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returns = k- simple capital gains model
- buy at p1 sell at p2 after n years
- k = (p2/p1)^(1/n)-1
- capital gains plus cash dividends at end of period
- buy at p1 sell at p2 after n years
- collect div dollars after n years
- k = ((p2+div)/p1)^(1/n)-1
- capital gains plus cash dividends at end of period with margin
- buy at p1 sell at p2 after n years
- collect div dollars after n years
- invest mp1 dollars of your money and borrow (1-m)p1 dollars at i%
- k = ((p2+div-i(1-m)p1)/mp1)^(1/n)-1
risk = uncertainty = lack of information: operationalized as standard deviation- assume that uncertain future returns may be represented by a gaussian distribution
- gaussian distribution defined 2 parameters: mean (E(k) and standard deviation (s(k)
- computing mean and std dev using historical returns
- mean = E(k) = sum(ki)/n
- stddev = sqrt (sum((ki-mean)(ki-mean))/(n-1))
- computing mean and std dev using probability estimates of future returns
- given probabilities p1, p2, p3, .... for returns k1, k2, k3,.....
- with the contstraint that sum(pi)=1
- mean = E(k) = sum(piki)
- stddev = sqrt (sum(pi(ki-mean)(ki-mean)))
- using the gaussian parameters to estimate probabilities
- to compute the prob. of k.LT.x or k.GT.x compute z
- z = (x-mean)/s
- z measures distance of x from the mean in units of s
- compute probability using tables or jamal's equation
arithmetic and geometric averaging of returns- question: what are returns earned per period on average for investing activities over n periods in which the investor earns returns of k1, k2, k3, ....kn
- answer: it depends on the type of investing activity
- type 1: invest $1 today and re-invest the entire portfolio at the end of each period
- value of the portfolio at the end of n periods = (1+k1)*(1+k2)*(1+k3)*....*(1+kn)
- average k per period = ((1+k1)*(1+k2)*(1+k3)*....*(1+kn))^(1/n) - 1
- this is called geometric averaging
- type 2: invest $1 at the beginning of each period and cash out portfolio at the end of each period
- so on the second period you will re-invest $1 not $(1+k1). this is the difference between type 2 and type 1
- in this case your average returns will be sum(ki)/n
- this is called arithmetic averaging
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