Value-at-Risk
Introduction
Risk assessment is essential in portfolio management. A portfolio manager has the duty of maximising returns of the portfolio of the assets for each level of risk. Alternatively, a portfolio manager should minimise the risk of the portfolio at any level of return of the portfolio. Several models are used in assessing the risk involved in a portfolio of investments. Value-at-risk is widely applied in determining portfolio risk. We explore the concept of value-at-risk and its usage in portfolio management.
It is a model that measures how much the value of a portfolio could fall over a specified period with a given probability or confidence level due to changes in market prices or interest rates. For instance, if the period under consideration is one month and the confidence level is 90%, the value-at-risk is the fall in the value of a portfolio that could happen with a 90% confidence level in the next trading month. A 90% confidence interval implies that losses more than the value-at-risk should occur less than 10% of the time. It is also referred to as the possible loss in the value of a portfolio from the normal market risk and not all kinds of risk.
Historical perspective of Value-at-risk model
The model was derived from the concept of Harry Markowitz’s portfolio theory. Although the model was developed a long time ago, it was not widely used until the mid-1990s. The model is mostly used by investment commercial banks to determine the potential loss in the values of the portfolios they trade as a result of unfavourable movements in the market over a specified period. Its application resulted from the several financial crises that rocked the financial services firms as well as the responses to those financial crises by the regulatory authorities. In response to the Great Depression which caused bank failures, the Securities and Exchange Commission was formed after the enactment of the Securities Exchange Commission Act. The commission then issued the first capital requirements for banks, in which banks were to ensure that their borrowings do not go beyond 2000% of their total equity capital. The introduction of derivative financial instruments and floating exchange rates increased risk thereby prompting the SEC to revise capital requirements.
In 1980, the commission introduced new requirements which tied the capital requirements of firms in the financial services sector to the potential losses that would occur with a 95% confidence level over a one month interval. Besides, the mid-1990s saw an increase in the volume of portfolios traded by investment and commercial banks. The volatility of these portfolios also increased in the mid-1990s. The above factors led to the growth in the application of value-at-risk model by investment and commercial banks as well as portfolio managers.
Description of the model
The definition of value-at-risk model highlights tow important components of the model. The two are the confidence level at which the market risk is evaluated and the period of time over which the risk is measured. The period used in the model is the holding period and should be short since the model assumes that the portfolio’s composition remains constant during the holding period. If a longer period is used with the model, the results will most likely be inaccurate since the composition of portfolios change over time. Therefore, a one-day holding period is more preferable than other longer holding periods. The model uses confidence level to express the risk in terms of percentiles. A value-at-risk measured at 98% is the potential amount of loss the portfolio is expected to surpass only 2% of the time. The loss obtained is the 98th percentile of the distribution of all potential losses on the portfolio.
Measurement of Value-at-risk
There are several approaches to measuring value-at-risk. Most of the valuation approaches use past data to determine the potential changes in the portfolio’s value in future periods. They are based on the assumption that past trends reflect future and potential movements in the value of portfolios. Three main approaches can be used to determine the value-at-risk. Each of the approaches is described below.
Variance-Covariance Method
Under this approach, historical volatility data and the correlation coefficient are used to determine the value-at-risk. The approach assumes normality, absence nonlinear non-linear positions and serial independence (Choudhry and Wong). The assumption of normality implies that all percentiles/confidence levels are multiples of known standard deviation. This implies that the standardized returns of the portfolio values follow a normal distribution. Portfolio returns themselves may not be normally distributed, but the standardized returns are. Standardised returns are determined by dividing the actual returns by the forecasted standard deviation (Choudhry and Wong). The determination of standardised returns shows that the approach is not focused on the actual size of returns but the size of the returns relative to the forecasted standard deviation.
The value-at-risk is determined by determining the standard deviation of the values of the portfolio over a given period. Serial independence implies that changes in the value of the portfolio in one day does not affect the expected changes in another day (Malz). Therefore, a portfolio manager can use the daily standard deviation to determine the standard deviation for a horizon using the formula below.
Standard deviation for the horizon = daily standard deviation ×√Number of days in the horizon
In determining the portfolio’s standard deviation, two approaches can be used; equally weighted moving average and exponentially weighted moving average methods. The equally weighted moving average method uses a fixed amount of historical to calculate the variance and the standard deviation of the portfolio. For instance, a portfolio manager can use the standard deviation of the value of the portfolio for the last 30 days if he/she believes that the most recent data gives a reasonable estimate of the portfolio’s standard deviation. The approach relies on recent data which may be a more accurate forecast of short-term volatility. However, using recent data reduces the sample size thereby increasing the sampling error in the determination of standard deviation and the value-at-risk. Besides, yesterday’s data may not accurately reflect today’s movements in portfolio values.
On the other hand, the exponentially weighted moving average method uses different standard deviations for the values of the portfolio in different periods (Rogers). It then assigns different weights to different periods with the most recent period assigned greater weights. This is because the approach believes that the most recent observations give a better reflection of the future variances than the past observations. Therefore, the weights assigned reduce exponentially.
After determining the mean portfolio value, the standard and correlation coefficient, the normal distribution is used to determine the value-at-risk.
Procedure of using variance-covariance method
The first step is to ensure that each cash flow is exposed to only one type of market variable. Therefore, the different types of risks are separated from each asset based on market variables. For instance, the return on a bond will be divided into principal and interest payments and further sub-divided by market factors such as interest and exchange rates. The cells with similar characteristics are then grouped together (Rogers). The third step involves determining the value at risk for each cell. Finally, the values at risk for all the cells are used to compute the portfolio value t risk using the variance-covariance matrix.
Illustration
A portfolio has two stocks A and B. Stock A has a total market value of $1,000 while B has a market value of $1,500. One day historical volatility (standard deviation) for the two stocks are 1% and 2% respectively. Historical data indicates that the correlation coefficient between the two is 1.23. Determine the value-at-risk of each asset and the portfolio at 99% over a ten-day holding period.
Solution
Value at risk = X × z × σ × √n
The Z score for 99% confidence level is 2.33
Value at risk, Stock A = 1,000 × 2.33 × 1% × √10 = $73.68
Value at risk, Stock B = 1,500 × 2.33 × 2% × √10 = $221.04
Portfolio value-at-risk = (RA2+ RB2+2CRA RB)
Where RA is the value-at-risk for stock A, RB is the value-at-risk for stock B and C is the correlation coefficient between the two stocks.
Portfolio value-at-risk = (73.682+ 221.042+(2 ×1.23 ×73.68 ×221.04))
= 94,351.54
= $307.17
Limitations of variance-covariance method
The approach assumes that the standardized returns are normally distributed. This is not always the case in reality. In most cases, the portfolio values and returns have outliers over a given period (Rogers). When there are outliers in the actual returns, the standardized returns will not be distributed normally hence the estimate of the value-at-risk will not be accurate. Besides, it is difficult and complex to compute the value at risk with asymmetric distributions.
Besides, there is always a standard error in the estimate of variances and covariance since the estimation is based on historical data. Firstly, only a sample of historical data is used in the determination of portfolio returns and variances (Alexander). This leads to sampling error hence the estimate of variances, standard deviation and covariance may not be accurate. Also, historical data may not reflect the changes that will occur in future. Inaccurate estimates of variances and covariance will lead to an inaccurate estimation of the value-at-risk even if the standardized returns obey the normality assumption.
The model is also limited by the non-stationary nature of variances and covariences of stocks in a portfolio over time. These variables change with changes in macroeconomic variables such as the exchange rate, interest rates, among other variables. Furthermore, the model only works where there is a linear relationship between portfolio positions and risk. If the portfolio includes options, the model may not apply since options do not have linear payoffs.
Mitigating the above challenges
The accuracy of the, model in estimating value at risk can be improved by refining sampling methods. This improves the accuracy of the estimated variance thereby enhancing the precision of the estimate for value-at-risk (Alexander). Besides, when variances change over time, estimates that allow the standard deviation to change over time such as the Autoregressive Conditional Heteroskedasticity can be used to enhance the accuracy of VaR. Finally, the model can be modified for use in non-linear situations such as portfolios that include options. The quadratic value-at-risk is suitable for such situations.
Historical simulation
This approach uses historical data to create a time series of returns of the portfolio and calculating the changes that might have occurred in each period. In this case, the series of price changes for every risk factor is obtained. The price changes are then used to generate a time series of the changes in the values of the portfolio. The series obtained is then sorted into percentiles. The value at risk is then given by the change in portfolio value corresponding to the desired confidence level.
Illustration
The above table shows the values of a portfolio in ten different periods. To assess the value at risk using historical simulation, the change in portfolio value is computed for each period. The profits/losses can be ranked in ascending orders follows:
The approach is simple since it does not require the determination of the correlation coefficients. However, it requires a large number of historical data points to ensure the accuracy of the model.
Limitations of historical simulation
The approach is more reliant on historical data than any other approach. This is a limitation since historical data is not always an accurate reflection of the future. The approach does not incorporate any assumptions on the distributions of returns. Besides, it gives equal weights to all data points hence, it is unrealistic (Alexander). This implies that when determining the VaR for 2017, the approach gives equal weights to value changes in 1990 and those in 2016. This is unrealistic since the most recent changes are more likely to predict future changes than changes that took place long ago. It is almost impossible to estimate new risks and assets using the approach since there is no historical data.
The accuracy of the historical simulation approach can be improved by assigning greater weights to recent past changes in portfolio values. Besides, the historical simulation should be combined with time series models which give a more accurate valuation of value at risk. Furthermore, volatility should be updated whenever there are changes. Sometimes historical volatility is far much lower than the current volatility hence, historical simulation without modifications give a lower value at risk than the actual value.
Monte Carlo Simulation Approach
Under this approach, a portfolio manager chooses a distribution that best describes the changes in portfolio values (Alexander). Random numbers are then generated based on the chosen distribution. The random numbers and the distribution are used to compute hypothetical profits or losses. The VaR is then determined as the loss at the required percentile or confidence level.
Conclusion
Value-at-risk model has become essential for portfolio managers especially commercial and investment banks. It helps in assessing portfolio risk by quantifying the amount of possible loss above the normal loss at a required confidence level. The accuracy of the estimate of value-at-risk depends on the data available and the approach of measurement applied. As shown above, Variance-Covariance and Monte Carlo simulation methods are effective in determining the value-at-risk. Historical simulation is likely to give an inaccurate estimate unless it is modified.
Works cited
Alexander, Carol. Market Risk Analysis. Chichester, England: John Wiley, 2008. Print.
Choudhry, Moorad and Max Wong. An Introduction To Value-At-Risk. Chichester, West
Sussex: Wiley, 2013. Print.
Malz, Allan M. Financial Risk Management. Hoboken, N.J.: Wiley, 2011. Print.
Rogers, Jamie. Strategy, Value And Risk: A Guide To Advanced Financial Management.
London: Palgrave Macmillan UK, 2013. Print.
