FRM candidates: please make sure you understand this idea (See Hull p. 284 for more).

Assumptions:

I just pulled price history for Adobe Systems (ADBE) for 30 trading days ending May 9th, 2008.

• Adobe's stock price = $40 (rounded up)
• The sample daily volatility (sample standard deviation) is 1.9%. To be specific, 1.9% is the sample standard deviation (volatility) of daily periodic returns over a thirty day period.
• The annual expected return on the stock = +15%

Questions:

• What is the annualized volatility (assume trading days = 250)?
• We assume daily periodic returns are normally distributed. What is the consequently assumed distribution of future stock prices?
• If we go forward two years, what is the expected (arithmetic) mean stock price? (continuous compounding)
• If we go forward two years, what is the expected (geometric) median stock price? (continuous compounding)

 

(don't peek until you try)

 

 

 

 

 

 

Answers:

The EditGrid below contains the calculations and the ADBE source data.

• To annualize, we apply the square root rule (variance scales with time). Annualized volatility = (1.9%)[SQRT[250]) = 30%
• Price levels are lognormally distributed. If LN(tomorrow's price/today's price) is normally distributed, then [tomorrow's price] is lognormally distributed.
• Mean future stock price = ($40)EXP[(15%)(2)]. That's the mean of the lognormal distribution. But the lognormal is not symmetrical: the median does not equal the mean.
• Per Hull, our expected geometric return is reduced by one half the variance (volatility erodes returns!). The geometric return = 15% - (30%^2)/2 = 10.5%.
• Median future stock price = ($40)EXP[(10.5%)(2)] = $49.35. See how this median must always be less than the mean in the lognormal distribution?

 

EditGrid:

No Comments  |  Login to add your comment

Login or Signup to Discuss