Chapter 14 - 3 Approaches for VaR

[Bill SD]

Probabilistic risk measures
The Acted notes (pg 15) that VaR can be calculated using 3 approaches: empirical (or historical), parametric and stochastic. Presume that Lam's 'Monte Carlo approach (page 216) is the same as the stochastic-simulation approach mentioned in Acted notes, although Lam omits the bootstrapping method.

Lam (page 217) also describes historical simulation as "uses historical data about actual price movements to generate scenarios, and then re-prices the portfolio according to these historical scenarios to generate the distribution of changes in portfolio value. Historical simulation is therefore similar to the Monte Carlo simulation approach, except that the changes in the risk factors are determined by historical experience, not chosen randomly."

Q1. Is Lam's historical simulation exactly the same as the empirical approach mentioned in Acted notes (even though the empirical description doesn't mention simulation and seems essentially to be a simple percentile of observed losses)? Or is it a more advanced version of the same thing.

Q2 Am I correct that both Practice Questions 14.6 and 14.9 are parametric approaches for VaR (Acted notes page 31-32) since they are assuming a normal distribution of losses and parameterising the expected annual volatility?

Q3: Are there empirical, parametric and stochastic approaches to calculating ruin probabilities? Or only for VaR, tVaR and expected shortfall?

Deterministic risk measures

Acted notes (pg 386) gives an example of the notional approach = risk weightings applied to the market value of assets and then compared to the value of liabilities in order to determine a notional (‘risk-adjusted’) financial position.

Q4: Under this approach would the liabilities also be discounted based on uncertainty -eg. lower discount rate for long-term, inflation linked liabilities in a foreign currency?

Acted notes (pg 387) defines "The factor sensitivity approach (ie sensitivity testing) determines the degree to which an organisation’s financial position (eg solvency or funding) is affected by the impact that a change in a single underlying risk factor (eg short-term interest rates) has on the value of assets and liabilities."

Q5: The Solvency II Standard formula includes factor sensitivities to calculate the VaR for certain risks such as equity (Acted notes Chapter 20 page 24). Would this be considered a deterministic VaR or are VaR measures always considered probabilistic?

Thanks in advance.

 

[Anna Bishop]

Hi Bill, I'm Anna, a different ActEd tutor, just giving Alvin a break from forum duties! Here are some answers:

Q1) For me, there's a difference between how we come up with the simulated values and then how we calculate the VaR. I'm wondering if this is where the confusion lies.

Let's say we're trying to model X = claim amounts on car insurance. We want to simulate 1000 values of X for our model.

We could simulate these historically (as per Lam, also known as bootstrapping in Module 15), or we could simulate using Monte Carlo simulation. Historical simulation involves recording our past claims over a given time period and then selecting 1000 of these - a bit like picking out 1000 raffle tickets from a big box (with replacement). MC simulation assumes that X has a distribution, eg Exp(5). To carry out this method, you simulate 1,000 random numbers between 0 and 1, and then use a method called inverse transform to turn them into 1,000 values from the Exp(5) distribution.

Either way, you end up with 1,000 values of X. Picture them in a spreadsheet listed from lowest to highest.

The empirical approach is then a way of calculating the VaR using these simulated values.

The parametric approach to calculating VaR is different, it is theoretical if you like (rather than based on simulated values). Under the parametric approach, you assign a distribution. For example, we might say X is Exp(5) again. To work out the 95% value at risk, you would solve F(x) = 0.95 and x would be the 95% VaR.

Q2) Yes

Q3) I would assume so Bill yes

Q4) My gut feel is that if it mentions the word 'value' then there would be some discounting involved. But then some liabilities, eg certain general insurance lines are valued at face value rather than discounted value.

Q5) VaR is a probabilistic measure. Again, perhaps useful to separate the modelling technique from the measure itself? A stress test is a deterministic modelling technique but VaR is a probabilistic measure.

Under SII, the equity stress is something like a 39% fall in the equity value. But someone somewhere will have calibrated this, ie thought about what level (ie 39%) the 99.5% fall should be set at.