Returns
Theoretical Returns
A finite probability space consists of a state space and a probability measure . The state sapce is a nonempty finite set, and the probability measure is a function that assigns to each element a number in so that .
A random variable is a real-valued function defined on . The expectation of is defined as
If we know there are different scenarios and corresponding returns of an asset in each of them, then we can use this formulat to calculate the expected returns.
Moments of returns
- Expected returns
- Variance of returns
- Standard deviation of returns
- (Standardized) Skewness of returns
- (Standardized) Kurtosis of returns
- Excess kurtosis (compared to Normal Distribution) = kurtosis - 3
Empirical Returns
- Price (simple) returns
- price log returns
- Total (simple) returns
- total log returns
- Dividend yield
- Cumulative total returns
- Cumulative price returns
Average Performances
Arithmetic average
Geometric average
Risk
Sample variance of returns
Standard deviation of returns (volatility)
(Standardized) Sample skewness of returns
(Standardized) Sample kurtosis of returns
Example:
we observe monthly returns , for the annual returns
the average return is
Assume the daily returns’ standard deviation is , then the annually returns’ standard deviation is about (assuming daily returns are i.i.d.)
Pricing
Dividend Discount Model
Since the gross return on a stock is
thus the price of a stock is
if the expected returns are constant, , then
when , we have
Assume the dividend growth rate is constant such that
therefore (Gordon Growt Model)
According to this model, the dividend yield is
The divident yield can reflect both discount rate and growth rate news.
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