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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)

      BUY AND HOLD FOR A LONG TERM

      The correctness of the buy and hold strategy can appear to be confirmed by a chart of just about all shares that have been quoted for a 15-year period on the London stock market. As just one example, the chart of Grand Metropolitan is shown in Figure 1.1. Taking the extremes of the chart, an investor could have bought Grand Met shares at 52p on 6th January 1978 and sold them on 31st January 1995 at 464p. If dealing costs are ignored, this represents a profit of 412p per share, i.e. a profit of 792% on the initial price.

      Figure 1.1 The Grand Metropolitan share price since 1978

      Unfortunately, dealing costs cannot be ignored, and the small investor suffers more than most as far as the level of costs is concerned. For the sake of argument, if we assume that a parcel of 1000 shares was purchased at 52p, then the dealing costs on such an amount would be approximately 2.5%. The selling costs would be approximately 1.5%. These percentages increase rapidly as the value of the deal falls below £1000 and decrease only slowly as the deal moves into the tens of thousands of pounds.

      Thus the dealing costs of buying 1000 shares at 52p would be about £13 and the selling costs of selling at 464p would be about £69. Now we can calculate a more realistic profit for the entire deal than the 792% we noted above:

       Buy 1000 shares at 52p Outlay = £520 + £13 = £533

       Sell 1000 shares at 464p Receipts = £4640 - £69 = £4571

       Actual gain = £4038

      This represents a gain of 757% on the outlay of £533. Therefore the dealing costs of this transaction have reduced the overall gain by some 35% over the 18-year period.

      If we are going to make this a realistic exercise, then there is one important aspect that is missing from this calculation. This concerns the dividends that would have been paid during the 18-year period for which the shares would have been held. To simplify matters, we can consider that Grand Metropolitan consistently paid a 5% dividend, year in and year out over this period.

      Since the average share price was halfway between 52p and 464p, i.e. 258p, we can estimate the cumulative dividend as:

       18 x 1000 x 258p x 5% = £2322

      This increases the actual gain from £4250 as calculated without dividends to £6572 with dividends. This now gives a gain of 1233% on the initial outlay of £533.

      Since we require some standard timescale over which to compare the gain from one situation with the gain from another, it is best to state this gain from investment in Grand Metropolitan shares as a percentage gain per annum. As we discussed above, it is not correct simply to divide the 1233% by 18 and use this as the annual gain. The annual gain has to be such that it compounds into a gain of 1233% over the 18-year period, so that we could compare it with that made by an investor who leaves his money and the accumulated interest in an interest-bearing account. If we do this, we find that the gain of 1233% equates to a gain of 15.5% per annum. This is superior to any gain that could have been made by depositing the money in the money market for one-year periods, year in and year out, and so appears to verify that long-term investment in shares is an excellent strategy.

      The profit from the position is made because the share price has risen more than sufficiently to offset the buying and selling costs and we have had the advantage of a number of dividends over the time period.

      MULTIPLE TRANSACTIONS OVER A LONG TERM

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