Finance Notes

This is a brief collection of some basic terms, names and methods related to finance.

Stock returns

Return specifications

Stock market returns could be described, by simply calculating the ratio between the price difference over a time period and the current price, i.e. \begin{align} R_t = \frac{p_{t+1}-p_t}{p_t}. \end{align} These returns don't add up over time, though, as e.g. a $5\%$ return for two consecutive days leads to a total return that is larger than $10\%$. Logarithmic returns in contrast, which are given by \begin{align} R^{\log}_t=\log(R_t+1)=\log(p_{t+1})-\log(p_t), \end{align} are additive in this sense and often more convenient to use in the analysis of stock returns.

Properties of returns

There are several properties that can usually be observed in stock market returns:

• (Volatility Clustering) Large (in absolute terms) stock market returns cluster in time. In mathematical terms, this means that the (time-varying) variance of a return process exhibits an autoregressive structure. Hence, one of the most famous means of modeling stock market returns are GARCH models.
• (Heavy tails) Even after accounting for the heteroscedasticity of the return process, its distribution most often still exhibits tails that are heavier than those of a normal distribution. This corresponds to a kurtosis that is larger than $3$.
• (Asymmetric distribution) One often observes that the distribution of returns is negatively skewed, resulting in a longer left tail. This means that extreme negative returns are more common than very large positive returns of the same size.
• (Leverage effect) Stock returns usually are negatively correlated with (future) volatility, i.e. negative returns lead to a higher volatility, while positive ones tend to decrease it.
• (Long-memory) The effect of shocks on the volatility of stock returns is commonly very persistent – e.g. more persistent than a GARCH model would suggest, which models the impact of shocks as exponentially decreasing over time.

Stock market indices

Stock market indices are (weighted) averages of a set of stocks, which are supposed to illustrate the overall performance of their market segment. Different types of stock indices can be distinguished:

• (Total return vs. net total return vs. price index) As the names suggest, price indices are solemnly based on the (weighted) stock prices, while total return indices also account for the financial returns associated with holding a share (dividends). The latter assumes that dividends received by the shareholders are fully reinvested, which is why total return indices are higher than their corresponding price index counterparts. As dividends are subject to taxation, there is a third type, which assumes that dividends are reinvested, but also subtracts a tax fee – "net total return" indices.
  Most of the commonly quoted indices are price indices (e.g. Dow Jones, S&P500, FTSE) with the notable exception of the German stock index DAX, which is a total return index.
• (Price- vs. capitalization-weighted indices) Furthermore, indices differ w.r.t. the weighting method used. While price-weighted indices (e.g. Dow Jones) only take into account the stock prices, capitalization-weighted indices (e.g. S&P500) weigh the different stocks proportionally to their market capitalization. Some are additionally weighted, s.t. only the public float capitalization is considered.