Finance 2: More background

1. Finance 2: More background

2. Varying compounding periods Different investment vehicles have different compounding periods: 1. Bonds: annual or semi-annual 2. Mortgages: monthly 3. Banks: daily or continuous If compounding occurs m times per year: r is an annual rate T=total # of compounding periods For single cash flow w/ continuous compounding:

3. Random Variable Setting-Sometimes we gather data on a few input variables to see how they affect an output variable. In other cases, we just gather data on a single variable to see what the data looks like (ex. Test scores, major league home runs, etc). For this sort of data collection, we call the variable a random variable and assume there is some random process behind the generation of the data. The process is assumed to be able to create an infinite amount of data, and when we gather data, we gather only a sample of all of the possible data.

4. Standard Deviation, Variance Standard deviation () is a measure of dispersion of data. Variance = 2 = (sample v all data) Two different random variables could be independent of one another (the values of one of the variables has no bearing on the values of the other variable). - ex a is tossing a six-sided die, and b is pulling a card from a 52 card deck. Or two variables could have some degree of dependence (the values of one rv are related to the values of the other) - a could be GM stock price, and b could be Motorola stock price (common factor is the economy or general market movement)

5. Covariance Covariance - is the measure that quantifies the degree to which two variables vary together. XY = Idea: X and Y could be positively correlated (move together) which would give positive differences from the mean together or negative diffs together. In either case, there will be lots of positive addends together making a pos covariance. The stronger the correlation, the more often this will happen, and the more pos values to sum. If X and Y are negatively correlated? (lots of negs) If there is no correlation? (equal #’s of pos/neg over time)

6. Mean of sum of rv’s If we sum two random variables, what do you think the sum mean of the sum will be?

7. Variance of sum of rv’s If we sum two random variables, what do you think the variance of the sum will be?

8. Correlation coefficient Note that the denom will always be positive. The num will be pos if pos correlation or neg if neg corr |Num| ≤ |denom| (takes much work to show) So -1 ≤ ≤1 || is greatest when |num| is greatest which occurs when you have either lots of pos addends (move together) or lots of negative addends (move opposite). If 1,-1 perfect correlation, if 0, no correlation.

9. Summary So now: var(a+b) = var(a) + var(b) + 2cov(a,b) cov(a,b) = abab cov(a,b) = 0 if a and b are independent (0 correlation) Generalizing to sums of any number of rv’s:

10. Why do we need this stuff? The measure of the risk of investment is the variability of the returns (or the measure of dispersion of the returns). Our measure of dispersion is the standard deviation which is the square root of the variance. In many cases, we will have multiple investments we are making, so we want to know the variability of the portfolio (the sum of all of the investments). Thus the need to find variances of sums of random variables.

11. iid random variables If random variables are independent and have the same distribution with the same mean and same variance, then we call the rv’s independent and identically distributed (iid). If each rv (X1,…,Xn) is iid w/ variance 2, see if you can determine the variance of the sum.

12. Mean of a linear transformation If X has mean xbar, find mean(a+bX)

13. Variance of linear transformation If var(X)= 2, see if you can find var(a+bX).

14. Variance of an average See if you can determine the variance of an average of n iid random variables w/ variance 2. The interesting thing about this is that as n, var 0. This tells us that if we add enough iid rv’s, we eliminate the variance --- important info for investing (though we never get iid rv’s in practice there.)