Page 15 of 46 4th July 2019 businessbusiness

Page 15 of 46 4th july 2019 businessbusiness

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Page 15 of 46 4th July 2019 - risk/content-section-0
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Investment risk 2.2 Calculating returns Our first question was: what is the mean expected total return for next year? We calculate this by taking each of the possible returns and weighting it by its relative probability. As our table is so simple and symmetrical, it is not difficult to see that the weighted mean return is 7% per annum. Our second question was: what is the degree of risk or uncertainty in this mean figure? In other words, how widely dispersed are the possible outcomes around the mean expected outcome? The most commonly used statistical measure of dispersion is the standard deviation. We need to be aware of some limitations in its use, but first here is how the calculation would look in the case of returns on Company X's shares. We shall use the notation E (R) to denote the mean expected return, which is 7% per annum in this case. Table 2 Calculation of standard deviation of actual total returns on a one-year investment in shares of Company X in each of the last 50 years A n n u al re tu r n R I Dis pe rsi on E (R ) − R i Sq ua re of dis pe rsi on [ E( R) − R I ] 2 Pr ob abi lity P I P i [ E(R ) − R i ] 2 Page 16 of 46 4th July 2019 - risk/content-section-0
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Investment risk 4 +3 9 0.2 1.8 6 +1 1 0.3 0.3 8 −1 1 0.3 0.3 10 −3 9 0.2 1.8 Sum of the squares of the probability-weighted dispersions (= variance) V = 4.2 Square root of variance ( V) = standard deviation The standard deviation is normally expressed in the same units as the expected return (in this case, per cent per annum) and is intuitively easier to understand than the variance, especially with a normal distribution where the possible outcomes are symmetrically dispersed around the mean. The particular usefulness of the standard deviation of normally distributed data is the way it divides up the data so that: 68.3% of the data points lie within one standard deviation on either side of the mean; 95.4% of the data points lie within two standard deviations on either side of the mean. In the case of an investment in shares of Company X, the calculation in Table 2 tells us that there is: a 68.3% probability that the return will lie between 4.95% (i.e. 7 – 2.05) and 9.05% (i.e. 7 + 2.05), and Page 17 of 46 4th July 2019 - risk/content-section-0
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Investment risk a 95.4% probability that it will lie between 2.9% (i.e. 7 – 2 x 2.05) and 11.1% (i.e. 7 + 2 x 2.05). The principal limitation on the use of standard deviation as a measure of dispersion in investment returns is that in the real world actual returns are not as neatly or symmetrically dispersed as they were in our Company X example. However, the shape of many long-run statistical series for investment returns is in fact quite close to the familiar bell-curve of the normal distribution. So for most practical purposes the standard deviation is as useful a measure as we are likely to find.
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