Suppose an investor has 60 percent accuracy in forecasting bull market and 80 percent accuracy in forecasting bear market (a 60–80 timer).
If the observation is bull market, then 0.60 (forecast accuracy for bull markets) is compared with a random number (between 0 and 1). If the random number is less than 0.60, which occurs with a 60 percent probability, then the market timer is assumed to have correctly predicted bull market and her return for that first observation is the market return. If the random number is greater than 0.60, then the market timer is assumed to have made an error and predicted bear market; her return for that observation is the risk-free rate.
Why the random number is less than 0.60 then the market timer is assumed to have correctly predicted bull market?
This scares me. I have (I believe) read the relevant sections on Monte Carlo simulation, yet I have absolutely no idea what you are talking about… Back to the books for me then!
The probability of getting a random number lesser than 0.6 is 60% if it has to be chosen between 0 and 1. And her bull market forecast accuracy is also 60%. So, these two can be compared. Similarly, the probabiiy of getting a random number greater than 0.60 is 40% which is also her probability of forecasting the bull market wrongly (1-0.60). So, again these can be compared.
If x is a random variable with a uniform distribution over [a, b], and c is any number in [a, b] (so a ≤ c ≤ b), then P(x < c) = (c − a) / (b − a).
Here, x is uniform on [0, 1], and c = 0.6, so P(x < 0.6) = (0.6 − 0) / (1 − 0) = 0.6 / 1 = 0.6.
I knew these two can be compared. I still didn’t get that why the random number is less than 0.60 then the market timer is assumed to have correctly predicted bull market?
We’re told that the market timer correctly predicts a bull market 60% of the time.
We’re trying to find a way to _ simulate _ that phenomenon; i.e., we want a process that is random (the market timer doesn’t know exactly when he’ll be correct), and that gives us a value of “true” or “correct” 60% of the time.
To get the random part, we use the computer’s (pseudo)random number generator. To get the “true” or “correct” 60%-of-the-time part, we institute a test that will be true 60% of the time and false 40% of the time.
That’s about it.
(Note that we’re not saying that the results we get _ will _ happen; as it’s random, we don’t know what will happen. All we’re saying is that the results we get will have the same statistics as what will happen, to the best of our ability to simulate that.)