PRM Exam 2 – Question Bank No.26

[Kelvin]

Hi,

Can someone explain the following question in regards to square root of time rule?
The correct answer is I & III. How would the realized volatility will be less than using the time rule. What is the intuitive thinking behind this? Thank you very much.

Kelvin

Which of the following statements are true?
I. The square-root-of-time rule for scaling volatility over time assumes returns on different days are independent
II. If daily returns are positively correlated, realized volatility will be less than that calculated using the square-root-of time rule
III. If daily returns are negatively correlated, realized volatility will be less than that calculated using the square-root-of-time rule
IV. If stock prices are said to follow a random walk, it means daily returns are independent of each other and have an expected value of zero

[Yemi Popoola]

Here is the answer to your question regarding- Realised volatility less than using the time rule:

1) Time rule assumes that volatility on different days are independent i.e. the change in stock price from yesterday does not affect the change in stock price today.

2) Time rule assumes identical distribution i.e. the mean and variance of the stock price remains the same on all days.

If the independent rule is violated (and it is always violated e.g. if there is a sharp rise in stock price yesterday, there might be another sharp rise today as investors try to cash in), then the realised volatility changes.

Now with negative correlation – this means that two days have a relationship where rise in volatility in one is more or less followed by fall in volatility in another i.e. The combined variance (and hence volatility) of the two will be less than the an constant volatility assumed under the Time rule (see assumption 2 above). Now I have explained it in the most basic terms and I hope it helps. Using the formula for combined variance i.e. (V1^2 + V2^2 +- 2COV12) should provide analytical insight. When the two are negatively correlated, the minus sign is used and the lowers the overall variance and hence the overall volatility. I hope this helps. Cheers.

[Kelvin]

Hi Yemi,

That totally makes sense to me. It basically means the one day increase offsets by the other day decrease(negatively correlated), hence the overall volatility is less than time rule volatility.

Thanks so much for your response.!

Kelvin

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