I didn’t get any responses on my “climate modeling” post even though I thought it was very good, nor did anyone respond to my brilliant post on “Intelligent Design”. Doesn’t anyone like the hard sciences any more? 😦

I suspect more people are interested in the technical side of the recent fiascoes on Wall Street and in Washington. But to warm you up, I’ll start with a math problem. Warning: having taken undergraduate finance courses will make you more likely to get this wrong.

There are 250 trading days in a year. You have 3 assets with the following return profiles:

Let a=(1.01)^3 = 1.030301. Each trading day the price of Asset A, with equal probability, either increases or decreases by a factor of a.

Asset B is just like asset A but uncorrelated with it.

Asset C either with probability 0.9 goes up by 1% (its price is multiplied by 1.01) or with probability 0.1 goes down by a factor of c = (1.01)^9 = 1.093685272684360901.

What is the annual expected return for A and C?

For each of A, A+B, C, A+C what is the approximate value of X such that this asset or combination of assets has a 95% chance of having lost at most X% of its value at the end of a year?

In Wall Street jargon, this is the “VaR” or “Value at Risk” — a VaR of 25% means that only 1 year in 20, on the average, should your asset have a loss of 25% or more.

This simple example will illustrate some essential mathematical blunders the “rocket scientists” on Wall Street made.

Bonus: I wrote down the exact value of c directly without using a calculator or spreadsheet (which don’t have enough digits anyway). How did I do that?

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OK, here is a hint on the math. The logarithm of the asset prices behaves more simply than the prices themselves, so looking at that first might give a clearer picture. In the long run these assets have a “lognormal” distribution. The key question is how long is “long”?

am i smoking something good, or did the original post have something like “up 3%” vs. “down 3%”?

if i’m remembering that correctly, then

thatwould have a virtually certain chance of tanking, since “down 3%” is a greater effect than “up 3%” (i.e., 0.97 times 1.03 is less than 1).ok, so i looked up some stuff … let me know if i’m off base:

for asset a or b, you can use “cumulative binomial”. for 250 trials, that hits 5% between 111 and 112 out of 250. for a worst case scenario we’ll take the 111.

that means up 111 times and down 139 times.

which means, what, -28 times log 1.030301, and then you have to un-log it by doing 10^answer

so … i get ~43% value remaining. so loss of at most 57%?

that sounds big.

i don’t know how to do the sum. for a + b can you divide something by rad 2?

in any case, all of this could be totally wrong … heh, considering i have now known what a binomial distribution is for exactly ten minutes. logs i can do from before.

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Bonus: I wrote down the exact value of c directly without using a calculator or spreadsheet (which don’t have enough digits anyway). How did I do thatbased on the look of “1.030301” it looks like you can just write out a shitload of rows of pascal’s triangle until you get to the one that starts with 1, 9; then just space them out every other decimal place and add them (with lots and lots of carryovers and overlaps).

I worded it carefully. There is a factor x = 1.030301 such that the price is either multiplied by x or divided by x. If I gave you a deal, heads you double your bet, tails you get half back, you’d take it, right? This the same thing, good for the player overall. It is not the same as an ADDITIVE gain or loss of an identical factor.

Your binomial distribution is correct, as is your Pascal’s triangle observation. If you do it to Asset C you get a similar answer, because a year is enough of a “long run” for these assets. Specifically, the 5% cutoff for a 90% event is 216 out of 250, corresponding to 216 “ups” and 34 “downs”, which gives a new price of 1.01^90 = 0.41, for a VaR of 59%, almost the same number. These lognormal distributions, when you take the log, have the same mean (zero), and the same standard deviation as well, so they are treated as EQUIVALENT assets in portfolios (before you start considering correlations with other assets).

But what about the short run? Suppose you just have a week with 5 trading days? Then your 5% cutoff for asset A is “0 up 5 down” for a loss of (1-1.030301^-5) = 14%, while your 5% cutoff for asset B is “2 up 1, 3 down 9” for a loss of (1-1.01^-25) =22%.

What about 1 day? Then the VaR is 3% and 9% for A and C.

Real assets can have 1-day moves much larger than 9%, and sometimes your estimation of their return profile will be wrong because the time window of historical data under conditions relevant to today is too short. This becomes even harder to model when you have to derive correlations between assets from historical data too. The world just changes too fast to get good data on these, and the simplifying assumptions financiers make always regularize and smooth things and end up underestimating fluctuations.

For A+B dividing by sqrt(2) would be great if you sold the portfolio and bought a new one each day that was rebalanced to equal amounts of each asset, another unrealistic assumption they make since trading has costs. But try to work it out where you are holding A+B and you’ll get a different answer.

As for the return: Asset A and Asset C have logarithms that behave the same way in the long run but the convergence is slow enough that the Assets themselves diverge.

A’s daily return factor is 0.5(1.01^3)+0.5(1.01^-3) = 1.000446. C’s is 0.9(1.01)+0.1(1.01^9) = 1.000458. Annualized these are gains of 11.78% and 12.12% respectively.