The initial value is the value of investment at the beginning of the period, usually the purchase price. The last value is the value of investment at the end of the time period.

Let’s consider the example of purchasing of two stocks. The price of first is $40 and second $50. If price of both goes up by $5, the absolute value of each investment will grow by the same amount. But we paid less for the first stock, therefore the rate of return per dollar of investment will be higher. Using the above formula we get for the first stock:

and for second stock:

As we see, in terms of percentage, the rate of return of the first stock is 12.5% and of the second is 10%. In these calculations we didn’t include transaction costs, dividends, etc., so our calculations were approximate. To get the exact value of investment return all the costs involved should be included in calculations. For example, let’s say we paid $4 purchasing commission for our stock and after one year we received $2 dividends. Purchase price was $40, last price was $45. Then,

If we sell investment we should include sell commission in calculations and if we consider time periods when we just hold investment, we won’t have transaction (buy/sell) commissions.

If we have several investments the overall rate of return will be computed by taking a weighted average of rates of individual investments where the weights are based on the initial investments:

where R(A), R(B), R(C), ... are rates of returns of investments with initial
values of A, B, C, ...

For example let’s consider the following investments with corresponding
rates of returns:

Initial Investment Rate of return $400.00 4% $250.00 7% $750.00 9% $500.00 15%The overall rate of return will be:

( $400.00 * 0.04 + $250.00 * 0.07 + $750.00 * 0.09 + $500.00 * 0.15 ) / $1900.00 = ( $16.00 + $17.50 + $67.50 + $75.00 ) / $1900.00 = $176.00 / $1900.00 = 0.0926 (9.26%)

where R(1), R(2), R(3), ... are returns for 1, 2, 3, ... time
periods (years, months, etc.) and N is the number of this periods.

For example, suppose the stock had the following rates of return:

Year Rate of Return 1 10% 2 15% 3 12% 4 8% 5 19%Then

The geometric mean is calculated by formula:

where RootN() is a N-th degree root (rest of the symbols have the same meaning as in arithmetic mean formula). For our stock from previous example we have:

Geometric Mean = Root5( (1+0.10)*(1+0.15)*(1+0.12)*(1+0.08)*(1+0.19) ) - 1 = = Root5( 1.10 * 1.15 * 1.12 * 1.08 * 1.19 ) - 1 = = Root5( 1.82087 ) - 1 = = 1.12734 - 1 = 0.12734 (12.73%)The geometric mean measures the compound rate of change of investment return while the arithmetic mean doesn't take into account compounding, it just gives average rate of return. When the rates of return for different time periods are the same, arithmetic and geometric means are equal. As returns become more varied, two measures become more different. Using arithmetic mean is recommended in case when rates of return for different periods doesn't differ too much, otherwise the geometric mean is more appropriate. For example, let's say we invested $100 for two year period. First year we lost $50(50%) and the second year we gain back $50(100%), then:

Year Rate of Return Year End Value 1 -50% $100*(1-50%) = $50 2 +100% $50*(1+50%) = $100 Arithmetic Mean = ( -50% + 100% ) / 2 = 25% Geometric Mean = SquareRoot( (1-0.5)*(1+1.0) ) - 1 = 0%For this case the arithmetic mean doesn't seem to be appropriate measure. We end up with same amount $100 after two years and our return is not 25%. The geometric mean shows proper value because it takes into account the fact, that when during second year our investment doubled, this increase applied on halved value ($50) of initial investment ($100).

The Sharpe's (proposed by William Sharpe) measure is calculated by formula:

where RP is average rate of return for portfolio during a period of
time, RF is risk free rate of return during the same period, StDev is
the standard deviation of returns which is used as a measure of risk of
portfolio. This measure can be interpreted as the excess return per
unit of risk. Portfolios with higher value of Sharpe's measure are
considered performing better.

For example let’s compute Sharpe's measure for following four stocks
(let's say risk free rate of return is 5%):

Stock Return Standard Deviation: Sharpe's Measure: A 8% 15% ( 8%-5%)/15% = 0.20 B 11% 7% (11%-5%)/ 7% = 0.86 C 15% 10% (15%-5%)/10% = 1.00 D 18% 19% (18%-5%)/19% = 0.68As we see, although stock D has higher return than stock C, due to high standard deviation of stock D, the performance of stock C by Sharpe's measure is much better.

Another widely used performance measure is Treynor's (proposed by Jack Treynor) measure, which is computed by formula:

As we see Treynor's measure is similar to Sharpe's measure except denominator, where standard deviation is replaced by systematic risk measure Beta, which reflects behavior of investment correlated to market index. More precisely, Beta represents the estimated change in the return of the investment for one unit change in the return of the market index. Let's compute Treynor's measure for stocks in previous example accepting values for Beta equal to 1.3, 1.7, 1.1, 0.8 for A,B,C and D stocks correspondingly:

Stock Return Beta: Treynor's Measure: A 8% 1.3 ( 8%-5%)/1.3 = 2.31 B 11% 1.7 (11%-5%)/1.7 = 3.53 C 15% 1.1 (15%-5%)/1.1 = 9.09 D 18% 0.8 (18%-5%)/0.8 = 16.25In this case stock D performs much better due to it's lower Beta.

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