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what is the theory behind rick van ness's risk calculation in the video about diversification?
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john bensin's answer covers the math, but i like the plain-english examples of the theory from william bernstein's fine book, the intelligent asset allocator.  at the author's web site, you can find the complete chapter 1 and chapter 2, though not chapter 3, which is the one with the "multiple coin toss" portfolio example i want to highlight. i'll summarize bernstein's multiple coin toss example here with some excerpts from the book.  (another top user, @joetaxpayer, has also written about the coin flip on his blog, also mentioning bernstein's book.) bernstein begins chapter 1 by describing an offer from a fictitious "uncle fred": imagine that you work for your rich but eccentric uncle fred. [...] he   decides to let you in on the company pension plan. [...] you must pick   ahead of time one of two investment choices for the duration of your   employment: certificates of deposit with a 3% annualized rate of return, or, a most peculiar option: at the end of each year uncle fred flips a coin. heads you receive a 30% investment return for that year, tails a   minus 10% (loss) for the year. this will be hereafter referred to as   "uncle fred’s coin toss," or simply, the "coin toss." in effect, choosing option 2 results in a higher expected return than option 1, but it is certainly riskier, having a high standard deviation and being especially prone to a series of bad tosses.  chapters 1 and 2 continue to expand on the idea of risk, and take a look at various assets/markets over time.  chapter 3 then begins by introducing the multiple coin toss example: time passes. you have spent several more years in the employ of your   uncle fred, and have truly grown to dread the annual coin-toss   sessions. [...] he makes you another offer. at the end of each year,   he will divide your pension account into two equal parts and conduct a   separate coin toss for each half [...] there are four possible   outcomes [...]: [...]   being handy with numbers, you calculate that your annualized return   for this two-coin-toss sequence is 9.08%, which is nearly a full   percentage point higher than your previous expected return of 8.17%   with only one coin toss. even more amazingly, you realize that your   risk has been reduced — with the addition of two returns at the   mean of 10%, your calculated standard deviation is now only 14.14%, as   opposed to 20% for the single coin toss. [...] dividing your portfolio between assets with uncorrelated results increases return while decreasing risk. [...]   if the second coin toss were perfectly inversely correlated   with the first and always gave the opposite result [hence, outcomes   1 and 4 above never occurring], then our return would always be 10%.   in this case, we would have a 10% annualized long-term return with zero risk! i hope that summarizes the example well.  of course, in the real world, one of the tricks to building a good portfolio is finding assets that aren't well-correlated, and if you're interested in more on the subject i suggest you check out his books (including the four pillars of investing) and read more about modern portfolio theory (mpt).
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