Parametric VaR - Single Exposure

Parametric Value at Risk (VaR) - Single Exposure

Introduction

VaR is a statistical measure that quantifies the risk in terms of monetary value for a stated portfolio over a specified time period. There are several types of VaR such as credit risk VaR, liquidity risk VaR, operational risk VaR and market risk VaR. Our focus for this and subsequent articles will be on market risk VaR.

There are several methodologies which can be used to measure (market risk) VaR.

  1. Parametric or variance covariance
  2. This is the method used for our discussion and primarily uses standard deviation, variance and covariance of market prices to estimate VaR.
  3. Historical
  4. Historical returns are used to fully revalue the portfolio and VaR at risk will be calculated based on a confidence interval.
  5. Monte Carlo
  6. VaR are estimated from simulating the market returns from historical standard deviations and correlation with full revaluation made on the porfolio. This is by far the slowest method of the three.

The Basic

User may download their own data via csv file. The only field name required is 'close'. Other fields in the following table - 'date', 'open', 'high' and 'low' are only for visual. 'return' column is calculated by the system.

Table 1: Apple stock historical data

Let's consider if we are holding a stock at current price of 100. What is our value at risk or in layman terms what the amount we expect to lose? For a single exposure we just need the volatility or standard deviation of the stock.

Our sample data is Apple stock from June 17, 2019 to June 17, 2020. For simplification we ignore dividend payment, stock splits, bonus issues etc. With the calculated volatility or standard deviation, the risk exposure is simply current market price of the stock (351.59 on June 17, 2020) multiplied by the daily volatility of 2.5393% which is equivalent to 8.93. If we want to add certain confidence to the number, say 95% confidence, we need to find the z-score. NORMSINV() is a standard function available for most spreadsheet such as Excel, Google Sheet and Numbers. NORMSINV(0.95) gives a value of 1.645 and we can recalculate:

\( \begin{align} \text{risk value} &= P\sigma1.645 \\ &=351.59\times 0.025393\times 1.645 \\ &=14.69 \\ \text{where }P &= \text{ Last Apple's stock price } \\ \sigma&=\text{ stock daily log return volatility or standard deviation} \end{align} \)

What does the number means? Before interpreting the number, it is best to note that we are assuming normal distribution of the log returns. Once that is spelled out, there are several ways the risk value can be interpreted:

  1. there is 95% chance or probability that we wont lose more than 14.69 on the next day.
  2. out of 100 working days, there will be 95 days in which the loss wont exceed the risk value.
  3. there is 5% chance or probability that we will lose more than the risk value
  4. out of 100 working days, there will be 5 days in which the loss will exceed the risk value

What if we interested in the risk value for holding Apple's stock for 10 days? How do we calculate and is there any adjustment to the formulation?. There are two ways to do it. The first method was explained in 'How To Calculate Volatility' , where we simply adjust the volatility by \(\sqrt{10}\). Hence;

\( \begin{align} \text{risk value} &= P\sigma\sqrt{10}1.645 \\ &=351.59\text{ x }0.025393\text{ x } 3.1623 \text{ x } 1.645 \\ &=46.44 \\ \end{align} \)

Now the interpretation of the number becomes:

  1. there is 95% chance or probability that we wont lose more than 46.44 for holding the stock for 10 days.
  2. out of 100 working days, there will be 95 days in which the loss wont exceed the 46.44.
  3. there is 5% chance or probability that we will lose more than the 46.44
  4. out of 100 working days, there will be 5 days in which the loss will exceed the 46.44

Another way to recalculate the risk value is by recalculating the volatility by recalculating the returns of the last 10 days. If we denote the last closing price as \(P_t\), the previous closing price as \(P_{t-1}\), then we are now comparing \(P_t\) against \(P_{t-10}\). The log returns, \(r_t\) = \(ln\frac{P_t}{P_{t-10}}\), \(r_{t-1}\) = \(ln\frac{P_{t-1}}{P_{t-11}}\) and so on.

Based on our sample data, the 10-day volatility is 6.1179%, a lower figure than the above method (1-day vol x \(\sqrt{10}\) yields 8.0300%). The difference is expected since the calculated 10-day volatility is an empirical study while the other is a statistical method. There is another simplication/assumption made but, we will leave it to reader to figure it out. Feel free to email support@faseries.com for the answer. The simplication is a necessity to avoid tedious mathematics during the process of calculating value at risk, at least for teaching/training purposes. Bear in mind however that in deriving value at risk, they are many more assumptions made and we will highlight them as we go along.

In our next part, we will look into calculating the risk of a portfolio with two assets/stocks and introduce correlations, variances and covariances.