Asset Allocation

Can someone explain how to interpret the Monte Carlo simulation diagram on page 236 volume 3

Don’t have the book on me but what is there to explain? Just know that Monte Carlo simulation runs independent simulations, ie thousands, based on Cash Flows, taxes, expected returns/volatility, etc and summarizes it in a chart/graph giving you say the probability of having \$5000 at end of year 10 or \$4000 and so on. You WILL NOT have to do a Monte Carlo simulation, but know its advantages and disadvantages and know that it is very sensitive to expected returns/vol assumptions.

if the comparison is to MVO, i wouldn’t say MC is very sensitive to expected returns/vol. MVO can be worse. i guess you don’t even feed MC directly with expected returns, instead you only need to define its distribution and let MC to take care of expectations. i guess advantages of MC should be 1) it’s a multi-period model; 2) it doesn’t require normality of return distribution, and of less assumptions; 3) it’s path dependent. 4) compuational wise, MVO is order of N square and MC is order of N. 5) MC produces not only expectaions but also their confidence intervals. … if one has only \$5000, he probably doesn’t need any model. but, if he ever needs one. i would say MC is not a bad choice. as long as the guy doesn’t mind to write some VB code in his excel, he can easily simulate monthly rebalance of his 5000 buck portfolio for 12 months with numerous trading rules, etc.

You definitely need to feed MC directly with expected returns, whether it calculates the return based of historical data set or not is another issue, but it either A)needs this data set to calculate an annualized rate of return or you need to provide one…also you can’t just take that annualized rate of return based off say the S&P for the last 10,20,30 years or so without adjusting it for market expectations. Also, you will need to supply the MC with std deviation via your direct input or a data set from which it can calculate (same data set as returns). Also, if there are no Cash Flows, the MVO and MC should spit out the same portfolio I believe.

MVO is very sensetivy to inputs, MC doesnt receive input as a single number (as MVO does), but rather a distribution of input parameters, and it will be sensitive to that, but i guess less sensetive then MVO, rand0m is right on this point. I also believe MC is much more computantionally intensive then MVO, if you want to produce any reliable results ofcourse. Rand0m can you please elaborate on your 4), as i to run montecarlo, if you have 2 assets and 10 periods, you need and you ran 10000 runs you will need 2 * 10,000 * 10 samplings.

I’ve used MC before and trust me it does use a “single” number to start out with but adjusts that number each path based on teh std deviation. 1st pass it might take that “single” number then next pass #2 it might adjust it by 1.2 std deviations, then next #3 might adjust it by -2.3 std deviations and so on. You are right in that it doesn’t only use say 12% for all paths and each year. Instead it takes the 12% as a starting point and adjusts it, so that might be where we are off a little as we are talking about 2 slightly different “single” numbers… And yes MVO is more sensitive than MC for the return/risk inputs.

here’s how you start one MC, assume stock price is log-normal (distribution) dSt = r*St*dt + sig*St*dWt (where dWt is standard brownian motion). the only input here are risk-free rate and vol. you can even assume sigma is some kind of distributions, then you get away with feeding volatility. you can then assume risk-free is another kind of distribution, then you stop feeding risk-free rate. real sigma or real risk-free rates or realized returns are all useful to calibrate the model along the way. but, the model is running fine without them. at least, that’s what the math says. in L2, we all learnt binomial tree. that is a visualized MC without path dependence. you give it a sigma and a risk free rate. it grows on its own, doesn’t it. so, bigwilly, i knew you used MC. although i didn’t, but i cannont turst you on this one.

The MC simulation that I work with runs off expected returns/vol, correlations, covariances, cash flows, etc… The computer does 1,000s of simulations each one tweaking either the returns or cashflow timings and amounts etc. So maybe your MC program works differently than mine

expected return should be something one would get out of the simulation. i really suspect the process you used was running MVO together with MC. i agree one would need to feed mc with variance/covariance, etc if he doesn’t think these parameters are random. the var/cov is indeed the sigma term in the calculus i cited earlier. but, people can use ARCH or GARCH or whatever to model it, in that case, you dont have to provide the matrix to start your simulation. of course, real var/cov, and others are useful to let simuation narrow down its “simulating” paths. btw, the only mc program i had is a 30 lines of vb code in my excel for my professor’s project many years back.

The program I have used, you can just provide say a data stream for say S&P, T-Bill, Oil, etc and it calculates everything for you and you can also override any of the calculations…say it comes back wtih a 14% expected return for S&P and you believe its closer to 12% you can adjust, etc. So how did we get on this topic anyways???

comp_sci_kid Wrote: ------------------------------------------------------- > I also believe MC is much more computantionally > intensive then MVO, if you want to produce any > reliable results ofcourse. Rand0m can you please > elaborate on your 4), as i to run montecarlo, if > you have 2 assets and 10 periods, you need and you > ran 10000 runs you will need 2 * 10,000 * 10 > samplings. ture, mc requires many simulation trials. but, for each trial, the computational resource needed is less than MVO. you can check on that. i pull that off from memory. besides, 10,000 MC trials for a two asset portfolio might be an over kill. someone must have done a pretty bad job in modelling. it doesn’t seem to converge!!