Thinking about Risk Presented on April 18, 2006 AllianzGI

2 What is Risk? Markowitz’s key insight: risk can be proxied by volatility. But in general, risk is not simply the volatility of cash return Example: comparing 1 month bills to 10 year zeros What is ‘risky’ depends on your economic circumstances, in particular on your liabilities Two steps for dealing with this issue: (1) find the minimum variance portfolio, which depends on your liabilities; then (2) calculate portfolio risk as the volatility around this minimum variance portfolio

3 Min. Variance Portfolio for Individual Investors The minimum risk portfolio is heavy in TIPS, which match the implicit liability of your future consumption stream Volatility of return, calculated around this TIPS-heavy portfolio, is risk There are different minimum risk portfolios for different individuals, so you and I may differ about which of two portfolios is the riskier The relative income hypothesis and “Keeping up with the Joneses” risk

4 Min. Variance Portfolio for Defined Benefit Pension Plans For a Pension Fund the usual objective is to maximize return/risk -- Information Ratio -- relative to plan liabilities Usually, liabilities are nominal cash payments Risk is the volatility of the plan’s surplus. The minimum risk portfolio defeases your liabilities. Or, more accurately, it comes as close to defeasing it as it can. In practice it is unlikely a truly zero risk portfolio can be found. Nominal bonds work, and TIPS work if you have COLA

5 Min. Variance Portfolio for Fund Managers Positions are managed relative to a benchmark Risk is fluctuations in return compared to the benchmark’s return To avoid risk completely, hold the benchmark portfolio, assuming it is investable You still have business risk if you retreat to the benchmark too often

6 Min. Variance Portfolio for Hedge Funds Absolute return orientation seems to imply cash is the benchmark But, these days hedge funds are normally measured against a peer group, an appropriate index of similar types of hedge funds The index is impossible to track. This means cash, with all its problems, is probably best

7 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 composition of the portfolio. They correspond to “minimize risk” then “add some more return, in exchange for bearing more risk.” We call them the passive (or hedging) portfolio and the active portfolio. There is corresponding passive risk and active risk. The active portfolio consists of two sub-portfolios, the beta portfolio and the alpha portfolio

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9 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 the minimum risk portfolio. Call this the “Beta Portfolio”. (2) Add Outperformance. Find managers who can beat quasi-efficient markets. Call their active positions –their active “bets” away from the minimum risk portfolio, in aggregate, the “Alpha Portfolio”.

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12 The Active Portfolio, Part 1: The Beta Portfolio Should be a portfolio of many different risk premiums. There is no justification in theory for focusing almost exclusively on the equity risk premium. Is there any justification in having beta risk at all? The Miller-Modigliani theorem says “probably not”. MM prediction: the stock market does not increase your share price when you take beta risks in your pension plan. Taxes may modify this conclusion

13 The Active Portfolio, Part 2: The Alpha Portfolio Is the collection of all the active “bets” taken by all your active managers, in aggregate. To reduce risk, it should be a portfolio of many different, independent alpha streams. This is sometimes called the Fundamental Law of Active Management. Don’t forget that alpha production is a zero sum game– in the best case Beware of beta being repackaged and offered by your manager as ‘pseudo alpha.’. If so, you need to adjust his results to find what his real alpha is, and how much alpha risk and beta risk you actually bear

14 The Hedging Portfolio: Importance of the Risk-Free Asset With a risk free asset, the variance of the hedging portfolio and all its co-variances are equal to zero This makes attributing the sources of risk easy: all the risk comes from the active portfolio With no risk free asset, you bear passive risk as well as the more familiar active risk Co-variances among active and passive risks may matter Partitioning risk by source is non-trivial

15  We may want to partition the risk in the portfolio into 3 sources: the passive or Hedging portfolio; the Beta portfolio; 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 with respect to the hedging portfolio, H, then multiply the result by H.  Second, repeat step 1 for B and A.  The sum of the three components is the total risk; the three components tell you which sources the risk is coming from  Notice that if H is risk-free, then the risk in the portfolio is all attributed to the active portfolio, as we would expect Where is you Risk Coming From?

16 Conclusions Are you measuring risk appropriately? Your liabilities matter and they define your risk free asset Start with this risk free asset, or at least the minimum variance portfolio, as the core of your portfolio To earn extra return, you may wish to add a “beta portfolio” to reap the harvest of risk premiums the market offers. But in theory, this will not help your stock price. [MM] (So, why are we doing this …?) If you think you can identify managers who can beat the market, then you can earn extra return by constructing a risky “alpha portfolio”. But in theory, this may not help your stock price. [MM again] Pension plans would do best to focus on defeasing liabilities