thebenchmark too often


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Min. Variance Portfolio for HedgeFunds

Absolute return orientation seems to imply cash is thebenchmark

But, these days hedge funds are normally measuredagainst a peer group, an appropriate index of similar types

of hedge funds

The index is impossible to track. This means cash, withall its problems, is probably best


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What are your Objectives?

More return and less risk

When you formally optimize a utility function, as in MPT,you get two additive terms that describe the compositionof the portfolio. They correspond to minimize risk then

add some more return, in exchange for bearing morerisk.

We call them the passive (or hedging) portfolio and theactive portfolio. There is corresponding passive risk and

active risk.

The active portfolio consists of two sub-portfolios, thebeta portfolio and the alpha portfolio


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Objective: More Return

There are two risky ways to get more return:

(1) Add Systematic Risk. Buy asset classes that offer

higher returns risk premiums compared to theminimum riskportfolio. Call this the Beta Portfolio.

(2) Add Outperformance. Find managers who can beat

quasi-efficient markets. Call their active positionstheiractive bets away from the minimum risk portfolio, in

aggregate, the Alpha Portfolio.


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The Active Portfolio, Part 1: TheBeta Portfolio

Should be a portfolio of many different risk premiums.

There is no justification in theory for focusing almostexclusively on the equity risk premium.

Is there any justification in having beta risk at all? TheMiller-Modigliani theorem says probably not.

MM prediction: the stock market does not increase yourshare price when you take beta risks in your pensionplan.

Taxes may modify this conclusion


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The Active Portfolio, Part 2: TheAlpha Portfolio

Is the collection of all the active bets taken by all youractive managers, in aggregate.

To reduce risk, it should be a portfolio of many different,independent alpha streams. This is sometimes called theFundamental Law of Active Management.

Dont forget that alpha production is a zero sum gamein the best case

Beware of beta being repackaged and offered by yourmanager as pseudo alpha.. If so, you need to adjust his

results to find what his real alpha is, and how much alpharisk and beta risk you actually bear


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The Hedging Portfolio: Importance of theRisk-Free Asset

With a risk free asset, the variance of the hedgingportfolio and all its co-variances are equal to zero

This makes attributing the sources of risk easy: all therisk comes from the active portfolio

With no risk free asset, you bear passive risk as well asthe more familiar active risk

Co-variances among active and passive risks maymatter

Partitioning risk by source is non-trivial


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We may want to partition the risk in the portfolio into

3 sources: the passive or Hedging portfolio; the Betaportfolio; and the Alpha portfolio. Recall the last two,taken together, constitute the active portfolio

To do so, proceed as follows:

First, take the partial derivative of volatility withrespect to the hedging portfolio, H, then multiply theresult by H.

Second, repeat step 1 for B and A.

The sum of the three components is the total risk; thethree components tell you which sources the risk iscoming from

Notice that if H is risk-free, then the risk in theportfolio is all attributed to the active portfolio, as we

would expect

Where is you Risk Coming From?


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Conclusions

Are you measuring risk appropriately? Your liabilitiesmatter and they define your risk free asset

Start with this risk free asset, or at least the minimumvariance portfolio, as the core of your portfolio

To earn extra return, you may wish to add a betaportfolio to reap the harvest of risk premiums the

market offers. But in theory, this will not help yourstock price. [MM] (So, why are we doing this ?)

If you think you can identify managers who can beatthe market, then you can earn extra return byconstructing a risky alpha portfolio. But in theory, this

may not help your stock price. [MM again]

Pension plans would do best to focus on defeasingliabilities

Allianz Case Study Risk Management

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