Boudoukh MAE

[Pennywenny]

Hello

I do not understand the calculation of the what is abbreviated as MAE in the notes.

For example on page 9 of the notes, in line 5, does anybody understand where the 1.32% figure for the MAE comes from?

Many thanks!

 

[David Harper CFA FRM]

Hi @Pennywenny

Good point, it's not explained at all; in my defense, it appears the Boudoukh reading does not actually explain it. I have tagged to insert an explain in non-urgent revision.

MAE in Boudoukh is the same MAPE found in the next reading (R34) Hull's "Incorporating Volatility Updating into the Historical Simulation Method for Value at Risk:"

Boudoukh, Richardson, and Whitelaw (1998) propose a mean absolute percentage error measure (MAPE) for measuring bunching. This is calculated as follows. For each period of 100 consecutive days for which estimates are made, the absolute difference between the actual number of tail events and the expected number of tail events is calculated. (For 5% tail events the expected number of tail events is 5; for 1% tail events the expected number of tail events is 1.) The measure is set equal to the mean of these absolute differences. MAPE is a combined measure of both bias and bunching. The impact of a bias in the measurement of tail events is clear. If the procedure for measuring tail events is biased so that in every 100-day period we observe two 1%-tail events then MAPE = 1. To see how the bunching component of the measure works consider the following example.

 

[Success2014]

Hi @Pennywenny

Good point, it's not explained at all; in my defense, it appears the Boudoukh reading does not actually explain it. I have tagged to insert an explain in non-urgent revision.

MAE in Boudoukh is the same MAPE found in the next reading (R34) Hull's "Incorporating Volatility Updating into the Historical Simulation Method for Value at Risk:"

In regards to MAPE, I am confusing myself w/ how to count actual versus predicted. Example in GARP books with 599 observations numbered 1 to 599. This supposedly allows 500 overlapping 100-day samples. If every 100th observation (observations 100, 200, 300, 400, and 500) are 1% tail events then every 100-day sample will contain exactly 1% tail event and MAPE will be 0. However, suppose bunched so observations are 100, 101, 300, 301, and 500 are tail events. He finds 198with no tail events, 104 with 1 tail event and 198 w/ 2 tail events...can someone explain the math behind this...I seem to be double counting or other when reconciling the numbers...thanks so much.

 

[David Harper CFA FRM]

Hi @Success2014 I count:

• 1-100: 1 sample: 1 tails event (100)
  
• 2-101 ... 100-199: 99 samples: 2 tail events (100 and 101)
• 101-200: 1 sample: 1 tail event (100)
• 102-201 ... 200-299: 99 samples: 0 tail events
• 201-300: 1 sample: 1 tail event
  
• 202-301 ... 300-399: 99 samples: 2 tails events (300 and 301)
• 301-400: 1 sample: 1 tail event (301)
• 302-401 ... 400-499: 99 samples: 0 tail events
• 401-500 ... 500-599: 99 samples: 1 tail event (100)

Such that,

• 198 samples of 0 tail events: 102-201 ... 200-299 and 302-401 ... 400-499 = 99*2 samples
  
• 104 sample of 1 tail event: 1-100, 101-200, 201-300, 301-400, 401-500 ... 500-599 = 1+1+1+1+100 = 104 samples
• 198 samples of 2 tail events: 2-101 ... 100-199 and 202-301 ... 300-399 = 98*2 = 99*2 samples