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If I have an opportunity of investment, let's call it investment (A), that costs $I$ in year 0 and gives me $CF_1$ in year 1, I will accept it only if $NPV>0$

$NPV = -I + \dfrac{CF_1}{1+k} > 0$

Now in order to discount the cash flows I have to choose $k$, the discount rate. $k$ will be the interest rate of an alternative investment with the same risk. Of course I don't choose a random investment as the alternative investment, but the best investment at the same risk, that is, the investment with the highest return but with the same risk.

This alternative investment, call it investment (B), is then at the efficient frontier. But how can investment (A) have a higher return then this alternative investment to begin with, given that investment (B) is at the efficient frontier?

Stated otherwise, If I choose investment (A), in year $0$ I will pay $I$, and after waiting one year, I will put $CF_1$ in my pockets.

If I choose investment (A), in year $0$ I will pay $I$, and after waiting one year, I will put $I(1+k)$ in my pockets.

I then choose (A) over (B) if the amount I receive after one year is greater in (A) than in (B), that is, if

$CF_1 > I(1+k)$

which is equivalent to the NPV condition.

But if investment (B) which return $k$ is at the efficient frontier, how can this last equation be satisfied at all? The best that (A) can do is give the same return as (B). So there should be no investments at all with $NPV>0$.

Or do I discount the cash flows with the rate of return of an alternative investment that is not at the efficient frontier? But in so doing I am ignoring an opportunity investment. I could invest the sum $I$ at that investment which is at the efficient frontier, and I would ignore that.

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    $\begingroup$ you're mixing frameworks. NPV calculations are commonly done with respect to cost of capital, not alternative investments (ie, if a given investment produces a net positive return, invest). comparisons (ie, A v. B) are made by evaluating NPV/IRR of each, and then comparing. MV/efficient frontier is a different paradigm. $\endgroup$ – Chris Oct 22 '19 at 21:57
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The assumption that the discount rate should be derived from the IRR of an alternative investment is not correct.

Commonly the WACC of the company (or the WACC of the funds needed for the investment if it is standalone) is used. If this is not available, you could make use of a combination of publicly available rates and some risk-adjustments: risk-free interest rate + inflation + risk adjustment.

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  • $\begingroup$ BUt the WACC of the company is the weighted average between the cost of capital and the cost of debt. The cost of capital can in turn be calculated by the CAPM, which takes into account the risk-free rate and the risk-premium: $k=r_f + \text{risk-premium}$. This is again the return that an investor can gain in the market at the same risk of the company, since the investor could replicate buying a stock in the company by buying risk-free bonds and the market index $\endgroup$ – raffaem Oct 23 '19 at 15:09
  • $\begingroup$ The EMH applies to public markets in securities. An entrepreneur can come up with a business idea with positive NPV, when valued at the WACC of his investors. Otherwise no one would ever start new companies, and no business would start a new product, people would only invest in existing activities. Finding an NPV >0 is not a market inefficiency, it is just a business opportunity. $\endgroup$ – Alex C Oct 24 '19 at 18:25
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Note that CF1 is a weighted average of possible future outcomes, about which it is possible for different investors to have different beliefs and risk preferences.

NPV = -I + sum(p(i) * CFi1) /(1+k) across i possible outcomes

If your beliefs about B give you confidence it's on the efficient frontier, then there is indeed no reason to buy A. B is already the best one can get for its risk profile. A might be as good as B. In which case, you would be indifferent between two different optimal alternatives. But if NPV(A)>NPV(B), then B cannot indeed be on the efficient frontier.

You buy A if your beliefs about A or B are different to someone else's, ie the market consensus. Whether or not, you, I or anyone else believes that the market is weak/semi-strong/strong efficient or not, all of us would recognise (and hopefully accept!) that belief in efficiency is far from universal across the market! Different investors will hold different views about both A and B. There would be no market to trade if they didn't!

Suppose for argument's sake that you preferred B and I preferred A, on account of differing views on the likelihood of different CF1 scenarios. The market is still efficient if neither of us has a persistent or structural advantage over each other. If we do this again and again, and we come out evens, neither of us has been able to buck the market. The market remains efficient, because neither of us have been able to generate or discern better prices ourselves.

Plus it is possible to have two alternative assets with the same E(CF1), the same level of aggregate risk, but different distributions of risk. Consider a choice between a fair 80:20 versus a 20:80 bet. Which one is "riskier"? Clearly, neither. There's a simple trade-off between the probability and magnitude of profit and loss. Some people might prefer the high chance, small win; others to tolerate the high chance of the small loss to avoid the chance of the big loss. Different people will have a different "k" here, which is a critical concept eg in insurance.

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