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December 31, 2018 - January 23, 2019
20% (the standard deviation),
We’ll take the expected market variability to be 19% annually5 or 5.5% monthly.6
All we are forcing with the simulation is that the average characteristics of a huge number of randomly generated outcomes will look something like what we expect.
keeping the average cost per share purchased below the average price per share.
The idea is not to run the simulation once and compare strategies but to do it many times, letting the randomness “average out” to get a better sense of how the strategies perform over a wide range of feasible outcomes.
Statistically we expect that the variability (as measured by the standard deviation) of random returns over time is related by the square root of the relative time.
隨機報酬的變異性與相對時間（長度）的平方根有關
only SQRT(), or 0.29,
一月是一年的1/12，所以相對時間的平方根：（1/12）^（0.5）= 0.29
use the “macro” feature of spreadsheets,
value averaging tends to outperform dollar cost averaging by over a 1% higher return,
Because value averaging operates by taking advantage of large moves in either direction, you might suspect that the relative return advantage of value averaging will increase with volatility.
the fact is that value averaging results in less investment exposure and value than dollar cost averaging portfolios when prices go way up, and in more investment exposure when prices go way down.
It is much more likely that the price of a $100 stock is outside a range of $80-$120 after a decade than after a month. This “total volatility” is a different concept than “annualized” or average volatility, which decreases over time (as you saw in previous chapters). As time increases, total volatility increases by the square root of time, but average volatility per period decreases by the square root of time. A simplified example: If the standard deviation of return over 1 year is 20%, what happens over a 4-year period? The square root of the time increase (four) is two. So total volatility
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總變動性是時間的平方根增加
平均變動性是時間的平方根減少
prices in financial markets possibly overreact.
如果實際上這是真實的話，這種不起眼且看似無害的觀察會帶來巨大的衝擊力。面對早期的理論，股票市場和其他金融市場價格隨機走勢，它飛得很快。
a series or daily, monthly, or annual rates of return on the market are uncorrelated, much like the sequential outcomes of the flip of a coin or the roll of the dice.
“Until recently, most financial economists agreed that stock returns are essentially unpredictable.”
As a result, few economists now believe that stock prices follow a strict random walk.
there is increasing evidence of mean reversion (defined below) in stock prices over periods of about two to four years.
Stock returns appear to be mean reverting.”6
Mean reversion means that the market overreacts in the short run but generally can be counted on to “correct itself” in the longer run.
investors who keep their heads in the face of bad news and big price drops stand to reap real benefits.
the market’s reactions dampened until its price level more closely reflected its fundamentally supportable level.
several regular episodes of commodity price overreaction followed by mean reversion.
the “market coin” is not a “fair” coin, because the market goes up more than down (as, of course, it should).
market returns from period to period will be independent, just like the coin flips above.
there still seem to be monthly overreactions, no matter which period we look at.
The year-to-year numbers display what we call mean reversion.
The persistence of short-term market returns seems to be seriously corrected over longer and longer periods.
The short-term persistence of market returns is phenomenal at the daily level—tomorrow’s return is quite likely to have the same characteristic as today’s return.
It clearly could not be coincidence that short-term overreactions so gradually and consistently turn into long-term mean reversion.
that formula strategies can almost automatically take advantage of market overreactions,
formula strategies such as value averaging guide you to take advantage of the potentially temporary high price by getting out (or of the low price by buying) before the overreaction is corrected.
the incremental return to value averaging could indeed be higher in real historical markets than in average simulated markets. This is particularly true when using 3-month or 4-month intervals for value averaging; it is less clear at monthly intervals.
the simulated market series were designed as a true random walk, without any overreacting or mean-reverting characteristics.
how often
“matching percentage” data
While value averaging during a mean-reverting period is good, it’s not as good as using a shorter period (that is also mean reverting).
It would seem that value averaging two, three, or four times a year would be reasonable possible strategies for milking the most out of the VA strategy.
The optimal strategy turned out (after the fact) to be a quarterly frequency for value averaging;
the market has a tendency (on average) to overreact in the short term and then to mean revert over some longer term.
Formula strategies that play on temporary possible market mispricings are a reasonable way of taking advantage of overreactions.
value averaging with a quarterly invest...
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In tests over various time periods, value averaging historically has worked very well when done at a frequency of 3, 4, or 6 times a year (every two, three, or four months).
Whereas this approach doesn’t seem as intuitive as simply looking at positive and negative returns, it is a bit easier to work with. No matter what the length of the period, the market return is just about as likely to be above average as below average; this more closely resembles a fair coin for each experiment. This is not true for the positive/ negative distinction, because the market tends to rise, especially over longer periods. In any case, the resulting conclusions are not sensitive to which technique is used.
The quarterly figures were not very sensitive to the time period used (pre-1958 versus post-1958).
The market overreaction presented in the original book doesn’t hold up as well today.
there was no compelling evidence of short-term overreaction in the two recent decades.
It could be that (due to technology?) cycles have shortened and market forces move faster to mean reversion.
examining persistence in market returns from 1990 to 2005 is that bad returns or below-average periods are not very persistent.
