Investments, tenth edition

   Value at Risk 

 The     value  at  risk    (denoted  VaR  to distinguish it from  Var,  the abbreviation for vari-

ance) is the loss corresponding to a very low percentile of the entire return distribution, 

for example, the 5th or 1st percentile return. VaR is actually written into regulation 

of banks and closely watched by risk managers. It is another name for  quantile  of a 

distribution. The quantile,  q,  of a distribution is the value below which lies  q % of the 

possible values. Thus the median is  q   5  50th quantile. Practitioners commonly estimate 

the 5% VaR, meaning that 95% of returns will exceed the VaR, and 5% will be worse. 

Therefore, the 5% VaR may be viewed as the best rate of return out of the 5%  worst-

case  future scenarios. 

 When portfolio returns are normally distributed, the VaR is determined by the mean and 

SD of the distribution. Recalling that  2 1.65 is the 5th percentile of the standard normal 

distribution (with mean  5  0 and SD  5  1), the VaR for a normal distribution is   

VaR(.05, normal) 5 Mean 2 1.65 SD  

 To obtain a sample estimate of VaR, we sort the observations from high to low. The VaR 

is the return at the 5th percentile of the sample distribution. Almost always, 5% of the 

 number of observations will not be an integer, and so we must interpolate. Suppose a 

sample comprises 84 annual returns so that 5% of the number of observations is 4.2. We 

must interpolate between the fourth and fifth observation from the bottom. Suppose the 

bottom five returns are   

225.03% 2 25.69% 2 33.49% 2 41.03% 2 45.64%  

 The VaR is therefore between  2 25.03% and  2 25.69% and would be calculated as   

VaR 5 225.69 1 0.2(25.69 2 25.03) 5 225.56%   

 Estimate the skew and kurtosis of the five rates 

in  Spreadsheet 5.2 . 

 CONCEPT CHECK 

5.7 

bod61671_ch05_117-167.indd   139

bod61671_ch05_117-167.indd   139

6/18/13   8:03 PM

6/18/13   8:03 PM

Final PDF to printer

---

140 

P A R T   I I

  Portfolio Theory and Practice