Why Long-Horizon Asset Growth Can Have Greater Variance and a Lower Mode
Summary
The document examines a spreadsheet that models portfolio wealth with a lognormal distribution derived from expected return and risk, then scales the log-return distribution across multiple years. The resulting wealth distributions spread out over time, while their modes fall below the starting value in the example. The question asks whether this conflicts with portfolio theory and observed asset growth.
The answer challenges the premise that variance should shrink as an investment horizon lengthens: a wider range of possible outcomes over longer periods is not inherently inconsistent with uncertainty. It also explains that a low return combined with high risk can produce a most likely outcome below the initial wealth because a lognormal distribution is asymmetric. The response offers conceptual interpretation rather than checking the spreadsheet’s parameter formulas or validating the model against market data. Its conclusions therefore address the stated distributional behavior, not whether the chosen inputs or assumptions accurately represent a real portfolio.
Key ideas
- A longer horizon can produce a wider distribution of possible wealth outcomes.
- A falling mode does not imply that the expected return is negative.
- The mode of a skewed lognormal distribution can lie below the starting wealth when risk is high relative to return.
- The response interprets the output but does not verify the spreadsheet formulas or empirical assumptions.
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Full text
# What's wrong with this asset growth simulation?
# What's wrong with this asset growth simulation?
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.
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?
## Answer by Michaël Le Barbier (score 4, accepted)
https://quant.stackexchange.com/a/9155
> 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.