The following question is a slightly reworded version of a real question in a real financial management company's risk questionnaire that was provided to someone locally. I've tried to be fair to the financial company while making their question a bit less vague, they actually had two similar questions which cover this issue in slightly different ways.

"You have the choice of placing your investment in one the following 5 portfolios, ranging from low to high risk. For each portfolio, you can assume the return over each consecutive year (edit: was 5 years) takes one of two possible values, with 50% probability of each outcome. Which portfolio would you prefer for your investment?

A: 50% chance of either +11% or -7%

B: 50% chance of either +17% or -10%

C: 50% chance of either +25% or -16%

D: 50% chance of either +37% or -27%

E: 50% chance of either +58% or -48%"

## 15 comments:

I'd choose C as the risk slope is lower for A/B/C, while C/D/E have the higher risk slope, likelihood is a coin flip throughout (0.5). The regression lines (A/B/C and D/E/F, respectively) are almost R^2=1.

Note to self: I'm a very poor gambler.

A/B/C line intercepts C/D/E line almost exactly at C. I don't know if that is the original intention. Drawing an A/E line places C furthest from the linear A/E line.

Well I made a small math error (like I actually know what I am doing, which I don't), but I'm sticking with C (although B now looks just slightly better).

>"I've tried to be fair to the financial company"

Everyone that uses any financial advisor in the UK for investment purposes has to take a risk questionnaire. Yes, it is repetitive asking the same questions in slightly different ways. I am not sure how much this is purely to check for consistency of answers or whether asking in different ways can legitimate provoke different answers.

Think there is probably a good chance different advisors have the same questionnaire and it wouldn't surprise me if they all were identical. So it is probably more a reflection of the FCA (regulator) than the firm.

Your numbers and the risk involved seem higher than I remember and probably less vague as you indicated.

Anyway, you have to be a risk seeker to prefer E to D. D is the marginally risk averse choice. Answers are clearly in order with A the most risk averse option. A 50% chance of loss seems rather high for the most risk averse category, the most risk averse person might prefer a certain 2% return.

Anyway, C seems a reasonable choice, not missing out on too much of the potential return while not sending the risk to really high levels. If acting as trustee looking after someone else money then A or B might be more sensible (defensible?) choice.

How many time you can toss the coin before you need to take the money out also plays a part but there are lots of other questions in the questionnaire covering that.

Not really sure why you are asking, it doesn't seem all that dumb for a risk questionnaire that is supposed to be checking/extracting what sort of preferences the client has between risk and return. Maybe it is as simple as wanting to compare your preferences to other people.

Ok, a hint.

What will the outcomes be after say 10 years of plan E? (ie 2 consecutive 5 year periods as above)

After 50 years?

The first you can do by direct calculation, the latter may be easiest with some random sampling.

You didn't say how many consecutive 5 years you intended at first. If there's only one period, then the expected return rises from 2% for A to 5% for E, but at the cost of increased risk. The risk-averse choice is A. None of the choices seem appealling, I'd rather invest my money myself thanks :-)

I didn't think about iterating it, because I kinda hoped that it would be "the same". And... it is. Your expected return for case E is just (1.05)^n. Your max gain becomes large and you max loss is most of your money, though.

Yes that's true in expectation. What about the distribution?

My assumption is that the question must be considered to be time-invariant, there is no logical reason to have a different answer in 2019 vs 2024. I'm thinking of this essentially from a mathematical rather than real-life perspective.

Ah, interesting. The distribution (which I did experimentally for case E, but which I presume is the same but for scaling for other cases; does case A... hmmm, appears to have the same peak as E, I wasn't expecting that) is sparse and bunched off towards the origin. I bet it's got a name that I once knew.

>"Yes that's true in expectation. What about the distribution?"

Yes, it is huge. With odd number of periods looks like 50% chance of loss. Even numbers more variable but your examples give 75% chance and 62.3% chances of loss. But so what? I said "you have to be a risk seeker to prefer E to D". A risk seeker is a thill seeker who likes the wild distribution. It doesn't look like a skewed distribution problem.

>"no logical reason to have a different answer in 2019 vs 2024"

Whether you want your money out in 2024 or 2029 or 2034 or whether you might change your plans about when you want the money out, may well play a role in how you choose to invest. Other questions in the questionnaire deal with that. You are just considering repeating the period to get to longer periods, it may not quite work like that but it doesn't seem an unreasonable first approximation and trying to do better is going to add complexity to the question and make it more difficult to easily give an answer.

>I bet it's got a name that I once knew.

Binomial?

Yes binomial!

A 1D Ransom Walk even (likelihoods all p=0.5). For large n (steps) the walk itself is Gaussian and via the LLN/CLT (I'm prettey sure it is the LLN more so than the CLT) it is exactly Gaussian.

I almost have the code to do this now (need to track not just the walk but also the percentages for each heads or tails step in a separate co-array).

I think the bottom line is this (asymptotically) ...

A: 50% chance of either +11% or -7% (1.11,0.93, 1.11*0.93=1.0323, mean compounding)

B: 50% chance of either +17% or -10% (1.17*0.9=1.053)

C: 50% chance of either +25% or -16% (1.25*0.84=1.05)

D: 50% chance of either +37% or -27% (1.37*0.73=1.0001)

E: 50% chance of either +58% or -48%" (1.58*0.48=0.9216)

B has the highest mean compounding rate of 1.053 (corresponding to a random walk ending where it starts (at zero). There are only odd or even steps (odd contains zero even contains either -1 or 1 (the steps closest to zero, which for large n is still Gaussian)).

None of the five are a double or nothing propositions, so the remaining amounts can never truly go to zero (within the limits of machine precision).

I would agree with WMC though, if I'm correct about the above, as 1.053 is only about 1%/year, it would be worth it if I were Philip J. Fry though (who paid for the 1000 year electricity bill though).

5 years was silly (I made that up myself to give a concrete time frame). I have now changed it to 1 year in the post.

Everett is just about there. Using logarithms helps (IMO). Will expand in longer post (but you can add comments in the meantime).

Well, that explains why maths grads can make so much money playing with the Black-Scholes equation in the city. Those who can't, give financial advice.

Multiple reinvestments are a random walk in log-space, since returns are multiplicative. So typical returns are (roughly) the logarithmic mean of the investment. But exp(0.5*[log(1+0.58) + log(1-0.48)]) ~ 0.9, so typical losses are 10% per reinvestment for strategy E. The probability distribution in log space is a Gaussian with a mean that decreases 10% with each reinvestment. So most investors will lose about 10% per year as they reinvest in E.

Mean returns (i.e. expected returns) are just the usual arithmetic mean, so 'on average' investors gain 5% per reinvestment for strategy E.

Basically the reason this happens is that the distribution of returns (not log returns) is extremely skewed, so most investors lose money, but a few lucky ones (in the tail) win big-time.

Still better than a casino (since expected returns still positive) and people seem to like those...

We have a winner :-) Guessing as to Unknown's identity, I'm not surprised....

JA,

If at all possible, teach me further father, in your follow on post.

I think I have it coded but need a bit more time and/or help (I say time because I will have to modify two pieces of code, one is done (hopefully), the other takes the binary output from the 1st to generate the statistics (I thought it would run as is, but not so lucky for now).

TIA :)

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