Читать книгу Millard on Channel Analysis - Brian Millard - Страница 6

Before we can proceed any further with a discussion of the merits of different investment strategies, we have to get clear in our minds the various ways in which we can calculate and compare gains (or losses) in investment capital. The most common way of calculating a gain is to express it as a percentage change from the starting value. Thus if an investor starts with £1000 and turns it into £2000 over a certain period of time then quite obviously he has made a gain of 100%. A different way of expressing the gain is to consider it as a factor by which the starting amount has to be multiplied. In this present example the investor has doubled his money, and therefore the gain factor is 2. If we deal with numbers that are not so round, then for example an investor turning £1000 into £1450 over a period of time will have made a gain of 45%, while the gain factor is 1.45. In this chapter we will be using both gain factors and percentage gains. It is easy to convert from percentages to gain factors and vice versa by the simple formulas:

Gain factor = (100 + percentage gain)/100

and

percentage gain = 100 x (gain factor - 1)

Now, of course, a gain in capital becomes meaningless without a timescale attached to it. An investor A who makes 100% on his starting capital over five years has not done as well as an investor B who makes the same gain in four years. The best way of comparing the two performances is to express them as gains (either gain factors or percentage gains will do) over the same time period, which in this case would conveniently be a year. One simple way of doing this would be to divide the total gain by the number of years. We would then find that investor A who doubled his money over five years would have made a gain of 20% per annum and investor B who took four years to do this would have made a gain of 25% per annum. The disadvantage of calculating gains in this way is that it ignores the ability to compound gains, i.e. to plough back into the next investment the total proceeds from the previous investment, both the original stake and any gain made from it. Throughout this chapter we will be adopting this approach of calculating gains as compound gains, i.e. as if they were made annually and reinvested.

Although such compound gains can be calculated from the percentage gain made each year, it is much easier to calculate them if we use gain factors, since we simply multiply the gain factors together to get the overall gain.

As an example, if we make a gain of 11% per annum, then this is the same as a gain factor of 1.11. To compute the gain over a number of years, say five, we simply multiply the gain factors together the number of times that we have years.

Thus 1.11 compounded for five years = 1.11 x 1.11 x 1.11 x 1.11 x 1.11

= (1.11)5

= 1.685

By our formula above, a gain factor of 1.685 is a percentage gain of 68.5% over five years. Note the difficulty of calculating the above if we tried to use percentages instead of gain factors. Scientific and financial calculators have a key which is usually labelled x ^y which makes this calculation easier than multiplying the numbers together the requisite number of times. In this case x is the gain factor, e.g. 1.11, and y is the number of years.

If the gains differ for each of the five years, then we still use the above method, but replace the value of 1.11 for that year by the appropriate gain factor. We cannot then use the x ^y key on the calculator, of course, since the x values are not all the same.

On a computer using BASIC, the line which gives the compounded gain, say G, from the annual gain, say A, is:

G=A^Y

where Y is the number of years.

Having shown how to compute an annual gain into a five-year gain, for example, we have to do the reverse of this to express the five-year gains of investors A and B as annual gains. Each of them made gains of 100%, i.e. gain factors of 2.0. Thus,

annual gain = 5th root of 2.0 for investor A

= 1.149

annual gain = 4th root of 2.00 for investor B

= 1.189

Thus if the gain is known for an n-year period, the annual gain is the nth root of this n-year gain. The problem with reducing a gain to a gain over a shorter time period is that most simple calculators only have square roots, and not nth roots. Some financial and all scientific calculators will have this facility, which is performed by a key which is usually labelled x ^1/y. In this case, x is the overall gain factor and y would be the number of years, or whichever period it is desired to reduce the gain to. With a computer, to get the annual gain A from a gain of G which has been obtained over a period of Y years there is a oneline program in BASIC using the EXP and LOG functions:

A = EXP(LOG(G)/Y)