Calculating rate of return of a long-term investment

 By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

06 September 2015

After reading last week’s column, Mike Maingi – the same one who asked the original question – followed up with a second one: “How did you arrive at 9.27 per cent annual return?”. As a reminder, the article was about the returns expected from a temporary shelter project. Mike’s investment club is considering putting one million shillings in it and we found that, after ten years, their investment will have grown to Sh2,427,900.

Without showing the calculation, I stated that this comes to about 9.27 per cent per annum average return. So now, Mike’s question is: how did I get that?

Suppose you put Sh100,000 in a bank account that pays one per cent compound interest per month. Now I must first warn that I am using one per cent per month in order to simplify the calculations. No bank in Kenya will pay you such a high rate!

At the end of the first month, your money will earn Sh1,000. Since it is compound interest, this Sh1,000 is added to you balance to make Sh101,000. In the second month, the one per cent interest is calculated on Sh101,000. This comes to Sh1,010, so the new balance is Sh102,010.

In the third month, the interest is calculated on Sh102,010 and it comes to Sh1,020.10. This is added to the previous balance of Sh102,010 to make a total of Sh103,030.10. Now let us stop there and find out what is going on mathematically.

One per cent is one divided by one hundred which is 0.01. So, the result of adding one per cent to any number is the same as multiplying it by 1.01. Thus, at the end of the first month, the balance in your account is 1.01 x Sh100,000 (=Sh101,000).

At the end of the second month, the balance is 1.01 x Sh101,000 (Sh102,010). That is; 1.01 x 1.01 x Sh100,000 – remember that Sh101,000 = 1.01 x Sh100,000. By similar logic, it should be clear that at the end of the third month, the total balance comes to 1.01 x 1.01 x 1.01 x Sh100,000 = Sh103,030.10. Try it out in your calculator and see.

Now, if some one said that he had invested Sh100,000 for three months and it rose to Sh103,030.10, how would you calculate the monthly compound interest rate? Simple: first divide the final amount (Sh103,030.10) by the starting figure (Sh100,000). The answer is 1.030301.

Secondly, since the money was invested for three months, find the cube root of 1.030301. The answer is 1.01. Finally, convert this to a percentage: it is 101 per cent. Therefore, the monthly rate is one per cent.

This is exactly what I did with Mike’s housing project. The final amount (Sh2,427,900) divided by the starting figure (Sh1,000,000) is 2.4279. Since this is returned after ten years, I calculated the tenth-root of 2.4279 (using a scientific calculator, of course!).

The answer is 1.092755. That is, 109.2755 per cent. Therefore, the annual return is about 9.28 per cent.