Market Risk and VaR Models: the parametric approach

Question 1

What is the question VaR tries to answer?

Answer 1

Question: which is the maximum loss that could be suffered in a given time horizon, such that there is only a very small probability, e.g. 1%, that the actual loss is then larger than this amount?
The answer to the question is based on 3 elements:
* Maximum potential loss that a position could suffer
* With a certain confidence level
* In a give time horizon
Pr(L > VaR) = 1-c

Question 2

What are the three main approaches to VaR?

Answer 2

1. Variance-covariance (parametric)
2. Historical Simulations
3. Monte Carlo Simulations

Question 3

What is the formula for parametric VaR?

Answer 3

VaR_i=MV_i * δ_i * σ_i * α

MV = Market Value
δ = Sensitivity (Modified duration/Beta)
σ = volatility
α = confidence level

Question 4

What are the three main alternatives for computing volatility used in VaR?

Answer 4

1. Historical Volatility
   * Simple
   * EWMA
2. Implied Volatility
3. GARCH models

Question 5

What is the preferred method for calculating volatility used in VaR?

Answer 5

Historical volatility, specifically EWMA. It gets rid of the echo effect and is available for every market factor.

Question 6

What is the echo effect?

Answer 6

The echo effect is a problem which arises when using simple historical volatility estimates. Echo effect describes the significant drops in estimated volatility when volatile events exit the observation window.

Question 7

What is the EWMA multiplier formula?

Answer 7

(1-λ)*λ^t

λ < 1, the higher the lambda the more weight given to the most recent observations

Question 8

What is the recursive formula for estimating volatility under EWMA?

Answer 8

σ²_n(EWMA) = λσ²_n-1+(1-λ)u²_n-1

Question 9

Does the no serial correlation assumption (σ_T=σ_d*sqrt(T)) for market factors’ returns hold in reality?

Answer 9

Not really, it only holds for very liquid markets from daily to weekly returns

Question 10

What is the scaling factor alpha for a 99% confidence level

Answer 10

2.323

A 99% confidence level implies that the possibility of losing more than the VaR we calculate is 1%

Question 11

What is the difference between the asset normal and the delta normal parametric VaR approach?

Answer 11

Asset normal:
* Normal distribution assumption for positions’ market values (prices)
Delta normal:
* Normal distribution assumption for market factors’ returns

The two approaches coincide if the sensitivity of position is linear

Question 12

What to do for calculating VaR of portfolio with correlations?

Answer 12

Map each position to its relevant market factor, by looking at equivalent broken down securities. Calculate overall VaR using the formula under the square root with each VaR squared plus 2 * The VaR of each security * the correlation for the 2 securities.

Question 13

What are the assumptions and limits of the variance-covariance approach

Answer 13

• Normal distribution assmuption of market factor returns
• Stability of variance-covariance approach
• Assumption of serial independence of market factor returns
• Linear sensitivity of position (linear payoff)

Question 14

What is a possible solution to the VaR normal distribution assumption?

Answer 14

1. Using a student t distribution of market factor returns. The lower the degrees of freedom the fatter the tails.
2. Mixture of normals: retruns are extracted by two normal distributions with the same mean but different variance. Volatility is a function of two factors: structural and cyclical so it should make sense.

Question 15

What is the solution to the assumption of linear sensitivity problem for VaR?

Answer 15

Delta-gamma approach:
VaR_i = MV_i * (δ_i * σ_i * α - γ/2 * α² * σ²_i)

Gamma in this case is the “modified convexity”, second order sensitivity