Calculate value at risk (VaR) for assets and portfolios with the new finvalrisk command. Determine the risk of investments using their historical performance, fitting a normal distribution to their returns, or specifying complex time-series models. This feature is a part of StataNow™.

How much we stand to lose from an investment is just as interesting a question as how much we stand to win. For risk-averse investors, potential losses may even carry more weight than potential profits in their decision making.

VaR is an intuitive approach to thinking about potential losses. It typically looks at low-probability, worst-case scenarios over a period of interest and answers the question: how much value could the investment lose?

The answer can be as simple as “the value we have observed this investment lose in the past under such worst-case scenarios” or as complex as fitting a statistical model to the data and forecasting the potential loss.

Stata's new finvalrisk command can estimate VaR percentiles from the historical return distribution, from a normal approximation to this distribution, and from a time-series model of the expected returns.

Let's see it work

Gambling borrowed cash: How bad could it be?

Every day, inexperienced investors find what they think is a great opportunity in the stock market and fall for the oldest temptation: they go to the bank and borrow money to buy stocks. After all, their cousin who is good at math told them that the company is a winner! What could possibly go wrong?

Think about their situation for a moment. They bought a risky asset with money that they do not actually have. If the asset grows in value, they are in luck: they sell, pay the bank back, and keep the difference. If the asset stays at its current value, they can at least sell it at the original price and pay back the full amount. But if the asset loses value, they will sell it at a lower price than they paid for it, which means that they will be paying the bank for that loss out of their own pocket.

Gamblers like these will be content to see any gains, even small ones, but they really do not want to see losses. They may not even have the cash to cover them. A VaR analysis can reveal how much trouble they may be getting themselves into.

Betting it all on AFH

We have end-of-month stock prices for 25 fictional companies.  We list the first five prices for five of the stocks:

. webuse finex
(Fictional stock price data)

. list aaa-das in 1/5


     
      aaa        afh        ard        cph        das 
  1. 
 10.00786    10.0088   10.00906   10.00645   10.00926 
  2. 
 9.967269   10.03054   10.30584   10.15102   10.31771 
  3. 
 9.889415   10.24619    10.2917   10.03707   10.30525 
  4. 
 10.02855    10.1483   10.55885    10.2759   10.35527 
  5. 
 10.02861   10.49291   10.18976    10.1349   10.31676 

The stock price of Apex Film Holdings (AFH) has been growing steadily over the past year.

. tsline afh in -12/l

But our newbie investor heard that AFH is about to acquire a very popular movie franchise. News of that big acquisition should push AFH's stock price up quickly over a short period of time, so the investor takes out a loan of $10,000 from the bank and buys AFH stock.

VaR analysis is carried out on returns data rather than price data, so let's compute the returns of AFH to get started.

. quietly finreturns afh, simple(r_afh) multiply(100)

These are simple returns multiplied by 100, so an example value of 1.5 means that the stock price grew by 1.5% in a month.

Historical returns and normal approximation

We first look at this asset's past performance.

. summarize r_afh, detail


                             AFH
      Percentiles      Smallest
 1%    -5.519961      -10.59864
 5%    -3.629369      -7.559971
10%    -2.248412      -7.178934       Obs                 779
25%    -.7257008      -6.951355       Sum of wgt.         779

50%     .7337448                      Mean           .6601578
                        Largest       Std. dev.      2.400168
75%     2.220452       7.148532
90%     3.613799       7.453087       Variance       5.760808
95%     4.442117       8.208542       Skewness      -.2865115
99%     6.612686       8.662755       Kurtosis       4.045699

AFH has had an average monthly return of 0.66%, which is the reported mean of this return series. We also see that AFH loses 5.52% or more of its value in a month 1% of the time (the 1st percentile), loses 0.73% or more of its value in a month 25% of the time (the 25th percentile), and gains 6.61% or more of its value in a month 1% of the time (the 99th percentile).

. histogram r_afh, title("Histogram of AFH returns") xline(-5.52 -0.72 6.61) 
     text(.19 -5 "1%") text(.19 -0.06 "25%") text(.19 7.3 "99%")
(bin=27, start=-10.59864, width=.71338499)

We say that the 25% VaR is 0.73%. The investor has a 25% probability of losing 0.0073 x $10,000 = $73 or more within a month. Similarly, the 1% VaR is 5.52%, so the investor has a 1% chance of losing 0.0552 x $10,000 = $552.

By default, the finvalrisk command computes these simple risk projections based on the empirical distribution of returns and computes a return based on a normal approximation to the return distribution. We specify the VaR percentiles of interest in the percentiles() option:

. finvalrisk r_afh, percentiles(1 25)

Value-at-risk percentiles

Number of obs = 779
Sample: 1955m2 thru 2019m12


             
 
 Empirical     Normal
 
 
 
1%           
 
                     
         AFH 
 
 -5.519961  -4.923469
 
 
 
25%          
 
                     
         AFH 
 
 -.7257008  -.9587312

An ARMA model for expected returns

Maybe our investor has some knowledge of time-series modeling and decides to base the estimation of VaR percentiles on a model of the returns rather than on the raw historical series. If the model is well specified, it will recover an expected return series clean of noise, that is, clean of fortuitous months where the return was unusually low or high because of events that do not represent the typical behavior of AFH returns.

Stata's lag-order selection tool for Autoregressive moving average (ARMA) models suggests that an ARMA(0,1) could be appropriate based on having the minimum BIC and HQIC:

. arimasoc r_afh
Fitting models (9): ......... done

Lag-order selection criteria

Sample: 1955m2 thru 2019m12                      Number of obs = 779


       Model 
 
        LL         df        AIC        BIC       HQIC
 
 
 
   ARMA(0,0) 
 
 -1786.898          2   3577.795   3587.111   3581.378
   ARMA(0,1) 
 
 -1782.691          3   3571.383   3585.357   3576.758
   ARMA(0,2) 
 
 -1782.362          4   3572.724   3591.356   3579.891
   ARMA(1,0) 
 
 -1782.944          3   3571.888   3585.863   3577.263
   ARMA(1,1) 
 
 -1782.017          4   3572.033   3590.665     3579.2
   ARMA(1,2) 
 
 -1780.594          5   3571.188   3594.478   3580.146
   ARMA(2,0) 
 
 -1782.619          4   3573.238    3591.87   3580.405
   ARMA(2,1) 
 
 -1780.679          5   3571.358   3594.648   3580.316
   ARMA(2,2) 
 
 -1780.435          6    3572.87   3600.818   3583.619

Selected (max) LL:   ARMA(2,2)
Selected (min) AIC:  ARMA(1,2)
Selected (min) BIC:  ARMA(0,1)
Selected (min) HQIC: ARMA(0,1)

And when we fit this model with the arima command, it indeed looks like a good fit:

. arima r_afh, ma(1) nolog

ARIMA regression

Sample: 1955m2 thru 2019m12                     Number of obs     =        779
                                                Wald chi2(1)      =      11.13
Log likelihood = -1782.691                      Prob > chi2       =     0.0008



             
 
                 OPG                                            
       r_afh 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]
 
 
 
r_afh        
 
                                                                
       _cons 
 
   .6601114   .0960543     6.87   0.000     .4718484    .8483743
 
 
 
ARMA         
 
                                                                
          ma 
 
                                                                
         L1. 
 
   .1074348   .0321978     3.34   0.001     .0443283    .1705413

      /sigma 
 
   2.385659   .0508852    46.88   0.000     2.285926    2.485392

Note: The test of the variance against zero is one sided, and the two-sided
      confidence interval is truncated at zero.

To obtain VaR percentile estimates based on this model, we type

. finvalrisk r_afh, percentiles(1 25) model(arima, ma(1))

Fitting ARIMA models ...

Value-at-risk percentiles

Model: ARIMA
Error family: Normal

Number of obs = 779
Sample: 1955m2 thru 2019m12


             
 
                            ARIMA      ARIMA
             
 
 Empirical     Normal  historical   one step
 
 
 
1%           
 
                                            
         AFH 
 
 -5.519961  -4.923469   -4.889843  -4.849744
 
 
 
25%          
 
                                            
         AFH 
 
 -.7257008  -.9587312   -.9490443  -.9089738

Note: One-step-ahead value at risk for 2020m1.

The ARIMA historical column in this output gives estimated percentiles for the sample. We can compare these directly with the empirical and normal percentiles we had seen before. The ARIMA-based 1% VaR of 4.89% is slightly better news than the empirical- and normal-based estimates. The estimated 25% VaR of 0.95% is also smaller than the empirical- and normal-based results.

The ARIMA one step column provides a forecast for the risk we can expect next month, that is, the month after the end of the return series. Our forecast says that the investor has a 1% probability of losing 4.85% or more in January 2020 and a 25% probability of losing 0.91% or more.

Modeling volatility

The residuals from estimating expected returns could also be interesting. In particular, the behavior of their variance over time can tell us whether we are in a period of turmoil (high volatility) or calm (low volatility) and how long these periods tend to last.

With the model(ewma) option, we can estimate VaR percentiles based on a simple model for the variance of AFH returns.

. finvalrisk r_afh, percentiles(1 25) model(ewma 0.95)

Fitting EWMA models ...

Value-at-risk percentiles

Model: EWMA
Error family: Normal
Shape parameter = 0.95

Number of obs = 779
Sample: 1955m2 thru 2019m12


             
 
                             EWMA       EWMA
             
 
 Empirical     Normal  historical   one step
 
 
 
1%           
 
                                            
         AFH 
 
 -5.519961  -4.923469    -5.03399  -4.000597
 
 
 
25%          
 
                                            
         AFH 
 
 -.7257008  -.9587312   -.9536346  -.6540177

Note: One-step-ahead value at risk for 2020m1.

EWMA stands for exponentially weighted moving-average model. The 0.95 in our specification is a parameter for the model we have fit to the variance.

The losses we now forecast for next month at both the 1% and the 25% probabilities are lower than our ARMA-based forecast. This could indicate that our ARMA forecast is biased towards pessimism because we are in a period of rather high volatility. The EWMA-based forecast accounts for the current volatility.

Modeling both expectation and variance of returns

We have plenty of data to estimate a more complex specification, so why not model both the expected returns and their volatility simultaneously? We can do this with autoregressive conditional heteroskedasticity (ARCH) models:

. arch r_afh, ma(1) arch(1) garch(1) nolog

ARCH family regression -- MA disturbances

Sample: 1955m2 thru 2019m12                     Number of obs     =        779
                                                Wald chi2(1)      =       6.25
Log likelihood = -1770.892                      Prob > chi2       =     0.0124



             
 
                 OPG                                            
       r_afh 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]
 
 
 
r_afh        
 
                                                                
       _cons 
 
   .7420881   .0901726     8.23   0.000     .5653531     .918823
 
 
 
ARMA         
 
                                                                
          ma 
 
                                                                
         L1. 
 
   .1020971   .0408531     2.50   0.012     .0220266    .1821676
 
 
 
ARCH         
 
                                                                
        arch 
 
                                                                
         L1. 
 
   .0985945   .0280868     3.51   0.000     .0435453    .1536436
             
 
                                                                
       garch 
 
                                                                
         L1. 
 
   .7707499   .0741664    10.39   0.000     .6253863    .9161135
             
 
                                                                
       _cons 
 
   .7516883   .3087694     2.43   0.015     .1465114    1.356865

We can ask for this or any other specification of Stata's arch command within finvalrisk:

. finvalrisk r_afh, percentiles(1 25) model(arch, ma(1) arch(1) garch(1))

Fitting ARCH models ...

Value-at-risk percentiles

Model: ARCH
Error family: Normal

Number of obs = 779
Sample: 1955m2 thru 2019m12



             
 
                             ARCH       ARCH
             
 
 Empirical     Normal  historical   one step
 
 
 
1%           
 
                                            
         AFH 
 
 -5.519961  -4.923469   -4.780788  -4.223384
 
 
 
25%          
 
                                            
         AFH 
 
 -.7257008  -.9587312   -.8646021  -.6755917

Note: One-step-ahead value at risk for 2020m1.

This ARCH-based, one-step forecast is more pessimistic than the EWMA forecast but more optimistic than the ARMA-based forecast above.

Looking at all of these results, the gambler concludes that there is at least a 25% probability of losing money the following month. But they also know that the monthly loss is unlikely to be large under typical circumstances, so perhaps they will be willing to place the bet, hoping the tip they heard about AFH comes true.