1.2: Statistics Overview Flashcards
What type of stocks tend to perform best over very long investment horizons (historically)?
Small stocks, but they also have periods of significant losses
What do investors demand to bear a give level of risk?
A risk premium (in terms of a higher expected return)
What does a probability distribution do?
Assign a probability (pR) that each possible return R will occur
What variance and standard deviation does a risk-free return have?
Zero
What is the variance of a return?
A measure of how spread out the distribution of the return is
What is a popular estimation approach when we don’t know the probability distribution?
To extrapolate from historical data. By counting the number of times realised return falls within a particular range, we can estimate the underlying probability distribution.
What does the mean return capture?
How much we make on average
What probability distribution do monthly stock returns follow?
Approximately a normal distribution
What is the realised return?
The return that actually occurs over a particular time period
Does the average return provide an estimate of the expected return?
Yes, it the probability distribution of returns is the same over time
What is the standard error?
The estimation error of a statistical estimate; the standard deviation of the average return
What are the necessary assumptions for how we calculate the standard error?
- The distribution of returns is identical each year
- Each year’s return is independent from prior years’ returns
What is the excess return and what does it measure?
The difference between the average return for an investment and the average return for treasury bills (risk-free investment).
Measures the average risk premium investors earn for bearing the risk of the investment
Is there a relationship historically between risk and return?
- For large portfolios, yes! Higher risk is rewarded with higher average return as a compensation
- For individual stocks, no.
What is an observed relationship between size and risk?
Larger stocks have lower volatility overall.
What are the two types of risk?
- Common risk: perfectly correlated and affects all securities
- Independent risk: uncorrelated and affects a particular security
What is diversification?
The averaging out of independent risks in a large portfolio
What does the risk of a portfolio depend on?
Whether the individual risks within it are common or independent. Independent risks are diversified in a large portfolio, while common risks are not
What is firm-specific risk?
Only affects the company itself = independent risk. Also called idiosyncratic, unique, or diversifiable risk.
What is systematic risk?
Affects the entire market = common risk. Also called undiversifiable, or market risk.
How does volatility depend on the size of the portfolio?
When risk is diversified, the volatility declines with the size of the portfolio, until only the systematic risk remains
What is the risk premium for diversifiable risk?
0.
Investors are not compensated for holding firm-specific risk, since they can “diversify it away”
Why do investors require a risk premium for holding systematic risk?
Because they are risk averse and otherwise would be better off selling their stocks and investing in risk-free bonds
What is the risk premium determined by?
The systematic risk of the security
What are portfolio weights?
The fraction of the total investment in the portfolio held in each individual investment in the portfolio
What is the total value of all the portfolio weights?
They always add up to 1. Individual weights can be negative (shorting or borrowing).
What determines how much risk is eliminated in a portfolio?
- The degree to which the stocks face common risks
- The degree to which their prices move together
What numbers is the correlation always between?
-1 and 1
What is the variance of a portfolio?
The sum of the covariances of the returns of all pairs of stocks in the portfolio multiplied by each of their portfolio weights
What can be said about the volatility of buying shares on margin (borrowing) compared to the volatility of the stock?
Your positions volatility will be higher than the volatility of the stock
What is the formula for the standard error of the estimated expected return?
SE = SD(individual risk) / sqrt(number of observations)
What is the formula for the 95% confidence interval for the expected return?
Historical average return +- (2*SE)
What is the formula for the estimated covariance between the returns of two stocks?
Cov(Ri, Rj) = (1 / (n-1)) * (SUMOF (Ri -meanRi)*(Rj - meanRj)
What is the formula for the correlation between the returns of two stock returns?
Corr(Ri, Rj) = Cov(Ri, Rj) / (SD(Ri) * SD(Rj) )
How can we calculate the portfolio variance for a large portfolio?
Using the box method.
Each box is calculated by:
Weight(i) * Weight(j) * Cov(Ri, Rj)
Then, all boxes are summed up to get the portfolio variance
