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Sorry if this is too basic, but I have this spreadsheet that simulates asset growth of a portfolio under a given return and risk using MPT.

Here is a plot of probability distribution of asset growth derived from it (return = 0.05, risk = 0.2). It shows that the variance becomes larger as you hold the asset longer.

enter image description here

This however, goes against the conventional wisdom that variance becomes smaller as you hold the portfolio longer. The simulation also shows that with return=0.05, risk=0.2, the most likely scenario (Mode) is that your asset will be at 100% on first year, 98% on fifth year and 95% on tenth year; i.e. the growth is negative. This also seems wrong.

However, I can't see what's wrong with the method. Here is what it's doing:

1) Derive mean asset growth as follows:
    μ = LN(m)-LN((s/m)^2+1)/2
    where m is the expected return and s is the expected risk of the portfolio

2) Derive standard deviation of asset growth as follows:
    σ = SQRT(LN((s/m)^2+1))

3) Derive asset growth distribution as follows:
    y = 1/x*NORMDIST(LN(x),μ*n,σ * SQRT(n),FALSE)
    where y is frequency, x is asset growth, n is year

Now when you plot (x,y) for year n, you get the aforementioned chart.

My questions are,

a) Is the method correct according to MPT?
b) Why does it differ from how actual asset growth behaves?

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1 Answer 1

up vote 3 down vote accepted

This however, goes against the conventional wisdom that variance becomes smaller as you hold the portfolio longer.

Which conventional wisdom says this? If the variance decreases with time, then the likelyhood of getting a return close to the expected return increases (Cecbycev's inequality). So you are telling me, I know more about the long-time future as about the short-time future. It sounds really weird to me.

i.e. the growth is negative. This also seems wrong.

Why does it seem wrong? If you have a very high risk and a low return, you will most probably make a loss—beacause the log-normal distribution is not symmetrical—and this is what your spreadhseet tells you. I do not understand why you are disatisfied or puzzled by this. Explore the spreadsheet by varying numbers.

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I read about "time diversification" on a few sources and just assumed the narrative was true. Good to confirm this was false... Thanks! –  Enno Shioji Oct 14 '13 at 23:19
    
You are welcome! As I added in my answer, an important point is that the lognormal distribution is not symmetrical. Mode is $\exp(m - s^2/2)$. What are your sources for “time diversification”? –  michipili Oct 15 '13 at 5:19
    
Here is an article about the phenomena: norstad.org/finance/risk-and-time.html#fallacy This goes inline with my experience; I remember reading that you should take higher risk if you are young because on the long term "risk cease to matter" on many broker's websites. Here is an example from the largest investment bank in Japan: nomura-am.co.jp/basicknowledge/thinks-exactly/… (it says "One can reduce risk by holding an asset long term") –  Enno Shioji Oct 15 '13 at 7:08
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