Question:

Three value at risk (VaR) methods are reviewed: Delta-normal, historical simulation and Monte Carlo. Which are true of the following? (answer could be zero, one, two or all three methods)

Among the VaR methods, which…

(i) Can conduct a full portfolio revaluation?
(ii) Requires a (parametric) distributional assumption?
(iii) Is LEAST appropriate for a portfolio that contains many embedded derivatives (e.g., options)?
(iv) Is computationally fast?
(v) Handles fat tails?
(vi) Is easy (simplest to implement)?
(vii) Suffers from sampling variation?
(viii) Is the most powerful?
(ix) Handles correlations?

Answer:

Here is Jorion’s typology. Note the essential trade-off is speed (local) versus accuracy (simulation):

Let’s summarize advantages and disadvantages of each.

Delta-normal Headline: Local valuation measures exposures with partial derivatives (first derivative = delta = duration, delta-gamma adds second derivative = convexity = gamma). Well-suited to large portfolios where optionality is not a major factor.

Delta-normal Advantages:
Easy to implement. Computationally fast. Can be run in real-time. Amenable to analysis (can run marginal and incremental VaR).

Delta-normal Disadvantages: Normality assumption violated by fat-tails (compensate by increasing the confidence interval). Inadequate for nonlinear assets

Historical Simulation Headline: nonparametric method that requires no distributional assumption (no specific assumption about the risk factors).

Historical Simulation Advantages: Simple to implement. No Covariance matrix. Can account for fat-tails
Does not require distributional assumption (e.g., normal). Can do full valuation. Allows for horizon choice. Intuitive.
No model risk (!)

Historical Simulation Disadvantages: Uses only one sample path (if history does not represent future, important tail events not captured). High sampling variation (data in tail may be small). Assumes stationary distribution (can be addressed with filtered simulation)

Monte Carlo Headline: Simulation method that generates random movements in the risk factors based on estimated parametric distributions (note: there is such thing as non parameteric Monte Carlo, but not within our scope; our scope is parameteric Monte Carlo)

Monte Carlo Advantages: Most powerful. Very flexible w/ unlimited scenarios. Handles fat tails. Handles nonlinearities. Incorporates passage of time (time-variation); e.g., including time decay of options.

Monte Carlo Disadvantages: Computationally intensive (need lots of computer and/or time). Can be expensive. Model risk, including price specification. Sampling variation

Answers:

(i) Can conduct a full portfolio revaluation?
Either Historical simulation or Monte Carlo simulation but not delta normal

(ii) Requires a (parametric) distributional assumption?
Delta normal and Monte Carlo but not historical simulation

(iii) Is LEAST appropriate for a portfolio that contains many embedded derivatives (e.g., options)?
Delta normal

(iv) Is computationally fast?
Delta normal. Monte Carlo’s drawback is “computational time” required

(v) Handles fat tails?
Historical simulation and Monte carlo

(vi) Is easy (simplest to implement)?
Both delta normal and historical simulation are considered “easy to implement.” Monte Carlo, on the other hand, is considered more powerful but more sophisticated

(vii) Suffers from sampling variation?
Both historical simulation and Monte Carlo

(viii) Is the most powerful? Monte Carlo

(ix) Handles correlations? Note the question is vague: do we mean cross (spatial) correlations or serial correlations. All three can incorporate correlation in one way or another.