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Investment Performance

In order to make intelligent investment decisions itís necessary to be able to evaluate different investments and compare them, or in other words, measure their performance. The rate of return is the simplest and most quoted measure of performance. It can be calculated for single or multiple periods of time. However, the rate of return doesnít take in account the risk associated with investment. In order to fully evaluate the investment itís necessary to adjust the rate of return for risk level.

Rate of Return for Single Period

The rate of return over single period of time can be computed by formula:

Rate of Return = (Last Value - Initial Value) / Initial Value = Change / Initial Value

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.

Letís 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:

($45 - $40) / $40 = 0.125

and for second stock:

($55 - $50) / $50 = 0.100

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 didnít 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, letís 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,

Initial Value = $40 - $4 = $36
Last Value = $45 + $2 = $47

Rate of Return = ( $47 - $36 ) / $36 = 0.306 (30.6%)

If we sell investment we should include sell commission in calculations and if we consider time periods when we just hold investment, we wonít 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:

Rate of Return( A,B,C,... ) = ( A*R(A) + B*R(B) + C*R(C) + ... ) / ( A+B+C+... ),

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

Rate of Return for Multiple Periods

There are two types and corresponding methods of calculating of average or mean rate of return for multiple periods - arithmetic mean and geometric mean. The Arithmetic mean is calculated by dividing the sum of the rates of returns for different periods of time (years, months, etc.) by the number of this time periods:

Arithmetic Mean = ( R(1) + R(2) + R(3) + ... ) / N,

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

Arithmetic Mean = ( 10% + 15% + 12% + 8% + 19% ) / 5 = 12.8%,

The geometric mean is calculated by formula:

Geometric Mean = RootN( (1+R(1))*(1+R(2))*(1+R(3))*...*(1+R(N)) ) - 1,

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).

Performance Adjusted for Risk

In order to compare different investments the rates of return is not sufficient. Two investments may have different rates of return but investment with higher return may be riskier than investment with lower return. For this reason it's more informative to have performance measure which reflects risk of investment. There are several widely accepted measures of return incorporating risk.
The Sharpe's (proposed by William Sharpe) measure is calculated by formula:

Sharpe's Measure = (RP - RF) / StDev,

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 letís 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% = 0.68
As 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:

Treynor's Measure = (RP - RF) / Beta

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 = 16.25
In this case stock D performs much better due to it's lower Beta.


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