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I am trying to determine an appropriate haircut for a basket of illiquid stocks that barely traded during the year. Can someone suggest me an approach to estimate the risk?

My dataset has a lot of missing dates. If I replace these blanks with the previous price, when I come up with a measure for the returns I obtain several days without change and a date with a huge change.

Thank you very much for any suggestion.

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Generally if they are missing a completely at random data in few places, you do not have to be worried.

I advice you to use one of the technics of imputation:
- Previous value - cannot be used in this case
- Educated Guessing - you have "knowledge" about the data, you can try to use some interpolation in your mind.
- Common-Point Imputation - try to average a missing value or do an interpolation if several values are missing.
- Regression analysis or moving average - use previous X values to predict the missing values

Personally I would go for the common-point imputation.

To see more, please visit:
7 Ways To Handle Missing Data

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You're going to have to do a lot of guesswork, obviously, so it's best to keep things mathematically simple. First off, choose a "certainty level" as some quantile $q$, perhaps around 0.9, and the corresponding normal variate $z=N^{-1}(1-q)$.

Start by figuring out how much time $T_i$ you think each position $N_i$ will take to liquidate if necessary. Then choose some high percentile (not necessarily $q$) like 95% and set your overall horizon $T$ to the 95th percentile of those liquidation times.

$$T = Q_{95}(\{T_i\})$$

Now for each stock, use the existing data to find a return beta to the SP500

$$r_i = \beta^{(i)} r_{SP} + \epsilon^{(i)}$$

The variance $\sigma_i^2$ of the error terms $\epsilon^{(i)}$ is the "idiosyncratic variance" for this stock.

To account for the fact that correlations go to 1.0 in times of market stress, we will use a haircut model where the SP500 and every idiosyncratic term in the portfolio has experienced a $q$-level loss at horizon $T$.

Ignoring interest rates, we value each stock with current price $S_i$ at

$$ P_i = S_i\exp\left[\beta^{(i)} \sigma_{SP} \sqrt{T} z + \sigma_{i} \sqrt{T} z - (\beta^{(i)} \sigma_{SP}^2 + \sigma_{i})T/2\right]$$

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  • $\begingroup$ Dear Brian, Thank you very much for your answer. Can you explain me a little bit more the following part: Now for each stock, use the existing data to find a return beta to the SP500: ri=β(i)rSP+ϵ(i). How do I estimate ri. In my case, I have one trading day today, the next one was 6 days ago, the previous ones 15 days ago. They occurred almost randomly... $\endgroup$ – Daniel Sep 9 '15 at 23:22

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