1. Statistics and Data Analysis Professor William Greene Stern School of Business Department of IOMS Department of Economics

2. Statistics and Data Analysis Part 11A – Lognormal Random Walks

3. Lognormal Random Walk The lognormal model remedies some of the shortcomings of the linear (normal) model. Somewhat more realistic. Equally controversial. Description follows for those interested.

4. Lognormal Variable If the log of a variable has a normal distribution, then the variable has a lognormal distribution. Mean =Exp[μ+σ 2 /2] > Median = Exp[μ] 30/46

5. Lognormality – Country Per Capita Gross Domestic Product Data 31/46

6. Lognormality – Earnings in a Large Cross Section 32/46

7. Lognormal Variable Exhibits Skewness The mean is to the right of the median. 33/46

8. Lognormal Distribution for Price Changes Math preliminaries: (Growth) If price is P 0 at time 0 and the price grows by 100Δ% from period 0 to period 1, then the price at period 1 is P 0 (1 + Δ). For example, P 0 =40; Δ = 0.04 (4% per period); P 1 = P 0 (1 + 0.04). (Price ratio) If P 1 = P 0 (1 + 0.04) then P 1 /P 0 = (1 + 0.04). (Math fact) For smallish Δ, log(1 + Δ) ≈ Δ Example, if Δ = 0.04, log(1 + 0.04) = 0.39221. 34/46

9. Collecting Math Facts 35/46

10. Building a Model 36/46

11. A Second Period 37/46

12. What Does It Imply? 38/46

13. Random Walk in Logs 39/46

14. Lognormal Model for Prices 40/46

15. Lognormal Random Walk 41/46

16. Application Suppose P 0 = 40, μ=0 and σ=0.02. What is the probabiity that P 25, the price of the stock after 25 days, will exceed 45? logP 25 has mean log40 + 25μ =log40 =3.6889 and standard deviation σ√25 = 5(.02)=.1. It will be at least approximately normally distributed. P[P 25 > 45] = P[logP 25 > log45] = P[logP 25 > 3.8066] P[logP 25 > 3.8066] = P[(logP 25 -3.6889)/0.1 > (3.8066-3.6889)/0.1)]= P[Z > 1.177] = P[Z < -1.177] = 0.119598 42/46

17. Prediction Interval We are 95% certain that logP 25 is in the interval logP 0 + μ 25 - 1.96σ 25 to logP 0 + μ 25 + 1.96σ 25. Continue to assume μ=0 so μ 25 = 25(0)=0 and σ=0.02 so σ 25 = 0.02(√25)=0.1 Then, the interval is 3.6889 -1.96(0.1) to 3.6889 + 1.96(0.1) or 3.4929 to 3.8849. This means that we are 95% confident that P 0 is in the range e 3.4929 = 32.88 and e 3.8849 = 48.66 43/46

18. Observations - 1 The lognormal model (lognormal random walk) predicts that the price will always take the form P T = P 0 e ΣΔ t This will always be positive, so this overcomes the problem of the first model we looked at. 44/46

19. Observations - 2 The lognormal model has a quirk of its own. Note that when we formed the prediction interval for P 25 based on P 0 = 40, the interval is [32.88,48.66] which has center at 40.77 > 40, even though μ = 0. It looks like free money. Why does this happen? A feature of the lognormal model is that E[P T ] = P 0 exp(μ T + ½σ T 2 ) which is greater than P 0 even if μ = 0. Philosophically, we can interpret this as the expected return to undertaking risk (compared to no risk – a risk “premium”). On the other hand, this is a model. It has virtues and flaws. This is one of the flaws. 45/46

20. Summary Normal distribution approximation to binomial Approximate with a normal with same mean and standard deviation Continuity correction Sums and central limit theorem Random walk model for stock prices Lognormal variables Alternative random walk model using logs 46/46