Mathematics Question

Please calculate the historic decile statistics: I have attached the excel sheet which needs to be filled out

i. average return (remember to divide the average by 100 to convert to decimal)

ii. variance

iii. standard deviation

iv. compound return

In the excel sheet comment on your findings. What do you find? How do small firms perform relative to large firms?

The Geometric (Time-Weighted) Average Return

The arithmetic average provides an unbiased estimate of the expected future return. But what does the time series tell us about the actual

performance of a portfolio over the past sample period? Let’s continue @ Example 5.4 using its very short sample period just to

illustrate. We will present results for meaningful sample periods later in the chapter.

Column F in I Spreadsheet 5.2 shows the investor’s “wealth index” from investing Sl in the portfolio at the beginning of the Page 132

first year. Wealth in each year increases by the “gross return.” that is. by the multiple (1 + HPR), shown in column E. The

wealth index is the cumulative value of Sl invested at the beginning of the sample period. The value of the wealth index at the end of the

fifth ver. $1.0275. is the terminal value of the Sl investment, which implies a 5-vear cumulative return of 2.75%.

An intuitive measure of performance over the sample period is the (fixed) annual HPR that would compound to the same terminal value

resulting from the sequence of actual returns in the time series. This is called the geometric or compound rate of return. Denote the

geometric rate by g, so that

Terminal value = (1 + r1) X (1 + 72) x -. X (1 + rs) = 1.0275

(1 + g)” = Terminal value = 1.0275 (cell F6 in Spreadsheet 5.2)

8 = Terminal value!

– 1 = 1.02751/5

1 = .0054 = .54% (cell F14)

(5.14)

Practitioners often call g a time-weighted (as opposed to dollar-weighted) average return to emphasize that each past return receives an

equal weight in the process of averaging. This distinction can be important because investment managers may experience significant

changes in funds under management as investors purchase or redeem shares. Rates of return obtained during periods when the fund is

large have a greater impact on Anal value than rates obtained when the fund is small. We discuss this distinction in @ Chapter 24 on

portfolio performance evaluation.

Notice that the geometric average return in L Spreadsheet 5.2, 54%, is less than the arithmetic average, 2.1%. The greater the volatility

in rates of return, the greater the difference between arithmetic and geometric averages. If returns come from a normal distribution, the

expected difference is exactly half the variance of the distribution, that is,

E (Geometric average) = E [Arithmetic average| – 1/20?

(5.15)

(A warning: I use @ Equation 5.15, you must express returns as decimals, not percentages.) When returns are approximately normal.

[ Equation 5.15 will be a good approximation.

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The discrepancy between the geometric and arithmetic averages arises from the asymmetric effect of positive and negative rates of

returns on the terminal value of the portfolio. (4 Example 5.5 illustrates this effect.