Analysis of Value at Risk (VaR) Methods in Financial Risk Management

Introduction

There are certain risks affecting the financial markets. Credit risk, Market risk, Operational Risk, strategic risk, financial risk are some examples of risk. One of the ways for risk management is Value at Risk. Value at risk (VaR) summarizes the total risk in a portfolio. It is commonly used for assessing possible losses by financial institution, portfolio managers and various corporate agencies. It is a tool to measure risk in an investment in normal market circumstances, over a given period says, day, month or year.

In other way, VaR is the dollar or percentage losses in portfolio (asset) value that will be equaled or exceed only p.100% of the time in the next ‘T’ days. A 1%, 5% and 10% VaR would be denoted by VaR(1%), VaR(5%) and VaR (10%).

Parameters for VaR Calculation

For instance, the VaR(5%) of $16000 indicates that there is a 5% i.e., p.100% chance that on any given day , the portfolio will experience a loss of $16000 or more or 95% i.

e., (1-p).100% chance that on any given day the portfolio will experience either a loss less than $15000 or a gain. If the changes in value of portfolio are normally distributed, it will be easier for the calculation of VaR from mean and standard deviation of the change in portfolio value during the time. Market risk capital is based on 10 day VaR i.e., 10 days ahead in the confidence level of 99%.Credit risk and operational risk is based on one-year VaR in the confidence level of 99.9%.

There are certain parameters needed to calculate VaR :

• Dollar Portfolio Value
• Time Horizon (T)
• Confidence Level (p)

VaR is expressed in dollars and defined by $VaR.

It implies probability of loss in dollar greater than dollar Value at risk ($VaR) has to be equal to p (Probability).

VaR is based on log returns and measures change in log returns rate in time T.

Pr (-RPF >VaR) = p  Pr (RPF < -VaR) = p. That is to say, Value of Risk is defined in terms of loss. Mathematically, we define loss as negative of return. If loss is greater than VaR, the return is smaller than – VaR.

The log return of an asset which has been daily continuously compounded is denoted by:

R t+1 = ln (St+1) – ln (St)

If the loss in the portfolio has a mean (μ) and standard deviation (σ), then

VaR = μ + ΣN-1 (p)

Here, p is the probability or confidence level, and N-1(.) is the inverse cumulative normal distribution. This can be calculated by using NORMSINV in Excel Spreadsheet.

Here, we have taken five stocks of Microsoft Corporation, Starbucks Corporation, Oracle Corporation, Goldman Sachs Group Inc and Sysco Corporation with ticker symbol MSFT_US, SBUX_US, ORACLE_US, GS_US and SYY_US respectively. These all are stock of US multinational Corporations. We extracted latest four years data of closing price of all these four stocks from Bloomberg Terminal. We collected 999 observations for each stock. We created portfolio of assets of five stocks. We calculate daily continuous return for each stock using log return formula in excel. We found VaR of portfolio on the basis of various methods or approaches like Historical Simulation, Weighted Historical Simulation, Risk Metrics and GARCH (1,1).

Calculation of Logarithmic return:

For each of the stocks, we calculated returns by using Logarithmic returns formula in excel spreadsheet,

Return (t) = ln (Pt / Pt-1)

We calculated return for all the stocks from the time series. We get positive as well as negative daily returns for all the stocks.

Calculation of Value of Portfolio:

Value of portfolio of ‘n’ stocks at time‘t’ is given by:

VPF, t = ∑_(i=1)^n Ni,t Si,t

Where,

Ni.t denotes number of shares of asset ‘i’ in portfolio at time‘t’.

Si,t denotes price of the asset ‘i’ in portfolio at time ‘t’.

Calculations of Return of Portfolio:

Here, we need pseudo log return which is calculated by:

RPF,.t = ln (VPF,t / VPF,t-1)

Where, ln is the natural log function in Excel.

The time series of returns of the assets should be arranged in ascending order i.e., from their lowest values to highest values.

Historical Stimulation, VaR(HS)

Historical stimulation calculates VaR using the past data to estimate what will happen in the future. It is purely based on historical data and can be calculated by using simple statistical tools. It doesn’t contain any parameters unlike any other methods. We just need to have historical data. Thus, it is known as non parametric approach. This method of VaR calculation is mostly famous in banking industry. Perigon and Smith (2006) surveyed among 60 banks of U.S., Canada and other international banks and found that seventy three percentages of such banks use historical stimulation for VaR calculation from 1996 to 2015. It is alternative to complicated methods of VaR calculation such as GARCH.

We need to accumulate a series of past daily returns, sort them in ascending order and look for the worst expected loss. Here we have calculated VaR for a portfolio using a one day time horizon, a 99% confidence level and 999 numbers of observations. We need to sort these returns and create histogram or frequency plots. Histogram plots actual loss experience. As historical simulation uses real data i.e., returns, it can capture real events which can’t be found out by other theoretical models. Here, each day are given equal weights. Historical simulation approaches use the actual percentiles of the observation period as value-at-risk measures. We uses PERCENTILE function in excel to calculate VaR on the basis of historical simulation.

For historical simulation, we need pseudo portfolio value on the basis of past stocks price but number of shares should be constant. This is given by:

VPF, t-1 = ∑_(i=1)^n Ni,t Si,t-1

If we need to calculate value of risk of portfolio at time‘t’, we need to fix a portfolio at time‘t-1’.

Once we get series of returns by ‘ [RPF, t+1- τ ]m τ=1’ , where m is the number of observations which says how far we are going to relate past data, we need to construct histogram (frequency diagram). Here for our calculation, m= 999. From the histogram, we need to find out the cut off point.

After generating all parameters like time horizon, confidence level, series of returns and value of portfolio, we calculate VaR as per historical simulation by using Percentile function in Excel. If confidence level is p then 100th percentile of the sequence of returns can be calculated as:

VaRPFt+1 = -Percentile ([RPF, t+1- τ ]m τ=1,100.p%)

Here, for every stock we have taken cut off point (p) as 0.01 and Ƞ as 0.99 i.e., 99% confidence level. We need 1th percentile if we want VaR at confidence level of 99%.

So we calculate Historical Simulation for last 22 days and for each day VaR, we consider series of returns at one day horizon.

After calculating VaR of portfolio by historical simulation, we calculate VaR for individual stocks by using historical simulation method.

The value of risk by historical simulation has been calculated from cell G1002 to G1023 for 22 days. The value of Risk at 10/23/2019 is calculated considering series of returns till 10/22/2019 and so on.

The VaR for portfolio is 2.66% for last 22 observations.

In case of MSFT_US, the VaR is approximately 3.85% for last 22 days observations.

In case of SBUX_US, the VaR is approximately 3.029%.

In regards of ORACLE_US, the VaR is approximately 4.245%.

In regards of SYY_US, the VaR is approximately 2.769%.

Weighted Historical Simulation, VaR (WHS)

The historical simulation doesn’t consider the weight. That is to say, equal weights are given to all observations. In this method of calculating Value at Risk, recent observations are given more weight since they indicates current volatility of the market and less weight is given to those observations which are further in the past. That is to say, recent observations are given more weight than past observations since they contribute more to the current market scenario. The future prediction depends on recent observations without ignoring past completely. The observations and their provided weights are assigned in the ascending order, from highest to the lowest. The non-parametric value is ‘τ’. As we go further back in past, τ increases. As τ increases, η has to be less than one. If ‘m’ number of sample is taken, τ goes from 1 to m and its sum will be equal to one. They are called exponentially declining weight .They are declining since η is less than one. Generally, we choose η between 0.95 and 0.99. We take hypothetical returns and sort them in ascending order and VaR (p.100%) is calculated by adding weight to those returns until we get our expected cut off.

Here, we have taken “τ” from 1 to 999 observations. As we move forward in the past, we can see value of τ increases.

For last 22 observations, starting from 10/23/2019 to 11/29/2019, we have calculated weighted historical simulation. For each observation we assign weight and calculate cumulative weighted value. The weighted historical simulation of five assets portfolio is 2.81% for last 22 days observations.

Risk Metric Model

Risk Metrics is one of the methods to calculate VaR of a portfolio by using data sets and distribution. Risk Metrics was actually established in 1994. Later on, around 1996, J.P. Morgan and Reuters partnered to develop the method- in effective way and make information widely available to professionals and the wider public. The main goal of risk metrics is to create transparency in market risks, establishing criteria to measure risk and providing clients with methods and techniques to manage risk. The Risk Metrics methodology for calculating the VaR assumes that a portfolio’s returns follow a normal distribution. It is better than historical stimulation which doesn’t use any parameters.

We need to get distribution of future returns based upon two components which are mean and price. In risk metric, information about consequence and likelihood is use to determine the overall level of risk. Here, we put weight on the past squared returns. The weighted average of today’s volatility and squared return is use to forecast the volatility of tomorrow. In risk metric method, we use forecasting method called exponential smoothing which says as we go backward in time ,the weight on past squared returns decline exponentially. The near one time series has more effect than the one farther away from the time series. Risk Metric Model is simplication of GARCH model.

GARCH

The GARCH model is the methodology to calculate VaR. It is Generalized Auto Regressive Conditional Heteroskedasticity. It is based on assumption that volatility of assets changes with respect to time. In order to predict future volatility, we use GARCH (1, 1). This method uses historical variances to predict future varainces. Past variance demonstrates future variances.

Statistically, GARCH (1, 1) is calculated by the following equation:

σ2t+1 = ω + αɛ2t + βσ2t

Where,

σ2 denotes variance,

ε denotes the residual squared,

t denotes time.

ω, α and β are statistical parameters which are estimated by the calculation of maximum likelihood estimation.

Suppose today is the time (t), the equation tells that the variance of tomorrow at time (t +1) is a function of

the square residual at time (t),

the variance of time (t),

the weighted average long-term variance

This can be also said that today’s variance is the function of yesterday’s variance.

GARCH (1, 1) recognizes one residual square and one variance square. This method focuses on the volatility clustering. Here, we assume that conditional variance ( σ2t+1) depends on value of ɛ. ɛ is such a variable having zero mean.

The risk measure is considered to be coherent if it satisfies certain conditions. There are four conditions that are needed to be satisfied in order to consider risk measure as coherent. They are:

• Monotonicity
• Sub-additivity
• Positive Homogeneity
• Translational invariance

Monotonicity is the condition which says higher is the expected return in the portfolio, lower is the risk. That means return is inversely proportional to risk.

Sub- additivity is the principle which says total assets of the portfolio cannot be more than sum of the risk of individual assets in the portfolio. It gives advantages of diversification benefits. If we have two assets in portfolio having particular risk, the risk of those assets will get diluted or diversified. So, the total risk of the assets will be less than sum of the individual risk of two assets. Positive homogeneity states that the size of the risk is directly proportional to the size of portfolio.

Translational invariance says that if we induce cash or sure capital in the portfolio then the risk associated with the portfolio decreases in proportion with the cash induce.

Generally, VaR is not the coherent risk measure since it is not sub-additive. VaR doesn’t give the benefit of diversification. VaR of a portfolio can be greater than the sum of VaRs of the individual stocks in the portfolio. This is because VaR is a quantile of profit and loss allocation rather than anticipation. Any financial institution uses the methodology of VaR for the measurement of risk without knowing the fact that it violates sub-additivity. Then, they may go for wrong investment strategy and fall in market risk. There is a special case when VaR becomes coherent risk measure. For normal districution, VaR becomes Coherent Risk measure since it becomes sub-additive. The value weighted VaR doesn’t cover sub additive property to be coherent. So it violates the coherernt risk measure. That’s why expected shortfall is considered as more prominent measure of calculating risk rather than VaR. Thus, many institutions have decided to use Expected Shortfall measure rather than VaR.

Conclusion

To summarize, VaR is the techniques to measure risk used by various financial institutions and corporate agencies. We have used different methodology for the calculations of VaR in excel spreadsheet. Historical simulation calculates risk on the basis of past series of returns and it doesn’t use any parameters. Weighted historical simulation measures risk on the basis of past returns but giving more weightage to recent observations since those observations can tell more about recent markets. Another more prominent method for risk management is Risk Metric Model. It is such statistical method which uses correlation and standard deviation of past returns to calculate today’s risk. GARCH (1, 1) uses three parameters “ω, α and β”, their value is calculated by maximum likelihood estimation in Excel.

VaR is not considering as coherent risk measure since it doesn’t satisfies the subadditivity condition. The risk measure for portfolio of two assets is greater than the sum of the risk measure of the individual aseets.

References

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