# Profitable reasons to understand growth rates

Two weeks ago, due principally to an acute case of writer's cramp following an absurdly long and torturous game of tennis, I missed the opportunity to contribute to Personal Finance's series of articles on financial literacy.

This was a great shame since my colleagues in the office had made an effort in providing me with some imaginative topics. "Where will you find the FTSE?" one wit asked. "At the end of your legsie" was the best he could come up with. My favourite was the definition of the Dow Jones: a Welsh-owned wooden barge used for crossing Dubai Creek.

Entertaining stuff, but I had already decided to write about my second-most favourite subject - the inestimable beauty of mathematics, and my pet hate - the inability of otherwise intelligent people to calculate a percentage growth rate.

Before writing on this subject, I tested it on a few friends and, without exception, they all said it was impossible to write an absorbing article about the calculation of percentage growth rates. (By the way you are wrong about my most favourite subject - it is, in fact, astrophysics.)

Never one to refuse a challenge, here I go.

If I tell you that a share price has moved from a value of US\$100 (Dh367) to \$120 in a year, all readers of The National would have no hesitation in telling me that the price had increased by 20 per cent per annum. Easy. But if the base price had been, say, \$130 and it had moved to \$169 over the same period, how would you calculate the percentage increase?

Most people would subtract the lower figure from the higher one to give \$39. They would then divide this by the base price of \$130 and multiply by 100 to give 30 per cent increase.

The easier option is to divide the new value by the old value to give the ratio 1:30, which shows immediately that there has been a 30 per cent increase. Are you still with me? Is this fascinating stuff? My friend Zee tells me that she would be yawning by now, but I think, unlike most of my readers, she fails to see the inner beauty of mathematics.

What we have done in the above analysis is convert from a simplistic additive approach (for example, looking at the difference in the two values) to a multiplicative model. Instead of looking at the difference in values from the old to the new and expressing this as percentage growth, we say that the old value has increased by a factor of 1.3 (or 130 per cent). This might not seem like a huge philosophical leap, but it helps enormously when the problems get more complicated, as the following example shows.

In the above example, we saw that the annual rate of growth was 30 per cent. If it continues at that rate, what will the share price be in two more years?

With the long-winded "additive" approach, we would calculate the growth in the second year at 30 per cent of \$169. This comes to \$50.70 and, when added to \$169, gives you a new value of \$219.70 at the end of the second year. In the third year, the incremental growth will be 30 per cent of \$219.70, which turns out to be \$65.91 and, when added to \$219.7, gives a final value of \$285.61.But in the multiplicative approach, all you have to do is calculate \$169 x 1.3 x 1.3 and you will get the same answer immediately.

When dealing with growth rates, stop thinking additively and start thinking multiplicatively.

I wonder how many readers have got this far. I fear that Zee has long since fallen asleep. That is her loss because I have saved the best 'til last ... Monte Carlo simulation! This is a mathematical technique used to solve probabilistic (or stochastic) models. It can be used to predict future scenarios for retirement. For example, if you make the following assumptions about an investor:

• Number of years to retirement: 15;
• Number of years during retirement: 30;
• Value of investment portfolio: \$300,000;
• Current saving rate: \$6,000pa;
• Annual investment growth before retirement: 6 per cent (net of inflation);
• Annual investment growth during retirement: 4 per cent (net of inflation).

This model assumes that the average investment growth rate is 6 per cent (net of inflation), while the investor is still working and saving. When he retires and is more cautious about his investment, the average growth rate falls to 4 per cent. Importantly, the risk (or volatility) also falls. In this example, I have assumed that volatility is 15 per cent while the investor is working. I will need a further 900 words to explain what this means, but suffice to say that it approximates the risk level of mature markets equities. And when he retires, the volatility falls to 7.5 per cent. This risk level equates to a bond-dominated portfolio.

The key question is: does the investor have enough money to retire for 30 years while drawing down at \$30,000a year? The Monte Carlo model answers this question by randomly generating future scenarios for investment growth rates and, using the "multiplicative" approach to the calculation of compound growth (as described above), to establish if there is enough money in the pot to last 30 years. In this example, it calculates that there is a 73 per cent chance of success. To improve this chance, the investor can increase his savings rate, retire later or lower his income requirements during retirement.

So there you are. A simple philosophical shift from additive to multiplicative thinking allows complex mathematical models to be taken in your stride. Wake up, Zee, it's all over.

Note: if you wish to try out this model yourself with your own parameters, you can do so at www.mondialdubai.com/retirementplanning