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P2 Focus Review: 7th of 8 (Investment Risk)

David Harper CFA FRM

David Harper CFA FRM
Staff member
P2 Focus Review 7th of 8: Investment Risk
  • Portfolio VaR
  • Var in Investment Managementt
  • Hedge Fund Strategies
  • Portfolio Construction, Evaluation and Attribution
Portfolio VaR
The focus is Jorion's brief but extremely dense Chapter 7 which consists largely of a set of related VaR metrics, all derived from the same mean-variance framework that informs n-asset portfolio variance. For candidate who are not familiar, it can seem like a whole new set of concepts, but Jorion is really just extending the two-asset portfolio variance in a few ways:
  • A two-asset portfolio becomes a n-asset portfolio rendered in matrix notation. It is superior to then perceive the 2-asset portfolio variance as just a special case of: [row vector of weights]*[covariance matrix]*[column vector of weights],
  • As before, variance is almost effortlessly translated into VaR with a simple scaling: VaR = SQRT[variance]*deviate; I like to think of this (parametric) VaR as "scaled" or "stressed" volatility.
  • Marginal VaR is introduced. Marginal VaR is the key plank underlying these metrics, and once again we won't be surprised to encounter another first partial derivative. In this case, the marginal volatility is the change in portfolio volatility with respect to a change in a component's weight, d[portfolio volatility]/d[weight(i)]. Put another way, what is the change to portfolio volatility if we increase the weight of a position? Because this first partial derivative concerns the impact on portfolio volatility, we only need to multiply by the deviate to achieve Marginal VaR.
For Tier 3 members who have time, I really do recommend reviewing my learning XLS ( http://www.bionicturtle.com/how-to/spreadsheet/2011.t8.b.2.-jorion-analytical-var/) which implements the two-currency portfolio that is the basis of Jorion's entire walk-through; I think it's the most efficient way to get a grasp on these set of metrics.

The testable focus appears to include:
  • The same two-asset portfolio foundation that appeared in Part 1, so don't skip this easy part. The general form is given by: VaR(P) = SQRT[VaR(1)^2 + VaR(2)^2 + 2*VaR(1)*VaR(2)*correlation]. Given this general form, you can easily find both the version for correlation = zero and perfect correlation =1.0. See http://www.bionicturtle.com/forum/threads/professor-jorion-chapter-7.6139
  • Marginal VaR, where the most likely useful formula is given by: Marginal VaR (i) = Portfolio $VaR / Portfolio $Value * beta (i, Portfolio); a difficult idea here is that beta(i,P) is the beta of the position with respect to the portfolio that includes beta, such that as the position increases, this beta tends toward one.
  • Component VaR really just re-formats unitless Marginal VaR into its dollar equivalent: Component VaR = marginal VaR * $Position. Important because Component VaRs sum to Portfolio VaR. Here is an analogy: As the sum of individual VaRs is greater than portfolio VaR which is equal to the sum of component VaRs, the sum of unexpected losses (UL) is greater than portfolio unexpected loss which is equal to the sum of risk contributions (RC).
  • Incremental VaR: GARP likes to quiz this. It's easy to get incremental VaR wrong because the question looks so easy, see http://www.bionicturtle.com/forum/threads/l2-2-18-portfolio-construction-invest.4191
Var in Investment Management
It's easy to miss the historically popular concept, Surplus at Risk (SaR), because it is under Funding Risk in the syllabus. Jorion provides an example on page 433 of Chapter 17. The basic notion is VaR but applied to a pension surplus which equals assets minus liabilities (S = A - L). So there are two ways this can be approached:
  • According to the volatility of a difference, as in VaR(surplus) = deviate*volatility(A-L), where notice we can apply the variance of a difference between two random variables, or
  • Per Jorion's illustration (page 433), where an ROA and ROL inform an expected growth in the surplus (so we can compute an expected future surplus), then a given asset volatility is scaled by the VaR deviate to return a surplus at risk (SaR). Specifically, Jorion's example is: current surplus is $100 million ($1.0 billion assets - $900 million liabilities) with expected growth (based on 9.5% ROA and 5.0% ROL) to $150 million. Given an input assumption of asset volatility = 9.4%, 99% surplus at risk (SaR) is computed by Jorion as: SAR = 9.4%*2.33*$1.0 billion assets = $218.68 million (~ 220 million). Now, clearly as discussion in our forum shows, there are at least three ways to characterize the SaR in this scenario (versus current surplus? versus expected surplus? versus zero surplus which is suggested by "shortfall"). For the exam, don't worry, it will be articulated, if it is quizzed at all.
I do think the final section (17.5) in Jorion is testable. I do think it's worth reviewing the example where Jorion illustrates "budgeting across active managers" by maximizing the information ratios (IRs) subject to a tracking error constraint (for some reason, Jorion's definition of tracking error is really what is called active return, and his tracking error volatility, TEV, is really what is called tracking error.)

Hedge Fund Strategies
I don't have shortcuts, sorry, I think each strategy needs to be reviewed carefully. The classic tip here is to study the strategies, not only for their mechanics, but as yield sought for specific, deliberate exposure bets to risk factors. If you have surplus time, here is the rare case when I would actually recommend the previously assigned in 2011, but discontinued in 2012, reading as perhaps superior to the current assignment (or, at least it is a very strong complement that is well worth your time): "Individual Hedge Fund Strategies (Chapter 5)" in Through the Alpha Smoke Screens: A Guide to Hedge Fund Return (2005) by Lars Jaeger.

Portfolio Construction, Evaluation and Attribution
There is a lack of strong historical sample for this sub-topic, except insofar that the FRM re-visits P1 RAPM measures. Based on the evidence, my mere opinion is:
  • The Litterman reading contains very little that is testable; if you will need this reading for the exam, I don't know why
  • In contrast, the new Bodie, Kane, Marcus reading is the key. It is low-hanging fruit (so be ready!) for the FRM to test familiar RAPMs: Sharpe’s measure, Treynor’s measure, Jensen’s measure and Information ratio.
  • Tracking error (TE) is highly testable. Please make sure you can calculate both an ex ante tracking error (given portfolio and benchmark volatilities and their return correlation) and ex post tracking error; e.g., http://www.bionicturtle.com/forum/threads/tracking-error.6481
  • Even the second half of the new Bodie chapter is classic source material for FRM testing: M^2, statistical significance using the t-statistic (highly testable!), application of Sharpe ratio, style analysis, and asset allocation