Question

In: Finance

Do you think share returns in reality are normally distributed or not? Give your reasons

Do you think share returns in reality are normally distributed or not? Give your reasons

Solutions

Expert Solution

Yes. Share returns in reality are normally distributed.

The reality is that the assumtion of a single volatility is not used. Simple derivatives are valued using their prices from the market, or some interpolation/extrapoaltion, and more complex derivatives are priced using PDEs/MonteCarlo, such that the market implied expected distribution of the underlying (including tails as fat as you like) is followed.

Most standard models assume stock returns are normally distributed even though everyone agrees that real-world returns have fat tails. We've all heard stories of hedge funds that went bankrupt cause something happened that their model called a "10 sigma event that should only happen once every billion years" and that's obviously a flawed model. My textbooks point this out but hand-wave it away like it's an unavoidable simplification.

Since 1950, the average annual return of the S&P 500 has been approximately 8% and the standard deviation of that return has been 12%. I want to look at monthly returns so let’s translate these to monthly

Monthly Expected Return = 8%/12 = 0.66%
Monthly Standard Deviation = 12%/(12^0.5) = 3.50%

More evidence of that is how the actual distribution of monthly S&P 500 returns is skinnier in its center than the normal distribution. The skinny middle and the fat tails imply that the normal distribution might not be the best describer of stock returns. Rather, there seem to be 2 regimes — a calm regime where we spend most of the time that is normally distributed (but with a lower volatility than 12%) and a regime with high volatility and terrible returns

It looks approximately normal but if we look to the left of the distribution, we can see the famous fat tails.The fat tails mean that extreme events occur more frequently in reality than what a normal distribution would predict.


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