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

Statistics and Data Analysis Part 11A – Lognormal Random Walks

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.

30/46 Lognormal Variable If the log of a variable has a normal distribution, then the variable has a lognormal distribution. Mean =Exp[μ+σ2/2] > Median = Exp[μ]

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

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

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

Lognormal Distribution for Price Changes 34/46 Lognormal Distribution for Price Changes Math preliminaries: (Growth) If price is P0 at time 0 and the price grows by 100Δ% from period 0 to period 1, then the price at period 1 is P0(1 + Δ). For example, P0=40; Δ = 0.04 (4% per period); P1 = P0(1 + 0.04). (Price ratio) If P1 = P0(1 + 0.04) then P1/P0 = (1 + 0.04). (Math fact) For smallish Δ, log(1 + Δ) ≈ Δ Example, if Δ = 0.04, log(1 + 0.04) = 0.39221.

35/46 Collecting Math Facts

36/46 Building a Model

37/46 A Second Period

38/46 What Does It Imply?

39/46 Random Walk in Logs

Lognormal Model for Prices 40/46 Lognormal Model for Prices

41/46 Lognormal Random Walk

42/46 Application Suppose P0 = 40, μ=0 and σ=0.02. What is the probabiity that P25, the price of the stock after 25 days, will exceed 45? logP25 has mean log40 + 25μ =log40 =3.6889 and standard deviation σ√25 = 5(.02)=.1. It will be at least approximately normally distributed. P[P25 > 45] = P[logP25 > log45] = P[logP25 > 3.8066] P[logP25 > 3.8066] = P[(logP25-3.6889)/0.1 > (3.8066-3.6889)/0.1)]= P[Z > 1.177] = P[Z < -1.177] = 0.119598

43/46 Prediction Interval We are 95% certain that logP25 is in the interval logP0 + μ25 - 1.96σ25 to logP0 + μ25 + 1.96σ25. Continue to assume μ=0 so μ25 = 25(0)=0 and σ=0.02 so σ25 = 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 P0 is in the range e3.4929 = 32.88 and e3.8849 = 48.66

44/46 Observations - 1 The lognormal model (lognormal random walk) predicts that the price will always take the form PT = P0eΣΔt This will always be positive, so this overcomes the problem of the first model we looked at.

45/46 Observations - 2 The lognormal model has a quirk of its own. Note that when we formed the prediction interval for P25 based on P0 = 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[PT] = P0exp(μT + ½σT2) which is greater than P0 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.

Summary Normal distribution approximation to binomial 46/46 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