Market risk calculation and interpretation

[AlexFrm]

Could someone explain what we have to do in the following situation:
If at time point 0 we buy bank assets and they droped by 20% then what type of distribution of assets returns we would have to use in the calculation of VAR at time point 1? As it occurred to me, the distribution of assets returns at point -20% is not equivalent to that at point 0%.

 

[David Harper CFA FRM]

Hi @AlexFrm Sometimes seemingly simple questions reveal layers of complexity! An "easy way" through this problem is to utilize VaR in percentage (return) terms, which we do tend to do. For example, if today P = $100.00, daily σ = 1.0%, and if the daily returns are normal, then our 95.0% normal VaR is given by 1.0%*1.65 = 1.65% and we can say, "the worst expected loss tomorrow is $1.65." Then tomorrow of the asset somehow drops by 10% to $90.00 and we ignore advice to update the VaR (the volatility will certainly increase!), we might say tomorrow that the worst expected loss is 1.65%*$90.00 = $1.48. I do not think this would be incorrect (and it illustrates why we use percentage returns and convert them to dollars, rather than just stay in dollars, i think) except that we are open to the criticism that we did not recalculate the 1.65% (in return terms) to update the volatility to something higher; for example, we can easily update the volatility with EMMA (λ = 0.94) such that the volatility updates to sqrt[0.94*1.0%^2 + 0.060*(-10%)^2] = 2.634%. In this way, we might alternatively say tomorrow that the 95.0% VaR is $90*2.634%*1.645 = $3.90. Notice how this update did not change the distributional assumption that the (arithmetic) returns are normally distributed?

But there are still more issues involved. It is unclear why would would change the distribution from T0 to T1? Faced with an extreme loss, there is always the question "was this just an unlucky tail outcome? or is our distribution mis-specified?" Theses are three different things: 1. the drop of 20%; 2. the volatility assumed; and 3. the distribution assumed. When we use 1.645 for 95% VaR, we assume normal distribution, but we do not need to, and in fact, the 20% drop suggests we need a heavy-tailed distribution. So I just wanted to remind that "VaR is just the quantile, given the probability distribution" and it is unclear why the asset drop would require any change at all in the distribution assumed. Finally, below I copy some interesting, hopefully relevant text, only because I do not want to suggest there is a single correct answer. We typically start learning with the assumption of static portfolios, but there is a whole topic of dynamic VaR (of which I actually consider your question to be related):

"IV.1.5.2 Static Portfolios: Market VaR measures the risk of the current portfolio over the risk horizon, and in order to measure this we must hold the portfolio over the risk horizon. A portfolio may be specified at the asset level by stating the value of the holdings in each risky asset. If we know the value of the holdings then we can find the portfolio value and the weights on each asset. Alternatively, we can specify the portfolio weights on each asset and the total value of the portfolio. If we know these we can determine the holding in each asset.

Formally, consider a portfolio with (long or short) holdings {n(1), n(2),..., n(k)} in k risky assets, so n(i) is the number of units long (n(i)> 0) or short (n(i) < 0) in the ith asset, and denote the ith asset price at time t by p(it). Then the value of the holding in asset i at time t is is n(i)p(it), and the portfolio value at time t is

P(t) = summation (from i = 1 to k): n(i)p(it)

We can define the portfolio weight on the ith asset at time t as

w(it) = n(i)p(it)/P(t)

In a long-only portfolio each n(i) > 0 and so P(t) > 0. In this case, the weights in a fully funded portfolio sum to one.

Note that even when the holdings are kept constant, i.e. the portfolio is not rebalanced, the value of the holding in asset i changes whenever the price of that asset changes, and the portfolio weight on every asset changes, whenever the price of one of the assets changes. So when we assume the portfolio is static, does this mean that the portfolio holdings are kept constant over the risk horizon, or that the portfolio weights are kept constant over the risk horizon? We cannot assume both. Instead we assume either

• no rebalancing – the portfolio holdings in each asset are kept constant, so each time the price of an asset changes, the value of our holding in that asset will change and hence all the portfolio weights will change; or
• rebalancing to constant weights – to keep the portfolio weights constant we must rebalance all the holdings whenever the price of just one asset changes.

Similar comments apply when a portfolio return (or P&L) is represented by a risk factor mapping. Most risk factor sensitivities depend on the price of the risk factor. For instance, the delta and the gamma of an option depend on the underlying price, and the PV01 of a cash flow depends on the level of the interest rate at that maturity. So when we say that a mapped portfolio is held constant, if this means that the risk factor sensitivities are held constant then we must rebalance the portfolio each time the price of a risk factor changes.
The risk analyst must specify his assumption about rebalancing the portfolio over the risk horizon. We shall distinguish between the two cases described above as follows:

• Static VaR assumes that no trading takes place during the risk horizon, so the holdings are kept constant, i.e. there is no rebalancing. Then the portfolio weights (or the risk factor sensitivities) will not be constant: they will change each time the price of an asset (or risk factor) changes. This assumption is used when we estimate VaR directly over the risk horizon, without scaling up an estimate corresponding to a short risk horizon to an estimate corresponding to a longer risk horizon. It does not lead to a tractable formula for the scaling of VaR to different risk horizons, as the next subsection demonstrates.
• Dynamic VaR assumes the portfolio is continually rebalanced so that the portfolio weights (or risk factor sensitivities, if VaR is estimated using a risk factor mapping) are held constant over the risk horizon. This assumption implies that the same risks are faced every trading day during the risk horizon, if we also assume that the asset (or risk factor) returns are i.i.d., and it leads to a simple scaling rule for VaR" --Carol Alexander, Vol IV pages 20-21