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The article deals with another utilization of the concept of the expected value, this time in investments. For illustration and training purposes we will pick up a few stocks and demonstrate not too difficult but the more enlightening calculation of investment expected return and risk, then we will form a portfolio and calculate its entire expected return and risk as well.

By way of introduction a holder of stocks is given three basic rights: 1) to participate in the control of a company, i.e. to participate in the annual general meeting, 2) to share company's profit, i.e. to be entitled for dividends if they are paid out, 3) to be entitled for share of liquidation balance in case the company is terminated.

For simplicity we will pick up only three companies: CEZ Group (one of the biggest Central European energy companies), Central European Media Enterprises Ltd and New World Resources Plc. and calculate both **individual and portfolio mean expected returns and risks**. (Please note that these companies are traded at the Prague Stock Exchange, Czech Republic, as this article is based on the original Czech article) Under no circumstance this can be considered a proper portfolio. First of all a good portfolio must be diversified geographically, by sectors and individual securities should not be positively correlated etc.

Investment experts argue about an appropriate number of stocks in the portfolio – the higher the number, the lower the risk (of course if well diversified), but according to the critics it diminishes the returns. Small investors are recommended to invest in low-cost index funds rather than to pick up individual stocks and to invest on regular basis (stock’s price averaging) rather than trying to time the market.

Anyway the theory of portfolio is not a subject of this article as it only aims to demonstrate the **calculation of the investment expected return and risk**. However I find it important to highlight this as well as that all prices and other estimates are random and illustrative only.

→ Expected Value in Gambling incl. Examples of House Edge Calculation

The calculations of the expected return and risk of the three above-mentioned companies, abbreviated as CEZ, CETV and NWRUK, are concluded based on the three following figures (or tables). The initial quotations (P_{0}) are taken from the exchange list as of 27 March 2012. The investment horizon is supposed to be a year, so that the returns are annual and thus there is no need to annualize them in case the investment horizon was shorter or longer than one year. The table headings have the following meaning:

- P
_{0}– the price of stocks on the day of investment - P
_{i}– future stock prices according to your assumptions or estimates - ER
_{i}– expected return as the difference between P_{i}and P_{0} - ER
_{i}(%) – expected return as a percentage, (ER_{i}÷ P_{0}) ˟ 100 - p
_{i}– the probability that the price would be P_{i}according to your estimate - ER
_{i}˟ p_{i}– per cent return ER_{i}weighted by the probability; the*mean expected return (ER)*is the sum of all weighted returns! - [ER– ER
_{i}]^{2}˟ p_{i}– this shows the variances (the dispersion) of individual returns ER_{i}from the mean expected return ER, therefore squared and, at last, weighted by the probability. By totaling these squared weighted variance we receive the dispersion and by taking the root of that we get a*standard deviation*.**The standard deviation is the representative of risk**– it says how much the real return can differ from the expected return with the 68% likelihood (so called the rule of one sigma – σ). By means of the standard deviation we can also figure out the top and bottom interval of the return → this is described in detail on the page Expected Return and Risk in Business.

For example as for the CEZ Group: The current price of the stocks is `CZK 802.00`

(1 USD = approx. 19 Czech koruna (crowns), 1 EUR = 25 CZK). Let us suppose first that the price goes down to `CZK 740.00`

. That represents the loss `CZK 62.00`

per share or `–7.73%`

. The probability that the e price goes down to CZK 740.00 we estimate to be `0.05`

(or 5%) only. Therefore we weight (i.e. multiply) the return 7.73% by the probability 0.05 and we arrive at the value `-0.39%`

.

We proceed similarly for the remaining price estimates, i.e. CZK 760.00 ... 860.00. If we make the total of all the returns weighted by their probabilities, we get the mean expected return `0.90%`

per annum. The return of the CEZ Group stocks is quite low in this case (or example). The returns of the remaining two stocks are higher, but, of course, the risks are higher too.

The risk is calculated as follows. On each row we deduct the estimated return from the mean expected return, then we square it and finally multiply by the probabilities. For example on the first row: `[0.90 – (–7.73)]`

, the second row: ^{2} ˟ 0.05 = 3.73`[0.90 – (–5.24)]`

and so on. By the total of all five rows we get the dispersion ^{2} ˟ 0.20 = 7.54`17.46%`

and by extracting the root of the dispersion we arrive at the standard deviation `4.18%`

, which represent the risk of deviance from the mean expected return 0.90%. If we add and deduct it to/from the mean expected return, we get the top and bottom interval of the expected return.

By the same way we get the characteristics for the stocks of the remaining two companies, i.e. CETV and NWR.

We will build a portfolio by putting the individual stocks and related calculations, i.e. the mean expected returns and risks, into the table below. Let us suppose that we have got CZK 100,000.00 (USD 5,000 or EUR 4,000) to buy stocks. The numbers of shares are shown in the table. The volume is the current market price times the number of shares bought. The volume also represents the weight of the stocks in the portfolio.

The mean expected returns and risks of individual stocks are taken from the three tables above. Now it is necessary to weight them by their volume, i.e. by their financial representation in the portfolio. The mean expected return of the entire portfolio is `7.91%`

(`0.90 ˟ 0.4103 + 10.63 ˟ 0.3016 + 15.04 ˟ 28.81 = 7.91%`

).

The risk is weighted in the same manner. The entire risk of the portfolio is `14.16%`

; now we can delimit the upper and lower interval as well.

→ Bet on Gold – Treatise on Gold and Money (16 July 2013)

→ Calculation and Determination of Variance, Standard Deviation and Confidential Intervals

→ All Articles on Probability in Gambling and Betting

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