The Power of Exponentials
Imagine you invest €100 in a long-term deposit at your bank, which promises you 2% fixed annual return. After one year:
€100 * 2% = €2
You have €2 profit. Or:
€100 * (100 + 2%) = €102
€100 * 102% = €102
Your initial investment grew from €100 to €102.
Percent is a number expressing a fraction of an amount of hundred units (In French, par cent literally means per hundred). Mathematically, 100 units of hundreths equals 1; As an example 24% only amounts to twenty four of the units and mathematically equals 0,24. As a consequence, the formulae above, can be rewritten as:
€100 * 0,02 = €2
Or, 2% of your initial investment, including your initial investment (100%) equals:
€100 * (1,00 + 0,02) = €102
STEP 1 - Understand Percentages
Step 2 -
Capitalise or distribute
Now imagine the banker asking you if you want to capitalise your interest profits, or you'd like to redistribute them. Capitalising your interests simply means that the banks pays you your interests in cash. Redistributing your interest means that the bank will use your interests and reinvest them on top of your principal amount. Your interest will be added to your principal amount and will, on its turn, generate additional interest.
Year 1 : €100 * 1,02 = €100 (invested) + €2 (profit)
Year 2 : €100 * 1,02 = €100 (invested) + €2 (profit)
Year 3 : €100 * 1,02 = €100 (invested) + €2 (profit)
Year 4 : €100 * 1,02 = €100 (invested) + €2 (profit)
Year 5 : €100 * 1,02 = €100 (invested) + €2 (profit)
After five years = €100 (invested) + 5 * €2 (profit) = €110
Year 1 : €100,00 * 1,02 = €100,00 (invested) + €2,00 (re-invest)
Year 2 : €102,00 * 1,02 = €102,00 (invested) + €2,04 (re-invest)
Year 3 : €104,04 * 1,02 = €104,04 (invested) + €2,08 (re-invest)
Year 4 : €106,12 * 1,02 = €106,12 (invested) + €2,12 (re-invest)
Year 5 : €108,24 * 1,02 = €108,24 (invested) + €2,16 (re-invest)
After five years = €100 * 1,02 ^ 5 (invested) = €110,41
Now although there might not seem to be much apparent difference between capitalisation and distribution in this case, the difference becomes (literally) exponentially more visible when the annual return rate increases.
Capitalisation of 10 years at 10% annual return on €100 = €300
Distribution of 10 years at 10% annual return on €100 = €672,72
In the graphs on the left, the difference between capitalisation and distribution between an annual return rate of respectively 2%, 10% and 50% is displayed. The graph below shows the difference between the final return after 10 years of distribution for an annual return rate of respectively 2%, 10% and 50%.
The key takeaways of the power of exponentials is that
Distribution can generate you substantially more wealth compared to capitalisation. Especially when taking into account inflation.
The power of exponentials is driven by two main factors (under distribution):
Time: The amount of time you can remain invested, has an exponential effect on the final value of your investment. The longer you can invest, the substantially higher your final investment value will be.
Rate of return: The rate of return has an exponential effect on the final value of your investment. The higher your rate, the substantially higher your final investment value will be.
Initial amount: The initial amount of your investment has a linear effect on the final value of your investment. Multiplying your initial amount by a factor, will have a same multiplication effect on the final value of your investment.
closing note -
50% loss is worse than 50% profit
Imagine you invest €100 in the stock market, and two scenarios happen. On day 1, in scenario one, your portfolio value increases with 50% to €150. In scenario 2, your portfolio value decreases with 50% to €50. After day 2 of trading, both portfolios however, are one again exactly valued at €100.
€100 * (100% + 50%) = €150
€150 * (100% - 33,33%) = €100
€100 * (100% - 50%) = €50
€50 * (100% + 100%) = €100
While both scenarios induce a simple 50% increase, 50% decrease respectively, note how in scenario 2, in order to re-attain €100, the portfolio value should increase with a 100% in order to get back to its original €100, while in scenario 1, only a 33,33% loss is required to re-attain €100.
Portfolio Value (PV) = (PVtoday - PVyesterday) / PVyesterday
The relative portfolio performance of today is based on the portfolio value of yesterday. In nominal terms, an X percentage of loss, is always more substantial in absolute terms compared to a profit. Think about it...
When you suffered a loss of 50%, your exposure went down with 50%, meaning that you have way less capital left to re-attain your previous height. In the other way, when you enjoy a 50% increase, your exposure went up with 50%, meaning that you are more exposed to potential losses.