(a) The S&P/ASX200 price index opened the year at 5,777 and closed at 6,120 by the end of the year. The equivalent accumulation index went from 56,240 to 64,425. What is the annual rate of return on each of these indices? Explain the difference.
(b) Using the approach covered in your textbook calculate the geometric average annual rate of return over five years given the following annual rates, year 1 = 5.10%, year 2 = 4.95%, year 3 = 4.83%, year 4 = 4.75% and year 5 = 4.70% . What is the arithmetic average? Explain the difference.
(a) Return on S&P/ASX200 price index = (Closing index - Opening Index) / Opening Index = ( 6120 - 5777 ) / 5777 = 5.937%
Return on equivalent accumulation index = (Closing index - Opening Index) / Opening Index = ( 64,425 - 56,240) / 56,240 = 14.55%
The difference in both the indices is due to the fact that &P/ASX200 price index is a price index. it considers only the prices of the stocks, and not the dividends and their reinvestment.
Thus, the equivalent accumulation index is shows higher returns than the price index, always, because it takes into account the price change, the dividends distributed, and the reinvestment of those dividends for the remaining period, thus higher returns, naturally.
(b) Given that r1 = 5.10% ; r2 = 4.95% ; r3 = 4.83% ; r4 = 4.75% and r5 = 4.70%
Arithmetic Average = (r1 + r2 + r3 + r4 + r5) / 5 = (5.10 + 4.95 + 4.83 + 4.75 + 4.70) / 5 = 4.866%
Geometric Average Annual Rate = [ (1+r1) (1+r2)(1+r3)(1+r4)(1+r5) ] 1/5 - 1
= [ (1.0510)(1.0495)(1.0483)(1.0475)(1.0470) ]1/5 - 1
=(1.2681524295 )1/5 - 1
=1.048659 - 1 = 0.048659 = 4.8659 %
We find that there is a slight difference between the two rates.
The difference is because Geometric average annual rate considers the compounding effect. Thus, it calculates the actual rate of return on investment.
The arithmetic return can be used as a quick estimate, but it is less accurate, and overstate or understate the returns, depending case by case.
In the given question, the difference is not huge, and hence, the difference is not clearly visible.
But if you take an example in which there are losses in one or two years, or if you take another example where the variance between the returns is high, then you will see how the arithmetic mean over or understates the returns, whereas geometric average gives the accurate result,