By Ntende John
I have seen some comments accusing me of inflating the future value of some investments I have made in treasury bonds and other assets. I am often surprised by these comments but on reflection I realize that not everyone has studied or understands these principles properly. So in this post I will try and explain the mathematics behind the numbers.
Let’s suppose you invest a lumpsum of ugx 10m in a unit trust paying a net interest of 10% per year and you leave the cash intact for 10 years without any additional contributions. How much do you think you will have after 10 years? To answer this question we have to think about the simple compounding formula:-
Pn = Po(1+r)^n; where Pn is the future value of the investment; Po is the initial lumpsum; r is the interest rate; and n is the number of years. So in our scenario the future value is 10*(1+.1)^10 which is 25.93m.
The same idea can applied to estimate the future value of an asset like land which tends to appreciate. This time we replace the interest rate with a growth rate. So if you buy a piece of land at 20m and it appreciates at 5% per year, then in 15 years the value will be estimated to be 20*(1+.05)^15 which gives you 41.6m.
You will notice that the future value of an asset directly correlates with the initial lumpsum and the interest rate or growth rate but is exponentially correlated with time. This point is important. The person who invests 10m gets much more than the person who invests 100k. However the person who invests for 30 years gets way much more than the person who invests for 5 years. This all because of the mathematics of compounding.
Things get a bit complicated when we now try to value a series of payments or what we call an annuity. So bear with me as we go down another mathematical rabbit hole. So imagine that we save 100k per month into a unit trust which pays 12% per year for 40 years. What would be the future value of such an investment? This time we modify the compounding formula to be:- Pn = Po((1+r)^n-1)/r; where Pn is the future value of the investment; Po is the series of payments; r is the interest rate; and n is the number of periods. So in this case Po is 100k; r is 1% since we are compounding monthly (12%/12); n is 40*12 which is 480 months; this gives us 100k*((1+.01)^480-1)/.01 which is 1.2bn.
So this sounds crazy; how can 100k turn into 1.2bn in 40 years. This is where the magic of compounding comes in. The future value formula is exponential and the value increases exponentially with time. That is why time in the market often beats timing the market. Things tend to compound with time including experience and knowledge.
Things get further complicated when there is growth in the annuity. Say you save 100k per month in the first year then you increase to 150k the next year. In this case the formula even get more complicated. This time we have Pn = Po((1+r)^n – (1+g)^n)/(r-g); where g is the growth rate. So imagine you save 1.2m per year (which is 100k per month) for 40 years in a unit trust paying 12% per year; but this time the amount you save increases by 5% per year. We end up with 1.2*((1.12)^40 – (1+.05)^40)/(.12-.05) which is 1.5bn.
Enough of the mathematics. Congratulations if you have reached this part. You don’t need to memorize the formulae. There are many free online compounding calculators you can use, but it is intuitive to understand how it works. So how can we take these insights and apply them to our personal finances?
The future value of an investment depends on the amount you invest. If that amount is zero don’t expect to find anything in the future. The larger the investment the larger the return, therefore invest as much as you can.
The future value value of an investment depends on the return or interest rate. Therefore seek the highest return possible with limited downside. Be careful of too much risk however which can bring your annuity to zero or even negative when you can’t service debt.
The value of an investment depends on growth. So find ways to increase the amounts you save and invest. Same things applies in business. Find ways to grow the bottom line.
The biggest impact on the value of your assets is time. Try to stay invested for as long as you can. Hold onto the land for as long as possible. Don’t liquidate your bonds or shares. Avoid withdrawing cash from unit trusts. Stick with the same business for a very long time. Stay married to the same person for a long time! Time is your biggest advantage when it comes to investing, so start early.
So I hope I have shed some more light into the underlying mathematics of annuity evaluation and I hope you find this helpful.
