bottom up and top down approach
07 Sep 2008
Feb
13
Value at Risk (VaR) for discrete, independent variables
by David Harper, CFA, FRM, CIPM
Philippe Jorion publishes an instructive article (Risk Management for Event-Driven Funds) in the current Financial Analyst Journal. He develops a risk metric for event-driven portfolios. A good example is a merger arb (aka., risk arb) strategy. These are unique as they have discontinuous, skewed distributions (deal outcome is binary and per-deal loss tends to exceeds gain). Even before he gets to realistic scenarios (correlated events), the setup is instructive because it shows how to estimate value at risk (VaR) for a discrete variable (deal succeeds or fails). I recreated one of his exhibits in the EditGrid spreadsheet below (you can open in Excel by selecting File > Export).
Only a few unrealistic assumptions are required; mainly, for this analysis, we assume events are uncorrelated:
- Probability of a failure (e.g., the merger doesn't happen so the stock, buoyed by the 'deal risk premium', collapses to its pre-announcement level): 15%
- Payoff under failure and success: -$15 and +$5
- Portfolio size: $100
The probability of success is calculated (1-failure) and so is expected payoff (weighted average). Then the columns are:
- No of deals: The same $100 portfolio is invested under various scenarios. First, into one deal, then five deals, up to 100 deals (i.e., $1 per deal). The numbers aren't realistic, of course
- Size of deals: Portfolio/No of deals
- Expected profit: always $2. The mean of the distribution is always $2. But VaR isn't about the mean. What is interesting is that, due to the discrete distribution, the VaR changes rapidly with the number of deals! This may at first glance appear to be diversification but it's rather the nature of discrete variables. When you don't have many events, your worst (expected) case is a un-smooth step function.
- No. of failures: the number of failures implied by a 95% value at risk (VaR). This can be found with the Excel/EditGrid function =CRITBINOM(). This function is analogous to the =NORMINV. It takes an alpha parameter which in this case is the VaR confidence (95%)
- Value at Risk (VaR): this calculates the expected payoff given the number of failures. The VaR is the negative of the dollar loss, so a $15 VaR refers to a $15 loss. After 40 deals, the VaR becomes negative: at this point, with 95% confidence, there is no expected loss.
- Actual confidence: I use the =BINOMDIST() to run the actual VaR. Why is it different than 95%? Due to the discrete variable. At each scenario, if we ran the function with only one less failure (failure - 1), =BINOMDIST() would return a value less than 95%. So, that wouldn't be enough failures to reach the confidence.
Here is the spreadsheet:
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