Monday 4 September 2017

Smart Portfolios: A post about a book, NN Taleb, and two conferences

September 18th is the official publishing date of my second book, "Smart Portfolios: A practical guide to building and maintaining intelligent investment portfolios (Harriman House, 2017)".

This blog post will give you some more information about the book, and more importantly help you decide if it's worth buying. I'll also let you know about a couple of forthcoming conferences where I will be talking about some of the key points (at Quantcon Singapore and QuantExpo Prague).

It is written in the form of an interview. As no other interviewer was available I decided to interview myself. If after reading this post you still want to buy the book then you can go to this link.

Shall we start with some easy questions?


What's your favourite colour?

I was hoping for a more highbrow interview than this. Niels Kaastrup-Larsen never asks such a trivial question.

Sorry. This is the first time I've ever interviewed such a well known and intelligent person.

No problem. Since I'm well known, intelligent, and also very easily flattered.

Perhaps instead I could ask you where the idea for the book came from?

That's a much better question. After leaving AHL in late 2013 amongst other things I was thinking about writing a book. I came up with an idea for a book which I was going to title "Black Magic". Once I had the cool title I had to decide what the book would actually be about. I proposed to the publisher (Harriman House) that I'd write something which would be subtitled something like: "Tales from the world of systematic hedge funds: How to invest and trade systematically".

After a long series of emails that got narrowed down to the shorter title "How to invest and trade systematically", and subsequently cut down further to "How to trade systematically". About eighteen months later "Systematic Trading" was published.

The obvious thing to do next was to write "How to invest systematically". Of course this is also a huge topic and I had to spend a fair bit of time thinking about what the focus of the book should be, and what ground it would cover that wasn't covered elsewhere. I also had to think of a more original title than "Systematic Investing".

How did you decide on the foc(us/i) of the book?

To some extent I wrote the book for myself: I wanted a framework for managing my long only investments; which included shares and ETFs, where I had to pay relatively high trading costs compared to my futures, where I allocated across multiple asset classes, and where there were real world problems like tax to worry about.

I then thought long and hard about what were the most important - and neglected - topics in investment books. I decided they were uncertainty and costs. These two ideas are actually linked, because costs are highly predictable, whereas almost everything else about financial returns is uncertain to varying degrees. It's important to make decisions with this firmly in mind.

Of course there is an overlap with Systematic Trading here because in that book I frequently emphasise the difficulty of knowing the future with any degree of certainty, and I also wrote an entire chapter on trading costs.

Like Systematic Trading I also wanted to publish something that was a complete framework. So the idea is you can use this book for almost any kind of unleveraged long-only investment (passive ETFs, individual shares and active funds), and it also covers a few different 'use cases'. Of course this makes the book pretty long. It's about 50% longer than "Systematic Trading", but the sticker price on the cover is the same (in GBP anyway) so it's actually better value.

So... if you like [winks] we can talk a bit more those key ideas of uncertainty and costs now.

Oh yes, sure. Perhaps you can talk a little more about uncertainty

In finance there are almost two opposing views. On the one hand there is Taleb who says "We don't know anything" and on the other you have almost the entire industry of quantitative finance that assumes we know everything with 3 decimal places of precision (obviously I'm exaggerating both viewpoints for effect).

The idea that we can't naively use the probability of past events to predict the future is hardly new; it goes back to Keynes and deeper into the past. In contrast in quant finance we normally assume that we can (a) know the model that generated financial returns data in the past (b) precisely measure the parameters of this model and (c) assume it will continue into the future.

The "Weak Taleb" attack on quant finance is an attack on (b); so "The casino is the only human venture I know where the probabilities are known... and almost computable... In real life you do not know the odds, you need to discover them... ” (Black Swan).

But we can make equally valid points that (a) is also untrue (there is no 'model' waiting to be found and measured); and that (c) is nonsense (the future will never be exactly like the past). A "Strong Taleb" attack would essentially make the points that: (a) there are no models [or at least none that are practically usable], (b) even if there were we couldn't ever know their parameters precisely, and (c) these models are unchanging into the future*.

* By the way for the purposes of this discussion a Markov state model is still a single "model" - not a way of dealing with models that could change in the future. 

This is all true - but extremely unhelpful. Nearly all the smart people in finance are aware of this problem, but mostly ignore it. In fact we probably just have to assume that there is a model, and we also have to assume that this model will work in the future. Or we might as well close our laptops and become non-systematic, "gut feel" discretionary investors and traders.

But it's quite straightforward to deal with the weak Taleb attack on point (b) and think about the accurate measurement of the past. First you need to get yourself away from the idea that there was only one past with one set of estimable parameters which are known with certainty. Past movements of financial markets are either [i] a random draw from an unknown distribution or [ii] just one of many possible parallel universes that could have happened or [iii] are realisations of some random hidden latent process. It's easier to model [i] but these ideas are functionally equivalent.

Quantifying the effect of this uncertainty of the past on parameter estimates is relatively trivial statistics 101. So for example if the mean of a return series is 5% a year, and the standard deviation 24%, and you have 36 years of data, then the estimation error for the mean is (24% / sqrt 36) or 4%, so the two standard deviation confidence interval is -3% to +13%. Even with a relatively long history of data that is a huge amount of uncertainty about what the modelled mean was in the past: and remember we're still making the quite strong assumptions that there really is a model generating the returns; which happens to be Gaussian normal; and which will remain unchanged in the future.

The key insight here is that there are different degrees of uncertainty. The confidence interval for a standard deviation in this case is much narrower: 18.4% to 29.6%. If we have more than one return series we can also estimate correlation; so for example between US bonds and stocks the confidence interval is around -0.1 to 0.2.

So we don't need to throw away all of our data; we can be a bit smarter and just calibrate how differently confident we can be in the individual estimates we draw from that data.

That's given me a headache! It sounds like you've written a very technical book on maths and/or philosophy...

Nothing of the sort! All the ideas are introduced in a very intuitive way (much simpler language than I've used above); and it's very much aimed at a non-quant but financially literate audience. The book is mostly about what practical use these findings have. Once you start thinking about the world in terms of quantified uncertainty you can still be a systematic, model based, investor; and you can simultaneously be a skeptical pupil of Taleb; but you can also still do some useful things.

So what practical problems do you address with this idea of (calibrated) uncertainty of the past?

The first main insight is that standard portfolio optimisation is partly junk. Of course everyone in finance knows this: but again there are two extreme views: "Complete junk - I don't believe in any of that nonsense and I'm just going to hold US tech stocks whose names begin with the letter A" or "Junk, but I'm going to use it anyway because what choice do I have?". But reality is more nuanced than either of these views.

The insight and intution behind Markowitz's work is extremely valuable - it's the baby in this particular bathwater. Though yes: estimates of risk adjusted returns have such huge past uncertainty they're mostly worthless. But estimates of volatility and correlation are more predictable and so have some value. So I address this question: how should you build portfolios given this knowledge?

The other main insight is that you shouldn't look at post-cost returns as you're subtracting apples (costs) from oranges (pre-cost returns). Pre-cost returns have huge estimation error. But costs are actually relatively predictable (unless you're trading so fast or in such size you affect the order book). A better approach is that the starting point for any decision should be that you go for the cheapest option unless the evidence strongly suggests - with some probability - that the more expensive option is better. I guess this is a Bayesian worldview, though I never use that term in the book.

Okay I get the hint. I think perhaps it would be good to talk about costs now

The first thing to say about costs is that although they're relatively predictable, they're not actually that easy to measure. Although there have been attempts to get funds to state the "total cost of ownership", in practice you have to make some educated guesses to work out likely costs of different forms of investment.

Once you have that information, what should you do? Anyone whose read my first book knows that costs are important when deciding how much, and what, to trade. But for long only investment there are a whole lot of other decisions where the notion of certain costs and uncertain returns is useful. For example should you buy a fund which is more expensive, but which has had - or should have - higher returns?

Another important point is that different kinds of investors have to worry about different kinds of costs. So relatively large investors have to worry about market impact. But for relatively small investors, especially those in the UK, the tyranny of minimum brokerage commissions is more important. A £10 commission on a £1,000 portfolio is 1%: quite a lot if you have realistic estimates of future returns. An important implication of this is that the right kind of portfolio will depend on how much capital you have to invest.

You've already talked about some common elements, but what would readers of Systematic Trading recognise in this book?

The main thing they will recognise is the idea of a top down, heuristic portfolio construction method which I call handcrafting in both books. The difference in Smart Portfolios is that I make it even simpler - all grouped assets have equal weights (once differential risk has been accounted for). 

In part two of the new book I also go into much more detail about how you'd practically build a cross asset portfolio using the top down handcrafting method: choosing appropriate ETFs, and where it makes sense to buy individual shares. 

Because of the emphasis on costs this would be done differently for smaller and larger investors. In particular larger investors can afford more diversification: smaller investors who buy too many funds will end up owning too many small chunks of things that they've had to pay multiple minimum commissions on. The advantage of a top down approach is it deals with this nicely: you just stop diversifying when it no longer makes sense (a decision based, naturally, on the certain costs and uncertain benefits of diversification). 

Earlier you talked about "different use cases"...

Glad to see you've been paying attention! Just as in Systematic Trading I realise that not everyone will sign up to the extremely pure dogma: in this case that risk adjusted returns are completely unpredictable. So the book also helps people who want to vary slightly from that central path, whilst limiting the damage they can do. These different use cases all appear in part three.

Firstly as you might expect I talk about systematic ways to forecast future returns. At the risk of being stereotyped one is a trend following model, the other is based on yields (so effectively carry). The point, as with Systematic Trading, isn't that these are the best ways to forecast the markets - they're just nicely familiar examples which most people are able to understand (and whose nuances I can explain). Unfortunately as with my first book a few people won't understand this and will pigeonhole me as a chartist / trend follower / technical trader...

Secondly I talk about using "gut feel" but in a systematic way. This is analogous to the "semi-automatic trader" in my first book. The idea being that some people will always think they can predict market returns / pick stocks; at least let's provide a framework where they can do limited damage.

Thirdly are people who are still convinced that active fund managers are the bees knees. I show them how to determine if this is true by looking through the prism of uncertain returns (perhaps higher realised alpha in the past) versus certain costs (higher management fees).

Finally there are the relatively recent innovations of Smart Beta; again more expensive than standard passive funds, but are they worth it? I also talk a bit about robo-advisors.

"Smart Beta": is that where the title of the book came from?

Sort of. It's an ironic title in that respect since you'll realise quite quickly I am pretty skeptical of Smart Beta at least in the guise of relatively expensive ETFs. Using systematic models to do the smart beta yourself is better, if you have sufficient capital.

But "Smart" actually sums up the book quite well (and yes, this is an ex-post rationalisation once I'd thought up the title. Deal with it). Smart for me means "Practical but theoretically well grounded".

So for example there are some technical books on things like Bayesian optimisation that deal with uncertainty, and other papers around trading costs. But if you introduce taxes into the mix you end up with really non tractable, non closed form models and it gets pretty unpleasant. This isn't the kind of the thing the average financial advisor can really use. Frankly even I don't use that kind of technical artillery when deciding if I should top up my pension fund.

And there is plenty of "backwoodsman" advice in less technical books that is either vague ("Don't trade too much"), overly simplistic ("Buy the cheapest passive funds") or worse isn't supported by theory ("Everyone should just own stocks").

What I tried to do in Systematic Trading, and continue in Smart Portfolios, is to provide some heuristic rules that are (a) as simple as possible and (b) theoretically correct, or at least supported by research. So for example one simple rule is "if you are paying a minimum brokerage commission of $1, you shouldn't invest in ETFs in units smaller than $300".

One, fair, criticism of my first book was that I didn't provide enough realistic examples. So I've probably gone overboard with them here in trying to make the book as accessible as possible.

A less fair criticism of Systematic Trading is that there weren't enough equations - which of course was deliberate! I've included some more here to aid clarity, but they are mostly extremely simple without an integral symbol in sight.

What about portfolio rebalancing?

Yes, that's another big topic where I try to use simple rules that are theoretically grounded. So there is the standard rebalancing method where you don't rebalance unless your positions are out of whack by a certain amount. But I introduce a simple method for calculating what "out of whack" is, which again depends on the cost level you face, which in turn depends on how much capital you have to invest.

Then there are other rules to deal with other common situations: rebalancing when you're using forecasting rules, the effect of taxes, changes in characteristics used to pick stocks, takeovers, and so on.

I really enjoyed Systematic Trading. Should I buy your second book?

It depends. "Smart Portfolios" is actually two books in one:

  • A practical discussion of the effects of estimation uncertainty on optimising portfolios
  • A complete handbook for long only investing in funds and shares 
So if you are a pure short term futures trader who already has a good understanding of statistical uncertainty then you'll probably find little of value in this book. It is definitely not "Systematic Trading 2: The Market Strikes Back". But feel free to buy it out of misplaced loyalty! Then give it to the guy or gal who manages your long only investments.

On the other hand if you read "Systematic Trading", and enjoyed it, but struggled to see how this related to your long only ETF or shares portfolio (with the exception of the "asset allocating investor" examples), then you should really find this book very useful.

Finally if you are in fact Taleb you should definitely read the second chapter of the book, but no more. After that I mostly assume that Gaussian Normal is a useful model when used properly, and you'd absolutely hate it. Although in my defence I do at least use "Kelly-Compatible" geometric means which penalise negative skew, rather than arithmetic means.

Is there anyone you'd like to thank?

Nine people were absolutely key in this book coming about. Stephen Eckett, top dog at Harriman House, commissioned the book. Craig Pearce spent months whipping my ramblings into marketable and readable condition. Riccardo Ronco and Tansu Demirbilek were brilliant reviewers. My third reviewer Tom Smith was also brilliant, but deserves a special mention as he also reviewed my first book; in both cases with no money changing hands (I suggested he pay me £500 for the privilege but this was greeted with derision). 

The other four people are my wife and children, who have had to put up with a distracted and absent minded husband and father for months on end. 

Any more books on the horizon?

Not immediately as I have a few other projects I'm working on which will take up most of my time over the next few months. But then I've got a couple of ideas. The first idea is to try and write "Systematic Trading and Investing for Idiots" (clearly a working title). Essentially a distillation of the methods and ideas in my first two books, but written for a wider retail audience. The second idea is to write something about the interaction of people and machines in the financial markets. With all the hype over AI in financial markets this might be an interesting book.

Are you doing any conferences in the near future where we can here more about your ideas?

Great question! [surreptitiously slips ten pound note to interviewer] 

At the end of this month I'm speaking in Singapore (at QuantCon) and then at the start of November in Prague (at a new event QuantExpo). Both of these events look to have a great lineup and I'd highly recommend them if you're within flying distance of either venue.

The talk I am giving at both venues will be about the impact of past uncertainty on the estimates used for portfolio optimisation: basically material covered in the first few chapters of the book. I'll also introduce some of the possible solutions to this problem. Many of these people will have seen before but I think it's good to understand specifically how they deal with uncertainty.

There might be other events coming up - keep an eye on my social media for news.

So finally: When and Where can people get your book?

It's officially published on the 18th September but currently available for pre-order. If you go to the website for the book at this link you'll get a link to my publishers page, which is the best place to buy it from my perspective (and currently the cheapest). The books website also has a lot more information about exactly what is in the book if you're still undecided. 


If you thought the (frankly incompetent) interviewer missed a key question then please feel free to comment below and I'll add the question (and answer it).


  1. if this book is as good as your last, congratulations in advance! hari p. krishnan

  2. Hi Rob,
    Looking forward to reading your new book!
    One of the natural questions (well, at least to me :) ), is that you mentioned once somewhere that you think it's probably not possible to make a living from trading only (i.e. to live a basic-comfortable life without the necessity to go to work every morning), so do you think that it might be possible to achieve that with a combination of passive investments and some active trading (and a “reasonable” amount of capital)., And can this book help with it?
    I guess this question might seem obvious\redundant\unanswerable, but maybe you have some thoughts on the subject, and your interviewer was not skillful enough to give you the opportunity to speak about them :) ...

    1. Indeed I myself live almost entirely off passive investments structured according to "Smart" principles. The problem with trading income, unlike dividends, it's very uneven and volatile and so not suited to having to meet regular bill payments.

  3. I must say that's a very fair price to allow me to grow my invaluable Carver collection.

  4. Really excited for the book!

    Is there anywhere within pysystemtrade that you can test the results using different buffering levels? eg what the results would be with a 5% buffer vs a 10% buffer?

  5. 2 questions about figure 32 on page 324 (Hi/Low momentum sharpe ratio distributions).

    1) Which timeseries did you use as input?
    2) If there are 100 monthly returns, of which 20 are classified as low-momentum (previous month 12mo sharpe<-.057), and 30 are classified as high-momentum (previous month 12mo sharpe>2), do you throw out the other 50 returns that are 'neutral' momentum? In other words you bootstrap low-momentum with 20 monthly returns and high-momentum with 30-monthly returns?

    Enjoying the book. But trying to replicate this plot I failed.

    1. 1) US monthly returns of stocks and bonds
      2) Yes exactly that.

    2. thanks for pointing me to that. let's see if I can reproduce your plots.

    3. I can reproduce the plots now using your data. I used S&P data starting from 1980. The plots do not overlap as much, hence my confusion.

    4. Phew! I guess this reinforces how tenous these effects are and how we shouldn't put too much weight on them!

  6. Hi Rob,

    Wonderful book!! I have a question regarding buffering. I know you simplified in this book, but in your other book it was the number of contracts * buffer. I visited the paper by your colleague and it can be quite dense. Is there a simplified version of calculating the optimal buffer and does that also apply to stocks as it does futures? If so, for example, using say a 10% buffer on 2 individual stocks. Holding all else constant except price, A is $5 and B is $10. If I have 100 shares of each with $1000 capital, in shares both have buffers of 10(100 * .1), however in dollar terms that is $50 for A and $100 for B and in percentages 5% buffer for A and 10% buffer for B. So economically the buffer for B is twice that of A. Now in general, lower priced stocks have larger bid asks than higher priced stocks so intuitively isn't this the opposite of what is desirable? We would actually want a larger buffer for more expensive to trade stocks(or lower priced stocks) relative to higher priced stocks?

    Thank you for all the work you've done.

    1. Unfortunately there is no simple method. You have to consider the difference between fixed and proportional transaction costs, the level of volatility, and the expected size of your alpha. That's why I use simple heuristics.

      Anyway in your example the buffer should be calculated as % of value, so both would have a 10% buffer.

    2. Ok. Just to clarify though for the futures its based on contracts, or (contracts +- contracts * .05, as opposed to weight in % +- 5%?

    3. Could be either. Weight is technically better, but I decided not to explain it in those terms to avoid making the book any longer.

    4. Great. I'm sorry just one more question regarding what I said below about the cube root. In the absence of estimating vol and the expected alpha the half width should be ((trans cost in %)^(1/3)) / 2?

    5. It's proportional to the cube root of transaction costs, but you can't specify it exactly without knowing something about vol.

  7. Just another note I understand it to be the cube root of the transaction cost, so does that mean a stock with a bid ask spread of 5bps has a full width of 7.9%(.0005^1/3), or for 100 shares, buffer is max 104(103.97) and min 96(96.03)?

  8. I noticed a good number of the more obscure US-based ETFs referenced in the diagrams of the book to have closed in the last two years (particularly a lot of the individual country ETFs, like CHNA); would you ever contemplate an update blog with open alternatives, where they exist?

    1. I got advice that to make the book more interesting I should quote specific tickers, but the downside is very obvious.

      The short answer is that it's a lot of work to do properly, so it's not something I could commit do doing on a regular basis. You can use or to find alternatives

      I'm actually considering setting up a paid-for service where you get a curated list of ETFs plus the tools to easily manufacture your own handcrafted portfolios without fiddling with spreadsheets, plus advice on optimal rebalancing. So watch this space.

  9. Personally I do not break down further then by region because in my opinion the high TER on country ETFs does not outway the additional benefit of diversification. (If I interpret your book correctly....). I like Vanguard ETFs. In Europe I use VEUR, VNRT, VAPX, VPJN, VFEM for equities.
    USA equivalents would be VGK, VTI, VPL, and VWO.

  10. For bonds as a European investor I use:
    VECP - Vanguard Europe Corporate Bonds
    VETY - Vanguard Europe Govt Bonds
    VUCP - Vanguard USA Corporate Bonds
    VUTY - Vanguard USA Treasury Bonds
    VEMT - Vanguard Emerging Market Bonds (govvies)

    From a US investor perspective I use:

    AGG/BND US Treasuries
    BNDX Intl govvies
    VCLT US corporate bonds
    EMB Emerging Market bonds
    EMCB Emerging Market corp bonds

    I admit I am biased towards Vanguard...

  11. Very nice indeed Rob. In my own vulgar language you are f****d as a stock picker and slightly less so as a systematic investor.

    1. I have noticed that your blog posts are normally more succinct than mine...

  12. Hi Rob,

    Congratulations on another great book which I'm currently working through. Very useful having the accompanying worked example spreadsheets but wondered is there any python code/notebooks in the offing ?

    Thanks, Stuart

    1. I have no specific plans since the new book isn't really aimed at python fluent people, but you can play with the portfolio optimisation code that litters the site in various places.

  13. Dear Robert,
    I bought and am carefully reading both of your books. Currently I invest in a portfolio according Kommer'book. It is a GDP-weighted world portfolio, but your approach makes me more sense.
    One question about the cost in your Smart Portfolios book because of my specific situation.
    I can and do use broker like you and discount German broker(flatex), where the commission structure is the same you use in your book, but
    on the one hand in my country(Hungary) I don't pay fixed commission, I pay 0.3-0.5% commission for ETFs and also for individual shares.
    On the other hand keeping my assets for minimum 5 years in an account in my own country I'm tax free. Otherwise the capital gain would be 15%.
    How would you handle this issue implementing your rules building portfolio?

    * how would you calculate the minimum amount to buy individual shares to get more granular diversification?
    * how would you calculate the cost holding specific ETF?
    * how this problem affects rebalancing cost?
    * how could I compare the costs holding portfolio at IB or in my own country?

    The size of the portfolio I manage myself is in the upper 6-figures €.

    1. With only % based costs in theory there is no lower limit before you invest in individual shares (at least until you consider rebalancing). But I wouldn't go crazy, I'd probably use a minimum of euros 1k in each share, or the rebalancing will be too high.

      There is plenty of detail in the book to answer the rest of your questions, in particular the effect of capital gains.

    2. Thank you for your answer. You are right I'm at Chapter 8, I have to keep reading.

  14. I noticed that in Chapter 1 of the new book, you discuss the difference between accumulation and distribution units, and say that there is no dividend tax paid on accumulation units. Are you sure about this? I've googled a bit, and it seems that the money reinvested in accumulation units is treated as a "notional distribution", and taxed as if it had been distributed.

    1. You may well be right. Pretty much anything specific I say about tax should be ignored and/or checked. Different jurisdictions apply different rules over different times. Only the general principles on reducing tax through smart rebalancing in part four are universally true.

  15. Hi Rob,

    Just out of curiosity, how much of your asset is allocated to "smart portfolio" and how much is allocated to "systematic trading". I'm doing some systematic trading, but not comfortable putting all the money in trading. So I wonder what breakdown would make sense. Obviously, this is a personal risk tolerance thing; but I wonder if there is a general advice.

    1. That's an excellent question. In terms of my risk allocation, 75% in 'smart portfolio's, 25% in systematic trading.

    2. Hi Rob - why don`t you have 100% of your risk allocation in systematic trading given it has (I assume) a higher expected Sharpe ratio that the smart portfolios?

    3. Another very good question. Basically trading is unsuitable for providing an income, as the returns you earn (essentially any profits over the HWM after tax) are lumpy and may be absent for years. As I rely on my investments for the bulk of my income it makes sense to put more money into dividend earning stocks and coupon paying bonds.

  16. Dear Rob, I have a question after reading your second book, and I don't know which post it belongs to, so i decided to write it here:

    In the book you distinguish between maximum GM and maximum Sharpe Ratio portfolios, this is a bit confusing to me, as I always thought that maximising Expected geometric mean of portfolio gives you Kelly-optimal allocation, which in turn maximizes Sharpe Ratio of the portfolio (this can be proven mathematically using linear algebra). Could you please elaborate on differences between max SR and max GM portfolios?


    1. If we take a specific example from my book; in chapter five the max GM portfolio is about 10-20% in bonds, but the max SR portfolio is about 60% bonds.

  17. Your algebra is wrong. Max GM is indeed Kelly optimal, but neither is max SR. GM = m - 0.5s^2 (good enough approximation) but SR = m/s (where m is the excess return, after deducting risk free rate; and s is the standard deviation).

    1. 1. Find w that maximizes GM = w'r - 0.5(w'*C*w) Taking derivatives and solwing for w(i) giwes me optimal w = C^(-1)*r

      2. Find w that maximizes SR = w'r / sqrt(w'*C*w) gives me the same optimal w = C^(-1)*r

      I used arithmetic mean in 2, should i use GM?

    2. Ok I worked out the problem. Your derivation is only applicable if there is unlimited leverage and short selling is allowed to achieve a particular risk target, since you don't account for the constraints in the portfolio (positive weights, summing to 1). In that case the maximum SR = maximum geometric mean (indeed with unlimited leverage maximum SR = maximum portfolio mean as well).


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