Volatile Projections: Part 2

In my last behemoth I postulated about getting some volatility into your projections of your fund, a useful tool in checking potential future variations in your progress towards a financial goal.  Be it Financial Independence or some other savings goal.

Normal Distribution
I started in Part 1 by using the uniform distribution as a base to generate random numbers from, which were then converted into a return.

The uniform distribution is only of limited use in this scenario.  A much more useful distribution is the Normal Distribution.

  A normal distribution has two parameters.

– The mean (in the example above this is 0).  We could set the mean to equal to the expected return of an asset or a portion of our portfolio.

– The Variance (the Standard Deviation squared).  The example above shows the impact of increasing the standard deviance (or variance).  The impact is to increase the probability of an individual sample being further away from the mean.

We can think of the range of returns that a fund/stock/asset could generate as following a normal distribution.  (This isn’t a true reflection of what historic distribution of returns actually look like, but it is a useful simple approximation.)

A LogNormal distribution
We could use a Normal distribution to generate a random number for our return over a given time period.  Alternatively we could use the LogNormal distribution to generate a random number for our projected fund value.  A subtle difference there, the first is generating a return and the second a fund value.

A function will be LogNormally distributed if the natural log of it is normally distributed.  WTF?  Essentially this relationship will let us look at a fund value/stock price as having the following relationship;

S1 = S0*exp(rT)

Where;
S1 = Fund value after one time step.
S0 = Fund value at the outset (where t = 0)
exp[]= is the natural exponential function
r = the expected rate of return (continuously compounded)
T = the time step (i.e. the time that has passed from S0 to S1)

This is really just another way of writing the standard projection we may normally use, but using the exponential function rather than the normal ‘to the power of’;

(fund value at t=0)*(1+rate of return)^number of time periods
E.g. Fund value at end of  two year = 1000*(1.05)^2
So…..if we are assuming that our returns are normally distributed then we can say that the fund value is LogNormally distributed.
S1/S0 = Exp(rT)
Taking natural logs (which removes the exponential);
Ln[S1/S0] = rT
With r (the return) being normally distributed we can see that our fund value is therefore LogNormally distributed.


Exponential instead of normal compounding
We are using the exponential function instead of raising (1+return) to the number of time periods as this lets us utilise the LogNormal property and makes the maths a bit simpler.  The exponential function assumes that the interest is continuously compounding.  This doesn’t actually happen in reality and will give us a slightly higher fund value at the end.

Normal compounding
>   1,000*(1.05)^30 = £4,322

Continuously compounding
>   1,000*EXP(0.05*30) = £4,482
That’s the maths out the way
The maths is interesting, but not essential.

Let’s use this property to project a fund while using the LogNormal distribution to generate our projected future fund values.

Using the following formula in sweet Excel;
Fund value at end of time period =Fund value at start * EXP((Expected Return*(Std deviation * NORMSINV(RAND()))*Timestep) 
Or;
=1000*EXP((0.05+(0.2*NORMSINV(RAND())))*(1/12)) 
Which is the fund value after one month of a fund starting at £1,000 with an expected annual return of 5% and volatility of 20% (standard deviation).  Copy it into excel and refresh it a few times and you should see the projected value at the end of one month bobble around a bit.  
So we just use this formula over and over, each step representing a month, replacing the £1,000 with the fund value calculated in the prior step.  If we do this 360 times we have a projected fund value at the end of 30 years.  We can run several simulations at the same time and have a look at the spread of results;
LogNormal Fund value Distribution – 10 runs
The darker line in black is the ‘base case projection’, so £1,000 projected to grow at 5% a year for 30 years.  =1,000*EXP(0.05*30)

So we can see some variation about the expected mean of £4,481.  I think where this comes in useful is looking at the potential downside, i.e. the worst case.  You could, for example run 1,000 simulations and see where the worst 2.5% of cases lie.  If this was borne out in reality, would it destroy your plans?  If so, maybe the plans need tightening up a bit.  Perhaps investing more in less volatile assets

That’s enough for now 

I will next build in the ability to add monthly contributions into the simulator, i.e. so it’s not just £1,000 projected, but can be updated to reflect our actual savings rate.

And the ability to contribute to multiple asset classes, so we can input the expected return and volatility of different asset classes and start to build up an asset allocation that represents our own portfolio.

Exciting times.

Mr Z

7 thoughts on “Volatile Projections: Part 2

  1. Cerridwen

    Don't you think that this is all ultimately an attempt to impose certainty on an uncertain world – even if it just amounts to confirming that uncertainty is all we can rely on 🙂

    "The maths is interesting" – maybe so, if you understand it. I'm just glad there's someone out there who does. Thanks for having a go at explaining it to me. When I'm retired I intend to stretch my brain exactly in this direction. 🙂

    Reply
  2. ermine

    I would be very wary of using a normal distribution for stock prices. They do not observe the central limit theorem which is predicated on the variables being mutually independent. Prices are set by human actions, and humans are ornery sorts who tend to panic and celebrate together, because we are social animals. The presumption of independence falls, the upshot is that stock market stats are fat-tailed. The panics come along and kill you because events that should be rare at three-sigma and more come along far more often than you expect.

    Proceed with caution and maybe a copy of Nasseem Taleb's The Black Swan! In particular, the excess number of three-sigma events over your prediction will make you overconfident against worst case scenarios, you will think they are less likely than they really are. Previous people who found this out the hard way include Long Term Capital Management, and a lot of banks in the 2008/9 financial crisis

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  3. Mr Zombie

    Haha – perhaps in a way. And it uses pseudo random numbers anyway, so it's not truly random in that sense 🙂

    It is interesting, but the more you look into it, the more complex it can get. And it's normally underpinned on simplifying assumptions…and that can be dangerous!

    Mr Z

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  4. Mr Zombie

    Hi Ermine,

    When I looked at Financial Economics an underpinning assumption is nearly always that investors are rational. Haha – about as wrong as it can get!

    I thought the normal distribution is useful here, only because it is simple and can easily be used in Excel. But should probably highlight that its not a great fit to actual data….

    Like you say, the assumption that returns will follow the Normal is dangerous as we don't see the crashes that happen and in general the projections will trudge on upwards. I suppose you could use this, coupled with a random shock (a drop in prices of 30-50% say) and see how your portfolio performs? In an attempt to try and keep things simple.

    We are using EGB2 distributions at work at the moment. As I understand it is a 4 parameter distribution that's far more malleable that the Normal distribution and can be fit to the data more accurately (by playing around with the skew and kurtosis). I don't work in that department at the moment but I quite fancy a stint there.

    Hmmm – perhaps I should highlight the limitations of using it more, besides there are some incredible Monte Carlo sims around that can be used. (At the expense of understanding exactly what's going on!)

    Is The Black Swan a good read? I tend to alter between an easy book and an 'academic' one. I'm just finishing one of the Game of Thrones books and on the look for another bool (so I can avoid trying to read the origin of species again).

    Mr Z

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  5. ermine

    You clearly know more about this than I do 😉 Appreciated you probably wanted to keep the post generally understandable, but the dark side of those fat tails need highlighting.

    I heard The Black Swan as an audiobook, which is a terrible way to read anything and often abridged so I don't know how it reads as a real book, though it was fascinating. I wouldn't describe it as 'academic', although your analytical background will make it an easier read!

    Reply

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