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If I have an opportunity of investment that costs I in year 0 and gives me CF_1 in year 1, I will accept it only if NPV>0

NPV = -I + (CF_1)/(1+k)

Now in order to discount the cash flows I have to choose k, the discount rate. k will be the interest rate of an investment with the same risk. Of course I don't have to choose a random investment, but the best investment at the same risk, that is, the investment with the highest return but the same risk. This alternative investment is then at the efficient frontier. But how can the investment I started from have a higher return then this alternative investment to begin with, given that this alternative investment is at the efficient frontier?

Stated otherwise, If I engage in the investment I started from, in year 0 I will pay I, and after waiting one year, I will put CF_1 in my pockets.

If instead I engage in the best alternative investment, after 1 year I will obtain I(1+k). Of course the NPV condition is at all equal to the condition CF_1 > I(1+k). But if the investment which gives me k is at the efficient frontier, how can this last equation be satisfied at all?

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    So basically you're asking "if the efficient frontier represents the best investment at every risk level, how can there be a better investment at a given risk level?" – D Stanley Oct 22 '19 at 20:49
  • Why do you think that k is the same as the best investment with the same risk? Answer: Because if it was greater then someone else would've already bought this investment before you did. So you should never invest, because someone else bought one first. However, you have the opportunity to use a different evaluation of k or CF_1 than everyone else. The discount rate is whatever people collectively accept, but maybe your preference is higher or lower. – user253751 Oct 23 '19 at 14:26
  • @user253751 does it means that basically whenever NPV>0 you have discovered a new point in the efficient frontier, better (=have a higher return) than the previous point at the same risk? – robertspierre Oct 23 '19 at 15:28
  • @rtrtrt That, or your "goodness function" (?) is different from someone else's. They don't think it's better, but you do. – user253751 Oct 24 '19 at 8:33
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You have not defined risk.

For cAPM, only systematic risk, or beta, is rewarded. If one investment has a high idiosyncratic risk and low beta while another has the same standard deviation, low idiosyncratic risk, and high beta; CAPM would assign a higher expected return to the latter despite having equal standard deviations.

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