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The Historical Simulation (HS)

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    There are three approaches to Value at Risk (VaR): analytical, Monte-Carlo and Historical simulation. It is a procedure for predicting the VaR by ‘simulating’ or constructing the cumulative distribution function (CDF) of assets returns over time. It uses historical data to assess the impact of market moves on a portfolio. Historical simulation works well even on uncorrelated variables Problem statement: Atlantic Investment Management firm had a portfolio which was heavily invested in Technology sector. Although the entire portfolio was invested in blue chip companies such as Google, Adobe and Microsoft, they were still highly exposed to market risk and sector risk because of low diversification. Atlantic investment firm needs to build a new portfolio which can eliminate or reduce the above-mentioned risk


    Accept the Risk: By accepting the risk, Atlantic Management fund will stay invested in its old portfolio of technology stocks, without making any changes to it. This strategy would require no active management and no additional cost. Since no hedging is done, no additional capital will be required. Higher returns come at higher cost. This strategy is flawed since investing in one particular sector exposes us to higher potential losses since we are not very well diversified. Solutions: Rebalancing and Diversification Historical Simulation.

    Evaluate potential Solutions

    Rebalancing and Diversification: In this strategy, Atlantic Investment firm will first have to diversify its portfolio and include stocks of non-technology company. Secondly, it will have to rebalance the weight in order to maximize the returns on the portfolio. This strategy. This strategy exploits the volatility of the assets in the portfolio to gain higher compounded returns, it also exploits the negative correlation to provide protection against the market risk or sector risk Historical Simulation: The biggest advantage of HS is, we do not have to make an assumption that the data is normally distributed with stable correlation to make sense out of it. HS can be done on any data set very easily and the results are also quite easy to interpret. Atlantic Investment Fund will need to run historical simulation to find out the optimal portfolio which will have a lower VaR and higher returns. HS has some limitations, since it considers past data the results are sometime not reflective of the current market. It does not always prepare the firm for future crisis. Implementation of solution: By using historical simulation Atlantic investment fund is trying to evaluate the best portfolio which has lowest VaR and highest expected returns. It has $3,00,000 to invest.

    •  Portfolio 1(Google, Adobe, Microsoft): Technology stocks with equal investment in each stock. This portfolio has a VaR of $11,430 and an expected return of about 15%
    •  Portfolio 2(Google, Boeing, Microsoft): Diversified portfolio with equal investment in each stock. This portfolio has a VaR of $10,420 and an expected return of 20%
    •  Portfolio 3(Google, Boeing, Microsoft, Pepsi): diversified portfolio with rebalanced weights in each stock. This portfolio has a VaR of $9,540 and an expected return of 25%

    Evaluation of outcome: By using historical simulation Atlantic Investment firm was able to identify the best portfolio. They should definitely invest in portfolio 3. This investment will help them to increase the expected return and reduce the Value at risk. They used diversification and rebalancing to build a portfolio and historical simulation to select the best portfolio. By doing this they came up with a beneficial strategy for the fund. Parametric approach: In parametric approach, the quantitative data requires assumption about the distributional characteristics about the population distribution. It is more powerful than non-parametric test when assumptions are met. Parametric approaches include GARCH, EWMA.

    Non-parametric approach: This approach works when data is not normally distributed. It is generally easy to compute but does not provide as powerful results as parametric approach. Non-parametric approach includes Historical simulation and Monte-Carlo simulation. Problem statement: Apple had returns ranging from -4.62% all the way up to 4.23% in the last 100 days. The analyst wants to evaluate future risk based on past data using historical simulation. There are some flaws in simple historical simulation, so the analyst needs to research more and utilize some more specific historical simulation techniques.


    Scenario analysis: We can use scenario analysis to estimate the risk based on past 100 days. But we cannot agree with the analyst since we don’t know the frequency of the extreme value, what if the extreme loss happened 100 days ago. In that case we won’t have enough data to get the accurate estimate risk. But this opinion is too subjective, it’s just an assumption. We should better use the existing historical data. So, we can use historical simulation, it’s a mathematical model, relatively rigorous, and its usage is relatively simple.

    Historical Simulation (HS): We have seen the definition of HS in previous case study. The problem with HS is that it assigns equal weight to all the returns in the entire data set, which is not right because as the information/news gets old, it becomes less relevant, so we assign lower weight to old information/news and higher weight to new information/news. Because of this inconsistency HS is limited in its accuracy. Secondly HS has a ghosting effect wherein the extreme and important observations are outcasted, because of these weaknesses we need to make further adjustments. Solutions: Age-weighted HS Volatility-weighted HS

    Evaluate potential Solutions

    Age-weighted HS: In this strategy, the returns are not assigned weights equally. Suitable choices of lambda parameter can help us to make VaR estimate more responsive to larger losses. Observations are given weight according to their age as them name implies, so that recent observation will have more weight than older ones. This strategy reduces ghosting effect. Volatility-weighted HS: This strategy directly accounts for volatility changes. Weights are assigned relative to the volatility. This strategy allows us to obtain VaR and ES estimates that can exceed the maximum loss in our historical data set: in periods of high volatility, historical returns are scaled upwards, and the HS series used in the procedure will have values that exceed actual historical losses.

    This is a major advantage over traditional HS, which prevents the VaR or ES from being any bigger than the losses in our historical data set. Implementation of solution. We can either use Age-weighted HS, wherein we will assign higher lambda to recent news or information and lower lambda to old news or information. Under volatility-weighted, if we want to assign the weights of statistics according to the time, we will use the EWMA. If we want to assign the weights of statistics according to volatility, we will use GARCH. Since we are limited on information regarding what approach the analyst took, we will try to evaluate the outcome of both the approaches.

    Evaluation of outcome

    Volatility-weighted HS works better than Age-weighted historical simulation because it not only eliminates the Ghost Effects and the Slow to Reflect Major Events, but also reduce the influence of overestimating or underestimating the VaR or ES as possible as it can. Volatility-weighted historical simulation produces near-term VaR estimates that are likely to be more sensitive to current market conditions. If the analyst has limited time, then he should choose Age-weighted historical simulation. If he has the ability to change all the old statistics into reasonable, adaptable new statistics in limited time, then he can choose Volatility-weighted historical simulation, which works better than Age-weighted historical simulation. It all depends on the ability of the analyst.


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