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.
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 = 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’;
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.
> 1,000*(1.05)^30 = £4,322
> 1,000*EXP(0.05*30) = £4,482
Let’s use this property to project a fund while using the LogNormal distribution to generate our projected future fund values.
|LogNormal Fund value Distribution – 10 runs|
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
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.