Investment Risk Management

Value at Risk (VaR) as part of the market risk of investment products.

Part 2 of the series risks of investment products.

Introduction

The volatility that was considered in part 1 exhibits weaknesses when used as sole risk measure. One weakness of volatility is that it does not differentiate between up or downward price movements. In the case of a loss or a crisis, volatility does also not provide information about the extent of the loss. To examine the risk of an instrument or portfolio further measures are therefore often used. This blog posts examines one of these key figures, namely Value at Risk (VaR) in more detail. It must be noted that Value at Risk is not only used in the field of market risk assessment but also in the fields of credit and liquidity risks.

VaR as indicator of the height of loss

The (market) Value at Risk estimates how much an investment (a single financial instrument or a portfolio of assets) might lose (with a given probability) in a certain time period such as one day. VaR has established itself as a standard risk measure in financial industries. The confidence level a is typically 95% or 99%, meaning that with a probability of 95 (or 99) percent of the cases, the loss does not exceed the VaR level.

As an example, we want to recapitulate Google’s chart between 2015 and 2019 (as in part 1) and to calculate the historical VaR of Google based on these data. The log-returns of Google are found below.

In the following graph the log returns (corrected to the mean values) are presented in a quantile plot. For this purpose, the logarithmic returns are ordered in ascending order. The first point corresponds to the value where 1% of the log returns are smaller, the second point to the value where 2% of the log returns are smaller and so forth. The red line highlights the a=0.05 section (with a confidence interval of 5 % ).

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The value of the return at the 5% point is -0.0246553. Using this value, and after correcting for the mean value we can calculate the VaR of the Google share for the first trading day in January 2020. Thereto we multiply the last traded value of the Google share (31.12.2019) with the exponential of -0.023916 which corresponds to our confidence interval of 5%.

This means that, the 1-day VaR of the Google share on the 31.12.2019 was USD 31.65. With a probability of 95%, the daily loss would not have been larger. Caution: VaR cannot state the extent of a loss if the VaR value is exceeded. This can be calculated using the Expected Shortfall (ES), which we will introduce in another blog post.

Calculation methods VaR

Variance-co-variance approach

DFor the calculation of VaR the term variance-covariance approach is often use as a synonym for the delta-normal-approach and matches the VaR concept originally developed by J.P Morgan. One proceeds from the assumption of a normal distribution of the underlying stochastic factors (in our case a normal distribution of the log return). For log returns, the VaR value in the currency of the asset is calculated by:

For sufficiently small periods of time, the expectation value of the log return is approximated with 0. The VaR value for our Google example calculated using the delta-normal approach is USD 32.52.

This approach is limited to linear instruments and is not suited to calculate the VaR of instruments with non-linear payoff functions or portfolios that contain such instruments.

HistoricalVaR

Contrary to the two previously presented models, there is no parametric model underlying the historical simulation. The approach corresponds to the one shown in the example:

1. 1. Calculation of the historical deltas of the relevant quantity (e.g. log returns) with the desired time interval (1 day, 10 days, …)
   2. Sorting the historical deltas. As an option one can carry out an adjustment to the mean values. ().
   3. Calculation of the quantile of the discrete distribution function (c) which corresponds to the used confidence interval.
   4. Calculation of the VaR by using c on today’s value of the asset.

Monte Carlo VaR

For the Monte Carlo VaR, N random values are drawn out of a parametrical distribution. These can be (present case) but would not have to be a normal distribution with a mean value and a standard deviation of the historical log return. After the draw and to obtain the VaR, the steps are carried out analogously to the historical VaR calculation.

1. Simulation of the deltas of the relevant quantity (e.g. log returns) by drawing out of the parametrical distribution.
2. Sorting the simulated deltas.
3. Calculation of the quantile of the discrete distribution function (c) which corresponds to the used confidence interval.
4. Calculation of the VaR by using c on today’s value of the asset.

The value of the Monte Carlo VaR is USD 31.14.

Other time horizons for the VaR

In the example above, we considered the 1-day VaR. How can we scale the VaR or calculate it for other time steps?

Calculation of the VaR with other time differences

Instead of calculating the log-returns on a daily basis, they can also be calculated on a monthly or yearly basis. To obtain sufficiently many data points, a «Moving Window» process should be used. In doing so, the time window (1 month, 1 year, …) is always moved by one day. The subsequent calculation follows the 1-day VaR approach. This method, even though it is algorithmically more complex, is to be preferred over a scaling of the 1-day VaR.

Scaling of the VaR

To scale the VaR the 1-day VaR is multiplied with the factor of time . This factor is derived from a scaling of the variance in standard normal distribution. Unlike the variance-covariance approach, which is based on the assumption of a normal distribution, this is not necessarily the case with a historical or Monte Carlo VaR.  in this case one must be careful when using the simple scaling method.

Conclusion

As beforementioned, VaR is an established risk assessment tool within financial industries that enables one to make a statement about the probability of a loss. However, it is necessary to view this risk assessment tool critically as VaR also exhibits certain weaknesses:

• Missing subadditivity: VaR is not a coherent risk measure (it is possible that the sum of the VaR values of sub-portfolios is smaller than the VaR of the whole portfolio) and the diversification effects are not always duly considered.
• Missing significance of the time series: to determine the distribution parameters or for the historical simulation, a time series is used. If this time series contains no crisis, the VaR might be severely underestimated in the case of an actual crisis. This can be dealt with by calculating the VaR using extreme value theory or the peak over threshold method.
• Missing information about amount of loss: VaR does not provide any information about the loss amount in cases where the VaR value is exceeded.

Literature: Wiley Finance: Financial Risk Manager Handbook, Phillip Jorion, GARP

Next Part: Product risk classification of financial products.