Analysis of financial data Anders Lundquist Spring 2010

Dependence and correlation, cont. Do the closing price for today depend (linearly) on yesterday’s closing price? r(x t,x t-1 )=0.989 Seems like a rather strong linear relationship

Scatterplot

Returns As we have noted previously, the closing price of a stock may not look very stationary, and it might be difficult to model it by some single probability distribution.

Time plot

Nordea, mean over time Constant?

Nordea, variance over time Constant?

Are Nordea closing prices normally distributed?

Returns If we instead of the prices themselves consider the change in price from one time point to another The price change takes the current level of the price as a starting point. Let x t denote the closing price at time t.

Returns Simple gross returnSimple net return Relative price change

Returns Simple returns are scale free. ”…for an investor a return is a scale free summary of the investment opportunity.” (Campbell et.al. 1997) One reason for analyzing returns.

Returns The other reason: returns often have more desirable statistical properties… Usually, the logarithm of the simple gross return is used. Called ”log returns”, ”continously compounded returns” (or just ”returns”).

Log returns Let y t denote the log return.

Time plot

Histogram

Numerical summaries Mean Minimum Q1 Median Q3 Maximum -0, , , , , ,14910 StDev Range IQR 0, , ,03732 Skewness Kurtosis 0,55 2,09

What about stationarity?

Stationarity, cont

Comment: The mean has a faster convergence rate when stationarity holds. It may also be the case that the variance is not constant – we will consider ARCH-models later in the course to take this into account.

Normality?

Correlation between log returns. Do the log return for today depend (linearly) on yesterday’s log return? r(y t,y t-1 )= Seems like a weak (almost non-existing) linear relationship

Scatterplot

What about two days ago?

Log returns Log returns appear fairly stationary They do not seem to depend on past observations Maybe the log returns are purely random?

Random log returns Implication: there is no better way to predict the log return than by simply using the expected log return. We estimate this with the average log return in our sample.

Random walks This is one kind of random walk model. More sophisticated random walk models are standard tools in e.g. option pricing (Black- Scholes) In that setting it is called geometric Brownian motion, and some fancy mathematics is used…

Efficient markets? Some economists claim that if the market is efficient, all price changes should be random! Price changes are only due to unanticipated events, which may be considered random in some sense.

Beating the market? Purely by chance, som investors will do better than others. Let’s assume that we have an efficient market where all stocks are equally likely to increase or decrease in price.

Only me… I choose one stock randomly. Then, P(stock price increases for five consecutive days) = (½) 5 = 1/32 Quite a small chance…

Many investors If instead 100 people choose one (different) stock randomly. Then, P(the stock price increases for five consecutive days for at least one person) = 1-P(no stock price increases for five consecutive days) = 1-(1-(1/32)) 100 =

Who to follow? Almost certainly, at least one person will buy a randomly selected stock that increases for five consecutive days, thus beating the market. The trick is to know who will be the lucky one…