Portfolio Selection: Efficient Diversification of Investments (Cowles Foundation Monograph: No. 16)
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Embracing finance, economics, operations research, and computers, this book applies modern techniques of analysis and computation to find combinations of securities that best meet the needs of private or institutional investors.
entries in column 3; column 5 presents the product of the entries in columns 2 and 4. The sum of column 5 is the average of the squared deviations from the average. The sum of column 5, in other words, is the variance of the series. The standard deviation is the square root of the variance. In the example under discussion the standard deviation equals V.0034 = .058. The relationship between returns and their standard deviation is illustrated in Figure 1. The asterisks in Figure 1 indicate annual
C)2 is obtained from the second by making replacements such as A2 for A- A and 2AB for AB + BA. THEOREM: If /?,?, and r are a triplet of random variables generated by any wheel, and if s is the random variable/? + q + r, then varfr) « var(/>) + cov(/?, 9) + cov(/?, r) + co\(q, p) + var(^) + cov(?, r) + cov(r,/>) + cov(r,f) + var(r), or, equivalently, var($) « var(/?) 4- var(?) -f var(r) + 2 cov(/?, ^) 4- 2 cov(/>, r) + 2 cov(^, r). The second version of varfa) follows immediately from the first
always equal to approximately 2.3026 times the common logarithm of the number. We shall let log(/4) represent the common logarithm of A; we shall let Ln(A) represent the natural logarithm of A. Both the common logarithm and the natural logarithm have the following extremely important properties: 1. The logarithm of A times B equals the logarithm of A plus the logarithm of B. More generally, the logarithm of Al • >42 ' ^3 * ' * ^n equals the logarithm of Al plus the logarithm of A2 plus the
return £ and variance of return var(r). For example, if £ = .1 and var(r) = .04, then the expected value of the natural logarithm of (1 + r) is approximately Ln(l.l) - [(1/2)(.04)/(1.1)2] = .095 - .017 -* .078. The anti-log of .078 is 1.08. The rate of growth over the long run may be expected to be about 8 % per annum. RETURN IN THE LONG RUN 125 The combination of expected return and variance which promises the greatest return in the long run is not necessarily the combination which best
expected return has properties similar to the curve relating £ and K. The chief difference is that the former is principally made of segments of hyperbolas rather than parabolas. If the analysis includes a security with zero variance, one segment of the curve relating efficient standard deviation to efficient expected return is a straight line, as illustrated by the segment starting at (£, a) = (0,0) in Figure 4 of Chapter II. The curve relating efficient standard deviation and efficient expected