Value at Risk (VaR) is a widely used risk measure of the risk of loss on a specific portfolio. For a given portfolio, probability and time horizon, VaR is defined as a threshold value such that the probability that the mark-to-market loss on the portfolio over the given time horizon exceeds this value is the given probability level.
Definition of expected shortfall
Expected shortfall (ES) is a risk measure, it is to evaluate the risk of a portfolio. It is an alternative to value at risk (Var) and is more accurate in the tail of the distribution. The "expected shortfall at q% level" is the expected return on the portfolio in the worst q% of the cases. ES evaluates the value of an investment focusing on the less profitable outcomes. For small values of q it focuses on the worst losses. On the other hand, unlike the discounted maximum loss even for lower values of q expected shortfall does not consider only the single most catastrophic outcome.
Similarities
Var and ES have some similarities. Some of them are:
Both of them are use for risk valuation
Both Var and ES can predict volatility too much ahead in time. The effectiveness, however, is not the same.
The estimates of VaR and ES are affected by an estimation error which is the natural sampling variability which happens due to the limited sample size. If the size of the sample becomes bigger the error will decrease.
When the estimation error is close to one -the underlying loss distribution is very close to normal- the relative standard deviation of Var and ES are almost equal.
Var and ES have tail risk under extreme value distributions
Differences
VaR and ES can have similarities but they also have important differences:
ES considers loss beyond the VaR and is seems to be sub-additive and in the same time VaR does not consider loss beyond the percentile and it is not sub-additive
The ES estimates are affected by how large and infrequent loss is realized in the specific sample, since ES considers the right tail of the loss distribution. On the other hand, the VaR estimates are less affected by large and infrequent loss than the ES estimates, since the VaR method does not consider loss beyond the VaR level. Hence when the loss distribution is more fat-tailed, the ES estimates become more varied due to the large loss, and their estimation error becomes larger than the estimation error of VaR and vice versa. Thus ES varies more than VaR at low default rates if it is estimated with the same sample size.
One problem with VaR when applied to discontinuous distributions is their sensitivity to small changes in the confidence level. In other words, they are not in general continuous with respect to the confidence level. On the other hand, ES is continuous with respect to confidence level.
The estimation methods used for standard VaR models do not measure extreme price movements. They assume that the returns of an asset follow a normal distribution. So they do not consider the fat-tailed distributions of actual returns, underestimate the extreme price movements.
VaR has tail risk when the loss underlying distributions overlap beyond the confidence level. In this case, VaR can be decreased by manipulating the tails of the loss distributions and these munipulations increase the possibility for extreme losses and may lead to a failure of risk management. This problem is very important when the portfolio loss is not linear and the distribution function of the loss is discontinuous
How EWMA provide time varying estimates
The Exponential Weighted Moving Average (EWMA) is defines the next period's variance as a weighted average of this period's variance and this period squared return.
σ2τ+1 = λσ2τ+1 + (1-λ) z2t
So by definition EWMA can considered that it captures time varying estimates because in a series data, each estimation of variance is affected by the previous day's estimation.
Strengths and weaknesses EWMA against GARCH
EWMA can be considered as a restrictive case of GARCH model. GARCH model include a variable more that EWMA. But except from this EWMA has some weakness
GARCH incorporates mean reversion of the returns but EWMA does not
EWMA places more emphasis in recent data and cannot capture the volatility clustering.
The GARCH model is more preferable from EWMA in terms of minimizing the number of exceeded observations in a back test
GARCH is more precise for long term predictions since expected volatility depends on current and long run volatility
But EWMA has some strengths comparing to GARCH model. EWMA is simpler than GARCH and EWMA is more preferable for ongoing revisions of volatility calculations. Moreover EWMA can provide as trustworthy results as GARCH with more limited data requirements.
Question 2
Strategies followed by LTCM
Swap-spread Strategy
The core strategy that LTCM followed is the swap-spread strategy. Swap-spread strategy is based on fixed- to-floating one currency interest rate swaps. In this situation an arbitrage opportunity can be created when the spread between the fixed rate that the fund lends is different from the floating rate the fund pays.
The core strategy that LTCM followed can is the swap spread. LTCM's profits were coming from the volatility of the swap spread. If the swap spread was getting bigger the value of the bond would increase and LTCM could make money by selling the bond before the maturity. If the swap spread decreased LTCM would add to the position, and if the T-bond rate, which would be used to finance the position, was not decreased more in relation to the Libor, LTCM would make profits.
Sometimes, when the swap spread was extremely wide, LTCM entered an interest rate swap by receiving fixed and paying Libor. In that case LTCM would enter in a repurchase agreement. The Fund would lend money and use as collateral the bond that wants to short. Afterwards, LTCM would sell the bond and repay the loan made to take the collateral.
The Yield-Curve Relative-Value Trade
Another strategy that LTCM were using was the Yield-curve relative-value trade. LTCM were using the variability of the forward rates. The Fund entered a swap trade using swaps on forward rates of different maturities. This trading strategy was based on the big variability of interest rates. The strategy would be adjusted that the fund is not exposed to the rise or fall of interest rates as well as not to care of the steepening or flattening of the yield curve.
Problems the strategies faced
In 1998 Russia devalued the ruble. This default affected the financial markets because many Russian banks and firms used their right on their derivative contracts that allowed them to terminate these contracts. The result was that many customers who had been using derivative contracts to hedge the risk of Russian rates, ending having contracts without value. This fact made the investors to choose to invest in high quality investments instead of less liquid and quality investments. This demand of high quality financial products cause a reduction of the spread of high and low quality investments. LTCM lost several billion due to this fact because had investing in the increase of this spread. Furthermore, Interest rates on T-bonds fell and in the same time the yields of the bonds that LTCM had invested did not fall as much as the rates of T-bonds. The result was that the losses LTCM had from the hedges not be offset by the gains on hedged bonds.
The Russian Default did not affected in great extend the developer economies but the emerging economies. Besides, LTCM did not hold any Russian bonds but the default affected countries that LTCM had invested. The fund had butterfly trade in Fraance and lost several billions due to the bad provision of French interest rates. The forward curve remained constant through that period and the specific strategy had negative results under those conditions.
Lessons learnt from LTCM crisis
We can learn some lessoms from LTCM crisis. These must be taken on consideration by various parties.
Traditional risk management models ignore the funding liquidity. So we should take into consideration credit risk, political risk or market disruptions when we take overleveraged positions when invest. We should make stress tests not only based on historical prices but looking at worst-case scenarios.
Moreover, investors as well as managers and policy makers should include the value of transparency in their estimations. In addition they should not invest in products considering only their credit rating.
There was moral hazard because the Fed would help LTCM. They should have let LTCM to be supported by Buffets help. The Fed should had intervene if there were really systematic risk.
