Vault-Prob/Stat

Question 1

You flip a coin 10 times. What is the probability of only three heads occuring? What is the probability of three heads occuring in a row?

Answer 1

• If you flip a coin 10 times, you will have total of 2^10 combinations. That is 1024
• Proability of 3 heads occuring. Order does not matter. Hence this is a combination problem
• (10 3) = 10! / (3! * 7!) = (10*9*8) / (3*2*1) = 120
• Answer 1 = 120/1024 = 11%
• Question 2: Here the order matters. Assume there are only three heads occur, no more.
  
  • A bit tricky:
  • Possible scenarios: HHHTTT…T, THHHTT..T
  • There are total of 8 such scenarios
  • Answer 2 = 8/1024

Question 2

What is linear regression?

Answer 2

Linear regression is the technique of estimating the unobserved population response y with the “Best” “Linear” “Unbiased” “Estimator” - BLUE

Question 3

A regression of a time series of monthly American Airline returns (RAA)on the variables fuel prices (FP) and average passenger revenue/mile (PR) results in the following equation:
RAA = 0.056 - 0.0036 FP + 0.0118 PR
where there are 120 data points, the standard error of the intercept is 0.035, the standard error of the coefficient of fuel price is 0.001 and the standard error of the passenger revenue/mile is 0.005. Which variables, if any, are significant?

Answer 3

Since n is reasonably large, we can use the rule of thumb that the computed t-value should be compared to tcritical = 2.0. For the intercept, the computed t-value is 0.056/.035=1.6 so the intercept is not significantly different from zero. For the coefficient of fuel price, the t-value is |- .0036|/0.001 = 3.6 so this coefficient is significant; the coefficient of passenger revenue/mile is also significant because 0.0118/0.005 = 2.36 > 2.

Question 4

If you boss claims that using more frequent data, such as daily data, will improve estimates for annual returns and standard deviations? What would I say to this?

Answer 4

• Improving the frequency of data does not improve precision
• I think the boss is implying that by looking at more data we will be increasing our estimate of the true, unobserved population mean, and the error [r - E(r)] will decrease
  
  • This is not true
• However, we would improve our estimate of standard error.

Question 5

A manager claims that he is been “beating the market” each year over the past ten years by 2%, assume that standard deviation of the market during this period has been 20%. What do you think of this claim?

Answer 5

• To evaluate if someone has truly beat the market, let’s evaluate the t-statistic
• Variable of interest is “Excess return”
  
  • Rc = 2%
  • In order for the “excess return” to be statistically significant, the excess return must be greater or equal to 2 times the standard deviation:
  • E(Rf - Rm) >= 2*std*(R_f - R_m)
    
    • = [2*std*(Rf - Rm) / sqrt(T)]

Or. t-stat = IR* Sqrt (y)

T-stat = Excess Return / [std / sqrt(n)]

Here, if we pick a t-stat of 2, that is about 95% confidence, there is 2.5% probability that the outperformance was due to good luck and true alpha of the manager is zero.