3.2.1 Understanding portfolio theory

The last section sets out the benefit of diversification when investing. You’ll now take this analysis further forward, by looking at how a portfolio can be constructed. This is called portfolio theory.

The animation introduces a key statistical concept which is important in investment management: ‘standard deviation’. This is a measure of the dispersion, or spread, of data around an average or mean number. The greater the standard deviation, the more varied or dispersed are the data around the average. The risk/return positions C & D shown at the end of this section’s animation will be explored further in the next section.

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MARTIN UPTON

This simple illustration demonstrates how portfolio theory works. Imagine you have an investment portfolio that contains two shares. Gelato makes ice cream, and Hotchoc, makes hot chocolate.

Suppose that there are four different types of weather Cold but sunny, Hot but rainy, Cold and rainy and finally Hot and sunny. Now suppose that over the last five years each type of weather period occurred 25% of the time. From this we can assume that each will have a 25% chance of occurring in the next five years. These 25% probabilities are shown in the table, next to the average monthly returns over the last five years for Gelato and Hotchoc.

Take a look at the hot and sunny period. The return for Gelato is expected to be 17.21% (obviously there were great ice cream sales), but a loss of 9.07% is expected for Hotchoc (so clearly not much call for hot chocolate). You can calculate the monthly expected return by multiplying each month’s outcome by its probability.

For Gelato this is 1 per cent. Here 0.25, the chance of that type of weather occurring in a particular period, is multiplied by -11.38%, which is the expected return in that period, and so on. Then, the annual (or yearly) equivalent return, is just the compound value of the monthly return.

For Gelato, each £1 invested earns a monthly return of 1 per cent. In the first month this would grow to £1 times 1.01. In the second month to £1 times 1.01 times 1.01, and so on for 12 months. Using this method, the annual equivalent return for Gelato is 12.7%. The annual expected return is the usual way to express returns on shares. Risk is measured by looking at the standard deviation (or dispersion) of returns. The more highly dispersed, or spread out, or volatile, the returns are, then the higher the risk involved in holding the share. These risk measures are also shown in the table: 13. % for Gelato and 8.2% for Hotchoc.

These measures can be displayed in a risk-return diagram. The vertical axis represents the expected return, and the horizontal axis represents risk. Point A shows what happens if the investor chooses to invest 100% of their portfolio in Hotchoc. You can see the risk the horizontal axis of the diagram. This measures at 8.2% the expected return can be read off the vertical axis, in Hotchoc’s case it is 4.4%. However, at point B, if the portfolio only holds Gelato shares, both risk and expected return are higher. The example of Gelato and Hotchoc provides a very good illustration of the risk-return trade-off. Hotchoc is safe, but has a low return, Gelato is riskier but has a higher return. If an investor wants to get higher returns then they have to take on higher risk. That’s where portfolio theory comes in. Using portfolio theory you can identify the best balance of shares in your investment portfolio to either reduce risk or maximise returns.

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Standard deviation can be calculated by following the steps below:

1. Obtain the average (or mean) of all the (daily) share prices over the period in question. To do this, add up all the daily recordings of the share price and divide by the number of those daily recordings.
2. For each share price, work out its difference from this mean and then square this difference. So if the difference is 4 pence, the square would be 16 and so on.
3. Once you have done the second step for all the share prices, add up all these ‘squares’.
4. Divide the total you get in the third step by the total number of share prices. So if the total of the ‘squares’ is 10000 and the number of share price recordings is 100, you end up with 10000/100 = 100.
5. Find the square root of the outcome from the fourth step. So if the outcome from the fourth step is 100, then the square root of this is 10 pence. This is the Standard deviation and it can either be expressed as an absolute number (i.e. 10 pence) or as a % of the mean share price.

The statistical meaning of a (that is, one) standard deviation is that it captures 68% of the individual recorded prices. So if the standard deviation in the example above is 10 pence, then 68% of the share prices are within 10 pence of the average (or mean) share price (either higher or lower).

Thsi information can also be downloaded in the supporting document How to calculate standard deviation and what does it mean? [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]