# Using investment portfolio risk to explain population biology

One of the problems that I frequently have to grapple with is how to emphasise the importance of protecting small populations to senior management who are primarily focussed on the economics of the business. So I delved into some population biology models and found a way to express population growth using some equations that are immediately recognisable to those more used to dealing with financial concerns. This has worked well in the past and (hopefully) may be of some use to those who may face a similar challenge.

You might think that in a situation where the average birth and death rates are equal that populations would remain stable but reproduction rates and survival rate are subject to random fluctuations. So while the average reproduction rate for the population might be 10, for any one particular individual the reproduction rate is a poison random variable. Similarly for any particular individual the probability of survival is also a random variable.

To calculate the size of a population (N) after a unit of time (t) we can use:

• N(t+1) = N(t).R(t) – deaths

Where

• N = Size of population
• R = the reproductive rate

This equation can also be applied to calculate the growth of a financial investment by simply substituting

• N = Dollar value of portfolio
• R = the interest rate received
• Deaths = fixed costs

But remember that reproduction rates – and interest rates – can vary over time. If you were to start with a portfolio of only \$10 dollars then a series of low rates can lead to going broke. But with a portfolio of \$10,000 the effect of a series of low rates is buffered by the relatively large size of the portfolio. To protect a small portfolio it is important that you take action to avoid a reduction in interest rate until the portfolio has grown to the point where the ratio of variance in interest rate is small in comparison to fixed costs. Or in terms of a small population it is necessary to avoid the introduction of any additional stresses until the ratio of natural variance is small in comparison to death rate.

Darwin Himself noted that ” .. rarity is the precursor to extinction. We can see that any form represented by few individuals will, during fluctuations in the seasons or in the number of its enemies run a good chance of utter extinction”.