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Question

  • There is an investor who is afraid of losing any of his seed money (initial investment).
  • Variance of investment returns is not a problem to him. He is willing to take variance as long as he does not lose his seed money.
  • The investor is afraid of losing any of his initial investment, even a little. So even if he earns a lot at the end of investment period, if he has to go through losing any of his initial investment (seed money) during the investment, he would not like this investment plan.
  • Which risk measure would be appropriate for this type of investor?

Example

  • Risk measures including variance, therefore, are not appropriate yardsticks for this type of investors. So I excluded variance, downside variance and sharpe ratios. ( I know sharpe ratio is a risk measure per se.)
  • Maximum Drawdown seems to work, but whether seed money is being lost or not does not take into its account. As such, I am not sure if MDD is approrpriate.
  • In the same sense, VaR and CVaR do not take into account the seed money
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  • $\begingroup$ I think you need some additional input here: if you are not willing to lose your initial investment, I understand that you do are comfy with any return above zero. Any return below zero is not acceptable to the investor. I think I got you wrong, though. $\endgroup$ Nov 7 '20 at 7:52
  • $\begingroup$ Which additional information would be needed? $\endgroup$
    – Eiffelbear
    Nov 7 '20 at 8:08
  • $\begingroup$ Probably I do not fully understand the description, yet. If they are afraid of loosing their investment, any desirable return distribution must be strictly positive.else, they might be looking at the probability of a loss in excess of 0, no? So are they afraid of loosing “all” their initial investment or “some”? $\endgroup$ Nov 7 '20 at 8:18
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    $\begingroup$ When you find an investment strategy with zero loss of losing my initial investment with unlimited potential upside, please let me know. $\endgroup$
    – will
    Nov 7 '20 at 9:50
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    $\begingroup$ Any trade you put on, in the moments after the trade, has an approximately 50% chance of being underwater. Probably the chances of a trade staying above 0 for every moment of the first day are very near to zero. So I think this investor just needs to hold cash in her domestic currency (maybe a risk free bank deposit if such a thing exists). $\endgroup$
    – StackG
    Nov 7 '20 at 9:57
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As many have suggested in the comments, it might be hard, if not impossible, to find an investment that gives positive returns with certainty. However, you might consider a metric such as

$$R=\underset{s\in [0,T]}{\min}r_{0,s}$$

where $r_{0,s}$ is the portfolio return between the initial investment point $0$ and $s$. $R$ gives the highest share of seed money lost between $0$ and $T$. Then you could try to find an investment that with high probability has a good $R$.

You can affect $R$ through dynamic trading. As in this paper, the optimal strategy seems to increase risk after positive returns. So you would start investing in the risk-free asset and gradually increase risk after that. This way it might in theory be possible to guarantee positive returns for initial investment, though not sure how well this works in practice.

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  • $\begingroup$ This is a good answer. Though you would probably have to use a very conservative quantile eg 0.01% otherwise the minimum among all possible returns will probably always eat into the initial investment. $\endgroup$ Nov 8 '20 at 9:04
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Since you yourself said the investor is willing to take variance/risk as long as she doesn't lose her seed money, then her objective function is not entirely a function of any risk measure describing some dispersion from a central tendency, neither the lower tail distribution. Instead, her objective function is a function of her cumulative returns being below $1$ at any point in time.

A risk measure is pointless for controlling such preferences and searching for one is totally missing the point (although you yourself did already rule out every possible risk measure out there so why keep asking?). Formulating an objective function with cumulative returns is pointless as well since cumulative returns are non-stationary and multi-modal.

All the investor can use is a perpetual stop-loss limit order that signals divestiture when portfolio price falls below the initial investment ("seed") price.

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  • $\begingroup$ Formulating an objective function with cumulative returns is pointless as well since cumulative returns are non-stationary and multi-modal. Neither point is correct. First, people derive utility form wealth, not returns, and the initial investment + cumulative returns do yield wealth. Second, stationarity or multimodality of the distribution of the argument of a utility function have nothing to do with the validity or sensibility of the utility function. $\endgroup$ Dec 19 '20 at 18:33
  • $\begingroup$ I never said anything about the stationarity and modality of a utility function. It is utility functions that are pointless since asset allocation can be carried out without making them the objective function $\endgroup$
    – develarist
    Dec 19 '20 at 22:55
  • $\begingroup$ Who said you did? While it is possible to work with asset allocation without an explicit utility function, it would usually yield nonsense results unless the implicit utility function is a sensible one. In any case, the sentence I cited above is incorrect. Nonstationarity and multimodality is not a problem for the objective of asset allocation (while it may or may not be a technical challenge in the implementation). $\endgroup$ Dec 20 '20 at 11:18
  • $\begingroup$ so you're saying that results obtained, in the presence of non-stationarity and multimodality, are qualified since these two issues are only a numerical obstacle only? $\endgroup$
    – develarist
    Dec 20 '20 at 13:12
  • $\begingroup$ What I am actually saying in my previous comments is that nonstationarity and multimodality are not a problem in the situation that your answer discusses. No, it is not a numerical problem at all. It is a modeling problem in the sense that an appropriate model needs to be built and an appropriate estimator chosen to account for nonstationarity; multimodality is a non issue in any case. $\endgroup$ Dec 20 '20 at 13:55

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